Quantum Signatures of Chaos [4th ed.] 978-3-319-97579-5, 978-3-319-97580-1

Table of contents :
Front Matter ....Pages i-xxvi
Introduction (Fritz Haake, Sven Gnutzmann, Marek Kuś)....Pages 1-14
Time Reversal and Unitary Symmetries (Fritz Haake, Sven Gnutzmann, Marek Kuś)....Pages 15-70
Level Repulsion (Fritz Haake, Sven Gnutzmann, Marek Kuś)....Pages 71-84
Level Clustering (Fritz Haake, Sven Gnutzmann, Marek Kuś)....Pages 85-109
Random-Matrix Theory (Fritz Haake, Sven Gnutzmann, Marek Kuś)....Pages 111-203
Supersymmetry and Sigma Model for Random Matrices (Fritz Haake, Sven Gnutzmann, Marek Kuś)....Pages 205-288
Ballistic Sigma Model for Individual Unitary Maps and Graphs (Fritz Haake, Sven Gnutzmann, Marek Kuś)....Pages 289-302
Quantum Localization (Fritz Haake, Sven Gnutzmann, Marek Kuś)....Pages 303-363
Classical Hamiltonian Chaos (Fritz Haake, Sven Gnutzmann, Marek Kuś)....Pages 365-407
Semiclassical Roles for Classical Orbits (Fritz Haake, Sven Gnutzmann, Marek Kuś)....Pages 409-510
Level Dynamics (Fritz Haake, Sven Gnutzmann, Marek Kuś)....Pages 511-589
Dissipative Systems (Fritz Haake, Sven Gnutzmann, Marek Kuś)....Pages 591-653
Back Matter ....Pages 655-659

Citation preview

Springer Series in Synergetics

Fritz Haake · Sven Gnutzmann  Marek Kuś

Quantum Signatures of Chaos Fourth Edition

Springer Complexity Springer Complexity is an interdisciplinary program publishing the best research and academic-level teaching on both fundamental and applied aspects of complex systems – cutting across all traditional disciplines of the natural and life sciences, engineering, economics, medicine, neuroscience, social and computer science. Complex Systems are systems that comprise many interacting parts with the ability to generate a new quality of macroscopic collective behavior the manifestations of which are the spontaneous formation of distinctive temporal, spatial or functional structures. Models of such systems can be successfully mapped onto quite diverse “real-life” situations like the climate, the coherent emission of light from lasers, chemical reaction-diffusion systems, biological cellular networks, the dynamics of stock markets and of the internet, earthquake statistics and prediction, freeway traffic, the human brain, or the formation of opinions in social systems, to name just some of the popular applications. Although their scope and methodologies overlap somewhat, one can distinguish the following main concepts and tools: self-organization, nonlinear dynamics, synergetics, turbulence, dynamical systems, catastrophes, instabilities, stochastic processes, chaos, graphs and networks, cellular automata, adaptive systems, genetic algorithms and computational intelligence. The three major book publication platforms of the Springer Complexity program are the monograph series “Understanding Complex Systems” focusing on the various applications of complexity, the “Springer Series in Synergetics”, which is devoted to the quantitative theoretical and methodological foundations, and the “SpringerBriefs in Complexity” which are concise and topical working reports, case-studies, surveys, essays and lecture notes of relevance to the field. In addition to the books in these two core series, the program also incorporates individual titles ranging from textbooks to major reference works.

Editorial and Programme Advisory Board Henry D.I. Abarbanel, Institute for Nonlinear Science, University of California, San Diego, USA Dan Braha, New England Complex Systems Institute and University of Massachusetts Dartmouth, USA Péter Érdi, Center for Complex Systems Studies, Kalamazoo College, USA and Hungarian Academy of Sciences, Budapest, Hungary Karl J Friston, Institute of Cognitive Neuroscience, University College London, London, UK Hermann Haken, Center of Synergetics, University of Stuttgart, Stuttgart, Germany Viktor Jirsa, Centre National de la Recherche Scientifique (CNRS), Université de la Méditerranée, Marseille, France Janusz Kacprzyk, System Research, Polish Academy of Sciences, Warsaw, Poland Kunihiko Kaneko, Research Center for Complex Systems Biology, The University of Tokyo, Tokyo, Japan Scott Kelso, Center for Complex Systems and Brain Sciences, Florida Atlantic University, Boca Raton, USA Markus Kirkilionis, Mathematics Institute and Centre for Complex Systems, University of Warwick, Coventry, UK Jürgen Kurths, Nonlinear Dynamics Group, University of Potsdam, Potsdam, Germany Linda Reichl, Center for Complex Quantum Systems, University of Texas, Austin, USA Ronaldo Menezes, Florida Institute of Technology, Computer Science Department, Melbourne, USA Andrzej Nowak, Department of Psychology, Warsaw University, Poland Hassan Qudrat-Ullah, York University, Toronto, Ontario, Canada Peter Schuster, Theoretical Chemistry and Structural Biology, University of Vienna, Vienna, Austria Frank Schweitzer, System Design, ETH Zurich, Zurich, Switzerland Didier Sornette, Entrepreneurial Risk, ETH Zurich, Zurich, Switzerland Stefan Thurner, Section for Science of Complex Systems, Medical University of Vienna, Vienna, Austria

Springer Series in Synergetics Founding Editor: H. Haken The Springer Series in Synergetics was founded by Herman Haken in 1977. Since then, the series has evolved into a substantial reference library for the quantitative, theoretical and methodological foundations of the science of complex systems. Through many enduring classic texts, such as Haken’s Synergetics and Information and Self-Organization, Gardiner’s Handbook of Stochastic Methods, Risken’s The Fokker Planck-Equation or Haake’s Quantum Signatures of Chaos, the series has made, and continues to make, important contributions to shaping the foundations of the field. The series publishes monographs and graduate-level textbooks of broad and general interest, with a pronounced emphasis on the physico-mathematical approach. More information about this series at http://www.springer.com/series/712

Fritz Haake • Sven Gnutzmann • Marek Ku´s

Quantum Signatures of Chaos Fourth Edition

123

Fritz Haake Universit¨at Duisburg-Essen Essen, Germany

Sven Gnutzmann School of Mathematical Sciences University of Nottingham Nottingham, United Kingdom

Marek Ku´s Center for Theoretical Physics Polish Academy of Sciences Warszawa, Poland

ISSN 0172-7389 ISSN 2198-333X (electronic) Springer Series in Synergetics ISBN 978-3-319-97579-5 ISBN 978-3-319-97580-1 (eBook) https://doi.org/10.1007/978-3-319-97580-1 Library of Congress Control Number: 2018953566 © Springer Nature Switzerland AG 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Für Gitta und Julia, Katya und Yarik, Asia und Kasia

Foreword to the First Edition

The interdisciplinary field of synergetics grew out of the desire to find general principles that govern the spontaneous formation of ordered structures out of microscopic chaos. Indeed, large classes of classical and quantum systems have been found in which the emergence of ordered structures is governed by just a few degrees of freedom, the so-called order parameters. But then a surprise came with the observation that a few degrees of freedom may cause complicated behavior, nowadays generally subsumed under the title “deterministic chaos” (not to be confused with microscopic chaos, where many degrees of freedom are involved). One of the fundamental problems of chaos theory is the question of whether deterministic chaos can be exhibited by quantum systems, which, at first sight, seem to show no deterministic behavior at all because of the quantization rules. To be more precise, one can formulate the question as follows: how does the transition occur from quantum mechanical properties to classical properties showing deterministic chaos? Fritz Haake is one of the leading scientists investigating this field, and he has contributed a number of important papers. I am therefore particularly happy that he agreed to write a book on this fascinating field of quantum chaos. I very much enjoyed reading the manuscript of this book, which is written in a highly lively style, and I am sure the book will appeal to many graduate students, teachers, and researchers in the field of physics. This book is an important addition to the Springer Series in Synergetics. Stuttgart, Germany February 1991

Hermann Haken

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Preface to the Fourth Edition

When Springer proposed to prepare a fourth edition, it didn’t take more than an instant to realize that I would not dispose of sufficient energy nor time to comply. This all the more so since it was clear to me that a little update here and there would not do. Nothing less than a thorough modernization would justify another edition, in my opinion. Younger companions would be needed to make the undertaking feasible. When Marek and Sven agreed to join in I was really happy, knowing that with their help the project could fly. Both had been members of my Essen group, as PhD student (SG) and postdoc (MK); both had contributed a lot to the group’s “quantum chaotic” endeavors and subsequently built their independent careers since. Marek’s expertise in group theory and level dynamics (of which he is a founding father) and Sven’s intimate knowledge of the nonlinear sigma model and the professional versatility of both in present-day mathematical physics made the cooperation a most pleasurable experience for me. And what an inversion of roles indeed, with me now learning from them. A new, “younger” book can now appear and I feel younger with it. Thanks a lot, my dear friends! Essen, Germany

Fritz Haake

Nearly all of the chapters of the previous edition have been worked over. The treatment of symmetries in the second one is rewritten more systematically, in particular in what concerns the non-standard symmetry classes. The supersymmetric sigma model for ensembles of random matrices is likewise updated in a separate chapter with particular emphasis on unitary maps. To reveal the real power of the supersymmetry technique, we have included a chapter on its application to individual chaotic dynamics and graphs. The conditions for a dynamical system to display universal spectral fluctuations become accessible there, like in the chapter on periodic-orbit theory.

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We have enjoyed discussions with Alexander Altland, Peter Braun, Tobias Micklitz, Uzy Smilansky, and Martin Zirnbauer. Essen, Germany Nottingham, UK Warszawa, Poland

Fritz Haake Sven Gnutzmann Marek Ku´s

Preface to the Third Edition

Nine years have passed since I dispatched the second edition, and the book still appears to be in demand. The time may be ripe for an update. As the perhaps most conspicable extension, I describe the understanding of universal spectral fluctuations recently reached on the basis of periodic-orbit theory. To make the presentation of those semiclassical developments self-contained, I decided to underpin them by a new short chapter on classical Hamiltonian mechanics. Inasmuch as the semiclassical theory not only draws inspiration from the nonlinear sigma model but actually aims at constructing that model in terms of periodic orbits, it appeared indicated to make small additions to the previous treatment within the chapter on superanalysis. Less voluminous but as close to my heart are additions to the chapter on level dynamics which close previous gaps in that approach to spectral universality. It was a pleasant duty to pay my respects to colleagues in our TransregioSonderforschungsbereich, Martin Zirnbauer, Alex Altland, Alan Huckleberry, and Peter Heinzner, by including a short account of their beautiful work on nonstandard symmetry classes. The chapter on random matrices has not been expanded in proportion to the development of the field but now includes an up-to-date treatment of an old topic in algebra, Newton’s relations, to provide a background to the Riemann–Siegel lookalike of semiclassical periodic-orbit theory. The chapters on level clustering, localization, and dissipation are similarly preserved. I disciplined myself to just adding an occasional reference to recent work and to cutting some stuff of lesser relative importance. There was the temptation to rewrite the introduction, to no avail. Only a few additional words here and there announce new topics taken up in the main text. So that chapter stands as a relic from the olden days when quantum chaos was just beginning to form as a field. Encouragement and help has come from Thomas Guhr, Dominique Spehner and Martin Zirnbauer and, as always, from Hans-Jürgen Sommers and Marek Ku´s.

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Preface to the Third Edition

I owe special gratitude to Alex Altland, Peter Braun, Stefan Heusler, and Sebastian Müller. They have formed a dream team sharing search and finding, suffering and joy, row and laughter. Essen, Germany August 2009

Fritz Haake

Preface to the Second Edition

The warm reception of the first edition, as well as the tumultuous development of the field of quantum chaos, has tempted me to rewrite this book and include some of the important progress made during the past decade. Now we know that quantum signatures of chaos are paralleled by wave signatures. Whatever is undergoing wavy space-time variations, be it sound, electromagnetism, or quantum amplitudes, each shows exactly the same manifestations of chaos. The common origin is nonseparability of the pertinent wave equation; that latter “definition” of chaos, incidentally, also applies to classical mechanics if we see the Hamilton–Jacobi equation as the limiting case of a wave equation. At any rate, drums, concert halls, oscillating quartz blocks, microwave and optical oscillators, and electrons moving ballistically or with impurity scattering through mesoscopic devices all provide evidence and data for wave or quantum chaos. All of these systems have deep analogies with billiards, much as the latter may have appeared of no more than academic interest only a decade ago. Of course, molecular, atomic, and nuclear spectroscopy also remain witnesses of chaos, while the chromodynamic innards of nucleons are beginning to attract interest as methods of treatment become available. Of the considerable theoretical progress lately achieved, the book focuses on the deeper statistical exploitation of level dynamics, improved control of semiclassical periodic-orbit expansions, and superanalytic techniques for dealing with various types of random matrices. These three fields are beginning, independently and in conjunction, to generate an understanding of why certain spectral fluctuations in classically nonintegrable systems are universal and why there are exceptions. Only the rudiments of periodic-orbit theory and superanalysis appeared in the first edition. More could not have been included here had I not enjoyed the privilege of individual instruction on periodic-orbit theory by Jon Keating and on superanalysis by Hans-Jürgen Sommers and Yan Fyodorov. Hans-Jürgen and Yan have even provided their lecture notes on the subject. While giving full credit and expressing my deep gratitude to these three colleagues, I must bear all blame for blunders.

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Preface to the Second Edition

Reasonable limits of time and space had to be respected and have forced me to leave out much interesting material such as chaotic scattering and the semiclassical art of getting spectra for systems with mixed phase spaces. Equally regrettably, no justice could be done here to the wealth of experiments that have now been performed, but I am happy to see that gap filled by my much more competent colleague Hans-Jürgen Stöckmann. Incomplete as the book must be, it now contains more material than fits into a single course in quantum chaos theory. In some technical respects, it digs deeper than general introductory courses would go. I have held on to my original intention though to provide a self-contained presentation that might help students and researchers to enter the field or parts thereof. The number of coworkers and colleagues from whose knowledge and work I could draw has increased considerably over the years. Having already mentioned Yan Fyodorov, Jon Keating, and Hans-Jürgen Sommers, I must also express special gratitude to my partner and friend Marek Ku´s whose continuing help was equally crucial. My thanks for their invaluable influence go to Sergio Albeverio, Daniel Braun, Peter Braun, Eugene Bogomolny, Chang-qi Cao, Dominique Delande, Bruno Eckhardt, Pierre Gaspard, Sven Gnutzmann, Peter Goetsch, Siegfried Grossmann, Martin Gutzwiller, Gregor Hackenbroich, Alan Huckleberry, Micha Kolobov, Pavel Kurasov, Robert Littlejohn, Nils Lehmann, Jörg Main, Alexander Mirlin, Jan Mostowski, Alfredo Ozorio de Almeida, Pjotr Peplowski, Ravi Puri, Jonathan Robbins, Kazik Rza¸z˙ ewski, Henning Schomerus, Carsten Seeger, Thomas Seligmann, Frank Steiner, Hans-Jürgen Stöckmann, Jürgen Vollmer, Joachim Weber, Harald Wiedemann, Christian Wiele, Günter Wunner, Dmitri Zaitsev, Kuba Zakrzewski, Martin Zirnbauer, Marek Zukowski, Wojtek Zurek, and, last but not least, Karol ˙ Zyczkowski. In part this book is an account of research done within the Sonderforschungsbereich “Unordnung und Große Fluktuationen” of the Deutsche Forschungsgemeinschaft. This fact needs to be gratefully acknowledged, since coherent long-term research of a large team of physicists and mathematicians could not be maintained without the generous funding we have enjoyed over the years through our Sonderforschungsbereich. Times do change. Like many present-day science authors I chose to pick up LATEX and key all changes and extensions into my little machine myself. As usually happens when learning a new language, the beginning is all effort, but one eventually begins to enjoy the new mode of expressing oneself. I must thank Peter Gerwinski, Heike Haschke, and Rüdiger Oberhage for their infinite patience in getting me going. Essen, Germany July 2000

Fritz Haake

Preface to the First Edition

More than 60 years after its inception, quantum mechanics is still exerting fascination on every new generation of physicists. What began as the scandal of non-commuting observables and complex probability amplitudes has turned out to be the universal description of the micro-world. At no scale of energies accessible to observation have any findings emerged that suggest violation of quantum mechanics. Lingering doubts that some people have held about the universality of quantum mechanics have recently been resolved, at least in part. We have witnessed the serious blow dealt to competing hidden-variable theories by experiments on correlations of photon pairs. Such correlations were found to be in conflict with any local deterministic theory as expressed rigorously by Bell’s inequalities. Doubts concerning the accommodation of dissipation in quantum mechanics have also been eased, in much the same way as in classical mechanics. Quantum observables can display effectively irreversible behavior when they are coupled to an appropriate environmental system containing many degrees of freedom. Even in closed quantum systems with relatively few degrees of freedom, behavior resembling damping is possible, provided the system displays chaotic motion in the classical limit. It has become clear that the relative phases of macroscopically distinguishable states tend, in the presence of damping, to become randomized in exceedingly short times; that remains true even when the damping is so weak that it is hardly noticeable for quantities with a well-defined classical limit. Consequently, a superposition (in the quantum sense) of different readings of a macroscopic measuring device would, even if one could be prepared momentarily, escape observation due to its practically instantaneous decay. While this behavior was conjectured early in the history of quantum mechanics, it is only recently that we have been able to see it explicitly in rigorous solutions for specific model systems. There are many intricacies of the classical limit of quantum mechanics. They are by no means confined to abrupt decay processes or infinitely rapid oscillations of probability amplitudes. The classical distinction between regular and chaotic motion, for instance, makes itself felt in the semiclassical regime that is typically associated with high degrees of excitation. In that regime, quantum effects like the xv

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discreteness of energy levels and interference phenomena are still discernible, while the correspondence principle suggests the onset of validity of classical mechanics. The semiclassical world, which is intermediate between the microscopic and the macroscopic, is the topic of this book. It will deal with certain universal modes of behavior, both dynamical and spectral, which indicate whether their classical counterparts are regular or chaotic. Conservative as well as dissipative systems will be treated. The area under consideration often carries the label “quantum chaos.” It is a rapidly expanding one and therefore does not yet allow for a definite treatment. The material presented reflects subjective selections. Random-matrix theory will enjoy special emphasis. A possible alternative would have been to make current developments in periodic-orbit theory the backbone of the text. Much as I admire the latter theory for its beauty and its appeal to classical intuition, I do not understand it sufficiently well that I can trust myself to do it justice. With more learning, I might yet catch up and find out how to relate spectral fluctuations on an energy scale of a typical level spacing to classical properties. There are other regrettable omissions. Most notable among these may be the ionization of hydrogen atoms by microwaves, for which convergence of theory and experiment has been achieved recently. Also, too late for inclusion is the quantum aspect of chaotic scattering, which has seen such fine progress in the months between the completion of the manuscript and the appearance of this book. This book grew out of lectures given at the universities of Essen and Bochum. Most of the problems listed at the end of each chapter have been solved by students attending those lectures. The level aimed at was typical of a course on advanced quantum mechanics. The book accordingly assumes the reader to have a good command of the elements of quantum mechanics and statistical mechanics, as well as some background knowledge of classical mechanics. A little acquaintance with classical nonlinear dynamics would not do any harm either. I could not have gone through with this project without the help of many colleagues and coworkers. They have posed many of the questions dealt with here and provided most of the answers. Perhaps more importantly, they have, within the theory group in Essen, sustained an atmosphere of dedication and curiosity, from which I keep drawing knowledge and stimulus. I can only hope that my young coworkers share my own experience of receiving more than one is able to give. I am especially indebted to Michael Berry, Oriol Bohigas, Giulio Casati, Boris Chirikov, Barbara Dietz, Thomas Dittrich, Mario Feingold, Shmuel Fishman, Dieter Forster, Robert Graham, Rainer Grobe, Italo Guarneri, KlausDieter Harms, Michael Höhnerbach, Ralf Hübner, Felix Israilev, Marek Ku´s, Georg Lenz, Maciej Lewenstein, Madan Lal Mehta, Jan Mostowski, Akhilesh Pandey, Dirk Saher, Rainer Scharf, Petr Šeba, Dima Shepelyansky, Uzy Smilansky, Hans-Jürgen ˙ Sommers, Dan Walls, and Karol Zyczkowski. Angela Lahee has obliged me by smoothening out some clumsy Teutonisms and by her careful editing of the manuscript. My secretary, Barbara Sacha, deserves a big thank you for keying version upon version of the manuscript into her computer.

Preface to the First Edition

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My friend and untiring critic Roy Glauber has followed this work from a distance and provided invaluable advice. I am grateful to Hermann Haken for his invitation to contribute this book to his series in synergetics, and I am all the more honored since it can fill but a tiny corner of Haken’s immense field. However, at least Chap. 8 does bear a strong relation to several other books in the series inasmuch as it touches upon adiabatic-elimination techniques and quantum stochastic processes. Moreover, that chapter represents variations on themes I learned as a young student in Stuttgart, as part of the set of ideas which has meanwhile grown to span the range of this series. The love of quantum mechanics was instilled in me by Hermann Haken and his younger colleagues, most notably Wolfgang Weidlich, as they were developing their quantum theory of the laser and thus making the first steps toward synergetics. Essen, Germany January 1991

Fritz Haake

Contents

1

Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1 13

2

Time Reversal and Unitary Symmetries . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Autonomous Classical Flows. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Spinless Quanta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Spin-1/2 Quanta .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4 Hamiltonians Without T Invariance . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5 T Invariant Hamiltonians, T 2 = 1 . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.6 Unitary Symmetries.. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.7 Kramers’ Degeneracy for T Invariant Hamiltonians, T 2 = −1.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.8 T Invariant Hamiltonians with T 2 = −1 and Additional Unitary Symmetries.. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.9 T Invariance with T 2 = −1 and no Unitary Symmetries .. . . . . . . . 2.10 Nonconventional Time Reversal . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.11 Stroboscopic Maps for Periodically Driven Systems . . . . . . . . . . . . . 2.12 Time Reversal for Maps . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.13 Canonical Transformations for Floquet Operators . . . . . . . . . . . . . . . . 2.14 Universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.15 Universality for the Kicked Top: The Level Spacing Distribution .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.16 Beyond Dyson’s Threefold Way . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.16.1 Spectral Mirror Symmetries .. . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.16.2 Universality in Non-standard Symmetry Classes . . . . . . 2.16.3 Hamiltonian Matrix Structures and Canonical Transformations for Non-standard Symmetries . . . . . . . . 2.16.4 Physical Realizations of Non-standard Symmetry Classes in Fermionic Systems . . . . . . . . . . . . . .

15 15 16 17 20 21 22 24 25 28 31 33 35 38 41 43 45 46 49 51 58

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2.16.5

Quantum Mechanical Realisation of Non-standard Symmetry Classes in Finite Dimensional Hilbert Spaces: Two Coupled Tops . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.16.6 Non-standard Universality for Two Coupled Tops . . . . . 2.17 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

61 66 68 69

3

Level Repulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 Symmetric Versus Nonsymmetric H or F . . . . .. . . . . . . . . . . . . . . . . . . . 3.3 Kramers’ Degeneracy .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.4 Universality Classes of Level Repulsion.. . . . . .. . . . . . . . . . . . . . . . . . . . 3.5 Nonstandard Symmetry Classes. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.5.1 The Chiral Symmetry Classes . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.5.2 Classes CI and BDIIIν . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.5.3 Classes BDν and C. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.6 Experimental Observation of Level Repulsion . . . . . . . . . . . . . . . . . . . . 3.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

71 71 72 74 77 78 79 80 80 81 82 83

4

Level Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 Invariant Tori of Classically Integrable Systems . . . . . . . . . . . . . . . . . . 4.3 Einstein–Brillouin–Keller Approximation .. . . .. . . . . . . . . . . . . . . . . . . . 4.4 Level Crossings for Integrable Systems . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5 Poissonian Level Sequences .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.6 Superposition of Independent Spectra . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.7 Periodic Orbits and the Semiclassical Density of Levels . . . . . . . . . 4.8 Level Density Fluctuations for Integrable Systems . . . . . . . . . . . . . . . 4.9 Exponential Spacing Distribution for Integrable Systems . . . . . . . . 4.10 Equivalence of Different Unfoldings . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

85 85 85 87 89 90 91 93 99 106 107 108 109

5

Random-Matrix Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2 Gaussian Ensembles of Hermitian Matrices . . .. . . . . . . . . . . . . . . . . . . . 5.2.1 Gaussian Orthogonal Ensemble .. . . .. . . . . . . . . . . . . . . . . . . . 5.2.2 Gaussian Unitary Ensemble . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2.3 Gaussian Symplectic Ensemble . . . . .. . . . . . . . . . . . . . . . . . . . 5.2.4 Arbitrary Matrix Dimension.. . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3 Riemann Geometry for Matrix Ensembles: Invariance of the Volume Element . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3.1 Change of Variables on a Manifold .. . . . . . . . . . . . . . . . . . . .

111 111 112 112 114 114 116 118 118

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5.3.2

6

Geometry for Matrix Ensembles: Invariance of the Volume Element d N H . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.4 More of Riemann Geometry for Matrix Ensembles: Eigenvalue Distributions.. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.5 Eigenvalue Distributions for the Non-standard Symmetry Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.6 Level Spacing Distributions (Wigner Surmizes) . . . . . . . . . . . . . . . . . . 5.7 Average Level Density (GUE) . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.8 Dyson’s Circular Ensembles . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.8.1 Circular Unitary Ensemble . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.8.2 COE and CSE . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.8.3 Poissonian Ensemble . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.9 Same Spectral Correlations in Circular and Gaussian Ensembles for N → ∞. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.10 Eigenvector Distributions .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.10.1 Single-Vector Density . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.10.2 Joint Density of Eigenvectors . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.11 Ergodicity of the Level Density . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.12 Asymptotic Level Spacing Distributions . . . . . .. . . . . . . . . . . . . . . . . . . . 5.13 Determinants as Gaussian Grassmann Integrals .. . . . . . . . . . . . . . . . . . 5.14 Two-Point Correlations of the Level Density . .. . . . . . . . . . . . . . . . . . . . 5.14.1 Two-Point Correlator and Form Factor . . . . . . . . . . . . . . . . . 5.14.2 Form Factor for the Poissonian Ensemble .. . . . . . . . . . . . . 5.14.3 Form Factor for the CUE . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.14.4 Form Factor for the COE . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.14.5 Form Factor for the CSE . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.15 Newton’s Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.15.1 Traces Versus Secular Coefficients... . . . . . . . . . . . . . . . . . . . 5.15.2 Solving Newton’s Relations . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.16 Selfinversiveness and Riemann–Siegel Lookalike . . . . . . . . . . . . . . . . 5.17 Higher Correlations of the Level Density .. . . . .. . . . . . . . . . . . . . . . . . . . 5.17.1 Correlation and Cumulant Functions . . . . . . . . . . . . . . . . . . . 5.17.2 Ergodicity of the CUE Form Factor.. . . . . . . . . . . . . . . . . . . . 5.17.3 Ergodicity of the CUE Two-Point Correlator.. . . . . . . . . . 5.17.4 Joint Density of Traces of Large CUE Matrices.. . . . . . . 5.18 Correlations of Secular Coefficients . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.19 Unfolding Spectra.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.20 Fidelity of Kicked Tops to Random-Matrix Theory .. . . . . . . . . . . . . . 5.21 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

132 133 133 136 137 141 151 156 156 158 158 161 165 167 167 170 172 174 174 176 179 183 185 191 193 200 202

Supersymmetry and Sigma Model for Random Matrices . . . . . . . . . . . . . 6.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2 Semicircle Law for the Gaussian Unitary Ensemble .. . . . . . . . . . . . . 6.2.1 The Green Function and Its Average.. . . . . . . . . . . . . . . . . . .

205 205 206 206

119 120 122 124 125 126 127 130 132

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6.2.2

6.3

6.4

6.5 6.6

6.7 6.8

6.9 6.10

An Aside: Complex Conjugation of Grassmann Variables.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2.3 The GUE Average . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2.4 Doing the Superintegral . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2.5 Two Remaining Saddle-Point Integrals .. . . . . . . . . . . . . . . . Superalgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3.1 Motivation and Generators of Grassmann Algebras . . . 6.3.2 Supervectors, Supermatrices . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3.3 Superdeterminants . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3.4 Complex Scalar Product, Hermitian and Unitary Supermatrices . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3.5 Diagonalizing Supermatrices .. . . . . . .. . . . . . . . . . . . . . . . . . . . Superintegrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4.1 Some Bookkeeping for Ordinary Gaussian Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4.2 Recalling Grassmann Integrals .. . . . .. . . . . . . . . . . . . . . . . . . . 6.4.3 Gaussian Superintegrals . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4.4 Some Properties of General Superintegrals.. . . . . . . . . . . . 6.4.5 Integrals over Supermatrices, Parisi–Sourlas–Efetov–Wegner Theorem .. . . . . . . . . . . . . . 6.4.6 Asymptotic Analysis of SuSy Integrals: Massive and Zero Modes .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . The Semicircle Law Revisited . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . The Two-Point Function of the Gaussian Unitary Ensemble . . . . . 6.6.1 Generating Function . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.6.2 Unitarity vs Pseudo-Unitarity and Superanalytic Hubbard-Stratonovich Transformation . . . . . . . . . . . . . . . . . 6.6.3 Efetov’s Sigma Model . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.6.4 Rational Parametrization . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Two-Point Functions of the Circular Ensembles . . . . . . . . . . . . . . . . . . 6.7.1 Generating Function . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Zero Dimensional Sigma Model for the CUE .. . . . . . . . . . . . . . . . . . . . 6.8.1 Color-Flavor Transformation .. . . . . . .. . . . . . . . . . . . . . . . . . . . 6.8.2 The Q Manifold .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.8.3 Riemann Geometry for Supermatrix Ensembles: ˜ . . . . . .. . . . . . . . . . . . . . . . . . . . Flat Measure d(Q) = d(Z, Z) 6.8.4 Proof of the Color-Flavor Transformation (6.8.2) .. . . . . 6.8.5 Evaluation of the Generating Function and the Two-Point Correlator for the CUE . . . . . . . . . . . . . 6.8.6 Evaluation of the Grassmann Integral G . . . . . . . . . . . . . . . . The Zero Dimensional Sigma Model for COE and CSE. . . . . . . . . . Universality of Spectral Fluctuations: Non-Gaussian Ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.10.1 Delta Functions of Grassmann Variables . . . . . . . . . . . . . . . 6.10.2 Generating Function . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

207 208 209 211 215 215 215 218 221 222 223 223 224 226 227 229 231 234 238 239 241 244 247 250 251 254 254 257 259 261 266 270 273 280 281 282

Contents

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6.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 286 References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 287 7

8

9

Ballistic Sigma Model for Individual Unitary Maps and Graphs . . . . . 7.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2 Generation Function for the Two-Point Correlator .. . . . . . . . . . . . . . . 7.3 Reduction to the Zero Dimensional Sigma Model and Condition of Validity .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.4 Perturbative Account of Fluctuations . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.4.1 Fluctuations on the Q Manifold .. . . .. . . . . . . . . . . . . . . . . . . . 7.4.2 Small Parameters . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.4.3 Semiclassical Limit . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.4.4 Conditions for Universal Behavior ... . . . . . . . . . . . . . . . . . . . 7.5 Quantum Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.5.1 Directed Graphs and Their Spectra .. . . . . . . . . . . . . . . . . . . . 7.5.2 The Sigma Model Approach to Spectral Statistics . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

289 289 290 291 292 292 293 294 296 297 297 299 302

Quantum Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2 Localization in Anderson’s Hopping Model .. .. . . . . . . . . . . . . . . . . . . . 8.3 The Kicked Rotator as a Variant of Anderson’s Model . . . . . . . . . . . 8.4 Lloyd’s Model.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.5 The Classical Diffusion Constant as the Quantum Localization Length.. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.6 Absence of Localization for the Kicked Top . .. . . . . . . . . . . . . . . . . . . . 8.7 The Rotator as a Limiting Case of the Top . . . .. . . . . . . . . . . . . . . . . . . . 8.8 Banded Random Matrices . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.8.1 Banded Matrices Modelling Thick Wires . . . . . . . . . . . . . . 8.8.2 Inverse Participation Ratio and Localization Length .. . 8.8.3 Sigma Model .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.8.4 Implementing the One Dimensional Sigma Model . . . . 8.9 Sigma Model for the Kicked Rotor . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.9.1 A Rotor Without Time Reversal Invariance . . . . . . . . . . . . 8.9.2 Inverse Participation Ratio . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.9.3 Sigma Model .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.9.4 Slow Modes . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

303 303 305 307 314 320 321 332 333 333 335 337 341 353 353 354 355 357 361 362

Classical Hamiltonian Chaos .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.2 Phase Space, Hamilton’s Equations and All That . . . . . . . . . . . . . . . . . 9.3 Action as a Generating Function . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.4 Linearized Flow and Its Jacobian Matrix . . . . .. . . . . . . . . . . . . . . . . . . . 9.5 Liouville Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

365 365 365 367 369 370

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9.6 9.7 9.8 9.9 9.10 9.11 9.12

Symplectic Structure .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Lyapunov Exponents.. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Stretching Factors and Local Stretching Rates . . . . . . . . . . . . . . . . . . . . Poincaré Map.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Stroboscopic Maps of Periodically Driven Systems .. . . . . . . . . . . . . Varieties of Chaos; Mixing and Effective Equilibration . . . . . . . . . . The Sum Rule of Hannay and Ozorio de Almeida . . . . . . . . . . . . . . . . 9.12.1 Maps .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.12.2 Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.13 Propagator and Zeta Function . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.14 Exponential Stability of the Boundary Value Problem .. . . . . . . . . . . 9.15 Sieber-Richter Self-Encounter and Partner Orbit.. . . . . . . . . . . . . . . . . 9.15.1 Non-technical Discussion . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.15.2 Quantitative Discussion of 2-Encounters . . . . . . . . . . . . . . 9.16 l-Encounters and Orbit Bunches . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.17 Densities of Arbitrary Encounter Sets . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.18 Concluding Remarks.. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.19 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

371 373 374 376 378 379 380 381 382 384 387 388 388 391 399 404 406 406 406

10 Semiclassical Roles for Classical Orbits. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.2 Van Vleck Propagator.. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.2.1 Maps .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.2.2 Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.3 Gutzwiller’s Trace Formula . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.3.1 Maps .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.3.2 Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.3.3 Weyl’s Law .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.3.4 Limits of Validity and Outlook .. . . . .. . . . . . . . . . . . . . . . . . . . 10.4 Lagrangian Manifolds and Maslov Theory .. . .. . . . . . . . . . . . . . . . . . . . 10.4.1 Lagrangian Manifolds . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.4.2 Elements of Maslov Theory .. . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.4.3 Maslov Indices as Winding Numbers .. . . . . . . . . . . . . . . . . . 10.5 Riemann-Siegel Look-Alike.. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.6 Spectral Two-Point Correlator.. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.6.1 Real and Complex Correlator . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.6.2 Local Energy Average . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.6.3 Generating Function . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.6.4 Periodic-Orbit Representation.. . . . . .. . . . . . . . . . . . . . . . . . . . 10.7 Diagonal Approximation . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.7.1 Unitary Class . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.7.2 Orthogonal Class. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.8 Off-Diagonal Contributions, Unitary Symmetry Class . . . . . . . . . . . 10.8.1 Structures of Pseudo-Orbit Quadruplets . . . . . . . . . . . . . . . .

409 409 409 411 416 420 420 425 431 432 434 434 441 445 450 456 457 458 460 461 466 466 468 468 471

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xxv

10.8.2 10.8.3

Diagrammatic Rules . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Example of Structure Contributions: A Single 2-Encounter . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.8.4 Cancellation of All Encounter Contributions for the Unitary Class . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.9 Semiclassical Construction of a Sigma Model, Unitary Symmetry Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.9.1 Matrix Elements for Ports and Contraction Lines for Links .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.9.2 Wick’s Theorem and Link Summation . . . . . . . . . . . . . . . . . 10.9.3 Signs .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.9.4 Proof of Contraction Rules, Unitary Case . . . . . . . . . . . . . . 10.9.5 Emergence of a Sigma Model .. . . . . .. . . . . . . . . . . . . . . . . . . . 10.10 Semiclassical Construction of a Sigma Model, Orthogonal Symmetry Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.10.1 Structures .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.10.2 Leading-Order Contributions.. . . . . . .. . . . . . . . . . . . . . . . . . . . 10.10.3 Symbols for Ports and Contraction Lines for Links .. . . 10.10.4 Gauss and Wick . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.10.5 Signs .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.10.6 Proof of Contraction Rules, Orthogonal Case . . . . . . . . . . 10.10.7 Sigma Model .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.11 Outlook .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.12 Mixed Phase Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.13 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11 Level Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.2 Fictitious Particles (Pechukas-Yukawa Gas). . .. . . . . . . . . . . . . . . . . . . . 11.3 Conservation Laws. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.4 Intermultiplet Crossings . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.5 Level Dynamics for Classically Integrable Dynamics . . . . . . . . . . . . 11.6 Two-Body Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.7 Ergodicity of Level Dynamics and Universality of Spectral Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.7.1 Ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.7.2 Collision Time . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.7.3 Universality . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.8 Equilibrium Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.9 Random-Matrix Theory as Equilibrium Statistical Mechanics . . . 11.9.1 General Strategy . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.9.2 A Typical Coordinate Integral .. . . . . .. . . . . . . . . . . . . . . . . . . . 11.9.3 Influence of a Typical Constant of the Motion . . . . . . . . . 11.9.4 The General Coordinate Integral .. . .. . . . . . . . . . . . . . . . . . . .

473 475 476 479 479 481 483 486 487 490 491 492 494 495 496 498 500 501 502 505 507 511 511 513 519 521 522 528 529 529 531 532 533 536 536 541 547 548

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Contents

11.9.5 Concluding Remarks . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.10 Dynamics of Rescaled Energy Levels . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.11 Level Curvature Statistics . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.12 Level Velocity Statistics . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.13 Dyson’s Brownian-Motion Model . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.14 Local and Global Equilibrium in Spectra . . . . . .. . . . . . . . . . . . . . . . . . . . 11.15 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

550 551 555 563 567 577 585 588

12 Dissipative Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.2 Hamiltonian Embeddings.. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.3 Time-Scale Separation for Probabilities and Coherences .. . . . . . . . 12.4 Dissipative Death of Quantum Recurrences .. .. . . . . . . . . . . . . . . . . . . . 12.5 Complex Energies and Quasi-Energies . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.6 Different Degrees of Level Repulsion for Regular and Chaotic Motion .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.7 Poissonian Random Process in the Plane . . . . . .. . . . . . . . . . . . . . . . . . . . 12.8 Ginibre’s Ensemble of Random Matrices. . . . . .. . . . . . . . . . . . . . . . . . . . 12.8.1 Normalizing the Joint Density . . . . . .. . . . . . . . . . . . . . . . . . . . 12.8.2 The Density of Eigenvalues . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.8.3 The Reduced Joint Densities . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.8.4 The Spacing Distribution .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.9 General Properties of Generators . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.10 Universality of Cubic Level Repulsion . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.10.1 Antiunitary Symmetries . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.10.2 Microreversibility . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.11 Dissipation of Quantum Localization .. . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.11.1 Zaslavsky’s Map . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.11.2 Damped Rotator.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.11.3 Destruction of Localization . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.12 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

591 591 591 597 600 609 611 614 616 617 619 621 622 628 632 632 634 640 640 643 646 649 652

Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 655

Chapter 1

Introduction

As is by now well known there are two radically different types of motion in classical Hamiltonian mechanics: the regular motion of integrable systems and the chaotic motion of nonintegrable systems. The harmonic oscillator and the Kepler problem show regular motion, while systems as simple as a periodically driven pendulum or an autonomous conservative double pendulum can display chaotic dynamics. To identify the type of motion for a given system, one may look at a bundle of trajectories originating from a narrow cloud of points in phase space. The distance between any two such trajectories grows exponentially with time in the chaotic case; the growth rate is the so-called Lyapunov exponent. For regular motion, on the other hand, the distance in question may increase like a power of time but never exponentially; the corresponding Lyapunov exponent can thus be said to vanish. When quantum effects are important for a physical system under study, the notion of a phase-space trajectory loses its meaning and so does the notion of a Lyapunov exponent measuring the separation between trajectories. In cases with a discrete energy spectrum, exponential separation is strictly excluded even for expectation values of observables, on time scales on which level spacings are resolvable. The dynamics is then characterized by quasi-periodicity, i.e. recurrences rather than chaos in the classical sense. The quasi-period, i.e. a typical recurrence time, is inversely proportional to a typical level spacing and must of course tend to infinity in the classical limit (formally h¯ → 0). Having lost the classical distinction between regular motion and chaos when turning to quantum mechanics, one is naturally led to look for other, genuinely quantum mechanical criteria allowing one to distinguish two types of quantum dynamics. Such a distinction, if at all possible, should parallel the classical case: As h¯ → 0, one group should become regular and the other chaotic. Intrinsically quantum mechanical distinction criteria do in fact exist. Some are based on the energy spectrum; others on the energy eigenvectors, or on the temporal evolution of suitable expectation values. © Springer Nature Switzerland AG 2018 F. Haake et al., Quantum Signatures of Chaos, Springer Series in Synergetics, https://doi.org/10.1007/978-3-319-97580-1_1

1

2

a

1 Introduction

b

1.0

< Jy >/j 0.5

0. 0

500

t

1000

0

500

t

1000

Fig. 1.1 Quasi-periodic behavior of a quantum mean value (angular momentum component for a periodically kicked top) under conditions of classically regular motion (a) and classical chaos (b). For details see Sect. 8.6

A surprising lesson was taught to us by experiments with classical waves, electromagnetic and sound. Classical fields show much the same signatures of chaos as quantum probability amplitudes. Quantum or wave chaos arises whenever the pertinent wave equation is nonseparable for given boundary conditions. The character of the field is quite immaterial; it may be real, complex, or vector; the wave equation may be linear like Schrödinger’s or Maxwell’s or nonlinear like the Gross– Pitaevski equation for Bose–Einstein condensates. In all cases, classical trajectories or rays arise in appropriate short wavelength limits; and these trajectories are mostly chaotic in the sense of positive Lyapunov exponents provided the underlying wave problem is nonseparable. Nonseparability is shared by the wave problem and the classical Hamilton–Jacobi equation ensuing in the short-wave limit and may indeed be seen as the fundamental condition for chaos. We shall discuss quantum (and wave) distinctions between “regular” and “irregular” motions at some length in the chapters to follow. It should be instructive to jump ahead a little and infer from Fig. 1.1 that the temporal quasi-periodicity in quantum systems with discrete spectra can manifest itself in a way that tells us whether the classical limit will reveal regular or chaotic behavior. The figure displays the time evolution of the expectation value of a typical observable of a periodically kicked top, i.e., a quantum spin J with fixed square, J 2 = j (j + 1) = const. In the classical limit,1 j → ∞, the classical angular momentum J /j is capable of regular or chaotic motion depending on the values of control parameters and on the initial orientation. Figure 1.1 pertains to values of the control parameters for which chaos and regular motion coexist in the classical phase space. Parts a and b of Fig. 1.1 refer to initial quantum states localized entirely within classically regular and classically chaotic parts of phase space, respectively. In both cases quasi-periodicity is manifest in the form of recurrences. Obviously, however, the

1 The classical limit of a periodically driven spin was realized by Waldner [1]; much later came the first realization of the quantum version in Jessen’s group [2].

1 Introduction

3

quasi-periodicity in Fig. 1.1a is a nearly perfect periodicity (“collapse and revival”) while Fig. 1.1b shows an erratic sequence of recurrences. We shall show later how one can characterize this qualitative difference in quantitative terms. In any event, the difference in question is intrinsically quantum mechanical; the mean temporal separation of recurrences of either type is proportional to j and thus diverges in the classical limit. In contrast to classical chaos, quantum mechanically irregular motion cannot be characterized by extreme sensitivity to tiny changes of initial data: Due to the unitarity of quantum dynamics, the overlap of two wave functions remains time-independent, |φ(t)|ψ(t)|2 = |φ(0)|ψ(0)|2 , provided the timedependences of φ(t) and ψ(t) are generated by the same Hamiltonian. However, an alternative characterization of classical chaos, extreme sensitivity to slight changes of the dynamics, does carry over into quantum mechanics, as illustrated in Fig. 1.2. This figure refers to the same dynamical system as in Fig. 1.1 and shows the timedependent overlap of two wave functions. For each curve, the two states involved originate from one and the same initial state but evolve with slightly different values (relative difference 10−4 ) of one control parameter; that (tiny!) difference apart, all control parameters are set as in Fig. 1.1, as are the two initial states used. The timedependent overlap remains close to unity at all times, if the initial state is located in a classically regular part of the phase space. For the initial state residing in the classically chaotic region, however, the overlap falls exponentially, down to a level of order 1/j. Such sensitivity of the overlap to changes of a control parameter is quite striking and may indeed serve as a quantum criterion of irregular motion. In more recently established jargon the behavior in question is called fidelity decay . In an experimental realization of the kicked top [2] fidelity decay as well as other distinctions of chaotic and regular motion have been observed. Somewhat more widely known are the following two possibilities for the statistics of energy levels (or quasi-energy levels for periodically driven systems). Generic classically integrable systems with two or more degrees of freedom have quantum levels that tend to cluster and are not prohibited from crossing when a parameter in the Hamiltonian is varied [4]. The typical distribution of the spacings of neighboring levels is exponential, P (S) = exp (−S), just as if the levels arose as the uncorrelated events in a Poissonian random process. Classically nonintegrable systems with their phase spaces dominated by chaos, on the other hand, enjoy the privilege of levels that are correlated such that crossings are strongly resisted [5–10]. There are three universal degrees of level repulsion: linear, quadratic, and quartic [P (S) ∼ S β for S → 0 with β = 1, 2, or 4]. To which universality class a given nonintegrable system can belong is determined by the set of its symmetries. As will be explained in detail later, anti-unitary symmetries such as time-reversal invariance play an important role. Broadly speaking, in systems without any antiunitary symmetry generically, β = 2; in the presence of an antiunitary symmetry one typically finds linear level repulsion; the strongest resistance to level crossings, β = 4, is characteristic of time-reversal invariant systems possessing Kramers’ degeneracy but no geometric symmetry at all. Clearly, the alternative between level clustering level repulsion belongs to the worlds of quanta and waves. It parallels, however, the classical distinction between

4

1 Introduction

Fig. 1.2 Time dependence of overlap of two wave vectors of kicked top, both originating from the same initial state but evolving with slightly different values of a control parameter. Upper and lower curves refer to conditions of classically regular and classically chaotic motion, respectively. For details see Sect. 8.6. Courtesy of Peres [3]

predominantly regular and predominantly chaotic motion. Figure 1.3 illustrates the four generic possibilities mentioned, obtained from the numerically determined quasi-energies of various types of periodically kicked tops; the corresponding portraits of trajectories in the respective classical phase spaces (Fig. 8.3) on p. 324

1 Introduction

5

Fig. 1.3 Level spacing distributions for kicked tops under conditions of classically regular motion β = 0 and classical chaos β = 1, 2, 4. The latter three curves pertain to tops from different universality classes and display linear β = 1, quadratic β = 2, and quartic level repulsion β = 4

will show the correlation between the quantum and the classical distinction of the two types of dynamics. An experimentally determined spectrum of nuclear, atomic, or molecular levels will in general, if taken at face value, display statistics close to Poissonian, even when there is no reason to suppose that the corresponding classical many-body problem is integrable or at least nearly integrable. To uncover spectral correlations, the complete set of levels must be separated into subsets, each of which has fixed values of the quantum numbers related to the symmetries of the system. Subsets that are sufficiently large to allow for a statistical analysis generally reveal level repulsion of the degree expected on grounds of symmetry. All three chaos related universal degrees of level repulsion, have by now been observed. linear and quadratic varieties have been observed experimentally to date. As for linear repulsion, the first data came from nuclear physics in the 1960s [8– 13]; mostly much later, confirmation came from microwaves [14–17], molecular [18] and atomic [19] spectroscopy, and sound waves [20, 21]. Figure 1.4 displays such results from various fields and provides evidence for the kinship of quantum and wave chaos. As regards quadratic level repulsion, it would be hopeless to look in nuclei; even though the weak interaction does break time-reversal invariance, that breaking is far too feeble to become visible in level-spacing distributions. Equally unwieldy are normal-size atoms with a magnetic field to break conventional time-reversal invariance. A homogeneous field would not change the linear degree of repulsion since it preserves antiunitary symmetry (generalized time-reversal); such a symmetry survives even in nonaligned electric and magnetic fields if both fields are homogeneous. Rydberg atoms exposed to strong and sufficiently inhomogeneous magnetic fields do possess quadratically repelling levels and will eventually reveal that property to spectroscopists. Microwave experiments again came to help [26– 28]; their showing quadratic repulsion was a most welcome achievement, all the

6

1 Introduction

Fig. 1.4 Level spacing distributions for (a) the Sinai billiard [10], (b) a hydrogen atom in a strong magnetic field [22], (c) an NO2 molecule [18], (d) a vibrating quartz block shaped like a three dimensional Sinai billiard [23], (e) the microwave spectrum of a three-dimensional chaotic cavity [24], (f) a vibrating elastic disc shaped like a quarter stadium [25]. Courtesy of H.-J. Stöckmann

more so since it once more underscored that wave chaos is not an exclusive privilege of quanta. See Fig. 1.5. Single atoms and molecules, free or in solid-state environments, have still been obstinate to reveal quartic repulsion, too. Again, the first observation in microwave graphs brought much relief [29–31]. See Fig. 1.5. In numerical treatments of systems with global chaos, an interesting predictability paradox arises. As indicated by a positive Lyapunov exponent, small

1 Introduction

7

Fig. 1.5 Left: Level-spacing distribution for microwave graphs with quadratic repulsion; smooth curve from GUE, dashed from GOE [31]. Right: Level-spacing distribution for microwave graphs with quartic repulsion; smooth curve from GSE, dashed from GUE [30, 31]. Courtesy of H.-J. Stöckmann

but inevitable numerical inaccuracies will amplify exponentially and thus render impossible long-range predictions of classical trajectories. Much more reliable by comparison can be the corresponding quantum predictions for mean values, at least in the case of discrete levels; quasi-periodic time evolution, even if very many sinusoidal terms contribute, is incapable of exponential runaway. A stupefying illustration of quantum predictions outdoing classical ones is given by the kinetic energy of the periodically kicked rotator in Fig. 1.6. The quantum mean (dotted) originates from the momentum eigenstate with a vanishing eigenvalue while the classical curve (full) represents an average over 250,000 trajectories starting from a cloud of points with zero momentum and equipartition for the conjugate angle. The classical average displays diffusive growth of the kinetic energy—which, as will be shown later, is tantamount to chaos—until time is reversed after 500 kicks. Instead of retracing its path back to the initial state, the classical mean soon turns again to diffusive growth. The quantum mean, on the other hand, is symmetric around t = 500 without noticeable error. Evidently, such a result suggests that a little caution is necessary with respect to our often too naive notions of classical determinism and quantum indeterminacy. Following a famous conjecture of Bohigas, Giannoni, and Schmit [10], certain universal features of spectral fluctuations in classically chaotic systems have been found to be well described by random-matrix theory [33, 34]. Based on a suggestion by Wigner, this theory takes random Hermitian matrices as models of Hamiltonians H of autonomous systems. Unitary random matrices are employed similarly for periodically driven systems, as models of the unitary “Floquet” operators F describing the change of the quantum state during one cycle of the driving;2 powers

2 Chaotic scattering and graphs make for equally interesting applications of unitary random matrices.

8

1 Introduction

Fig. 1.6 Classical (full curve) and quantum (dotted) mean kinetic energy of the periodically kicked rotator, determined by numerical iteration of the respective maps (see Chap. 8 for details). The quantum mean follows the classical diffusion for times up to some “break” time and then begins to display quasi-periodic fluctuations. After 500 kicks the direction of time was reserved; while the quantum mean accurately retraces its history, the classical mean reverts to diffusive growth, thus revealing the extreme sensitivity of chaotic systems to tiny perturbations (here round-off errors). For details see Chaps. 8, 12. Courtesy of Graham and Dittrich [32]

F n of the Floquet operator with n = 1, 2, 3, . . . yield a stroboscopic description of the driven dynamics under consideration. There are four important classes for both Hermitian and unitary random matrices, one related to classically integrable systems and three to nonintegrable ones. The four classes in question illustrate the quantum distinction between regular and irregular dynamics discussed above: their eigenvalues (energies in the Hermitian case and quasi-energies, i.e., eigenphases in the unitary case) cluster and repel according to one of three universal degrees, respectively. The matrix ensembles with level repulsion can be defined by the requirement of maximum statistical independence of all matrix elements within the constraints imposed by symmetries. We shall present a review of random-matrix theory in Chap. 5. Since several excellent texts on this subject are available [33–35], neither completeness nor rigor will be attempted. We shall rather keep to an introductory style and intuitive arguments. When fancier machinery is employed for some more up-to-date issues, it is patiently developed rather than assumed as a prerequisite. The so called sigma model, originally a product of field theory for disordered systems and rapidly adapted to chaotic dynamics [36], has become an important theoretical tool for analyzing spectral statistics of ensembles of (full, sparse, or banded) random matrices. More recently, it has even allowed to investigate individual dynamics and to clarify conditions under which universal level repulsion á la RMT or quantum localization. Correspondingly, we had to develop that technique in some depth, and now indeed it permeates several chapters. A certain class of periodically driven systems of which the kicked rotator is a prototype displays an interesting quantum anomaly. Like all periodically driven

1 Introduction

9

systems, those in question are generically nonintegrable classically. Even under the conditions of fully developed classical chaos, however, the kicked rotator does not display level repulsion. The reason for this anomaly can be inferred from Fig. 1.6. The kinetic energy, and thus the quantum mechanical momentum uncertainty, does not follow the classical diffusive growth indefinitely. After a certain break time the quantum mean enters a regime of quasi-periodic behavior. Clearly, the eigenvectors of the corresponding Floquet operator then have an upper limit to their width in the momentum representation. They are, in fact, exponentially localized in that basis, and the width is interpretable as a “localization length”. Two eigenvectors much further apart in momentum than a localization length have no overlap and thus no matrix elements of noticeable magnitude with respect to any observable; they have no reason, therefore, to stay apart in quasi-energy and will display Poissonian level statistics. A theoretical understanding of the phenomenon has been generated by Grempel, Fishman, and Prange [37] who were able to map the Schrödinger equation for the kicked rotator onto that for Anderson’s one-dimensional tight-binding model of a particle in a random potential. Actually, the potential arrived at in the map is pseudorandom, but that restriction does not prevent exponential localization, which is rigorously established only for the strictly random case. It is amusing to find that number-theoretical considerations are relevant in a quantum context here. Quantum localization occurs only in the case where a certain dimensionless version of Planck’s constant is an irrational number. Otherwise, the equivalent tight-binding model has a periodic potential and thus extended eigenvectors of the Bloch type and eigenvalues forming continuous bands. A treatment of the kicked rotor will form part of Chap. 8 on quantum localization. Ensembles of banded random matrices will also be treated in that chapter; these display a transition from localized to “generic” delocalized states as the band width grows. We could not get away without a short chapter on classical mechanics, since a fine discovery on chaotic trajectories has proven the clue to understanding universality (and conditions thereof) of quantum spectral fluctuations: long periodic orbits do not exist as mutually independent individuals but rather come in bunches with arbitrarily small orbit-to-orbit action differences. Under weak resolution the topological distinction between the orbits in a bunch is blurred such that a bunch looks like a single orbit. The construction principle of bunches is related to close self-encounters of an orbit where two or more orbit stretches run mutually close for a time span much longer than the inverse Lyapunov rate . More than a century ago, the great Henri Poincaré came to within a hair’s bredth of that phenomenon when [38] writing “Etant données . . . une solution particulière quelconque de ces équations, on peut toujours trouver une solution périodique (dont la période peut, il est vrai, être très longue), telle que la différence entre les deux solutions soit aussi petite que l’on veut, pendant un temps aussi long qu’on le veut. D’ailleurs, ce qui nous rend ces solutions périodiques si précieuses, c’est qu’elles sont, pour ainsi dire, la seule brêche par où nous puissions essayer de pénétrer dans une place jusqu’ici réputée inabordable”.

10

1 Introduction

A separate chapter will be devoted to the semiclassical approximation for chaotic dynamics, Gutzwiller’s periodic-orbit theory. The basic trace formulas for maps and autonomous flows will be derived and discussed. As a beautiful application we shall describe the recently established semiclassical explanation of universal spectral fluctuations. Long periodic orbits, with periods of the order of the Heisenberg time TH , will be found “at work”. The latter time is related to the mean level spacing Δ, the energy scale on which universal spectral fluctuations arise, as TH = 2π h¯ /Δ; inasmuch as the mean spacing is small in the sense Δ ∝ h¯ f with f the number of degrees of freedom (note f ≥ 2 for chaos), the Heisenberg time diverges as TH ∝ h¯ −f +1 in the limit h¯ → 0. The success of periodic-orbit theory on that time scale is quite remarkable in view of the infamous exponential proliferation of periodic orbits with increasing period. The key to success was provided by a new insight into classical chaotic dynamics,the just mentioned phenomenon of orbit bunching. Considerable emphasis will be given to an interesting reinterpretation of randommatrix theory as “level dynamics” in Chap. 11. Such a reinterpretation was already sought by Dyson but could be implemented only recently on the basis of a discovery of Pechukas [39]. The fate of the N eigenvalues and eigenvectors of an N × N Hermitian matrix, H = H0 +λV , is in one-to-one correspondence with the classical Hamiltonian dynamics of a particular one dimensional N-particle system upon changing the weight λ of a perturbation V [39–41]. This fictitious system, now often called Pechukas–Yukawa gas, has λ as a time, the eigenvalues En of H as coordinates, the diagonal elements Vnn of the perturbation V in the H representation as momenta, and the off-diagonal elements Vnm related to certain angular momenta. As explained in Chap. 11, random-matrix theory now emerges as a result of standard equilibrium statistical mechanics for the fictitious N-particle system. Moreover, the universality of spectral fluctuations of chaotic dynamics will find a natural explanation on that basis. The fictitious N-particle model also sheds light on another important problem. A given Hamiltonian H0 +λV may have H0 integrable and V breaking integrability. The level dynamics will then display a transition from level clustering to level repulsion. We shall see that such transitions can be understood as equilibration processes for certain observables in the fictitious N-particle model. A related phenomenon is the transition from one universality class to another displayed by a Hamiltonian H (λ) = H0 + λV when H0 and V have different (antiunitary) symmetries. An ad hoc description of such transitions could previously be given in terms of Dyson’s Brownian-motion model, a certain “dynamic” generalization of random-matrix theory. As will become clear below, that venerable model is in fact rigorously implied by level dynamics, i.e., the λ-dependence of H0 + λV , provided the energy scale is reset in a suitable λ-dependent manner. In the end, we turn to dissipative systems. Instead of Schrödinger’s equation, we face master equations of Markovian processes. Real energies (or quasi-energies) are replaced with complex eigenvalues of generators of time translations while the density operator or a suitable representative, like the Wigner function, takes over the role of the wave function. One would again like to identify genuinely

1 Introduction

11

quantum mechanical criteria to distinguish two types of motion, one becoming regular and the other chaotic in the classical limit. There is a general qualitative argument, however, suggesting that such quantum distinctions will be a lot harder to establish for dissipative than for Hamiltonian systems. The difference between, say, a complicated limit cycle and a strange attractor becomes apparent when phase-space structures are considered over several orders of magnitude of action scales. The density matrix or the Wigner function will of course reflect phase-space structures on action scales upward of Planck’s constant and will thus indicate, with reasonable certainty, the difference between a strange and a simple attractor. But for any representative of the density operator to reveal this difference in terms of genuinely quantum mechanical features without classical meaning, it would have to embody coherences with respect to states distinct on action scales that are large compared to Planck’s constant. Such coherences between or superpositions of macroscopically distinct states are often metaphorized as Schrödinger cat states. In the presence of even weak damping, such superpositions tend to decohere so rapidly in time to mixtures that observing them becomes difficult if not impossible. An illustration of the dissipative death of quantum coherences is presented in Fig. 1.7. As in Fig. 1.1a, column a in Fig. 1.7 shows quantum recurrences for an angular momentum component of a periodically kicked top without damping. The control parameters and the initial state are kept constant for all curves in the column and are chosen such that the classical limit would yield regular behavior. Proceeding down the column, the spin quantum number j is increased to demonstrate the rough proportionality of the quasi-period to j (which may be thought of as inversely proportional to Planck’s constant). The curves in column b of Fig. 1.7 correspond to their neighbors in column a in all respects except for the effect of weak dissipation. The damping mechanism is designed so as to leave j a good quantum number, and the damping constant is so small that classical trajectories would not be influenced noticeably during the times over which the plots extend. The sequences of quantum mechanical “collapses and revivals” is seen to be strongly altered by the dissipation, even though the quasi-period is in all cases smaller than the classical decay time. We shall argue below that the lifetime of the quantum coherences is shorter than the classical decay time and also shorter than the time constant for quantum observables not sensitive to coherences between “macroscopically” distinct states by a factor of the order of 1/j ∼ h¯ . Similarly dramatic is the effect of dissipation on the erratic recurrences found under conditions of classical chaos. Obviously, dissipation will then tend to wipe out the distinction between regular and erratic recurrences. The quantum localization in the periodically kicked rotator also involves coherences between states that are distinct on action scales far exceeding Planck’s constant. Indeed, if expressed in action units, the localization length is large compared to h¯ . One must therefore expect localization to be destroyed by dissipation. Figure 1.8 confirms that expectation. As a particular damping constant is increased, the time dependence of the kinetic energy of the rotator tends to resemble indefinite classical diffusion ever more closely.

12

1 Introduction

Fig. 1.7 Quantum recurrences for periodically kicked top without (left column) and with (right column) damping under conditions of classically regular motion. For details see Chap. 12

We ought to confess right away which important parts of ‘quantum chaos’ we have given short shrift or, worse, stayed away from entirely: ‘chaotic’ wave functions, scars, rogue waves, scattering, transport, quantum ergodicity, many-body systems and, perhaps most regrettably, the zeros of Riemann’s zeta function and its elusive dynamical-system counterpart.

References

13

Fig. 1.8 Classical (uppermost curve) and quantum mean kinetic energy of periodically kicked rotor with damping. The stronger the damping, the steeper the curves. In all cases, the damping is so weak that the classical curve is still indistinguishable from that without dissipation. Courtesy of Dittrich and Graham [32]

References 1. 2. 3. 4. 5. 6. 7.

F. Waldner, D.R. Barberis, H. Yamazaki, Phys. Rev. A 31, 420 (1985) S. Chaudhury, A. Smith, B.E. Anderson, S. Ghose, P.S. Jessen, Nature 461, 768 (2009) A. Peres: Quantum Theory: Concepts and Methods (Kluwer Academic, New York, 1995) M.V. Berry, M. Tabor, Proc. R. Soc. Lond. A 356, 375 (1977) G.M. Zaslavskii, N.N. Filonenko, Sov. Phys. JETP 8, 317 (1974) M.V. Berry, M. Tabor, Proc. R. Soc. Lond. A 349, 101 (1976) M.V. Berry, Les Houches Session XXXVI 1981, in Chaotic Behavior of Deterministic Systems, ed. by G. Iooss, R.H. Helleman, R. Stora (North-Holland, Amsterdam, 1983) 8. O. Bohigas, R. Haq, A. Pandey, Nuclear Data for Science and Technology, ed. by K. Böckhoff (Reidel, Dordrecht, 1983) 9. G. Bohigas, M.-J. Giannoni, Mathematical Andrandom Computational Methods in Nuclear Physics. Lecture Notes in Physics, vol. 209 (Springer, Berlin/Heidelberg, 1984) 10. O. Bohigas, M.J. Giannoni, C. Schmit, Phys. Rev. Lett. 52, 1 (1984) 11. T.A. Brody, J. Flores, J.B. French, P.A. Mello, A. Pandey, P.S.M. Wong, Rev. Mod. Phys. 53, 385 (1981) 12. N. Rosenzweig, C.E. Porter, Phys. Rev. 120, 1698 (1960) 13. H.S. Camarda, P.D. Georgopulos, Phys. Rev. Lett. 50, 492 (1983) 14. M.R. Schroeder, J. Audio Eng. Soc. 35, 307 (1987) 15. H.J. Stöckmann, J. Stein, Phys. Rev. Lett. 64, 2215 (1990) 16. H. Alt, H.-D. Gräf, H.L. Harney, R. Hofferbert, H. Lengeler, A. Richter, P. Schart, H.A. Weidenmüller, Phys. Rev. Lett. 74, 62 (1995) 17. H. Alt, H.-D. Gräf, R. Hofferbert, C. Rangacharyulu, H. Rehfeld, A. Richter, P. Schart, A. Wirzba, Phys. Rev. E 54, 2303 (1996) 18. T. Zimmermann, H. Köppel, L.S. Cederbaum, C. Persch, W. Demtröder, Phys. Rev. Lett. 61, 3 (1988); G. Persch, E. Mehdizadeh, W. Demtröder, T. Zimmermann, L.S. Cederbaum: Ber. Bunsenges. Phys. Chem. 92, 312 (1988) 19. H. Held, J. Schlichter, G. Raithel, H. Walther, Europhys. Lett. 43, 392 (1998) 20. C. Ellegaard, T. Guhr, K. Lindemann, H.Q. Lorensen, J. Nygård, M. Oxborrow, Phys. Rev. Lett. 75, 1546 (1995) 21. C. Ellegaard, T. Guhr, K. Lindemann, J. Nygård, M. Oxborrow, Phys. Rev. Lett. 77, 4918 (1996) 22. A. Hoenig, D. Wintgen, Phys. Rev. A 39, 5642 (1989) 23. M. Oxborrow, C. Ellegaard, Proceedings of the 3rd Experimental Chaos Conference, Edinburgh (1995) 24. S. Deus, P.M. Koch, L. Sirko, Phys. Rev. E 52, 1146 (1995) 25. O. Legrand, C. Schmit, D. Sornette, Europhys. Lett. 18, 101 (1992) 26. U. Stoffregen, J. Stein, H.-J. Stöckmann, M. Ku´s, F. Haake, Phys. Rev. Lett. 74, 2666 (1995)

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1 Introduction

27. P. So, S.M. Anlage, E. Ott, R.N. Oerter, Phys. Rev. Lett. 74, 2662 (1995) ˙ 28. O. Hul, S. Bauch, P. Pakoñski, N. Savytskyy, K. Zyczkowski, L. Sirko, Phys. Rev. E 69, 056205 (2004) 29. C.H. Joyner, S. Müller, M. Sieber, Europhys. Lett. 107, 50004 (2014) 30. A. Rehemanjiang, M. Allgaier, C.H. Joyner, S. Müller, M. Sieber, U. Kuhl, H.-Stöckmann, Phys. Rev. Lett. 107, 064101 (2016) 31. A. Rehemanjiang, M. Richter, U. Kuhl, H.-J. Stöckmann, Phys. Rev. E 97, 022204 (2018) 32. T. Dittrich, R. Graham, Ann. Phys. 200, 363 (1990) 33. M.L. Mehta, Random Matrices (Academic, New York, 1967); 2nd edition 1991; 3rd edition (Elsevier, 2004) 34. C.E. Porter (ed.), Statistical Theory of Spectra (Academic, New York 1965) 35. T. Guhr, A. Müller-Groeling, H.A. Weidenmüller, Phys. Rep. 299, 192 (1998) 36. K. Efetov: Supersymmetry in Disorder and Chaos (Cambridge University Press, Cambridge, 1997) 37. S. Fishman, D.R. Grempel, R.E. Prange, Phys. Rev. Lett. 49, 509 (1982); Phys. Rev. A 29, 1639 (1984) 38. J.H. Poincaré, Les Méthodes nouvelles de la mécanique céleste (Gauthier-Villard, Paris, 1899); (reed. A. Blanchard, Paris, 1987) 39. P. Pechukas, Phys. Rev. Lett. 51, 943 (1983) 40. T. Yukawa, Phys. Rev. Lett. 54, 1883 (1985) 41. T. Yukawa, Phys. Lett. A 116, 227 (1986)

Chapter 2

Time Reversal and Unitary Symmetries

2.1 Autonomous Classical Flows A classical Hamiltonian system is called time-reversal invariant if from any given solution x(t), p(t) of Hamilton’s equations an independent solution x (t ), p (t ), is obtained with t = −t and some operation relating x and p to the original coordinates x and momenta p. The simplest such invariance, to be referred to as conventional, holds when the Hamiltonian is an even function of all momenta, t → −t , x → x , p → −p , H (x, p) = H (x, −p) .

(2.1.1)

This is obviously not a canonical transformation since the Poisson brackets {pi , xj } = δij are not left intact. The change of sign brought about for the Poisson brackets is often acknowledged by calling classical time reversal anticanonical. We should keep in mind that the angular momentum vector of a particle is bilinear in x and p and thus odd under conventional time reversal. The motion of a charged particle in an external magnetic field is not invariant under conventional time reversal since the minimal-coupling Hamiltonian (p − (e/c)A)2 /2m is not even in p. Such systems may nonetheless have some other, nonconventional time-reversal invariance, to be explained in Sect. 2.10. Hamiltonian systems with no time-reversal invariance must not be confused with dissipative systems. The differences between Hamiltonian and dissipative dynamics are drastic and well known. Most importantly from a theoretical point of view, all Hamiltonian motions conserve phase-space volumes according to Liouville’s theorem, while for dissipative processes such volumes contract in time. The difference between Hamiltonian systems with and without time-reversal invariance, on the other hand, is subtle and has never attracted much attention in the realm of classical physics. It will become clear below, however, that the latter difference plays an important role in the world of quanta [1–3].

© Springer Nature Switzerland AG 2018 F. Haake et al., Quantum Signatures of Chaos, Springer Series in Synergetics, https://doi.org/10.1007/978-3-319-97580-1_2

15

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2 Time Reversal and Unitary Symmetries

2.2 Spinless Quanta The Schrödinger equation ˙ ih¯ ψ(x, t) = H ψ(x, t)

(2.2.1)

is time-reversal invariant if, for any given solution ψ(x, t), there is another one, ψ (x, t ), with t = −t and ψ uniquely related to ψ. The simplest such invariance, again termed conventional, arises for a spinless particle with the real Hamiltonian H (x, p) =

p2 + V (x) , V (x) = V ∗ (x) , 2m

(2.2.2)

where the asterisk denotes complex conjugation. The conventional reversal is t → −t , x → x , p → −p , ψ(x) → ψ ∗ (x) = Kψ(x) .

(2.2.3)

In other words, if ψ(x, t) solves (2.2.1) so does ψ (x, t) = Kψ(x, −t). The operator K of complex conjugation obviously fulfills K2 = 1 ,

(2.2.4)

i.e., it equals its inverse, K = K −1 . Its definition also implies K (c1 ψ1 (x) + c2 ψ2 (x)) = c1∗ Kψ1 (x) + c2∗ Kψ2 (x) ,

(2.2.5)

a property commonly called antilinearity. The transformation ψ(x) → Kψ(x) does not change the modulus of the overlap of two wave functions, |Kψ|Kφ|2 = |ψ|φ|2 ,

(2.2.6)

while the overlap itself is transformed into its complex conjugate, Kψ|Kφ = ψ|φ∗ = φ|ψ .

(2.2.7)

The identity (2.2.7) defines the property of antiunitarity which implies antilinearity [1] (Problem 2.4). It is appropriate to emphasize that we have defined the operator K with respect to the position representation. Dirac’s notation makes this distinction of K especially obvious. If some state vector |ψ is expanded in terms of position eigenvectors |x,  |ψ =

dx ψ(x)|x ,

(2.2.8)

2.3 Spin-1/2 Quanta

17

the operator K acts as  K|ψ =

dx ψ ∗ (x)|x ,

(2.2.9)

i.e., as K|x = |x. A complex conjugation operator K can of course be defined with respect to any representation. It is illustrative to consider a discrete basis and introduce   K |ψ = K ψν |ν = ψν∗ |ν . (2.2.10) ν

ν

Conventional time reversal, i.e., complex conjugation in the position representation, can then be expressed as K = U K

(2.2.11)

with a certain symmetric unitary matrix U, the calculation of which is left to the reader as Problem 2.5. Unless otherwise stated, the symbol K will be reserved for complex conjugation in the coordinate representation, as far as orbital wave functions are concerned. Moreover, antiunitary time-reversal operators will, for the most part, be denoted by T . Only the conventional time-reversal for spinless particles has the simple form T = K.

2.3 Spin-1/2 Quanta All time-reversal operators T must be antiunitary T ψ|T φ = φ|ψ ,

(2.3.1)

because of (1) the explicit factor i in Schrödinger’s equation and (2) since they should leave the modulus of the overlap of two wave vectors invariant. It follows from the definition (2.3.1) of antiunitarity that the product of two antiunitary operators is unitary. Consequently, any time-reversal operator T can be given the so-called standard form T = UK ,

(2.3.2)

where U is a suitable unitary operator and K the complex conjugation with respect to a standard representation (often chosen to be the position representation for the orbital part of wave functions). Another physically reasonable requirement for every time-reversal operator T is that any wave function should be reproduced, at least to within a phase factor, when

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2 Time Reversal and Unitary Symmetries

acted upon twice by T , T 2 = α , |α| = 1 .

(2.3.3)

Inserting the standard form (2.3.2) in (2.3.3) yields1 U KU K = U U ∗ K 2 = U U ∗ = α, i.e., U ∗ = αU −1 = αU † = α U˜ ∗ . The latter identity once iterated gives U ∗ = α 2 U ∗ , i.e., α 2 = 1 or T 2 = ±1 .

(2.3.4)

The positive sign holds for conventional time reversal with spinless particles. It will become clear presently that T 2 = −1 in the case of a spin-1/2 particle. See also Problem 2.9. The conventional time-reversal operation for a spin-1/2 results from requiring that T J T −1 = −J

(2.3.5)

holds not only for the orbital angular momentum but likewise for the spin. With respect to the spin, however, T cannot simply be the complex conjugation operation since all purely imaginary Hermitian 2 × 2 matrices commute with one another. The more general structure (2.3.2) must therefore be considered. Just as a matter of convenience, we shall choose K as the complex conjugation in the standard representation where the spin operator S takes the form S = h2¯ σ ,       01 0 −i 1 0 σx = , σy = , σz = . 10 i 0 0 −1

(2.3.6)

The matrix U is then constrained by (2.3.5) to obey T σx T −1 = U Kσx KU −1 = U σx U −1 = −σx T σy T −1 = U Kσy KU −1 = −U σy U −1 = −σy

(2.3.7)

T σz T −1 = U Kσz KU −1 = U σz U −1 = −σz , i.e., U must commute with σy and anticommute with σx and σz . Because any 2 × 2 matrix U can be represented as a sum of Pauli matrices, we can write U = ασx + βσy + γ σz + δ .

(2.3.8)

1 Matrix transposition will always be represented by a tilde, while the dagger † will denote Hermitian conjugation.

2.3 Spin-1/2 Quanta

19

The first of the equations (2.3.7) immediately gives α = δ = 0; the second yields γ = 0, whereas β remains unrestricted by (2.3.7). However, since U is unitary, β must have unit modulus. It is thus possible to choose β = i whereupon the timereversal operation reads T = iσy K = eiπσy /2 K .

(2.3.9)

This may be taken to include, if necessary, the time reversal for the orbital part of wave vectors by interpreting K as complex conjugation both in the position representation and in the standard spin representation. The operation (2.3.9) squares to minus unity, in contrast to conventional time reversal for spinless particles. Indeed, T 2 = iσy Kiσy K = (iσy )2 = −1. If one is dealing with N particles with spin 1/2, the matrix U must obviously be taken as the direct product of N single-particle matrices, T = iσ1y iσ2y . . . iσNy K   π  Sy σ1y + σ2y + . . . + σNy K = exp iπ = exp i K , (2.3.10) 2 h¯ where Sy now is the y-component of the total spin S = h¯ (σ 1 + σ2 + . . . + σ N )/2. Equivalently, one may write   Jy T = exp iπ K h¯

(2.3.11)

where Jy = Ly + Sy is the total angular momentum. This follows immediately from [Sy , Ly ] = 0 and ei2πLy /h¯ = 1 (as Ly /h¯ has only integer values eigenvalues). In the following we will choose (2.3.11) to be the general form of the conventional time-reversal operator in the position representation. The square of T depends on the number of particles according to

+1 N even T = −1 N odd . 2

(2.3.12)

We shall occasionally refer to “kicked tops”, dynamical systems involving only components of an angular momentum J = (Jx , Jy , Jz ) as dynamical variables. The square J 2 is then conserved and the Hilbert space can be chosen as the 2j + 1 dimensional space with J 2 = j (j + 1) spanned by the eigenvectors |j, m of, say, Jz with m = −j, −j + 1, . . . , j . The conventional time reversal operator is then given by (2.3.11) as T = eiπJy /h¯ K with K|j, m = |j, m and squares to +1 or −1 when the quantum number j is integer and half-integer, respectively.

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2 Time Reversal and Unitary Symmetries

2.4 Hamiltonians Without T Invariance If there is no time-reversal invariance and there are also no geometric symmetries or any other symmetries represented by unitary operators (see Sect. 2.6) then any Hermitian matrix (Hμν )∗ = H˜ μν = Hνμ , is a permissable Hamiltonian. We will shortly proceed to identify the subclasses of Hermitian matrices to which timereversal invariant Hamiltonians belong in the following sections. At this stage let us introduce the concept of “canonical transformations” as the largest class of unitary operators which leave a class of permissable Hamiltonians invariant: if H is an permissable Hamiltonian, so is V H V † for all canonical transformations V . Coming back to Hamiltonians unrestricted by antiunitary symmetries in a Hilbert space of dimension N, the canonical transformations is the Lie group U(N) of unitary N × N matrices. Indeed if H = H † and H = V H V † it is easy to check that H = H † for any V ∈ U(N). A few more remarks concerning the time evolution operators of permissable Hamiltonians is in place. The time evolution operators U (t) = e−iH t /h¯ generated by all complex Hermitian Hamiltonians form the Lie group U (N). The Hamiltonians themselves can be associated with the generators X = iH of the Lie algebra u(N), the tangent space to the group U(N). Moreover, a general complex Hermitian N ×N Hamiltonian can be diagonalized by a unitary transformation Hμν =

N  λ=1

† Uμλ Eλ Uλν

=

N 

∗ Uμλ Eλ Uνλ .

(2.4.1)

λ=1

The diagonalizing transformation U is not unique even if one arranges the energies in an increasing way, Eλ+1 ≥ Eλ . Indeed, if U is multiplied with a diagonal unitary matrix from the right U = U diag (e−iφ1 , . . . e−iφN ) then U will also diagonalize H (and keep the order of the eigenvalues). The diagonal unitary matrices form a subgroup U(1)N of U(N). If all eigenvalues are different, U(1)N is indeed the maximal subgroup of U(N) that leaves the diagonal matrix containing the eigenvalues unchanged. If there are degeneracies, the corresponding subspace allows for more general transformations. However, in the absence of unitary symmetries degeneracies can only occur accidentally, i.e., degeneracies will be destroyed by any permissable change of physical parameters or coupling constants entering the Hamiltonian. Right multiplication of a matrix U ∈ U(N) with V ∈ U(1)N defines an equivalence relation: U ∼ U if and only if there is V ∈U(1)N such that U = U V . By identifying all equivalent matrices one arrives at the coset space U(N)/U(1)N (the set of equivalence classes). For all of these reasons, complex Hermitian Hamiltonians are said to form the “unitary symmetry class”. Alternatively, this symmetry class is sometimes denoted as class A (see Sect. 2.16 for more details).

2.5 T Invariant Hamiltonians, T 2 = 1

21

2.5 T Invariant Hamiltonians, T 2 = 1 Next we consider Hamiltonians H such that there is an antiunitary operator T with [H, T ] = 0 , T 2 = 1 .

(2.5.1)

and T can be defined without reference to particular values of the real physical parameters (coupling constants but also physical constants such as h¯ or a particle mass) that may enter the Hamiltonian. In this case H can always be given a real matrix representation and such a representation can be found without diagonalizing H . As a first step toward proving the above statement, we demonstrate that, with the help of an antiunitary T squaring to plus unity, T invariant basis vectors ψν can be constructed. Take any vector φ1 and a complex number a1 . The vector ψ1 = a1 φ1 + T a1 φ1

(2.5.2)

is then T invariant, T ψ1 = ψ1 . Next, take any vector φ2 orthogonal to ψ1 and a complex number a2 . The combination ψ2 = a2 φ2 + T a2 φ2

(2.5.3)

is again T invariant. Moreover, ψ2 is orthogonal to ψ1 since ψ2 |ψ1  = a2∗ φ2 |ψ1  + a2 T φ2 |ψ1   ∗ = a2 T 2 φ2 |ψ1 = a2 φ2 |ψ1 ∗ = 0 .

(2.5.4)

By so proceeding we eventually arrive at a complete set of mutually orthogonal vectors. If desired, the numbers aν can be chosen to normalize as ψμ |ψν  = δμν . With respect to a T invariant basis, the Hamiltonian H = T H T is real, Hμν = ψμ |H ψν  = T ψμ |T H ψν ∗ ∗ = ψμ |T H T 2 ψν ∗ = ψμ |T H T ψν ∗ = Hμν .

(2.5.5)

Note that the Hamiltonians in question can be made real without being diagonalized first. It is therefore quite legitimate to say that they are generically real matrices. In the absence of any other unitary symmetries any real symmetric N × N matrix H = H˜ = H ∗ is a permissable Hamiltonian of a time-reversal invariant system with T 2 = 1. The real orthogonal matrices form the Lie group O(N) ⊂ U(N) of N × N matrices V that satisfy V V˜ = 1 and V = V ∗ . They leave the set of real symmetric Hamiltonians invariant. Indeed, if H = H˜ = H ∗ it is straightforward to check that H = V H V † also satisfies H = H˜ = H ∗ for any V ∈ O(N). As

22

2 Time Reversal and Unitary Symmetries

the largest subset of U(N) that leaves the real symmetric Hamiltonians invariant, the group O(N) forms the corresponding canonical transformation. Moreover a real symmetric Hamiltonian can be diagonalised with an orthogonal matrix O ∈ O(N),

i.e. Hμν = N O E μλ λ Oνλ . By requiring that the energies are ordered Eλ+1 ≥ λ=1 Eλ and choosing det O = 1 (this reduces the group of orthogonal matrices O(N) to the subgroup SO(N) of special orthogonal matrices) the diagonlization matrix becomes essentially unique—only accidentally degenerate eigenvalues leave (again) some choice in the corresponding subspace. For these reasons quantum systems with a time reversal invariance and T 2 = 1 (and no other symmetries) are called the “orthogonal symmetry class” or sometimes AI (see Sect. 2.16). It may be worthwhile to look back at Sect. 2.2 where it was shown that the Schrödinger equation of a spinless particle is time-reversal invariant provided the Hamiltonian is a real operator in the position representation. The present section generalizes that previous statement. The time evolution operators for the orthogonal class can be characterized from the point of view of Lie groups, in analogy to U (t) = e−iH t /h¯ ∈ U(N) for the unitary class. In the orthogonal symmetry class H = H˜ implies that the evolution matrix is symmetric as well, U (t) = U˜ (t). Any symmetric unitary matrix U = U˜ ∈ U(N) may be written as U = V V˜ for some V ∈ U(N). The existence of such a V can be shown by diagonalising U = W DW † where W ∈U(N) and D ∈ U(1)N is a diagonal unitary matrix. Then U = U˜ implies that W = W ∗ ∈ O(N) and U = W D W˜ = V V˜ for V = W D 1/2 . The matrix V is however not unique: if U = V V˜ one can take any orthogonal matrix O ∈ O(N) to define V = V O and gets U = V V˜ . For each fixed V there is an equivalence class of such V ’s. Therefore, the time evolution operators U (t) in question live in the coset space U(N)/O(N), at all times t.

2.6 Unitary Symmetries A system with Hamiltonian H is called invariant under a unitary operator U if [U, H ] = 0

or equivalently U H U † = H .

(2.6.1)

Then H is also invariant under U n for any integer n. If we can find a U (without reference to the eigenbasis of H and independently of the values of physical constants relevant for the Hamiltonian) then we call U a unitary symmetry operator. The set of all unitary symmetry operators forms a group—the symmetry group G of H . If G= U(1) ≡ {eiα } then the symmetry group is called trivial as every Hamiltonian is invariant under such global gauge transformations. We then say that H has no unitary symmetries (meaning no non-trivial symmetries). In order to exclude the trivial global gauge transformations let us redefine the symmetry group as the set of all special unitary symmetries (i.e. det U = 1). A non-trivial G may contain a finite number of operators or form a continuum (in that case G is a Lie group).

2.6 Unitary Symmetries

23

Many unitary symmetries have a geometric origin. For instance an atom with a nucleus of charge Ze and Z electrons is invariant under any rotations in space. These form the Lie group SO(3) of special orthogonal transformations in three dimensions which is augmented in quantum mechanics to its double cover SU(2) (generated by the total angular momentum operators Jx , Jy , and Jz ) in order to include rotation of half-integer spins. Invariance under rotations implies that the symmetry group G contains a subgroup equivalent to SU(2) (in our example there are additional permutation symmetries related to exchange of electrons). When considering Hamiltonians in absence or presence of an antiunitary symmetry Sect. 2.4 (unitary class A) and Sect. 2.5 (orthogonal class AI), we always assumed that there are no unitary symmetries present. Indeed, if there are any such symmetries they usually imply that there is a basis in which all permissible Hamiltonians (all invariant under the same symmetry group G) acquire a blockdiagonal form. For instance for the rotation group one can go into an eigenbasis of the operators J2 = Jx2 +Jy2 +Jz2 and Jz and all Hamiltonians will be block-diagonal (coupling only states with the same eigenvalues of J2 and Jz ). If there is no anti-unitary time-reversal operator then each of the blocks is a complex Hermitian matrix and by reducing the description to one block we arrive back at the discussion of the unitary class in Sect. 2.4. Similar reductions are possible for finite symmetry groups without Hermitian generators. More care has to be taken when reducing a system with an anti-unitary symmetry with T 2 = 1 with respect to a unitary symmetry group G. If we reduce the system by block-diagonalising the Hamiltonian H and all symmetry operators G then some blocks may still be invariant under T . Reducing the attention to one of these blocks we may use T to find a basis of the subspace such that the block becomes a real symmetric matrix in the orthogonal class. However, in general there will be blocks that are not left invariant by T . If we reduce our attention to such a block, no antiunitary symmetry is left and one expects the block to be in the unitary class. A full discussion of this requires representation theoretic background for which we refer to the literature [4, 5]. Instead we consider one example which gives the general idea in a nutshell—further examples how unitary symmetries affect the symmetry classification of quantum systems under time-reversal operators will be given in the following sections of this chapter. Assume a particle in the (x, y)-plane with integer (or no) spin in a real potential V (x, y) that is invariant under rotations by an angle 2π/3 in the plane (and has no further symmetries). The Hamiltonian is H = p2 /2m + V (x, y), invariant under the conventional time-reversal operator T ≡ K, i.e. [H, T ] = 0. The symmetry group is then G= {1, R, R 2 } where the unitary operator R = ei2πJz /3h¯ = ei2π(xpy −ypx )/3h¯ rotates by 2π/3, √ 1 3 y RxR = − x − 2 2 √ 3 1 † Rpx R = − px − py 2 2 †

√ 1 3 RyR = x− y 2 2 √ 3 1 px − py . Rpy R † = 2 2 †

(2.6.2)

24

2 Time Reversal and Unitary Symmetries

As R 3 = 1 the eigenvalues of R are eil2π/3 where l = 0, ±1. In the corresponding eigenbasis H becomes block-diagonal ⎞ ⎛ H−1 0 0 H = ⎝ 0 H0 0 ⎠ 0 0 H1

(2.6.3)

and we can now ask what effect T -invariance has one each of the blocks. By assumption T = K = T −1 was the conventional time-reversal operator of complex conjugation (in the position representation). From T xT −1 = x and T pT −1 = −p we find T RT −1 = R or [T , R] = 0. So if we take any eigenstate φ of R its timereverse will also be an eigenstate of R: Rφ = ei2πl/3φ

⇒ RT φ = e−i2πl/3T φ .

(2.6.4)

If l = 0 then T φ and φ are both eigenstates of R with unit eigenvalue. So we can reduce T to that block and use [H0, T ] = 0 to find a basis of the corresponding subspace that turns H0 into a real symmetric matrix. However, if l = ±1 then φ and T φ belong to different blocks and we cannot reduce T to one of the blocks on their own. The block H1 (or H−1 ) will thus have no further restrictions and be described by a complex Hermitian matrix in the unitary class. The two blocks H1 and H−1 are however not independent! Diagonalising H1 leads us to eigenstates H φ = Eφ with Rφ = ei2π/3 φ but then T φ is also an eigenstate H T φ = ET φ with the same energy but RT φ = e−i2π/3T φ. Alternatively (without referring to diagonalisation) one may choose the bases of the three subspaces such that T φ = φ for all basis states in the l = 0 subspace and the bases for l = 1 and l = −1 are obtained from one another by the time reversal operator. In this basis T and H take the form ⎛

⎞ 0 0 K T = ⎝0 K 0⎠ K 0 0

⎛ ∗ ⎞ H1 0 0 and H = ⎝ 0 H0 0 ⎠ 0 0 H1

(2.6.5)

where H0 is real symmetric (orthogonal class AI ) and H1 complex Hermitian (unitary class A).

2.7 Kramers’ Degeneracy for T Invariant Hamiltonians, T 2 = −1 Consider a Hamiltonian invariant under a time reversal T , i.e. [H, T ] = 0 .

(2.7.1)

2.8 T Invariant Hamiltonians with T 2 = −1 and Additional Unitary Symmetries

25

If ψ is an eigenfunction with eigenvalue E, so is T ψ. As shown above, we may choose the equality T ψ = ψ without loss of generality if T 2 = +1 (with some caveats if there are additional unitary symmetries). Here, we propose to consider time-reversal operators squaring to minus unity, T 2 = −1 .

(2.7.2)

In this case, ψ and T ψ are orthogonal, ∗  ψ|T ψ = T ψ|T 2 ψ = −T ψ|ψ∗ = −ψ|T ψ = 0 ,

(2.7.3)

and therefore all eigenvalues of H are doubly degenerate. This is Kramers’ degeneracy. It follows that the dimension of the Hilbert space must, if finite, be even. This fits with the result of Sect. 2.3 that T 2 = −1 is possible only if the number of spin-1/2 particles in the system is odd; the total-spin quantum number s is then a half-integer and 2s + 1 is even. In the next two sections, we shall discuss the structure of Hamiltonian matrices with Kramers’ degeneracy, first for the case with additional unitary symmetries of geometric origin and then for the case in which there are no additional unitary symmetries.

2.8 T Invariant Hamiltonians with T 2 = −1 and Additional Unitary Symmetries As an example of a class of time-reversal invariant Hamiltonians with T 2 = −1 and an additional unitary symmetry one may consider an additional unitary parity operator such that [Rx , H ] = 0 , [Rx , T ] = 0 , Rx2 = −1 .

(2.8.1)

The corresponding unitary symmetry group is G = {1, Rx , Rx2 , Rx3 } with Rx2 ≡ −1. This could be realized, for example, by a rotation through π about, say, the x-axis, Rx = exp (iπJx /h¯ ); in a system with half-integer spin. As the eigenvalues of Jx are half-integers this ensures Rx2 = −1 and T = eiπJy /h¯ K is the conventional timereversal operator for half-integer spin. To reveal the structure of the matrix H , we use an eigenbasis of the parity operator Rx (following analogous steps as in the example in Sect. 2.6). As Rx2 = −1 we have two (degenerate) eigenvalues +i and −i and we may choose an eigenbasis basis such that Rx |n± = ±i|n± .

(2.8.2)

26

2 Time Reversal and Unitary Symmetries

Moreover, since T changes the parity, Rx T |n± = T Rx |n± = ∓iT |n± ,

(2.8.3)

the basis can be organized such that T |n± = ±|n∓ .

(2.8.4)

This implies that the dimension is either even or infinite. For simplicity let us assume that the dimension is 2N. The Hamiltonian H , the time-reversal operator T and the parity Rx then fall into four N × N blocks H =

  H+ 0 , 0 H−

 T =



 0 −K , K 0

Rx =

 i 0 0 −i

(2.8.5)

where the forms of T and Rx follow directly from the properties of the basis and the form of H follows from [H, Rx ] = 0. Moreover, the two blocks H+ and H− are not independent. Time-reversal invariance implies H = T H T −1 , or         ∗ H+ 0 0 −K 0 K H+ 0 H− 0 = . = 0 H− 0 H− 0 H+∗ K 0 −K 0

(2.8.6)

So H+ = H−∗ = H˜ − . At this point Kramers’ degeneracy emerges: Since they are the transposes of one another, H + and H − have the same eigenvalues. Moreover, they are in general complex and thus have U (N) as their group of canonical transformations (unitary class A). It is illustrative to restrict the class of Hamiltonians further such that the symmetry group is enlarged by one further parity Ry with 

   Ry , H = 0 , Ry , T = 0 , Rx Ry + Ry Rx = 0 , Ry2 = −1

(2.8.7)

which might be realized as Ry = exp (iπJy /h¯ ). The anticommutativity of Rx and Ry implies that the unitary symmetry group has now eight elements G = {±1, ±Rx , ±Ry , ±Rx Ry }. It also immediately tells us that Ry changes the Rx parity, just as T does, Rx Ry |m± = ∓iRy |m±. It follows that Ry has a off-diagonal block-matrix form. Each off-diagonal block is some unitary matrix (since Ry is unitary) and Ry2 = −1 implies  Ry =

0 V −V † 0

 (2.8.8)

where V is a unitary N × N matrix. The condition T Ry T −1 = Ry restricts V to be symmetric V = V˜ = KV † K. We may proceed and require [H, Ry ] = 0

2.8 T Invariant Hamiltonians with T 2 = −1 and Additional Unitary Symmetries

27

which leads us to H+ = V KH− KV † which identifies Tˆ+ = V K as a new (non-conventional) time-reversal symmetry operator in the reduced N-dimensional subspace. Observing Tˆ+2 = 1 we may now follow the standard procedure for timereversal invariant Hamiltonians with T 2 = 1 (see Sect. 2.5) to choose a new basis within that subspace that makes H+ (and thus also H− = H˜ + ) real symmetric (orthogonal symmetry class, AI ). In hindsight we may have noticed right away that Tˆ = Ry T is an additional (non-conventional) time-reversal invariance of H with Tˆ 2 = Ry T Ry T = Ry2 T 2 = 1. As a final illustration of the cooperation of conventional time-reversal invariance with geometrical (unitary) symmetries, the case of full isotropy, [H, J ] = 0, deserves mention. The appropriate basis here is |αj m with h¯ m and h¯ 2 j (j + 1) the eigenvalues of Jz and J 2 , respectively. The Hamiltonian matrix then falls into blocks given by 

j=

1 2

,

      αj m|H |βj m = δjj δmm α H (j,m)  β , 3 , 2

5 2

, . . . , m = ± 12 , ± 32 , . . . , ±j .

(2.8.9)

It is left to the reader as Problem 2.10 to show that (1) due to T invariance, for any fixed value of j , the two blocks with differing signs of m are transposes of one another, H (j,m) = H˜ (j,−m) ,

(2.8.10)

and thus have identical eigenvalues, and (2) invariance of H under rotations (especially [H, Jx ± iJy ] = 0) makes all blocks with equal values of j and different values of m equal, H (j,m) = H (j,m ) , in an appropriate basis. The two statements above imply that the blocks H (j,m) are all real and thus have the orthogonal transformations as their canonical transformations. The same can be derived as well for T 2 = 1 with full isotropy—so any time-reversal invariance with full isotropy always leads to blocks in the orthogonal class. To summarize, we have seen how the presence of a time-reversal invariance with T 2 = −1 with an additional unitary symmetry group leads to reduced systems that may be either in the unitary class A (e.g. for a half-integer spin system with one parity) or the orthogonal class AI (e.g. for a half-integer spin system with two parities or full rotational isotropy). An altogether new symmetry class with canonical transformations that are unitary and symplectic will be encountered if T 2 = −1 with no additional unitary symmetries in Sect. 2.9. This will be called the symplectic symmetry class or class AII. In general the reduction of a timereversal invariant system with either T 2 = 1 or T 2 = −1 with an additional unitary symmetry group leads to block-diagonal Hamiltonians (where some blocks

28

2 Time Reversal and Unitary Symmetries

may not be independent). Each of these blocks then belongs to one of the three symmetry classes A, AI or AII, depending on whether the corresponding subspace 2 = ±1 which only acts inside the has a reduced time-reversal invariance Tˆred corresponding subspace. It has only recently been realised that there is a very general representation theoretic setting which gives a complete and systematic picture how the three symmetry classes may appear during the reduction process. We refer to the literature [4, 5] for further details.

2.9 T Invariance with T 2 = −1 and no Unitary Symmetries When a time reversal with T 2 = −1 is the only symmetry of H, it is convenient to adopt a basis of the form |1 , T |1 , |2 , T |2 , . . . |N , T |N .

(2.9.1)

[Note that in (2.8.5) another ordering of states |n+ andT |n+ = |n− was chosen.] Sometimes we shall write |T n for T |n and T n| for the corresponding Dirac bra. For the sake of simplicity, the Hilbert space is again assumed to have the finite dimension 2N. If the complex conjugation operation K is defined relative to the basis (2.9.1), the unitary matrix U in T = U K takes a simple form which is easily found by letting T act on an arbitrary state vector

T |ψ

(ψm+ |m + ψm− |T m) , m   ∗ |T m − ψ ∗ |m ψm+ = m− m

|ψ =

.

(2.9.2)

Clearly, in each of the two-dimensional subspaces spanned by |m and |T m, the matrix U, to be called Z from now on, takes the form Zmm =

  0 −1 ≡ τ2 1 0

(2.9.3)

while different such subspaces are unconnected, Zmn = 0 for m = n .

(2.9.4)

The 2N × 2N matrix Z is obviously block diagonal with the 2 × 2 blocks (2.9.3) along the diagonal. In fact it will be convenient to consider Z as a diagonal N × N matrix, whose nonzero elements are themselves 2 × 2 matrices given by (2.9.3).

2.9 T Invariance with T 2 = −1 and no Unitary Symmetries

29

Similarly, the two pairs of states |m, T |m, and |n, T |n give a 2 × 2 submatrix of the Hamiltonian   m|H |n m|H |T n ≡ hmn . (2.9.5) T m|H |n T m|H |T n The full 2N × 2N matrix H may be considered as an N × N matrix each element of which is itself a 2 × 2 block hmn . The reason for the pairwise ordering of the basis (2.9.1) is, as will become clear presently, that the restriction imposed on H by time-reversal invariance can be expressed as a simple property of hmn . As is the case for any 2 × 2 matrix, the block hmn can be represented as a linear combination of four independent matrices. Unity and the three Pauli matrices σ may come to mind first, but the condition of time-reversal invariance will take a nicer form if the anti-Hermitian matrices τ = −iσ are employed, 

     0 −i 0 −1 −i 0 τ1 = , τ2 = , τ3 = −i 0 1 0 0 +i

3 τi τj = k=1 εij k τk , τi τj + τj τi = −2δij

(2.9.6)

where the Levi-Civita symbol is defined as εij k = 1 if (ij k) is an even permutation of (123) and εij k = −1 if (ij k) is an odd permutation of (123) and εij k = 0 else. (μ) Four coefficients hmn , μ = 0, 1, 2, 3, characterize the block hmn , (0) hmn = hmn 1 + hmn · τ .

(2.9.7)

Now, time-reversal invariance gives   hmn = T H T −1 mn   = ZKH KZ −1 mn   ∗ −1 = ZH Z mn

=

−τ2 h∗mn τ2

  ∗ ∗ ∗ τ2 = −τ2 h(0) 1 + h · τ mn mn ∗

∗ = h(0) mn 1 + hmn · τ ,

(2.9.8)

(μ)

which simply means that the four amplitudes hmn are all real: ∗

(μ) h(μ) mn = hmn .

(2.9.9)

30

2 Time Reversal and Unitary Symmetries

Matrices of size 2 × 2 with the property (2.9.9) are called “quaternion real”.2 Mathematically they are considered as extension of the complex numbers to a number system with three ‘imaginary units’—indeed, in the present context it would make sense to consider the 2 × 2 real quaternion submatrices as numbers such that the dimension of H would indeed be N × N. Note that the property (2.9.9) does look nicer than the one that would have been obtained if we had used Pauli’s triple σ instead of the anti-Hermitian τ . The Hermiticity of H implies the relation hmn = h†nm ,

(2.9.10) (μ)

which in turn means that the four real amplitudes hmn obey (0) h(0) mn = hnm (k) h(k) mn = −hnm , k = 1, 2, 3 .

(2.9.11)

It follows that the 2N × 2N matrix H is determined by N(2N − 1) independent real parameters. All of the above considerations may be summarized by introducing the notion of the symplectic transpose or Z-transpose AZ of a 2N-dimensional matrix A, ˜ . AZ = Z A˜ Z˜ = −Z AZ

(2.9.12)

Z

This Z-transpose shares the properties AZ = A, det AZ = det A, trAZ = trA and (AB)Z = B Z AZ with the standard transpose. For two vectors x and y the symplectic product is defined as ωZ (y, x) = y · Zx. It is straightforward to show that the symplectic transpose is the appropriate generalization for this symplectic product as ωZ (y, Ax) = ωZ (AZ y, x). A matrix that is equal to its symplectic transpose A = AZ is called self-dual, symplectic symmetric or Z-symmetric. Now the condition that H is time-reversal invariant with respect to T just means that H , apart from being Hermitian, is also Z-symmetric HZ = H = H†

(2.9.13)

which notion we thus recognize as synonymous to quaternion real. With the structure of the Hamiltonian now clarified, it remains to identify the canonical transformations that leave this structure intact. To that end we must find the subgroup of unitary matrices that preserve the form T = ZK of the time-reversal operator. In other words, the question is to what extent there is freedom in choosing

2 Historically they have been introduced by William Rowan Hamilton in the nineteenth century and applied to classical mechanics (Gauss had discovered them before but not published the work).

2.10 Nonconventional Time Reversal

31

a basis with the properties (2.9.1). The allowable unitary basis transformations S have to obey ˜ = SS Z T T = ST S −1 = SZKS −1 = SZ SK

⇒

SS Z = 1 .

(2.9.14)

So the unitary matrices S obey S −1 = S † = S Z and they form the Lie group Sp(2N) (often written as USp(2N)) of unitary symplectic 2N × 2N matrices. Unfortunately the conventional notations are somewhat confusing as Sp(2N, R) or Sp(2N, C) refer to groups of real and complex symplectic matrices which need not be unitary (indeed Sp(N) =U(2N) ∩ Sp(2N, C) ), see Problem 2.12. The symplectic transformations just found are in fact the relevant canonical transformations since they leave a quaternion real Hamiltonian quaternion real. To show this consider H = SH S −1 with a real quaternion (Hermitian and selfdual) H . Then Z

H = S −1 H Z S Z = SH S −1 = H Z

(2.9.15)

which shows that H is again Hermitian and Z-symmetric. In fact this is true for all H = H Z = H † if and only if S −1 = S † = S Z . It is also obvious that a self-dual Hermitian matrix is diagonalized by a unitary symplectic matrix. Time reversal invariant Hamiltonians with T 2 = −1 are said to form the “symplectic symmetry class” or class AII. The pertinent time evolution operators U (t) = e−iH t /h¯ obey U (t) = U (t)Z , they are thus unitary and self-dual. Now let V ∈ U(2N) be an arbitrary unitary 2N × 2N matrix, then clearly U = V Z V is unitary and self-dual. It also works the other way: if U = U Z is a unitary symplectic 2N × 2N matrix then one can find a V ∈ U(2N) such that U = V Z V . However, V is not unique: if S is unitary symplectic (S Z S = 1, S ∈ Sp(2N)) then V and SV form the same self-dual unitary matrix (SV )Z SV = V Z S Z SV = V Z V = U . This shows that the unitary self-dual matrices form a coset space U(2N)/Sp(2N).

2.10 Nonconventional Time Reversal We have defined conventional time reversal (2.3.11) by requiring T xT −1 = x T pT −1 = −p T JT

−1

(2.10.1)

= −J

and, for any pair of states, T φ|T ψ = ψ|φ T 2 = ±1 .

(2.10.2)

32

2 Time Reversal and Unitary Symmetries

The motivation for this definition is that many Hamiltonians of practical importance are invariant under conventional time reversal, [H, T ] = 0. An atom and a molecule in an isotropic environment, for instance, have Hamiltonians of that symmetry. But, as already mentioned in Sect. 2.1, conventional time reversal is broken by an external magnetic field. In identifying the canonical transformations of Hamiltonians from their symmetries in Sects. 2.5–2.9, extensive use was made of (2.10.2) but none, as the reader is invited to review, of (2.10.1). In fact, and indeed fortunately, the validity of (2.10.1) is not at all necessary for the above classification of Hamiltonians according to their group of canonical transformations. Interesting and experimentally realizable systems often have Hamiltonians that commute with some antiunitary operator obeying (2.10.2) but not (2.10.1). There is nothing strange or false about such a “nonconventional” time-reversal invariance: it associates another, independent solution, ψ (t) = T ψ(−t), with any solution ψ(t) of the Schrödinger equation, and is thus as good a time-reversal symmetry as the conventional one. An important example is the hydrogen atom in a constant magnetic field [6, 7]. Choosing that field as B = (0, 0, B) and the vector potential as A = B × x/2 and including spin-orbit interaction, one obtains the Hamiltonian H =

 e2 B 2  2 e2 eB p2 2 − − x + f (r)LS . + y (Lz + gSz ) + 2m r 2mc 8mc2

(2.10.3)

Here L and S denote orbital angular momentum and spin, respectively, while the total angular momentum is J = L + S. This Hamiltonian is not invariant under conventional time reversal, denoted by T0 for the rest of this section. Instead there is invariance under the non-conventional antiunitary symmetry T = eiπJx /h¯ T0 .

(2.10.4)

If spin is absent, T 2 = 1, whereas T 2 = −1 with spin. In the subspaces of constant Jz and J 2 , one has the orthogonal transformations as the canonical group in the first case (Sect. 2.2) and the unitary transformations in the second case (Sect. 2.8 and Problem 2.11). When a homogeneous electric field E is present in addition to the magnetic field, the operation T in (2.10.4) ceases to be a symmetry of H since it changes the electric-dipole perturbation −ex · E. But T = RT0 is an antiunitary symmetry where the unitary operator R represents a reflection in the plane spanned by B and E. Note that the component of the angular momentum lying in this plane changes sign under that reflection since the angular momentum is a pseudo-vector. While the Zeeman term in H changes sign under both conventional time reversal and under the reflection in question, it is left invariant under the combined operation. The electricdipole term as well as all remaining terms in H are symmetric with respect to both T0 and R such that [H, RT0 ] = 0 indeed results.

2.11 Stroboscopic Maps for Periodically Driven Systems

33

As another example Seligman and Verbaarschot [8] proposed two coupled oscillators with the Hamiltonian H =

1 2

 2 p1 − a x23 +

1 2

 2 p2 + a x13

+ α1 x16 + α2 x26 − α12 (x1 − x2 )6 .

(2.10.5)

Here, too, T0 invariance is violated if a = 0. As long as α12 = 0, however, H is invariant under T = eiπL2 /h¯ T0

(2.10.6)

and thus representable by a real matrix. The geometric symmetry T T0−1 acts as (x1 , p1 ) → (−x1 , −p1 ) and (x2 , p2 ) → (x2 , p2 ) and may be visualized as a rotation through π about the 2-axis if the two-dimensional space spanned by x1 and x2 is imagined embedded in a three-dimensional Cartesian space. However, when a = 0 and α12 = 0, the Hamiltonian (2.10.5) has no antiunitary symmetry left and therefore is a complex matrix. (Note that H is a complex operator in the position representation anyway.)

2.11 Stroboscopic Maps for Periodically Driven Systems Time-dependent perturbations, especially periodic ones, are characteristic of many situations of experimental interest. They are also appreciated by theorists inasmuch as they provide the simplest examples of classical nonintegrability: Systems with a single degree of freedom are classically integrable, if autonomous, but may be nonintegrable if subjected to periodic driving. Quantum mechanically, one must tackle a Schrödinger equation with an explicit time dependence in the Hamiltonian, ˙ ih¯ ψ(t) = H (t)ψ(t) .

(2.11.1)

The solution at t > 0 can be written with the help of a time-ordered exponential



−i U (t) = exp h¯



t





 (2.11.2)

dt H (t ) 0

+

where the “positive” time ordering requires   A(t)B(t ) A(t)B(t ) + = B(t )A(t)

if t > t if t < t

.

(2.11.3)

34

2 Time Reversal and Unitary Symmetries

Of special interest are cases with periodic driving, H (t + nτ ) = H (t) , n = 0, ±1, ±2, . . . .

(2.11.4)

The evolution operator referring to one period τ, the so-called Floquet operator U (τ ) ≡ F ,

(2.11.5)

is worthy of consideration since it yields a stroboscopic view of the dynamics, ψ(nτ ) = F n ψ(0) .

(2.11.6)

Equivalently, F may be looked upon as defining a quantum map, ψ ([n + 1]τ ) = F ψ(nτ ) .

(2.11.7)

Such discrete-time maps are as important in quantum mechanics as their Newtonian analogues have proven in classical nonlinear dynamics. The Floquet operator, being unitary, has unimodular eigenvalues (involving eigenphases alias quasi-energies) and mutually orthogonal eigenvectors, F Φν = e−iφν Φν ,   Φμ |Φν = δμν .

(2.11.8)

We shall in fact be concerned only with normalizable eigenvectors. With the eigenvalue problem solved, the stroboscopic dynamics can be written out explicitly, ψ(nτ ) =



e−inφν Φν |ψ(0) Φν .

(2.11.9)

ν

Monochromatic perturbations are relatively easy to realize experimentally. Much easier to analyse are perturbations for which the temporal modulation takes the form of a periodic train of delta kicks, H (t) = H0 + λV

+∞ 

δ(t − nτ ) .

(2.11.10)

n = −∞

The weight of the perturbation V in H (t) is measured by the parameter λ, which will be referred to as the kick strength. The Floquet operator transporting the state vector from immediately after one kick to immediately after the next reads F = e−iλV /h¯ e−iH0 τ/h¯ .

(2.11.11)

2.12 Time Reversal for Maps

35

The simple product form arises from the fact that only H0 is on between kicks, while H0 is ineffective “during” the infinitely intense delta kick. It may be well to conclude this section with a few examples. Of great interest with respect to ongoing experiments is the hydrogen atom exposed to a monochromatic electromagnetic field. Even the simplest Hamiltonian, H =

e2 p2 − − Ez cos ωt , 2m r

(2.11.12)

defies exact solution. The classical motion is known to be strongly chaotic for sufficiently large values of the electric field E : A state that is initially bound (with respect to H0 = p2 /2m − e2 /r) then suffers rapid ionization. The quantum modifications of this chaos-enhanced ionization have been the subject of intense discussion. See Ref. [9] for the early efforts and for a brief sketch of the present situation Sect. 8.1. A fairly complete understanding has been achieved for both the classical and quantum behavior of the kicked rotator [10], a system of quite some relevance for microwave ionization of hydrogen atoms. The Hamiltonian reads +∞  1 2 ˜ H (t) = δ(t − nτ ) . p + λ cos Θ 2I n = −∞

(2.11.13)

The classical kick-to-kick description is Chirikov’s standard map [11]. In Chap. 8 (on quantum localization) we shall treat that prototypical system with the powerful method of the superanalytic sigma model. Somewhat richer in their behavior are the kicked tops for which H0 and V in (2.11.10) and (2.11.11) are polynomials in the components of an angular momentum J . Due to the conservation of J 2 = h¯ 2 j (j + 1), j = 12 , 1, 32 , 2, . . . , kicked tops enjoy the privilege of a finite-dimensional Hilbert space. The special case H0 ∝ Jx , V ∝ Jz2

(2.11.14)

can be realized experimentally in various ways to be discussed in Sect. 3.6.

2.12 Time Reversal for Maps It is easy to find the condition which the Hamiltonian H (t) must satisfy so that a given solution ψ(t) of the Schrödinger equation ˙ ih¯ ψ(t) = H (t)ψ(t)

(2.12.1)

36

2 Time Reversal and Unitary Symmetries

yields an independent solution, ˜ ψ(t) = T ψ(−t) ,

(2.12.2)

where T is some antiunitary operator. By letting T act on both sides of the Schrödinger equation, − ih¯

∂ T ψ(t) = T H (t)T −1 T ψ(t) ∂t

(2.12.3)

or, with t → −t, ih¯

∂ T ψ(−t) = T H (−t)T −1 T ψ(−t) . ∂t

(2.12.4)

For (2.12.4) to be identical to the original Schrödinger equation, H (t) must obey H (t) = T H (−t)T −1 ,

(2.12.5)

a condition reducing to that studied previously for autonomous dynamics. For periodically driven systems it is convenient to express the time-reversal symmetry (2.12.5) as a property of the Floquet operator. As a first step in searching for that property, we again employ the formal solution of (2.12.1). Distinguishing now between positive and negative times, ψ(t) =

U+ (t)ψ(0) ,

t>0

U− (t)ψ(0) ,

t 0 and propose to consider ˜ ψ(t) = T ψ(−t) = T U− (−t)ψ(0) = T U− (−t)T −1 T ψ(0) .

(2.12.7)

˜ If H (t) is time-reversal invariant in the sense of (2.12.5), this ψ(t) must solve the original Schrödinger equation (2.12.1) such that U+ (t) = T U− (−t)T −1 .

(2.12.8)

The latter identity is in fact equivalent to (2.12.5). The following discussion will be confined to τ -periodic driving, and we shall take the condition (2.12.8) for t = τ. The backward Floquet operator U− (−τ ) is then simply related to the forward one. To uncover that relation, we represent U− (−τ ) as a product of time evolution

2.12 Time Reversal for Maps Fig. 2.1 Discretization of the time used to evaluate the time-ordered exponential in (2.12.9)

37 Δt –tn

–t1 0

t1

tn Time

operators, each factor referring to a small time increment,

   i −τ U− (−τ ) = exp − dt H (t ) h¯ 0 − = eiΔt H (−tn)/h¯ eiΔt H (−tn−1)/h¯ . . . . . . eiΔt H (−t2)/h¯ eiΔt H (−t1)/h¯ .

(2.12.9)

As illustrated in Fig. 2.1, we choose equidistant intermediate times between t−n = −τ and tn = τ with the positive spacing ti+1 − ti = Δt = τ/n. The intervals ti+1 − ti are assumed to be so small that the Hamiltonian can be taken to be constant within each of them. Note that the positive sign appears in each of the n exponents in the second line of (2.12.9) since Δt is defined to be positive while the time integral in the negatively time-ordered exponential runs toward the left on the time axis. Now, we invoke the assumed periodicity of the Hamiltonian, H (−tn−i ) = H (ti ), to rewrite U− (−τ ) as U− (−τ ) = eiΔt H (0)/h¯ eiΔt H (t1)/h¯ . . . . . . eiΔt H (tn−2)/h¯ eiΔt H (tn−1)/h¯ = U+ (τ )†

(2.12.10)

which is indeed the Hermitian adjoint of the forward Floquet operator. Now, we can revert to a simpler notation, U+ (τ ) = F, and write the time-reversal property (2.12.8) for t = τ together with (2.12.10) as a time-reversal “covariance” of the Floquet operator T F T −1 = F † = F −1 .

(2.12.11)

This is a very intuitive result indeed: The time-reversed Floquet operator T F T −1 is just the inverse of F . An interesting general statement can be made about periodically kicked systems with Hamiltonians of the structure (2.11.10). If H0 and V are both invariant under some time reversal T0 (conventional or not), the Hamiltonian H (t) is T0 -covariant in the sense of (2.12.5) provided the zero of time is chosen halfway between two successive kicks. The Floquet operator (2.11.11) is then not covariant in the sense (2.12.11) with respect to T0 , but it is with respect to T = eiH0 τ/h¯ T0 .

(2.12.12)

38

2 Time Reversal and Unitary Symmetries

The reader is invited, in Problem 2.13, to show that the Floquet operator defined so as to transport the wave vector by one period starting at a point halfway between two successive kicks is T0 -covariant. We have stated before that in order to qualify for a symmetry, a unitary or anti-unitary operator should be available without reference to particular values of physical constants. In the present context this means that one can choose the zero time such that H0 does not appear in the time-reversal operator as is the case in the present example. Once that is established it may be practically more convenient to stick to a different convention for the zero time and, e.g. use (2.12.12) as the symmetry if the zero time is such that the kick arises at the end of the period. This is also the case for the periodically kicked rotator defined by (2.11.13), where F has an antiunitary symmetry of the type (2.12.12) and the reference to H0 disappears by choosing a different zero time. Analogously the Floquet operator of the kicked top (2.11.14) is covariant with respect to T = eiH0 τ/h¯ K where K is complex conjugation in the standard representation of angular momenta in which Jx and Jz are real and Jy is imaginary.

2.13 Canonical Transformations for Floquet Operators The arguments presented in Sects. 2.4, 2.5, 2.8, and 2.9 for Hermitian Hamiltonians carry over immediately to unitary Floquet operators. We shall assume a finite number of dimensions throughout. We will also, at first, assume that there either are no unitary symmetries (or that any such have been used to reduce the analysis to an appropriate subspace that cannot be reduced further). First, such irreducible N × N Floquet matrices without any T covariance have U (N) as their group of canonical transformations. Indeed, any transformation from that group connects admissible Floquet matrices, preserves eigenvalues, unitarity, and normalization of vectors. The proof is analogous to that sketched in Sect. 2.4. Next, when F is T covariant with T 2 = 1, one can find a T invariant basis in which F is symmetric. In analogy with the reasoning in (2.5.5), one takes matrix elements in T F † T −1 = F with respect to T invariant basis states (such that T |ψν  = |ψν  = T −1 |ψν ),       Fμν = ψμ T F † T −1 ψν = T ψμ T F † ψν  = F † ψν |ψν = Fνμ . (2.13.1) It is worth recalling that time-reversal invariant Hamiltonians were also found, in Sect. 2.5, to be symmetric if T 2 = 1. Of course, for unitary matrices F = F˜ does not imply reality. The canonical group is now O(N), as was the case for [H, T ] = 0, T 2 = 1. It is easy to see that O(N) ⊂ U(N) is the largest subgroup such that F = F˜ and F = OF O † implies F = F˜ for all O ∈ O(N). F |φν  = e−iφν |φν ,

and thus F † |φν  = eiφν |φν 

(2.13.2)

2.13 Canonical Transformations for Floquet Operators

39

and let T act on the latter equation: e−iφν T |φν  = T F † T −1 T |φν  = F T |φν  .

(2.13.3)

The orthogonality of |φν  and T |φν , following from T 2 = −1, has already been demonstrated in (2.7.3). The Hilbert space dimension must again be even. By assumption no other unitary symmetries are present and we may employ the basis (2.9.1), thus giving the time-reversal operator the structure T = ZK .

(2.13.4)

The restriction imposed on F by T covariance can be found by considering the 2×2 block   m|F |n m|F |T n Fmn = T m|F |n T m|F |T n (0) 1 + f mn · τ = fmn

(2.13.5)

which must equal the corresponding block of T F † T −1 . In analogy with (2.9.8),     fmn = T F † T −1 = ZKF † KZ −1 mn mn   = − Z F˜ Z = −τ2 f˜nm τ2 mn   (0) (0) = −τ2 fnm 1 + f nm · τ˜ τ2 = fnm 1 − f nm · τ .

(2.13.6)

Now, the restrictions in question can be read off as (0) (0) fmn = fnm

f mn = −f nm .

(2.13.7) (μ)

They are identical in appearance to (2.9.11) but, in contrast to the amplitudes hmn , (μ) the fmn are in general complex numbers. The pertinent group of canonical transformation is the symplectic group defined by (2.9.14) since SF S −1 is T covariant if F is. Indeed, reasoning in parallel to (2.9.15), ˜ −1 T SF S −1 T −1 = ZKSF S −1 KZ −1 = ZS ∗ KF K SZ = SZKF KZ −1 S −1 = ST F T −1 S −1  † = SF † S −1 = SF S −1 .

(2.13.8)

40

2 Time Reversal and Unitary Symmetries

So far we have assumed that there are no additional unitary symmetries or that the system has been reduced completely. On the other hand, if unitary symmetries are still present their discussion completely parallels that of autonomous systems. It may not hurt to repeat a bit: For a reducible system the time-reversal covariant Floquet matrix becomes block-diagonal in an appropriate basis (along with all the unitary symmetry operators) and in each block one needs to consider whether a time-reversal operator can be defined that only acts within that block. In some blocks time-reversal symmetry can be broken, in other blocks one may need to redefine the time-reversal operator whose square can be independent of the original time-reversal operator of the reducible system. We conclude this section with a few examples of Floquet operators from different universality classes, all for kicked tops. These operators are functions of the angular momentum components Jx , Jy , Jz and thus entail the conservation law J 2 = h¯ 2 j (j + 1) with integer or half-integer j . The latter quantum number also defines the dimension of the matrix representation of F as (2j + 1). The simplest top capable of classical chaos, already mentioned in (2.11.14), has the Floquet operator [12, 13] F = e−iλJz /(2j +1)h¯ e−ipJx /h¯ . 2

2

(2.13.9)

Its dimensionless coupling constants p and λ may be said to describe a linear rotation and a nonlinear torsion. (For a more detailed discussion, see Sect. 8.6.) The quantum number j appears in the first unitary factor in (2.13.9) to give to the exponents of the two factors the same weight in the semiclassical limit j  1. This simplest top belongs to the orthogonal universality class: Its F operator is covariant with respect to generalized time reversal T = eipJx /h¯ eiπJy /h¯ T0

(2.13.10)

where T0 is the conventional time reversal. An example of the unitary universality class is provided by 2

F = e−iλ Jy /2j h¯ e−iλJz /(2j +1)h¯ e−ipJx /h¯ . 2

2

2

(2.13.11)

Finally, the Floquet operator F = e−iV e−iH0 , H0 = λ0 Jz2 /j h¯ 2 ,

(2.13.12)   V = λ1 Jz4 /j 3 h¯ 4 + λ2 (Jx Jz + Jz Jx ) /h¯ 2 + λ3 Jx Jy + Jy Jx /h¯ 2

is designed so as to have no conserved quantity beyond J 2 (i.e., in particular, no geometric symmetry) but a time-reversal covariance with respect to T = e−iH0 T0 .

2.14 Universality

41

Now, since T 2 = +1 and T 2 = −1 for j integer and half integer, respectively, the top in question may belong to either the orthogonal or the symplectic class.

2.14 Universality One of the most striking experimental observations (backed up by a large amount of numerical findings) on chaotic systems is the non-trivial universality of certain statistical properties showing up in the quantum (quasi-)energy spectra and/or the pertinent eigenfunctions. Universal means system independent, identical behavior for all dynamics within each of several universality classes. These classes differ by their symmetries, and in fact the Wigner-Dyson classification in terms of antiunitary symmetries provides the three most well-known universality classes (orthogonal, unitary, and symplectic). In the preceding sections of the present chapter we had actually presented symmetry rather than universality classes, and for good reason. One may think of a universality class as of a subclass of a symmetry class, the subclass admitting only dynamics sharing the same non-trivial statistical properties (see the next section for first examples). We reserve the specification of precise conditions for universal quantum behavior for later chapters, contenting ourselves here with some crude qualitative characterizations and empirical statements. Classical chaos invariably arises when the classical Hamiltonian motion on some manifold is non-integrable. At least two degrees of freedom are necessary for autonomous systems while a single freedom can suffice in the presence of driving periodic in time. However, only in the rarest of cases one then finds chaos everywhere in phase space, without stable periodic orbits embedded in islands of regular motion. Universal quantum features can still appear, provided the overall phase space volume of these regular islands is negligibly small compared to the volume of the chaotic region (the so-called chaotic sea). An important case is constituted by disordered systems like independent electrons moving in a metallic or semiconducting solid where irregularly placed impurities or lattice defects act like random scatterers. If such scatterers were modelled as finite-size hard balls a classical electron could be expected to move chaotically, and even full chaos should arise for a density of balls large enough to suppress trajectories running unscattered in between bounces against the exterior boundary. Randomly placed point scatterers, on the other hand, are invisible classically. They can, however, affect quantum waves and even entail the universal behavior one usually encounters under conditions of classical chaos. One may then say that there is quantum chaos without classical chaos, and because of that the limiting situation of point scatterers is highly interesting. Popular models of disorder involve Hamiltonians containing parameters which are not assigned precise values but only some probability distribution. One then

42

2 Time Reversal and Unitary Symmetries

faces ensembles rather than individual systems and aims at calculating ensemble averages of suitable quantities. Random-matrix theory, the topic of Chap. 5 proceeds most radically, modelling Hamiltonians or Floquet operators such that all matrix elements are considered random, restricted only by symmetries according to the pertinent symmetry class. Suitable probability densities for the matrix elements are employed. We shall see the pertinent ensembles turn out to yield the correct, if phenomenological, description of universal quantum behavior. It is interesting that universal spectral fluctuations can often be observed within an individual quantum system. Instead of averaging over system parameters at a fixed energy E one then averages over different parts of a large spectrum to yield statistically significant signatures of universality. Such quantum universality (with respect to spectral fluctuations) of dynamics that are fully chaotic classically was first conjectured in the early 1980s (see, e.g. [14]). The conjecture is mostly referred to as due to Bohigas, Giannoni, and Schmit. Within random-matrix theory one can show that the spectral fluctuations of one large random matrix indeed display the same universality as the whole ensemble, with probability 1; but note that probability 1 leaves a lot of room for exceptions. That will be the topic of Sects. 5.11, 5.17.2, and 5.17.3. More to the point is the progress made over the last decade towards understanding of the semiclassical mechanisms that build up universality and the conditions thereof, see Chaps. 7 and 10. Apart from trying to understand quantum chaos in realistic systems (e.g. atomic physics) theoreticians have used a large number of models to analyse and understand quantum chaos in the semiclassical regime. To mention but a few in roughly historical order: the kicked rotator, quantum billiards, the kicked top, the quantum cat map, the quantum baker map, and quantum graphs. Some of these models seem quite distant from experiment at first sight. However, as most questions in quantum chaos are of a fundamental character and often deal with either generic or universal behaviour it is most appropriate for the theoretician to use the kind of model that exhibits a phenomenon in the most simple way available. Such models will feature prominently in this book. In particular, we have already introduced kicked tops for all three symmetry classes above and will keep coming back to them. It is appropriate to add that universality in quantum chaos results from quantum interference. Analogous interference effects can be observed for many classical waves (e.g. electromagnetic or acoustic ones). One may then speak of wave disorder and wave chaos. The analogy often goes as far as that the same mathematical equations describe either a quantum chaotic or a wave chaotic setup. Many quantum chaotic effects have indeed been observed in such analogous wave chaos experiments, see the discussion in Chap. 1 and for optical experiments [15]. To illustrate the wealth of chaotic phenomena we shall meet with in this book we point out that there are model systems that display full classical chaos but no universal quantum signatures thereof. Examples are the cat map [16, 17] and arithmetic-chaos billards on surfaces of constant negative curvature [18–20].

2.15 Universality for the Kicked Top: The Level Spacing Distribution

43

2.15 Universality for the Kicked Top: The Level Spacing Distribution Consider a spectrum of N eigenenergies (or eigenphases) {En }N n=0 which has been ordered En ≤ En+1 . We can then define N − 1 level spacings ΔEn = En+1 − En

for n = 1, . . . , N − 1.

(2.15.1)

We can then define the mean level spacing ΔE =

N−1 1  EN − E1 ΔEn = N −1 N −1

(2.15.2)

n=1

and we will assume here for simplicity that the mean level spacing of a (sufficiently max large) subspectrum {En }nn=n min does not differ appreciably from the one defined with the total spectrum. The statistical quantity ΔE is not a universal quantity. It depends on the particular units that are used and can easily be changed in experiment. It turns out that the distribution of the level spacings is a universal property for chaotic or disordered quantum systems if measured in units of the mean level spacing. We thus introduce3 ΔSn =

ΔEn

(2.15.3)

ΔE

and consider the staircase function IN (S) =

1 #{ΔSn ≤ S} . N −1

(2.15.4)

This is the integrated level spacing distribution, i.e. the number of level spacings ΔSn that are smaller than S divided by the total number of spacings. Universality implies that IN (S) has a limiting function 

s

lim IN (S) = I (S) ≡

N→∞

P (S )dS

(2.15.5)

0

as N → ∞ which only depends on the symmetry class of the system (unitary, orthogonal or symplectic) and no other system specific parameters. We have also introduced the corresponding level spacing distribution P (S). The functional form of P (S) for each universality class will be calculated in Chap. 5. At this stage it is sufficient to focus on the degree of level repulsion. For sufficiently small S the

3 This

rescaling is often referred to as spectral unfolding; it will show up repeatedly in this book.

44

2 Time Reversal and Unitary Symmetries

universal level spacings obey P (S) ∼ cS β

for some constant c = 0 .

(2.15.6)

The exponent β gives the degree of level repulsion. For the three universality classes of quantum chaos and disorder they take the values β = 1 for orthogonal symmetry, β = 2 for unitary symmetry and β = 4 for symplectic symmetry. Thus levels of chaotic and disordered quantum systems repel with a degree that depends on the underlying symmetry. This is in stark contrast to the lack of level repulsion in a completely uncorrelated sequence of eigenvalues which has a Poisson distribution P (S) = e−S ∼ 1 − S (such that β = 0). We will see later that quantum systems that contain sufficient symmetries to make the classical dynamics fully integrable obey Poissonian statistics.4 In practice, when we want to compare a given spectrum to the universal predictions the limit N → ∞ may not be available due to experimental or numerical restrictions. Universality can nevertheless show up when one builds histograms of the available data. The bin width has to be chosen in a balanced way such that it is sufficiently small to resolve the theoretical prediction (i.e. the bin width must be much smaller than unity) but large enough that each bin contains many level spacings. In Sect. 2.13 variants of the kicked top have been introduced for each symmetry class. By choosing the parameters appropriately one can ensure that the corresponding classical dynamics is practically fully chaotic. The kicked top with Floquet operator F defined in Eq. (2.13.9) belongs to the orthogonal class. By choosing the coupling constants such that the corresponding classical map shows chaos and diagonalizing F , the level spacing distribution has been shown [12, 13] to obey linear level repulsion, as confirmed in Fig. 2.2. Analogously the Floquet operator F defined by Eq. (2.13.11) belongs to the unitary class and show quadratic level repulsion in the spectrum of quasi-energies. The quadratic level repulsion is shown clearly in Fig. 2.2 (third box). There, the angular momentum quantum number is chosen as j = 500 and the parameters of the map are p = 1.7, λ = 10, and λ = 0.5. Finally, the Floquet operator F defined in Eq. (2.13.12) is time-reversal invariant with T 2 = 1 for integer angular momentum quantum number j where the Hilbert space dimension N = 2j + 1 is odd. For half integer j the Hilbert pace dimension is even and T 2 = −1. So, in the two cases F belongs either to the orthogonal or the symplectic class. If parameters are chosen to ensure classical chaos one observes either linear (see Fig. 2.2, second box) or quartic level repulsion (see Fig. 2.2, fourth box). The alternative repulsion behaviour is most striking given that the two Floquet operators only differ by choosing j = 500 and j = 499.5, all other parameters

4 If one uses all symmetries of an integrable system to reduce the system to irreducible blocks this often amounts to complete diagonalization. The Poissonian nature of the spectrum only appears if such a reduction is not performed.

2.16 Beyond Dyson’s Threefold Way

45

Fig. 2.2 Level spacings distributions for an uncorrelated Poissonian spectrum (classically regular motion) with no level repulsion β = 0 and for kicked tops under conditions of classical chaos β = 1, 2, 4. The latter three histograms pertain to tops from different symmetry classes and display linear β = 1, quadratic β = 2, and quartic level repulsion β = 4. The smooth lines are theoretical predictions for each case: P (S) = e−S for β = 0 and the Wigner surmises of random-matrix theory for β = 1, β = 2, and β = 4, see Sect. 5.6

being the same (λ0 = λ1 = 2.5, λ2 = 5, and λ3 = 7.5). The difference in the degree of level repulsion is obvious. Such a strong reaction of the degree of level repulsion to a change as small as one part per thousand is really rather remarkable. No quantity with a well-defined classical limit could respond so dramatically [21, 22]. For comparison Fig. 2.2 also shows a completely uncorrelated spectrum as seen generically under conditions of classical regular motion (independent of the symmetry class). Classical chaos has an obvious strong and non-trivial effect on level-repulsion and the whole distribution. We will discuss level repulsion in more detail in Chap. 3.

2.16 Beyond Dyson’s Threefold Way We have been concerned with the orthogonal, unitary, and symplectic symmetry classes of Hamiltonians or Floquet operators. They form what is often called the Wigner-Dyson symmetry classes of quantum systems. Building on earlier work of Wigner on random matrices Dyson [23] deduced that classification from group theoretical arguments about complex Hermitian (or unitary) matrices. Much of the present book builds on Dyson’s scheme. During the nineties of the last century, work on the low-energy Dirac spectrum in chromodynamics [24] and on low-energy excitations in disordered superconductors has highlighted universal behavior not fitting Dyson’s threefold way. In particular, Altland, Zirnbauer and their colleagues [25–29] have argued that seven further symmetry classes exist and pointed to realizations in solid-state physics. We will refer to them as non-standard symmetry classes as opposed to the three

46

2 Time Reversal and Unitary Symmetries

Wigner/Dyson classes. The complete classification corresponds to one given by Cartan for Riemannian symmetric spaces.5 The notation class A for the unitary, AI for the orthogonal, and AII for the symplectic symmetry class is based on this Cartan scheme. Here we will introduce the additional symmetry classes by including spectral mirror symmetries and the operators that define them following Ref. [30, 31]. For a discussion of the ten symmetry classes in terms of Riemannian symmetric spaces see Ref. [25, 28]. More recently, the topic of nonstandard symmetry classes has reemerged in the relation of d dimensional topological insulators/superconductors to Anderson localization in d − 1 dimensions [32–38]. A more complete overview may be found in the review [39]. Similar ideas for quantum optical systems are discussed in [40]. A short discussion of the non-standard classes is in order. In its full glory their description requires a many-body or quantum-field theoretic setting of interacting electrons. Our aim is to introduce the reader to the conceptual essence without even trying to do justice to advanced field-theoretic topics or the underlying mathematics. Our approach suffices if applied to the effective one-particle Hamiltonian of noninteracting fermions or for finite dimensional systems.

2.16.1 Spectral Mirror Symmetries An energy spectrum is symmetric around an energy E0 if for every eigenvalue E0 + E there is another eigenvalue at E0 − E. The symmetry point E0 may or may not be itself an eigenenergy. Without loss of generality we may set E0 = 0 by redefining the Hamiltonian H → H = H − E0 . Note that a spectral mirror symmetry for

5 A symmetric space is a Riemannian manifold M with global invariance under a distance preserving geodesic inversion (sign change of all normal coordinates reckoned from any point on M). The curvature tensor is then constant. The scalar curvature can be positive, negative, or zero. The positive-curvature case deserves special interest since the pertinent compact symmetric spaces house the unitary quantum evolution operators. The set of evolution operators (or unitary matrices, in a suitable irreducible representation) can be shown to form a symmetric space, where the matrix inversion U → U −1 yields geodesic inversion w.r.t. the identity as a distance preserving transformation (isometry), with Tr (U −1 dU )2 as the metric. As the name suggests a Riemannian symmetric space M is a metric space with a high degree of symmetry. In two dimensions the plane, the sphere or the hyperbolic plane are well-known examples. In general, symmetric spaces are best thought of as generalizations of these to higher dimensions that share the stated properties. They come in three varieties: Euclidean, compact, or non-compact corresponding to a vanishing curvature (like the plane), a positive curvature (like the sphere), or a negative curvature (like the hyperbolic plane). Formally, a Riemannian symmetric space is a smooth Riemannian manifold such that geodesic reflection at any point is an isometry. This property ensures that one can extend any finite geodesic curve at both ends by using the isometries of the space. Though not immediately obvious this implies that one can represent a Riemannian symmetric space as a homogeneous coset space M ≡ G/H . Here G is the connected component of the full isometry group that contains the identity and H the subset of G that leaves some point x invariant (known as the stabiliser of x). For a symmetric space the stabiliser group is the same at any point.

2.16 Beyond Dyson’s Threefold Way

47

a spectrum not bounded from above implies non-boundedness from below. That fact apparently leads to thermodynamic instability. But there is no problem if we consider either a finite dimensional system or an effective one-particle Hamiltonian for non-interacting Fermions where E0 = 0 is the Fermi energy and states with energies E < 0 are occupied. We shall consider systems whose spectral mirror symmetry is due to some spectral symmetry operator. Any symmetry operation on Hilbert space is either represented by a unitary operator P or an anti-unitary operator C. Now take any eigenstate |ν such that H|ν = E|ν. It is obvious that P or C lead to a symmetric spectrum if either HP|ν = −EP|ν or HC|ν = −EC|ν. This condition on any eigenstate means that the Hamilton operator anti-commutes with either a unitary or an anti-unitary symmetry operator, [P, H]+ = 0 or [C, H]+ = 0. Spectral mirror symmetries may coexist with time-reversal invariance. If P is a unitary spectral mirror symmetry in a time-reversal invariant system there also exists an anti-unitary spectral symmetry operator C ≡ PT that anti-commutes with the Hamiltonian. Similarly, a system with both types of spectral mirror symmetries is also time-reversal invariant with respect to T ≡ PC. Assuming that the system has been reduced completely with respect to any unitary symmetries we know that C, P and T commute,6 [C, P] = [C, T ] = [P, T ] = 0 .

(2.16.1)

Following the same idea as for time-reversal symmetry on can show that an antiunitary spectral symmetry C obeys C 2 = ±1 .

(2.16.2)

6 This can be achieved by redefining the symmetry operators appropriately. To do so we take a timereversal operator T obeying [H, T ] = 0 and T 2 = s1 = ±1 and an anti-unitary spectral mirror symmetry operator C with [H, C ]+ = 0 and C 2 = s2 = ±1. Then P1 = C T and P2 = T C are two, not necessarily equal, unitary spectral mirror symmetry operators satisfying [H, P1/2 ]+ = 0. The four products P12 , P22 , P1 P2 and P2 P1 are all unitary and commute with the Hamiltonian. Since we assume the Hamiltonian fully reduced, all four of them must be a trivial multiplication with a phase factor. Two products are determined by the properties of C and T , namely P1 P2 = s1 s2 = P2 P1 , phase factors indeed. Not yet fixed are the phases for the remaining two products. We denote those by P12 = eiα1 and P22 = eiα2 and have P1 = s1 s2 P2−1 = s1 s2 eiα2 P2 . No choice of the phase α2 annuls the commutator [C , T ] = P1 − P2 = (s1 s2 eiα2 − 1)P2 for both values of s1 s2 = ±1. But redefining the time-reversal operator as T → T = eiφ T we retain the properties [H, T ] = 0 and T 2 = s1 and find [C , T ] = (s1 s2 eiα2 −iφ − eiφ )P2 . That commutator does vanish for φ = α2 /2 if s1 s2 = 1 and φ = (α2 + π)/2 for s1 s2 = −1. With P = C T = T C we then have [P , C ] = [P , T ] = [C , T ] = 0 for the redefined symmetry operators.

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2 Time Reversal and Unitary Symmetries

Next consider any unitary spectral mirror symmetry P that anticommutes with H , i.e. PH = −H P. Then P 2 is a unitary operator that commutes with H — indeed P 2 H = −PH P = H P 2. As we have assumed that no (non-trivial) unitary symmetries are left P 2 = eiα must be a global gauge transformation. However Pe−iα/2 is a unitary spectral mirror symmetry if P is. If there is no timereversal invariance (and thus also no anti-unitary mirror symmetry) we may choose P without loss of generality such that P 2 = 1. In the presence of time-reversal invariance the additional constraint [P, T ] = 0 implies [P 2 , T ] = 0 or (details will be given later in Sect. 2.16.3) P 2 = ±1 .

(2.16.3)

The seven non-standard symmetry classes are obtained by all possible combinations of spectral mirror symmetries and time-reversal invariance. First, there are three non-standard symmetry classes that are not time reversal invariant: either there is a unitary operator with P 2 = 1 or an anti-unitary with C 2 = ±1. In time-reversal invariant systems, as P = CT it is sufficient to consider just anti-unitary mirror symmetries C 2 = ±1 along with time reversal symmetries T 2 = ±1 This leads to four symmetry classes that combine time-reversal symmetry with spectral mirror symmetry: if T 2 = 1 either C 2 = 1 (P 2 = 1) or C 2 = −1 (P 2 = −1); if T 2 = −1 either C 2 = −1 (P 2 = 1) or C 2 = 1 (P 2 = −1). Some symmetry classes may be further divided into subclasses according to the number of generic ‘zero modes’, i.e., the number of vanishing energy eigenvalues for a generic member of the symmetry class. The number of zero modes is given by the socalled topological index ν as ν (2ν) in symmetry classes without (with) Kramer’s degeneracy. We will see in Sect. 2.16.3 how that index arises. For historical reasons these seven classes have been split into two groups. The first group is given by the three chiral classes—the ones that have a unitary mirror symmetry with P 2 = 1. Their importance has first been observed in investigations of Dirac fermions in Quantum Chromodynamics where the spectral symmetry is related to chirality. For this reason we will call P a chiral symmetry operator though in general P need not be related to chirality. The four remaining classes have mainly been discussed in connection to mesoscopic disordered superconductors or superconducting-normalconducting hybrid systems where the anti-unitary mirror symmetry is connected to electron-hole conjugation. For this reason we call C a charge conjugation symmetry operator, though again, in general C need not be related to charge conjugation at all. We will discuss some of these classes in more detail in the following sections. The ‘tenfold way’ is summarised in Table 2.1. The table also shows the corresponding symmetric spaces which can be recognised as equivalent to the spaces of time evolution operators for the corresponding symmetry classes in Hilbert spaces of appropriate size.

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Table 2.1 The ten quantum symmetry classes Symmetry class A AI AII AIIIν (ν ∈ N) BDIν (ν ∈ N) CIIν (ν ∈ N) C CI BDν (ν = 0, 1) BDIIIν (ν = 0, 1)

Unitary Orthogonal Symplectic Chiral unitary Chiral orthogonal Chiral symplectic Anti-pseudo-chiral unitary a Anti-chiral orthogonal a Pseudo-chiral unitary a Anti-chiral symplectic a

T

P

C

0 +1 −1 0 +1 −1 0 +1 0 −1

0 0 0 +1 +1 +1 0 −1 0 −1

0 0 0 0 +1 −1 −1 −1 +1 +1

Symmetric space U(N) U(N)/O(N) U(2N)/Sp(N) U(2N+ν)/U(N+ν)×U(N) SO(2N+ν)/SO(N+ν)×SO(N) Sp(2N+ν)/Sp(N+ν)×Sp(N) Sp(N) Sp(N)/U(N) SO(2N+ν) SO(4N+2ν)/U(2N+ν)

The first column gives the Cartan classification of the symmetry class. The second column gives the verbal name of the symmetry class. If a symmetry class obeys time-reversal symmetry or a spectral mirror symmetry the entry ±1 in the corresponding column indicates that the symmetry operator squares to ±1. The entry 0 means absence the corresponding symmetry. The occurence of generic zero modes is indicated by attaching the topological index ν to the symbol of the symmetry class. The sixth column gives the corresponding Riemannian symmetric space (of compact type) a These names have been invented by us in order to extend the well-established names of the other symmetry classes

2.16.2 Universality in Non-standard Symmetry Classes In the preceding sections we have given some evidence for universal spectral fluctuations in the three Wigner-Dyson symmetry classes. The non-standard symmetry classes have a symmetry point in their spectra which we have chosen as E = 0. One may expect (and we will show it to be the case) that this symmetry influences spectral correlations for energies sufficiently close to the symmetry point. In order to test that expectation we are not allowed to perform a spectral average like the one introduced in Sect. 2.14; such an average would wash out any special behaviour at E = 0. Instead, like it or not, we must average over an ensemble of spectra. To that end we imagine the dynamical system in question to allow for variation of some parameters x = (x1 , x2 , . . . ) without changing the symmetry of the Hamiltonian H = H (x). A set of realizations {x1, . . . , xN } with N  1 and the pertinent spectra will make for a suitable ensemble. On the other hand, finite dimensional systems may allow for varying the Hilbert space dimension N to generate an ensemble without otherwise changing the Hamiltonian. Towards characterizing the announced difference of spectral fluctuations near and far away from the symmetry point at E = 0 we introduce the staircase function    N  min El (xn ) > Eo 1 , θ s− I˜Eo (s) = N ΔE n=1

(2.16.4)

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2 Time Reversal and Unitary Symmetries

with Eo > 0 an “observation point” and ΔE the mean level spacing in our ensemble of spectra, evaluated at Eo . For given s the function N I˜Eo (s) gives the number of spectra in which the distance between the observation point Eo and the first energy eigenvalue above is less than sΔE. The corresponding density is formally given by d ˜ P˜Eo (s) = IE (s) . ds o

(2.16.5)

To turn P˜Eo (s) and I˜Eo (s) into effectively smooth functions of s, we send N → ∞. Assuming, momentarily, Eo located far from the symmetry point at E = 0 (much further away than ΔE) we look at the range of small s, 0 < s < 1. In the largeN limit under study the effectively smooth I˜Eo (s) must be the expected (for our ensemble) number of levels in the interval [Eo , Eo + sΔE], simply because the distinction of the symmetry point will not be felt far away. Now, the mean level density being 1/ΔE the expected number of levels in the said interval of length sΔE is nothing but s such that we have I˜Eo (s) = s. For the derivative of that smoothened staircase we get P˜Eo (0) = 1

for

Eo  ΔE ,

N → ∞.

(2.16.6)

The foregoing discussion also applies to the Wigner-Dyson classes and one indeed observes P˜Eo (s) → 1 as s → 0 there, at least as long as Eo is kept away from the end points of Wigner’s semicircles; there is no symmetry point in the spectra to make for exceptional behavior. However, in the non-standard symmetry classes exceptional behavior does arise near the symmetry point E = 0. In particular, upon setting the observation point on the symmetry point, Eo = 0, one often finds P˜0 (s) ∼ cs α

with

c = 0 .

(2.16.7)

The exponent α measures the degree of repulsion from the centre of the spectral mirror symmetry. It is analogous to the exponent β that measures repulsion between eigenvalues. The value of α characterises the universality classes: it vanishes in the three Wigner-Dyson classes but is a non-zero integer in most universality classes within the non-standard symmetry classes. It is interesting that some symmetry classes (e.g. al the chiral classes) contain multiple universality classes that depend on the algebraic form how the symmetry is realised. We shall turn to evaluations of the exponents α and β in the subsequent chapter.

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2.16.3 Hamiltonian Matrix Structures and Canonical Transformations for Non-standard Symmetries In appropriate bases the standard symmetry classes correspond to Hamilton matrices that are complex Hermitian for the unitary class, real symmetric for the orthogonal class, and self-dual (or, equivalently, real quaternionic) in the symplectic class. In the non-standard classes the additional anti-commutation relation leads to more detailed structure and new canonical transformations. We shall now endeavor to develop this structure in full glory, not without warning the reader that a bit of patience will be needed. In fact, some readers might be content with just studying the derivation of canonical transformations and matrix structures for the chiral classes presented immediately below. The Three Chiral Symmetry Classes (AIIIν , BDIν , CIIν ) with P 2 = 1 We start with the chiral unitary class AIIIν where no time-reversal invariance exists and the unitary spectral mirror symmetry operator P (chirality operator) obeys P H P † = −H and P 2 = 1. Hence P −1 = P † = P and P can only have the eigenvalues 1 and −1. Without loss of generality we may assume that the multiplicity of −1 is N ≥ 1 and the multiplicity of 1 is N + ν with ν ≥ 0.7 The total Hilbert space dimension is then 2N + ν. In the eigenbasis of P we then have   1N+ν 0 P= 0 −1N

(2.16.8)

where 1N and 1N+ν are the unit matrices with dimension N and N + ν, and the zeros refer to rectangular matrices with vanishing entries. It is easy to see that Hamiltonians fulfilling the requirement P H P † = −H are of the form  H =

0 V† V 0

 ,

(2.16.9)

with V a complex rectangular matrix of dimension N × (N + ν). Note that V is at most of rank N and thus there are at least ν vectors of dimension N + ν that are annihilated by V . These vectors may be extended to (2N + ν)-dimensional eigenvectors of H with vanishing energy. We now identify ν as the topological quantum number that we have already mentioned in Sect. 2.16.1. The pertinent canonical transformations have the form   U1 0 S= , (2.16.10) 0 U2

the mulitplicity of −1 is larger than the multiplicity of 1 just replace P by −P . The case N = 0 implies that P = 1 and hence H = 0. A non-trivial class thus requires N ≥ 1.

7 If

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2 Time Reversal and Unitary Symmetries

with U1 ∈ U(N + ν) and U2 ∈ U(N) arbitrary unitary matrices. Indeed, if H has the form (2.16.9), so has H = SH S † with V replaced by V = U2 V U1† . We may thus identify the set of canonical transformations with the product U(N + ν)×U(N). Finally, for the time evolution operator U (t) = e−iH t /h¯ the chiral spectral mirror symmetry implies PU (t)P = U (t)† . In general, unitary matrices obeying PU P = U † are obtained by taking an arbitrary unitary matrix W ∈ U(2N + ν) and setting U = W PW † P. Indeed then, U † = PW PW † = PU P. But W is not unique: substituting W = Wˆ S with S ∈ U(N)× U(N + ν) yields U = W PW † P = Wˆ SPS † Wˆ † P = Wˆ P Wˆ † P because SPS † = P. The matrices U obeying PU P = U † thus form the coset space U(2N + ν)/U(N + ν)× U(N) (which is also a Riemannian symmetric space; see proceeding footnote). Next, let us consider the chiral orthogonal class BDIν where we have an additional anti-unitary time reversal operator that obeys T 2 = 1 and the two commutation relations [H, T ] = 0 and [P, T ] = 0. We continue to work in the basis where P is diagonal with multiplicity N for the eigenvalue −1 and multiplicity N + ν for the eigenvalue 1. We claim that we can choose this basis such that T |ψ = |ψ which just means that T ≡ K is the complex conjugation operator with respect to this basis. Indeed, for any eigenstate P|ψ = σ |ψ (where σ = ±1) [P, T ] = 0 implies T |ψ is also an eigenstate with the same eigenvalue σ . We can then construct a T -invariant basis in the same way as for the standard orthogonal class. The form of the Hamiltonian matrix in the chiral orthogonal class can now be found using the two conditions T H T = H and PH P = −H . In our chosen basis we already know that the first condition implies that H = H˜ is real symmetric while the second implies the structure (2.16.9), now  H =

0 V˜ V 0

 (2.16.11)

with an arbitrary real rectangular matrix V of dimension N ×(N +ν). Proceeding as in the chiral unitary case (replacing unitary matrices by orthogonal ones) we identify the set of canonical transformations S that leave the form (2.16.14) invariant as the product of two orthogonal groups S ∈ O(N + ν)×O(N). The unitary evolution operators U (t) = e−iH t /h¯ in the chiral orthogonal class obey U = U˜ and U † = PU P. In order to reveal the symmetric space behind these conditions it is convenient to introduce the matrix   0 1 Λ = N+ν , (2.16.12) 0 i1N a unitary square root of the chirality operator P = Λ2 obeying Λ−1 = Λ† = Λ∗ . By setting U = ΛOΛ† the new matrix O is unitarily equivalent to U . The conditions ˜ = 1, and O˜ = POP. U † U = 1, U = U˜ and U † = PU P translate to O = O ∗ , OO In short we have a real orthogonal matrix O ∈ O(2N + ν) with the additional constraint O˜ = POP. In fact, the home of the matrix O is the connected subgroup

2.16 Beyond Dyson’s Threefold Way

53

SO(2N + ν) (containing the identity) since only members thereof can be reached with time evolution operators U = e−iH t /h¯ = ΛOΛ† . Special orthogonal matrices that obey the additional constraint may be constructed from an arbitrary special orthogonal matrix W by setting O = W P W˜ P which is again analogous to the chiral unitary case. Along the same line as there we now identify the set of time evolution operators in the chiral unitary class as the symmetric space SO(2N + ν)/SO(N + ν)× SO(N). Remains the chiral symplectic class CIIν where an additional anti-unitary timereversal operator T exists that obeys [H, T ] = 0 for all Hamiltonians in the class and [P, T ] = 0 as well as T 2 = −1. Again we choose a basis where P is diagonal. As in the orthogonal case we see that T leaves the two eigenspaces of P invariant: if |ψ is an eigenstate of P with eigenvalue σ = ±1 then so is T |ψ. The only difference is that now |ψ and T |ψ are orthogonal. Without loss of generality we may then choose the basis states as in the standard symplectic case, see Eq. (2.9.1). The time reversal operator has the form T = KZ where K is complex conjugation with respect to the basis chosen and Z is the matrix given in (2.9.3) and (2.9.4); recall that Z defines the symplectic structure and the symplectic transpose. It follows that the multiplicities of the eigenvalues σ = ±1 of P must be even. Accordingly we write     12(N+ν) 0 Z2(N+ν) 0 P= and Z = . (2.16.13) 0 −12N 0 Z2N The two conditions T H T −1 = H and PH P −1 = −H then imply that the Hamiltonians matrices in the chiral symplectic class have the form 

0 VZ H = V 0

 (2.16.14)

−1 where V is an arbitrary complex 2N × 2(N + ν) matrix and V Z = Z2(N+ν) V˜ Z2N is its symplectic transpose. Following steps analogous to the chiral orthogonal case one shows that the corresponding canonical transformations S are of the form (2.16.10) where now U1 ∈ Sp(N + ν) and U2 ∈ Sp(N) are unitary symplectic matrices. Moreover, again in complete analogy to the chiral orthogonal case (replacing the standard transpose by the symplectic transpose in each step), one finds that the time evolution operators span the symmetric space Sp(2N + ν)/Sp(N + ν)×Sp(N).

The Two Classes CI and BDIIIν with P 2 = −1 and T 2 = ±1 We move on to the next two classes with a unitary spectral mirror symmetry P that squares to −1, P 2 = −1. In Sect. 2.16.1 we already mentioned that we can always assume P 2 = 1 if there is no anti-unitary time-reversal symmetry. So P 2 = −1 is only relevant for time-reversal invariant systems. Before constructing the canonical Hamiltonian matrices let us shortly discuss why P 2 = −1 is the only choice that gives anything new in the presence of time-reversal symmetry T . Let us assume that all unitary

54

2 Time Reversal and Unitary Symmetries

symmetries have been used to reduce the Hilbert space to invariant subspaces. As P 2 is a unitary symmetry it must be trivial, that is a global gauge transformation P 2 = eiα . We know that T commutes with both P and the Hamiltonian H . Let us consider the eigenvalues ±eiα/2 of P. Now P = T PT −1 implies that ±e−iα/2 are eigenvalues as well. Thus either ±eiα/2 = ±1 and P 2 = 1 or ±eiα/2 = ±i and P 2 = −1. Above we have already described the three chiral classes with P 2 = 1. Now let us describe the Hamiltonian matrix structure for the two classes with P 2 = −1 and T 2 = ±1. No established name exists for these symmetry classes apart from their Cartan classifications CI and BDIIIν . Given the minus sign one may refer to them as anti-chiral orthogonal and anti-chiral symplectic. We start with the anti-chiral orthogonal class CI and choose an eigenbasis of P. If |ψ is an eigenstate of the spectral mirror symmetry, so is T |ψ. Indeed, P|ψ = iσ |ψ with σ = ±1, then PT |ψ = T P|ψ = −σ i|ψ. This implies that the multiplicities of ±i are the same and that one may choose a basis where P =i

  1N 0 0 −1N

 and T = K

0 1N 1N 0

 ≡ KΣx

(2.16.15)

where K is complex conjugation with respect to the chosen basis. The resulting structure for the Hamilton matrix H is (2.16.9) with a complex symmetric N × N matrix V . Moreover, H is also symmetric with respect to the Σx transpose, H = H Σx ≡ Σx H˜ Σx−1 . It is easily checked that the corresponding canonical transformations have the form  ∗  U1 0 S= (2.16.16) 0 U1 where U1 ∈ U (N) is an arbitrary unitary matrix of size N × N. Finally the unitary time evolution operators span a space of unitary matrices that obey U † = PU P −1 and U = U Σx ≡ Σx U˜ Σx−1 . Combining the two conditions to   U † = PΣx U˜ (PΣx )−1 ≡ Z U˜ Z −1 where Z = 10N −10N reveals that U ∈ Sp(N) is a unitary symplectic matrix with the additional requirement that U , like H , is also symmetric with respect to the Σx -transpose U = U Σx . Note that the matrix Z here can be brought into the form (2.9.3) and (2.9.4) used to define the symplectic structure by a simple permutation of the basis states. Now choosing an arbitrary W ∈ Sp(N) one may easily verify that U = W Σx W is both unitary symplectic U Z U = 1 (because (W Σx )Z = (W Z )Σx ) and satisfies the additional condition U = U ΣX by construction. Moreover setting W = S Wˆ where S of the blockdiagonal form (2.16.16) a short calculation shows U = W Σx W = Wˆ Σx Wˆ . Note that S Z S = 1 and thus (2.16.16) embeds the group U(N) as a subgroup of Sp(N). Hence we identify the set of time evolution operators as the symmetric space Sp(N)/U(N). Now why do we call the ensemble in question orthogonal anti-chiral, in spite of the fact that the orthogonal group does not seem to enter the canonical transformation? One good reason comes from random-matrix theory: the universal

2.16 Beyond Dyson’s Threefold Way

55

spectral fluctuations in the orthogonal anti-chiral class go over to the ones of the orthogonal class as one moves away from the spectral symmetry point at E = 0. Moreover, there is an algebraic reason that we do want to show the reader as it provides an alternative canonical form of the Hamiltonian matrix. Let us consider a change of basis brought about by the unitary symmetric matrix e−iπ/4 X= √ 2



1N −i1N −i1N 1N



= X˜ .

(2.16.17)

In the new basis a short calculation gives the transformed unitary spectral mirror symmetry and time-reversal operator as Pˆ = XPX† =



0 −1N 1N 0

 (2.16.18)

and Tˆ = XT X† = XΣx XK ≡ K, .

(2.16.19)

The alternative canonical form of the Hamiltonian matrix is then a real symmetric matrix of the form   B A Hˆ = XH X† = ; (2.16.20) A −B here A = A˜ and B = B˜ are real symmetric matrices, the real and imaginary parts of the complex symmetric matrix V = A + iB that entered the original canonical form. Finally, the canonical transformations in the new basis are Sˆ = XSX† =



C D −D C

 (2.16.21)

where C and D are the real and imaginary parts of the unitary matrix U1 = C + iD. Note that U1† U1 = 1 implies S˜ˆ Sˆ = 1. The canonical transformations Sˆ are thus a representation of the group U(N) in the real (special) orthogonal matrices SO(2N). So it is fair to consider the canonical transformations as a restricted subset of the orthogonal transformations, and this is the algebraic justification for calling the symmetry class CI anti-chiral orthogonal. Finally, we turn to the symplectic anti-chiral class BDIIIν (for ν = 0 just denoted by DIII) where the unitary spectral symmetry obeys P 2 = −1 and the time-reversal operator obeys T 2 = −1. In analogy to the orthogonal anti-chiral case we may choose a basis where P =i

  0 12N+ν 0 −12N+ν

(2.16.22)

56

2 Time Reversal and Unitary Symmetries

and  T =K

0

12N+ν

−12N+ν 0

 ≡ KZ .

(2.16.23)

We write the dimension of the blocks in the foregoing two matrices as 2N + ν with ν = 0, 1 in order to distinguish even and odd multiplicity of the eigenvalues of P; note that we waived the index 2N + ν on the unitary matrix Z, to avoid undue ugliness. The conditions PH P −1 = −H and T H T −1 = H (or, equivalently, H = H Z = Z H˜ Z −1 ) imply that the canonical form of the Hamiltonian matrix is (2.16.9) with a complex anti-symmetric matrix V of dimension (2N + ν) × (2N + ν). It is now time to reveal the integer ν as the topological quantum number counting the number of zero modes. To that end let us consider the spectrum of V which in general consists of complex numbers. The spectrum satisfies P (λ) = 0 where the characteristic polynomial P (λ) = det(λ − V ) = det(λ − V˜ ) = det(λ + V ) = (−1)2N+ν P (−λ) is an even function of λ for ν = 0 and an odd function if ν = 1. This implies that H has at least one zero mode if ν = 1 and generally none for ν = 0. Our notation is thus consistent with the definition of ν as the topological quantum number. The canonical transformations S are of the form (2.16.16) with U1 ∈ U(2N + ν). The reference to symplectic in the name anti-chiral symplectic class comes again from the observation that universal properties of random-matrix ensembles in this class go over to the ones of the standard symplectic class when one is far away from the spectral symmetry point E = 0. To give an additional algebraic foundation to our naming BDIIIν symplectic anti-chiral we note that the canonical ˜ † = S † . We may consider the canonical transformations also obey S Z = Z SZ transformations as a representation of U(2N + ν) in the unitary symplectic matrices Sp(2N + ν). It remains to consider the unitary time evolution operators U (t) = e−iH t /h¯ . The conditions U † = T U T −1 = PU P −1 reduce U to being self-dual (symmetric with respect to the Z-transpose) U = U Z = Z U˜ Z −1 and U † = U Σx = Σx U˜ Σx−1 . The latter restriction essentially means that U becomes real in an appropriate basis. So let us consider the unitary basis transformation eiπ/4 Xˆ = iX = √ 2



1N −i1N −i1N 1N



= X˜ .

(2.16.24)

which satisfies Xˆ 2 = Σx = Σx−1 , Xˆ = X˜ˆ and Xˆ Z = iX† . With ˆ Xˆ † . Uˆ = XU

(2.16.25)

a short calculation shows that Uˆ ∗ = Uˆ ∈ SO(4N + 2ν) is real orthogonal and also obeys Uˆ Z = Uˆ . The corresponding symmetric space can be found by writing ˆ Xˆ † = XSX† as Uˆ = O Z O where O ∈ SO(4N + 2ν) is arbitrary. Now let Sˆ = XS in (2.16.21) be a canonical transformation represented in the new basis as a subset of

2.16 Beyond Dyson’s Threefold Way

57

SO(4N + 2ν). Setting O = Sˆ Oˆ shows that O and Oˆ define the same time evolution matrix U = O Z O = Oˆ Z Oˆ and we identify the set of time evolution matrices with the symmetric space SO(4N + 2ν)/U(2N + ν). Let us also note the alternative standard forms   0 −12N+ν † ˆ P = XPX = ≡Z (2.16.26) 12N+ν 0 for the spectral mirror symmetry, ˆ Tˆ = XZK Xˆ † = i



 12N+ν K ≡ iΣx K = −KiΣx 0

(2.16.27)

  C D ˆ H =i D −C

(2.16.28)

0

12N+ν

for the time-reversal operator and

where C = −C˜ and D = −D˜ are real anti-symmetric (such that H is imaginary and anti-symmetric). They are related to the complex anti-symmetric matrix V that defines the original canonical form by V − V † = −2C and V + V † = 2iD. The Two Classes C and BDν with C 2 = ±1 and no Time-Reversal Symmetry With all classes that have a unitary spectral symmetry operator accounted for, only two symmetry classes remain to be explained. There may be a spectral mirror symmetry due to CH C −1 = −H with an anti-unitary operator C. Indeed in four of the five non-standard symmetry classes discussed above one may choose C = PT . Only the chiral unitary class has no time-reversal symmetry and thus no anti-unitary spectral mirror symmetry. The remaining two classes only have an antiunitary spectral mirror symmetry while time-reversal symmetry is broken. They are distinguished by the property C 2 = 1 for class BDν (for ν = 0 often just denoted as D) or C 2 = −1 for class C. Any other case C 2 = eiα = ±1 can be excluded following the analogous discussion of the anti-unitary time-reversal operator. Both classes are straightforward to describe. We start with class BDν where C 2 = 1. We may choose a basis such that C = K is just complex conjugation. The condition CH C −1 = −H then becomes H ∗ = −H or H = iA

(2.16.29)

where A = −A˜ is real and anti-symmetric. The Hilbert space dimension may be even or odd. In the even case there are generally no zero modes while in the odd case there is at least one vanishing energy eigenvalue due to the form of Hamiltonian. We thus write the Hilbert space dimension as 2N + ν with the topological quantum number ν = 0 for even dimension and ν = 1 for odd dimension. The canonical

58

2 Time Reversal and Unitary Symmetries

transformations S that leave the form of H invariant (connect real anti-symmetric matrices) are just the real orthogonal matrices O ∈ SO(2N + ν). The time evolution operators U (t) = e−iH t /h¯ = eAt /h¯ are real and orthogonal with no further restrictions. They thus span the whole group SO(2N + ν). There is no established name for this class apart from the Cartan classification BDν . We suggest pseudochiral unitary class BDν : pseudo-chiral to refer to the anti-linear spectral mirror symmetry and unitary because the universal spectral properties in random-matrix theory far away from the symmetry energy resemble those of the unitary class. Note that SO(2N + ν) ⊂ U(2N + ν), so this class may rightly be considered as a subset of the unitary class. The last class to discuss is the class C where C 2 = −1. As we already know we may find a basis such that  C=

 0 −1N K ≡ ZK 1N 0

(2.16.30)

and the Hilbert space dimension must be even. Note that CH C −1 = −H is equivalent to the condition H = −H Z . The corresponding structure of the Hamiltonian matrix is   A B H = (2.16.31) B ∗ −A∗ where A = A† is a complex Hermitian N × N matrix and B = B˜ is complex symmetric. One quickly checks that a unitary transformation S preserving the condition H = −H Z must obey S Z = S † such that S ∈ Sp(N) is unitary symplectic. As for the time evolution operator U (t) = e−iH t /h¯ one also finds U Z = U † . The set {U (t)} thus spans the group Sp(N) ⊂ U(2N). This class also lacks any established name apart from the Cartan classification C. One might speak of the anti-pseudo-chiral unitary class C. Entitled to a good rest are the brave ones having laboured through this algebraic ordeal of Hamiltonian matrix structures for non-standard symmetries.

2.16.4 Physical Realizations of Non-standard Symmetry Classes in Fermionic Systems Physical realizations of non-standard symmetries for any type of particle require a field-theoretic setting. In standard quantum mechanics the Hilbert space of a freely moving particle is inifinite dimensional and has an energy spectrum that is not bounded from above. Non-standard symmetry classes have a spectrum that is mirror symmetric and thus cannot be bounded from but one side. Thermodynamic instability threatens and must somehow done away with. The field-theoretic setting

2.16 Beyond Dyson’s Threefold Way

59

with Fermionic particles opens up the realization of non-standard symmetry classes, both relativistic and non-relativistic. In the relativistic case the non-interacting oneparticle states are described by the Dirac equation and the negative-energy states of the Dirac equation physically correspond to positive-energy excitations of antiparticles. In the non-relativistic setting one may consider electronic excitations of a solid state. Here the negative energy states physically correspond to hole-like excitation with a positive charge and a positive excitation energy. In superconductors electron- and hole-like excitations are coupled and lead to non-trivial realizations of non-standard symmetry classes. We will discuss this case here in some detail to complete the picture. Later, in Sect. 2.16.5 we will give an example of non-standard symmetry classes in physical models with a finite Hilbert space (in that case the energy spectrum is always bounded from both below and above). Let us now deal with the Fermionic setting and consider normal-superconducting hybrid structures. These are solid-state structures where some regions are normal conductors while other regions are superconducting (some regions may also be filled with isolators). These can be realized as Andreev billiards where a normal conductor of some finite shape has superconductors attached to some parts of the boundary while other parts of the boundary may be an interface to an isolator. A prominent effect distinguishing such hybrid structures from all-normal electronic billiards is Andreev scattering [41]: An electron leaving the normal conductor to enter a superconductor may there combine with another electron of (nearly) opposite velocity to form a Cooper pair. A hole with velocity (nearly) opposite to the lost electron must then enter the normal conductor and retrace the path of the lost electron, a small angular mismatch apart which is due to the small energy mismatch  of a quasiparticle relative to the Fermi energy EF . The hole picks up a scattering phase π/2 − φ where φ is the phase of the superconducting order parameter at the interface. In a certain sense (that does not contradict energy conservation) an electron of energy  is scattered into a hole of energy −. Due to Andreev scattering a non-vanishing Cooper-pair amplitude forms within the normal conductor, close to the interface with each superconductor, as the following rough argument indicates. When the hole “created” by Andreev scattering retraces the path of the original electron back to the interface with the same superconductor and becomes retroreflected as an electron again the coherent succession of two electrons appears like a Cooper pair. An observable consequence is a gap, called Andreev gap in the excitation spectrum. A solid-state realization of the chiral unitary class AIII is a disordered tightbinding model on a bipartite lattice with broken time reversal invariance, such as the random flux problem [42]. Moreover, the class AIII arises from BdG Hamiltonians with invariance under time reversal and spin rotation about the z axis in spin space [32]. For further solid-state applications of the chiral classes see [41]. Applications in chromodynamics have been discussed by Verbaarschot [24]. The simplest description of the many-electron problem arises in the meanfield approximation which yields an effective single-particle theory. The pertinent second-quantized Bardeen-Cooper-Schrieffer (BCS) Hamiltonian involves annihilation operators cα and creation operators cα† . The index α accounts for, say, N orbital

60

2 Time Reversal and Unitary Symmetries

single-particle states as well as two spin states such that α = 1, 2, . . . 2N. Electrons being Fermions these operators obey the anticommutation rules cα cβ† + cβ† cα = δαβ .

(2.16.32)

 1 1 hαβ cα† cβ + Δαβ cα† cβ† + Δ∗αβ cβ cα ; 2 2

(2.16.33)

The Hamiltonian then reads H =

 αβ

herein the matrix h accounts for normal motion due to kinetic energy, single-particle potential, and possibly magnetic fields; the order-parameter matrix Δ brings in superconduction and coupling of electrons with holes. Hermiticity of H and Fermi statistics restrict the 2N × 2N-matrices h and Δ as hαβ = h∗βα ,

Δαβ = −Δβα .

(2.16.34)

It is convenient to write the BCS Hamiltonian as row×matrix×column,    1 h Δ c + const , (2.16.35) H = c† , c −Δ∗ −h˜ c† 2 with const = 12 Tr h, so as to associate the BCS-Hamitonian H with a Hermitian 4N × 4N matrix   h Δ H = (2.16.36) −Δ∗ −h˜ known as the Bogolyubov-deGennes (BdG) Hamiltonian. The “physical space” spanned by the orbital and spin states is thus enlarged by a two-dimensional “particle-hole space”.8 The restrictions (2.16.34) take the form H = −Σx H ∗ Σx = −CH C −1

(2.16.37)

where C = Σx K is an anti-unitary spectral mirror symmetry operator with C 2 = 1. If no time-reversal operator exists that commutes with H and no unitary symmetries are present either we now identify the BdG Hamiltonian as non-standard symmetry class BD0 . In the presence of time-reversal symmetry or additional

8 No extra states are introduced here even though the BdG jargon does invite such misunderstanding; the 2N single-electron states acted upon by the matrix h may have their energies above or below the Fermi energy; BdG jargon terms them “particle states” and even indulges in speaking ˜ the BdG hole states are in fact identical copies of about “hole states” acted upon by the matrix −h; the BdG particle states.

2.16 Beyond Dyson’s Threefold Way

61

unitary symmetries classes C, CI and BDIII can be obtained by the appropriate reductions. For details we refer to the literature [26].

2.16.5 Quantum Mechanical Realisation of Non-standard Symmetry Classes in Finite Dimensional Hilbert Spaces: Two Coupled Tops Let us now consider two coupled quantum tops with respective angular momentum operators L = (Lx , Ly , Lz ) and M = (Mx , My , Mz ). They satisfy the standard commutation relations   klm Lm , [Mk , Ml ] = i klm Mm , [Lk , Ml ] = 0 [Lk , Ll ] = i m∈{x,y,z}

m∈{x,y,z}

(2.16.38) where k, l ∈ {x, y, z} and klm is the totally anti-symmetric Levy-Civita tensor with xyz = 1. They generate the group SU(2) × SU(2) and we assume that the Hilbert space V is an irreducible representation of this group such that both angular momenta have the same total angular momentum j . The dimension is then N = dim V = (2j + 1)2 and any state |ψ ∈ V obeys   L2 |ψ = L2x + L2y + L2z |ψ = j (j + 1)|ψ = M2 |ψ

(2.16.39)

The standard basis {|m1 , m2 } with m1 , m2 ∈ {−j, −j + 1, . . . , j } ≡ Zj consists of the common eigenstates of Lz and Mz , Lz |m1 , m2  = m1 |m1 , m2 ,

Mz |m1 , m2  = m2 |m1 , m2  .

(2.16.40)

We will discuss the particular Hamiltonian H =

λ1 λ2 Lx Mx (Lz + Mz ) + j + 1/2 (j + 1/2)2

(2.16.41)

where λ1 and λ2 are coupling constants. Feinberg and Peres [43, 44] introduced this as a paradigmatic model for quantum chaos and showed that the corresponding classical dynamics at energy E = 0 is fully chaotic for λ1 = 1 and λ2 = 4. The non-standard symmetry of this model has been analyzed only recently [45]. Indeed it is straightforward to recognize that P = eij α Rx(1) (π) ⊗ Ry(2) (π) = eij α−iπLx −iπMy

(2.16.42)

is a unitary spectral mirror symmetry operator. Here j α is a real phase to be (1) (2) considered later, Rx (π) rotates the first top by π around the x-axis, and Ry (π) rotates the second top by π around the y-axis.

62

2 Time Reversal and Unitary Symmetries

For the full symmetry classification of the model we need to consider all unitary symmetries and time-reversal invariance. Let us start with the unitary symmetries. The Hamiltonian (2.16.41) is invariant under exchange of the two tops and under a simultaneous rotation of both tops by an angle π around the z-axis. Let us denote the corresponding unitary quantum operators by U1 and U2 and define them through their action in the standard basis U1 |m1 , m2  =|m2 , m1 

(2.16.43a)

U2 |m1 , m2  =eiπ(2j −m1 −m2 ) |m1 , m2 

(2.16.43b)

for m1 , m2 ∈ Zj . So U1 describes the exchange of tops. It has eigenvectors |m1 , m2  ± |m2 , m1  with eigenvalues ±1 and thus obeys U12 = 1. It acts on angular momentum operators as U1 Lk U1† = Mk

and U1 Mk U1† = Lk .

(2.16.44)

The operator U2 is already diagonal in the standard basis. Note that we have chosen U2 = ei2j π Rz (π) = ±Rz (π) where Rz (π) = Rz(1) (π) ⊗ Rz(2) (π) = e−iπ(Lz +Mz ) describes the simultaneous rotation of both tops. The additional scalar factor does not change the action on angular momentum operators, U2 Lx U2† = −Lx ,

U2 Ly U2† = −Ly ,

U2 Lz U2† = Lz ,

(2.16.45a)

U2 Mx U2† = −Mx ,

U2 My U2† = −My ,

U2 Mz U2† = Mz .

(2.16.45b)

The extra phase factor does however affect the eigenvalues eiπ(2j −m1−m2 ) = ±1. This implies U22 = 1 as well. The two operations clearly commute, [U1 , U2 ] = 0, and have the common eigenstates ∝ (|m1 , m2  ± |m2 , m1 ). Their action on angular momentum operators implies that they both leave the Hamiltonian invariant U1 H U1† = U2 H U2† = H .

(2.16.46)

The Hilbert space therefore separates into four invariant orthogonal subspaces V = V++ ⊕ V+− ⊕ V−+ ⊕ V−−

(2.16.47)

and the Hamiltonian becomes block-diagonal in the appropriately ordered common eigenbasis ⎛

⎞ 0 0 H++ 0 ⎜ 0 H+− 0 0 ⎟ ⎟ . H =⎜ ⎝ 0 0 H−+ 0 ⎠ 0 0 0 H−−

(2.16.48)

2.16 Beyond Dyson’s Threefold Way

63

For later use it will be useful to describe the subspaces in a little more detail. The common eigenbasis of U1 and U2 in the invariant subspaces are given by |m1 , m2 , ++ =

|m1 , m1 

if m1 = m2 ,

|m1 ,m2 +|m √ 2 ,m1  2

if m2 > m1 ,

(2.16.49a)

where m1 ∈ Zj , m2 ≥ m1 and 2j − m1 − m2 is even; |m1 , m2  + |m2 , m1  √ 2

|m1 , m2 , +− =

(2.16.49b)

where m1 ∈ Zj , m2 > m1 and 2j − m1 − m2 is odd; |m1 , m2  − |m2 , m1  √ 2

|m1 , m2 , −+ =

(2.16.49c)

where m1 ∈ Zj , m2 > m1 and 2j − m1 − m2 is even; |m1 , m2  − |m2 , m1  √ 2

|m1 , m2 , −− =

(2.16.49d)

where m1 ∈ Zj , m2 > m1 and 2j − m1 − m2 is odd. The corresponding dimensions are N++ =

if j = 1, 2, . . . is integer,

(j + 1)2 (j

+ 1)2

−

1 4

if j = 1/2, 3/2, . . . is half-integer,

(2.16.50a)

N+− =(2j + 1)(j + 1) − N++ ,

(2.16.50b)

N−+ =N++ − 2j − 1, v

(2.16.50c)

N−− =N+−

(2.16.50d)

and obey N = (2j + 1)2 = N++ + N+− + N−+ + N−− as required. Before moving on let us also state the obvious but relevant fact that the standard basis {|m1 , m2 } and the common eigenbasis {m1 , m2 , ±, ±} are related by a real orthogonal transformation. Now with all unitary symmetries taken into account to fully reduce the Hamiltonian, let us consider time reversal symmetry. As is well known, the standard basis where Lz and Mz are diagonal makes for real symmetric matrices representing Lx , Mx , Lz , and Mz , while the matrices of Ly and My are purely imaginary and antisymmetric. Let us thus define the anti-unitary operator T by its action on an arbitrary state expanded in the standard basis T

 m1 ,m2 ∈Zj

αm1 ,m2 |m1 , m2  =

 m1 ,m2 ∈Zj

∗ αm |m1 , m2  , 1 ,m2

(2.16.51)

64

2 Time Reversal and Unitary Symmetries

such that T = K is nothing but complex conjugation and thus squares to the identity, T 2 = 1. Moreover, considering matrix elements we immediately see T Lx T −1 =Lx ,

T Ly T −1 = − Ly ,

T Lz T −1 =Lz ,

(2.16.52a)

T Mx T −1 =Mx ,

T My T −1 = − My ,

T Mz T −1 =Mz .

(2.16.52b)

The operator T is thus an unconventional time-reversal operator (the conventional time reversal operator changes the sign of all angular momentum components) and the Hamiltonian is invariant under time-reversal [T , H ] = 0 .

(2.16.53)

This time-reversal invariance carries over to all invariant subsystems because a real orthogonal matrix has been used to obtain the block-diagonal form of H . We will use the same symbol T for the induced operation in the subspaces (complex conjugation with respect to the common basis {|m1 , m2 , ±, ±}) and thus have [H++ , T ] = 0,

[H+− , T ] = 0,

[H−+ , T ] = 0,

and [H−− , T ] = 0 . (2.16.54)

Moreover, we know that commutativity of the mirror symmetry and time reversal operators, [P, T ] = 0, can always be arranged. In our case, this is achieved by fixing the phase left open in (2.16.42) as α = π. Indeed, T PT = e−i(j α+πLx −πMy ) and for this to equal ei(j α−πLx −πMy ) we must have ei(2j α−2πLx ) = 1; in the latter expression we can imagine Lx diagonalized; obviously, then, both for integer and half-integer values of j the desired property is secured by α = π. Our mirror symmetry operator is thus elevated to P = eiπ(j −Lx −My ) . By the same token we find P2 =

1

if j = 1, 2, . . . is integer

−1 if j = 1/2, 3/2, . . . is half-integer .

(2.16.55)

A final property of the mirror symmetry operator we shall need presently is the action in the standard basis, P|m1 , m2  = (−1)j −m2 | − m1 , −m2  ,

(2.16.56)

which can be proven using standard properties of Wigner’s D-matrices. The ten-fold symmetry classification allows for three ‘orthogonal’ classes with invariance under a time-reversal operation obeying T 2 = 1: the standard orthogonal class AI and the two non-standard classes BDI and CI. A superficial glance at Table 2.1 might suggest to discard the Wigner-Dyson class AI since the entry ‘0’ for in the column for unitary mirror symmetry operator means absence of P, while our Hamiltonian (2.16.41) does anticommute with the unitary spectral mirror symmetry operator P given above. Such a conclusion would

2.16 Beyond Dyson’s Threefold Way

65

be overhasty, however. The chirality operator (2.16.42) is defined globally, that is in the whole Hilbert space V , and we must scrutinize its action in the invariant subspaces to avoid erroneous guesses. In fact, we propose to show that all three orthogonal classes are realized in the different invariant subspaces. We proceed to checking what operators our ‘global’ mirror symmetry operator induces in the four invariant orthogonal subspaces V±± of the Hamiltonian (see (2.16.47)). In the basis {|m1 , m2 ; ±±} where H is block-diagonal (see (2.16.48)) it is straightforward to see that the matrix of the spectral mirror symmetry operator takes the form ⎛ ♣ ⎜0 P =⎜ ⎝0 0

0 0 0 ♥

0 0 ♠ 0

⎞ 0 ♦⎟ ⎟ 0⎠

(2.16.57)

0

where the four non-zero entries, all unitary, are symbolized by suits. We read off ♣

V++ −→ V++ ,

♠

V−+ −→ V−+ ,

♦

♥

V+− −→ V−− ,

V−− −→ V+− . ♣

Before drawing conclusions we would like to explain V++ −→ V++ . To that end considering P|m1 , m2  ∝ | − m1 , −m2  + (−1)m2 −m1 | − m2 , −m1  and using 2j − m1 − m2 even we must show that the difference m2 − m1 is even. For j integer, the sum m1 + m2 is even which is possible only if m1 and m2 are either both even or both odd; both cases give even m2 − m1 . For half-integer j we write j = ν + 12 , m1,2 = μ1,2 + 12 and repeat the forgoing reasoning to arrive at μ2 −μ1 = m2 − m1 even. Done. The other three mappings are arrived at similarly. We conclude that the global operator P only induces chirality operators inside the subspaces V++ and V−+ wherein they commute with the pertinent blocks of the Hamiltonian. To draw further conclusions about what happens in these two subspaces we return to a more serious notation, ♣ = P++ and ♠ = P−+ . Let 2 us first assume that j is integer. We then have P±+ = 1 and H++ and H−+ both belong to the chiral orthogonal class BDI. The eigenvalues of P±+ can only be ±1. Using (2.16.56) it is straightforward to see that for odd j the dimension N++ = (j + 1)2 of V++ is even and the eigenvalues ±1 appear with exactly the same degeneracy N++ /2. For even j however the dimension N++ is odd and the degeneracies cannot match exactly. One can show that the difference is always one in this case which implies that there is at least one zero mode. To summarize, the topological quantum number (which counts the number of zero modes) satisfies ν++ = 0 if j is even and ν++ = 1 if j is odd. In the invariant subspace V−+ the situation is just the other way around: whenever j is odd N−+ is odd and ν−+ = 1 while for even j one has ν−+ = 0. Now let us assume that j is half2 = −1 and H integer. Then P±+ ±+ both belong to the anti-chiral orthogonal class CI. The eigenvalues of P±+ are now ±i. The dimensions N±+ are always even for

66

2 Time Reversal and Unitary Symmetries

half-integer j and one can check, using (2.16.56), that the eigenvalues ±i have the same degeneracy N±+ /2. As regards the remaining two subspaces V−− and V+− which are interconnected by the entries ♦ and ♥ in (2.16.57), we conclude that for any j the corresponding Hamiltonians H+− and H−− belong to the orthogonal class AI. So indeed, our coupled tops do allow for all three orthogonal symmetry classes.

2.16.6 Non-standard Universality for Two Coupled Tops In this section we give a numerical example of universal features in the nonstandard symmetry classes that are absent in the standard classes. Figure 2.3 shows the integrated distribution I˜0 (s) of the first positive energy eigenvalue for two coupled tops with the Hamiltonian (2.16.41).9 Setting λ1 = 1 and λ2 = 4 to ensure that the classical dynamics is chaotic one may numerically find the spectrum for many values of the spin quantum number j independently in each invariant subspace as introduced in the proceeding subsection and compare it to the predictions of random-matrix theory [45]. The data for Fig. 2.3 include spectra for j = 40.5, 41, . . . 600. For each value of j and each of the four subspaces 60 energy eigenvalues near E = 0 were obtained and used to find the mean level spacing. The smallest positive energy eigenvalue was then recorded in units of the numerically obtained mean level spacing. Repulsion or attraction of the first positive energy level (relative to the symmetry point E = 0) is measured by the behaviour I˜0 (s) ∼ cs α+1 for small s. In the standard symmetry classes no repulsion or attraction arises, as characterized by the values α = 0 and c = 1. Attraction implies α = 0 and c > 1 while repulsion means α > 0 for some c = 0 (or α = 0 with c < 1). Such behavior does arise for the non-standard symmetry classes, as we shall show in the subsequent Chap. 3 by calculating the pertinent exponents α and prefactors c. In the proceeding subsection we have shown that the reduced Hamiltonians of our two coupled tops belong to the standard symmetry class AI, and to the non-standard symmetry classes BDI0 , BDI1 , and CI depending on the angular momentum quantum number j and the symmetry of the invariant subspace. Figure 2.3 now reveals universal behavior of the integrated distribution of the first positive energy level, in agreement with what the pertinent symmetry classes demand.

prefer to employ the integrated variant I˜0 (s) rather than the distribution P˜0 (s) as it gives a clearer picture. Resolving the distribution P˜0 (s) is always obstructed by the necessary binning and requires a larger number of numerical realizations.

9 We

2.16 Beyond Dyson’s Threefold Way

67

a)

b)

1

1

0.9

0.9 BDI0

AI

0.7

0.6

0.6

0.5

0.4

0.3

0.3

0.2

0.2

0

0.1

BDI1 /CI 0

1

s

0

2

c)

1

2

1

2

s

1

0.9

0.9 BDI0

0.8

AI

0.7

0.7

0.6

0.6

0.5

0.4

0.3

0.3

0.2

0.2 0.1

BDI1 /CI 0

1

s

2

AI

0.5

0.4

0.1

BDI0

0.8

I˜0 (s)

I˜0 (s)

BDI1 /CI 0

d)

1

0

AI

0.5

0.4

0.1

BDI0

0.8

0.7

I˜0 (s)

I˜0 (s)

0.8

0

BDI1 /CI 0

s

Fig. 2.3 Integrated distribution I˜0 (s) of the first positive energy eigenvalue (on the scale of the mean level spacing). The black lines show the universal predictions of Gaussian random-matrix ensembles for the symmetry classes AI, BDI0 , BDI1 and CI. Graph (a) collects the data for reduced Hamiltonians in the non-standard class BDI0 where the attraction of the lowest positive energy eigenvalue to E = 0 can be seen from the linear behaviour for small S where the slope is visibly larger than one (the two curves correspond to different symmetries in the reduced subspaces). Graph (b) shows analogous data for reduced Hamiltonians in class BDI1 —the quadratic behaviour for small s is consistent with α = 1. Graph (c) shows the corresponding behaviour for reduced Hamiltonians in class CI and mirrors the behaviour in class BDI1 which has identical universal predictions (for this measure). Eventually graph (d) collects the data from all reduced Hamiltonians in the standard symmetry class AI where no attraction and repulsion is expected and the data is consistent with that expectation (the 4 lines correspond to the two different symmetries of the invariant subspaces where integer and half-integer spins have been collected in separate data sets)

Each graph in Fig. 2.3 collects all data for one of the four cases. One can clearly see that AI shows the expected behaviour α = 0 and c = 1. In class BDI0 the universal prediction is a repulsion (α = 0 and a known value c > 1). For the classes BDI1 and CI the universal predictions coincide and show repulsion with α = 1 (and the same values for c).

68

2 Time Reversal and Unitary Symmetries

2.17 Problems 2.1 Consider a particle with the Hamiltonian H = (p − (e/c)A)2 /2m + V (|x|) where the vector potential A represents a magnetic field B constant in space and time. Show that the motion is invariant under a nonconventional time reversal which is the product of conventional time reversal with a rotation by π about an axis perpendicular to B. Give the general condition for V (x) necessary for the given nonconventional T to commute with H . 2.2 Generalize the statement in Problem 2.1 to N particles with isotropic pair interactions. 2.3 Show that KxK −1 = x, KpK −1 = −p, and KLK −1 = −L, where L = x × p is the orbital angular momentum and K the complex conjugation defined with respect to the position representation. 2.4 (a) Show that antiunitary implies antilinearity. (b) Show that antilinearity and |Kψ|Kφ|2 = |ψ|φ|2 together imply the antiunitarity of K. # 2.5 Show that Uμν = Uνμ = dxμ|x ν|x, U † = U −1 , for K = U K˜ where K and K˜ are the complex conjugation operations in the continuous basis |x and the discrete basis |μ, respectively. 2.6 Show that for spin-1 particles, time reversal can be simply complex conjugation. 2.7 Show that the unitary matrix U in T = U K is symmetric or antisymmetric when T squares to unity or minus unity, respectively. 2.8 Show that φ|ψ = T φ|T ψ∗ for T = U K with U † = U −1 . 2.9 Use the associative law T T 2 = T 2 T for the antilinear operator of time reversal to show that the unimodular number α in T 2 = α must equal ±1. 2.10 Show that time-reversal invariance with T 2 = ±1 together with full isotropy implies that the canonical transformations for all subspaces (see Sect. 2.8) are given by the orthogonal ones. 2.11 Find the group of canonical transformations for a Hamiltonian obeying [T , H ] = 0, T 2 = −1 and having cylindrical symmetry. 2.12 Show that the symplectic matrices S defined by SZ S˜ = Z form a group. 2.13 Let H0 and V commute with an antiunitary operator T . Show that T F T −1 = F † with F = e−H0 τ/2h¯ e−ikV /h¯ e−iH0 τ/2h¯ . 2.14 What would be the analogue of H (t) = T H (−t)T −1 if the Floquet operator were to commute with some T0 ? 2.15 Show that the eigenvectors of unitary operators are mutually orthogonal. 2.16 Show that U (N) is canonical for Floquet operators without any T covariance.

References

69

2.17 Show that U (N) ⊗ U (N) is canonical for Floquet operators with T F T −1 = F † , T 2 = −1, [Rx , F ] = 0, [T , Rx ] = 0, Rx2 = −1. 2.18 Show that O(N)⊗O(N) is canonical if, in addition to the symmetries in Problem 2.16, there is another parity Ry commuting with F and T but anticommuting with Rx . 2.19 Give the group of canonical transformations for Floquet operators in situations of full isotropy.

References 1. E.P. Wigner, Group Theory and its Applications to the Quantum Mechanics of Atomic Spectra (Academic, New York 1959) 2. C.E. Porter (ed.), Statistical Theories of Spectra (Academic, New York 1965) 3. M.L. Mehta, Random Matrices (Academic, New York, 1967); 2nd edition 1991; 3rd edition (Elsevier, 2004) 4. C.H. Joyner, S. Müller, M. Sieber, Europhys. Lett. 107, 50004 (2014) 5. B. Gutkin, J. Phys. A 40, F761 (2007) 6. D. Delande, J.C. Gay, Phys. Rev. Lett. 57, 2006 (1986) 7. G. Wunner, U. Woelck, I. Zech, G. Zeller, T. Ertl, F. Geyer, W. Schweitzer, H. Ruder, Phys. Rev. Lett. 57, 3261 (1986) 8. T.H. Seligman, J.J.T. Verbaarschot, Phys. Lett. A 108, 183 (1985) 9. G. Casati, B.V. Chirikov, D.L. Shepelyansky, I. Guarneri, Phys. Rep. 154, 77 (1987) 10. G. Casati, B.V. Chirikov, F.M. Izrailev, J. Ford, Stochastic Behavior in Classical and Quantum Hamiltonian Systems, ed. by G. Casati, J. Ford. Lecture Notes in Physics, vol. 93 (Springer, Berlin/Heidelberg, 1979) 11. B.V. Chirikov, Preprint no. 267, Inst. Nucl. Physics Novosibirsk (1969); Phys. Rep. 52, 263 (1979) 12. F. Haake, M. Ku´s, R. Scharf, Z. Physik B 65, 381 (1987) 13. M. Ku´s, R. Scharf, F. Haake, Z. Physik B 66, 129 (1987) 14. O. Bohigas, M.J. Giannoni, C. Schmit, Phys. Rev. Lett. 52, 1 (1984) 15. J.P. Woerdman, J. Dingjan, M.P. van Exter, Coherence and Quantum Optics VIII, ed. by N.P. Bigelow, J.H. Eberly, C.R. Stroud, I.A. Walmsley (Springer, Boston, 2003), p. 321 16. J.P. Keating, Nonlinearity 4, 309 (1991) 17. J.P. Keating, F. Mezzadri, Nonlinearity 13, 747 (2000) 18. E.B. Bogomolny, B. Georgeot, M.J. Giannoni, C. Schmit, Phys. Rep. 291, 219 (1997) 19. P. Braun, F. Haake, J. Phys. A 43, 262001 (2010) 20. P. Braun, arXiv:1508.02075v1[nlin.CD] (2015) 21. F. Haake, M. Ku´s, R. Scharf, Fundamentals of Quantum Optics II, ed. by F. Ehlotzky. Lecture Notes in Physics, vol. 282 (Springer, Berlin/Heidelberg, 1987) 22. R. Scharf, B. Dietz, M. Ku´s, F. Haake, M.V. Berry, Europhys. Lett. 5, 383 (1988) 23. F. Dyson, J. Math. Phys. 3, 1199 (1962) 24. J. Verbaarschot, Phys. Rev. Lett. 72, 2531 (1994) 25. M.R. Zirnbauer, J. Math. Phys 37, 4986 (1996) 26. A. Altland, M.R. Zirnbauer, Phys. Rev. B 55, 1142 (1997) 27. A. Altland, B.D. Simons, M.R. Zirnbauer, Phys. Rep. 359 283 (2002) 28. M.R. Zirnbauer, arXiv:math-ph/0404058v1 25 (2004) 29. P. Heinzner, A. Huckleberry, M. Zirnbauer, Commun. Math. Phys. 257, 725 (2005)

70 30. 31. 32. 33.

2 Time Reversal and Unitary Symmetries

S. Gnutzmann, B. Seif, Phys. Rev. E 69, 056219 (2004) S. Gnutzmann, B. Seif, Phys. Rev. E 69, 056220 (2004) A.P. Schnyder, S. Ryu, A. Furusaki, A.W.W. Ludwig, Phys. Rev. B 78, 195125 (2008) S. Ryu, A.P. Schnyder, A. Furusaki, A.W.W. Ludwig, New. J. Phys. 12, 065010 (2010); arXiv:0905.2029[cond-mat.mes-hall] (2009) 34. H. Schomerus, M. Marciani, C.W.J. Beenakker, Phys. Rev. Lett. 114, 166803 (2015) 35. J.C.Y. Teo, C.L. Kane, Phys. Rev. B 82, 115120 (2010) 36. A. Kitaev, AIP Conference Proceedings, vol. 1134, no. 1 (American Institute of Physics, New York, 2009), pp. 22–30 37. C.W.J. Beenakker, J.M. Edge, J.P. Dahlhaus, D.I. Pikulin, S. Mi, M. Wimmer, Phys. Rev. Lett. 111, 037001 (2013) 38. D. Bagrets, A. Altland, Phys. Rev. Lett. 109, 227005 (2012) 39. C.W.J. Beenakker, Rev. Mod. Phys. 87, 1037 (2015) 40. T. Kitagawa, M.A. Broome, A. Fedrizzi, M.S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, A.G. White, Nat. Commun. 3, 882 (2012) 41. B.D. Simons, A. Altland, Theories of Mesoscopic Physics. CRM Series in Mathematical Physics (Springer, New York, 2001) 42. A. Furusaki, Phys. Rev. Lett. 82, 604 (1999); and references therein 43. M. Feingold, A. Peres, Physica D 9, 433 (1983) 44. M. Feingold, N. Moiseyev, A. Peres, Phys. Rev. A 30, 509 (1984) 45. Y. Fan, S. Gnutzmann, Y. Liang, Phys. Rev. E 96, 062207 (2017)

Chapter 3

Level Repulsion

3.1 Preliminaries In the previous chapter, we classified Hamiltonians H and Floquet operators F by their groups of canonical transformations. Now we propose to show that orthogonal, unitary, and symplectic canonical transformations correspond to level repulsion of, respectively, linear, quadratic, and quartic degree [1, 2]. The different canonical groups are thus interesting not only from a mathematical point of view but also have distinct measurable consequences. It is a fascinating feature of quantum mechanics that different behavior under time reversal actually becomes observable experimentally. Resistance of levels to crossings is a generic property of Hamiltonians and Floquet operators just as nonintegrability is typical for classical Hamiltonian systems with more than one degree of freedom. (In fact, as will become clear in Chap. 4, Hamiltonians with integrable classical limits and more than one degree of freedom do not display level repulsion.) Therefore, knowing that the degree of level repulsion is 1, 2, or 4 for some system implies nothing more than (1) some information about the symmetries and (2) that the system is classically nonintegrable. Conversely, the universality of spectral fluctuations calls for explanation. To reveal the phenomenon of repulsion, the levels of H or F must be divided into multiplets; each such multiplet has fixed values for all observables except H or F from the corresponding complete set of conserved quantities. Levels belonging to different multiplets have no inhibition to cross when a parameter in H or F is varied. Such intermultiplet crossings are in fact well known, e.g., from the Zeeman levels pertaining to multiplets of different values of the total angular momentum quantum number j (Fig. 3.1). The levels within one multiplet typically avoid crossings, as illustrated in Fig. 3.2 for a kicked top. A simple explanation of level repulsion was given by von Neumann and Wigner in 1929 [3]. In fact, the present chapter will expound and generalize what is nowadays often referred to as the von Neumann–Wigner theorem. © Springer Nature Switzerland AG 2018 F. Haake et al., Quantum Signatures of Chaos, Springer Series in Synergetics, https://doi.org/10.1007/978-3-319-97580-1_3

71

72

3 Level Repulsion

Fig. 3.1 Level crossings between different Zeeman multiplets

E

E2 E1

0

1.0

2.0 magnetic field

Fig. 3.2 Dependence of the quasi-energies of a kicked top on some control parameter λ. For λ  3, the classical motion is predominantly regular, whereas classical chaos prevails for λ > 3. Note that revel repulsion is much more pronounced in the classically chaotic range

3.2 Symmetric Versus Nonsymmetric H or F When two levels undergo a close encounter upon variation of a parameter in a Hamiltonian, their fate can be studied by nearly degenerate perturbation theory. Assuming that each of the two levels is nondegenerate, one may deal with a twodimensional Hilbert space spanned by approximants |1, |2 to the corresponding eigenvectors. By diagonalizing the 2 × 2 matrix   H11 H12 H = , ∗ H H12 22

(3.2.1)

one obtains the approximate eigenvalues E± =

1 2

(H11 + H22) ±

$

1 4

(H11 − H22)2 + |H12 |2 .

(3.2.2)

3.2 Symmetric Versus Nonsymmetric H or F

73

An important difference between Hamiltonians with unitary and with orthogonal canonical transformations becomes manifest at this point. In the first case, the Hamiltonian (3.2.1) is complex. The discriminant in (3.2.2), D=

1 4

(H11 − H22)2 + (Re {H12 })2 + (Im {H12 })2 ,

(3.2.3)

is thus the sum of three nonnegative terms. When orthogonal transformations are canonical, on the other hand, the matrix (3.2.1) is real symmetric, Im {H12} = 0, whereupon the discriminant D has only two nonnegative contributions. Clearly, by varying a single parameter in the Hamiltonian, the discriminant and thus the level spacing |E+ − E− | can in general be minimized but not made to vanish (Fig. 3.3). To make a level crossing, E+ = E− , a generic possibility rather than an unlikely exception, three parameters must be controllable when unitary transformations apply, whereas two suffice in the orthogonal case. The resistance of levels to crossings should therefore be greater for complex Hermitian Hamiltonians than for real symmetric ones. The above argument can likewise be applied to Floquet operators. The most general unitary 2 × 2 matrix with unit determinant reads  F =

α + iβ γ + iδ −γ + iδ α − iβ

 , with

α2 + β 2 + γ 2 + δ2 = 1 .

(3.2.4)

Apart from a common phase factor, which can be set equal to unity by a proper choice of the reference point of the eigenphases of F, the four parameters α, β, γ , δ

E

5

0

–5

–10

0

10 λ

Fig. 3.3 Hyperbolic form of a typical avoided crossing of two levels

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3 Level Repulsion

are real. Three of these parameters, say β, γ , and δ, may be taken as independent. The unimodular eigenvalues of F are $ e± = α ± i β 2 + γ 2 + δ 2 .

(3.2.5)

Again, the number of nonnegative additive contributions to the discriminant is generally three, unless the additional condition that F be symmetric, γ = 0, reduces that number to two. One might argue (correctly!) that due to the unitarity condition, α 2 + β 2 + γ 2 + 2 δ = 1, it suffices to control the single parameter 1 − α 2 . However, control over β 2 + γ 2 + δ 2 allows one to achieve E+ = E− in the Hamiltonian case as well. What counts in this argument is that the discriminants in (3.2.2) and (3.2.5) can be controlled by the same number of independent parameters. We shall refer to this number as the codimension of a level crossing. To summarize, both for Hamiltonian and Floquet operators, the number n of controllable parameters necessary to enforce a level crossing is three or two depending on whether H and F are unrestricted (beyond Hermiticity and unitarity, respectively) or are symmetric matrices. These two possibilities correspond to the alternatives of unitary and orthogonal canonical transformation groups: n=

2

orthogonal

3

unitary

.

(3.2.6)

It remains to be shown that the codimension n of a level crossing yields the degree of level repulsion as n − 1 (Sect. 3.4).

3.3 Kramers’ Degeneracy When there is an invariance of H or a covariance of F, T H T −1 = H or T F T −1 = F † ,

(3.3.1)

under some antiunitary time-reversal transformation that squares to minus unity, T 2 = −1 ,

(3.3.2)

each level is doubly degenerate. The close encounter of two levels must therefore be discussed for a four-dimensional Hilbert space [4]. Four approximants to the eigenvectors can be chosen and ordered as in (2.9.1), i.e., as |1, T |1, |2, T |2. The 4 × 4 matrix H or F can then be represented so that it consists of four 2 × 2 blocks hmn or fmn ; see (2.9.5) and (2.13.5). The symmetry (3.3.1), (3.3.2) yields /μ) (μ) the restrictions (2.9.11) and (2.13.7) for the amplitudes hmn and fmn , respectively.

3.3 Kramers’ Degeneracy

75

Barring, for the moment, any geometric symmetry, there are no further restrictions. In a slightly less belligerent notation, the corresponding 4 × 4 Hamiltonian reads ⎛

⎞ α+β 0 γ − iσ −ε − iδ ⎜ 0 α + β ε − iδ γ + iσ ⎟ ⎟ H =⎜ ⎝ γ + iσ ε + iδ α − β 0 ⎠ −ε + iδ γ − iσ 0 α−β

(3.3.3)

with six real parameters α, β, γ , δ, ε, σ. Rather than invoking (2.9.11), it is of course also possible to verify the structure (3.3.3) directly by using T 2 = −1 and the ordering of the four basis vectors given above. For instance,  1|H |T 1 = 1|T H T −1 |T 1 = 1|T H |1 = −T 1|H |1∗ = −1|H |T 1 = 0 .

(3.3.4)

Since T 2 = −1 the quartic secular equation of (3.3.3) is biquadratic and yields the two double roots $ E± = α ± β 2 + γ 2 + δ 2 + ε 2 + σ 2 . (3.3.5) Similarly, the 4 × 4 Floquet matrix F also has the structure (3.3.3), but by virtue of its unitarity, the five parameters iβ, iγ , iδ, iε, iσ become real, and α remains real; actually, α may be considered to be given in terms of the other five parameters since unitarity requires α 2 + (iβ)2 + (iγ )2 + (iδ)2 + (iε)2 + (iσ )2 = 1 .

(3.3.6)

As in (3.2.4), we have chosen a phase factor common to all elements of F so as to make the two unimodular eigenvalues $ e± = α ± i (iβ)2 + (iγ )2 + (iδ)2 + (iε)2 + (iσ )2

(3.3.7)

complex conjugates of one another. For both H and F discriminants appear in the eigenvalues (3.3.5) and (3.3.7) which have five nonnegative additive contributions. The codimension of a level crossing is thus n=5

symplectic .

(3.3.8)

As for the orthogonal and unitary cases, it will become clear presently that the value n = 5 is larger by one than the degree of the level repulsion characteristic of Hamiltonian and Floquet operators whose group of canonical transformations is symplectic.

76

3 Level Repulsion

It is instructive to check that the number n reduces to 3 and 2 when geometric symmetries are imposed so as to break H or F down to block diagonal form with, respectively, U (2) and O(2) as canonical transformations of the 2 × 2 blocks on the diagonal. For instance, by assuming a parity Rx with the properties (2.8.1), one may choose the four basis vectors as parity eigenstates such that Rx |1 = −i|1 , Rx |T 2 = −i|T 2 Rx |2 = i|2 , Rx |T 1 = i|T 1 .

(3.3.9)

It follows that 1|H |2 = 1|Rx H Rx−1 |2 = 1|Rx H Rx |2 = −1|H |2 = 0 , i.e., γ =σ =0, but there are no further restrictions and so n = 3. Alternatively, by rearranging the order of the basis states to |1, |T 2, |2, |T 1, the matrix H indeed becomes block diagonal ⎛

⎞ α + β −ε − iδ 0 0 ⎜−ε + iδ α − β 0 0 ⎟ ⎟ . H =⎜ ⎝ 0 0 α − β ε + iδ ⎠ 0 0 ε − iδ α + β

(3.3.10)

Both of the nonzero blocks have fixed parity. Time reversal connects the two blocks but has no consequences within a block. Clearly, then, U (2) is the canonical group, and n = 3 for each block. An additional parity Ry obeying (2.8.7) allows one to endow the basis with Ry |1 = |T 1 , Ry |T 1 = −|1 Ry |2 = |T 2 , Ry |T 2 = −|2

(3.3.11)

whereupon 1|H |T 2 = −T 1|H |2, i.e., δ = 0. The matrix H becomes real, the canonical group of the 2 × 2 blocks is O(2), and the index n is reduced to 2.

3.4 Universality Classes of Level Repulsion

77

3.4 Universality Classes of Level Repulsion It is intuitively clear that an accidental level crossing becomes progressively less likely, the larger the number of parameters in H or F necessary to enforce a degeneracy. This number of parameters, or the “codimension of a level crossing”, it was found in the preceding sections, takes on values characteristic of the group of canonical transformations, ⎧ ⎪ ⎪ ⎨2 orthogonal (3.4.1) n = 3 unitary ⎪ ⎪ ⎩5 symplectic . Now we shall show that the degree of level repulsion expressed by the level spacing distribution P (S) is determined by n according to P (S) ∝ S n−1 = S β for S → 0 .

(3.4.2)

Linear, quadratic, and quartic level repulsion is thus typical for Hamiltonians and Floquet operators with, respectively, orthogonal, unitary, and symplectic canonical transformations. Following common practice, we have introduced the exponent of level repulsion as β = n − 1. The limiting behavior (3.4.2) follows from the following elementary argument [5, 6]. The spacing distribution for a given spectrum can be defined as P (S) = δ(S − ΔE)

(3.4.3)

where ΔE stands for a distance between neighboring levels and the angular brackets  . . .  mean an average over all ΔE. For S → 0, i.e., for S smaller than the average separation, the level pairs contributing to P (S) correspond to encounters sufficiently close for ΔE to be representable by nearly degenerate perturbation theory. In a unified notation, such spacings can be written as ΔE =

)

x2 =

$ x12 + . . . + xn2 .

(3.4.4)

Here ΔE may refer to energies, as in (3.2.2) and (3.3.5), or to quasi-energies. Of course, the level pairs contributing to P (S) for a fixed small value of S will generally possess different values of the n parameters x. The average over the level spectrum can therefore be understood as an average over x with a suitable weight W (x). The latter function may not be easy to obtain, but fortunately its precise form does not matter for the asymptotic behavior of  P (S) =

 )  d n xW (x)δ S − x 2 .

(3.4.5)

78

3 Level Repulsion

By simply rescaling the n integration variables x → Sx,  P (S) = S β

 )  d n x W (Sx) δ 1 − x 2 .

(3.4.6)

The power law (3.4.2) thus applies, provided that the weight W (x) is neither zero nor infinite at x = 0. A finite nonvanishing value of W (0) must, however, be considered generic since the dictates of symmetries are already accounted for in the codimension n = β + 1 of a level crossing. The full probability distribution P (S) in 0 ≤ S < ∞ is not as easily found as its asymptotic form for small S. It will be shown below that P (S) can be constructed on the basis of the theory of random matrices or by using methods from equilibrium statistical mechanics. Trying to establish the spacing distribution P (S) and the degree of level repulsion for some dynamical system is a sensible undertaking only if the number of levels is large: Only in that “semiclassical” situation can a smooth histogram be built up even for small S where P (S) itself is small; but then indeed P (S) becomes “selfaveraging” for a single spectrum.

3.5 Nonstandard Symmetry Classes The seven new symmetry classes introduced at the end of the previous chapter exhibit a somewhat richer variety of level repulsion. The reader will recall that due to either a unitary or anti-unitary spectral mirror symmetry the spectra are symmetric about E = 0. Sufficiently far away from E = 0 the spectrum behaves according to the three standard symmetry classes where level repulsion between neighboring energy levels is measured by the exponent β in the small-s behavior of the level spacing distribution, P (S) ∝ S β . Near E = 0 an additional effects sets in— repulsion or attraction of individual energy eigenvalues from the symmetry point E = 0. This additional effect disappears for energies much larger than the mean level spacing. The small-E behavior may be measured by an exponent α in the distribution P˜0 (s) ∝ s α (for s 1) of the lowest positive energy s = E1 /ΔE (in units of the mean level spacing). Each symmetry class is characterized by a value of the exponent α which usually also depend on the topological quantum number (the number of zero modes). Table 3.1 gives a complete overview of the values of the exponents β and α in each symmetry class. The exponent α for each symmetry class may be derived straightforwardly in full analogy to our derivation of the exponent β. That is, we can determine α by working with the smallest matrices H allowed by the pertinent symmetry class.

3.5 Nonstandard Symmetry Classes Table 3.1 Summary of level repulsion exponents β and α in the ten quantum symmetry classes

79 Symmetry class A AI AII AIIIν (ν ∈ N) BDIν (ν ∈ N) CIIν (ν ∈ N) C CI BDν (ν = 0,1) BDIIIν (ν = 0,1)

Unitary Orthogonal Symplectic Chiral unitary Chiral orthogonal Chiral symplectic Anti-pseudo-chiral unitary Anti-chiral orthogonal Pseudo-chiral unitary Anti-chiral symplectic

β 2 1 4 2 1 4 2 1 2 4

α 0 0 0 1+2ν ν 3+4ν 2 1 2ν 1+4ν

3.5.1 The Chiral Symmetry Classes We start with the chiral unitary class AIIIν and consider the Hamilton matrix ⎞ ⎛ 0 0 . . . 0 z1 ⎜ 0 0 ... 0 z ⎟ ⎜ 2 ⎟ ⎟ ⎜ (3.5.1) H = ⎜. . . . . . . . . . . . . . . ⎟ ⎟ ⎜ ⎝ 0 0 . . . 0 zν+1 ⎠ ∗ z1∗ z2∗ . . . zν+1 0 where zj ∈ C (j = 1, . . . , ν + 1). This is of the required canonical form (2.16.9) with N = 1 and topological index ν and has the smallest admissible dimension 2 + ν. By its construction, $ H comes with the ν-fold degenerate eigenvalue E = 0 ν+1 2 and the pair E = ± j =1 |zj | . We thus need to set n = 2ν + 2 real parameters to zero in order to drive the first positive eigenvalue to zero. So n = 2ν + 2 plays the same role as the codimension for a level-crossing in the previous discussion and following analogous steps one finds the exponent α = n − 1 = 2ν + 1. In the orthogonal chiral class BDIν we consider the same Hamiltonian matrix (3.5.1) with real entries zj ∈ R. The codimension reduces to n = ν + 1 and the repulsion exponent becomes α = ν. For the remaining symplectic chiral class CIIν we replace each entry in (3.5.1) by a quaternion 2 × 2 matrix     ξj∗ −ζj ξj ζj ∗ and zj → . (3.5.2) zj → −ζj∗ ξj∗ ζj∗ ξj The matrix dimension thus becomes 4 + 2ν. Note that Kramers’ degeneracy applies to this class. The spectrum now contains the 2ν-fold degenerate eigenvalue E = 0 $ ν+1 2 2 and the two-fold degenerate eigenvalues E = ± j =1 |ξj | + |ζj | . There are $ 2 2 n = 4ν + 4 real parameters in j |ξj | + |ζj | such that we find the exponent α = n − 1 = 4ν + 3.

80

3 Level Repulsion

3.5.2 Classes CI and BDIIIν The anti-chiral orthogonal class CI makes for the simplest case of all as the canonical form of lowest non-trivial dimension is just   0 z H = ∗ (3.5.3) z 0 with eigenvalues E = ±|z|. So n = 2 and α = 1. In the anti-chiral symplectic class BDIIIν we have to consider the cases ν = 0 and ν = 1 separately. For ν = 0 the canonical form of lowest non-trivial dimension is ⎛

0 ⎜0 H =⎜ ⎝0 z∗

0 0 −z∗ 0

0 −z 0 0

⎞ z 0⎟ ⎟ 0⎠ 0

(3.5.4)

which has doubly degenerate eigenvalues E ± |z|. So again n = 2 and α = 1 ≡ 1 + 4ν. In the case ν = 1 the dimension needs to be twice an odd number. The lowest dimensional non-trivial case is ⎛ ⎞ 0 0 0 0 z1 z2 ⎜ 0 0 0 −z 0 z ⎟ 1 3⎟ ⎜ ⎜ ⎟ ⎜ 0 0 0 −z2 −z3 0 ⎟ H =⎜ (3.5.5) ⎟ . ∗ ∗ ⎜ 0 −z1 −z2 0 0 0 ⎟ ⎜ ∗ ⎟ ⎝z1 0 −z3∗ 0 0 0 ⎠ z2∗ z3∗ 0 0 0 0 A bit of algebra shows that all eigenvalues ) are two-fold degenerate (Kramers’ degeneracy) and given by E = 0 and E = ± |z1 |2 + |z2 |2 + |z3 |2 . We have n = 6 independent real parameters in the positive eigenvalue and thus α = 5 = 1 + 4ν.

3.5.3 Classes BDν and C In the pseudo-chiral unitary class BDν we must again distinguish the two cases ν = 0 and ν = 1. For ν = 0 the canonical form of the Hamiltonian in lowest non-trivial dimension is   0 ix H = (3.5.6) −ix 0

3.6 Experimental Observation of Level Repulsion

81

with one real parameter x ∈ R. The eigenvalues are E = ±x. So n = 1 and α = 0 ≡ 2ν. For ν = 1 one considers the case ⎛

⎞ 0 ix1 ix2 H = ⎝−ix1 0 ix3 ⎠ −ix2 −ix3 0

(3.5.7)

with $ real parameters x1 , x2 , x3 ∈ R. This has eigenvalues E = 0 and E = ± x12 + x22 + x32 such that n = 3 and α = 2 ≡ 2ν. Finally, for the anti-pseudo-chiral class C we consider the Hamiltonian 

x z H = ∗ z −x

 (3.5.8)

with one real parameter ) x ∈ R and one complex parameter z ∈ C. The two eigenvalues are E = x 2 + |z|2 with n = 3 and α = 2.

3.6 Experimental Observation of Level Repulsion Systematic statistical analyses of complex energy spectra first became popular among nuclear physicists half a century ago. Upon being sorted into histograms, the spacings between highly excited neighboring levels of nuclei revealed linear level repulsion and thus time-reversal invariance of the strong interaction mainly responsible for nuclear structure [1, 2, 7]. While the electroweak interaction does not enjoy that symmetry, it is far too weak to be detectable in spectral fluctuations on the scale of a mean level spacing in complex nuclei. There are rather fewer experimental verifications of linear level repulsion in electronic spectra. Evidence was found for NO2 molecules [8] and Rb atoms [9]. While the nuclear, atomic, and molecular data just mentioned are now generally accepted as quantum manifestations of chaos, the somewhat less difficult to observe linear repulsion between eigenfrequences of microwave resonators with sufficiently irregular shape [10–15] requires explanation in the framework of classical wave theory rather than quantum mechanics. In fact, the Helmholtz equation for any of the components of the electromagnetic field within a resonator involves the very same differential operator, ∇ 2 +k 2 with k denoting the wave vector, as does Schrödinger’s equation for a free particle in a container; as long as the boundary conditions at the walls do not mix different components of the electromagnetic field, there is a complete mathematical equivalence between the quantum and the classical wave problem. The chaos of which one sees the quantum or wave signatures is that of a “billiard” with the shape of the “box” in question, at least if the boundary conditions express specular reflection of the point particle that idealizes the billiard ball.

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3 Level Repulsion

The first of the microwave experiments just mentioned [10] was actually meant to simulate sound waves of air in a concert hall and was done in complete innocence of the jargon of present-day chaology; the simulation of sound by microwaves was preferred to “listening” since concert halls tend to have overlapping rather than separated resonances and are thus not too well suited for picking up spacing statistics. It is amusing to realize, though, that audio engineers do go for wave chaos when designing the boundaries of concert halls. Acoustic chaos has also been ascertained experimentally as linear repulsion of the elastomechanical eigenfrequencies of suitably shaped discs [16] and quartz blocks [17, 18]. Inasmuch as vibrating crystals are anisotropic and support longitudinal as well as transverse sound waves, these experiments suggest that the distinction between chaotic and regular waves arises quite independently of the character of the medium and the detailed form of the pertinent wave equation. The common origin of quantum and wave chaos may be seen in the nonseparability of the wave equation, just as we may attribute chaos in classical mechanics to the nonseparability of the Hamilton–Jacobi equation. If we see the Hamilton–Jacobi equation as the shortwave limit of wave theories, we can consider nonseparability as the universal chaos criterion. Neither quadratic nor quartic level repulsion has been observed to date in nuclei, atoms, and molecules. It might be possible to break time-reversal invariance for Rydberg atoms with strongly inhomogeneous magnetic fields and thus realize the quadratic case. In the absence of such experiments on quantum systems, the observation of quadratic repulsion in microwave resonators with broken timereversal invariance was a most welcome achievement [19–22]. The interplay of time reversal invariance with geometric symmetries may bring about quadratic level repulsion. In fact, quadratic repulsion was predicted in [23] and observation with microwaves reported in [24, 25], a threefold rotational symmetry bringing about GUE statistics. Another example are constant-width billiards, as predicted in [26] and experimentally confirmed with microwaves [27]. More recently, GSE statistics with quartic level repulsion in microwave graphs (without half-integer spin!) was revealed in [28]. Such graphs were then designed and the prediction confirmed [29, 30]. Even extensions of microwave experiments to the chiral classes are under way and will soon be published [31].

3.7 Problems 3.1 Show that a unitary 4 × 4 matrix with Kramers’ degeneracy can be given in the form ⎛ ⎞ α + iβ 0 σ + iγ δ − iε ⎜ 0 α + iβ δ + iε −σ + iγ ⎟ ⎟ F =⎜ ⎝−σ + iγ −δ + iε α − iβ ⎠ 0 −δ − iε σ + iγ 0 α − iβ

References

83

with the six real parameters α, β, . . . obeying α 2 + β 2 + γ 2 + δ 2 + ε2 + σ 2 = 1. Use basis vectors |1, |T 1, |2, |T 2, and T 2 = −1. 3.2 Show that the matrix F from Problem 3.1 has n = 3 if it is invariant under a parity Rx with [T , Rx ] = 0, Rx2 = −1. Moreover, show that n = 2 if a second parity holds with [T , Ry ] = 0, Rx Ry + Ry Rx = 0, Ry2 = −1. What is the structure of F if full isotropy holds? 3.3 Verify that nearly degenerate perturbation theory typically gives a hyperbolic form to a level crossing. 3.4 In the orthogonal case, adjacent energy levels can be steered to crossing in a two-dimensional parameter space. Show that the two energy surfaces are connected at the degeneracy like the two sheets of a double cone (Berry’s diabolo [5, 6]).

References 1. C.E. Porter (ed.), Statistical Theories of Spectra (Academic, New York, 1965) 2. M.L. Mehta, Random Matrices (Academic, New York, 1967); 2nd edition 1991; 3rd edition (Elsevier, 2004) 3. J. von Neumann, E.P. Wigner, Phys. Z. 30, 467 (1929) 4. R. Scharf, B. Dietz, M. Ku´s, F. Haake, M.V. Berry, Europhys. Lett. 5, 383 (1988) 5. M.V. Berry, Les Houches, Session XXXVI, 1981, in Chaotic Behavior of Deterministic Systems, ed. by G. Iooss, R.H.G. Helleman, R. Stora (North-Holland, Amsterdam, 1983) 6. T.A. Brody, J. Floris, J.B. French, P.A. Mello, A. Pandey, S.S.M. Wong, Rev. Mod. Phys. 53, 385 (1981); Appendix B 7. O. Bohigas, R.U. Haq, A. Pandey, Nuclear Data for Science and Technology, ed. by K.H. Böchhoff (Reidel, Dordrecht, 1983) 8. T. Zimmermann, H. Köppel, L.S. Cederbaum, C. Persch, W. Demtröder, Phys. Rev. Lett. 61, 3 (1988); G. Persch, E. Mehdizadeh, W. Demtröder, T. Zimmermann, L.S. Cederbaum: Ber. Bunsenges. Phys. Chem. 92, 312 (1988) 9. H. Held, J. Schlichter, G. Raithel, H. Walther, Europhys. Lett. 43, 392 (1998) 10. M.R. Schroeder, J. Audio Eng. Soc. 35, 307 (1987) 11. H.-J. Stöckmann, J. Stein, Phys. Rev. Lett. 64, 2215 (1990) 12. H. Alt, H.-D. Gräf, H.L. Harney, R. Hofferbert, H. Lengeler, A. Richter, P. Schart, H.A. Weidenmüller, Phys. Rev. Lett. 74, 62 (1995) 13. H. Alt, H.-D. Gräf, R. Hofferbert, C. Rangacharyulu, H. Rehfeld, A. Richter, P. Schart, A. Wirzba, Phys. Rev. E 54, 2303 (1996) 14. H. Alt, C. Dembowski, H.-D. Gräf, R. Hofferbert, H. Rehfeld, A. Richter, R. Schuhmann, T. Weiland, Phys. Rev. Lett. 79, 1026 (1997) 15. A. Richter, Emerging Applications of Number Theory, ed. by D.A. Hejhal, J. Friedman, M.C. Gutzwiller, A.M. Odlyzko. IMA vol. 109 (Springer, New York, 1998), p. 109 16. O. Legrand, C. Schmit, D. Sornette, Europhys. Lett. 18, 101 (1992) 17. C. Ellegaard, T. Guhr, K. Lindemann, H.Q. Lorensen, J. Nygård, M. Oxborrow, Phys. Rev. Lett. 75, 1546 (1995) 18. C. Ellegaard, T. Guhr, K. Lindemann, J. Nygård, M. Oxborrow, Phys. Rev. Lett. 77, 4918 (1996) 19. U. Stoffregen, J. Stein, H.-J. Stöckmann, M. Ku´s, F. Haake, Phys. Rev. Lett. 74, 2666 (1995)

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3 Level Repulsion

20. P. So, S.M. Anlage, E. Ott, R.N. Oerter, Phys. Rev. Lett. 74, 2662 (1995) ˙ 21. O. Hul, S. Bauch, P. Pakoñski, N. Savytskyy, K. Zyczkowski, L. Sirko, Phys. Rev. E 69, 056205 (2004) 22. H.-J. Stöckmann, Quantum Chaos, An Introduction (Cambridge University Press, Cambridge, 1999) 23. F. Leyvraz, C. Schmit, T.H. Seligman, J. Phys. A 29, L575 (1996) 24. C. Dembowski, H.-D. Gräf, A. Heine, H. Rehfeld, A. Richter, C. Schmit, Phys. Rev. E 62, 4516(R) (2000) 25. R. Schäfer, M. Barth, F. Leyvraz, M. Müller, T.H. Seligman, H.-J. Stöckmann, Phys. Rev. E 66, 016202 (2002) 26. B. Gutkin, J. Phys. A 40, F761 (2007) 27. B. Dietz, T. Guhr, M. Miski-Oglu, A. Richter, Phys. Rev. E 90, 022903 (2014) 28. C.H. Joyner, S. Müller, M. Sieber, Europhys. Lett. 107, 50004 (2014) 29. A. Rehemanjiang, M. Allgaier, C.H. Joyner, S. Müller, M. Sieber, U. Kuhl, H.-Stöckmann, Phys. Rev. Lett. 107, 064101 (2016) 30. A. Rehemanjiang, M. Richter, U. Kuhl, H.-J. Stöckmann, Phys. Rev. E 97, 022204 (2018) 31. H. Stöckmann, Private communication

Chapter 4

Level Clustering

4.1 Preliminaries Here we want to consider classical autonomous systems with f degrees of freedom and 2f pairs of canonical variables pi , qi . We shall meet with invariant tori, caustics, and Maslov indices and proceed to the semiclassical torus quantization à la Einstein, Brillouin and Keller (EBK) and its modern variant, a periodic-orbit theory. The latter will allow us to understand why there is no repulsion but rather clustering of levels for generic integrable systems with two or more degrees of freedom; the density of level spacings therefore usually takes the form of a single exponential, P (S) = e−S , a behavior known as the Berry-Tabor conjecture [1]. Single-freedom systems, if autonomous, are always integrable and do not respect any general rule for their spacing statistics; we shall postpone a discussion of their behavior to Chap. 11 (level dynamics). Readers with more curiosity about EBK than can be satisfied here are referred to Percival’s review [2] or the book by Brack and Bhaduri [3]. Mathematically inclined readers may taste the flavor of serious mathematical work on the Berry-Tabor conjecture in Marklov’s reports [4] and [5] and references therein. A semiclassical treatment of classically chaotic dynamics will follow in Chap. 10.

4.2 Invariant Tori of Classically Integrable Systems The dynamics in question have f independent constants of the motion Ci (p, q) with vanishing Poisson brackets: * + ∂Ci ∂Cj ∂Ci ∂Cj Ci , Cj = − =0. ∂p ∂q ∂q ∂p

© Springer Nature Switzerland AG 2018 F. Haake et al., Quantum Signatures of Chaos, Springer Series in Synergetics, https://doi.org/10.1007/978-3-319-97580-1_4

(4.2.1)

85

86

4 Level Clustering

Each phase-space trajectory p(t), q(t) thus lies on an f -dimensional surface, the embedded in the 2f -dimensional phase space. If that surface is smooth, compact, and connected, it has the topology of a torus which may be revealed by a suitable canonical transformation . A nontrivial example is provided by a point mass moving in a plane under the influence of an attractive central potential V (r). The energy E and the angular momentum L are two independent constants of the motion and define a twodimensional torus in the four-dimensional phase space. A phase-space trajectory starting on such a torus will keep winding around it forever. In general, trajectories will not be closed; only for the harmonic [V (r) ∼ r 2 ] and the Kepler [V (r) ∼ 1/r] potentials are all trajectories closed. For other potentials, periodic orbits are possible but constitute, as will become clear presently, a subset of measure zero in the set of all trajectories. When projected onto the two-dimensional configuration space, every trajectory of the present model remains between two circles, which are concentric with respect to the center of force. These circles, called caustics, are the singularities of the projection of the 2-torus onto the configuration plane. When touching a caustic the trajectory has a momentarily vanishing radial momentum, i.e., a turning point of its radial libration. For a circular “billiard table,” every configuration-space orbit also remains in between two concentric circles of which only the inner one is a caustic, however; the outer one is a boundary at which the particle bounces off rather than coming to rest momentarily. Note that no caustic is ever encountered in a rotation for which an angle increases indefinitely, where all other coordinates, if any, remain constant. Classical motion on an f -torus embedded in a (2f )-dimensional phase space is conveniently described in terms of f pairs of action and angle variables Ii , Θi . The actions characterize the f -torus via Ii =

1 2π

, p dq > 0 ,

(4.2.2)

Γi

where Γi is one of the f independent irreducible loops around the torus. Being properties of invariant tori, the Ii are constants of the motion. By a suitable canonical transformation, the actions Ii become new momenta with angles Θi as their conjugate coordinates. The Hamiltonian, a constant of the motion itself, can be expressed as a function of the f actions and does not depend on the angles. Hamilton’s equations thus give Θ˙ i =

∂H ∂Ii

≡ ωi ,

Θi (t) = ωi t + Θi (0) .

(4.2.3)

4.3 Einstein–Brillouin–Keller Approximation

87

The frequencies ωi are the angular velocities with which the phase space point travels around the torus. The angle Θi changes by 2π as the loop Γi is completed once. When the frequencies ωi are incommensurate, the trajectory starting from Θ(0) will tend to fill the torus densely as time elapses. We might even speak of ergodic motion in that case since time averages (of suitable quantities) will tend to ensemble averages with uniform probability density on the torus. Periodic orbits result, on the other hand, when the ωi are related rationally, i.e., when all ωi are integral multiples of a certain fundamental frequency ω0 , ω = Mω0 , Mi integer .

(4.2.4)

Such an orbit closes in on itself after the fundamental period 2π/ω0 , having completed M1 trips around the loop Γ1 , M2 around Γ2 etc. The corresponding torus is called rational. Every rational torus accommodates an f -parameter continuum of closed orbits related to one another by shifts of the initial angles Θ(0). Rational tori are as exceptional among the tori in phase space as the rational numbers among the reals.

4.3 Einstein–Brillouin–Keller Approximation To solve the Schrödinger equation of a classically integrable system in the semiclassical limit, one may use the ansatz [2, 3, 6–10] ψ(q) = a(q)eiS(q)/h¯ .

(4.3.1)

By demanding that ψ(q) be single valued in the q, one obtains a quantum condition for each of the f actions,   Ii = mi + 14 αi h¯ ,

(4.3.2)

with nonnegative integer quantum numbers mi and the so-called Maslov indices αi . The ith Maslov index αi equals the number of caustics encountered along the ith irreducible loop Γi around the classical torus with respect to which the action Ii was defined. The quantization prescription (4.3.2) is often referred to as semiclassical or torus quantization. The quantum condition on the action originates from the phase change in the semiclassical wave function along the loop Γi , ΔS = h¯

, Γi

p dq . h¯

(4.3.3)

88

4 Level Clustering

Another contribution to the total change of phase may come from the amplitude a(q), which can be shown to diverge at caustics. Such a singularity means, of course, that the ansatz (4.3.1) breaks down near a caustic. It was a brilliant idea of Maslov to switch to momentum space when q approaches a caustic and to ˜ replace (4.3.1) with a similar ansatz for ψ(p). Since a caustic in q-space is ˜ not a caustic in p-space, no singularity threatens ψ(p) where the semiclassical form (4.3.1) of ψ(q) runs into trouble. Replacing the latter by the Fourier transform ˜ of ψ(p) near the q-space caustic (see Eq. (10.4.2)), one finds the total phase increment picked up along Γi as 2π Ii π ΔS = − αi . h¯ h¯ 2

(4.3.4)

That change must be an integral multiple of 2π for the wave to close in on itself uniquely, i.e., not to destroy itself by destructive interference. By expressing the Hamiltonian H (I ) in terms of the quantized actions (4.3.2), one obtains the semiclassical approximation for the energy levels Em = H (I m ) = H

  m + 14 α h¯ .

(4.3.5)

In integrable systems, these semiclassical levels usually give an excellent approximation for sufficiently high degrees of excitation. Their accuracy is often surprisingly good for low energies, too, and in some exceptional cases even the whole spectrum is reproduced rigorously. The f -dimensional harmonic oscillator, for instance, has the Hamiltonian   f   1 2 1 2 2 H = p + mωi qi = ωi Ii . 2m 1 2 i =1

(4.3.6)

i

Since a libration encounters two turning points in every period, one must set αi = 2 and thus obtain    Em = (4.3.7) h¯ ωi mi + 12 , mi = 0, 1, 2, . . . i

which is indeed the exact result. Quite amusingly, the contribution of the Maslov indices gives the correct zero-point energy. As a second example, the reader is invited to check that the Hamiltonian of the hydrogen atom can be expressed in terms of the radial, polar, and azimuthal actions as H =−

me4 . 2(Ir + IΘ + Iφ )2

(4.3.8)

4.4 Level Crossings for Integrable Systems

89

The radial and polar motions are librations and the azimuthal motion is a rotation, so one has αr = αΘ = 2 and αφ = 0 and thus the semiclassical energies Em = −

me4 2h¯ 2 n2

, n = mr + mΘ + mφ + 1

(4.3.9)

which is again exact. Finally, a particle in an f -dimensional box is described by f 1  2 H = pi , 0 ≤ qi ≤ ai . 2m

(4.3.10)

i =1

The squared momenta pi2 are f independent constants of the motion and the actions read   ai  0 |pi | 1 . (4.3.11) Ii = dq|pi | + dq (−|pi |) = ai 2π 0 π ai Reexpressed in the actions, the Hamiltonian H =

π 2  Ii2 2m ai2

(4.3.12)

i

yields the semiclassical levels Em =

π 2 h¯ 2  m2i 2m ai2 i

which again coincide with the exact ones, except that the quantum numbers m1 start from 1 instead of 0 [3]. For further examples and a more systematic treatment, the reader is referred to Ref. [3].

4.4 Level Crossings for Integrable Systems It is easy to see that the levels obtained by torus quantization generally are not inhibited from crossing when f ≥ 2; the codimension of a crossing is n = 1. To this end, one imagines that the Hamiltonian and thus the asymptotic eigenvalues    Em (k) = H h¯ m + 14 α , k ,

(4.4.1)

90

4 Level Clustering

depend on a single parameter k. In an f -dimensional space with axes m1 , m2 , . . . , mf , the quantum numbers in (4.4.1) are represented by the points of a primitive cubic lattice where all components of the lattice vectors are nonnegative integers. In this space, the function H of f continuous variables m defines a continuous family of energy surfaces. An allowed energy eigenvalue arises for every such energy surface intersecting one of the discrete lattice points, say m∗ . In general, an energy surface intersecting m∗ will not intersect any other lattice point. There is a whole bundle of energy surfaces through m∗ , labelled by the parameter k, however, and by properly choosing k, a special surface can, in general, be picked which does intersect a second lattice point. The restriction “in general” of the foregoing statement excludes funny exceptions like the harmonic oscillator for which the energy surface is a hyperplane. Barring such exceptional surfaces, one concludes that a level crossing can be enforced by controlling a single parameter in the Hamiltonian. It follows from the arguments of Sect. 3.4 that the spacing density typically approaches a non-zero constant at zero spacing, P (S) → P (0) = 0 for S → 0 .

(4.4.2)

This is in contrast to the power-law behavior typical of nonintegrable systems. The foregoing reasoning does not apply to a single ladder or “multiplet” of levels arising if we let one of the f quantum numbers run while keeping all others fixed or, equivalently, if we have f = 1 to begin with. The parametric changes possible for such single multiplets will be discussed in Sect. 11.5.

4.5 Poissonian Level Sequences Before proceeding to show that the generic integrable system has an exponential distribution of its level spacings, it will be useful to introduce some probabilistic concepts. In particular, we intend to demonstrate here that exponentially distributed level spacings allow an interpretation of the sequence of levels as a random sequence without any correlation between the individual “events.” Let us assume that the spectrum is already unfolded to unit mean spacing of neighboring levels everywhere, S¯ = 1. Then, we employ, as an auxiliary concept, the conditional probability g(S)dS of finding a level in the interval [E + S, E + S + dS] given one at E. By virtue of the assumed homogeneity of the level distribution, g(S) does not depend on E. The probability density P (S) of finding the nearest neighbor of the level at E in dE at E + S is evidently the product of g(S)dS with the probability that there is no other level between E and E + S, 

∞

P (S) = g(S) S

dS P (S ) .

(4.5.1)

4.6 Superposition of Independent Spectra

91

Differentiating with respect to S we get ∂g ∂P = ∂S ∂S



∞ S

  ∂g −g P , dS P (S ) − gP = g −1 ∂S

(4.5.2)

a differential equation solved by P (S) ∝ g(S)e−

#S 0

dS g(S )

.

(4.5.3)

For a Poissonian level sequence, now, there are no correlations at all between the individual levels. Thus, the conditional probability g(S) is a constant, and P (S) = C exp (−AS). Normalizing and scaling so that S¯ = 1, one obtains the exponential distribution P (S) = e−S .

(4.5.4)

As a by-product of the above reasoning, the result is that the conditional probability density g(S) behaves like a power g(S) → S β

(4.5.5)

with β = 0, 1, 2, 4 for, respectively, integrable systems and nonintegrable systems with orthogonal, unitary, and symplectic canonical transformations.

4.6 Superposition of Independent Spectra In this section we shall present an important limit theorem which will help us to understand level spacings of integrable systems: L independent spectra, each assumed to have a mean level density unity its own spacing distribution, superpose to a spectrum with exponentially distributed spacings in the limit L → ∞. The arguments presented below go back in essence to Rosenzweig and Porter [11] and Berry and Robnik [12]. To get an understanding it is convenient to employ the probability E(S) that an interval of length S is empty of levels. That “gap probability” is intimately related to the spacing distribution P (S). In fact, assuming the mean level distance unity (see footnote below), the spacing distribution P (S) can be obtained as the second derivative of the gap probability E(S), P (S) =

∂ 2E . ∂S 2

(4.6.1)

To see this, let S = x+y, and consider the difference E(x+y)−E(x+y+Δx) which is the probability that x +y is empty and the neighboring interval Δx not empty. For

92

4 Level Clustering

small Δx, the probability that Δx contains two or more levels is of second or higher order in Δx. Therefore, −[∂E(x + y)/∂x]Δx is the probability that x + y is empty and Δx contains one level. Reasoning analogously, [∂ 2 E(x +y)/∂x∂y]ΔxΔy is the probability that x+y is empty and the surrounding intervals Δx and Δy each contain one level. The latter probability, on the other hand, may be expressed as the product of the “single-event” probability Δx and the conditional probability P (S)Δy with S = x + y; note that in the units used, the mean density of levels is unity. The gap probability and identity (4.6.1) will feature again in Chaps. 5 and 11. Now we consider L independent ladders of levels. For each of them, μ = 1, 2, . . . , N, we assume that a constant mean spacing 1/μ can be arranged by a suitable possibly non-linear rescaling of the energy axis and a smooth spacing distribution Pμ (S) can be defined,1 the latter normalized as 

∞

 dS Pμ (S) = 1

∞

and

0

dS S Pμ (S) =

0

1 . μ

(4.6.2)

The probability of finding no level of the μth ladder in an interval of length S is, in accordance with (4.6.1), 



∞

Eμ (S) = μ

∞

dx S



∞

dyPμ (y) = μ

x

dx(x − S)Pμ (x) .

(4.6.3)

S

It follows that Eμ (S) falls off monotonically from Eμ (0) = 1 to Eμ (∞) = 0. Therefore the initial slope is Eμ (0) = −μ and the small-S behavior Eμ (S) = 1 − μ S + . . . .

(4.6.4)

Due to the assumed independence of the ladders, the joint probability that there is no level from any ladder in an interval of length S is just the product E(S) =

L -

Eμ (S) = exp



μ=1

ln Eμ (S) .

(4.6.5)

μ

Now, we assume that L  1 and that the μ all have roughly equal magnitudes, μ ∝ 1/L and normalize for convenience such that the total density is unity, L 

μ = 1 .

(4.6.6)

μ=1

1 The condition for such “unfolding” of a spectrum to be possible will be explained and practical ways of achieving it will be given in Sect. 4.10.

4.7 Periodic Orbits and the Semiclassical Density of Levels

93

m2

m

m m1

Fig. 4.1 Adjacent energy eigenvalues of integrable systems may lie far apart in quantum number space if the number of degrees of freedom is larger than one

The joint probability E(S) falls off more rapidly by a factor of the order L than the typical single-ladder probability Eμ (S). Therefore, it is legitimate to replace the Eμ (S) on the right-hand side of (4.6.5) by the first two terms of their Taylor series (4.6.4). The exponential form for E(S) results in the limit L → ∞, E(S) → e−S .

(4.6.7)

It is hard to resist speculating, by naive appeal to the limit theorem just established, that the semiclassical spectrum (4.3.5) must have, for f ≥ 2 and generic functions H (I ), an exponential spacing distribution. After all, the semiclassical levels tend to follow one another quite erratically with respect to the direction of the vector m, as is obvious from Fig. 4.1. There is, no doubt, an element of truth in such naive reasoning. However, the only reasonably sound derivation of the exponential spacing distribution for integrable systems known at present is lengthy and technical and does not use the limit theorem (4.6.7).

4.7 Periodic Orbits and the Semiclassical Density of Levels The starting point for deriving the spacing distribution is the density of semiclassical energy levels (E) =



     δ E − H h¯ m + 14 α

(4.7.1)

{mi >0}

which will be shown to be representable as a sum of contributions from all classical periodic orbits.

94

4 Level Clustering

The presentation will closely follow Berry and Tabor’s original one [1, 13]. Note that the density (E) is normalized here such that its integral over an energy interval ΔE gives the number of levels contained in the interval (rather than the fraction of the total number of atoms). To pursue the goal indicated it is convenient to rewrite (4.7.1) as     α  (E) = . (4.7.2) d f I δ [E − H (I )] δ f I − h¯ m + 4 I >0 m The previous restriction on the f quantum numbers m > 0 is dropped here in favor of the restriction I > 0 on the f -fold integral. Next, we employ Poisson’s summation formula, +∞ 

ei2πMx =

+∞ 

δ(x − m) ,

(4.7.3)

m = −∞

M = −∞

to replace the f -dimensional train of delta functions by an f -fold sum over exponentials, (E) =

+∞ 1  −iπα·M/2 e h¯ f M = −∞  × d f I δ [E − H (I )] ei2πM·I /h¯ ,

(4.7.4)

I >0

and each of the f summation variables Mi runs through all integers. The M = 0 term in the series representation (4.7.4), 1 (E) ¯ ≡ f h¯

 d f I δ [E − H (I )] ,

(4.7.5)

I >0

is the venerable Thomas–Fermi result, also known as Weyl’s law, which we shall come back to in Sect. 5.7 [To see that (4.7.5) follows from (5.7.4) note that h¯ −f = h−f (2π)f and that H (I ) is independent of the f angle variables Θ conjugate to the actions I .] As already indicated in the symbol (E), ¯ the Thomas–Fermi level density (4.7.5) may be considered a local spectral average of (E), that does not reflect local fluctuations in the level sequence such as local clustering or repulsion. An important and intuitive interpretation of (4.7.5) becomes accessible through its integral, the average level staircase σ¯ (E) =

1 h¯ f

 d f I Θ [E − H (I )] , I >0

(4.7.6)

4.7 Periodic Orbits and the Semiclassical Density of Levels

95

which counts the number of levels below E as the phase-space volume “below” the corresponding classical energy surface divided by h¯ f ; each quantum state is thus assigned the phase-space volume h¯ f . The M = 0 terms in (4.7.4) describe local fluctuations in the spectrum with scales ever finer as |M| increases. Defining 

e−iπα·M/2 M

(4.7.7)

d f I δ [E − H (I )] ei2πM·I /h¯ .

(4.7.8)

Δ(E) = (E) − (E) ¯ =

M=0

one has the Mth density fluctuation M =

1 h¯ f

 I >0

To evaluate the I integral, we introduce a new set of orthogonal coordinates as integration variables, ξ0 , ξ1 , . . . , ξf −1 , the “zeroth” one of which measures the perpendicular distance from the energy surface, and the remaining f − 1 variables parameterize the energy surface (Fig. 4.2). The integral then takes the form  M =

d

f −1

ξe

i2πM·I (ξ)/h¯

 dξ0

δ(ξ0 ) |∂H /∂ξ0|

(4.7.9)

where it is understood that d f −1 ξ contains the Jacobian of the transformation of the integration variables and the vector ξ lies in the energy surface so as to have the Fig. 4.2 Energy surface in action space

I3 ξ0

ξ1 I

ξ2

I2

I1

96

4 Level Clustering

f − 1 components ξ1 , . . . , ξf −1 . By noting that 

∂H ∂ξ0

 ξ0 =0

= |∇I H (I )|H = E = |ω (I (ξ ))|

(4.7.10)

is the length of the f -dimensional frequency vector, one obtains  M =

d f −1 ξ

ei2πM·I (ξ )/h¯ . |ω(I (ξ ))|

(4.7.11)

Without sacrificing consistency with the semiclassical approximation for the energy levels, the remaining integral over the energy surface can be evaluated in the limit h¯ → 0 in which the phase 2πM · I (ξ )/h¯ is a rapidly oscillating function of ξ . The integral will thus be negligibly small unless there is a point ξ M on the energy surface for which the phase in question is stationary, M·

∂I = 0 for ξ = ξ M , i = 1, 2, . . . , f − 1 . ∂ξi

(4.7.12)

Since the f −1 vectors ∂I /∂ξi are all tangential to the energy surface, the stationaryphase condition (4.7.12) requires the lattice vector M to be perpendicular to the energy surface. An equally significant consequence of the vector ξ lying in the energy surface is ∂H ∂I ∂I = ∇I H · =ω· =0, ∂ξi ∂ξi ∂ξi

(4.7.13)

i.e., the orthogonality of the frequency vector to the energy surface. It follows from (4.7.12) and (4.7.13) that the lattice vector M and the frequency vector ω must be parallel at the points of stationarity of the phase 2πI · M/h¯ , ω1 : ω : . . . : ωf = M1 : M2 : . . . Mf .

(4.7.14)

Because the Mi are integers, one concludes that the ωi must be commensurate and the corresponding tori, defined by I M = I (ξ M ), are rational. Obviously, a nonnegligible Mth density fluctuation M corresponds to periodic orbits on the rational torus I M . The lattice vector M defines the topology of the closed orbits inasmuch as an orbit obeying (4.7.14) closes in on itself after M1 periods 2π of Θ1 , M2 periods of Θ2 , etc. Figure 4.3 illustrates closed orbits of simple topologies (Mr , Mφ ) for a two-dimensional potential well, Mr counting the number of librations of the radial coordinate and Mφ the number of revolutions around the center before the orbit closes. When a lattice vector M induces a nonnegligible density fluctuation M so do all its integer multiples since these also fulfill the stationary-phase condition (4.7.12). Therefore, it is useful to introduce the primitive version μ of M such that the

4.7 Periodic Orbits and the Semiclassical Density of Levels

97

Fig. 4.3 Closed orbits of simple topologies in a two-dimensional configuration space

components μi are relatively prime, together with its multiples M = qμ, where q is a positive integer. The density fluctuation qμ corresponds to a closed orbit which is just the primitive orbit μ traversed q times. The total action along a closed orbit M = qμ, S(M) = 2πM · I M ,

(4.7.15)

is obviously q times the action along the primitive orbit μ, S(M) = qS(μ) . It remains to write the stationary-phase approximation to M . The general formula for the stationary-phase approximation of a multiple integral reads  d n xeif (x) Φ(x) =

 xs

.

(2π)n |det(∂ 2 f (x)/∂x

i ∂xj )|x=x s

Φ(x s )eif (x

s )+iβπ/4

(4.7.16)

where x s are the points of stationary phase, defined by ∂f/∂xis = 0, and β(x s ) is the difference in the number of positive and negative eigenvalues of the n × n matrix ∂ 2 f/∂xi ∂xj .

98

4 Level Clustering

The determinant in (4.7.16) deserves special comment. According to (4.7.11), in this case it is (f − 1) × (f − 1) and reads 

2π ∂ 2I det M· ∂ξi ∂ξj h¯



 =  ≡

2πq h¯

f −1

2πq|μ| h¯

  ∂ 2I det μ · ∂ξi ∂ξj

f −1

  K Iμ ,

(4.7.17)

where    μ ∂ 2I K I = det μ ˆ · ∂ξi ∂ξj

(4.7.18)

is the scalar curvature of the energy surface at the point I M . To appreciate this interpretation of K, let us consider f = 2; the vector dI /dξ1 is tangential to the energy surface and d(dI /dξ1 ) = (d 2 I /dξ12 )dξ1 is the increment of that tangent vector along dξ1 ; the larger the value of μˆ · d 2 I /dξ12 , the stronger the curvature of the energy surface (Fig. 4.4). For f = 3, it is easy to see that K is the Gaussian curvature of the then two-dimensional energy surface. The final result for the Mth density fluctuation now reads  M =

h¯ q|μ|

(f −1)/2

exp{i[qS(μ)/h¯ + πβ(μ)/4]} . |ω(I μ )| · |K(I μ )|1/2

(4.7.19)

When summing up according to (4.7.7), one must realize that (4.7.14) implies that the Mi = qμi all have the same sign because all ωi are nonnegative. Moreover, from (4.7.11) ∗ M = −M ,

Fig. 4.4 Curvature of the energy surface

(4.7.20)

4.8 Level Density Fluctuations for Integrable Systems

99

and thus the total density fluctuation takes the form Δ(E) = 2h¯ −(f +1)/2



     −1 1/2   |μ|(f −1)/2 ω I μ K I μ 

μ>0

×

∞  q =1

q

(f −1)/2



β(μ)π qS(μ) qμ · απ + cos − 2 4 h¯

 . (4.7.21)

This remarkable formula expresses the semiclassical level density fluctuations in terms of purely classical quantities, the latter related to periodic orbits. Berry and Tabor [13] followed the synthesis of the level density for the Morse potential in two dimensions according to (4.7.21) by accounting for more and more closed orbits. Starting from the smooth Thomas–Fermi background (4.7.5), ever finer variations of (E) become visible as more closed orbits are included. In the limit, because all such orbits are allowed to contribute, (E) develops a delta-function peak at each energy level Ei . It is quite remarkable that the periodic-orbit result (4.7.21) breaks down for the harmonic oscillator since the energy surfaces for this “pathological” system are planes with zero curvature. In fact, the harmonic oscillator was already excluded in Sect. 4.4 for the same reason.

4.8 Level Density Fluctuations for Integrable Systems Once more, we assume it possible to rescale the semiclassical energy levels Em = H (I = mh ¯)

(4.8.1)

to uniform mean density (See Sect. 5.11 for conditions allowing such unfolding; note also that we have dropped the Maslov indices in (4.8.1); they may be imagined eliminated by a shift of the origin in m space; in the semiclassical limit, mi  αi , they are quite unimportant anyway). The particular unfolding procedures to be explained in Sect. 5.11 are less suitable for the present purpose than one designed by Berry and Tabor [1] which endows the rescaled levels em with the homogeneity property eβm = β f em , β > 0 .

(4.8.2)

The construction of the em proceeds in two steps. First, one replaces Planck’s constant h¯ with a continuous variable h and follows the energy levels Em (h) = H (mh) as h varies. Their intersections with a fixed reference energy E define a sequence of discrete values hm . The mapping of the original energy levels Em (h¯ )

100

4 Level Clustering

E

9 8 6

75

4

3

2

1

9 8 7 6 5 4 3 2 1 1

2

h/h

3

Fig. 4.5 Reshuffling of energy levels in the Berry–Tabor rescaling

onto the numbers hm will in general be nonlinear (Fig. 4.5) and may even reshuffle the ordering. A second nonlinear mapping gives the rescaled levels as  em =

1 hm

f 

 d I Θ [E − H (I )] = f

I >0

h¯ hm

f σ¯ (E)

(4.8.3)

where σ¯ (E) is the Thomas–Fermi or Weyl form of the average level staircase, normalized such that σ¯ (E)/N → 1 for E → ∞. The homogeneity (4.8.2) follows trivially from E = H (mhm ) = H (βmhm /β). It also follows that hm ≥ h¯ and em ≤ σ¯ (E) for Em ≤ E.

(4.8.4)

The rescaled levels em have the density (e) =



δ(e − em )

(4.8.5)

m>0

the local spectral average of which is obtained when the summation over m is replaced with an f -fold integration,  (e) ¯ = m>0

d f mδ(e − em ) ,

(4.8.6)

as was explained in Sect. 4.7. [The reader may recall the derivation of (4.7.5) from (4.7.2) with the help of Poisson’s summation formula.] By virtue of the homogeneity property (4.8.2), the average density is a constant. Indeed, by changing

4.8 Level Density Fluctuations for Integrable Systems

101

the integration variables in (4.8.6) as m = α 1/f x with arbitrary positive α,  (e) ¯ =α 

x>0

=

  d f xδ e − eα 1/f x = α

df x δ

e

x>0

α



− ex = ¯



e

x>0

α

d f xδ (e − αex ) (4.8.7)

.

The constant , ¯ however, must have the value unity since the rescaled average staircase  e de (e ¯ ) = e ¯ (4.8.8) 0

grows from 0 to σ¯ (E) as e grows from 0 to σ¯ (E), so that σ¯ (E) = ¯ σ¯ (E) gives ¯ = 1 .

(4.8.9)

The remainder of the argument parallels that given in Sect. 4.7. Poisson’s summation formula is invoked again to write the density fluctuations as Δ(e) = (e) − 1 =



(4.8.10)

M ,

M=0

 M =

d f mδ(1 − em )ei2πM·me

1/f

.

(4.8.11)

In the Mth density fluctuation, the “standard” energy surface em = 1 occurs in the delta function, and the current energy e is elevated, with the help of the homogeneity (4.8.2), into the phase factor. Again introducing orthogonal coordinates ξ = (ξ, . . . , ξf −1 ) on the energy surface and using the stationary-phase approximation, M =

  exp [i2πM · me1/f − iπ(f − 1)/4]  , |∇m em | · |det(M · (∂ 2 m/∂ξi ∂ξj )|1/2 e(f −1)/2f m=mM

(4.8.12)

provided there is a point mM on the energy surface at which the phase is stationary M·

∂m =0; ∂ξi

(4.8.13)

otherwise, if there is no solution mM to (4.8.13), the Mth density fluctuation is negligible.

102

4 Level Clustering

In a spectral region near some energy e0 , the density fluctuation M displays oscillations of the form M (e) = AM (e0 ) exp {i [KM (e0 )(e − e0 ) + ΦM (e0 )]}

(4.8.14)

with an amplitude ⎡

⎤ 1   2 m − 2  ∂ −(f −1)/2f  e ⎦ AM (e0 ) = ⎣|∇m em |−1 det M 0 ∂ξ ∂ξ  i

j

,

(4.8.15)

m = mM

a “wave number” KM (e0 ) =

2πM · mM 1−1/f

,

(4.8.16)

f e0

and a phase 1/f

ΦM (e0 ) = 2πM · mM e0

−

π (f − 1) . 4

For a fixed large energy e0 , the phase ΦM varies more rapidly (by a factor e0 ) than the wave number KM when one of the components of the vector M changes by unity. Consecutive phase jumps, taken modulo 2π, will tend to fill the interval [0, 2π] randomly. Therefore, the total density fluctuation Δ = (e) − 1 may be expected to depend quite erratically on e − e0 ; correspondingly, the density–density correlation function S(e0 + τ, e0 ) =

1 Δσ



+Δσ/2 −Δσ/2

dσ Δ(e0 + τ + σ )Δ(e0 + σ )

(4.8.17)

should decay rapidly with increasing τ. The averaging interval Δσ must be large enough to contain many levels but small enough to allow approximation (4.8.14). To check on the expected behavior of S, we insert (4.8.10), (4.8.14) into (4.8.17). Only the “diagonal” terms in the double sum over two vectors M, M survive. For these, M = −M , K−M = KM , Φ−M = −ΦM .

(4.8.18)

It follows that S(e0 + τ, e0 ) =

 M

A2M eiKM τ .

(4.8.19)

4.8 Level Density Fluctuations for Integrable Systems

103

For convenience, now let us consider the Fourier transform of S with respect to the variable τ,  +∞ ˜ 0 , K) = 1 dτ e−iKτ S(e0 + τ, e0 ) S(e 2π −∞  = A2M δ (K − KM ) .

(4.8.20)

M

In the limit of large energies, e0 → ∞, both AM and KM are, as already mentioned, smooth functions of M. Thus no appreciable error results when the sum over M is replaced by an integral, ˜ 0 , K) = S(e

 d f MA2M δ (K − KM ) .

(4.8.21)

As a first step toward evaluating the M-integral, the length of the frequency vector in the amplitude AM must be reexpressed as   ˆ |∇m em |m = mM = f/ mM · M

(4.8.22)

ˆ = M/|M|. Indeed, the energy surface relevant for the gradient in question where M is e = 1, and a nearby energy surface is characterized by the point mM (1 + ε), with a small real number ε, at which 1 + Δe = e(1+ε)mM = (1 + ε)f emM = (1 + ε)f ≈ 1 + f ε .

(4.8.23)

ˆ · εm such that the On the other hand, the distance between the two surfaces is M ˆ quotient Δe/εM · m gives (4.8.22). Combining (4.8.15), (4.8.22), and (4.8.21) one gets the Fourier transformed density–density correlation function as ˜ 0 , K) = S(e  ˆ 2 δ(K − 2πM · mM )/f e1−1/f (mM · M) 0 df M . (f −1)/f 2 f −1 2 ˆ f e0 |M| |det(∂ m · M/∂ξi ∂ξj )|m = mM

(4.8.24)

A most remarkable simplification results now if one changes the integration 1−1/f variables: M → Ke0 M; the integral is independent of both e0 and K, ˜ 0 , K) = const . S(e

(4.8.25)

The fact that S˜ is independent of the energy e0 is not really a big surprise since the rescaling of the energy (4.8.3) was designed so as to make the spectrum statistically homogeneous. The lack of dependence of S˜ on K, however, implies merely the

104

4 Level Clustering

expected rapid falloff of S(τ ) with increasing |τ | : ˜ S(τ ) = 2π S(0)δ(τ ).

(4.8.26)

˜ A little more geometry reveals that the prefactor 2π S(0) of the delta function is unity. ˆ with the The f integration variables M in (4.8.24) may be chosen as |M| and M unit vector determined in terms of f − 1 angles; the delta function allows one to perform the |M| integral, S˜ =

1 2πf



ˆ d f −1 M

ˆ · mM M . ˆ |det(∂ 2 m · M/∂ξ i ∂ξj )|m = mM

(4.8.27)

ˆ of M determines a direction m Now every direction M ˆ of m = |m|m. ˆ Switching to the f − 1 integration variables m, ˆ S˜ =

1 2πf



ˆ d f −1 m

ˆ m| ˆ |∂ M/∂ ˆ |mm ˆ · M| ˆ · (∂ 2 mm/∂ξ |det(M ˆ i ∂ξj )|m = mM

.

(4.8.28)

As already mentioned in Sect. 4.7, the determinant in the denominator of (4.8.28) is related to the curvature of the energy surface. To establish an interpretation more useful in the present context, recall that the f −1 vectors t i = ∂m/∂ξi are tangential ˆ is orthogonal to the energy surface, whereas M ˆ · ∂m = M ˆ · ti = 0 . M ∂ξi

(4.8.29)

This identity holds independently of ξ so that the increments of M and t i along an infinitesimal dξ are constrained to obey ˆ =0. ˆ i + tidM Mdt

(4.8.30)

A previous stipulation was that the f −1 coordinates ξi be orthogonal to one another such that an arbitrary infinitesimal vector dm on the energy surface obeys dm =

t i dξi ,

(dm)2 = t 2i dξi2 i

(4.8.31)

i

Now, it is convenient to require that the ξi are also locally Cartesian, dm2 =

 i

dξi2 ⇔ t 2i = 1 ,

(4.8.32)

4.8 Level Density Fluctuations for Integrable Systems

105

i.e., to make the t i unit vectors. Then, the identity (4.8.30) gives 2 ˆ ˆ ˆ ˆ ∂ t i = −tˆi ∂ M = − ∂ M i ˆ ∂ m =M M ∂ξi ∂ξj ∂ξj ∂ξj ∂ξj

(4.8.33)

ˆ i denotes the ith Cartesian component of the vector d M. ˆ The determinant where d M in question,    ∂M  ˆi ∂ 2m ∂M  ˆ  ˆ det M = det =  ,  ∂ξ  ∂ξi ∂ξj ∂ξj

(4.8.34)

is thus revealed as the Jacobian for a transformation from the f − 1 angular ˆ to the f − 1 Cartesian coordinates ξ . coordinates M The integral (4.8.28) now takes the form 

  ˆ ˆ ˆ m |∂ M/∂ m| d f −1 m ˆ m ˆ ·M ˆ |∂ M/∂ξ | m>0 ˆ        1 ˆ m  ∂ξ  , = ˆ m ˆ ·M d f −1 m  2πf ∂m ˆ

S˜ =

1 2πf

(4.8.35)

which allows for the following geometric interpretation. The infinitesimal d f −1 m ˆ is a surface element on the f -dimensional unit sphere and d f −1 m|∂ ˆ ξˆ /∂m| is a Cartesian surface element on the energy surface e = 1 with the vectorial representation ˆ This vector has the component d f −1 m|∂ξ ˆ ·m d f −1 m|∂ξ ˆ /∂ m| ˆ M. ˆ /∂ m| ˆ M ˆ along m, ˆ which is simply the differential area on the energy surface “above” the element d f −1 m ˆ of the unit sphere. Written with the help of the polar representation of the energy surface, that latter element is d f −1 m[m( ˆ m)] ˆ f −1 , and the integral in (4.8.35) takes the form  1 ˜ d f −1 m ˆ [m(m)] ˆ f . (4.8.36) S= 2πf m>0 ˆ The physical meaning of (4.8.36) becomes apparent when the (f − 1)-fold integral is blown up to the f -fold integral: 1 S˜ = 2π



  d f m Θ m − m(m) ˆ .

(4.8.37)

m>0

One recognizes the average level staircase σ¯ (e) evaluated at e = 1. Since σ¯ (e) = e, the final result reads 1 . S˜ = 2π

(4.8.38)

106

4 Level Clustering

In view of (4.8.26), the density–density correlation function becomes S(τ ) = δ(τ ) .

(4.8.39)

The local fluctuations of the level density at different energies bear no correlations, just as if the levels followed one another as the independent events of a Poisson process.

4.9 Exponential Spacing Distribution for Integrable Systems Now, we propose to demonstrate that the correlation function S(τ ) is related to the conditional probability density g(τ ) of finding a level in the interval [e + τ, e + τ + dτ ], given one at e. It was shown in Sect. 4.5 that this conditional probability density determines the spacing distribution P (σ ) according to P (σ ) = const g(σ )e−

#σ 0

dτg(τ )

(4.9.1)

.

Denoting the local spectral average by the brackets  . . . , let us consider 1 − δ(τ ) + S(τ ) 3 = 1 − δ(τ ) + = −δ(τ ) + =







4 δ(e0 + τ − em ) − 1



m

45 δ(e0 − em ) − 1

m

δ(e0 + τ − em )δ(e0 − em )

m,m

δ(e0 + τ − em )δ(e0 − em ) .

(4.9.2)

m=m

The little calculation in (4.9.2) uses  = 1. By writing out the spectral average explicitly, one obtains 1 − δ(τ ) + S(τ ) 4  + Δσ   2 1 dσ δ(e0 + τ + σ − em ) δ (τ − em + em ) . = Δσ − Δσ m 2 m (=m)

(4.9.3) The sum over m gives the probability density of having a level at a distance τ away from em and the curly bracket averages over all levels em in the interval Δσ around e0 . The function considered in (4.9.3) is therefore the probability density for finding

4.10 Equivalence of Different Unfoldings

107

a level in [e +τ, e +τ +dτ ] given one at e, where e may lie anywhere in the spectral range Δσ around e0 , g(τ ) = 1 − δ(τ ) + S(τ ) .

(4.9.4)

Together with S(τ ) = δ(τ ), which is typical for integrable systems, one has g(τ ) = 1

(4.9.5)

and thus the exponential distribution of level spacings P (τ ) = e−τ .

(4.9.6)

4.10 Equivalence of Different Unfoldings The unfolding employed in Sect. 4.8 makes the rescaled levels homogeneous with degree f in the f quantum numbers m. That homogeneity was of critical importance at several stages of the argument. For one thing, all geometric considerations were referred to the single standard energy surface e = 1; moreover, the oscillatory dependence of the Mth density fluctuation M on the energy could be made manifest. The question arises whether the exponential spacing distribution is really a general property of integrable systems with f ≥ 2 and not simply an artefact due to a peculiar unfolding. In Sect. 5.19 we will argue that the most natural unfolding involves the average level staircase σ¯ (E) as e = σ¯ (E) .

(4.10.1)

In practical applications, the average density is often used to rescale e = E (E) ¯ .

(4.10.2)

This unfolding is only locally equivalent to (4.10.1), i.e., with respect to spectral regions within which the density ¯ is practically constant. The unfoldings (4.10.1), (4.10.2) do not in general provide homogeneity of and e . An exception arises only degree f to the rescaled semiclassical levels em m for integrable Hamiltonians which themselves are homogeneous in the f actions I (see Problem 4.6). For all other integrable systems, it is not immediately obvious that (4.10.1), (4.10.2) also yield exponentially distributed spacings. An important property of (4.10.1) is the monotonicity of e (E) since this ensures that ei > ej provided Ei > Ej . The density-based unfolding (4.10.2) need not strictly have that monotonicity but does in almost all cases of practical relevance. Not so for Berry and Tabor’s unfolding used in Sect. 4.8! It was in fact pointed out

108

4 Level Clustering

that the original levels Em may be reshuffled in their order by the rescaling (4.8.3) rather than just squeezed together or stretched apart to secure uniform mean spacing. Could the em owe their exponential spacing distribution to such reshuffling? We cannot formally prove that (4.10.1) and (4.10.2) yield exponentially distributed spacings whenever (4.8.3) does so. Nonetheless, a reasonably convincing argument may be drawn from the following example. Consider a particle with f = 2 subjected to a harmonic binding in one coordinate and free in a finite interval with respect to the second coordinate. Then, the Hamiltonian takes the form H = αI1 + βI22

(4.10.3)

and the semiclassical levels read (again dropping the Maslov index for the oscillator part!) 2 2 Em = α hm ¯ 1 + β h¯ m2 .

(4.10.4)

Clearly, intersections of levels become possible once Planck’s constant is replaced by a continuous variable h. Therefore, the Berry–Tabor levels em will not have precisely the same ordering as the Em . The possible reshuffling, however, is not due to any externally imposed randomness; rather, one is facing the fact that a single free parameter in H suffices to enforce a level crossing, i.e., a generic property of integrable systems. All effective randomness in the sequence of the em thus appears entirely due to the assumed integrability of the Hamiltonian H.

4.11 Problems 4.1 Give H (I ) for the Kepler problem. 4.2 Show that the hydrogen spectrum is as pathological (nongeneric) as that of the harmonic oscillator. Give other examples of nongeneric spectra of integrable systems. 4.3 Show that the probability density for finding the kth neighbor of a level in the distance increment [S, S+dS], for a stationary Poissonian “process” is Pk (S) =

S k−1 −S e . (k − 1)!

For k = 2 the so-called semi-Poissonian distribution results which is usually written as P (S) = 4Se−2S , so as to secure S = 1; see Ref. [14]. 4.4 Show that the conditional probability g(S) defined in Sect. 4.5 is related to Dyson’s two-level cluster function (Sect. 5.11) Y (S) = 1 − g(S), provided that the spectrum is homogeneous.

References

109

4.5 Prove rigorously that L independent spectra with Eμ (S) = (1/L2 ) exp (−LS) superpose to yield a spectrum with exponentially spaced levels. 4.6 Show that the Thomas–Fermi level staircase is homogeneous of degree f for integrable systems whose Hamiltonian is a homogeneous function of the f actions I . Give the explicit form of σ¯ (E). 4.7 In which sense may the frequencies ωi defined by (4.2.2), (4.2.3) be taken to be positive? (see the remark on (4.7.14) following (4.7.19)).

References 1. 2. 3. 4.

M.V. Berry, M. Tabor, Proc. R. Soc. Lond. A 356, 375 (1977) I.C. Percival, Adv. Chem. Phys. 36, 1 (1977) M. Brack, R.K. Bhaduri, Semiclassical Physics (Addison-Wesley, Reading, MA, 1997) J. Marklov, Proceedings of the 3rd European Congress of Mathematics, Barcelona 2000. Progress in Mathematics, vol. 202 (Birkhäuser, Basel, 2001), pp. 421–427 5. J. Marklov, Proceedings of the XIII International Congress on Mathematical Physics, London 2000 (International Press, Boston, 2001), p. 359 6. V.P. Maslov: Théorie des Perturbations et Méthodes Asymptotiques (Dunod, Paris, 1972) 7. V.P. Maslov, M.V. Fedoriuk, Semiclassical Approximation in Quantum Mechanics (Reidel, Boston, 1981) 8. J.B. Delos, Adv. Chem. Phys. 65, 161 (1986) 9. J.-P. Eckmann, R. Sénéor, Arch. Ration. Mech. Anal. 61, 153 (1976) 10. R.G. Littlejohn, J.M. Robbins, Phys. Rev. A 36, 2953 (1987) 11. N. Rosenzweig, C.E. Porter, Phys. Rev. 120, 1698 (1960) 12. M.V. Berry, M. Robnik, J. Phys. A: Math. Gen. 17, 2413 (1984) 13. M.V. Berry, M. Tabor, Proc. R. Soc. Lond. A 349, 101 (1976) 14. E.B. Bogomolny, U. Gerland, C. Schmit, Phys. Rev. E 59, R1315 (1999)

Chapter 5

Random-Matrix Theory

5.1 Preliminaries A wealth of empirical and numerical evidence suggests universality for local fluctuations in quantum energy or quasi-energy spectra of systems that display global chaos in their classical phase spaces. Exceptions apart, all such Hamiltonian matrices of sufficiently large dimension yield the same spectral fluctuations provided they have the same group of canonical transformations (see Chap. 2). In particular, the level spacing distribution P (S) generally takes the form characteristic of the universality class defined by the canonical group. Most notable among the exceptions barred by the term “untypical” are systems with “localization” that will be discussed in Chap. 8. Conversely, “generic” classically integrable systems with at least two degrees of freedom tend to display universal local fluctuations of yet another type, Poissonian, that we have discussed in Chap. 4. The aforementioned universality is the starting point for the theory of random matrices (RMT). After early success in reproducing universal features in spectra of highly excited nuclei, that theory was boosted into even higher esteem when the connection of “integrable” and “chaotic” with different types of universal spectral fluctuations was spelled out by Bohigas, Giannoni, and Schmit [1], with important hints due to Berry and Tabor [2], McDonald and Kaufman [3], Casati, ValzGris, and Guarneri [4], and Berry [5]. The classic version of random-matrix theory deals with three Gaussian ensembles of Hermitian matrices, one for each group of canonical transformations. Any member of an ensemble can serve as a model of a Hamiltonian. Similarly, there are three ensembles of random unitary matrices to represent Floquet or scattering matrices. “Poissonian” ensembles of diagonal matrices with independent, random, diagonal elements are often used to model integrable Hamiltonians. Even systems with localization have been accommodated in their own “universality class” of banded random matrices that will be dealt with in Sect. 8.8.

© Springer Nature Switzerland AG 2018 F. Haake et al., Quantum Signatures of Chaos, Springer Series in Synergetics, https://doi.org/10.1007/978-3-319-97580-1_5

111

112

5 Random-Matrix Theory

Random-matrix theory phenomenologically represents spectral fluctuations such as those expressed in the level spacing distribution or in correlation functions of the density of levels by suitable ensemble averages. The immense usefulness of RMT lies in the fact that it yields closed-form results for many spectral characteristics. The extent to which an individual Hamiltonian or Floquet operator can be expected to be faithful to the RMT averages is open to discussion. A partial answer to that question is provided by a certain ergodicity property of the various ensembles. Explanations of the success of random-matrix theory will be presented in Chap. 7 (ballistic sigma model), Chap. 10 (periodic-orbit theory), and Chap. 11 (level dynamics). This chapter mostly accounts for classic elementary material but also pays tribute to the seven non-standard symmetry classes [6]. We shall also include a brief introduction to and some applications of Grassmann analysis. The latter thread will be spun further toward “superanalysis” in Chap. 6. There, random-matrix theory will be taken up in terms of what is called the supersymmetric sigma model. For more extensive treatments and greater mathematical rigor, the reader may consult Refs. [7, 8]. A useful review is [9]. For the sake of notational convenience, we set h¯ = 1 throughout this chapter.

5.2 Gaussian Ensembles of Hermitian Matrices We shall first present the Wigner/Dyson ensembles for the smallest non-trivial matrix dimensions. The surprisingly simple generalization to arbitrary matrix size will be reserved for Sect. 5.2.4.

5.2.1 Gaussian Orthogonal Ensemble The construction of the Gaussian ensembles will be illustrated by considering real symmetric 2 × 2 matrices with O(2) as their group of canonical transformations (If reflections are to be excluded, the group would is SO(2)). What we are seeking is a probability density P (H ) for the three independent matrix elements H11 , H22 , H12 normalized as 

+∞ −∞

 dH11dH22 dH12P (H ) ≡

dH P (H ) = 1 .

(5.2.1)

Two requirements suffice to determine P (H ). First, P (H ) must be invariant under any canonical, i.e., orthogonal transformation of the two-dimensional basis, dH P (H ) = dH P (H ) , H = OH O˜ , O˜ = O −1 .

(5.2.2)

5.2 Gaussian Ensembles of Hermitian Matrices

113

Second, the three independent matrix elements must be uncorrelated. The function P (H ) must therefore be the product of three densities, one for each element, P (H ) = P11 (H11 )P22 (H22 )P12 (H12) .

(5.2.3)

The latter assumption can be reinterpreted as one of minimum-knowledge input or of maximum disorder. To exploit (5.2.2) and (5.2.3), it suffices to consider an infinitesimal change of basis,  O=

1 −Θ Θ 1

 (5.2.4)

,

for which H = OH O˜ gives = H − 2ΘH H11 11 12 = H + 2ΘH H22 22 12 = H + Θ(H − H ) . H12 12 11 22

(5.2.5)

Factorization and the invariance of P (H ) yield

 d ln P11 d ln P22 − 2H12 P (H ) = P (H ) 1 − Θ 2H12 dH11 dH22  d ln P12 −(H11 − H22 ) . dH12

(5.2.6)

Since the infinitesimal angle Θ is arbitrary, its coefficient in (5.2.6) must vanish, 1 d ln P12 2 − H12 dH12 H11 − H22



d ln P22 d ln P11 − dH11 dH22

 =0.

(5.2.7)

This gives three differential equations, one for each of the three independent functions Pij (Hij ) since each Pij has its own exclusive argument Hij . The solutions are Gaussians and have the product    2 2 2 P (H ) = C exp −A H11 − B (H11 + H22 ) . + H22 + 2H12

(5.2.8)

Of the three integration constants, B can be made to vanish by appropriately choosing the zero of energy, A fixes the unit of energy, and C is determined by normalization. Without loss of generality, then, P (H ) can be written as P (H ) = Ce−A Tr H . 2

(5.2.9)

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5 Random-Matrix Theory

5.2.2 Gaussian Unitary Ensemble The discussion of complex Hermitian Hamiltonians with U (2) as the group of canonical transformations proceeds analogously. There are four real parameters H11 , H22, Re {H12}, Im {H12 } to be dealt with now, and they are all assumed statistically independent. The probability density P (H ) thus factorizes as in (5.2.3) ∗. but with P12 as a function of Re {H12 }, Im {H12 } or, equivalently, of H12 and H12 The normalization (5.2.1) is modified such that P12 is integrated over the whole complex H12 plane, 

+∞

−∞

dH11 dH22d 2 H12 P (H ) = 1

(5.2.10)

where d 2 H12 = dRe H12 dIm H12 . The three functions P11 (H11 ), P22 (H22 ), and ∗ ) are determined by demanding that P (H ) is invariant under unitary P12 (H12 , H12 transformations of the matrix H . Up to an inconsequential phase factor, the general infinitesimal change of basis is represented by the 2 × 2 matrix U = 1 − iε · σ

(5.2.11)

with ε an infinitesimal vector and σ the triple of Pauli matrices. This U shifts the Hamiltonian by dH = −i [ε · σ , H ] .

(5.2.12)

Now, the analogue of (5.2.7) reads       ∂ ln P12 d ln P22 d ln P11 + εx + iεy H12 − + (H11 − H22 ) ∗ dH11 dH22 ∂H12 +2εz H12

∂ ln P12 − c.c. = 0 . ∂H12

(5.2.13)

Again, three differential equations can be extracted from this identity and the solutions are all Gaussians. If the zero of energy is chosen appropriately, the probability density P (H ) once more takes the form (5.2.9).

5.2.3 Gaussian Symplectic Ensemble Let us turn finally to Hamiltonians with Kramers’ degeneracy and with no geometric invariance. The smallest Hilbert space is now four dimensional. The 4 × 4

5.2 Gaussian Ensembles of Hermitian Matrices

115

Hamiltonian is most conveniently written in quaternion notation, H =

  h11 h12 , h21 h22

(5.2.14)

where each hij is a 2 × 2 block, representable as a superposition of unity and the triplet τ = −iσ [see (2.9.7)]. Due to (2.9.9), (2.9.11), H is determined by six real (0) (0) (μ) amplitudes h11 , h22 , h12 , all of which are taken to be independent random numbers with a probability density of the form     μ  μ (0) (0) 63 P (H ) = P11 h11 P22 h22 μ = 0 P12 h12 , # (0) (0) 6 (μ) 1 = dh11 dh22 3μ = 0 dh12 P (H ) .

(5.2.15)

To find the six respective probability densities, it suffices to require invariance of P (H ) when H is subjected to an arbitrary infinitesimal symplectic transformation. Such a change of basis is represented by   1−ξ ·τ α S= −α 1 + ξ · τ

(5.2.16)

with α an infinitesimal angle and ξ an infinitesimal real vector; of course, (5.2.16) is meant in quaternion notation, and the matrices τ = −iσ are as defined in (2.9.6). The 2 × 2 blocks of the Hamiltonian acquire the increments (0)

dh11 = −dh22 = 2αh12   (0) (0) dh12 = α h(0) − h 22 11 + ξ · h12 − 2h12 ξ · τ .

(5.2.17)

Now, we invoke P (H + dH ) = P (H ) for arbitrary α and ξ and thus obtain  2h(0) 12

d ln P11 (0)

dh11

−

d ln P22 (0)

dh22



(0)   (0) d ln P12 + h(0) − h =0, 22 11 (0) dh12

(0)

h(i) 12

d ln P12 (0)

dh12

(i)

− h(0) 12

d ln P12 (i)

dh12

= 0 for i = 1, 2, 3 .

(5.2.18)

The reader should note that the first of these identities has the same structure as (5.2.7) and (5.2.13). Since each of the six functions we are seeking has its own exclusive argument, the identities (5.2.18) imply six separate differential equations. As in the cases considered previously, the solutions are all Gaussians. With a proper

116

5 Random-Matrix Theory

choice for the zero of energy, their product1

 3        (μ) 2 (0) 2 (0) 2 h12 P (H ) = C exp − 2A h11 + h22 + 2

(5.2.19)

μ=0

again gives the density P (H ) in the form (5.2.9). To summarize, three different ensembles of random matrices follow from demanding (1) invariance of P (H ) under the three possible groups of canonical transformations and (2) complete statistical independence of all matrix elements. In view of the Gaussian form of P (H ) and the three groups of canonical transformations, these ensembles are called Gaussian orthogonal, Gaussian unitary, and Gaussian symplectic. Although P (H ) has been constructed here for the smallest possible dimensions, the result P (H ) = C e−A Tr H , 2

(5.2.20)

holds true independently of the dimensionality of H , see the next subsection and Refs. [7, 8]. Of course, in both P (H ) and in the integration measure, the appropriate number of independent real parameters must be accounted for; that number is N(N + 1)/2 and N 2 , respectively, for the real symmetric and complex Hermitian N × N matrices and N(2N − 1) for the quaternion real 2N × 2N matrices. The invariance of the ensembles under the appropriate canonical transformations is not fully established before the invariance of the differential volume element in matrix space is shown, see Sect. 5.3.

5.2.4 Arbitrary Matrix Dimension The generalization to arbitrary matrix size N will here be expounded for the GOE of N × N real symmetric matrices, Hij = Hj i . As above, we shall invoke the invariance of the ensemble under orthogonal transformations, ˜ , P (H ) = P (OH O)

O O˜ = 1 ,

(5.2.21)

1 The reader’s attention is drawn to the overall factor 2 in the exponent of (5.2.19); it stems from Kramers’ degeneracy. When dealing with the GSE, some authors choose to redefine the trace operation as one half the usual trace, so as to account for only a single eigenvalue in each Kramers’ doublet; correspondingly, these authors take the determinant as the square root of the usual one. Throughout this book we will keep the standard definitions.

5.2 Gaussian Ensembles of Hermitian Matrices

117

as well as the statistical independence of the matrix elements Hij with i < j , P (H ) =

N i=1

Pii (Hii )

 -

 Pij (Hij ) .

(5.2.22)

i 0 |tm |2 |tn |2 = |tm |2 |tn |2 + |tm+n |2 + |tm−n |2 − 2|tmax(m,n) |2 . The relative covariance between two traces thus comes out as ⎧ N−m−n ⎪ for N − n ≤ m ≤ N ⎪ ⎨ mn |tm |2 |tn |2 N−m+n − 1 = − Nmin(N,n) for N < m ≤ N + n ⎪ ⎪ |tm |2 |tn |2 ⎩0 otherwise

(5.17.14)

(5.17.15)

where without loss of generality m > n > 0 is assumed. Statistical independence of the two traces would entail vanishing relative cross-correlation, and that is in fact the behavior prevailing in most ranges of m, n. Otherwise, where the crosscorrelation does not vanish, it at least turns out small, i.e., of order 1/N; more precisely, −1/N ≤ |tm |2 |tn |2 /(|tm |2 |tn |2 ) − 1 ≤ 0. We may conclude from the above investigations that the traces tn of CUE matrices behave, at least as far as moments of orders 1 to 4 are concerned, as if they were independent Gaussian random variables to within corrections of relative weight 1/N. That same statement holds true for matrices from the COE and the CSE; in the latter case the precision is only O(1/ ln N) when n is near half the Heisenberg time nCSE = N; the interested reader is referred to Ref. [48] for these H cases and to Ref. [49] for higher order moments in the case of the CUE. Equipped with |tm |2 |tn |2 and |tm |4 , we can estimate the CUE variance (5.17.10) of the time-averaged form factor. Since the cross-correlation (5.17.15) of two traces is nonpositive for m = n, we get an upper limit for Var|tn |2 / |tn |2  by dropping the “off-diagonal” terms with n = n in (5.17.10). For the diagonal terms with n > 0, we get an estimate from the second line of (5.17.11), |tn |4 − (|tn |2 )2 ≤ (|tn |2 )2 since 2|tn |2 ≥ |t2n |2 . It follows that Var|tn |2 / |tn |2  ≤

1 Δn

Δn→∞

−→

0.

(5.17.16)

5.17 Higher Correlations of the Level Density

179

The locally time averaged form factor |tn |2 / |tn |2 Δn exhibits ever weaker fluctuations about its CUE mean, as the averaging time window Δn grows large. The fluctuations allowed by the CUE variance Var|tn |2 / |tn |2  ≤ 1/Δn show up at any fixed time n as the underlying unitary matrix ranges throughout the CUE. Most remarkably, the estimate (5.17.16) is independent of the running time n around which the averaging window is located. We may say that in the limit 1/Δn → 0 exceptional unitary matrices with persisting finite fluctuations have measure zero in the CUE. Conversely, universal behavior with vanishing fluctuations prevails with measure unity. Of course, individual dynamics that are classically integrable have a form factor very different from the CUE mean even if non-invariant under time reversal. Well, within the set of all classical the subset of integrable ones has measure zero. Much as we love random-matrix theory, we had better admit that it does not provide criteria for its own applicability.

5.17.3 Ergodicity of the CUE Two-Point Correlator The question we propose to address now is analogous to the one just answered for the form factor: After which modifications does the two-point correlator of the level density of an individual CUE matrix F (or the Floquet matrix of a dynamical system from the same symmetry class) stand a chance to exhibit universal behavior, with small deviations from the CUE mean? Given the spectrum of eigenphases of such a matrix, {φi , i = 1 . . . N}, we have

the level density (φ) = N −1 i δ(φ − φi ). The product of two such densities is a double sum over products of two Dirac deltas. Two smoothing operations are necessary to produce a plottable function that can be compared to its CUE mean. As one such operation offers itself an average over the center phase φ − φ ≡ ψ,  2π  eπ   dψ  eπ  1   e2π  ψ+ − φi + φj ; δ  ψ− = 2π N N 2πN N 0 i,j

(5.17.17) here the ‘spacing variable’ e is referred to the mean level spacing as a unit, φ − φ ≡ e 2π N . The second operation must smoothen the remaining train of deltas to a plottable function. The number of deltas with i = j being N(N − 1) and their locations distributed over a 2π-interval one may either broaden the deltas to, say, Lorentzians 2π 2π of width larger than N 2 but smaller than the mean level spacing N ; or else one averages over a suitable e-interval; both methods lead to equivalent results [50]. We shall here stick to smoothing by integration and consider the first primitive of the single-matrix cluster function  e de y(e ) (5.17.18) y (1)(e) = 0

180

5 Random-Matrix Theory

with the cluster function arising from the above two-point function in the usual way, see Sect. 5.14.1: Subtract the squared mean density to get a cumulant, subtract the delta function located at e = 0 stemming from ‘self correlation’ of levels (i = j ), renormalize by dividing out the squared mean density, change the overall sign to get y(e) =

∞  1 2πne 2  N − |tn |2 cos + 2 . N N N

(5.17.19)

n=1

To arrive at the foregoing expression we represented the (2π-periodic) Dirac delta inφi . The CUE mean we by a Fourier series and invoked the traces tn = ie obviously have y(e) =

N  2πne 1 2  N − n cos + 2 . N N N

(5.17.20)

n=1

We shall be concerned with the first primitive y (1)(e) =

∞ 1  N − |tn |2 2πne e + sin N πN n N

(5.17.21)

N 1 N −n 2πne e + sin . N πN n N

(5.17.22)

n=1

and its CUE mean y (1)(e) =

n=1

It is worth noting some properties of the cluster function y(e) and its integral y (1)(e) which are valid for any spectrum. First, the cluster function is even and periodic in e, y(e) = y(−e) ,

y(e + N) = y(e) .

(5.17.23)

The first primitive inherits the property y (1)

N 2

   + e + y (1) N2 − e = 1

as is easily checked, and that means that y (1) definition (5.17.21) entails y (1)(0) = 0

y (1)

N  2

=

1 2

N 2

 +e −

y (1)(N) = 1

(5.17.24) 1 2

is odd in e. The

(5.17.25)

as universal points for any spectrum. In between these universal points the function y (1)(e) in general differs from spectrum to spectrum. However, we shall see in Sect. 5.20 that Floquet operators of classically fully chaotic kicked tops have

5.17 Higher Correlations of the Level Density

181

integrated cluster functions y (1)(e) coinciding with the pertinent ensemble averages y (1)(e), CUE, COE, or CSE, up to fluctuations vanishing as N → ∞.   We proceed to showing that the CUE variance Var y (1) (e) 2  = y (1)(e) − y (1)(e) of that primitive is of the order N1 for all values of the spacing variable e. As a first step we realize that the additive term Ne in (5.17.21) and (5.17.22) cancels. We thus have, with the abbreviation sn = sin 2πne N ,   Var y (1)(e) ≡ V (e)

(5.17.26)

  ∞   sm sn  −2 = (πN) N − |tm |2 N − |tn |2 mn m,n=1

 N  sm sn (N − m)(N − n) . − mn m,n=1

So we are back with the fourth moments |tm |4 and |tm |2 |tn |2 , see (5.17.11) and (5.17.14), for the diagonal and off-diagonal terms in the foregoing double sums. The former read Vdiag = (πN)

−2

∞ 2   2 sm 4 − |t |2 |t | m m m2

m=1

= (πN)

−2

 [N/2]  m=1

+

sin2 2πme N + m2 ∞ 

m=N+1

N  m=[N/2]+1

sin2 2πme N m2

sin2

N − 2m  2πme  1+ N m2

   2 N −N .

(5.17.27)

For the second equation in the above chain we have invoked (5.17.11) and marked the subleading terms, of relative order N1 , in red. The leading terms reflect Gaussian statistics of the traces involved, as  discussed  in the previous section. The off-diagonal terms in Var y (1)(e) = V can be written as Voff

2 = (πN)2

 1...N  m>n

+

 sm sn  |tm |2 |tn |2 − mn mn not both≤N  m>n

(5.17.28)

   sm sn  |tm |2 |tn |2 − N |tm |2 + |tn |2 + N 2 . mn

For their evaluation a glance at Fig. 5.5 is helpful. The various regions in the mn-plane are depicted where the normalized covariance (5.17.15) vanishes or is

182

5 Random-Matrix Theory

n

Fig. 5.5 Regions in the mn-plane where the normalized covariance (5.17.15) vanishes (shaded) or gives off-diagonal contributions to the CUE  variance Var y (1) (e)

4 Voff

0 3 Voff

2 Voff

N

0

1 Voff

2 Voff

N

0 m

1 , stems from the first line nonzero. Four contributions to voff arise. The first, voff in (5.17.28), the other three from the second line. They read

1 Voff =−

2 =− Voff

3 Voff =−

4 Voff =−

2 (πN)2 2 (πN)2 2 (πN)2 2 (πN)2

N 

m−1 

sm sn (m + n − N) mn

m=[N/2] n=N−m+1 2N 

N 

m=N+1 n=m−N+1 2N 

m−1 

m=N+1 n=N+1 ∞ 

(5.17.29)

sm sn (N − m + n) mn

sm sn (N − m + n) mn

m−1 

m=2N+1 n=m−N+1

sm sn (N − m + n) mn

A quick argument applicable for large N reveals the correctness of our above claim about the smallness  of the fluctuations of the integrated correlator throughout the CUE, Var y (1)(e) = Vdiag + Voff = O( N1 ). The large-N behavior can be # q/N

captured by replacing the sums over m and n by integrals as qm=p → p/N dμ. The claimed behavior then springs up   immediately for any e ∈ [0, N], N being the period N of y (1)(e) and Var y (1)(e) . In fact, our result 4    i Var y (1) (e) = Vdiag + Voff i=1

(5.17.30)

5.17 Higher Correlations of the Level Density

183

is much stronger than the foregoing argument. It is valid for any value of the matrix dimension N. The above sums are easily evaluated numerically, for any nonsymmetric unitary N × N matrix F with N up to several thousands and for all values of the spacing variable e within the period N. The results are put together in a separate section at the end of the present chapter.  We should mention that the same result for Var y (1)(e) has been established in Ref. [50] along completely different lines, on the basis of an expression for the level density correlators of all orders, exact for all N if only for the CUE [51].

5.17.4 Joint Density of Traces of Large CUE Matrices A powerful theorem about Toeplitz determinants, due originally to Szegö and Kac and extended by Hartwig and Fischer [52], helps to find the marginal and joint distributions of the traces tn of CUE matrices with finite “times” n in the limit, as the dimension N goes to infinity [45]. In that limit, the finite-time traces will turn out to be statistically independent and to have Gaussian distributions with the means and variances already determined above, tn = 0, |tn |2 = n. Once more we invoke the ubiquitous theorem (5.12.4) for the CUE mean of a symmetric function of all eigenphases, again for that function a product. The integral over the phase φi can then be pulled into the ith row of the determinant in (5.12.4), whereupon the average becomes the Toeplitz determinant N -

f (φm ) = det(fl−m ) ≡ T ({f }) ,

m=1

l, m = 1, 2, . . . , N ,

(5.17.31)

the elements of which are the Fourier coefficients 

2π

fm = 0

dφ imφ e f (φ) 2π

(5.17.32)

of the function f (φ). The Hartwig–Fischer theorem assigns to the determinant the asymptotic large-N form T ({f }) = exp(Nl0 +

∞ 

nln l−n )

(5.17.33)

n=0

imφ . The with the ln the Fourier coefficients of ln f (φ), i.e., ln f (φ) = ∞ n=−∞ ln e function f (φ) must meet the following four conditions for the (1)

theorem to hold: 2 < ∞, f (φ) = 0 for 0 ≤ φ < 2π, (2) arg f (2π) = arg f (0), (3) ∞ |n||f | n n=−∞ and (4) ∞ n=−∞ |fn | < ∞.

184

5 Random-Matrix Theory

For a first application, we consider the marginal distribution of the nth trace tn whose Fourier transform is the characteristic function (5.14.11) and involves   i f (φ) = exp − (ke−inφ + k ∗ einφ ) . 2

(5.17.34)

Provided that n remains finite as N → ∞, that function fulfills all conditions of the theorem (see below for an explicit check), and the only nonvanishing Fourier ∗ = −ik/2. The asymptotic form of the coefficients of its logarithm are ln = l−n 2 Toeplitz determinant thus reads e−nk /4 and yields the density of the nth trace as its Fourier transform as the Gaussian already anticipated several times, P (tn ) =

1 −|tn |2 /n e . πn

(5.17.35)

The foregoing reasoning is easily extended to the joint density of the first

N imφ 6n 2 l ) whose Fourier transform n traces P (t1 , . . . , tn ) = m=1 δ (tm − l=1 e P˜ (k1 , . . . , kn ) is once more of the form (5.17.31) with the function f (φ) from (5.17.34) generalized to the sum 8

9 n i  −imφ ∗ imφ f (φ) = exp − (km e + km e ) . 2

(5.17.36)

m=1

∗ = −ik /2 with The nonvanishing Fourier coefficients of ln f (φ) are now lm = l−m m m = 1, . . . , n and entail a limiting form of the characteristic function which is just the product of the marginal ones met above. The joint density then comes out as the announced product of marginal distributions

  n  1 2 P (t1 , . . . , tn ) = exp − |tm | /m . n!π n

(5.17.37)

m=1

The result obviously generalizes to the joint density of an arbitrary set of finitetime traces. The previously resulting hints of statistical independence and Gaussian behavior of all finite-time traces in the limit N → ∞ are thus substantiated. To appreciate the importance of the restriction to finite n and to avoid a bad mathematical conscience, it is well to verify the aforementioned conditions of the Hartwig–Fischer theorem on the function f (φ) in (5.17.36). The first two of them are clearly fulfilled since i ln f (φ) is real, continuous, and (2π)-periodic. The third and fourth conditions are met since the derivative f (φ) is square integrable (provided n remains finite!), 

2π 0

dφ|f (φ)|2 =

∞  m=−∞

m2 |fm |2 = π

n  m=1

m2 |km |2 < ∞ .

(5.17.38)

5.18 Correlations of Secular Coefficients

185

2 Since condition (3),

∞ |m| ≤ |m|2 in the foregoing Parseval inequality, we

check ∞ |m||f | < ∞. Finally, Cauchy’s inequality yields m m=−∞ |fm | = |f0 |+

m=−∞1 1 2 1/2 2 1/2 ( m=0 |mfm | ) ; due to Parsefal’s m=0 | m ||mfm | ≤ |f0 | + ( m=0 | m | )

inequality and the convergence of m=0 1/m2 , we find condition (4) met as well. For finite dimension N, the independence as well as the Gaussian character of the traces are only approximate; both properties tend to get lost as the sum of the orders (times) of the traces involved in a set increases. This is quite intuitive a finding since already all traces for a single unitary matrix are uniquely determined by the N eigenphases such that only N/2 traces are linearly independent.

5.18 Correlations of Secular Coefficients We propose here to study the autocorrelation of the secular polynomial of random matrices from the circular ensembles [44, 45, 53]. These correlations provide a useful reference for the secular polynomials of Floquet or scattering matrices of dynamic systems and their semiclassical treatment. In particular, we shall be led to an interesting quantum distinction between regular and chaotic motion. A starting point is the secular polynomial of an N × N unitary matrix F , det(λ − F ) =

N 

n

(λ) AN−n

n=0

N = (λ − e−iφi ) ,

(5.18.1)

i=1

which enjoys the self-inversiveness discussed in the preceding subsection. An autocorrelation function of the secular polynomial for any of the circular ensembles may be defined as β

PN (ψ, χ) =

-

(e−iφi − e−iψ )(eiφi − eiχ ) ,

(5.18.2)

i=1...N

with the overbar indicating an average over the circular ensemble characterized by the level repulsion exponent β. For the orthogonal and unitary circular ensembles, this correlation function may be identified with det(F − e−iψ )(F † − eiχ ). In the symplectic case, however, the definition (5.18.2) accounts only for one level per 2  N n = Kramers’ doublet such that we have det(F − λ) = n=0 (−λ) aN−n 6N 1 −iφi − λ)2 and thus P 4 (ψ, χ) = [det(F − e−iψ )(F † − eiχ )] 2 . i=1 (e N Due to the homogeneity of the circular ensembles, the correlation function in question depends on the two phases ψ and χ only through the single unimodular variable x = ei(χ−ψ) ; thus it may be written in terms of the auxiliary function f (φ, x) = (e−iφ − x)(eiφ − 1)

(5.18.3)

186

5 Random-Matrix Theory

as β

PN (x) =

-

f (φi , x) =

i

N 

x n |An |2 .

(5.18.4)

n=0

In writing the last member of the foregoing equation, we have once more invoked the homogeneity of the circular ensembles to conclude that Am A∗n = δmn |An |2 .

(5.18.5)

Obviously, our correlation function may serve as a generating function for the β ensemble variances of the secular coefficients, |An |2 = n!1 d n PN (x)/dx n |x=0 . β

The correlation function PN (x) and the variances |An |2 are most easily calculated in the Poissonian case β = 0,  PN0 (x)

2π

=

|An |2 =

0

  N . n

dφ f (φ, x) 2π

N = (1 + x)N , (5.18.6)

The treatment of the CUE, COE, and CSE proceeds in parallel to the calculation of the characteristic function of the traces in the preceding section. As in (5.14.5), we must average a product, now with factors f (φi , x) rather than exp(−i cos nφi ). For the CUE, that replacement changes (5.14.11) into  PN2 (x) = det 

2π

0

dφ i(m−n)φ e f (φ, x) 2π



= det (1 + x)δ(m − n) − δ(m − n + 1) − xδ(m − n − 1) =

N 

xn .



(5.18.7)

n=0

The CUE variance of the secular coefficient thus comes out independent of n, |An |2 = 1 ,

(5.18.8)

and that independence is in marked contrast to the binomial behavior (5.18.6) found for the Poissonian case. Similarly, for the orthogonal and symplectic ensembles, we are led to Pfaffians β of antisymmetric matrices Aβ , PN (x) = Pf(Aβ (x))/Pf(Aβ (0)). In the orthogonal

5.18 Correlations of Secular Coefficients

187

case where for simplicity we assume even N, one finds (5.14.23) replaced with A1mm (x)

 2π 1 = dφdφ sign(φ − φ )f (φ, x)f (φ , x)ei(mφ+m φ ) 4πi 0   1 (1 + x)2 1 − ) δm,−m = + x( (5.18.9) m m+1 m +1  1 1  − (1 + x) − δm,−m +1 + x δm,−m −1 m m +

1 x2 δm,−m +2 + δm,−m −2 m−1 m+1

where again the underlined index runs in unit steps through the half-integers in 0 < |m| ≤ (N − 1)/2. The analogue of (5.14.33) in the symplectic case reads A4mm (x) = (m − m )

 0

2π

dφ i(m+m )φ e f (φ, x) 2π

(5.18.10)

  = (m − m ) (1 + x)δ(m + m ) − δm,−m −1 − xδm,−m +1 with 0 < |m| ≤ (2N − 1)/2. The evaluation of the Pfaffian is a lot easier in the symplectic case than in the orthogonal one since A1 is13 ‘pentadiagonal’ but A4 is only ‘tridiagonal.’ Moreover, the 2N × 2N matrix A4 falls into four N × N blocks, and the two diagonal ones vanish such that the Pfaffian Pf(A4 (x)) equals, up to the sign, the determinant of either off-diagonal N × N block. One reads off the following recursion relation for the correlation function PN4 (x) = Pf(A4 (x))/Pf(A4 (0)) in search, 4 PN4 (x) = (1 + x)PN−1 (x) − x

(2N − 2)2 P 4 (x) ; (2N − 1)(2N − 3) N−2

(5.18.11)

this in turn is easily converted into a recursion relation for the variances , (N)

(N−1) 2 1 (N−1) | + |An−1 |2 −

|An |2 = |An

n

(2N − 2)2 (N−2) 2 |A | . n(2N − 1)(2N − 3) n−1

(5.18.12)

13 To remove the following quotation marks, think of the rows of the determinant swapped pairwise, the first with the last, the second with the last but one, etc.

188

5 Random-Matrix Theory (N)

(N)

With A0 = 1 for all naturals N, one finds the solution (with An stripped back down to An ) |An

|2

 2  −1 N 2N = n 2n

typographically

. N1

≈

N . πn(N − n)

(5.18.13)

Finally mustering a good measure of courage, we turn to the orthogonal case, i.e., the Pfaffian of the “pentadiagonal” antisymmetric matrix A1m,m (x) given in (5.18.10). A little convenience is gained by actually removing those quotation marks in the way indicated in the footnote, i.e., by looking at the matrix Am,m ≡ A1m,−m ; recall that the underlined indices take the half-integer values −M, −M + 1, . . . , M with M = (N − 1)/2; the determinants of A and A1 can differ at ) most in their sign which√is irrelevant for our correlation function PN1 (x) = det A1 (x)/ det A1 (0) = det A(x)/ det A(0). The key to the latter square root is to observe that the N × N matrix Am,m (x) fails to be the product BC of the two tridiagonal matrices Bm m = −

1+x x 1 δm,m −1 + δm,m +1 δm,m + m m+1 m−1

Cm m = −(1 + x)δm,m + xδm,m −1 + δm,m +1

(5.18.14)

only in the two elements in the upper left and the lower right corner. As a remedy, we momentarily extend the matrix dimension to N + 2, bordering the matrices A, B, C by frames of single rows and columns as ⎞ 1 1 1 ... 1 1 1 ⎟ ⎜ (−M − 1)−1 0 ⎟ ⎜ ⎟ ⎜ 0 0 ⎟ ⎜ ⎟ ⎜ .. .. B =⎜ ⎟, . B . ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 0 ⎟ ⎜ −1 ⎝ x(M + 1) ⎠ 0 0 0 0 ... 0 0 1 ⎛

⎛

−M − 1 x 0 . . . 0 0 ⎜ 0 ⎜ ⎜ 0 ⎜ ⎜ .. C =⎜ C . ⎜ ⎜ ⎜ 0 ⎜ ⎝ 0 0 0 0 ... 0 1

⎞ −M − 1 −M ⎟ ⎟ ⎟ −M + 1 ⎟ ⎟ .. ⎟, . ⎟ ⎟ M−1 ⎟ ⎟ M ⎠ M+1

(5.18.15)

(5.18.16)

5.18 Correlations of Secular Coefficients

189

⎞ −M − 1 0 0 . . . 0 0 0 ⎜ 1 0 ⎟ ⎟ ⎜ ⎟ ⎜ 0 ⎟ 0 ⎜ ⎜ .. ⎟ .. A =⎜ ⎟ . ⎟. A . ⎜ ⎟ ⎜ ⎜ 0 ⎟ 0 ⎟ ⎜ ⎝ 0 ⎠ 0 0 0 0 ... 0 1 M +1 ⎛

(5.18.17)

A = B C . The The bordered matrix A now enjoys the product structure in full,  6 M+1 C B and det A = various determinants obviously obey det = m=−M−1 m det −(M + 1)2 det A such that upon drawing the square root of det A , we get . PN1 (x)

=

det A(x) = (−1)? det A(0)



2 N +1

2 det C .

(5.18.18)

It remains to evaluate the (N + 2) × (N + 2) determinant det C . Since the first column contains just a single non-zero element, we reduce the dimension to N + 1, ⎛

−(1 + x) x ... 0 ⎜ 1 −(1 + x) . . . 0 ⎜ ⎜ 0 1 ... 0 ⎜ ⎜ det C .. .. .. ⎜ = . . ... . −(M + 1) ⎜ ⎜ ⎜ 0 0 ... x ⎜ ⎝ 0 0 . . . −(1 + x) 0 0 ... 1

⎞ −M −M + 1 ⎟ ⎟ −M + 2 ⎟ ⎟ ⎟ .. ⎟. . ⎟ ⎟ M −1 ⎟ ⎟ M ⎠ M +1

Here the original N × N matrix C appears in the upper left corner. Starting with the uppermost one, we now add to each row the sum of all of its lower

n neighbors such that in the elements of the new last column the sums f (n) = m=−M−1 m appear and the determinant reads ⎛ ⎞ f (M) −x 0 . . . 0 ⎜ 1 −x . . . 0 f (M − 1) ⎟ ⎜ ⎟ ⎜ 0 1 . . . 0 f (M − 2) ⎟ ⎜ ⎟ ⎜ . . ⎟ det C .. .. ⎟. . . =⎜ . . . . . . . ⎜ ⎟ M+1 ⎜ ⎟ ⎜ 0 0 . . . 0 f (−M + 1) ⎟ ⎜ ⎟ ⎝ 0 0 . . . −x f (−M) ⎠ 0 0 . . . 1 f (−M − 1)

190

5 Random-Matrix Theory

We finally annul the secondary diagonal with unity as elements: Beginning at the top, we add to each row the x1 -fold of its immediate upper neighbor. The arising determinant equals the product of its diagonal elements, det C = (M +

2M+1 1)(−x)2M+1 n=0 ( x1 )n f (−M − 1 + n). Recalling M = (N − 1)/2 with N even, we get from (5.18.18) the correlation function in search as PN1 (x) =

N  n=0

 n(N − n) xn 1 + N +1

(5.18.19)

and read off the mean squared secular coefficients for the COE, |An |2 = 1 +

n(N − n) . N +1

(5.18.20)

Let us summarize and appreciate the results for the four circular ensembles. The mean squared secular coefficients (5.18.6), (5.18.8), (5.18.13), (5.18.20) are special cases of   N Γ (n + 2/β)Γ (N − n + 2/β) β (5.18.21) |An |2 = Γ (2/β)Γ (N + 2/β) n for the appropriate values of the repulsion exponent β. Actually, the latter general formula has been shown [45] to be valid for arbitrary real β, provided the joint density of eigenvalues (5.8.10) is so extended. β

With the family relationship between the |An |2 pointed out, the differences should be commented on as well. For fixed n = 0, N, we face a growth of that function with decreasing β. That trend is not surprising; an increase of the degree of level repulsion implies a tendency toward equidistant levels. For β → ∞, one must expect a perfectly rigid spectrum14 according to the secular equation λN − 1 = 0, i.e., An = 0 except for n = 0, N. Conversely, the Poissonian statistics arising for β → 0 entails the weakest spectral stiffness and thus the largest mean squared   secular coefficients. For a large dimension N, the Poissonian limit |A0n |2 = Nn β

is exponentially larger than the three other distinguished |An |2 with β = 1, 2, 4. The difference in question may in fact be seen as one of the quantum criteria to distinguish regular (β = 0) from chaotic dynamics (β = 1, 2, 4), as will become clear in the following section where a detailed comparison of random matrices and Floquet matrices of dynamical systems will be presented; see in particular Fig. 5.11.

14 Indeed, generalizing the joint distribution of eigenvalues (5.4.1) to arbitrary real β = n − 1, one 2 is led to a Wigner distribution Pβ (S) = AS β e−BS with A and B fixed by normalization and S = 1 which for β → ∞ approaches the delta function δ(S − 1).

5.19 Unfolding Spectra

191 β

The correlation function PN (x) itself is due some attention. To give it a nice appearance, we return to (5.18.4) and realize that due to the symmetry of the |an |2 β under n ↔ N − n, we may switch from PN (x) to a function [53] PN (e−i2πη/N )eiπη

N

β

C (η) = β

β

PN (1)

=

2 e−i2πη(n/N−1/2)

N 2 n=0 |an |

n=0 |an |

(5.18.22)

β

which is real for real η; note the normalization to CN (0) = 1. Using our results for the variances |An |2 and doing sums as integrals, ⎧  N 2πη ⎪ cos ≈ exp − ⎪ ⎪ N ⎪   ⎪ ⎨3 1 ∂ 2 sin πη 1 + π 2 ∂η2 πη C β (η) = 2 ⎪ sin πη ⎪ ⎪ πη ⎪ ⎪ ⎩ J0 (πη)

2π 2 η2 N



for β = 0 for β = 1

.

(5.18.23)

for β = 2 for β = 4

Checking for the small-η behavior, we find that the decay of the correlation function proceeds faster, the larger the level-repulsion exponent β, and actually much more slowly in the Poissonian case than in any of the three cases corresponding to classical chaos.

5.19 Unfolding Spectra It may be well to digress from random-matrix theory for a moment and to discuss the problem of how to extract a local average level density (and indicators of spectral fluctuations) from a given sequence of measured or calculated levels. That problem was already touched upon in Sects. 2.15, 4.10, and 5.14 and will here coped with more thoroughly. Our present goal is analogous to that of finding a smoothed density of particles (and indicators of density fluctuations like the form “factor”) for a gas or a liquid. Easiest to cope with are spectra (or systems of particles) that appear homogeneous on energy (or length) scales exceeding a certain minimal range ΔEmin : the number ΔN of levels (degenerate ones counted with their degree of degeneracy) per energy interval ΔE yields the average density ¯ = ΔN/(ΔEN) provided ΔE ≥ Emin . More often, however, one encounters “macroscopically” inhomogeneous spectra (analogous to compressible fluids with density gradients) and then faces the problem of having to distinguish between local fluctuations in the level sequence and a systematic global energy dependence of the average density. When defining (E) ¯ =

ΔN 1 ΔE N

(5.19.1)

192

5 Random-Matrix Theory

one must take an energy interval ΔE comprising many levels, ΔN  1, so that local fluctuations do not prevent the ratio ΔN/ΔE from being smooth in E. On the other hand, ΔE must appear extremely tiny with respect to the scale on which (E) ¯ varies systematically. For the obvious conflict to be reasonably resolvable, a self-consistent separation of energy scales must be possible. By requiring that the average density (E) ¯ should vary little over an interval of the order of the local mean level spacing 1/(E)N, ¯ one obtains the self-consistency condition   ¯ (E)

2 N (E) ¯ .

(5.19.2)

In practice, the following procedure has proven viable. One considers the convolution of the density (E) = (1/N) N i = 1 δ(E − Ei ) with the Gaussian g(E) =

1 2 2 √ e−E /Δ , Δ π

N 1  g(E − Ei ) Δ (E) = N

(5.19.3)

(5.19.4)

i =1

and chooses the width Δ by minimizing the mean square deviation between the level staircase σ (E) =

1  Θ(E − Ei ) N

(5.19.5)

i

and the integral of Δ (E),  σΔ (E) =

E −∞

dE Δ (E ) .

(5.19.6)

If an optimal value of Δ can be found in this way, one may make the identification ¯ and identify σΔ (E) with the average level staircase σ¯ (E). Δ (E) = (E) Fortunately, the mean square deviation between σ (E) and σΔ (E) often depends weakly on Δ for Δ in the range of several typical level spacings; in such cases the self-consistency condition (5.19.2) is well obeyed. The concept of level spacing fluctuations (and of other measures of local fluctuations in the spectrum) requires revision when the average density (E) is energy-dependent. For instance, inasmuch as the terms “level clustering” and “level repulsion” are meant to describe local fluctuations, it would be quite inappropriate to describe a spectral region with high ¯ as one with less repulsion than a region with smaller . ¯ Indeed, before it even begins to make sense to compare local fluctuations from two spectral regions with different average densities, one must secure uniformity of the average spacing throughout the spectrum by local changes of the unit of energy.

5.20 Fidelity of Kicked Tops to Random-Matrix Theory

193

In stating, at the end of Sect. 5.6, that the Wigner distributions (5.6.2) hold for the three Gaussian ensembles of random matrices not only for the smallest dimensions possible but also, to a satisfactory degree of accuracy, in the limit N → ∞, it was tacitly assumed that the levels rescale to a uniform average density with the help of Wigner’s semicircle law. A natural way of “unfolding” a spectrum to uniform density can be found as follows: One looks for a function f (E) such that the rescaled levels ei = f (Ei )

(5.19.7)

have unit mean spacing, the mean evaluated with respect to energy intervals [E − ΔE/2 , E + ΔE/2]. The corresponding rescaled interval goes from f (E − ΔE/2) to f (E + ΔE/2). The requirement 1= =

Δe 1 ΔN = ΔN [f (E + ΔE/2) − f (E ΔE ¯ ΔN f (E) = f (E)/(E)N

− ΔE/2)]

(5.19.8)

yields, upon integration, the function f (E) as N times the average level staircase  σ¯ (E) =

E −∞

dE (E ¯ ) .

(5.19.9)

The rescaled levels are thus ei = N σ¯ (Ei ) .

(5.19.10)

The unfolding (5.19.10) is certainly natural, but it is not the only one possible. For many applications, another unfolding, ¯ i) , ei = Ei N (E

(5.19.11)

is convenient. The two rescalings (5.19.10), (5.19.11) are strictly equivalent only in the (uninteresting) case of homogeneous spectra [(E) ¯ = const]; but with respect to spectral regions in which the energy dependence of (E) is sufficiently weak to be negligible, there is practically no difference between (5.19.10) and (5.19.11).

5.20 Fidelity of Kicked Tops to Random-Matrix Theory Periodically kicked tops are unsurpassed in their faithfulness to random-matrix theory, and that fact will be put into view here. As already mentioned several times, tops can be designed so as to be members of any universality class.

194

5 Random-Matrix Theory

The Floquet operators employed for the unitary class read    τy Jy2 τz Jz2 − iαz Jz exp −i − iαy Jy F = exp −i 2j + 1 2j + 1   τx Jx2 × exp −i − iαx Jx . 2j + 1 

(5.20.1)

Indeed, if all torsion constants τi and rotational angles αi are non-zero and of order unity, no time-reversal invariance (nor any geometric symmetry) reigns even approximately and the classical dynamics is globally chaotic, tiny islands of regular motion apart. To secure an antiunitary symmetry as time-reversal invariance while keeping global classical chaos, we just have to erase torsion and rotation with respect to one axis; the resulting Floquet operator then pertains to the orthogonal class, as discussed in Sect. 2.13. Finally, simple symplectic tops are attained by choosing a representation of half-integer j for 8

τ1 Jz2 τ3 (Jx Jy + Jy Jx ) τ2 (Jx Jz + Jz Jx ) F = exp −i −i −i 2j + 1 2(2j + 1) 2(2j + 1)   τ4 Jz2 , × exp −i 2j + 1

9

(5.20.2)

since no geometric symmetries are left while the appearance of only two unitary factors secures an antiunitary symmetry.15 Figure 5.6 shows the spacing distributions P (S) and their integrals I (S) = #S 0 dS P (S ) for tops with the Floquet operators just listed. The graphs pertaining to the unitary (τz = 10, τy = 0, τx = 4, αz = αy = 1, αx = 1.1), orthogonal (τz = 10, αz = 1, αy = 1, all others zero), and Poissonian (τz = 10, αz = 1, all others zero) cases were obtained for j = 1000. The symplectic case is a little more obstinate for large matrix dimensions for which reason the pertinent graph was constructed by averaging over the distributions obtained from all half-integers j between 49.5 and 99.5, keeping the control parameters fixed to τ1 = 10, τ2 = 1, τ3 = 4, τ4 = 2.1. In all cases, agreement with the predictions of random-matrix theory is good. Incidentally, for the databases involved, the Wigner surmises (5.6.2) suffice as representatives of random-matrix theory. The fidelity of the kicked top to RMT as regards the distribution P (S) of nearest-neighbor spacings lends support to the celebrated conjecture of Bohigas, Giannoni, and Schmit [1], as do many more dynamics with classical phase spaces overwhelmingly filled with chaotic motion. Moreover, P (S) as given by a histogram 15 The reader might (and should!) wonder whether the “orthogonal” version of (5.20.1) goes “symplectic” for half-integer j . It does not. The reason is T 2 = +1 for all j in that case since two components of an angular momentum can simultaneously be given real representations.

5.20 Fidelity of Kicked Tops to Random-Matrix Theory

195

P(S)

I(S)

S Fig. 5.6 Spacing distributions P (S) and their integrals I (S) for kicked tops with the Floquet operators (5.20.1) and (5.20.2), with rotational angles and torsion constants as given in the text. Hilbert space dimension 2j + 1 = 2001 for the Poissonian (β = 0), orthogonal (β = 1), and unitary (β = 2) cases; in the symplectic case (β = 4), distributions for all half-integer j between 49.5 and 99.5 were superimposed while keeping the coupling constants τi and thus the classical dynamics unchanged

with all bins richly populated should be a good indicator of universal spectral fluctuations: since a single bin collects level pairs from mutually independent spectral regions one may imagine contributing pairs to come from independent two dimensional Hilbert spaces; the reasoning of Chap. 3 should therefore apply. In other words, P (S) should be a selfaveraging quantity for which the CUE average reliably describes an individual system. When trying to build the two-point cluster function from a set of 2j + 1 quasienergies, one must first worry about how to generate a smooth function from the sum over products of two delta functions provided by the naive definition y(φ, e) = 1 −

 2π 2  N

i=j

δ2π (φ +

  πe πe − φi )δ2π φ − − φj . N N

(5.20.3)

We encountered a similar problem in proving the ergodicity of the two-point cluster function for the CUE in Sect. 5.17.3. Like there, we smoothen the delta functions by subjecting y(φ, e) to both an average over the center phase and an integral over e.

196

5 Random-Matrix Theory

The first of these smoothing operations gives16 

2π

y(e) = 0

 2πe  2π  dφ y(φ, e) = 1 − 2 − φi + φj δ2π 2π N N

(5.20.4)

i=j

and the second produces the first primitive of the cluster function 

e

y (1) (e) = 0

de y(e ) = e −

 1   e2π + φi − φj . Θ N N

(5.20.5)

i=j

For this quantity, Fig. 5.7 reveals fine agreement between tops and the randommatrix predictions for the Poissonian, unitary, orthogonal, and symplectic universality classes. We find y (1) as self-averaging as the spacing distribution, and it should be, for much the same reasons. The small deviations of the integrated correlator y (1)(e) for a single top from the CUE average visible in Fig. 5.7 look like weak noise. That noise persists for larger e, as Fig. 5.8 shows for the first half period, 0 ≤ e ≤ N2 ; in the second half period, N2 ≤ e ≤ N the integrated correlator is given by the symmetry y (1)(e) + y (1)(N − e) = 1 about e = N2 , see (5.17.24). On the scale of that figure, the correlation decay for e ≥ 0 and the mirror symmetric revival at e ≤ N look instantaneous for the chosen dimension, N = 2j + 1 = 19.201. Most importantly, the figure reveals the fluctuations of the top’s y (1)(e) about the CUE mean as homogeneous in e outside the windows of correlation decay and revival. A numerical study [50] of the changeof the fluctuation strength with the dimension N reveals the variance Var y (1)(e) ∝ N1 . In both the size and the homogeneity of the fluctuations of y (1)(e) the universal ergodicity of y (1)(e) manifest itself, as established for the CUE in Sect. 5.17.3. It may be appropriate to display the CUE variance determined there, on an e-interval covering one full period of the integrated correlator, so as to picturewise reveal decay and revival of the autocorrelation of the level density: Remarkably, in the windows of correlation decay and revival the fluctuations of y (1)(e) fall much below the otherwise stationary noise level, see Fig. 5.8; this is due to the existence of universal points at e = 0, N/2, N where all unitary matrices have coinciding values of y (1)(e), see (5.17.25). As an alternative to the second one of the two smoothing operations (going to the integrated correlator, see (5.20.5)) one may work with the correlator y(e) itself and complexify the variable e. The N(N − 1) ∼ N 2 spacings φi − φj = 0 intervening in (5.20.4) all live in the interval [0, 2π) and have a mean difference ∼ 2π/N 2 . 2π The delta peaks in y(e) should therefore be smeared out if Im Ne  N 2 ; on the other hand, that imaginary part must not wash away the systematic dependence on Ree and must therefore be smaller than unity, independent of N. With that, we can expect, on the basis of the fluctuations left √ after smoothing by integration, that y(e) will exhibit fluctuations (rms) of order 1/ N outside the windows of correlation 16 The

reader is invited to check the equivalence of (5.17.21) and (5.20.4).

5.20 Fidelity of Kicked Tops to Random-Matrix Theory

197

y (1) (e)

y (1) (e)

Fig. 5.7 Integrated two-point cluster function y (1) (e) according to (5.20.5) for kicked tops (rugged curves) with Hilbert space dimensions and coupling constants as for Fig. 5.6. The full curves represent the predictions of the pertinent circular ensembles 0.5

N π 2 Var y (1) (e)



1

y (1) (e)

0.5 0.1 0

10 0

0.1

e/N

20

e

0.5

Fig. 5.8 (left) y (1) (e) as in Fig. 5.7 but now N = 19.201. Plot extends over half a period now. Insert shows blown up noise stretch. (Note that the abscissa is now in units of e/N.) (right) CUE variance of y (1) (e) of left part of figure. Plot extends over a little more than the period N = 20

decay and revival; inside those windows the fluctuations will even be smaller and tend to vanish as e → 0 mod ( N2 ). The results of Ref. [50] fully confirm the foregoing reasoning. A glance at Fig. 5.9 suggests more delicacies in the contrast between the form factor |tn |2 of a single top and the mean over the pertinent circular ensemble, here the CUE. As already argued in Sect. 5.17.2, the form factor derived from a single spectrum performs a rather unruly dance about the ensemble mean in its n dependence. No disobedience to random-matrix theory is to be lamented, however,

198

5 Random-Matrix Theory

Fig. 5.9 (a) Time dependence of form factor of unitary kicked top (dots) with Hilbert space dimension and control parameters as for (5.6). The full curve depicts the CUE form factor (5.14.16). In good agreement with random-matrix theory, about 40% (87%) of all dots lie within a band of one (two) standard deviation(s) (5.20.6) around the CUE mean, the respective bands are delimited by the shortdashed (longdashed) lines (b) Histogram for the normalized form factor τ = |tn |2 /|tn |2CUE for unitary kicked top with j = 500 and coupling constants as for (5.6). The full curve depicts the prediction of random-matrix theory based on the Gaussian distribution (5.17.34) of the traces

since that theory itself predicts such fluctuations [48]. The shortdashed lines in the graphs of part (a) of Fig. 5.9 define a band of one standard deviation −1 $  2 2 Δ(n, β) = |tn | |tn |4 − |tn |2

(5.20.6)

around the CUE mean, the latter shown as the full curve. Inasmuch as the underlying matrix dimension is large, the asymptotic Gaussian behavior found in Sect. 5.17.4 1 3 can be expected to prevail. Thus, we may expect the fraction e− 2 − e− 2 ≈ 0.3834 2 of all points |tn | to lie within that band and actually find the fraction 0.3966 for the top. Similarly, the abscissa and the longdashed curve surround a band of two standard deviations which should contain 86.47% of all points and does 87.0%. To further corroborate the obedience of the top to random-matrix theory, we display,  1/2 in part (b) of Fig. 5.9, a histogram for the distribution of |tn |2 / |tn |2CUE which indeed closely follows the full curve provided by the Gaussian distribution of the traces. Figure 5.10 presents a check on the ergodicity of the form factor discussed in Sect. 5.17.2. Part (a) of the figure pertains to the “unitary” top specified above (here with j = 500) and shows a local time average |tn |2 /|tn |2  taken over a time window Δn = 20 ≈ N/50. The fluctuations from one n to the next are much smoothed compared to the non-averaged form factor in Fig. 5.9; they define a band whose width should be compared with the CUE variance Var|tn |2 /|tn |2  estimated in (5.17.16). In fact, the CUE bound 1/Δn of (5.17.16) is respected by the top: Part (b) of Fig. 5.10 shows that bound as the full curve and the numerically found width

5.20 Fidelity of Kicked Tops to Random-Matrix Theory

199

Fig. 5.10 (a) Time dependence of local time average of normalized form factor |tn |2 /|tn |2CUE  according to (5.17.9) for unitary kicked top with j = 500 and coupling constants as for (5.6); the time window is Δn = 20 ≈ (2j + 1)/50 (b) The full curve gives the upper bound 1/Δn of CUE variance of |tn |2 /|tn |2CUE  according to (5.17.16) vs. the time window Δn; the dots depict the standard deviation from the mean of |tn |2 /|tn |2CUE  for the same unitary top as in (a), determined from clouds as in (a)

of the fluctuations of the local time average for various values of the time window Δn as dots. The statistics of secular coefficients of Floquet matrices remain to be scrutinized. A simple indicator is the n dependence of the coefficients an whose mean squared moduli were evaluated in Sect. 5.20. Remarks similar to those above on the n dependence of the traces tn apply here: A single Floquet matrix produces a sequence of secular coefficients fluctuating so wildly that the underlying systematic n dependence becomes visible only after some careful smoothing. As done for the traces above, one could average the |an |2 over some interval Δn or over different dimensions N and plot versus n/N, while keeping rotational angles and torsion constants fixed. Instead, Fig. 5.11 is based on averages over small volume elements of control parameter space, while the dimension N ∝ 1/h¯ remains fixed. All four universality classes are seen to yield n dependences of |an |2 not dissimilar to the averages over the pertinent circular ensemble. In particular, the integrable case has its |an |2 so overwhelmingly larger than any of the other three classes that we may indeed see that difference as one of the quantum indicators of the alternative regular/chaotic. The term “not dissimilar” needs to be qualified: The differences in the circular-ensemble averages are appreciably larger for the secular coefficients than for any of the other quantities shown above. Worse yet, the differences change when the underlying volume element in control parameter space is shifted. The reason for this strong nonuniversality of the secular coefficients can be understood on the basis of the semiclassical periodic-orbit theory to be discussed in Chap. 10. Periodic orbits of period n will turn out to determine the nth trace tn and through Newton’s formulae (5.15.8) the secular coefficients am with all m ≥ n; system specific properties show up predominantly in short periodic orbits with periods not much larger than unity which do not affect the large-m traces tm but can and indeed do affect the large-m coefficients am .

200

an

5 Random-Matrix Theory 2

β=0

an

β=1

2 15

4.1014

10 2.1014

0

5 10

20 n

30

40

2 an 1.5

50 β=2

0

1

0.5

0.5 10

20 n

30

40

50

20 n

30

40

0

50 β=4

2 an 1.5

1

0

10

10

20 n

30

40

50

Fig. 5.11 Mean squared moduli of secular coefficients |an |2 for the circular ensembles (dashes) and for kicked tops (dots) vs n. The averages for the tops are based on 20,000 unprejudiced random points in control parameter space from the intervals τz ∈ [10, 15], αz ∈ [0.6, 1.2], αy ∈ [0.6, 1.2] for the orthogonal top; for the unitary top, αx = 1.1, αy = αz = 1, τy = 0, and τx , τz ∈ [10, 25]; and for the symplectic top, τ1 ∈ [15, 20], τ4 ∈ [10, 15], τ2 = 1, τ3 = 4. All cases involve spectra of 51 eigenvalues

To summarize, no greater faithfulness to random-matrix theory could be expected than that met in tops. The importance of these systems as standard models of both classical and quantum chaotic behavior will be further underscored in Chap. 8 by showing that they comprise, in a certain limit, the prototypical system with localization, the kicked rotator. A rough characterization of that limit at which we shall arrive there is that a top will begin to show quantum localization as soon as an appropriate localization length is steered to values smaller than the dimension 2j + 1.

5.21 Problems 5.1 Show that S in (5.2.16) is indeed a symplectic matrix. 5.2 Show that the most general unitary 2 × 2 matrix can be specified with the help of four real parameters α, ε as U = exp (iα + iε · σ ). Why does a single parameter suffice to diagonalize a real symmetric 2 × 2 matrix? How many are needed for a complex Hermitian 2 × 2 matrix? 5.3 Show the invariance of the differential volume element in matrix space for the GSE. Proceed analogously to the case of the GUE and the GOE, write the GSE

5.21 Problems

201

matrix H for N = 2 as a 6 × 6 matrix, and calculate the Jacobian for symplectic transformations in a way allowing generalization to arbitrary N. 5.4 Present an argument demonstrating that Wigner’s semicircle lawholds for all three of the Gaussian ensembles. 5.5 Calculate the local mean level density for the spin Hamiltonian H = −kj/2 + pJz +kJz2 /2j with −j ≤ Jz ≤ +j and j  1, where k and p are arbitrary coupling constants. Compare with Wigner’s semicircle law. 5.6 Calculate the mean value and the variance of the squared moduli of the components y of the eigenvectors of random matrices for all three ensembles. 5.7 Calculate the ensemble mean for the density of levels defined in (5.11.3) by using the eigenvalue distribution of the CUE. 6

5.8 Calculate the ensemble mean of (1/N) 1i ... N 1j (...=N i) Θ(|φj − φi | − s) using (5.12.4). What meaning does the resulting probability density have? 5.9 Generalize (5.12.6) to P (S) = (∂ 2 E/∂S 2 )/2 for a constant but nonunit mean density of levels . 5.10 Show that a real antisymmetric matrix a of even dimension can be brought to block diagonal form with the blocks (5.13.18) along the diagonal by a real orthogonal transformation L, LL˜ = 1, L = L∗ . Hint: Why are the eigenvalues imaginary and why must they come in pairs ±iai ? What can be said about the two eigenvectors pertaining to a pair ±iai of eigenvalues? Reshuffle the diagonalizing unitary matrix by suitably combining columns so as to make it orthogonal. What can finally be done to enforce det L = 1? Why is det a a perfect square and positive? What changes if the dimension is odd? 5.11 Prove the representation (5.13.20) for a Pfaffian. 5.12 Evaluate the normalization constant N4 as defined in (5.8.10) of the joint density of eigenphases of the CSE. Proceed as done for the COE in Sect. 5.14.4 by keeping track of all proportionality factors, powers of i apart. The result is N4 = π N N! (2N − 1)!! 5.13 Leisure permitting, the reader will want to check the Fourier transforms from the form factor to the cluster function in (5.14.16), (5.14.30), and (5.14.36); only the orthogonal case requires ambition. A more worthwhile exercise, quasi-mandatory for those intending to acquire later a semiclassical understanding of random-matrix type behavior of generic dynamic systems, is to prove the following: (1) If the (2n)th derivative K (2n) (τ ) of an even real function K(τ ) has a delta-function singularity,  K (2n) (τ ) = a δ(τ − τ0 ) + δ(τ + τ0 ) + . . ., that singularity contributes to the #∞ Fourier transform as −∞ dτ K(τ ) cos 2πeτ = (−1)n 2a(2πe)−2n cos 2πeτ0 + . . ..   (2) A logarithmic singularity in K (2n) (τ ) = a ln |τ −τ0 |+ln |τ +τ0 | +. . . goes hand #∞ in hand with −∞ dτ K(τ ) cos 2πeτ = (−1)n+1 a(2π)−2n |e|−(2n+1) cos 2πeτ0 +. . ..

202

5 Random-Matrix Theory

Use this to explain the leading nonoscillatory and oscillatory terms in the cluster functions, checking, in particular, correctness of the coefficients. 5.14 Show that the roots of a self-inversive polynomial are either unimodular or come in pairs such that λi λ∗i = 1. This is an easy task for N = 2, and you may be satisfied with that. 5.15 Evaluate the variance of the time-averaged form factor (5.17.10) for the Poissonian ensemble, and thus show that the inequality (5.17.16) holds here as well. 5.16 Solve the moment–cumulant relations (5.15.4) to express the general cumulant Cn in terms of the moments M1 , . . . Mn and vice versa, after the model of the solutions (5.15.10) of Newton’s relations.

References 1. O. Bohigas, M.J. Giannoni, C. Schmit, Phys. Rev. Lett. 52, 1 (1984); J. de Phys. Lett. 45, 1015 (1984) 2. M.V. Berry, M. Tabor, Proc. R. Soc. Lond. A 356, 375 (1977) 3. S.W. McDonald, A.N. Kaufman, Phys. Rev. Lett. 42, 1189 (1979) 4. G. Casati, F. Valz-Gris, I. Guarneri, Lett. Nouvo Cimento 28, 279 (1980) 5. M.V. Berry, Ann. Phys. (USA) 131, 163 (1981) 6. A. Altland, M.R. Zirnbauer, Phys. Rev. B 55, 1142 (1997) 7. M.L. Mehta, Random Matrices (Academic, New York, 1967); 2nd edition 1991; 3rd edition (Elsevier, 2004) 8. C.E. Porter (ed.), Statistical Theories of Spectra (Academic, New York, 1965) 9. T. Guhr, A. Müller-Groeling, H.A. Weidenmüller, Phys. Rep. 299, 189 (1998) 10. M.A. Stephanov, J.J.M. Verbaarschot, T. Wettig, arXiv:hep-ph/0509286v1 11. E.P. Wigner, Proceedings of the 4th Canadian Mathematical Congress, Toronto, p. 174 (1959) 12. F.J. Dyson, J. Math. Phys. 3(1), 140, 157, 166 (1962) 13. M. Ku´s, J. Mostowski, F. Haake, J. Phys. A: Math. Gen. 21, L1073–1077 (1988) 14. F. Haake, K. Zyczkowski, Phys. Rev. A 42, 1013 (1990) 15. M. Feingold, A. Peres, Phys. Rev. A 34, 591 (1986) 16. B. Mehlig, K. Müller, B. Eckhardt: Phys. Rev. E 59, 5272 (1999) 17. C.E. Porter, R.G. Thomas, Phys. Rev. 104, 483 (1956) 18. T.A. Brody, J. Floris, J.B. French, P.A. Mello, A. Pandey, S.S.M. Wong, Rev. Mod. Phys. 53, 385 (1981) 19. M.V. Berry, J. Phys. A 10, 2083 (1977) 20. V.N. Prigodin, Phys. Rev. Lett. 75, 2392 (1995) 21. H.-J. Stöckmann, Quantum Chaos, An Introduction (Cambridge University Press, Cambridge, 1999) 22. A. Pandey, Ann. Phys. 119, 170–191 (1979) 23. M.L. Mehta, Commun. Math. Phys. 20, 245 (1971) 24. F.J. Dyson, J. Math. Phys. (N.Y.) 3, 166 (1962) 25. F.J. Dyson, Commun. Math. Phys. 19, 235 (1970) 26. B. Dietz, PhD thesis, Essen (1991) 27. B. Dietz, F. Haake, Z. Phys. B 80, 153 (1990) 28. F.J. Dyson, Commun. Math. Phys. 47, 171–183 (1976) 29. B. Riemann, Monatsberichte d. Preuss. Akad. d. Wissensch., Berlin 671 (1859) 30. H.M. Edwards, Riemann’s Zeta Function (Academic, New York, 1974)

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31. H.L. Montgomery, Proc. Sympos. Pure Math. 24, 181 (1973) 32. A.M. Odlyzko, Math. Comput. 48, 273 (1987) 33. M.V. Berry, J.P. Keating, Supersymmetry and Trace Formulae: Chaos and Disorder, ed. by J.P. Keating, I.V. Lerner (Plenum, New York, 1998) 34. P. Bourgade, J.P. Keating, Chaos, ed. by B. Duplantier, S. Nonnenmacher, V. Rivasseau. Progress in Mathematical Physics, vol. 66 (Birkhäuser, Basel, 2013) 35. J. Keating, Random Matrices and Number Theory: Some Recent Themes. Stochastic Processes and Random Matrices: Lecture Notes of the Les Houches Summer School, vol.104, July 2015 (2017) 36. C. Itzykson, J.-B. Zuber, Quantum Field Theory (McGraw-Hill, New York, 1890) 37. F.A. Berezin, Method of Second Quantization (Academic, New York, 1966) 38. F.A. Berezin, Introduction to Superanalysis (Reidel, Dordrecht, 1987) 39. K. Efetov, Supersymmetry in Disorder and Chaos (Cambridge University Press, Cambridge, 1997) 40. I. Newton: Universal Arithmetic (1707); see e.g., W.W. Rouse Ball: A Short Account of the History of Mathematics, 4th edition (1908) or www.maths.tcd.ie/pub/HistMath/People/ Newton/RouseBall/RB_Newton.html 41. A. Mostowski, M. Stark, Introduction to Higher Algebra (Pergamon, Oxford, 1964) 42. M. Marden, Geometry of Polynomials (American Mathematical Society, Providence, 1966) 43. E. Bogomolny, O. Bohigas, P. Leboeuf, Phys. Rev. Lett. 68, 2726 (1992); J. Stat. Phys. 85, 639 (1996) 44. M. Ku´s, F. Haake, B. Eckhardt, Z. Physik B 92, 221 (1993) ˙ 45. F. Haake, M. Ku´s, H.-J. Sommers, H. Schomerus, K. Zyckowski, J. Phys. A: Math. Gen. 29, 3641 (1996) 46. M.V. Berry, J.P. Keating, J. Phys. A 23, 4839 (1990) 47. E. Bogomolny, Comments At. Mo. Phys. 25, 67 (1990); Nonlinearity 5, 805 (1992) 48. F. Haake, H.-J. Sommers, J. Weber, J. Phys. A: Math. Gen. 32, 6903 (1999) 49. H.-J. Sommers, F. Haake, J. Weber, J. Phys. A: Math. Gen. 31, 4395 (1998) 50. P. Braun, F. Haake, J. Phys. A: Math. Gen. 48, 135101 (2015) 51. J.P. Conrey, D.W. Farmer, M.R. Zirnbauer, arXiv:math-ph/0511024v2 (2007) 52. R.E. Hartwig, M.E. Fischer, Arch. Ration. Mech. Anal. 32, 190 (1969) 53. U. Smilansky, Physica D 109, 153 (1997)

Chapter 6

Supersymmetry and Sigma Model for Random Matrices

6.1 Preliminaries As we have seen in Chap. 5, representing determinants as Gaussian integrals over anticommuting alias Grassmann variables makes for great simplifications in computing averages over the underlying matrices. Superanalysis allows to go further, representing (products of) quotients of (secular) determinants of Hamiltonians H or unitary Floquet matrices F as “mixed” Gaussian integrals, ordinary ones for denominators and Grassmannian ones for numerators. Such combinations of secular determinants can be used as ‘generating functions’ for correlators of the densities of eigenvalues of H or F ; their representation by Gaussian superintegrals offers great convenience for performing averages over suitable ensembles of Hermitian or unitary matrices. The aim of this section is to provide a self-contained introduction to the supersymmetry technique. Interested readers will find more extensive treatments in Refs. [1–4]. We shall start with a homeopathic dose and derive Wigner’s semicircle law for the GUE. No superhocus-pocus beyond trading a determinant for a Grassmann integral will be exercised. The subsequent introduction of supervectors, supermatrices, and superdeterminants (in short, superalgebra) and finally of differentiation and integration will appear as a welcome compaction of notation. Nontrivial supersymmetry first arises when we proceed to the two-point cluster function of the GUE and there encounter the celebrated nonlinear supermatrix sigma model. Greater ease will be met with and more care will be taken in establishing the sigma model for Dyson’s circular ensembles. Inasmuch as this introduction to the superanalytic sigma model is concerned with no more than the ‘classic’ Wigner-Dyson ensembles we just recover results already obtained by more elementary methods in Chap. 5. The power of the sigma model will reveal itself when we extend it to non-Gaussian ensembles in Sect. 6.10, ensembles of banded matrices, and even individual dynamics, graphs (Chap. 7) and the kicked rotor (Chap. 8) © Springer Nature Switzerland AG 2018 F. Haake et al., Quantum Signatures of Chaos, Springer Series in Synergetics, https://doi.org/10.1007/978-3-319-97580-1_6

205

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6 Supersymmetry and Sigma Model for Random Matrices

6.2 Semicircle Law for the Gaussian Unitary Ensemble This section may serve as a warm-up which impatient readers might prefer to skip.

6.2.1 The Green Function and Its Average We obtain the density of levels as the trace of the energy-dependent propagator alias Green function G(E) =

N 1 1 1  1 Tr = N E −H N E − Ei

(6.2.1)

i=1

by letting the complex energy variable E approach the real axis from below, E → E − i0+ ≡ E − , and taking the imaginary part, (E) =

1 Im G(E − ); π

(6.2.2)

the well-known identity E1− = P E + iπδ(E) is at work here. In view of the intended GUE average by superintegrals, we must extract G(E) from a quotient of two determinants. The generating function Z(E, j ) =

det (E − H ) det i(E − H ) = det (E − H − j ) det i(E − H − j )

(6.2.3)

helps since it yields  1 ∂Z(E, j )  G(E) = .  N ∂j j =0

(6.2.4)

The reason for sneaking in the factors i in the last member of (6.2.3) will be given momentarily. One proves the identity (6.2.4) most simply by writing out Z(E, j ) in the eigenbasis of H . To familiarize the reader with the machinery to be used abundantly later, it is well to also consider an alternative proof through the following steps:     1 ∂  = ∂ exp{−Tr ln (E − H − j )}   ∂j det (E − H − j ) 0 ∂j 0   1 ∂ exp{−Tr ln (E − H ) + j Tr + O(j 2 )} = ∂j E −H 0

6.2 Semicircle Law for the Gaussian Unitary Ensemble

207

  ∂ 1 1 2  exp{ j Tr + O(j )} = det (E − H ) ∂j E −H 0 =

1 1 Tr . det (E − H ) E − H

(6.2.5)

Now, we represent the determinant in the numerator of the quotient in (6.2.3) by a Gaussian Grassmann integral à la (5.13.1) and the inverse determinant by a Gaussian integral with ordinary commuting integration variables as in (5.13.21),   N d 2 zk ∗ Z(E , j ) = dηk dηk (6.2.6) π k=1 ⎫ ⎧ N  ⎨  ⎬   ∗ ∗ × exp i Hmn + (j − E − )δmn zn + ηm zm (Hmn − E − δmn )ηn . ⎭ ⎩ −

m,n=1

The reader will recall that the kth ordinary double integral goes over the whole complex zk plane while the Grassmann integrals need no limits of integration. A combination of both types of integrals as encountered above for the first time is called a superintegral. At any rate, it can be seen that the previously sneaked in factors i ensure convergence of the ordinary Gaussian integral in (6.2.6) since the real part of −iE − is negative. All is well prepared for the GUE average Z   N   2z  d k ∗ ∗ ∗ (E − − j )zm dηk dηk exp − i Z(E − , j ) = zm + E − ηm ηm π m k=1

  ∗ z + η∗ η ) × exp i Hmn (zm n n m

(6.2.7)

m,n

now that the random matrix elements Hmn appear linearly in the exponential to be averaged. It is worth noting that the coefficient of Hmn in that exponential is a “commuting variable” because it is bilinear in the Grassmannian variables.

6.2.2 An Aside: Complex Conjugation of Grassmann Variables Whether or not one wants to define complex conjugation for Grassmannians is, for the purposes of this book, a matter of taste [5]. If refraining one confines the notions of reality, Hermitian conjugation, and unitarity to the numerical parts of ‘mixed’ numbers and matrices. All calculations

208

6 Supersymmetry and Sigma Model for Random Matrices

below run through without difficulty as an interested reader is invited to check in a second round of study. One may, however, define complex conjugation of the Grassmannian generators ηk and ηk∗ introduced in (5.13.2) by (ηk )∗ = ηk∗ ,

(ηk∗ )∗ = −ηk ,

(ξ η)∗ = ξ ∗ η∗ .

(6.2.8)

With this definition of complex conjugation extended to the whole Grassmann ∗ ∗ ∗ ∗ ∗ ∗ algebra one secures the reality of ηk∗ ηk since (ηk∗ η k ) = (ηk ) ηk = −ηk ηk = ηk ηk . ∗ ∗ Both possibilities allow to say that the sum m,n Hmn (zm zn + ηm ηn ) in the second exponent in is real; for possibility this is because

(6.2.7)

the second

∗ η )∗ = ∗ (−η η∗ ) = ∗η . ( Hmn ηm Hmn Hnm ηn∗ ηm = Hmn ηm n m n n Unless stated otherwise we shall adhere to the notion of complex conjugation of Grassmannians according to (6.2.8) and thus retain the familiar notions of Hermitian conjugation and unitarity for matrices with mixed entries.

6.2.3 The GUE Average We recall from Chap. 5 that the diagonal matrix elements Hmm and the pairs ∗ for 1 ≤ n ≤ m ≤ N are all statistically independent Gaussian Hmn , Hnm = Hmn random variables with zero means; the variances may be written as 2 = |H |2 = Hmm mn

λ2 N

(6.2.9)

where λ is a real number setting

the unit of energy. The linear combination m,n Hmn Mnm with any fixed nonrandom Hermitian matrix M is a real Gaussian random variable with the property ei

Hmn Mnm

1

= e− 2 (

Hmn Mnm )2

λ2

= e− 2N

|Mmn |2

.

(6.2.10)

∗ z + η∗ η , we obtain Applying this to the average in (6.2.7) with Mmn = zm n m n

  ∗ z + η∗ η ) exp i Hmn (zm n m n m,n

 λ2  ∗ ∗ = exp − (zm zn + ηm ηn )(zn∗ zm + ηn∗ ηm ) 2N m,n

(6.2.11)

 λ2     2  ∗ 2 ∗ = exp − |zm |2 − ηm ηm + 2 zm ηm zn∗ ηn . 2N m m m n

6.2 Semicircle Law for the Gaussian Unitary Ensemble

209

The GUE average is done, and with remarkable ease at that, as we may be permitted to say. The expense incurred is the remaining 4N-fold superintegral in (6.2.7) to whose evaluation we now turn.

6.2.4 Doing the Superintegral The most serious obstacle seems to be the non-Gaussian character of the aver∗ , η , η∗ }. But that age (6.2.11) with respect to the integration variables {zm , zm m m impediment can be removed by the so-called Hubbard–Stratonovich transformation of which we need to invoke three variants, 2

e

λ − 2N

e λ2

e− N

λ2 2N



2 m |zm |



∗ m ηm ηm

2 2

∗ ∗ m,n zm ηm zn ηn

7 = 7

N λ2

 

∞ −∞

da − N a 2 +ia m |zm |2 √ e 2λ2 , 2π

N ∞ db − N2 b2 −b m ηm∗ ηm , (6.2.12) √ e 2λ λ2 −∞ 2π 

∗ λ2 − N σ ∗ σ +iσ ∗ m zm ηm +i n zn ηn∗ σ = . dσ ∗ dσ e λ2 N =

The first two of these identities are elementary Gaussian integrals; in particular, ∗η the appearance of Grassmannians is not a problem since the bilinear form ηm m commutes with everything. It will not hurt to explain the third a little. To that end, we ∗ η by checking first convince ourselves of the Grassmann of Σ ≡ zm m

character 2 ∗ ∗ ∗ z∗ is symmetric that its square vanishes: Indeed, Σ = mn zm zn ηm ηn = 0 since zm n and ηm ηn antisymmetric in the summation indices. The left-hand side of the third 2 2 of the above identities may thus be written as exp[− λN Σ ∗ Σ] = 1 − λN Σ ∗ Σ = 1+

λ2 ∗ N ΣΣ ;

but to that very form the right-hand side is also easily brought by  2 # first expanding the integrand and then doing the integrals, r.h.s. = λN dσ ∗ dσ −    2 # 2 N ∗ σ σ − 12 (σ ∗ Σ + Σ ∗ σ )2 = − λN dσ ∗ dσ σ ∗ σ ( λN2 + ΣΣ ∗ ) = 1 + λN ΣΣ ∗ . λ2 Multiplying the three identities in (6.2.12), we get the GUE average (6.2.11) replaced by a fourfold superintegral    dadb − N (a 2 +b2 +2σ ∗ σ ) ∗ z + η∗ η ) = dσ ∗ dσ e 2λ2 exp i Hmn (zm n m n 2π m,n  ∗ ∗ ∗ ia|zm |2 − bηm × exp ηm + iσ ∗ zm ηm + izm ηm σ . (6.2.13) m

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6 Supersymmetry and Sigma Model for Random Matrices

Upon feeding this into the generating function (6.2.7), we may enjoy that the (4N)fold integral over the {zk , zk∗ , ηk , ηk∗ } has become Gaussian. Moreover, all of the N fourfold integrals over zk , zk∗ , ηk , ηk∗ with k = 1, . . . , N are equal such that their product equals the Nth power of one of them,  Z(E − , j ) =  ×

dadb ∗ − N (a 2 +b2 +2σ ∗ σ ) dσ dσ e 2λ2 2π

(6.2.14)

  N d 2z ∗ dη dη exp −i (E − −j − a)|z|2 + (E − −i b) η∗η−σ ∗z∗ η−zη∗ σ . π

To integrate over η∗ and η we proceed in the by now familiar fashion and expand the integrand, 

   dη∗ dη exp −i (E −i b) η∗η − σ ∗z∗ η − zη∗ σ  =  =

(6.2.15)

 dη∗ dη −i (E − i b) η∗ η − 12 z∗ z (σ ∗ ηη∗ σ + η∗ σ σ ∗ η)   dη∗ dη η∗ η −i (E −i b) + z∗ zσ ∗ σ

= i (E −i b) − z∗ zσ ∗ σ . For the remaining double integral over z to exist, we specify the complex energy as slightly below the real energy axis, E = E − , and benefit from the foresight of sneaking the factors i into the last member of (6.2.3); the integral yields 

d 2z [i (E − − i b) − z∗ zσ ∗ σ ] exp[−i(E − − j − a)|z|2] π =

σ ∗σ E − − ib + , (6.2.16) E − − j − a (E − − j − a)2

whereupon the generating function takes the form  Z(E − , j ) =

 dadb ∗ N dσ dσ exp − 2 (a 2 + b 2 + 2σ ∗ σ ) (6.2.17) 2π 2λ  N  N E− − i b σ ∗σ × 1 + . E− − j − a (E − − i b)(E − − j − a)

6.2 Semicircle Law for the Gaussian Unitary Ensemble

211

Once in the mood of devouring integrals, we let those over σ and σ ∗ come next; as usual, we expand the integrand, 

N   N ∗  σ ∗σ dσ dσ exp − 2 σ σ 1 + − λ (E − i b)(E − − j − a)      N Nσ ∗ σ = dσ ∗ dσ 1 − 2 σ ∗ σ 1 + − λ (E − i b)(E − − j − a)   1 1 , =N 2 − − (E − i b)(E − − j − a) λ ∗

(6.2.18)

and are left with an ordinary twofold integral over the commuting variables a, b,    E − − i b N N 2 dadb 2 exp − 2 (a + b ) =N (6.2.19) 2π 2λ E− − j − a   1 1 × 2− − . (E − i b)(E − − j − a) λ 

Z(E − , j )

Differentiating according to (6.2.4), we proceed to the averaged Green function   E − − i b N N 2 dadb 2 exp − 2 (a + b ) = 2π 2λ E− − a

 1 N +1 N × − − . E − a λ2 (E − − i b)(E − − a) 

G(E − )

(6.2.20)

6.2.5 Two Remaining Saddle-Point Integrals Each of the two terms in the curly bracket above yields a double integral which can in fact be seen as a product of separate single integrals; each of the latter can be done by the saddle-point method in the limit of large N. We take on the one over a first and consider  ∞ ) ˆ daA(a)e−Nf (a) ≈ 2π/Nf (a)A( ˆ a)e ˆ −Nf (a) (6.2.21) −∞

where A(a) is independent of N, f (a) = a 2 /2λ2 + ln(E − − a),

(6.2.22)

and aˆ is the highest saddle of the (modulus of the) integrand for which the path of steepest descent (and thus constant phase of the integrand) in the complex a plane

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6 Supersymmetry and Sigma Model for Random Matrices

can be continuously deformed back into the original path of integration without crossing any singularity; should there be several such saddles, degenerate in height, one has to sum their contributions [6]. The saddle-point equation f (a) = 0 is quadratic and has the two solutions a± =

E− 2

  $ 1 ± 1 − ( E2λ− )2

(6.2.23)

for which the square root is assigned a branch cut in the complex energy plane along the real axis from −2λ to +2λ. (1) ( E2 )2 < λ2 : The small imaginary part of E − requires, for E > 0, evaluating the square root on the lower lip of the cut such that upon dropping ImE − , we have the saddle points a± = E/2 ∓ i{λ2 − (E/2)2 }1/2 . They are mutual complex conjugates and lie in the complex a plane on the circle of radius λ around the origin. The integrand in (6.2.20) vanishes for a → ±∞ on the real axis and has no singularities at finite a except for a pole at a = E − , right below the real axis. The original path of integration along the real axis can be deformed so as to climb along a path of constant phase of the integrand over the saddle at a− , without crossing the said pole; a path of constant phase from −∞ to +∞ over the saddle at a+ also exists but cannot be deformed back to the original path without crossing the pole. Thus, only the saddle at a− contributes.  (2) ( E2 )2 > λ2 : Expanding the square root for  → 0+ we have a± = (E/2) 1 ± ) )    1 − (2λ/E)2 −i(/2) 1±1/ 1 − (2λ/E)2 . Again, the saddle point a+ lies below the real axis while a− lies above such that only the latter can contribute. Thus, for all cases we conclude that 

∞ −∞

. daA(a)e−Nf (a) ≈

2π A(a− )e−Nf (a− ) , Nf (a− )

(6.2.24)

where the phase of the square root in the denominator is defined by the steepest descent direction of integration across the saddle point. A little surprise is waiting for us in the integrals over b in (6.2.20), 

∞ −∞

dbB(b)e−Ng(b)

with

g(b) = b2 /2λ2 − ln(E − − ib) .

(6.2.25)

Now b = −iE is a zero of the integrand rather than)the location of a singularity. Saddles are encountered for ib± = a± = (E − /2) 1± 1 − (2λ/E − )2 . The saddle points b± are obtained by a clockwise (π/2)-rotation of the a± about the origin in the complex a plane. Two cases again arise: (1) (E/2)2 < λ2 : The saddles lie symmetrically to the imaginary axis of the ) complex a-plane at b± = −iE/2 ∓ λ2 − (E/2)2 . They are degenerate in height since Reg(b+ ) = Reg(b− ). Since the integrand has no singularity at any

6.2 Semicircle Law for the Gaussian Unitary Ensemble

213

finite point b, the path of integration can be deformed so as to climb uphill from −∞ to b+ , then descend to the zero at −iE, climb again to b− , and finally descend toward +∞. That deformed path may be chosen as one of constant phase, except at the zero of the integrand where no phase can be defined, such that, the phase jumps upon passing through the exceptional point. 2 (2) (E/2)2 > saddle points are purely imaginary, b± =   λ :)In that case both −i(E/2) 1 ± 1 − (2λ/E)2 , and b+ is further away from the origin than b− ; see Fig. 6.1. The saddle at b+ is higher than that at b− since Reg(b+ ) < Reg(b− ), and one might thus naively expect only b+ to be relevant. But surprisingly, b+ does not enter at all, as the following closer inspection reveals. The phase of the integrand, i.e., Img(x + iy) = xy/λ2 + arctan[x/(E + y)], vanishes along the imaginary axis. That path of the vanishing phase passes through both saddles and the zero of the integrand; along it the integrand has a minimum at b− and a maximum at b+ . Another path of constant vanishing phase goes through each saddle; along the one through b− , the modulus of the integrand has a maximum at b− ; that latter path extends to ±∞ + i0 and makes b− the only contributing saddle. No excursion from b− to b+ can be steered toward +∞ + i0 without climbing higher after reaching b+ , except for the inconsequential excursion returning to b− along the imaginary axis. To

Fig. 6.1 Contour plot of the exponent −g(b) from (12.2.25). The saddles at b± are marked as ⊕ and &. Height is indicated in shades of grey, √ ranging from black (low) to white (high). Left for case (1), i.e., (E/2)2 < λ2 , with E = 1, λ = 1/ 2. The two saddles are of equal height. The path of integration climbs up the valley from the left through b− and descends to the zero of exp(−Ng(b)) on the imaginary axis, then again uphill to √b+ , and therefrom √ down the valley toward +∞. Right for case (2), i.e., (E/2)2 > λ2 , with E = 2 + 0.1, λ = 1/ 2. We recognize that the saddle at b+ lies higher than that at b− and also see the logarithmic “hole” below b+ . As described in the text the path of integration climbs up the valley from the left toward the saddle at b− and descends on the other side into the valley to the right

214

6 Supersymmetry and Sigma Model for Random Matrices

summarize the discussion of the b-integral, we note that



∞ −∞

dbB(b)e−Ng(b) ≈

⎧ ⎪ ⎪

⎪ ⎪ ⎪ ⎨

ˆ + ,b− b=b

⎪ ⎪ ⎪ $ ⎪ ⎪ ⎩

$

ˆ 2π ˆ −Ng(b) B(b)e ˆ Ng (b)

2π −Ng(b− ) Ng (b− ) B(b− )e

for ( E2 )2 < λ2 for ( E2 )2 > λ2 . (6.2.26)

Combining the a- and b-integrals, we first treat the case labelled (2) above, i.e., (E/2)2 > λ2 . Since only the saddles at a− and b− contribute, we get G(E − )

 1 N +1 N ≈ − − 2π(E − − a− ) λ2 (E − a− )(E − − ib− ) .    (2π)2 exp −N f (a− ) + g(b− ) . × 2 N f (a− )g (b− )

(6.2.27)

However, due to a− = ib− we have f (a− ) + g(b− ) = 0 such that the exponential factor equals unity. Similarly, f (a− ) = g (b− ) = λ−2 − (E − −a− )−2 whereupon  −1/2 {. . .} = 1 + O(1/N). the curly bracket above cancels as N −1 f (a− )g (b− ) But since corrections of relative order 1/N are already made ) in the saddle-point +−1 *  . approximation, G(E − ) = (E − − a− )−1 ≈ (E −/2) 1 + 1 − (2λ/E − )2 *  − Setting  = 0, we finally get the purely real result G(E ) ≈ (E/2) 1 + ) +−1 1 − (2λ/E)2 . Thus, the average density vanishes, (E) ≈ 0 for (E/2)2 2 >λ . In the hope of a finite mean density of levels, we finally turn to case (1), (E/2)2 < 2 λ . According to the above findings, we confront contributions from the two pairs of saddles {a− , b− } and {a− , b+ }. The first pair is dealt with in full analogy with the discussion of ) the corresponding pair {a− , b− } under case (2) and furnishes G(E − )  −1 with E/2−i λ2 − (E/2)2 . The pair a− , b+ is incapable of providing anything of asymptotic weight: The argument is replaced with   2 of 2the exponential2in (6.2.27)  −N f (a− ) + g(b+ ) = −N a− /(2λ ) + ln a+ − a+ /(2λ2 ) − ln a− which is ∗ , such that the exponential remains unimodular purely imaginary due to a− = a+ however large N. Collecting all other to the model + * + *N-dependent factors according of (6.2.27), we get N −1 . . . = N −1 N/λ2 − (N + 1)/(a+ a− ) = −1/Nλ2 , i.e., an overall weight of order 1/N. Indeed then, only the pair {a− , b− } contributes in the limit N → ∞, just as in case (2). It remains to collect the limiting form of the averaged Green function,

lim

N→∞

G(E − )

=

⎧ ⎨

√1 (E/2)(1+ 1−(2λ/E)2 ⎩ √1 E/2−i λ2 −(E/2)2

for

( E2 )2 > λ2

for

( E2 )2 < λ2 ,

(6.2.28)

6.3 Superalgebra

215

and, after taking the imaginary part, Wigner’s semicircle law, lim (E) =

N→∞

0 1 πλ2

)

λ2 − (E/2)2

for

( E2 )2 > λ2

for

( E2 )2 < λ2 .

(6.2.29)

Needless to say, the edges of the semicircular level density at E = ±2λ are rounded off at finite N, as may be studied by scrutinizing next-to-leading-order corrections. The importance of the semicircle law is somewhat indirect: It must be scaled out of the energy axis before the predictions of random-matrix theory for spectral fluctuations take on their universal form to which generic dynamical systems are so amazingly faithful.

6.3 Superalgebra 6.3.1 Motivation and Generators of Grassmann Algebras A compaction of our notation is indicated before we can generalize the above discussion of the mean density of levels (E) to two-point correlation functions like (E)(E ). We need to generalize some basic notions of algebra to accommodate both commuting and anticommuting variables as components of vectors and elements of matrices. We had previously (see Sect. 5.13) introduced Grassmann variables η1 , . . . , ηN with the properties ηi ηj + ηj ηi = 0 and shall stick to these as the “generators of a 2N -dimensional Grassmann algebra”. Whenever convenient, we shall, as already done in the previous section, complement these N generators by N more such, ∗ , so as to have a 22N -dimensional Grassmann algebra. It is sometimes η1∗ , . . . , ηN convenient (although never necessary) to define complex conjugation as (ηi )∗ = ηi∗ and (ηi∗ )∗ = −ηi .

6.3.2 Supervectors, Supermatrices We shall have to handle (2M)-component vectors with M commuting components S1 , . . . , SM and M anticommuting components χ1 , . . . , χM ,   S Φ= , χ

⎛

⎞ S1 ⎜ ⎟ S = ⎝ ... ⎠ , SM

⎛

⎞ χ1 ⎜ ⎟ χ = ⎝ ... ⎠ . χM

(6.3.1)

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6 Supersymmetry and Sigma Model for Random Matrices

Quite intentionally, we have denoted the components χi with a Greek letter differing from the one reserved for the generators ηi since it will

be necessary

to allow the χi to be odd functions of the generators like, e.g., χi = j cij ηj + j kl cij kl ηj ηk ηl . . . with commuting coefficients c. Similarly, the commuting character of the Si does not prevent the appearance of η’s but restricts the Si to be even in the η’s, like Si =

zi + j k zij k ηj ηk +. . .. If a verbal distinction between commuting numbers without and with additions even in the η’s is indicated we shall call the former numerical; an addition even in the Grassmannians is a nilpotent quantity1; the coefficients c and z in the above characterizations of anticommuting χ’s and commuting S’s are meant to be numerical. In somewhat loose but intuitive jargon, we shall sometimes refer to commuting variables as Bosonic and anticommuting ones as Fermionic. At any rate, a vector of the structure (6.3.1) is called a supervector. If the need ever arises, one can generalize the definition of a supervector such as to allow for the number of elements of the Bosonic and Fermionic subvectors to be different, MB for S and MF for χ. Moreover, MF need not coincide with the number N of generators of the Grassmann algebra. Unless explicitly stated otherwise, we shall in the following take MB = MF = M. Supermatrices are employed to relate supervectors linearly. We introduce such 2M × 2M matrices as  F =

 aσ , ρ b

 FΦ =

    aσ S aS + σ χ = . ρ b χ ρS + bχ

(6.3.2)

For F Φ to have the same structure as Φ, where the first M components are Bosonic and the last M ones are Fermionic, of the four M × M blocks in F , the matrices a and b must have Bosonic elements while σ and ρ must have Fermionic elements. It is customary to speak of a as the Bose–Bose block or simply the Bose block, of b as the Fermi–Fermi or simply the Fermi block, of σ as the Bose–Fermi block, and of ρ as the Fermi–Bose block. For products of supermatrices, the usual rules of matrix multiplication must hold. (If MB = MF , a is MB × MB , b is MF × MF , while the two blocks with Fermionic elements are rectangular matrices, σ an MB × MF one and ρ an MF × MB one.) We need to define transposition for supermatrices. Since it is desirable to retain the usual rule (F1 F2 )T = F2T F1T , we must modify the familiar rule for transposing purely Bosonic matrices to 

F

T

T  aσ a˜ = = ρ b −σ˜

 ρ˜ ; b˜

(6.3.3)

As done previously in this book, we reserve the tilde to denote the familiar matrix transposition and shall use the superscript T for transposition of supermatrices that

1 A nilpotent quantity has vanishing powers with integer exponents upward of some smallest positive one.

6.3 Superalgebra

217

includes the unfamiliar minus sign in the Fermi–Bose block. A further price for retaining the usual rule for transposition of products is that double transposition is not the identity operation,  (F T )T =

 a −σ = σ3BF F σ3BF = F , −ρ b

(6.3.4)

where σ3BF is the diagonal supermatrix that is equal to the identity matrix in the Boson-Boson block and minus the identity in the Fermi-Fermi block. Our convention for the transpose also implies      aσ S2 = Φ2T F T σ3BF Φ1 (Φ1 , F Φ2 ) ≡ Φ1T F Φ2 ≡ S˜1 χ˜ 1 χ2 ρ b

(6.3.5)

where we have naturally extended the convention of the transpose to the supervectors Φ1/2 . A useful extension of the notion of a trace to supermatrices is the supertrace StrF = Tra − Trb =

M 

(amm − bmm ) .

(6.3.6)

m=1

The minus sign in front of the trace of the Fermi block is beneficial by bringing about the usual cyclic invariance for the supertrace, StrF1 F2 = StrF2 F1 ,

(6.3.7)

since StrF1 F2 = Tr(a1 a2 + σ1 ρ2 − ρ1 σ2 − b1 b2 ) = Tr(a1 a2 + σ1 ρ2 + σ2 ρ1 − b1b2 ) is obviously symmetric in the labels 1 and 2. It follows that StrMF M −1 = StrF where M is another supermatrix and M −1 is its inverse. A most welcome consequence of the cyclic invariance of the supertrace is the following identity for the logarithm of a product, Str ln F G = Str ln F + Str ln G

(6.3.8)

whose proof parallels that of the corresponding identity for ordinary matrices. Sketching that proof provides an opportunity to recall the definition of the logarithm as a series, ∞    (−1)n ln F = ln 1 + (F − 1) = − (F − 1)n , n

(6.3.9)

n=1

up to analytic continuation for F outside the range of convergence. To display unity as the reference matrix about which the expansion goes we momentarily introduce

218

6 Supersymmetry and Sigma Model for Random Matrices

F = 1 + f, G = 1 + F −1 g such that F G = 1 + f + g. Thus, the identity to be proven may be written as Str ln(1 + f + xg) = Str ln(1 + f ) + Str ln[1 + x(1 + f )−1 g] ?

(6.3.10)

where x is a real parameter which can obviously be sneaked in without gain or loss of generality, but with profit in convenience. Expanding the x-dependent logarithms about the unit matrix, we transform our identity to −

∞  (−1)n n=1

n

?

Str(f + xg)n = Str ln(1 + f ) −

∞  (−1)n n=1

n

Str



n x g . 1+f

(6.3.11)

When differentiating with respect to x, we benefit from the cyclic invariance of the trace and rid ourselves of the factors (1/n) in the nth terms of the expansions, such that the expansions take the forms of geometric series, 1 1 1 ? g= g. 1 + f + xg 1 + (1 + f )−1 xg 1 + f

(6.3.12)

But now the question mark can be dropped since the two sides do equal one another due to (ab)−1 = b−1 a −1 . We can also wave good-bye to the question mark in (6.3.10); the functions of x on the two sides have coinciding derivatives for all x and take the same value for x = 0. Needless to say, our proof of (6.3.8) applies to traces of logarithms of products of ordinary matrices as well.

6.3.3 Superdeterminants The definition of a superdeterminant SdetF to be given presently is meant to preserve three familiar properties of usual determinants: (1) a supermatrix F and its transpose F T should have the same superdeterminant, SdetF = SdetF T , (2) the superdeterminant of a product should be the product of the superdeterminants of the factor supermatrices, SdetF1 F2 = SdetF1 SdetF2 ; and (3) for a supermatrix, the logarithm of its superdeterminant should equal the supertrace of its logarithm, ln SdetF = Str ln F . We shall find these requirements met when the superdeterminant is defined as the ratio of usual determinants   aσ SdetF = Sdet if det b0 = 0 (6.3.13) = det(a − σ b−1 ρ)/ det b ρb

6.3 Superalgebra

219

where b0 denotes the numerical part of the Fermi block b. If det b0 = 0, the superdeterminant of F does not exist. Its inverse may, however, be given a meaning, (SdetF )−1 = det(b − ρa −1 σ )/ det a

if

det a0 = 0 ,

(6.3.14)

provided the numerical part a0 of the Bose block is nonsingular. If both a0 and b0 are nonsingular, the compatibility of the two definitions must be proven. No work is required to that end if F is block diagonal with σ = ρ = 0, so that the superdeterminant is the ratio of the determinants of the Bose and Fermi blocks, SdetF =

det a det b

for

σ = ρ = 0,

(6.3.15)

a case of some importance for applications. The compatibility proof for general F is a good opportunity to shape up for less trivial adventures to follow later. Under the assumption mentioned, we may rewrite (6.3.13) as SdetF = det(1 − a −1 σ b−1 ρ) det a/ det b and (6.3.14) as (SdetF )−1 = det(1 − b−1 ρa −1 σ ) det b/ det a such that we need to show the equality of the two usual N × N determinants det(1 − a −1 σ b−1 ρ) and −1  or, equivalently, the equality of Tr ln(1 − a −1σ b−1 ρ) with det(1 − b−1ρa −1 σ ) −Tr ln(1 − b−1 ρa −1 σ ). To that end, we expand both logarithms and need to verify that Tr(a −1 σ b−1 ρ)n = −Tr(b−1 ρa −1 σ )n .

(6.3.16)

But this can be checked by rewriting the l.h.s. in two steps. The first relies on the cyclic invariance of the usual trace, l.h.s. = Trσ [b−1 ρ(a −1 σ b−1 ρ)n−1 a −1 ]. No less easy even though a little unfamiliar is the second step. Again under the protection of the trace operation, we want to change the order of the factors σ and [. . .]; but since both factor matrices are now Fermionic, a minus sign results, Trσ [. . .] = −Tr[. . .]σ , simply because the matrix elements of σ and [. . .] anticommute; the Fermionic character of [. . .] follows from the fact that the latter matrix involves an odd number of Fermionic factors ρ and σ ; with the minus sign generated by shifting σ to the right, we have indeed arrived at the equality (6.3.16) and thus at the compatibility of (6.3.13) and (6.3.14). The proof of the equality of the superdeterminants of F and F T , SdetF = SdetF T

(6.3.17)

is a nice aside left to the reader. We shall, however, pause to show that indeed ln SdetF = Str ln F .

(6.3.18)

220

6 Supersymmetry and Sigma Model for Random Matrices

To that end, we write the general supermatrix F as a product, 

    aσ a0 1 a −1 σ = , ρ b 0 b b−1 ρ 1

F =

(6.3.19)

and use (6.3.8) for the supertrace of its logarithm,     a0 1 a −1 σ . Str ln F = Str ln + Str ln −1 b ρ 1 0b

(6.3.20)

The first of the supertraces appearing on the r.h.s. is simply evaluated as     det a a0 ln a 0 , Str ln = Str = Tr ln a − Tr lnb = ln 0b 0 ln b det b

(6.3.21)

whereas for the second, we must invoke once more the definition of the logarithm as a series; momentarily abbreviating as ν = a −1 σ, μ = b−1 ρ,  Str ln

 n   ∞ (−1)n−1 1ν 0ν Str . = μ1 μ0 n n=1

But only even-order terms of the series have nonvanishing supertraces such that   ∞ ∞   1 1  (νμ)n 0 n n Str = − − Tr(μν) =− Tr(νμ) 0 (μν)n 2n 2n n=1

=−

∞  n=1

n=1

1 Tr(νμ)n = Tr ln(1 − νμ) = ln det(1 − νμ) . n

Putting (6.3.21) and (6.3.22) into (6.3.20), we get Str ln F  σ b−1 ρ)/ det b = ln SdetF , the desired result. Finally, the product rule Sdet(F1 F2 ) = (SdetF1 ) (SdetF2 )

(6.3.22)  = ln det(a −

(6.3.23)

follows from the above as ln SdetF1 F2 = Str ln F1 F2 = Str ln F1 + Str ln F2 = ln SdetF1 + ln SdetF2 = ln SdetF1 SdetF2 . We kindly invite the reader to put to use the superknowledge thus far acquired to interpret and prove the following useful formula A B  Sdet = sdet(A) sdet(D − CA−1 B) . CD

(6.3.24)

6.3 Superalgebra

221

6.3.4 Complex Scalar Product, Hermitian and Unitary Supermatrices As we have already seen when deriving the semicircle law in the previous section, it is sometimes desirable to associate complex conjugates with Grassmann variables. Our present goal of setting up a suitable superalgebraic framework means that we need to define complex conjugation of supervectors. Well then, we shall work with Φ=

  S , χ

Φ∗ =

 ∗ S , χ∗

(Φ ∗ )∗ =



 S , −χ

(6.3.25)

whereby we simply double the number of generators of the Grassmann algebra formed by the Fermionic components to 2MF . As already done in Sect. 6.2, the definition of complex conjugation is completed by requiring that (χi∗ )∗ = −χi , (χi χj )∗ = χi∗ χj∗ , and (χi + χj )∗ = χi∗ + χj∗ . It follows that χi∗ χi is real. However, χi + χi∗ is not real since its complex conjugate is −χi + χi∗ . We shall employ the scalar product (Φ1 , Φ2 ) =

M 

∗ ∗ (S1i S2i + χ1i χ2i )

(6.3.26)

i=1

which enjoys the familiar property (Φ1 , Φ2 )∗ = (Φ2 , Φ1 ). The Hermitian conjugate or “adjoint” of a supermatrix F is obtained by taking both the complex conjugate and the transpose, F † = (F T )∗ = (F ∗ )T ;

(6.3.27)

this is useful because of (F Φ1 , Φ2 ) = (Φ1 , F † Φ2 ). We should note that in contrast to twofold transposition, twofold Hermitian conjugation is the identity operation, 

F

††

 ††  † ∗†  aσ a ρ˜ a −σ ∗∗ = = = =F. −σ˜ ∗ b† −ρ ∗∗ b ρ b

(6.3.28)

A Hermitian supermatrix can be defined as usual as a supermatrix equalling its own adjoint; it has the structure  H = H† =

 a ρ† ρ b

where

a † = a, b† = b .

(6.3.29)

Unitary supermatrices preserve the scalar product of supervectors, (U Φ1 , U Φ2 ) = (Φ1 , Φ2 ) ,

U U † = U †U = 1 .

(6.3.30)

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6 Supersymmetry and Sigma Model for Random Matrices

6.3.5 Diagonalizing Supermatrices Special attention is due to Hermitian supermatrices. The eigenvalue problem H Φ = hΦ yields coupled homogeneous equations for the Bosonic and Fermionic components S and χ or, after eliminating of one of them, 1 ρ)S = 0 b−h

(6.3.31)

1 ρ † )χ = 0 . a−h

(6.3.32)

(a − h − ρ † (b − h − ρ

The special structure of these equations forbids the appearance of too many eigenvalues: the first makes sense only if the numerical part of b − h has a nonvanishing determinant and thus provides MB eigenvalues for the Bose block; similarly, the second requires det(a − h)num = 0 and gives MF eigenvalues for the Fermi block. To reveal this structure explicitly it is well to consider the simplest case MB = MF = 1 for which (6.3.31) may be rewritten as the quadratic equation h2 − (a + b)h + ab − ρ ∗ρ = 0; one of the two solutions, hBghost = b − ρ ∗ ρ/(a − b), must be discarded since hBghost − b has a vanishing numerical part, i.e., is nilpotent. Similarly, (6.3.32) has the spurious solution hFghost = a + ρρ ∗ /(a − b) such that we are left with one eigenvalue each for the Bose and Fermi blocks, hB = a + ρ ∗ ρ/(a − b) ,

hF = b + ρ ∗ ρ/(a − b) , for MB = MF = 1 . (6.3.33)

No eigenvalue exists for the supermatrix under scrutiny if a = b. An eigenvalue is called positive if its numerical part is positive. Momentarily staying with MB = MF = 1, we may wonder which unitary supermatrix diagonalizes H ; it is H = U diag(hB , hF )U −1 with U=

  η 1 + ηη∗/2 , 1 − ηη∗/2 η∗

U −1 =

  1 + ηη∗/2 −η −η∗ 1 − ηη∗/2

(6.3.34)

where η = −ρ ∗ /(a − b), η∗ = +ρ/(a − b), and U −1 = U † . Obviously again, for ρ ∗ = 0, nilpotency of a − b would preclude diagonalizability. Our expressions for the eigenvalues hB and hF , the matrices U and U −1 , and for η and η∗ remain valid when b is continued to arbitrary complex values; they are in fact most often used for imaginary b. Even though H then ceases to be Hermitian and U to be unitary, U still diagonalizes H . However, unitarity of U can be restored by redefining complex conjugation for Grassmannians such that η and η∗ remain complex conjugates with b an arbitrary complex number (See Problems 6.5 and 6.6).

6.4 Superintegrals

223

6.4 Superintegrals 6.4.1 Some Bookkeeping for Ordinary Gaussian Integrals To properly build up the notion of superintegrals as integrals over both commuting and anticommuting variables, it is well to recall the familiar ordinary Gaussian integral   M d 2 Si i=1

π

M    exp − Si∗ aij Sj = i,j =1

1 , det a

(6.4.1)

where the range of integration is the full complex plane for each of the M complex integration variables Si and the M × M matrix a must have a positive real part for convergence. The differential d 2 Si is meant as dReSi dImSi . For maximal convenience in changing integration variables, it is sometimes advantageous to think of the differential volume element as an ordered antisymmetric product, the so-called wedge product,2 d 2 Si = dReSi ∧ dImSi = −dImSi ∧ dReSi = (2i)−1 dSi∗ ∧ dSi = −(2i)−1 dSi ∧ dSi∗ ; the variables Si and Si∗ , as well as their differentials dSi and dSi∗ , may then, with some caution, be regarded as independent such that the complex Si -plane is tiled with area elements dSi∗ ∧ dSi = −dSi ∧ dSi∗ . One bonus of this bookkeeping device lies in the possibility of changing integration variables in the sense of independent analytic continuation in both Si and Si∗ , so as to give up the kinship between Si and Si∗ as mutual complex conjugates. The simplest such change shifts one, but not the other, of these variables by a constant complex number. Thus, e.g., 

d 2 S e−S

∗ (S+c)

=π.

(6.4.2)

For any change of the integration variables, the artifice of the wedge product readily yields the familiar transformation of the differential volume element, M i=1

d 2 Si = det

M M  S∗, S  ∂(S ∗ , S) - 2 d z = J d 2 zi . i ∂(z∗ , z) z∗ , z i=1

(6.4.3)

i=1

2 Readers not previously familiar with the wedge product of differentials of commuting variables have every right to be momentarily confused. They have the choice of either spending a quiet hour with, e.g., Ref. [7], or simply ignoring the following remark and convincing themselves of the correctness of (6.4.2) in some other way, most simply by expanding the integrand in powers of c.

224

6 Supersymmetry and Sigma Model for Random Matrices

In particular, for a linear transformation which keeps the Si∗ unchanged but reshuffles the Si ,    S 1 . (6.4.4) aij Sj ⇒ J zi = = z det a j

In fact, (6.4.1) can be seen as an application of (6.4.4): With the latter transformation introduced in the Gaussian integral, we read Si∗ and zi as complex conjugates and  # # M  ∗ M get (2πi)−1 dS ∗ ∧ dz exp(−S ∗ z) = π −1 d 2 Se−S S = 1. With these remarks in mind, we may change the integration variables in (6.4.1)

as Si → Si − j (a −1 )ij zj , Si∗ → Si∗ − j zj∗ (a −1 )j i to get   M M M     d 2 Si exp − Si∗ aij Sj exp (Si∗ zi + zi∗ Si ) (det a) π i,j =1

i=1

≡



i=1

M M    ∗ ∗ (Si zi + zi Si ) = exp zi∗ (a −1 )ij zj . exp

(6.4.5)

i,j =1

i=1

The angular brackets (. . .) in the foregoing identity suggest a change of perspective, hopefully not too upsetting to the reader3; they are a shorthand for the average of (. . .) w.r.t. the normalized Gaussian distribution appearing in the first member. In fact, we may read the quantity in (6.4.5) as the moment generating function for that Gaussian. A nice application is Wick’s theorem, which we find by differentiating w.r.t. the auxiliary variables {zi , zi∗ } and subsequently setting these to zero; the simplest cases read Si Sj∗  = (a −1 )ij Si Sj Sk∗ Sl∗ 

=

(6.4.6)

Si Sk∗ Sj Sl∗  + Si Sl∗ Sj Sk∗ 

= (a −1 )ik (a −1 )j l + (a −1 )il (a −1 )j k , and the general case is offered to the interested reader as Problem 6.7.

6.4.2 Recalling Grassmann Integrals By now, we have often used the Grassmann analogue (5.13.1) of (6.4.1),   M  1...M   ∗ dχn dχn exp − χi∗ bij χj = det b , n=1

3 Truly

(6.4.7)

ij

daring readers might even enjoy contemplating the kinship of (6.4.5) as well as its superanalytic generalizations below (see (6.4.9), (6.4.16)) with the Hubbard–Stratonovich transformation.

6.4 Superintegrals

225

which holds for any M × M matrix b and, like all Grassmann integrals, does not require boundaries to become definite. To tune in to what is to follow, we propose once more to prove that integral representation of det b, in a way differing from the

original one of Sect. 5.13. Changing integration variables as ρi = j bij χj while holding on to the χ ∗ ’s and employing the Jacobian (5.13.17), J

  χ = det b ρ

-

⇒

dχi∗ dχi = det b

i

-

dχi∗ dρi ,

(6.4.8)

i

we see the Gaussian integral (6.4.7) yielding the determinant det b, up to the M  # M # dχ ∗ dρ(1 − χ ∗ ρ) = 1. inconsequential factor dχ ∗ dρ exp(−χ ∗ ρ) = change of integration variables in (6.4.7), χi → χi −

The−1fruit of another ∗ → χ ∗ − η∗ (b −1 ) with constant Grassmannians η , η∗ , (b ) η , χ ij j ji i i j j j i i is worthy of respectful consideration, (det b)

−1

 M

 dχn∗ dχn

1...M  1...M   exp − χi∗ bij χj + (χi∗ ηi + ηi∗ χi )

n=1

≡

 exp

M 

ij

(χi∗ ηi + ηi∗ χi )

i=1



= exp

i

 1...M 

ηi∗ (b−1 )ij ηj .

(6.4.9)

ij

We confront the Grassmannian analogue of the moment generating function (6.4.5) that invites pushing the analogy to ordinary Gaussian integrals toward “means” like χi χj∗ . By differentiating (6.4.9) w.r.t. to some of the η’s and η∗ ’s and setting all of these “auxiliary” parameters to zero thereafter, we get χi  = χi∗  = 0 ,

χi χj∗  = (b−1 )ij =

∂ ln det b ; ∂bj i

(6.4.10)

the latter identity is, of course, an old friend from determinantology (see, e.g., (8.4.5)) and may show that the “probabilistic” perspective in which we are indulging here does have useful implications, its admittedly frivolous appearance notwithstanding. Upon taking more derivatives, we encounter the Grassmannian version of Wick’s theorem, e.g., χi χj χk∗ χl∗  = −χi χk∗ χj χl∗  + χi χl∗ χj χk∗  .

(6.4.11)

It is well to note that the different powers of det a and det b in the Jacobians (6.4.4) and (6.4.8) and thus in the Gaussian integrals (6.4.1) and (6.4.7) determine the starting point of superanalysis in random-matrix theory.

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6 Supersymmetry and Sigma Model for Random Matrices

6.4.3 Gaussian Superintegrals The simplest Gaussian superintegral is just the product of the Gaussian integral representations for det b and 1/ det a,   −1 det b a0 = Sdet (6.4.12) 0b det a   M    d 2 Si ∗ = dηi dηi exp − Si∗ aij Sj + ηi∗ bij ηj , π i=1

ij

which we hurry to compact with the help of Φ = 

S  η

and F =

a 0 0b

as

dΦ ∗ dΦ exp(−Φ † F Φ) = (SdetF )−1 .

(6.4.13)

We have already used this in Sect. 6.2 for the generating function Z in (6.2.3) and (6.2.6). There, as again below, we profit from having lifted the matrices a, b to an exponent which makes subsequent averages over these matrices easy to implement, provided that their elements are independent Gaussian random numbers. We should not forget the existence condition a + a † > 0. Once and for all, we have introduced here the differential volume element for the integration over a supervector Φ, dΦ ∗ dΦ =

M d 2 Si i=1

π

dηi∗ dηi =

M dS ∗ ∧ dSi i

i=1

2πi

dηi∗ dηi .

(6.4.14)

If ever needed, one may allow for the numbers MB and MF of Bosonic and Fermionic components of Φ to be different. We need to talk about transformations of integration variables. As long as we stick to separate transformations for Bosonic and Fermionic variables (i.e., forbid mixing of the two types) and further restrict ourselves to linear reshufflings of the Fermionic variables, we just assemble the respective Jacobians (6.4.4) and (6.4.8) as J

 S ∗ , S, η∗ , η  z∗ , z, χ ∗ , χ

= det

 ∂(S ∗ , S) ∂(χ ∗ , χ)  ∂(z∗ , z) ∂(η∗ , η)

= Sdet

∂(S ∗ , S, η∗ , η) . ∂(z∗ , z, χ ∗ , χ)

(6.4.15)

The final form of the Jacobian, often called the Berezinian, also holds for transformations mixing Bosonic and Fermionic variables; of course, only such transformations are to be admitted which lead to MB new Bosonic and MF Fermionic variables; in particular, the new Bosonic variables may have additive pieces even in the old Fermionic ones. For a proof of the general validity of the last expression for the Berezinian, we refer the reader to Berezin’s book [8].

6.4 Superintegrals

227

The compact version (6.4.13) of our superintegral is actually more general than the original, (6.4.12), inasmuch as it does not require the supermatrix F to be block diagonal with vanishing Bose–Fermi and Fermi–Bose blocks. To see this, we simply replace the integration variables Φ by Ψ = F the  ΦΦ while leaving = (SdetF )−1 ; in Φ ∗ unchanged. The Berezinian of that transformation is J Ψ # the remaining integral dΦ ∗ dΨ exp(−Φ † Ψ ) we may, in the sense of analytic continuation already alluded to in the previous subsections, reidentify Ψ with the complex conjugate of Φ, whereupon the integral takes the form (6.4.12) with F = 1 and is therefore equal to unity. Of course, the Bose–Bose block FBB of F must obey † FBB + FBB > 0 for the integral (6.4.13) to exist. It is both interesting and useful to extend Wick’s theorem to superanalysis, i.e., to “moments” of a “normalized Gaussian distribution” (Sdet F ) exp(−Φ † F Φ). In analogy with our above reasoning for the Bosonic and Fermionic momentgenerating functions (6.4.5) and (6.4.9), we avail ourselves of a superanalytic one by −1 ) Ψ , Φ ∗ → changing integration variables in (6.4.13) as Φi → Φi − 2M ij j j =1 (F i

2M ∗ ∗ −1 Φi − j =1 Ψj (F )j i to get  (SdetF )

dΦ ∗ dΦ exp(−Φ † F Φ) exp (Φ † Ψ + Ψ † Φ)

≡ exp (Φ † Ψ + Ψ † Φ) = exp(Ψ † F −1 Ψ ) .

(6.4.16)

By differentiating w.r.t. the auxiliary variables {Ψi , Ψi∗ }, we obtain the superanalytic generalization of (6.4.6) and (6.4.11), Φi Φj∗ Φk Φl∗  = Φi Φj∗ Φk Φl∗  + (−1)? Φi Φl∗ Φk Φj∗  = (F −1 )ij (F −1 )kl + (−1)? (F −1 )il (F −1 )kj ;

(6.4.17)

the unspecified sign factor is −1 if an odd number of commutations of Fermionic variables is involved in establishing the order of indices ilkj from ij kl and +1 otherwise. Needless to say, that identity encompasses the purely Bosonic (6.4.6) and the purely Fermionic (6.4.11) as special cases.

6.4.4 Some Properties of General Superintegrals One more issue comes up with changes of integration variables in integrals like 

 I=

d MB S

d MF χf (S, χ) ,

(6.4.18)

R

where d MB S = dSMB . . . dS2 dS1 , d MF χ = dχMF . . . dχ2 dχ1 and where the MB Bosonic integration variables Si , as well as their range of integration R, are

228

6 Supersymmetry and Sigma Model for Random Matrices

purely numerical, i.e., contain no even nilpotent admixtures. According to (5.13.7) and (5.13.8), I is well defined provided that the ordinary integral remaining converges, once the Grassmannian one is done. We may, however, think of a change of the integration variables which brings in new Bosonic variables with nilpotent admixtures even in the Grassmannians χi , formally S = S(z, χ). The question arises as to what the original integration range R turns into and what is to be understood by the new Bosonic part of the differential volume element. Berezin realized that consistency with the previously established rule (6.4.15) of changing integration variables requires us simply to ignore the nilpotent admixtures of the zi everywhere in the integrand, in d MB S, as well as in the new integration range, provided that the function f (S, χ) and all of its derivatives vanish on the boundary of the original range R. Referring the interested reader to [9] for the general case, we confine ourselves here to sketching the consistency proof for MB = 1, MF = 2. Then, the integral (6.4.18) takes the form 



b

I=

dS

dχ2 dχ1 f (S, χ)

(6.4.19)

a

where S, a, b are purely numerical. The transformation S = z + Z(z)χ1 χ2

(6.4.20)

has the Jacobian dS/dz = 1 + Z (z)χ1 χ2 . Accepting Berezin’s rule that the new boundaries should still be a, b, we need to find the conditions for the integral I˜ =





b

dz

dχ2 dχ1

a

dS f (z + Z(z)χ1 χ2 , χ) dz

(6.4.21)

to equal I in (6.4.19). Expanding the integrand, we get f (z + Z(z)χ1 χ2 , χ) = (z,0) f (z, χ) + ∂f ∂z Z(z)χ1 χ2 , and thus I˜ =





b

dz a



=I+

dχ2 dχ1 b

dz a

∂f (z, 0) dS  f (z, χ) + Z(z)χ1 χ2 dz ∂z

∂f (z, 0)Z(z) ∂z

= I + f (b, 0)Z(b) − f (a, 0)Z(a) . Obviously, now, I˜ = I provided f = 0 at the boundaries S = a, b. More caution is indicated when the integrand does not vanish at the boundaries of the range R, as the following example shows. Let the integrand f (S, χ) in (6.4.19) be g(S + χ1 χ2 ) = g(S) + g (S)χ1 χ2 . The integral comes out as I = g(b) − g(a). If one insists in shifting the Bosonic integration variable to z = S + χ1 χ2 , one should

6.4 Superintegrals

229

rewrite the integral by introducing unit step functions,  dS dχ2 dχ1 g(S + χ1 χ2 )Θ(b − S)Θ(S − a)



∞



−∞ ∞



−∞ ∞

I = = =

 dz dχ2 dχ1 g(z)Θ(b − z + χ1 χ2 )Θ(z − a − χ1 χ2 )     dz dχ2 dχ1 g(z) Θ(b−z) + δ(b−z)χ1 χ2 Θ(z−a) − δ(z−a)χ1χ2

−∞

= g(b) − g(a) .

(6.4.22)

Fortunately, we shall be concerned mostly with integrands that vanish, together with all of their derivatives, at the boundary of the integration range R and in that benign case, one enjoys the further rules 

 d



MB

S

R

d MF χ

∂ (fg) = 0 , ∂Si

d MF χ

∂ (fg) = 0 , ∂χi

 d MB S

R

(6.4.23)

which are useful for integration by parts.

6.4.5 Integrals over Supermatrices, Parisi–Sourlas–Efetov–Wegner Theorem We shall have ample opportunity to deal with integrals over manifolds of nonHermitian supermatrices  Q=

 a ρ∗ ρ ib

(6.4.24)

with real ordinary numbers a, b and a pair of Grassmannians ρ, ρ ∗ . In particular, for integrands of the form f (StrQ, StrQ2 ) that vanish for a → ∞, b → ∞, we shall need the identity  I =

dQ f (Str Q, Str Q2 ) = f (0, 0)

where

dQ =

dadb ∗ dρ dρ . 2π (6.4.25)

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6 Supersymmetry and Sigma Model for Random Matrices

This is mostly referred to as the Parisi–Sourlas–Efetov–Wegner (PSEW) theorem, in acknowledgment of its historical emergence4 [1, 10–13]. An important special case is    1 2 dQ exp − Str Q = 1 , (6.4.26) 2 and the correctness of that latter identity is obvious from StrQ2 = a 2 + b2 + 2ρ ∗ ρ. # An immediate generalization is dQ exp (−c Str Q2 ) = 1 with an arbitrary positive parameter c. To prove the general case (6.4.25), we need to recall from Sect. 6.3.5 that Q can be diagonalized as Q = U diag(qB , iqF )U −1 with U given in (6.3.34), but η = −ρ ∗ /(a − ib), η∗ = ρ/(a − ib). The eigenvalues read qB = a +

ρ∗ρ , a − ib

iqF = ib +

ρ∗ρ . a − ib

(6.4.27)

To do the superintegral in (6.4.25), it is convenient to change integration variables from a, b, ρ, ρ ∗ to qB , qF and the Grassmannian angles η, η∗ . Evaluating the Berezinian for that transformation, we get the measure dQ =

dqB dqF dη∗ dη . 2π(qB − iqF )2

(6.4.28)

The integral in search thus takes the form  I =

dqB dqF dη∗ dηf (qB − iqF , qB2 + qF2 ) 2π(qB − iqF )2

(6.4.29)

which is discomfortingly undefined: the Bosonic integral diverges due to the pole in the measure while the Grassmannian integral vanishes. The following trick [13, 14] helps to give a meaning to the unspeakable 0 · ∞. One sneaks the factor exp (−cStr Q2 ) into the integrand and considers the c-dependence of the integral  I (c) =

dQ f (Str Q, Str Q2 ) exp (−c Str Q2 ) 

=

(6.4.30)

dqB dqF dη∗ dηf (qB − iqF , qB2 + qF2 ) exp {−c (qB2 + qF2 )} 2π(qB − iqF )2

which taken at face value, is as undefined for any value of c as for c = 0. However, the situation improves for the derivative dI (c)/dc if we differentiate under the integral. Due to qB2 + qF2 = (qB + iqF )(qB − iqF ), the pole of the integrand at the origin of the (qB − qF )-plane now cancels such that the still vanishing Grassmann 4 Useful generalizations, in particular to higher dimensions, and a watertight proof can be found in Ref. [13].

6.4 Superintegrals

231

integral enforces dI (c)/dc = 0. We conclude that I (c) itself is independent of c and may thus be evaluated√at, say, c → ∞. The latter limit is accessible after the transformation Q → Q/ 2c. Since the Berezinian of that latter transformation equals unity, 

√ 1 dQ f [Str Q/ 2c, Str Q2 /(2c)] exp (− Str Q2 ) c→∞ 2  1 = f (0, 0) dQ exp (− Str Q2 ) = f (0, 0) . 2

I = lim

(6.4.31)

6.4.6 Asymptotic Analysis of SuSy Integrals: Massive and Zero Modes In the following we will often consider integrals of the form Z=

 M d 2 zk dηk∗ dηk −S(z∗ ,z,η∗ ,η;s) e π

(6.4.32)

k=1

S(z∗ , z, η∗ , η; s)

with a commuting function of its arguments. There are 2M real commuting and 2M anti-commuting integration variables. For each k = 1, . . . , M we refer to (zk∗ , zk , ηk , ηk∗ ) as one mode such that M gives the number of distinct modes. The argument s = (s1 , . . . , sK ) refers to K additional real parameters (in our contexts often called source variables) and the integral is normalized such that Z = 1 for s = 0. In all cases of interest here a variant of the Parisi-Sourlas-EfetovWegner theorem applies at s = 0 such that I (0) = e−S0 where S0 = S(0, 0, 0, 0; 0). The normalization then implies S0 = 0. For s = 0 the Parisi-Sourlas-Efetov-Wegner theorem generally does not apply. One sometimes refers to this fact by saying that the parameters s ‘break the supersymmetry’ of the integral. In our setting the function Z will be a generating function and the interesting information for us is contained in some derivative of Z with respect to the parameters s. If the number M of modes is small one may often perform the integral directly. We will however encounter large M or even M → ∞. As an instructive toy case we now consider the function ∗

∗

∗

S(z , z , η , η; s) =

M 

Sk (zk∗ , zk , ηk∗ , ηk ; s)

(6.4.33)

k=1

such that the integral is a product of M independent integrals Z=

M k=1

 Zk

with

Zk =

d 2 zk dηk∗ dηk −Sk (z∗ ,zk ,η∗ ,ηk ;s) k k e . π

(6.4.34)

232

6 Supersymmetry and Sigma Model for Random Matrices

As already indicated in the above formulas a single SuSy breaking parameter (K = 1) will be sufficient for the present discussion.5 Before we continue with a choice of the functions Sk let us discuss the appropriate variant of the Parisi-Sourlas-Efetov-Wegner theorem. Assume that F (x) is a real function (with existing derivative F (x)) of the real variable x with the # ∞ first −F (x) properties F (0) = 0 and 0 e dx = C < ∞ (i.e. the integral exists and Fk (x) → ∞ as x → ∞). It is then straightforward to show 

d 2 zdη∗ dη −F (|z|2 +η∗ η) e = π



 d 2 zdη∗ dη −F (|z|2 )  e 1 − F (|z|2 )η∗ η π



∞

=

e−F (x)F (x)dx = e−F (0) = 1 .

0

(6.4.35) We now go back, motivate a specific choice for Sk and consider the asymptotic regime s → 0, M → ∞. In all relevant cases that double limit may be written in terms of a single large parameter N by setting s = /N and M ∝ N d with d ≥ 0 (d = 0 implies that the number of modes remains finite). We pick   Sk (zk∗ , zk , ηk∗ , ηk ; s = /N) =(N + 2) log 1 + |zk |2 + η∗ η   − N log 1 + (1 − mk )(|zk |2 + e−/N η∗ η) (6.4.36) where mk is a non-negative parameter that we will call the mass of the k-th mode. The corresponding integral becomes  N d 2 zk dηk∗ dηk 1 + (1 − mk )(|z|2 + e−/N η∗ η) π (1 + |z|2 + η∗ η)N+2  ∞  (1 + (1 − mk )R)N =1 + (1 − mk )N 1 − e−/N dR (1 + R)N+2 0 

Zk () =

(6.4.37)

The remaining integration is with respect to R ≡ |zk |2 . As N → ∞ the integral gives the asymptotic result 

∞ 0

(1 + (1 − mk )R)N dR = (1 + R)N+2

5 Later



∞

dRe−Nmk R (1 + O(1/N)) =

0

we will usually encounter the case of two parameters, K = 2.

1 + O(N −2 ) Nmk (6.4.38)

6.4 Superintegrals

233

such that Zk () =

(1 − mk ) + O(N −2 ) . Nmk

(6.4.39)

This result can obviously not be extended to zero-modes, i.e., modes with vanishing mass mk = 0. In that case the integral reduces to   Zk () = 1 + N e/N − 1 0

∞

dR

  1 /N = 1 + N e − 1 = 1 +  + O(1/N) . (1 + R)2

(6.4.40) We see that the zero-mode responds with a term of order one to breaking the supersymmetry while the massive modes respond with of order 1/N  terms  in our M d Zk  dZ  example. If we are interested in the derivative d  = then k=1 d  =0 =0 this expression is dominated by the zero-modes. Individual massive modes are suppressed by an order 1/N. If M remains finite as N → ∞ then one may usually neglect the massive modes altogether. However, massive modes need to be retained in the crossover regime where a zero-mode gains a mass by changing some coupling parameter g. In this case there is at least one mass, say mj = mj (g) depends continuously on the coupling parameter g, and we may assume that mj (0) = 0 while mj (g) > 0 for g > 0. As long as mj (g) 1/N one may not neglect the corresponding mode and the asymptotic analysis above does not apply. One speaks of a soft mode. When mj (g)  1/N the mode is no longer soft and may eventually be neglected. If the number of modes M ∝ N d grows (d > 0), one needs to be more careful with neglecting massive modes for two reasons. The first is that an increasing number of modes and corresponding masses may introduce (many) soft modes that cannot be neglected (and the above asymptotic analysis does not apply). The second is that (even if there are no soft modes at all) the number of massive modes may grow so fast that any contribution from zero modes is dominated by the collective k) contribution k:mk =0 (1−m Nmk . The asymptotic analysis that leads to this collective contribution remains intact. This collective contribution may dominate, for instance, if d > 1 such that M/N = N d−1 → ∞ if there is only a finite number of zero modes and the massive modes are bounded from below mk > const > 0 (and obey mk = 1). What can we learn from this toy example for more general cases? We will encounter modes that are coupled, modes that have a more complicated structure (e.g. an individual mode may be represented by a supermatrix or a supervector of higher dimension), and the explicit form of the function we used above may be different. In these cases one may often identify zero-modes by a saddle-point

234

6 Supersymmetry and Sigma Model for Random Matrices

analysis of the origin. In our case S(z∗ , z, η ∗ , η; s = /N) =

(6.4.41)

M  (Nmk + 2)(|zk |2 + η∗ η) + (1 − mk )η∗ η + O( 2 , |zk |4 , |zk |2 η∗ η) k=1

such that a saddle-point approximation is suitable for Nmk  0. Indeed the result we stated for massive modes is equivalent to a saddle-point approximation. If Nmk is not large (zero-modes and soft modes) then the saddle-point approximation is not valid and we need to integrate this mode in more detail (as we have done for the zero-modes). In the general case one may always introduce modes (by a suitable linear transformation) such that the quadratic terms are diagonal S(z∗ , z, η∗ , η; s = /N) =

M 

Nμk (|zk |2 + ηk∗ ηk ) + λk η∗ η + . . . .

(6.4.42)

k=1

Note that the scaling of each mode is ambiguous as replacing (zk∗ , zk , ηk∗ , ηk ) → (czk∗ , czk , cηk∗ , cηk ) does not change the integral. What is relevant is the ratio μk /λk which one may identify as the relevant mass terms such that μk /λk  1/N are massive modes that may be calculated in a saddle-point approximation.6 In many cases the collective leading saddle-point contributions are negligible and one may just reduce the integral to the zero modes and soft modes. Note however that a rigorous justification for neglecting massive modes is usually very hard. Indeed collective contributions beyond the leading Gaussian approximation become increasingly complicated as the modes are coupled among each other leading to an often uncontrollable increase of terms.

6.5 The Semicircle Law Revisited To illustrate the conciseness of the superanalytic formulation, we propose to resketch the derivation of the semicircle law using the newly established language. The average generating function (6.2.7) can be written as the superintegral  Z(E − , j ) =

  dΦ ∗ dΦ exp − iΦ † F Φ = Sdet−1 F ,

F = Eˆ ⊗ 1N − 1ˆ ⊗ H − Jˆ ⊗ 1N

(6.5.1)

our example case μk /λk = (mk + 2/N)/(1 − mk ) which gives the same condition for massive modes as we have stated in terms of mk .

6 In

6.5 The Semicircle Law Revisited

235

  6 d 2 Si ∗ over the 2N-component supervector Φ = Sη with dΦ ∗ dΦ = N i=1 π dηi dηi . The three summands in the 2N × 2N matrix F have been written as direct products of 2 × 2 and N × N matrices, a hat distinguishes the former,   − 1 0 ˆ E=E = E − 1ˆ , 01

  j 0 ˆ J = . 00

(6.5.2)

The GUE average (6.2.11) reads   Nλ2   1 ˜2 , Str Q exp{i Φ † 1ˆ ⊗ H Φ} = exp − (Φ † 1ˆ ⊗ H Φ)2 = exp − 2 2  

∗ ∗ 1 z z z η

m m m

m m m (6.5.3) Q˜ = ∗ ∗ η z η N m m m m m ηm and yields  Z(E − , j ) =

  λ2 ˜ 2  . dΦ ∗ dΦ exp − NStr i(Eˆ − Jˆ)Q˜ + Q 2

(6.5.4)

This demands a Hubbard-Stratonovich transformation for us to come back to a Gaussian superintegral over the supervector Φ. The resulting representation (6.2.14) of the average generating function can be seen as a fourfold superintegral over the elements of a 2 × 2 supermatrix  ∗ a σ Qˆ = , σ ib

Str Qˆ 2 = a 2 + b 2 + 2σ ∗ σ ,

ˆ = dQ

dadb ∗ dσ dσ , 2π

(6.5.5)

and takes the form  Z(E − , j ) =  =  =

ˆ e dQ

−

N Str Qˆ 2 2λ2



dφ ∗ dφ e−iφ

† (E− ˆ Q− ˆ Jˆ)φ

N

ˆ2

ˆ e− 2λ2 Str Q Sdet−N (Eˆ − Qˆ − Jˆ) dQ N

(6.5.6)

   ˆ exp − NStr 1 Qˆ 2 + ln(Eˆ − Qˆ − Jˆ) . dQ 2λ2

Here the original 4N-fold superintegral over the components of the supervector Φ and their conjugates could be replaced  by  the Nth power of a fourfold integral over the two-component supervector φ = ηz . While in Sect. 6.2 we proceeded by doing the two Grassmann integrals over the skew elements of Qˆ rigorously and reserved a saddle-point approximation for the integral over the diagonal elements, here we shall treat all variables equally and

236

6 Supersymmetry and Sigma Model for Random Matrices

propose a saddle-point approximation for the whole superintegral  Z(E − , j ) =

ˆ Jˆ) ˆ e−N A(Q, dQ ,

ˆ Jˆ) = Str A(Q,

 1 Qˆ 2 + ln (Eˆ − Qˆ − Jˆ) . 2 2λ

(6.5.7)

Before embarking on that adventure, it is well to pause by remarking that the foregoing integral must equal unity if we set Jˆ = 0,  Z(E − , 0) =

ˆ

d Qˆ e−N A(Q,0) = 1,

(6.5.8)

since unity is indeed the value of a block diagonal superdeterminant whose Bose and Fermi blocks are identical, c.f. (6.2.3) or (6.4.12). This is relevant since due to (6.2.4) we eventually want to set j and thus the matrix Jˆ equal to zero,  G(E − ) =

 ˆ 1 J ˆ ˆ Str e−N A(Q,0) . dQ j Eˆ − Qˆ 

(6.5.9)

Now, the spirit of the saddle-point approximation, we may appeal to the assumed largeness of N and pull the preexponential factor out of the integral and evaluate it at ˆ Then, the remaining superintegral the saddle-point value Qˆ 0 of the supermatrix Q. yields unity, and we are left with 

1 Jˆ G(E − ) = Str j Eˆ − Qˆ 0

 .

(6.5.10)

To find Qˆ 0 , we must solve the saddle-point equation 

ˆ 1 ˆ 0) = Str δ Qˆ Q − δA(Q, ˆ ˆ λ2 E−Q

 =0

⇒

1 Qˆ 0 = . ˆ λ2 E − Qˆ 0

(6.5.11)

ˆ 0 diagonalizable, we may start looking for diagonal solutions; there is Assuming Q indeed no undue loss of generality in that since all other solutions are accessible from diagonal ones by unitary transformations. For diagonal saddles, however, the saddle-point equation becomes an ordinary quadratic equation and yields two solutions (c.f. (6.2.23)) $   Q± = (E − /2) 1 ± 1 − (2λ/E − )2 .

(6.5.12)

Precisely as in Sect. 6.2 we must evaluate the square root in Q± = a± = ib± on the lower lip of the cut along the real axis from −2λ to +2λ in the complex energy

6.5 The Semicircle Law Revisited

237

plane. Concentrating immediately on (E/2)2 < λ2 , i.e., case (1) of Sect. 6.2.5 ) Q± = a± = ib± = E/2 ∓ i λ2 − (E/2)2 . Four possibilities arise for the diagonal 2 × 2 matrix in search,   Q+ 0 , 0 Q+

  Q+ 0 , 0 Q−

  Q− 0 , 0 Q+

  Q− 0 . 0 Q−

(6.5.13)

According to (6.5.10), (6.5.11), each of the first two would contribute Q+ /λ2 , and the last two Q− /λ2 to the average Green function. Therefore, the first two must be discarded right away since the imaginary part of Im G(E− ) = π(E) must be positive. This corresponds with our discarding the saddle a+ in Sect. 6.2.5, arguing that the original contour of integration could not be continuously deformed so as to pass through that saddle without crossing a singularity. The question remains whether both of the last two contribute or only one of them and then which. To get the answer, we scrutinize the Gaussian superintegral over the fluctuations around the ˆ 0) of the action, i.e., the second-order term in saddle. The second variation δ 2 A(Q, ˆ ˆ ˆ ˆ δ Q of A(Q0 + δ Q, 0) − A(Q0 , 0), reads 1 δ A(Qˆ 0 , 0) = Str 2 2

8

ˆ 2 (δ Q) (δ Qˆ Qˆ 0 )2 − λ2 λ4

9 (6.5.14)

and must be evaluated for the two candidates for Qˆ 0 . Denoting by δ 2 A−− the result for diag(Q− , Q− ) and by δ 2 A−+ that for diag(Q− , Q+ ) (for simplicity at E = 0), 1 ˆ 2 = 1 (δa 2 + δb2 + δσ ∗ δσ ) , Str (δ Q) 2 λ λ2 1 = 2 (δa 2 + δb2 ) . λ

δ 2 A−− = δ 2 A−+

(6.5.15)

Obviously now, the second saddle does not contribute at all (to leading order in N) # since dδ Qˆ exp(−Nδ 2 A+− ) = 0, such that we arrive at G(E − ) =

$  1 − 2 − (E − /2)2 E /2 + i λ λ2

for

(E/2)2 < λ2 ,

(6.5.16)

the result already established by a less outlandish calculation in Sect. 6.2; see (6.2.28). The reader is invited to go through the analogous reasoning for (E/2)2 > λ2 . At any rate, the semicircle law follows by taking the imaginary part. Incidentally, it is quite remarkable that the nonscalar saddle diag(Q− , Q+ ) does not contribute since it would come with a whole manifold of companions: Indeed, a ˆ † with any Qˆ solving the saddle-point equation gives rise to further solutions U QU † ˆ ˆ unitary transformation U , provided that Q = U QU , i.e., provided that Qˆ is not proportional to the unit matrix. As a final remark on the saddle diag (Q− , Q+ ), we mention that it corresponds to the saddle {a− , b+ } of Sect. 6.2.5 which there, too,

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6 Supersymmetry and Sigma Model for Random Matrices

did not contribute in leading order, due to the vanishing of the preexponential factor at the saddle. Should the previously uninitiated suffer from a little dizziness due to our use of the saddle-point approximation for a superintegral, cure might come from Problem 6.9.

6.6 The Two-Point Function of the Gaussian Unitary Ensemble We proceed to calculate Dyson’s two-point cluster function for the Gaussian unitary ensemble and show that it equals its counterpart for the circular unitary ensemble (see Sect. 5.14),  YGUE (e) = YCUE (e) =

sin πe πe

2 .

(6.6.1)

As in Sect. 6.2, we shall start with the Green function G(E) = N −1 Tr(E − H )−1 but now must consider the GUE average of the product of two such, G(E1 )G(E2 ). We shall see (in the beginning of the next Subsection) that the average behaves differently, depending on whether the two complex energies lie on the same side or on different sides of the real axis. In the former case, the average product tends to the product of averages, N→∞

G(E1 + iδ)G(E2 + iδ) −→ G(E1 + iδ) G(E2 + iδ) ,

(6.6.2)

while in the latter case G(E1 + iδ)G(E2 − iδ) tends, apart from normalization, to a function of the single variable e = (E1 − E2 )N

E + E  1 2 , 2

(6.6.3)

i.e., the energy difference measured in units of the mean level spacing at the center energy. Anticipating the validity of the factorization (6.6.2) and employing (6.2.2), we can immediately check the connected density-density correlator,    Δ(E1 )Δ(E2 ) = (E1 ) − (E1 ) (E2 ) − (E2 ) , to be given by the real part of the connected version ΔG(E1 − i0+ )ΔG(E2 + i0+ ) of the averaged product of two Green functions, Δ(E1 )Δ(E2 ) =

1 Re ΔG(E1 − i0+ )ΔG(E2 + i0+ ) . 2π 2

(6.6.4)

6.6 The Two-Point Function of the Gaussian Unitary Ensemble

239

6.6.1 Generating Function The product of two Green functions can be obtained from the generating function Z(E1− , E2+ , E3 , E4 ) =

det(E3 − H ) det(E4 − H ) det(E1− − H ) det(E2+ − H )

= (−1)N

det i(E3 − H ) det i(E4 − H ) det i(E1− − H ) det(−i)(E2+ − H )

(6.6.5)

by differentiating as G(E1− )G(E2+ ) =

1 ∂ 2 Z(E1− , E2+ , E3 , E4 )  .  E1 =E3 ,E2 =E4 N2 ∂E1 ∂E2

(6.6.6)

This is analogous to (6.2.3) and (6.2.4), and the proof parallels that in (6.2.5): Each differentiation brings about one factor G and the determinants cancel pairwise once we set E1 = E3 , E2 = E4 . We note the symmetry of the generating function under the exchange E3 ↔ E4 which we shall refer to as Weyl symmetry. In the end, we shall be interested in (the limit of) real energy arguments. For the moment it is imperative, however, to place the energy arguments E1− , E2+ of the spectral determinants in the denominator away from and on different sides of the energy axis. Such precaution is not necessary for E3 , E4 which may be arbitrary complex. The reader is invited to check that E + and E − respectively pertain to the retarded and advanced Green function. Silly as it may appear to have sneaked various factors ±i in the last member of (6.6.5) (since they obviously cancel against the factor (−1)N ), such precaution pays, similarly as in Sect. 6.2, once we employ the Gaussian superintegral (6.4.12) to represent quotients of determinants, Z=(−1)

 ×

 N

N

i=1

   d 2 z1i ∗ dη1i dη1i exp i z1† (H −E1− )z1 + η1† (H −E3 )η1 π

N d 2 z2i i=1

π

 ∗ dη2i dη2i

  exp i − z2† (H −E2+ )z2 + η2† (H −E4)η2 , (6.6.7)

with Bosonic vectors z1 , z2 and Fermionic vectors η1 , η2 , all having N components. Indeed, the i’s now conspire to ensure convergence of the Bosonic Gaussian integrals, just as in (6.2.6).

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6 Supersymmetry and Sigma Model for Random Matrices

To save space, we introduce the (4N)-component supervector ⎛ ⎞ ⎛ ⎞ z1 Φ1 ⎜z2 ⎟ ⎜Φ2⎟ ⎟ ⎜ ⎟ Φ=⎜ ⎝η1⎠ = ⎝Φ3⎠ η2

“Bose-Fermi notation”

(6.6.8)

Φ4

such that the differential volume element dΦ ∗ dΦ comprises the one in the preceding integral. Writing the energy variables with their infinitesimal imaginary offsets as Eα + i(−1)α 0+ , α = 1, 2, 3, 4 and employing the symbols Lα where L1 = L3 = L4 = −L2 = 1 to accommodate the funny looking convergence ensurer L2 = −1 in the exponent, we can write the generating function in the compact form  Z = (−1)

N

 4 dΦ dΦ exp i ∗

Φ † Lα α=1 α

   α + H − Eα − i(−1) 0 Φα .

(6.6.9)

All is set now for the average over the GUE; the superhocus-pocus has served to move the random Hamiltonian matrix H upstairs into an exponent. Recalling the variances (6.2.9) and the Gaussian average of exponentials (6.2.10), in the spirit of (6.2.11) we have  2 1  † = exp − Φα Lα H Φα α=1 2 9 8 2 1 † † † † (6.6.10) = exp − z1 H z1 − z2 H z2 + η1 H η1 + η2 H η2 2

  4 exp i





Φα† Lα H Φα

   λ2  ∗ ∗ = exp − Lα Lβ Φαm Φαn Φβn Φβm . 2N m,n α,β

Once at work down-sizing bulky expressions, why not continue with 4 × 4 matrices L = diag(1, −1, 1, 1) ,

Eˆ = diag(E1− , E2+ , E3 , E4 ) ,

N  ∗ ˜ αβ = 1 Q Φαm Φβm N

(6.6.11)

m=1

and write the averaged generating function analogously to (6.5.4) as   2   ˜ 2 . ˆ = (−1)N dΦ ∗ dΦ exp NStr − iEˆ QL ˜ − λ (QL) Z(E) 2

(6.6.12)

The supermatrix Q˜ = Q˜ † is Hermitian and has a non-negative Bose–Bose block.

6.6 The Two-Point Function of the Gaussian Unitary Ensemble

241

It remains to do the 8N-fold superintegral in (6.6.12) which, unfortunately, contains a quartic expression in the integration variables in the exponent. Taking a short breath, it will be good to realize that at the corresponding stage of the calculation of the mean Green function in Sect. 6.2, we had to employ the Hubbard– Stratonovich transformation (HST) to return to Gaussian integrals. We here need to use the following superanalytic generalization of the HST,     1  Nλ2 2 ˜ ˜ ˜ Str QLQL = − exp − dQ exp − Str (QL) − iqL QL . 2 2λ2 M4 (6.6.13) 

6.6.2 Unitarity vs Pseudo-Unitarity and Superanalytic Hubbard-Stratonovich Transformation The exponential in the foregoing superintegral (6.6.12) would be invariant under the transformation ˜ † Q˜ → T QT

(6.6.14)

where the 4 × 4 supermatrix T satisfies T † LT = L

(6.6.15)

if ReEˆ were proportional to the 4 × 4 unit matrix. Of course, we need to allow for different energy arguments. Therefore, the “pseudo-unitary” (aka hyperbolic) symmetry (6.6.14) and (6.6.15) of Q cannot hold rigorously. But since the cumulant function in search decays to zero on an energy scale of the order of the mean level spacing which for the normalization of H chosen is of the order 1/N, the relevant differences between the energies Eα are very small, O(1/N), and the pseudounitary symmetry does indeed hold to leading order in N. We shall refer to supermatrices T with the property T † LT = L as pseudounitary. The reader is invited to check that these matrices form a group (see Problem 6.11) ˜ The intended proof of the HST will involve the fact that the 4×4 supermatrix QL in (8.8.16) can be diagonalized by a pseudounitary matrix. That diagonalizability ˜ as arises due to the Hermiticity Q˜ = Q˜ † and the positivity of the Bose block of Q, we proceed to showing next. Since L is diagonal and has the 2×2 unit matrix in the Fermi block, we may save space and labor and still retain the essence of the proof by first restricting ourselves to the toy problem of ordinary 2 × 2 matrices. For these, it is easy to see that the manifold M2 of Hermitian matrices Q = Q† for which QL with L = diag (1, −1)

242

6 Supersymmetry and Sigma Model for Random Matrices

is pseudounitarily diagonalizable is M2 :

 ∗ ac Q=Q = , c b †

 2

|c|
0, we det S > 0. Direct calculation yields S † LSL = 10 01 √ can renormalize as T = S/ det S such that T is indeed pseudounitary, T † LT = L. Positive matrices Q fall in our manifold M2 : Positivity of Q requires a > 0, b > 0, ab > |c|2 and is compatible with |c|2 < (a + b)2 /4 since ab ≤ (a + b)2 /4 = (a − b)2 /4 + ab. To proceed from the toy problem to the manifold M4 of 4 × 4 supermatrices diagonalizable by pseudounitary transformations, we take two steps. The first is to let Q = Q† be 4×4 and block diagonal with the Bose block from M2 . Then, the toy argument goes through practically unchanged where T is also block diagonal; the † toy problem teaches us that the Bose block of T is pseudounitary, TBB LBB TBB = LBB ; the Fermi block TF F is unitary since the Hermitian 2 × 2 matrix (QL)F F = QF F LF F = QF F can be diagonalized by a unitary 2 × 2 matrix. Then, the 4 × 4 matrix T itself is pseudounitary in the sense T † LT = L. The second step of elevation amounts to showing that a 4 × 4 matrix QL can be brought to block diagonal form by a pseudounitary matrix T . Still sticking to the Bose-Fermi notation we show that the BF and FB blocks can be made to vanish.

6.6 The Two-Point Function of the Gaussian Unitary Ensemble

243

The pseudo-unitary transformation doing the job reads  1 − γ †γ k ) γ † , T = −γ k 1 − γ kγ †  ) 1 − kγ † γ ) −kγ † † T = . γ 1 − γ kγ † )



(6.6.18)

Here, the typographically less voluminous name k is given to the 2 × 2 block LBB = diag(1, −1) ≡ k. Moreover, γ and γ † denote a pair of 2 × 2 matrices with four independent anticommuting entries each, viz., γij and (γ † )ij = γj∗i , such that γj∗i is complex conjugate to γj i . There are eight independent parameters in T and T † ; these are all Grassmannians and just right in number to make the equations (T † QLT )BF = (T † QLT )FB = 0 generically solvable. Pseudounitarity, on the other hand, is built into (6.6.18). As block diagonality is thus achieved, the Bose and Fermi blocks and their eigenvalues will have changed only by acquiring nilpotent additions; such additions cannot foul up the positivity of the Bose block since positivity of an eigenvalue is a property of its numerical part. Finally, we turn to the Hubbard–Stratonovich transformation needed to turn the integral over the supervectors Φ, Φ ∗ in (8.8.16) into a Gaussian one. We cannot uncritically employ the usual formula which would involve an unrestricted integral over an auxiliary 4 × 4 supermatrix. The correct procedure was pioneered in a rather different context by Schäfer and Wegner [15], later adapted to superanalytic needs by Efetov [1], and comprehensively discussed in Ref. [2]; it amounts to restricting the integration range for the auxiliary supermatrix to the manifold M4 . To explain the essence of the idea, it suffices once more to consider ordinary 2 × 2 matrices and prove the integral identity 

1 ˜ 2 2 tr(QL)

exp −



 =−

M2

  ˜ dQ exp − 12 tr(QL)2 + i trQLQL

(6.6.19)

for matrices Q˜ ∈ M2 ; as usual, the integration measure is dQ = dadbd 2c/(2π)2 . The minus sign signals that the foregoing integral is far from being a usual Gaussian ˜ to diagonal form by a suitable pseudounitary one. For the proof, we first bring QL −1 ˜ transformation, T QLT = diag(q+ , q− ). Subsequently, when T is absorbed in the integration variable Q, neither the integration range nor the measure is changed. The remainder of the proof is a straightforward calculation, 



∞

da

−∞

=π





∞

db

−∞ ∞

da

−∞



  d 2 c exp − 12 (a 2 + b 2 − 2|c|2) + i(aq+ − bq− )

2 |c|2 0; therefore, the reordering undertaken is known as the “advanced-retarded notation”.

6.6.4 Rational Parametrization We eventually want to make contact with the semiclassical treatment of individual quantum dynamics and therefore switch to the notation used in the pertinent literature and in Sect. 10.6.3, e1− = −eB ,

e2+ = eA ,

e3 = eD ,

e4 = −eC .

(6.6.35)

We note that eA as well as eB now have positive imaginary parts. The Weyl symmetry of the generating function now takes the form Z(eA , eB , eC , eD ) = Z(eA , eB , −eD , −eC ) .

(6.6.36)

We correspondingly rewrite Eq. (6.6.33)  Z0 =

i

dμ(T ) e− 2 Str [diag(−eB ,eD ,eA ,−eC ) Q(T )] ,

Q(T ) ≡ T ΛT −1 ,

(6.6.37) (6.6.38)

apologizing for the abuse of notation which hopefully will not cause confusion because the previous Q(T ) will never again show up. The new one enjoys the properties Q(T )2 = 1

and

Str Q(T ) = 0 .

(6.6.39)

The integration range dμ(T ) is a coset space, the group of pseudo-unitary 4 × 4 supermatrices deprived of the subgroup U(2)×U(1, 1), that subgroup comprising the pseudo-unitaries commuting with Λ.

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6 Supersymmetry and Sigma Model for Random Matrices

We proceed to parametrizing the transformations as  T =

1B B˜ 1



 ,

T

−1

=

˜ −1 ˜ −1 −B(1 − BB) (1 − B B) ˜ − B B) ˜ −1 (1 − BB) ˜ −1 −B(1

 (6.6.40)

where the explicit 2 × 2 structure is in the “advanced-retarded sector” (AR) with entries in the “Bose-Fermi sector” (BF). These entries will be written as  B=

 xμ , νy

B˜ =



x∗ ν∗ μ∗ −y ∗



 = kB † ,

k=

1 0 0 −1

 .

(6.6.41)

The sadle-point manifold thus results as ⎛

1+B B˜ 1−B B˜

Q(T ) = T ΛT −1 = ⎝

˜ −1 −2B(1 − BB)

˜ − B B) ˜ −1 2B(1

˜ 1−BB

− 1+BB ˜

⎞ ⎠

(6.6.42)

˜ and the integral over the manifold becomes one over the BF matrices B, B,  Z0 = −

  ˜ ˜  ˜ exp i Str diag(eA , eC ) 1 + BB + diag(eB , eD ) 1 + B B . d(B, B) ˜ 2 1 − BB 1 − B B˜ (6.6.43)

Further below (next section), we shall reveal the integration measure as flat, ˜ = dμ(T ) = d(B, B)

d 2x d 2y ∗ dμ dμdν ∗ dν , π π

(6.6.44)

and the integration range for the two Bosonic integrals to be |x|2 < 1 and |y|2 < ∞ .

(6.6.45)

With that, the superintegral in (6.6.43) is surprisingly easy to do. The key is the ˜ both have the eigenvalues observation that the 2 × 2 matrices B B˜ and BB lB = |x|2 + μμ∗ +

(xν ∗ − y ∗ μ)(x ∗ ν − yμ∗ ) |x|2 + |y|2 + μμ∗ − νν ∗

lF = −|y|2 + νν ∗ +

≥ 0,

(6.6.46)

(xν ∗ − y ∗ μ)(x ∗ ν − yμ∗ ) ≤ 0; |x|2 + |y|2 + μμ∗ − νν ∗

(6.6.47)

the Bose-Bose eigenvalue lB is positive while the Fermi-Fermi one, lF , is negative. ˜ and pseudo-unitary for B B, ˜ The diagonalizing matrices are unitary for BB ˜ = D −1 BBD ˜ A−1 B BA =



lB 0 0 lF

 ,

(6.6.48)

6.6 The Two-Point Function of the Gaussian Unitary Ensemble

249

and read   −iφB   ∗ 0 e η 1 + ηη2 , A= ∗ 0 e−iφF η∗ 1 − ηη2 η=− τ =−

|x|2

D=

  ∗ 1 − τ 2τ τ , ∗ −τ ∗ 1 + ηη2

xν ∗ − y ∗ μ , + |y|2 + μμ∗ − νν ∗

(6.6.49)

x ∗ μ + yν ∗ . |x|2 + |y|2 + μμ∗ − νν ∗

The foregoing A is a product whose right factor is diagonal and thus not needed for the diagonalization; the purpose of that matrix rather is to provide the matrix −1 A−1 BD, “singular values”, A √ which √ is also diagonal, with positive √ BD = √ −1 ˜ diag(+ lB , + −lF ); on the other hand, D BA = diag(+ lB , − −lF ); the phases φB , φF needed for the purpose in question differ from the phases of the original Bosonic elements x, y of B by suitable nilpotent additions (which will not be of further relevance). Now, the eight new variables lB , lF , φB , φF , η, η∗ , τ, τ ∗ can ˜ The be employed as integration variables instead of the original elements of B, B. Berezinian whose calculation we forgo yields the integration measure dlB dlF dφB dφF dη∗ dηdτ ∗ dτ , 4π 2 (lB − lF )2

˜ = d(B, B)

(6.6.50)

and the integration ranges become 0 < lB ≤ 1 ,

−∞ < lB < 0 ,

0 ≤ φB , φF ≤ 2π .

(6.6.51)

In terms of the new variables the integral (6.6.43) takes the form, after doing the trivial phase integrals, 



1

Z0 = 0



−∞

dlB

dlF

dη∗ dηdτ ∗ dτ

0

i 1 + l

1 (lB − lF )2

 1 + lF (eC + eD ) (6.6.52) 2 1 − lB 1 − lF  i  1 + lB 1 + lF  × exp − (eA − eC )τ τ ∗ + (eB − eD )ηη∗ . 2 1 − lB 1 − lF

× exp

B

(eA + eB ) −

The rational functions of lB , lF appearing in the two foregoing exponents represent, ˜ BB. ˜ A bit troublesome is the singularity of of course, the rational functions of B B, the integrand at lB = lF = 0, due of course to the measure (6.6.50). We shall take care of that presently.

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6 Supersymmetry and Sigma Model for Random Matrices

We first do the Grassmann integrals and find the remaining integrand for the integrals over lB , lF freed of the said singularity, 

1

Z 0 = (eA − eC )(eB − eD )  ×

0 0 −∞

i 1+lB dlB 2 1−lB (eA +eB ) e (1 − lB )2

(6.6.53)

1+l dlF − 2i 1−lF (eC +eD ) F e +... . (1 − lF )2

The remaining Bosonic integrals become elementary in terms of the integration 1+lF B variables nB = 1+l 1−lB , nF = 1−lF and yield Z0 =

i (eA − eC )(eB − eD )  i (eA +eB −eC −eD ) e2 − e 2 (eA +eB +eC +eD ) + . . . . (eA + eB )(eC + eD ) (6.6.54)

The dots . . . stand for the additive contribution of the singularity mentioned. That contribution is determined quite elegantly by realizing that Z 0 can be regarded as a function of the three independent variables (eA + eB ), (eA − eC ), (eB − eD ); the fourth combination, (eC + eD ), then becomes a linear combination of the three variables chosen as independent. Taking the derivative of Z 0 in (6.6.52) w.r.t., say, (eA +eB ), an integrand arises which is no longer singular at nB = nF = 0 since one power of (nB − nF ) cancels, and therefore that derivative is correctly obtained from the foregoing expression (6.6.54) without the correction . . .. The latter correction can thus be seen as an integration constant K(eA − eC , eB − eD ) independent of i eA + eB . That integration constant is fixed as K = e 2 (eA +eB −eC −eD ) by imposing the Weyl symmetry (6.6.36) whereupon we get Z0 =

(eA + eD )(eB + eC ) i (eA +eB −eC −eD ) e2 (eA + eB )(eC + eD ) −

(eA − eC )(eB − eD ) i (eA +eB +eC +eD ) . e2 (eA + eB )(eC + eD )

(6.6.55)

The two-point cumulant (5.14.16) is thus recovered.

6.7 Two-Point Functions of the Circular Ensembles In Sect. 5.14 we had already determined the two-point functions of the density of levels for all three of Dyson’s circular ensembles, using more traditional methods including Grassmann integrals. Why do we come back to these two-point functions in the framework of the supersymmetric sigma model? Why indeed, since this more modern path will turn out more complicated than the traditional one!

6.7 Two-Point Functions of the Circular Ensembles

251

The reason is the immense gain in flexibility towards generalizations. Higherorder correlations of the level density are amenable immediately. Most importantly, extensions beyond random-matrix theory will come into reach. In particular, we shall be led to conditions under which individual systems will display universal spectral fluctuations. Modifications covering autonomous (rather than periodically driven ones) dynamics, quantum graphs, systems with quantum localization are possible and will be taken up. We recall the definition of the ‘connected’ (cumulant type) two-point function R(e) =

> =  πe   N 2  2π 2 πe   ρ φ− − ρ φ+ N N 2π N

∞ 2   2 2πne = 2 , |tn | cos N N

(6.7.1)

tn = TrF n ,

n=1

where ρ(φ) = μ δ2π (φ−φμ ) is the density of eigenphases {φμ , μ = 1, 2, . . . , N} of a unitary N × N Floquet matrix F , normalized by dividing out the mean density N ρ¯ = 2π . The angular brackets · demand an average over the pertinent circular ensemble. The function R(e) is the real part of a complex correlator ∞ 2   2  i 2π ne |tn | e N . N2

C(e) =

(6.7.2)

n=1

The present R(e) is related to the cluster function used in the previous chapter as R(e) = δ(e) − Y (e).

6.7.1 Generating Function We shall approach the two-point function of the level density via a generating function, = Z(a, b, c, d) =

det(1 − c U ) det(1 − d U † ) det(1 − a U ) det(1 − b U † )

> (6.7.3)

which depends on four complex variables a, b, c, d. The moduli |a|, |b| are taken (infinitesimally) smaller than unity while c, d remain unrestricted. By its definition, the generating function enjoys the following properties: Z(a, b, a, b) = 1

normalization

(6.7.4)

 1 1 Z(a, b, c, d) = (cd)N Z a, b, , d c

Weyl symmetry .

(6.7.5)

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6 Supersymmetry and Sigma Model for Random Matrices

Two more useful properties of the generating function must be mentioned. For one, it will become clear a bit further below that Z can depend only on the bilinear combinations ab, cd, bc, and ad; only three of these, however, are independent such that we are free to choose the following combinations as independent variables, c = ei+ /N , a

ab = ei2πe/N , with |ab| < 1 and ac ,

d b

and

d = ei− /N b

(6.7.6)

arbitrary complex. Second, we have Z − 1 ∝ (a − c)(b − d) .

(6.7.7)

The latter proportionality is readily verified as follows. First setting b = d we have Z(a, b, c, b) = det(1 − aU )/ det(1 − cU ). Then employing the identity det(·) = exp Tr ln(·) in powers of U we arrive at the structure

and expanding m Z(a, b, c, b) = 1+ ∞ m=1 cm TrU  with coefficients cm not worth further scrutiny. But the homogeneity of the circular ensembles entails the identities TrU m  = 0 for all non-zero integers m. We conclude Z(a, b, c, b) = 1. The same reasoning reveals Z(a, b, a, d) = 1 and we have confirmed the validity of Z − 1 ∝ (a − c)(b − d). Therefore, the complex correlator can be extracted from Z by any of the following prescriptions  2ab  ∂ ∂ Z  c d 2 a=b=c=d=eiπ e/N N   = − 2 ∂+ ∂− Z  iπ e/N

C(e) =

,± =0

a=b=e

=

(6.7.8)

 2ab Z−1  .  N 2 (a − c)(b − d) a=b=c=d=eiπ e/N

It is now obvious that we loose nothing by restricting our considerations to small ∗ ≈ 0. The variable e can eventually also be taken as real in the real phases ± = ± sense Im e ↓ 0. To reveal the correctness of the preceding prescriptions we note ∂c det(1 − cU ) = ∂c eTr ln(1−cU ) = −Tr ∂d det(1 − dU † ) = −Tr and have



=

2 N2

(6.7.9)

U† det(1 − dU † ) 1 − dU †

2ab ∂ ∂ Z a=b=c=d=eiπ e/N N2 c d in powers of U and U †

expanding  Z a=b=c=d=eiπ e/N

U det(1 − cU ) 1 − cU

=

2 N2



 cU cU †  Tr 1−cU Tr 1−cU †  c=eiπ e/N . Upon

and using tn = TrU n we get 2ab ∂∂ N2 c d iπ(m+n)e/N ∗ tm tn . The homogeneity of the m,n=1 e

∞

6.7 Two-Point Functions of the Circular Ensembles

253

circular ensembles entails tm tn∗  = δmn |tm |2  such that we indeed arrive at the complex correlator given in (6.7.2). As a preparation for introducing the sigma model we write the generating function as a Gaussian superintegral, Z=

#

d(ψ, ψ ∗ )

(6.7.10)  

∗ ∗ ∗ × exp N k,l=1 − ψ+,B,k (δkl − aUkl )ψ+,B,l − ψ−B,k (δkl − bUlk )ψ−,B,l  ∗ ∗ − ψ+,F,k (δkl − cUkl )ψ+,F,l − ψ−,F,k (δkl − dUlk∗ )ψ−,F,l ; here, the 4N integration variables comprise 2N pairs of mutually complex con∗ jugate ordinary (Bosonic) variables ψ±,B,k , ψ±,B,k and 4N mutually independent ∗ Grassmannians ψ±,F,k , ψ±,F,k . The first index, λ = +, −, distinguishes the two ‘columns’ in the definition (6.7.3) of Z which can be associated with forward (+) and backward (−) time evolution; we shall speak of the two dimensional advanced (−)/retarded (+) space AR when referring to that index. The second index, s = B, F, distinguishes denominator and numerator in (6.7.3); since the two cases are respectively represented by Bosonic and Fermionic variables we speak of a two dimensional Bose-Fermi (BF) space. Finally, the third index, k = 1, 2, . . . N, pertains to the Hilbert space of the quantum dynamics (QD) wherein the Floquet matrix operates. The flat integration measure reads d(ψ, ψ ∗ ) =

N d 2 ψ+,B,k d 2 ψ−,B,k ∗ ∗ dψ+,F,k dψ+,F,k dψ−,F,k d ψ−,F,k . π π

k=1

(6.7.11) The mentioned restrictions on a, b secure the existence of the Bosonic integrals in (6.7.10). We may lump the integration variables into four supervectors ψ± and ∗ and introduce the diagonal BF matrices ψ± eˆ+ =

a

 c

,

eˆ− =

b

 d

(6.7.12)

and compact the superintegral (6.7.10) to  Z=

       ∗T ∗T 1 − eˆ+ U ψ+ − ψ− 1 − eˆ− U † ψ− . d(ψ, ψ ∗ ) exp − ψ+ (6.7.13)

We may say that the quadratic form of the supervectors in the exponential is diagonal in AR and BF but not in QD, the latter fact due to the appearance of the Floquet matrix U .

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6 Supersymmetry and Sigma Model for Random Matrices

6.8 Zero Dimensional Sigma Model for the CUE 6.8.1 Color-Flavor Transformation Our next task will be to do the average over the CUE. A remarkable identity, called color-flavor transformation (sometimes generalized Hubbard-Stratonovich transformation) [16] trades that average against an integral over two 2 × 2 supermatrices operating in BF, Z=

Z Z  BB ZF ZFB ZFF

and

Z˜ Z˜ = ˜ BB ZFB

Z˜ BF  . Z˜ FF

(6.8.1)

The said identity reads, with four supervectors ψ1 , ψ1 , ψ2 , ψ2 ,  U(N)

  dU exp ψ1T U ψ2 + ψ2T U † ψ1  =

(6.8.2)

  ˜ 1 + ψ2∗T Zψ2 , ˜ sdet(1 − ZZ) ˜ N exp ψ1T Zψ d(Z, Z)

with sdet denoting the superdeterminant in BF. It is obviously possible to inter˜ The U -integral on the lhs is over the CUE (the group U(N) of change Z ↔ Z. unitary N × N matrices, that is). On the rhs, the integration range is defined by the restrictions ∗ Z˜ BB = ZBB ,

∗ Z˜ FF = −ZFF ,

∗ ZBB ZBB < 1

(6.8.3)

and the integration measure is normalized as 

˜ sdet(1 − ZZ) ˜ N = 1. d(Z, Z)

(6.8.4)

˜ The Z-Z-integral on the rhs of (6.8.2) is a product of N factors of the same form, one for each value of the QD index on the supervectors ψ and ψ ∗ , simply because the quadratic form in the exponent is diagonal in QD. The integration measure is flat, defined in analogy to (6.7.11). More mathematically inclined readers might enjoy going through the proof of that flatness in Ref. [17]. The label ‘color-flavor transformation’ originates from quantum chromodynamics (QCD) [16], a field so far outside the scope of this book that we shall not even try to argue that the nomenclature is intuitive in its field of birth. Rather, we just state that in our present context ‘color’ (C) refers to the QD index whose N values make for C = N colors. In QCD parlance, the matrix U on the lhs of (6.8.2) couples the ‘four fields’ ψ via their color degrees of freedom; on the rhs the coupling is via the matrices Z and Z˜ which in the context of QCD would refer to coupling the fields via their flavor degree of freedom.

6.8 Zero Dimensional Sigma Model for the CUE

255

The integral transformation (6.8.2) is a special case of a more general color-flavor transformation that allows for FB Bosonic and FF Fermionic flavors. We have given and derived the case FB = FF = 1. For later use let us quickly summarise the more general case FB = FF = F that will be used later. For this one considers fields ψ that carry a flavor index f = 1, . . . , F , in addition to the color index n = 1, . . . , N and the BF index s = B, F . This flavor index also enters the supermatrices Z and Z˜ which now have dimension 2F × 2F . One may still write the identity in the form (6.8.2). For clarity let us write out the exponents on both sides in detail. On the lhs the exponent becomes N F    s=B,F f =1 n,n =1

† ψ1,nf s Unn ψ2 ,n f s + ψ2,nf s Unn ψ1 ,n f s

(6.8.5)

while the exponent on the rhs explicitly reads F  N 

 s,s =B,F

f,f =1

∗ ˜ ψ1,nf s Zf s,f s ψ1 ,nf s + ψ2,nf s Znf s,f s ψ2 ,nf s .

(6.8.6)

n=1

Further below, when dealing with the ensembles COE and CSE we shall meet with two Bosonic and two Fermionic flavors F = 2; when applying the sigma model to individual maps rather than ensembles of such we shall encounter large values of F and even consider the limit F → ∞. Interestingly, the proof of the color-flavor transformation to be presented in Sect. 6.8.4 runs through for any number F of flavors. We defer the proof of the color-flavor transformation to Sect. 6.8.4 below. Here, we proceed to using that transformation for doing the CUE average in the generating function (6.7.13). Making appropriate choices for the four supervectors ψ1 , ψ2 , ψ1 , ψ2 in (6.7.13)8 and doing the U -integral we have  Z=

d(ψ, ψ ∗ )



˜ sdet(1 − ZZ) ˜ N d(Z, Z)

(6.8.7)

  ∗T ∗T ∗T ∗T ˜ Zψ+ . ψ+ − ψ− ψ− + ψ+ eˆ+ Z eˆ− ψ− + ψ− × exp − ψ+ The exponential appearing here can be further compacted by introducing a 2 × 2 matrix in AR, M=

 1 −eˆ Z eˆ  + − ; −Z˜ 1

(6.8.8)

8 Different such choices are possible and lead to different but of course equivalent appearances of eˆ± .

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6 Supersymmetry and Sigma Model for Random Matrices

it is to be noted that M acts like unity in QD and is 4 × 4 in AR⊗BF. For the integral over the ψ, ψ ∗ (both in AR⊗BF⊗QD) we get # Gaussian   supervectors ∗ ∗T −N d(ψ, ψ ) exp − ψ Mψ = (SdetM) = exp(−NStr ln M) , where the Nth power of the 4 × 4 superdeterminant SdetM arises due to the structure of M just mentioned. We so arrive at the generating function  Z=  ≡

˜ Sdet(1 − ZZ) ˜ N Sdet−N d(Z, Z)

 1 −eˆ Z eˆ  + − , −Z˜ 1

(6.8.9)

˜ e−N S , d(Z, Z)

˜ + Str ln S = − Str ln(1 − ZZ)

 1 −eˆ Z eˆ  + − , −Z˜ 1

which in fact constitutes the (‘zero dimensional’) supersymmetric sigma model, the principal goal of the present chapter. The qualifier ‘zero dimensional’ somewhat cryptically points to the fact that the ‘action’ S is structureless in QD. (We shall in a later chapter meet with more general sigma models which do carry structure in QD.) Two other forms of the foregoing ‘action’ S are worth noting. First, the 4 × 4 superdeterminant SdetM can be written as 2 × 2 in BF with the help of the A B = sdet(A) sdet(D − CA−1 B) , wherein Sdet and sdet identity (6.3.24), Sdet CD are, respectively, 4 × 4 and 2 × 2. The action S in (6.8.9) then takes the alternative form ˜ ˜ + Str ln(1 − eˆ+ Z eˆ− Z) S = −Str ln(1 − ZZ)



(6.8.10)

This latter form immediately reveals the correctness of the claim made further above about the independent variables of Z, see (6.7.6). Indeed, the variables a, b, c, d abZ adZ  BB BF which does appear exclusively in the building block eˆ+ Z eˆ− = bcZFB cdZFF enjoy the claimed properties. Moreover, the representation (6.8.10) will be the ˜ starting point for our evaluation of the Z-Z-integral further below. Particularly helpful for recognizing the mathematical structures behind the sigma model is a last rewriting of the action as a supertrace in AR⊗BF,   ˜ S = Str ln (1 − E BF )Q(Z, Z)Λ + 1 + E BF ,  1 Z −1   ˜ = 1Z 1 0 Q(Z, Z) ≡ T ΛT −1 , Z˜ 1 0 −1 Z˜ 1  eˆ 0  + ; E BF = 0 eˆ−

(6.8.11) (6.8.12)

6.8 Zero Dimensional Sigma Model for the CUE

257

the matrix Q and its ‘basic configuration’ Λ should be read as 2 × 2 in AR whose entries are 2 × 2 in BF.9 We note that the integration variables Z, Z˜ exclusively appear in the supermatrix Q which obeys Q2 = 1 ,

Str Q = 0 .

(6.8.13)

In fact, one may consider the elements of Q as the basic integration variables and is thus immediately led to asking what supermanifold Q lives on. Before proceeding to that question we would like to make contact with the previous section where the “rational parametrization” (6.8.12) was introduced after ˜ doing the saddle-point approximation (with the BF matrices Z, Z˜ called B, B) which involved the limit N → ∞. In the present situation that parametrization automatically appeared through the exact color-flavor transformation and allows to keep N finite. The names Z, Z˜ will consistently be reserved for the constituting matrices in exact procedures.

6.8.2 The Q Manifold According to the definition in (6.8.11) every supermatrix Q is generated from its basic configuration Λ by conjugation with a transformation T . Obviously, the manifold of Q’s is unchanged when the transformation is changed as T → T h with h = diag(h+ , h− ) any invertible 4 × 4 supermatrix from AR⊗BF that is block diagonal in AR, simply because all such h commute with Λ. Therefore, the manifold of Q’s is smaller than the group G = Gl(2|2) of invertible 4 × 4 supermatrices (acting in AR⊗BF, in our context) of which the h’s form the subgroup H = Gl(1|1) × Gl(1|1). Rather, the Q manifold is the coset space G/H . Incidentally, the transformations T are in G/H as well, for the same reason. One could even write Q more generally, in a coordinate free manner, as Q = gΛg −1 A B  with g ∈ G. Setting g = C D , the entries being 2 × 2 in QD, and assuming A and D invertible one returns to the ‘rational’ parametrization in terms of Z and Z˜  −1  A 0  −1 −1 ˜ through g = CA1−1 BD1 0 D and Z = CA , Z = BD . A last remark on the Q manifold is in order. Due to the properties of T and Q, the most general invariance of Q is given by T → T ,

T =

 1 Z , Z˜ 1

Z = (AZ + B)(CZ + D)−1 ,

T =

 1 Z  , Z˜ 1

(6.8.14)

˜ ˜ −1 . Z˜ = (C + D Z)(A + B Z)

9 This ordering of matrices in AR⊗BF is often called ‘AR notation’. Unless noted explicitly otherwise we shall stick to that ordering.

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6 Supersymmetry and Sigma Model for Random Matrices

A B  The matrices A, B, C, D intervening must allow for invertibility of C D ; a further ˜ ˜ restriction is that Z and Z and Z, must range in the domain (6.8.3). # Z , like −N S requires an integration range for the Bosonic ˜ The superintegral d(Z, Z)e variables {ZBB , ZFF , Z˜ BB , Z˜ FF }, as was already necessary in the color-flavor transformation (6.8.2). There we had noted the restrictions (6.8.3). Equivalently, we may characterize the integration range as the Riemannian submanifold MB × MF of G/H where10 MB = U(1, 1)/U(1) × U(1) ∼ = H2

(two−hyberboloid,

MF = U(2)/U(1) × U(1) ∼ = S2

(two−sphere,

BB sector) (6.8.15)

FF sector) ;

here, denotes  group of pseudo-unitary 2 × 2 matrices obeying  U(1,1)  0 the 0  † U = 10 −1 . To see the equivalence of the latter stipulation with (6.8.3) U 10 −1 (and for all discussions of integration ranges in general) we may momentarily disregard all Grassmannian  variablesand order all 4 × 4 matrices in BF⊗AR as block diagonal, m = diag mBB , mFF with entries that remain 2 × 2 in AR. That ordering is often called ‘Bose-Fermi notation’. The matrix Λ, in particular, then 0 reads Λ = diag σ3 , σ3 with Pauli’s σ3 = ( 10 −1 ) operating in AR, while the Q’s thus admitted are restricted by −1 QBB = TBB σ3 TBB

∼ =

MB ,

−1 QFF = TFF σ3 TFF

∼ =

MF .

(6.8.16)

Since all diagonal AR matrices commute with σ3 the transformations TBB (TFF ) are equivalent to TBB hBB (TFF hFF ) with arbitrary invertible 2 × 2 matrices hBB (hFF ). If TFF hFF is in U(2) the TFF span the coset space U(2)/U(1)×U(1); similarly, if TBB hBB is in U(1,1) the TBB span the coset space U(1,1)/U(1)×U(1). The reader is kindly invited to explore a more direct way to the manifold MB × MF by writing out the numerical parts of T in the Bose-Fermi notation and find out why TFF and TBB respectively act in effect unitarily and pseudo-unitarily. To conclude our discussion of the Q manifold, we want to point out that G/H is G-invariant: with every g ∈ G the product gQ is again a member of G/H . That statement entails the most general changes of the transformation T . To see that we look at the product  A B  1 Z   A + B Z˜ AZ + B   A + B Z˜  0 = = T , C D Z˜ 1 C + D Z˜ CZ + D 0 CZ + D  1 Z  (6.8.17) T = Z˜ 1

gT =

10 The

equivalence of MF with the two-sphere and of MB with the two-hyperboloid will be shown in Sect. 6.8.5.

6.8 Zero Dimensional Sigma Model for the CUE

259

with the general invariance of the Q manifold Z = (AZ + B)(CZ + D)−1 ,

Z˜ = (D Z˜ + C)(B Z˜ + A)−1 .

(6.8.18)

The G-invariance of the coset space G/H entails the G-variance of the measure ˜ which we now propose to reveal as flat, thus substantiating the d(Q) = d(Z, Z) claim made when introducing the color-flavor transformation above.

6.8.3 Riemann Geometry for Supermatrix Ensembles: Flat ˜ Measure d(Q) = d(Z, Z) Generalizing the considerations of Sect. 5.3 to supermatrices we would like to ˜ on the Q manifold. Starting with establish the flatness of the measure d(Z, Z) Q = T ΛT −1 we write the ‘squared length element’ in G/H as  2 Str(dQ)2 = Str T −1 dT , Λ .

(6.8.19)

The commutator showing up here is readily shown to be 

T

−1

 dT , Λ =



˜ −1 dZ −2(1 − Z Z) 0 ˜ −1 d Z˜ 0 2(1 − ZZ)

 (6.8.20)

such that the squared length element becomes ˜ −1 d Z(1 ˜ − Z Z) ˜ −1 dZ , Str(dQ)2 ∝ str(1 − ZZ)

(6.8.21)

up to a numerical factor which can be established in the end by normalizing the measure. ˜ −1 and (1 − Z Z) ˜ −1 diagonalized, We now imagine the 2 × 2 matrices (1 − ZZ) ˜ −1 = ARA−1 , (1 − Z Z)

˜ −1 = DRD −1 , (1 − ZZ)

R=

0  , 0 RFF

R

BB

and note in passing that this diagonalization implies singular-value decompositions ˜ diagonal. Not fortuitously, the latter decompositions making A−1 ZD and D −1 ZA are a special case of the general invariance of the Q manifold (6.8.18) (with B = C = 0). The squared length element then becomes ˜ ≡ strRdY Rd Y˜ ≡ str dW d W˜ . Str(dQ)2 ∝ strR(A−1 dZD)R(D −1 d ZA) (6.8.22)

260

6 Supersymmetry and Sigma Model for Random Matrices

Momentarily employing a shorthand with dW and Y four-component supervectors and Rˆ a diagonal 4 × 4 supermatrix we can write ⎛ ⎞ ⎛ dWBB RBB 0 ⎜ dWFF ⎟ ⎜ 0 −RFF ⎟ ⎜ dW = ⎜ ⎝ dWBF ⎠ = ⎝ 0 0 dWFB 0

⎞⎛ ⎞ 0 0 dYBB ⎟ ⎜ 0 0 ⎟ ⎟ ⎜ dYFF ⎟ = RdY ˆ . 0 ⎠ ⎝dYBF ⎠ RBB dYFB 0 0 − RFF

ˆ Obviously now, the transformation dW = RdY has unit Berezinian, sdetRˆ = 1. ˆ Y˜ with unit Berezinian. Analogously, we get d W˜ = Rd We proceed to the transformations dZ → dY, d Z˜ → d Y˜ and their Berezinians. Starting with A−1 dZD = dY we write that mapping as a sequence of two, D with dZ = A−1 dZ. Writing the first step componentwise, dY dY στ =

= dZ dZ D , we face two transformations, one for each value of the BF index σ. σ μ μτ τ 2 The combined Berezinian is (sdetD) . Similarly the second-step transformation has the Berezinian (sdetA)−2 such that the transformation dZ → dY has the Berezinian 2 ˜ ˜ ( sdetD sdetA ) . On the other hand, the mapping d Z → d Y has A and D swapped and thus the reciprocal Berezinian. The overall mapping dZ → dY, d Z˜ → d Y˜ , we conclude, has unit Berezinian. ˜ is established: the squared length element The flatness of the measure d(Z, Z) Str dQ2 = str dW d W˜ involves only (components of) independent differentials and obviously has a constant diagonal metric matrix and thus the product of the independent differentials as the infinitesimal volume element d(W, W˜ ). The same ˜ since the Berezinian for W, W˜ → Z, Z, ˜ as we have just seen, holds true for d(Z, Z) equals unity. But hold it! The quantity Str(dQ)2 could not be an honest-to-goodness ‘squared length element’ if it did not have a positive numerical part. We must show this to be the case. To that end we momentarily consider the numerical parts of all supermatrices in play. In particular,  x 0   |x|eiφB 0  , = 0 |y|eiφF 0y  x ∗ 0   |x|e−iφB  0 = = 0 −y ∗ 0 −|y|e−iφF

Z (num) = Z˜ (num)

(6.8.23)

become diagonal to begin with. We can therefore write D (num) =

1 0 01

,

A(num) =

 eiφB 0  0 eiφF

(6.8.24)

6.8 Zero Dimensional Sigma Model for the CUE

261

such that dY (num) = (A−1 ZD)(num) =

 dZ

BB e

0

−iφB

(num) 0 , dZFF e−iφF (num) 0

 ˜ iφB ˜ (num) = d ZBB e d Y˜ (num) = (D −1 ZA) 0 d Z˜ FF eiφF  dY ∗ 0 (num) BB . = ∗ 0 −dYFF

(6.8.25) (6.8.26)

We now conclude that the numerical part of the squared length element  2  ∗ 2 ∗ (num) Str(dQ)2(num) ∝ RBB dYBB dYBB + RFF dYFF dYFF is indeed positive. We also recognize the restrictions (6.8.3) as necessary for the overall consistency of our rational parametrization of Q.

6.8.4 Proof of the Color-Flavor Transformation (6.8.2) Closely following Zirnbauer’s proof in Ref. [16] we start by introducing a (super)algebra of operators defined as [cαi , cβ ](s) ≡ cαi cβ − (−1)|α|·|β| cB cαi = δαβ δ ij j†

j†

j†

[cαi , cβ ](s) ≡ cαi cβ − (−1)|α|·|β| cβ cαi = 0 j

j

j

(6.8.27)

[cαi† , cβ ](s) ≡ cαi† cβ − (−1)|α|·|β| cβ cαi† = 0 . j†

j†

j†

Here the upper indices i, j = 1 . . . N refer to color (QD, that is). The lower indices α, β are composite and refer to AR and BF and, if necessary, a further flavor index f = 1, 2, . . . F . For the application to the CUE average no flavor summation index is necessary such that F = 1. When dealing with the COE and the CSE we shall encounter F = 2 and when introducing the sigma model for individual quantum maps we must reckon with large F . Most importantly, an eventual flavor index f will behave like a silent witness to the following proof of the integral transformation (6.8.2) and therefore needs no further mention. The symbol |α| stands for |α| = 0 if α = {±, B} and |α| = 1 if α = {±, F}. We may somewhat loosely speak of annihilation and creation operators of Bosonic and Fermionic ‘auxiliary particles’. The ‘supercommutator’ [·, ·](s) is the usual commutator except if the two operators involved are both Fermionic, and in that case the supercommutator is the usual anticommutator.

262

6 Supersymmetry and Sigma Model for Random Matrices

The above operators act in a Fock space whose vacuum is defined by i† i c+s |0 = c−s |0 = 0 ,

i† i 0|c+s = 0|c−s = 0,

s = B, F .

(6.8.28)

One may imagine the (+) states all filled and the (-) states all empty. For the Fermions that state is the familiar Fermi sea but for the Bosons it is highly unusual and even looks suspicious. Zirnbauer [16] motivates the choice and checks consistency; we shall just use the above ‘vacuum’ and the Fock space erected thereon. The Fock space contains a subspace, to be called ‘flavor sector’ or ‘color neutral sector’, whose member states include as many (+) particles as (−) holes, 

cαi† cαj |flavor state ≡ C ij |flavor state = 0 .

(6.8.29)

α

The vacuum belongs to that flavor sector, C ij |0 =



j

i† c−s c−s |0 = δ ij (−1 + 1) = 0 .

(6.8.30)

s

j The ‘color operators’ (flavor singlets) C ij = α cαi† cα we here meet with form the Lie algebra gl(N) of N × N matrices, as is clear from their commutators [C ij , C kl ]s = [C ij , C kl ] = δ j k C il − δ il C j k . Conversely, the flavor operators (color singlets) Fαβ = supercommutation rules

(6.8.31)

N i

cαi† cβi obey the

[Fαβ , Fγ δ ]s = Fαβ Fγ δ − (−1)(|α|+|β|)(|γ |+|δ|)Fγ δ Fαβ = δβγ Fαδ − (−1)(|α|+|β|)(|γ |+|δ|)δαδ Fβγ

(6.8.32)

and form the Lie superalgebra gl(2, 2) of 4 × 4 supermatrices which in our case pertain to AR⊗BF. The flavor sector can be shown [16] to be irreducible inasmuch as all of its states can be generated from one reference state, say the vacuum, by multiple action of flavor operators Fαβ . When presenting and discussing the color-flavor transformation above we had already incurred the supergroup G≡Gl(2|2) of invertible 4 × 4 supermatrices, its subgroup H ≡Gl(1|1)×Gl(1|1) of supermatrices h = diag(h+ , h− ) that are block diagonal in AR, as well as the coset space G/H . Now, in equipping ourselves for the proof of the color-flavor transformation, we need the Lie supergroup Gl(2|2) corresponding to the Lie superalgebra gl(2,2) formed by the flavor operators Fαβ .

6.8 Zero Dimensional Sigma Model for the CUE

263

In particular, we shall have to employ the Fock state representation of group elements g ∈ G,   Tg = exp cαi† (ln g)αβ cβi ;

(6.8.33)

here and below we employ the summation convention for all indices, color and flavor; moreover, we stipulate that the Grassmann variables in the supermatrix ln g j anticommute with the Fermionic operators in the set {cαi† , cβ }. Due to its foregoing definition, Tg acts in the flavor sector of the Fock space. We skip proving the (nearly obvious) fact that the Tg form a supergroup homomorphic to G, i.e., Tg Tg = Tgg .

(6.8.34)

The subgroup H =Gl(1|1)×Gl(1|1) is represented by   i† i† i i (ln h+ )st c+t + c−s (ln h− )st c−t Th = Tdiag(h+ ,h− ) = exp c+s     i† i† i i = exp c+s exp c−s . (ln h+ )st c+t (ln h− )st c−t

(6.8.35)

We note the commutativity of the two exponentials in the second line of the foregoing equation. It immediately follows that the left exponential acts like unity on the vacuum. to act like  We also readily find the right  exponential  i† i |0 = exp N(−1)|s|+1 (ln h ) exp c−s (ln h− )st c−t = exp − Nstr ln h− = − ss −N sdet (h− ) such that Th |0 = sdet−N (h− )|0 ≡ μ(h)|0

and

0|Th−1 = μ(h)−1 0| . (6.8.36)

We may thus say that the vacuum carries a one dimensional representation μ of the subgroup H . We now consider the operator  P = d(Q) Tg |00|Tg−1 . (6.8.37) G/H

Here, the integral over the coset space G/H is well defined, the integrand being −1 a function on G/H . Indeed, we have Tgh |00|Tgh = Tg Th |00|Th−1 Tg−1 = Tg |00|Tg−1 for all h ∈ H . The operator P commutes with all transformations Tg0 ∈ G,  Tg0 P = 

G/H

= G/H

d(Q)Tg0 Tg |00|Tg−1 d(Q)Tg0 g |00|Tg−1

(6.8.38)

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6 Supersymmetry and Sigma Model for Random Matrices

 = 

G/H

= G/H

d(Q)Tg |00|T −1 −1

g0 g

d(Q)Tg |00|Tg−1 Tg0 = P Tg0 .

The G-invariance of the measure d(Q) was used here. Recalling the irreducibility of the flavor sector we can argue that the integral defining the operator P covers the flavor sector of our Fock space. The fact that P commutes with all Tg0 then means, by Schur’s lemma, that P is proportional to unity on the flavor sector. We propose to show that P is the identity on the flavor sector by showing that the constant of proportionality is unity. To that end we choose a parametrization of the coset space G/H slightly different from the one in (6.8.11) but still hold ˜ That means we look for subgroup elements h = on to the local coordinates Z, Z. ˜ ˜ ˜ The elements g(Z, Z) ˜ ∈G diag h+ (Z, Z), h− (Z, Z) that are functions of Z, Z. to be specified must have the form ˜ = g(Z, Z)



1Z Z˜ 1



 ˜ h+ (Z, Z) 0 . ˜ 0 h− (Z, Z)

(6.8.39)

˜ can be used to demand Our freedom to pick the functions h± (Z, Z) ˜ , ˜ −1 = g(−Z, −Z) g(Z, Z)

(6.8.40)

and that choice uniquely fixes 

˜ − 12 Z(1 − ZZ) ˜ − 12  (1 − Z Z) ˜ − 12 ˜ − Z Z) ˜ − 12 (1 − ZZ) Z(1     ˜ + 12 0 1Z 10 (1 − Z Z) = . ˜ − 12 01 0 Z˜ 1 (1 − ZZ)

˜ = g(Z, Z)

(6.8.41)

We shall come to rejoice in that choice before long. The pertinent Fock-space operator reads  i†   1 i†  i i ˜ Tg(Z,Z) ˜ = exp c+s Zst c−t exp 2 c+s (ln(1 − Z Z))st c+t    i†  i† i i ˜ ˜ st c+t × exp − 12 c−s (ln(1 − ZZ)) . st c−t exp c−s Z

(6.8.42)

  Here, we have used ln 10 Z1 = ln Z and exploited the commutativity of the four exponentials. That commutativity also reveals the second and the fourth exponential to act like unity on the vacuum and the third exponential to give a superdeterminant,

6.8 Zero Dimensional Sigma Model for the CUE

265

similarly to what happened above in (6.8.36),   i† N i ˜ Tg(Z,Z) ˜ |0 = exp c+s Zst c−t |0 × sdet 2 (1 − ZZ) ≡ |Z .

(6.8.43)

But now look: due to the choice (6.8.40) we get, without new calculation, 0|T −1

˜ g(Z,Z)

  N i† i ˜ = 0|Tg(−Z,−Z) ˜ = sdet 2 (1 − ZZ)0| exp − c+s Zst c−t ≡ Z| , (6.8.44)

and that is enjoyable indeed. We can now return to the operator P . Since the invariant measure on G/H ˜ we have expressed in the coordinates Z, Z˜ is d(Z, Z)  P =

MB ×MF

−1 ˜ d(Z, Z)T ˜ |00|T g(Z,Z)

˜ g(Z,Z)

 =

MB ×MF

˜ d(Z, Z)|ZZ| . (6.8.45)

The # vacuum expectationNvalue of P is now readily seen to be unity, 0|P |0 = ˜ sdet(1 − ZZ) ˜ d(Z, Z) = 1, due to our normalization (6.8.4) of the integration measure. We also conclude that P acts like zero on all states that are not color singlets; in other words, P projects the Fock space onto the flavor sector. The final tool we need to prove the color-flavor transformation are the (nonnormalized) coherent states  i† i   ∗j j   i† i ∗j j  j∗ i ψ+s + ψ−t c−t |0 = exp c+s ψ+s exp ψ−t c−t |0 ≡ |ψ+s , ψ−t  , exp c+s (6.8.46) j†

j∗

i i and c−t with eigenvalues ψ+s and −ψ−t , respectively. Their right eigenstates of c+s duals are

 j† j   i∗ i  j i∗ , ψ−t | = 0| exp ψ+s c+s exp c−t ψ−t , ψ+s j

(6.8.47) j

i† i∗ left eigenstates of c+s and c−t with respective eigenvalues ψ+s and −ψ−t . We recall that the Grassmann variables occurring here anticommute with the pertinent Fermionic operators. These Bose-Fermi coherent states span the whole Fock space. They can be projected on the flavor sector by making a unitary rotation in color j j† i i space, cA → cA U j i , cA → (U j i )∗ cA , and averaging over all such rotations, j∗

i , ψ−t  = P |ψ+s

 U(N)

  j† i∗ j dU exp (U j i )∗ c+s ψ+s + ψ−t c−t U j i |0 .

(6.8.48)

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6 Supersymmetry and Sigma Model for Random Matrices

Such equipped, we prove the color-flavor transformation (6.8.2) through the following few lines:    i∗ i i∗ ˜ i ˜ sdet(1 − ZZ) ˜ N exp ψ+s d(Z, Z) Zst ψ−t + ψ−s Zst ψ+t  =

˜ sdet(1 − ZZ) ˜ N d(Z, Z)

 i†   i∗ i j† i  i exp c+s |0 × 0| exp ψ+s c+s − c−t ψ−t Zst c−t     i† ˜ i† i i i∗ j † × 0| exp − c−s exp c+s Zst c+t ψ+s + ψ−t c−t |0   i∗ i  i† i  j† i  i∗ j † ˜ = 0| exp ψ+s d(Z, Z)|ZZ| exp c+s c+s − c−t ψ−t ψ+s + ψ−t c−t |0  i† i   i∗ i j† i  i∗ j † P exp c+s = 0| exp ψ+s c+s − c−t ψ−t ψ+s + ψ−t c−t |0   i† ij j  i∗ i j† i  j†  i∗ = exp c+s 0| exp ψ+s c+s − c−t ψ−t U ψ+s + ψ−t (U j i )∗ c−t |0 

dU(N)

= dU(N)

 i∗ ij j j† i∗ exp ψ+s U ψ+s + ψ−t (U j i )∗ ψ−t .

The first equality in the foregoing chain is due to the eigenvalue properties of the Bose-Fermi coherent states. The next two equations recognize the projector P onto the flavor sector. In the forth equation we have employed the action (6.8.48) of P on a coherent state, and in the last equation once more the eigenvalue properties of a coherent state. We are done.

6.8.5 Evaluation of the Generating Function and the Two-Point Correlator for the CUE We turn to evaluating the superintegral for the generating function  Z=

MB ×MF

˜ e−N S , d(Z, Z)

(6.8.49)

 ˜ + str ln(1 − eˆ+ Z eˆ− Z) ˜ , S = − str ln(1 − ZZ) see (6.8.9), (6.8.10) [17]. To that end we write the intervening BF matrices as in the ˜ see (6.6.41) with complex preceding section where Z, Z˜ carried the names B, B, ∗ ∗ entries x, y and Grassmannians μ, ν, μ , ν . (We had already noted the numerical parts of Z, Z˜ in (6.8.23).) The flat integration measure then reads 2 2 ˜ = d x d y dμ∗ dμdν ∗ dν . d(Z, Z) π π

(6.8.50)

6.8 Zero Dimensional Sigma Model for the CUE

267

Our first step will be to reveal the manifolds MF = U(2)/U(1) × U(1) and MB = U(1, 1)/U(1)×U(1) as two-sphere S 2 and two-hyperboloid H 2 , respectively. This is an elementary exercise in Lie algebra which some readers may want to skip. Starting with the sphere we note that U(2) consists of matrices of the form   α β with 0 ≤ ψ < 2π and the complex entries α, β restricted as eiψ −β ∗ α∗

|α|2 +|β|2 = 1; the latter restriction is enforced by α = cos θ2 eif , β = sin θ2 ei(f +φ) with 0 ≤ θ ≤ π and 0 ≤ φ, f < 2π. Upon identifying all of those matrices differing by the overall real phase ψ we arrive at the coset space U(2)/U(1). The further restriction to U(2)/U(1) × U (1) means identifying all matrices differing in the relative phase between the two rows, i.e. in f , such  that only the two   cos θ2 sin θ2 eiφ α β parameters θ, φ remain, −β ∗ α ∗ = . The resulting coset θ −iφ θ − sin

e cos 2 2 S parametrized 2

by the polar angle space is obviously isomorphic to the sphere θ and the azimuth φ or the complex stereographic projection variable z = tan θ2 eiφ . In terms of the latter we get the coset space spanned by matrices of the form )

1 1 + |z|2



1 z −z∗ 1

 ∈ U(2)/U(1) × U(1) .

(6.8.51)

The natural measure for the sphere and thus for the coset space is d 2y

d 2z

sin θ dθdφ 4π

=

, to be identified with π in (6.8.50). Indeed, the matrix element y in π(1+|z|2 )2 Z must be identified with the stereographic projection variable z. The sphere is covered as y = z covers the whole complex plane (Fig. 6.2).

1

cosh 2t

x3

x3 1

θ

θ 2

θ 2 -1

sin θ z = tan θ2

-1

sinh t

z = tan 2θ = tanh 2t

Fig. 6.2 Stereographic projections of sphere S 2 (left) and (upper half of) hyperboloid H 2 (right). Plots in planes of constant azimuth. For S 2 , projection onto equatorial plane from south pole. For H 2 , projection from center onto plane tangential to pericenter

268

6 Supersymmetry and Sigma Model for Random Matrices

The geometric interpretation of the coset space U(1, 1)/U(1) × U(1) is reached  analogously. The group U(1, 1) is spanned by matrices of the form eiψ βα∗ αβ∗ with two complex entries restricted by pseudounitarity as |α|2 − |β|2 = 1 or α = cosh 2t eif , β = sinh 2t ei(f +φ) with 0 ≤ t < ∞. Dropping the overall phase factor we get the coset space U(1, 1)/U(1), and by dropping the relative phase f we arrive at U(1, 1)/U(1) × U(1). Instead of by t, φ that latter set can be parametrized by the complex variable z = eiφ tanh 2t which ranges in the unit disk |z|2 < 1. The pertinent matrices have the form   1 1 z ) ∈ U(1, 1)/U(1) × U(1) . (6.8.52) 1 − |z|2 z∗ 1 The Cartesian coordinates x1 =

2Re z , 1−|z|2

x2 =

2Im z , 1−|z|2

x3 =

1+|z|2 1−|z|2

are related as

+ − = −1, z3 > 0. The coset space in question thus indeed turns out isomorphic to the upper sheet H 2 of the hyperboloid and z a complex stereographic projection variable which can now be identified with the entry x in Z. The upper sheet of H 2 is covered as the projection variable sweeps the interval |x| = |z| < 1. We could now try to do the superintegral giving the generating function Z using the independent matrix elements in Z, Z˜ as integration variables. More convenient, however, is another parametrization reached by the singular-value decomposition of Z, Z˜ already used in the proceeding section and alluded to in our excursion to Riemann geometry in Sect. 6.8.3,  √l  √l 0  0  B B ˜ = √ √ ≡ Z , D −1 ZA ≡ Z˜ . A−1 ZD = 0 −lF 0 − −lF (6.8.53) x12

x22

x32

˜ ZZ ˜ The supermatrices A, D which diagonalize the products Z Z, as  lB also 0 −1 −1 ˜ ˜ A Z ZA = D ZZD = with the real eigenvalues already given 0 lF in (6.6.46), as were the matrices A and D in (6.6.49). As in Sect. 6.6.4 the eight new variables lB , lF , φB , φF , η, η∗ , τ, τ ∗ ) can be ˜ The employed as integration variables instead of the original elements of Z, Z. pertinent integration measure was given in (6.6.50). Using the singular-value decomposition (6.8.53) we can write the action S in (6.8.53) S = ln

1 − lF 1 − ablB + ln + str ln(1 − m) 1 − lB 1 − cdlF

(6.8.54)

with the (nilpotent) matrix   (6.8.55) m = diag (1 − ablB)−1 , (1 − cdlF )−1   ˜ − Z eˆ+ + (a − c)(b − d)ZΔ ˜ − ZΔ+ , × (a − c)Z˜ Z eˆ− Δ+ + (b − d)ZΔ  −τ τ ∗ τ   ηη∗ eη  , Δ , e = ei(φB −φF ) . = (6.8.56) Δ+ = − τ ∗ −τ τ ∗ −e∗ η∗ ηη∗

6.8 Zero Dimensional Sigma Model for the CUE

269

To arrive at this form of S we invoked the singular-value decomposition (6.8.53) and used the identities A−1 eˆ+ A = eˆ+ + (a − c)Δ+ and D −1 eˆ− D = eˆ− + (b − d)Δ− . It is to be noted that the matrices m and Δ± are nilpotent. We can forget the unimodular factors e, e∗ : they disappear when we change integration variables as eη → η and e∗ η∗ → η∗ . The integrals over the phases φB,F then just give the factor 4π 2 and we can write the generating function  Z = lim

→0 



G=



1

dlB



−∞ −

dlF

   1 (1 − lB )(1 − cdlF ) N G , (lB − lF )2 (1 − lF )(1 − ablB)

dη∗ dηdτ ∗ dτ e−Nstr ln(1−m)

(6.8.57) (6.8.58)

where from here on the matrix Δ− is understood purged of the said phase factors. Why have we cut out an infinitesimal neighborhood of lB = lF = 0 from the integration range? According to the transformation (6.8.53) the ‘point’ lB = lF = 0 corresponds to Z = Z˜ = 0 where the integrand has no singularity in terms of the parametrization (6.6.41), such that the integral is immune against cutting out the -neighborhood. Moreover, the integral exists independently of the chosen integration variables and must therefore be correctly given by the limiting procedure in (6.8.57). On the other hand, the transformation to polar variables (6.8.53) does bring in a singularity: indeed, the factor (lB − lF )−2 diverges at lB = lF = 0 while #1 # −∞ the matrix m vanishes. Therefore, the integral 0 dlB 0 dlF [. . .] differs from Z = # −∞ lim→0 dlB − dlF [. . .] by a finite term due to the singularity of the integrand in the ‘polar coordinates’. No extra work is needed to determine the contribution of that singularity since it is automatically fixed by the property Z − 1 ∝ (a − c)(b − d), see (6.7.7), as unity. We thus have 



1

Z −1= 0



−∞

dlB

dlF 0

   1 (1 − lB )(1 − cdlF ) N G (lB − lF )2 (1 − lF )(1 − ablB )

(6.8.59)

and can even conclude that the factor (a − c)(b − d) in Z − 1 must come from the Grassmann integral G in the foregoing superintegral. An elementary if lengthy calculation (see the next subsection) gives that Grassmann integral as   N(lB − lF ) 1 − abcdlBlF + N(ablB − cdlF ) G = (a − c)(b − d) . (1 − ablB )2 (1 − cdlF )2

(6.8.60)

We can now circumvent doing the remaining integral over lB and lF by invoking the Weyl symmetry (6.7.5). We rewrite that symmetry for the quantity J (ab, cd) =

(1 − ab)(1 − cd) (Z − 1) (a − c)(c − d)

(6.8.61)

270

6 Supersymmetry and Sigma Model for Random Matrices

which depends only on the two variables ab and cd and find   (a − c)(c − d)  1  1  J (ab, cd) + (cd)N J ab, = (cd)N − 1 − (cd)N J ab, . (1 − ab)(1 − cd) cd cd

(6.8.62) Herein, the rhs depends only on the variables ab and cd while the lhs is proportional to (a − c)(c − d). So both sides vanish. We conclude J (ab, cd) = 1 − (cd)N and arrive at the generating function  (a − c)(c − d)  Z = 1 + 1 − (cd)N = ZCUE (1 − ab)(1 − cd)

(6.8.63)

and, through (6.7.8), the complex two-point correlator CCUE (e) =

e2πie − 1 2N 2 sin2

πe N

.

(6.8.64)

That correlator is periodic in the phase e with period N. Much smaller is the scale on which C(e) is independent of N, according to N sin(πe/N) ∼ πe 2π ie and C(e) ∼ e2(πe)−1 2 , in agreement with the correlator of the Gaussian unitary ensemble. That behavior arises in windows of correlation decay and revival around e = 0, N, 2N, . . .. Outside these windows the correlator is of the order N12 .

6.8.6 Evaluation of the Grassmann Integral G An way of doing the Grassmann integral G is by differentiation, # economic dη∗ dηdτ ∗ dτ (·) = ∂η∗ ∂η ∂τ ∗ ∂τ (·), since the chain and product rules prove helpful.11 It is worthwhile to spell out the product rule since a peculiarity of Grassmann calculus must be pointed at. Due to the anticommutativity of Grassmannians we must distinguish between “even” and “odd” functions A(τ, τ ∗ , η, η∗ ). Even ones additively contain only bilinear terms and/or the quadruple of the arguments while odd functions contain only linear or trilinear terms. The product rule can then be written as  ∂τ A(τ, τ ∗ , η, η∗ )B(τ, τ ∗ , η, η∗ ) = Aτ (τ ∗ , η, η∗ )B(τ ∗ , η, η∗ ) ± A(τ ∗ , η, η∗ )Bτ (τ ∗ , η, η∗ )

11 This

subsection is lifted from Ref. [17], word by word.

(6.8.65)

6.8 Zero Dimensional Sigma Model for the CUE

271

where the plus (minus) sign refers to even (odd) A. The absence of the argument τ on the right-hand side of the forgoing rule and the alternative ± constitute the announced peculiarity. We promise to pedantically adhere to the notation thus introduced, while evaluating G: the absence of any of the four arguments from any function means either that the argument has been removed by differentiation (as indicated by the corresponding index) or that a cofactor is differentiated (whereupon in the function in question the argument must be replaced with0 as well). Writing D(τ, τ ∗ , η, η∗ ) = sdet 1 − m(τ, τ ∗ , η, η∗ ) we have G = ∂η∗ ∂η ∂τ ∗ ∂τ D(τ, τ ∗ , η, η∗ )−N . In differentiating we respect that the superdeteminant D(τ, τ ∗ , η, η∗ ) is even, to get  G = ∂η∗ ∂η ∂τ ∗ − NDτ (τ ∗ , η, η∗ )D(τ ∗ , η, η∗ )−(N+1)  = ∂η∗ ∂η − NDτ ∗ τ (η, η∗ )D(η, η∗ )−(N+1)

(6.8.66)

− N(N + 1)Dτ (η, η∗ )Dτ ∗ (η, η∗ )D(η, η∗ )−(N+2) ; in the last step we have used that Dτ (τ ∗ , η, η∗ ) is odd. Continuing in this vein we get a sum of 15 terms, all with the four derivatives distributed over up to four superdeterminants. However, eleven of these terms vanish due to Dτ = Dτ ∗ = Dη = Dη∗ = 0. Indeed, a glance at the definitions of the matrices Δ± in (6.8.56) reveals that upon setting three of the four Grassmannians to zero, one of these matrices must vanish while the other becomes purely off-diagonal, such that the supertrace str ln(1 − m) vanishes; moreover, that supertrace is devoid of the bilinear summands τ η and τ ∗ η∗ and we conclude Dητ = Dη∗ τ ∗ = 0. We finally use D = D(0, 0, 0, 0) = 1 and are left with   G = −NDη∗ ητ ∗ τ + N(N + 1) Dτ ∗ τ Dη∗ η + Dη∗ τ Dητ ∗ .

(6.8.67)

We must now inspect the 2 × 2 superdeterminant D(τ, τ ∗ , η, η∗ ). Momentarily dropping arguments we write  D = sdet

1 − mBB −mBF −mFB 1 − mF F

 =

1 − mBB mBF mFB − . 1 − mF F (1 − mF F )2

(6.8.68)

By its definition (6.8.55), the matrix m has even diagonal entries and odd offdiagonal ones. Furthermore, all four entries are nilpotent. In particular, mnF F = 0 for n > 2, and therefore we can expand as D − 1 = −mBB + mF F − mBB mF F + m2F F − mBF mFB (1 + 2mF F ) . (6.8.69) That expansion must equal D(τ, τ ∗ , η, η∗ ) = 1 +Dτ ∗τ τ τ ∗ +Dη∗ η ηη∗ +Dη∗ τ τ η∗ + Dητ ∗ τ ∗ η + Dη∗ ητ ∗ τ τ τ ∗ ηη∗ , with the coefficients appearing in (6.8.67); note that

272

6 Supersymmetry and Sigma Model for Random Matrices

we are back to the promised pedantry. To determine these coefficients we need to inspect the matrix m in detail. Momentarily writing δ+ = a − c, and δ− = b − d we have    mBB (τ, τ ∗ , η, η∗ ) = (1 − ablB)−1 lB − δ+ bτ τ ∗ + δ− aηη∗ − δ+ δ− τ τ ∗ ηη∗  ) − δ+ δ− −lB lF τ ∗ η    mF F (τ, τ ∗ , η, η∗ ) = (1 − cdlF )−1 lF − δ+ dτ τ ∗ + δ− cηη∗ − δ+ δ− τ τ ∗ ηη∗  ) − δ+ δ− −lB lF τ η∗    mBF (τ, τ ∗ , η, η∗ ) = (1 − ablB)−1 lB δ+ b + δ− ηη∗ τ )    + −lB lF δ− c − δ+ τ τ ∗ η    mFB (τ, τ ∗ , η, η∗ ) = (1 − cdlF )−1 lF δ+ d + δ− ηη∗ τ ∗ )    + −lB lF δ− a − δ+ τ τ ∗ η∗ . Only the diagonal elements and the product mBF mFB of the off-diagonals contain terms bilinear in the Grassmannians. We read out the second derivatives Dτ ∗ τ =

(a − c)(blB − dlF ) , (1 − ablB )(1 − cdlF ) Dη∗ τ = −Dητ ∗

(b − d)(alB − clF ) , (1 − ablB )(1 − cdlF ) √ (a − c)(b − d) −lB lF =− (1 − ablB )(1 − cdlF ) Dη∗ η = −

(6.8.70)

and their combination Dτ ∗ τ Dη∗ η + Dη∗ τ Dητ ∗ = −

(a − c)(b − d)(ablB − cdlF )(lB − lF ) . (1 − ablB)2 (1 − cdlF )2

(6.8.71)

The fourth derivative, in turn, receives contributions from all terms on the right-hand side of the expansion (6.8.69), Dη∗ ητ ∗ τ =

(a − c)(b − d)(1 + cdlF )(lB − lF ) . (1 − ablB )(1 − cdlF )2

(6.8.72)

Putting together Eqs. (6.8.71) and (6.8.72) we get the announced result (6.8.60) for the Grassmann integral G.

6.9 The Zero Dimensional Sigma Model for COE and CSE

273

6.9 The Zero Dimensional Sigma Model for COE and CSE Let us move on to using the supersymmetry approach for the two ensembles of Dyson’s where time reversal invariance reigns, the orthogonal (COE) and symplectic (CSE) circular ensembles. We again start from the generating function Z(a, b, c, d) defined in (6.7.3), but now · refers to the COE or CSE average. Following similar steps as for the CUE with a few crucial additional ingredients we will be able to derive a zero-dimensional nonlinear σ -model for both ensembles. We will focus on the new ideas that enter the derivation and keep repetition to a minimum. As before we will follow closely the original work of Zirnbauer [16]. One key ingredient will be to replace the COE and CSE averages by averages over the whole unitary group according to Eqs. (5.8.20) and (5.8.21): when averaging over U ∈ COE (resp. U ∈ CSE) we may write U = V V˜ (resp. U = V V Z ) and average V over the unitary group U(N) (resp. U(2N))—or equivalently CUE (with ˜ appropriate choice of dimension).  0 1  As a reminder, V is the usual transpose while ) is the symplectic transpose or Σ-transpose of V . V Σ = Σ V˜ Σ˜ (with Σ = −1 0 As the derivation of the σ -model for the two ensembles follows basically the same steps with the only difference that they use a different transpose we will write the identities (5.8.20) and (5.8.21) as f (U )CXE = f (V V X )CUE≡U(gN)

(6.9.1)

where CXE is either COE or CSE, and the X-transpose V X will be the standard transpose in the orthogonal and the Σ-transpose in the symplectic case, or V X = XV˜ X˜

(6.9.2)

˜ where X = 1 for the COE and X = Σ for the CSE. In either case X−1 = X, a property to be used frequently in the following. Moreover we write the matrix dimension of the unitary group as gN where g = 1 in the orthogonal and g = 2 in the symplectic case. With the replacement U → V V X the generating function (6.7.3) becomes 5    † det 1 − cV V X det 1 − dV X V †   ZCXE (a, b, c, d) =   † det 1 − aV V X det 1 − bV X V † 3

.

(6.9.3)

V ∈U(gN)

The matrices U and their representatives V V X live in what was previously and will again be called the quantum dynamical Hilbert space QD which is taken as N dimensional for the COE and 2N dimensional for the CSE. The next step in the derivation of the CUE σ -model was to write each determinant as a Gaussian superintegral over commuting and anticommuting variables. This would however lead to an exponent which is quadratic in the unitary matrix V and inhibit the use of the color-flavor transformation for the average over the unitary group. A useful

274

6 Supersymmetry and Sigma Model for Random Matrices

trick is to double the dimension according to  √    √ 1 − aV √ X det 1 − aV V X = det ≡ det(1 − aW ) − aV 1

(6.9.4)

in all four determinants before introducing the Gaussian superintegral.  √ It is aesthetically convenient to introduce an extra X-transpose det 1 − bW †   √ † in the advanced sector. This surely does not change the = det 1 − bW X corresponding determinant. The doubling of dimension in (6.9.4) introduces an auxiliary degree of freedom with components x = 1, 2. We are now ready to write the determinants as a Gaussian superintegral over supervectors ψ and ψ ∗  ZCXE =

∗T ψ −ψ ∗T ψ + − −

d(ψ, ψ ∗ ) e−ψ+

= > ∗T 1/2 ∗T 1/2 X † eψ+ eˆ+ W ψ+ +ψ− eˆ− W ψ−

; CUE

(6.9.5) the products in the exponents are written in matrix notation with summation understood to extend over QD, BF, and the auxiliary space. The foregoing superintegral is basically the same as (6.7.13), with the following modifications: the matrices eˆ± are replaced by the square roots    √a  √b 1/2 1/2 √ √ , eˆ− = (6.9.6) eˆ+ = c d which operate, as in the previous treatment of the CUE, in the two dimensional ∗ is now 4gN (with 2gN Bose-Fermi space BF. The dimension of ψ± and ψ± commuting and as many anti-commuting entries). The main difference however lies in the auxiliary structure of W due to the dimension doubling. Using that structure in the exponent within · we get †

∗T ∗T ψ+ eˆ+ W ψ+ + ψ− eˆ− W X ψ− = 1/2

1/2

(6.9.7) †

∗T ∗T ∗T ∗T ψ1,+ eˆ+ V ψ2,+ + ψ2,+ eˆ+ V X ψ1,+ + ψ1,− eˆ− V X ψ2,− + ψ2,− eˆ− V † ψ1,− 1/2

1/2

1/2

1/2

and see that the exponent in the generating function (6.9.5) is linear in V , V † , V X , † and V X . If we want to apply the color-flavor transformation only V and V † are † allowed. We can get rid of the undesirables V X and V X by invoking χ T Aξ = BF T T ξ A σ3 χ for a supermatrix A and supervectors ξ and χ to write 1/2 ˜ BF ∗ ∗T 1/2 X T ψ2+ eˆ+ V ψ1+ = ψ1+ Xeˆ+ V Xσ 3 ψ2+ , 1/2 ∗T 1/2 X† T ˜ 3BF ψ ∗ ; ψ1− eˆ− V ψ2− = ψ2− Xeˆ− V † Xσ 1−

(6.9.8)

6.9 The Zero Dimensional Sigma Model for COE and CSE

275

The replacements (6.9.8) allow us to combine the advanced and retarded contributions involving W to the exponent in the superintegral (6.9.5) as †

∗T ∗T ψ+ eˆ+ W ψ+ + ψ− eˆ− W X ψ−      eˆ 1/2V 0 ψ 2,+ ∗T T + = ψ1,+ ψ1,+ X 1/2 ˜ ∗ σ3BF Xψ 0 eˆ+ V 2,+      V † eˆ 1/2 ψ 0 1,− ∗T ψ T X − + ψ2,− 1/2 2,− ˜ ∗ σ3BF Xψ 0 V † eˆ− 1,− 1/2

1/2

(6.9.9)

∗T ∗T ≡ χ1,+ eˆ+ V χ2,+ + χ2,− V † eˆ− χ1,− ; 1/2

1/2

∗ , χ , here, we have compacted the notation by introducing four new vectors χ1+ 2+ ∗ χ2− , and χ1− ; their components are distinguished by a two-valued index which we choose to denote as m =↑, ↓. The space spanned by these two components is customarily called ‘TR’ (for time reversal). It is convenient to accompany these four TR vectors by four more, ∗ , χ , and χ ∗ , defined such that the components can be compactly χ1+ , χ2+ 2− 1− written as

χx,±,↑ =ψx,±

∗ ˜ x,± χx,±,↓ =σ3BF Xψ

(6.9.10a)

∗ ∗ χx,±,↑ =ψx,±

∗ ˜ x,± . χx,±,↓ =Xψ

(6.9.10b)

∗ are linearly dependent, It is then easily verified χx,± and χx,±



∗ χx,± = τ χx,±

with

τ=

0 σ3BF X X˜ 0

 .

(6.9.11)

That linear dependence (6.9.11) means that there are just as many independent χ, χ ∗ as ψ, ψ ∗ . For later use let us note τ 2 = σ3BF

and τ 3 = τ −1 = τ T = τ˜ .

(6.9.12)

Furthermore, we may rewrite the quadratic form in the first exponential in the Gaussian superintegral (6.9.5) ∗T ∗T ψ+ ψ+ + ψ− ψ− =

 1  ∗T ∗T ∗T ∗T χ1,+ χ1,+ + χ2,+ χ2,+ + χ1,− χ1,− + χ2,− χ2,− 2 (6.9.13)

such that we can now express the whole exponent in the Gaussian superintegral (6.9.5) in terms of the newly introduced vectors. As that exponent is explicitly linear in V and V † the stage is prepared to perform the color-flavor transformation.

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6 Supersymmetry and Sigma Model for Random Matrices

In analogy to (6.8.2) we have > = ∗T eˆ 1/2 V χ ∗T † 1/2 χ1,+ 2,+ +χ2,− V eˆ− χ1,− + e  =



V ∈U(gN)

˜ sdet 1 − ZZ ˜ d(Z, Z)

gN

∗T

1/2

eχ1,+ eˆ+

1/2 ∗T Zχ ˜ 2,+ Z eˆ− χ1,− +χ2,−

.

(6.9.14)

The color-flavor transformation is now meant for gN colors and F = 2 flavors as proven in Sect. 6.8.4; the new flavor index is m =↑, ↓; the flavor space is nothing but TR. Therefore, the supermatrices Z and Z˜ are 4 × 4 matrices in BF⊗TR. We now point to the fact that with (6.9.13) and (6.9.14) the whole integrand in the integral representation (6.9.5) of the generating function ZCXE has become a product of two factors, one for each value of the auxiliary index x = 1, 2 labelling the vectors χx,± . We now change integration variables from (ψ, ψ ∗ ) to χ↑ , χ↓ thus χ↑ ,χ↓ picking up the Berezinian J ( ψ,ψ ∗ ) = 1. We thus write  ZCXE =

˜ sdet(1 − ZZ) ˜ gN d(Z, Z)

 d(χ↑ , χ↓ )

(6.9.15)

  1/2 ∗T 1/2 ∗T ˜ × exp χ1+ Zχ2+ eˆ+ Z eˆ− χ1− + χ2− × exp (1)

 1  ∗T ∗T ∗T ∗T χ1+ χ1+ + χ1− χ1− + χ2+ χ2+ + χ2− χ2− 2 (2)

= ZCXE ZCXE with (1)

(2) ZCXE



 1  ∗T 1/2 1/2 ∗T ∗T χ1+ + χ1− χ1− 2χ1+ eˆ+ Z eˆ− χ1− + χ1+ 2   1  ∗T ˜ ∗T ∗T = d(χ2↑ , χ2↓ ) exp 2χ2− χ2+ + χ2− χ2− . (6.9.16) Zχ2+ + χ2+ 2

ZCXE =

d(χ1↑ , χ1↓ ) exp

The exponentials in both of these two factors enjoy a certain symmetry which is best revealed by looking at the scalar product χμ∗T χμ where μ (like μ ) momentarily comprises the auxiliary quasispin index x = 1, 2 and the AR index with values + and −. The scalar product involves summation over the QD (color), TR (flavor), and BF indices. We have χμ∗T χμ = (τ χμ )T χμ ; on the other hand, by pulling χμ∗T to the right of χμ we get χμ∗T χμ = χμT σ3BF χμ∗ = χμT τ˜ 2 τ χμ = (τ χμ )T χμ and conclude (τ χμ )T χμ = (τ χμ )T χμ ,

(6.9.17)

i.e. symmetry under χμ ↔ χμ . That symmetry is a tautology for μ = μ and ˜ To reveal thus for the four terms in the exponents in (6.9.16) not involving Z, Z.

6.9 The Zero Dimensional Sigma Model for COE and CSE

277

˜ we the corresponding property of the remaining two terms (which do contain Z, Z) consider χμ∗T Mχμ with M a supermatrix in QD⊗BF⊗TR. Analogous reasoning gives (τ χμ )T Mχμ = (τ χμ )T M τ χμ

(6.9.18)

with the ‘τ -transpose’ M τ of M M τ = τ M T τ −1 .

(6.9.19)

For use below we note the quickly verified properties (M τ )τ = M, sdetM = sdetM τ , and strM = strM τ . The symmetry (6.9.18) allows to symmetrize the two terms in the exponents in (6.9.16) involving Z, Z˜ as ∗T ˜ ˜ 2,+ + (τ χ2,+ )T Z˜ τ χ2,− . 2χ2,− Zχ2,+ = (τ χ2,− )T Zχ

(6.9.20)

∗T 2χ1+ eˆ+ Z eˆ− χ1− = (τ χ1,+ )T eˆ+ Z eˆ− χ1,− + (τ χ1,− )T eˆ− Z τ eˆ+ χ1,+ . 1/2

1/2

1/2

1/2

1/2

1/2

Once the latter identities are inserted in (6.9.16) the integrands therein are not only manifestly symmetric but also contain only the independent integration variables χx,± . We are thus set to do the Gaussian superintegrals involved. Before undertaking that final task we would like to point to an important ˜ While the τ simplification of the τ -transposes of the supermatrices Z and Z. transpose (6.9.19) for arbitrary 4 × 4 matrices M ∈ TR⊗BF contains flavor, Bose-Fermi, and color indices we note that the τ -transpose of Z and Z˜ may be written in terms of a reduced τ matrix  ⎧ ⎪ 0 σ3BF ⎪ ⎪ in the orthogonal case ⎪ ⎨ 1 0  τ0 =  (6.9.21) ⎪ 0 iσ3BF ⎪ ⎪ ⎪ in the symplectic case. ⎩ −i 0 With this we may write the τ -transpose of Z as well as Z τ = τ0 Z T τ0−1 purely in the TR and BF degrees of freedom of Z. The same applies by analogy to Z˜ τ . Of course, in the orthogonal case X = 1 such that τ = τ0 anyway (modulo tensor products with identity matrices). In the symplectic cases we have X = Z and we basically replace Z → i when going from τ to τ0 . This translates the property X2 = 1 (orthogonal case) and X2 = Z 2 = −1 (symplectic case) into the space of the supermatrices. We thus see that the properties T 2 = ±1 of the time-reversal operator lead to a different natural τ -transposition in the corresponding supermatrix spaces.

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6 Supersymmetry and Sigma Model for Random Matrices

Last, we turn to the Gaussian integrals (6.9.16) for the generating function ZCXE and write    1 (2) (6.9.22) ZCXE = d(χ2 ) exp − (τ χ2 )T mχ2 2 T , χ T ), where we have collected advanced and retarded components as χ2T = (χ2,+ 2,− and introduced the matrix   1 Z˜ τ m= ˜ (6.9.23) Z 1

which is 2 × 2 in AR. As m is diagonal in the gN-dimensional color space QD the integral separates further into a product of gN equivalent Gaussian integrals. The reader is kindly invited to check12  −gN/2 (2) ZCXE = sdet 1 − Z˜ τ Z˜ .

(6.9.24)

In analogy the integral over the fields χ1 gives −gN/2  (1) ZCXE = sdet 1 − Z τ eˆ+ Z eˆ−

(6.9.25)

τ = eˆ . Altogether we arrive at the exact expression where we used eˆ± ±

 ZCXE =

 gN ˜ sdet 1 − ZZ ˜ d(Z, Z)

(6.9.26)

−gN/2  −gN/2  sdet 1 − Z˜ τ Z˜ , × sdet 1 − Z τ eˆ+ Z eˆ− valid for arbitrary dimension N. There are 16 independent real commuting and 16 ˜ independent anticommuting variables contained in the matrices Z and Z. The appearance of exponents ∝ N in the foregoing integrand invites a saddlepoint approximation for large N. To find the saddle configurations of Z, Z˜ it is well to recall that our generating function ZCXE (a, b, c, d) allows to get the twopoint correlator C(e) of the density of energy levels through suitable derivatives (see (6.7.8) and (6.7.6)). Therefore, and in order to get a reasonable limit of C(e) as N → ∞ we may restrict the range of the variables a, b, c, d to a O( N1 )

12 Recall from Sect. 5.13 that ordinary (over commuting variables, that is) real Gaussian integrals lead to inverse square roots of determinants while such integrals over anti-commuting variables give Pfaffians (which also square to determinants). It is straightforward to generalize these two cases to ‘real’ Gaussian super-integrals where one accordingly arrives at square roots of superdeterminants. See also Sect. 5.14.4 for an application of Pfaffians.

6.9 The Zero Dimensional Sigma Model for COE and CSE

279

neighborhood of unity. The BF matrices eˆ± then are close to the unit matrix, eˆ± = 1 + O(1/N). The above superintegral takes the leading-order form  ZCXE =

˜

˜ p(Z, Z) ˜ e−(gN/2)s(Z,Z) d(Z, Z)

(6.9.27)

˜ and s(Z, Z) ˜ independent of N and the preexponential p(Z, Z) ˜ fixed with p(Z, Z) by the O(1/N) terms in eˆ± . Saddles are determined by the exponent   ˜ ˜ = −2Tr ln(1 − ZZ) ˜ + ln(1 − Z τ Z) + ln(1 − Z˜ τ Z) s(Z, Z) through

∂s ∂Z

=

∂s ∂ Z˜

(6.9.28)

= 0. The easily found solution Z˜ = Z τ

(6.9.29)

gives the configuration exclusively contributing to the integral in the limit N → ∞. The independent variables are thereby reduced in number to 16. Of these, 8 are Bosonic and 8 Fermionic. The asymptotic generating function  ZCXE ∼  ∼

  ⎞ gN 2 ˜ sdet 1 − ZZ  ⎠ d(Z) ⎝ sdet 1 − Z˜ eˆ+ Z eˆ− ⎛

(6.9.30)

d(Z)e−(gN/2)S ,

  ˜ + ln(1 − Z τ eˆ+ Z eˆ− ) S = −str ln(1 − ZZ)

(6.9.31)

looks like the CUE one, c.f. (6.8.9) and (6.8.10), except for the replacement N → gN/2, the dimension doubling of the supermatrices (due to the time reversal space TR), and the saddle-point condition Z˜ = Z τ . The latter restriction amounts to a difference of status: the foregoing result for ZCXE is valid only asymptotically, for large N, while the CUE counterpart is exact for any value of N. As we have done for the CUE in Sect. 6.8.1 we can finally rewrite the foregoing action S so as to accommodate the integration variable Z in a larger supermatrix Q. With no difference in argument we get the analogue of (6.8.11)   ˜ S = Str ln (1 − E BF )Q(Z, Z)Λ + 1 + E BF ,    1 Z −1 ˜ = 1Z 1 0 Q(Z, Z) ≡ T ΛT −1 , Z˜ 1 0 −1 Z˜ 1  eˆ 0  + . E BF = 0 eˆ−

(6.9.32)

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6 Supersymmetry and Sigma Model for Random Matrices

The ‘central object’ of the sigma model, Q, again enjoys the properties Q2 = 1 and StrQ = 0; it is now 8 × 8 and lives in (a submanifold of) AR⊗TR⊗BF. We could write out the analogues of Sect. 6.8.2 and describe these manifolds for the COE and CSE but prefer to refer the reader to Zirnbauer’s original work [16] and references therein. Finally, we note that the above sigma model representations for ZCXE can be evaluated by adapting the arguments of Sect. 6.8.5. The respective cluster functions come out as in Chap. 5 where much simpler techniques were shown to work. Readers having labored through the present chapter must be congratulated for their persistence. They may look forward to extensions of the sigma model to other types of problems where more elementary methods do not apply. It is instructive to parametrize the 4 × 4 supermatrix Z in terms of four 2 × 2 supermatrices, ZC , ZD , Z˜ C and Z˜ D that make up the blocks of Z in TR as13  ⎧ ⎪ ZD ZC ⎪ ⎪ ⎪ ⎨ Z˜ T σ BF Z˜ T D  Z=  C 3 ⎪ ZC Z ⎪ D ⎪ ⎪ ⎩ T BF T ˜ −ZC σ3 Z˜ D  ⎧ BF T ⎪ ⎪ Z˜ D σ3 ZC ⎪ ⎪ ⎨ Z˜ T ZD C τ ˜  Z =Z=  ⎪ ˜ D −σ BF Z T Z ⎪ C 3 ⎪ ⎪ ⎩ ˜ T ZD ZC

in the orthogonal case; in the symplectic case; (6.9.33) in the orthogonal case; in the symplectic case.

The difference between orthogonal and symplectic case is a single sign. It turns out however that this sign changes the whole integral in a very fundamental way: this sign is the signature of the different geometry described by Z in the two cases.

6.10 Universality of Spectral Fluctuations: Non-Gaussian Ensembles Hackenbroich and Weidenmüller presented an alternative derivation of the two-point function of ensembles of random matrices [18] which does not assume Gaussian

13 The

block ZD is called diffusion mode and ZC the Cooperon mode, names invented in the disordered systems community [3]. In the CXE sigma model these two modes are coupled as they expressed as blocks in a larger supermatrix. If one breaks time-reversal symmetry continuously the diffusion mode remains massless but the Cooperon mode acquires a mass. If that mass is sufficiently large (greater than 1/N) one may neglect the Cooperon mode and what is left is the standard CUE result (i.e. time-reversal is fully broken).

6.10 Universality of Spectral Fluctuations: Non-Gaussian Ensembles

281

statistics for the matrix elements14 Hij . These ensembles are characterized by matrix densities of the form P (H ) ∝ exp[−NTr V (H )] ,

(6.10.1)

constrained only such that moments exist. The appearance of the trace secures invariance of the ensemble w.r.t. the desired “canonical” transformations, be these unitary, orthogonal, or symplectic. Here, we shall confine ourselves to the unitary case of complex Hermitian matrices H . By choosing V (H ) = H 2 , we would be back to the Gaussian ensemble for which the mean density of levels is given by the semicircle law. For general V (H ), that latter law does not reign, but after rescaling the energy axis in the usual way, e = EN(E), we shall recover the same twopoint function as for the GUE above. This will be no surprise for the reader who has studied level dynamics and appreciated, in particular, the arguments of Sect. 11.10. Universality of spectral fluctuations is at work here. Once we have ascertained the independence of the two-point function Y (e) of the function V (H ) and invoke the ergodicity à la Pandey (see Sect. 5.14.2), we shall have gone a long way toward understanding why a single dynamical system with global chaos in its classical phase space has universal spectral fluctuations. As a preparation to the reasoning of Hackenbroich and Weidenmüller, we need to familiarize ourselves with delta functions of Grassmann variables.

6.10.1 Delta Functions of Grassmann Variables For Grassmann as for ordinary variables, the delta function is defined by Lehmann et al. [24]  (6.10.2) f (η0 ) = dη δ(η − η0 )f (η) . With f (η) = f0 + f1 η, we check the simple representations δ(η − η0 ) =

1 i

 dσ eiσ (η−η0 ) = η − η0 .

(6.10.3)

Neither representation suggests drawing anything peaked, but who would want to draw graphs for functions of Grassmann variables anyway. A more serious comment is that δ(aη) = aδ(η) for an arbitrary complex number a; in particular, δ(−η) = −δ(η). The latter two properties are well worth highlighting since they contrast to δ(ax) = |a|−1 δ(x) for real a. The first of the representations in (6.10.3)

14 For

recent progress on the universality problem within RMT see Refs. [19–23].

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6 Supersymmetry and Sigma Model for Random Matrices

is obviously analogous to the familiar Fourier integral representation of the ordinary # delta function, δ(x) = (2π)−1 dk exp ikx. Thus equipped, we can proceed to delta functions of supermatrices. Starting with the simplest case, 2 × 2 supermatrices of the form Q=

 ∗ aη , η ib

(6.10.4)

we define δ(Q − Q ) as the product of delta functions of the four matrix elements ∗

δ(Q − Q ) = δ(a − a )δ(b − b )δ(σ ∗ − σ )δ(σ − σ )

(6.10.5)

and note the integral identities    1 ˆ exp iNStrQ(Q ˆ dQ − Q ) 2π    ˆ f (Q ) = dQd Qˆ exp iNStrQ(Q − Q ) f (Q) ;

δ(Q − Q ) =

(6.10.6)

all integration measures are defined here according to our convention dQ = (dadb/2π)dσ ∗dσ ; note that the factor N in the exponent is sneaked in at no cost, for the sake of later convenience. When generalizing to 2n × 2n supermatrices, replacing a in (6.10.4) by an n×n Bose √ block etc., we still accompany each Bosonic differential increment with a factor 1/ 2π and therefore must replace the factor 2 (2π)−1 in the first of the identities (6.10.6) by (2π)−n to get the correct power of 2π required by the Fourier-integral representation of the ordinary delta function. The second of the identities (6.10.6) remains intact for any n.

6.10.2 Generating Function We return to the generating function (6.6.9) for the two-point function and average over the non-Gaussian ensemble (6.10.1),  ˆ = dH P (H )Z(E) ˆ Z(E)   ˆ = (−1)N dΦ ∗ dΦ dH P (H ) exp[i Φ † L(H − E)Φ] .

(6.10.7)

The admitted non-Gaussian character of P (H ) forbids us from resorting to the Hubbard–Stratonovich transformation employed in Sect. 6.6.1. We still enjoy the unitary invariance P (H ) = P (H ) with H = U H U † and U an arbitrary unitary N × N matrix. Due to that invariance, the integrand of the foregoing supervector

6.10 Universality of Spectral Fluctuations: Non-Gaussian Ensembles

283

integral is, after the ensemble average, a “level-space scalar”, i.e., it depends on the vector Φ and its adjoint only through N 

∗ Φαi Φβi Lβ = N Q˜ αβ Lβ ;

(6.10.8)

i=1

note that the 4 × 4 supermatrix Q˜ was already encountered in (6.6.11). Instead of Hubbard–Stratonovich, now we use the second of the delta function identities (6.10.6) for 4 × 4 supermatrices and f (Q) = 1 to write15   ˆ = (−1)N dΦ ∗ dΦ dQd Qˆ exp[iNStr Q(Q ˆ ˜ ˆ Z(E) − QL)]exp[i Φ † L(H − E)Φ] = (−1)

N

= (−1)

N



 ˆ ˆ ˆ dQd Q exp[iNStr QQ] dΦ ∗ dΦexp[i Φ † L(H − Eˆ − Q)Φ]



 −1 ˆ ˆ Sdet L(H − Eˆ − Q) dQd Qˆ exp(iNStr QQ)

  −1 ˆ ˆ Sdet (H − Eˆ − Q) = dQd Qˆ exp(iNStr QQ) 

ˆ exp Str ln(H − Eˆ − Q) ˆ . = dQd Qˆ exp(iNStr QQ)

(6.10.9)

Even though the reader should be at peace by now with compact notation, it may be well to spell out that the superdeterminant above is 4N × 4N and that within it we must read H as the tensor product of the N × N random matrix H in level space with the 4 × 4 unit matrix in superspace, whereas conversely Eˆ and Qˆ are 4 × 4 matrices in superspace and act like the N × N unit matrix in level space. At this point, we may imagine the matrix H diagonalized in level space and do the ensemble average with the help of the joint density of levels P (E) = N exp[−N

 i

V (Ei )]

(Ei − Ej )2

(6.10.10)

i 1 would accompany similar terms in Sc .

7.4.3 Semiclassical Limit We now represent the supermatrices representing fluctuations, Zd and Z˜ d (x) modes by Wigner functions Zd (x), Z˜ d (x), with x designating the pertinent phase-space variables. Then # # the tracelessness (7.4.3) is expressed by phase-space integrals, dxZd(x) = dx Z˜ d(x) = 0, and the QD trace of a product of Zd and Z˜ d becomes # the integral trQD AZ˜ d BZd = dxAZ˜ d (x)BZd (x) with A, B arbitrary # BF matrices. The integration range for all x integrals is the phase-space volume dx = Ω. The stroboscopic time evolution is accounted for by a propagator F , U Z˜ d U † → F Z˜ d (x) .

(7.4.6)

The stage is thus set for implementing the semiclassical limit. To within corrections of higher than first order in h¯ the Wigner function propagator becomes the classical Frobenius-Perron operator [13]. The latter describes decay of the classical phase-space density towards ergodic equilibrium, described by the constant-inphase-space Wigner function Ω1 obeying F Ω1 = Ω1 .

7.4 Perturbative Account of Fluctuations

295

Different types of classical decay are possible [14]. It may here suffice to look at the case of purely exponential decay, with complex frequencies λμ called Pollicott-Ruelle resonances. These come with normalizable biorthogonal right and left eigenfunctions of F , to be denoted as c˜μ (x) (right) and cμ (x) (left). The

eigenvalues of F are e−λμ . We expand as Z˜ d (x) = μ z˜ μ c˜μ (x), the latter sum excluding the ergodic stationary eigenfunction of F . The free-fluctuations and the coupling parts of the action then become sums over resonances, Sd =



  Sdμ = str z˜ μ zμ − e−λμ z˜ μ eˆ− zμ eˆ+ ,

(7.4.7)

Scμ ,

(7.4.8)

μ

Sc =

 μ

Scμ =



  e−λμ · ,

μ

   · = str z˜ μ eˆ− zμ eˆ+ −

1 1 − B˜ Bˆ

˜ z˜ μ (1 − BB)

1 1 − Bˆ B˜

 ˜ μ eˆ+ , eˆ− (1 − B B)z

here and below the supertrace refers to BF only. It is important to note that the sum over resonances does not include the stationary phase-space density, the constantin-phase-space 1/Ω. In fact, what we have called ‘fluctuations’ up to now can and will hence be referred to as ‘decaying modes’. We now turn to the generating function Z given by the integral (7.2.8). The inte˜ Zd , Z˜ d . gration variables therein fall into independent subsets pertaining to B, B, Expanding the integrand in powers of the coupling and dropping terms of higher than first order in Sc we get  Z=

˜ e−N S0 d(B, B)



  d(Zd , Z˜ d ) e−Sd 1 − Sc + . . .

(7.4.9)

  ≡ Z0 Zd 1 − Sc  + . . . where Z0 is the generating function of the zero dimensional sigma model, Zd the factor exclusively contributed by the decaying modes, Zd =

- 1+ μ

 e−λμ (a − c)(b − d) , (1 − abe−λμ )(1 − cde−λμ )

(7.4.10)

and Sc  the relative correction due to the coupling. Like the full generating function Z and the mean-field one Z0 , the free-decay part Zd is now seen to deviate from unity by a summand proportional to (a − c)(b − d). The mean coupling must therefore also obey Sc  ∝ (a − c)(b − d) .

(7.4.11)

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7 Ballistic Sigma Model for Individual Unitary Maps and Graphs

7.4.4 Conditions for Universal Behavior The fruits of our perturbative labor for the correlator C(e) can now be harvested, using (6.7.8). The forgoing results for the generating function entail C(e) − C0 (e) =



2ei2e/N ∂+ ∂− Zd ± =0 N2

−



2ei2e/N ∂+ ∂− Sc ± =0 N2

(7.4.12)

 where C0 (e) denotes the CUE correlator and we have used Sc ± =0 = 0. The first of the two corrections to the universal form C0 (e) is fixed by (7.4.10) as the sum over resonances 

2ei2e/N ∂c ∂d Zd  =0 N2 ±

=

2 N2

ei2e/N−λμ μ (1−ei2e/N−λμ )2

∼

2 N2

e−λμ μ (1−e−λμ )2 .

(7.4.13)

We see that (1) the underlying classical dynamics must be endowed with a finite gap,   Δg = minλμ =0 1 − e−λμ > 0 .

(7.4.14)

and (3) the remaining sum over resonances must be finite, 

sinh−2 (λμ ) < ∞ .

(7.4.15)

μ

The latter two conditions are necessary for universal behavior, to within corrections of order 1/N 2 . The second correction term in the correlator can be shown to be no larger than O( N12 ) as well, under no further restriction on the dynamics. For the somewhat lengthy calculation leading to that result we refer the reader to Ref. [4]. We conclude that the conditions (7.4.14) and (7.4.15) are necessary and sufficient for the decaying-mode corrections to the universal two-point correlator to be of the order 1/N 2 . Interestingly, the CUE correlator takes values of that same order outside the periodically repeated windows of correlation decay and revival. Even more importantly, the system specific noise visible in the smoothed correlator (smoothed either by integration or by an imaginary part of the spacing variable 1/N Ime 1, see Sect. 5.20 or [12]) are of the order √1 (for the root N

mean square deviation), much larger than the N12 -corrections (to the correlator) due to the decaying modes. Of course, the √1 -noise R(e) could also be established in N the framework of the ballistic sigma model by determining the four-point function of the level density.

7.5 Quantum Graphs

297

7.5 Quantum Graphs 7.5.1 Directed Graphs and Their Spectra A quantum graph G is defined by a metric graph and a wave equation in the form of an eigenequation for a self-adjoint operator. We allow for V vertices and N edges (aka bonds) such that each edge e is attached to one vertex at either side and has length Le . We only consider directed connected graphs. For each edge a coordinate 0 ≤ xe ≤ Le and a wave function φe (xe ) are introduced. While for general quantum graphs one considers the Schrödinger equation one each edge we adopt, for simplicity, the first-order wave equation − iφe (xe ) = kφe (xe ) ,

e = 1, 2, . . . , N

(7.5.1)

with the local solution φe (xe ) = ae eikxe ; the amplitudes ae are (at this point) undetermined. Inasmuch as the sign of k fixes a direction on each edge we can speak of incoming and outgoing edges for each vertex. The wave equation needs to be accompanied by matching conditions at the vertices. Indeed, we would like to view the wave equation as the eigen-equation for a momentum operator that locally takes the form of the first-derivative operator on each edge (multiplied by the imaginary unit). To make that momentum operator self-adjoint [15], one must stipulate that each vertex has (1) as many incoming as outgoing edges and (2) a unitary scattering matrix assigned to it. A scattering matrix expresses the wave amplitudes on the outgoing edges as linear superpositions of the incoming amplitudes. Note that such a construction generally does not allow for time-reversal symmetry (i.e. an antiunitary operator commuting with the momentum operator). Solutions to the wave equation (7.5.1) with the matching conditions only exist for a discrete set of wave numbers—the momentum spectrum. That spectrum is fixed by the “characteristic equation” det(1 − U (k)) = 0 ,

U (k) = ST (k)

(7.5.2)

where U (k), S, and T (k) are three unitary N × N matrices which act on the wave function amplitudes on each edge. The matrix T (k)ee = δee eikLe is a diagonal matrix that contains the phase difference at the two ends of the edge, φe (Le ) = eikLe φe (0), while the scattering matrix S harbors all matching conditions: if there is a vertex such that e is an incoming edge and e an outgoing edge then See is the element of the scattering matrix at that vertex; otherwise, if the two edges are not connected through a vertex one has See = 0. Thus S encodes all matching conditions and also the connectivity of the graph. The matrix U (k) is called the quantum map and fully defines the quantum graph (connectivity, scattering amplitudes at vertices, and phases for the transport along edges). According to the characteristic equation, k belongs to the spectrum if there is an eigenfunction such

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7 Ballistic Sigma Model for Individual Unitary Maps and Graphs

that φe (0) = e U (k)ee φe (0) = e See eikLe φe (0), i.e. if there is a stationary vector under the quantum map. * +N Let eiθ! (k) !=1 be the set of the unimodular eigenvalues of the quantum map. The phases θ! (k) are continuous functions of k. It is easy to see [10] that these ! (k) functions are strictly increasing dθdk > 0, and that the spectrum of the graph is given by k!,n such that θ! (k!,n ) = 2πn (n ∈ Z). Note that we are now dealing* with+ two different spectra: the infinite momentum spectrum of the quantum graph k!,n n∈Z,!=1,...,N , and the parametric N-dimensional quasi-energy spectrum {θ! (k)} of the quantum map which depends on k as a parameter. We further specify our graph by not allowing for loops (edges starting and ending at the same vertex) or ‘parallel’ edges (i.e. there is at most one directed edge between any two vertices). Most importantly, we choose all edge lengths rationally independent. One reason why quantum graphs have become a paradigmatic model for quantum chaos (and especially spectral statistics thereof) is that both the sigma-model and semiclassical approaches can be introduced in a much cleaner way than for more general systems. We will introduce the semiclassical approach to spectral statistics in Chap. 10. It is based on the Gutzwiller trace formula that express the fluctuations in the density of states in a semiclassical approximation as a sum over periodic orbits of the corresponding classical Hamiltonian dynamics. For quantum graphs Kottos and Smilansky [6, 7] have derived an exact formula completely analogous to the Gutzwiller trace formula. Let us discuss this trace formula shortly before coming

back to the σ -model approach. For a directed graph with total length L = e Le the

Kottos-Smilansky trace formula for the density of (momentum) levels ρ(k) = !,n δ(k − k!,n ) may be written as ∞

ρ(k) =

 1 d L L − Im ln det(1 − U (k)) = + !p Arp eirk!p . 2π π dk 2π p

(7.5.3)

r=1

The summations in the last expressions are over all primitive periodic orbits p and their repetitions r. A periodic orbit in a directed graph is a cyclic sequence (e1 , e2 , . . . , en ) ≡ (e2 , . . . , en , e1 ) of directed edges such that ej +1 is outgoing from the vertex where ej ended (including the case j = n where en+1 = e1 so that the edges form a closed path). Orbits differing only in their starting points on one and the same closed path are considered equivalent, and a primitive periodic orbit is one that cannot be considered as a repetition of a shorter one. Each repetition of a periodic orbit contributes with a complex amplitude !p Arp and an additional

phase factor ei!p k where !p = nj=1 Lej is the length of the primitive orbit and 6 Ap = nj=1 Sej+1 ej is the product of all scattering amplitudes along the orbit. In Chap. 10 we will meet the Gutzwiller trace formula where the sum is over primitive periodic orbits of the corresponding Hamiltonian dynamics and their repetitions, the amplitude Ap becomes a classical stability amplitude, and the phase k!p is replaced by the classical action of the periodic orbit in units of h. ¯ In the same chapter

7.5 Quantum Graphs

299

we will learn how the trace formula may be used to understand universal spectral statistics in terms of action correlations in double sums over periodic orbits. The equivalent approach for quantum graphs (here in terms of length correlations) has been pioneered by Kottos and Smilansky on the level of the diagonal approximation [6, 7] and later by Schanz, Berkolaiko and Whitney for off-diagonal contributions [16, 17]. For an introduction to the periodic-orbit approach for quantum graphs and a derivation of the Kottos-Smilansky trace formula we refer to [10].

7.5.2 The Sigma Model Approach to Spectral Statistics Let us now consider the spectral statistics of quantum graphs along the lines of the previously treated Floquet maps. To highlight the analogy of the quantum map for directed graphs to the Floquet maps of periodically driven dynamics we shall speak of the N dimensional habitat of U, S, T as QD. One can show [10] that the spectral statistics of the quantum graph under study is equivalent to the k-averaged spectral statistics of the quantum map—in certain situations even with complete rigour [18]. Therefore, we shall employ that kaverage. As in the proceeding sections we describe two-point correlations in terms of a generating function, 3

5  det (1 − cU (k)) det 1 − dU (k)†  ;  Z(a, b, c, d) = det (1 − aU (k)) det 1 − bU (k)†

(7.5.4)

k

the k-average 1 F (k)k := lim K→∞ K



K/2 −K/2

F (k)dk ,

(7.5.5)

contains the center-phase average previously used in the Floquet map setting. However, the k-average is much stronger than just center phase since Z(a, b, c, d) depends on the parameter k via N unimodulars {eikLe }N e=1 . In fact, the map k → ikL ikL 1 N (e ,...,e ) describes a ‘trajectory’ on an N-dimensional torus, k playing the role of a time. If the trajectory is ergodic on the torus with uniform measure we may replace the k-average (formally a time average) by an average over the torus (the corresponding phase-space average). The trajectory is indeed ergodic in our case of rationally independent edge lengths Le (‘frequencies’). We can therefore replace kLe → ϕe and  (·)k →

dN ϕ (·) . (2π)N

(7.5.6)

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7 Ballistic Sigma Model for Individual Unitary Maps and Graphs

The generating function thus involves an N-fold phase average,   d N ϕ det (1 − cT (ϕ)S) det 1 − dS † T (−ϕ)  , (2π)N det (1 − aT (ϕ)S) det 1 − bS † T (−ϕ)

 Z(a, b, c, d) =

(7.5.7)

where T (ϕ)ee = δee eiϕe is now a diagonal matrix of random phases. Most interestingly, the edge lengths have disappeared altogether, due to the N-fold phase average; this is a first glimpse of universality. In order to see universality in the sense of the BGS conjecture we now move on and look for the conditions for the so called unistochastic ensemble [19] T (ϕ)S to be equivalent to the CUE. The construction of the sigma model proceeds as in Sect. 7.2 and again leads to  Z(a, b, c, d) =

˜ ˜ e−S (Z,Z) d(Z, Z)

(7.5.8)

    ˜ = −str ln 1 − ZZ ˜ ˜ † eˆ− Z eˆ+ . S(Z, Z) + str ln 1 − S ZS

(7.5.9)

Comparing with Eq. (7.2.9) we see the graph scattering matrix S taking the role of the Floquet matrix U ; the matrix T (ϕ) has disappeared through the phase average (7.5.6). The only other difference to Floquet maps which in fact amounts to a considerable simplification is hidden in the compact notation. For individual quantum maps, the supermatrices Z, Z˜ were full matrices and the integral involved 4N 2 real commuting (and as many anti-commuting) degrees of freedom. For quantum graphs, the supermatrices Z, Z˜ are block-diagonal and we are left with only 4N real commuting (and as many anti-commuting) degrees of freedom in the remaining integral. The origin the simplification lies in the N-fold phase average (7.5.6) in the generating function (7.5.7). Each edge on the graph has a separate phase average and thus affords its own color-flavor transformation (single flavor) with 2 × 2 supermatrices Ze and Z˜ e in BF. All edges together give rise to the block diagonal Zee = δee Ze and Z˜ ee = δee Z˜ e . In order to see how classical dynamics sneaks in and universal behavior arises we introduce the mean fields Bss =

N 1  Ze,ss , N e=1

N 1  ˜ B˜ ss = Ze,ss N

(7.5.10)

Z˜ b,ss = B˜ ss + δ Z˜ b,ss .

(7.5.11)

e=1

and deviations therefrom through Zb,ss = Bss + δZb,ss ,

The ‘mean-field approximation’ which neglects the fluctuations gives the zero dimensional sigma model. For the same reasons as in Sect. 7.4 we now choose the ‘on-Q fields’ Zd and Z˜ d rather than the ‘additive fluctuations’ δZ and δ Z˜ as basic representatives

7.5 Quantum Graphs

301

of non-universal corrections, using the relations (7.4.2) and requiring the tracelessness (7.4.3) of the on-Q fields. The further treatment of the non-universal corrections then parallels the one given for general Floquet maps, with the graph specific simplifications due to the block diagonality of Zd and Z˜ d . In particular, the coupling part of the action (7.4.4) reduces to Sc =



 |See |2 str Z˜ de eˆ− Zde eˆ+

(7.5.12)

ee

−

1 1 − B˜ Bˆ

˜ Z˜ de (1 − BB)

1 1 − Bˆ B˜

 ˜ de eˆ+ + . . . . eˆ− (1 − B B)Z

As a most remarkable consequence of the block diagonality of the fluctuations the quantum dynamics is represented by the edge-to-edge transition probabilities |See |2 = |U (k)ee |2 = Fee

(7.5.13)

which are independent of the wave number k. The quantum dynamics thus acquires a classical appearance, and exactly so. (Recall that for Floquet maps we had to introduce classical dynamics through the approximation (7.4.6).) Interestingly, the classical dynamics ruling the small fluctuations on our connected directed graph is stochastic rather than Hamiltonian. We are facing a Markov

process since the matrix F is bistochastic, i.e., e Fee = e Fee = 1. The uniform probability distribution P = 1/N is invariant under the action of the e

map F , that is e Fee Pe = N −1 e Fee = 1/N = Pe . If we assume that all processes e → e allowed by the connectivity of the graph have some finite probability, the map F is even ergodic, and the uniform distribution is the unique stationary mode with its eigenvalue 1 of F non-degenerate. All other eigenvalues e−λμ have modulus below one, |e−λμ | < 1; the possibly complex rates thus have positive real parts, Re λμ > 0. The time scale on which ergodicity becomes apparent in the dynamics is given by the inverse of the classical gap Δg = minλμ =1 |1 − e−λμ | ,

(7.5.14)

analogous to the gap of the ergodic Frobenius-Perron operator in the Floquet map setting (which latter, however, does not depend on N). As already found for Floquet maps, the gap Δg is the central quantity that governs the strength of deviations from universality in spectral correlations of quantum graphs. We should note that only the decaying eigenmodes are admitted in Zd , Z˜ d , the stationary one being barred by the tracelessness of the decaying fields. The final result (7.4.12) for the complex correlator C(e) of our treatment of general Floquet maps can now be taken over but requires a slightly refined interpretation. The gap Δg need not stay fixed and may even close for a sequence of quantum graphs (in contrast to a Floquet map) with growing N. Moreover, the number of resonances is exactly N (again in contrast to a Floquet map where the

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7 Ballistic Sigma Model for Individual Unitary Maps and Graphs

number of resonances is ignorant of N). In particular, the first of the two correction terms in (7.4.12) can be estimated as  2ei2e/N   2 1   ∂c ∂d Zd ± =0  ≤ .  2 N N Δ2g

(7.5.15)

To obtain the upper limit we have replaced the sum over resonances by the factor N. Now a gap remaining finite gives a correction of the order N1 , rather than the 1 correction found for Floquet maps. In the graph case, the correction becomes N2 even larger when the gap Δg closes as N → ∞, say according to the power law Δg = N −ν . The correction under discussion then vanishes asymptotically only as long as ν < 1/2. Again, we should recall that our analysis is valid only within the windows of correlation decay and revival where C(e) remains finite for N → ∞. The second term in the correction (7.4.12) can be estimated similarly.

References 1. S. Müller, S. Heusler, P. Braun, F. Haake, A. Altland, Phys. Rev. E 72, 046207 (2005) 2. K.B. Efetov, Supersymmetry in Disorder and Chaos (Cambridge University Press, Cambridge, 1997) 3. S. Müller, S. Heusler, A. Altland, P. Braun, F. Haake, New J. Phys. 11, 103025 (2009); arXiv:0906.1960v2 4. A. Altland, S. Gnutzmann, F. Haake, T. Micklitz, Rep. Prog. Phys. 78, 086001 (2015) 5. J. Müller, A. Altland, J. Phys. A 38, 3097 (2005); J. Müller, T. Micklitz, A. Altland: http:// arxiv.org/abs/0707.2274 (2007) 6. T. Kottos, U. Smilansky, Phys. Rev. Lett. 79, 4794 (1997) 7. T. Kottos, U. Smilansky, Ann. Phys. 274, 76 (1998) 8. S. Gnutzmann, A. Altland, Phys. Rev. Lett. 93, 194101 (2004) 9. S. Gnutzmann, A. Altland, Phys. Rev. E 72, 056215 (2005) 10. S. Gnutzmann, U. Smilansky, Adv. Phys. 55, 527 (2007) 11. V.E. Kavtsov, A.D. Mirlin, Sov. Phys. JETP Lett. 60, 656 (1994) 12. P. Braun, F. Haake, J. Phys. A 48, 135151 (2015) 13. W.P. Schleich, Quantum Optics in Phase Space (New York, Wiley, 2001) 14. P. Gaspard, Chaos, Scattering and Statistical Mechanics (Cambridge University Press, Cambridge, 1998) 15. R. Carlson, Trans. Am. Math. Soc. 351, 4069 (1999) 16. G. Berkolaiko, H. Schanz, R. Whitney, Phys. Rev. Lett. 88, 104101 (2002) 17. G. Berkolaiko, H. Schanz, R. Whitney, J. Phys. A 36, 8373 (2003) 18. G. Berkolaiko, G.B. Winn, Trans. Am. Math. Soc. 362, 6261 (2010) 19. G. Tanner, J. Phys. A 34, 8485 (2001)

Chapter 8

Quantum Localization

8.1 Preliminaries This chapter will focus mainly on the kicked rotator that displays global classical chaos in its cylindrical phase space for sufficiently strong kicking. The chaotic behavior takes the form of “rapid” quasi-random jumps of the phase variable around the cylinder and “slow” diffusion of the conjugate angular momentum p along the cylinder. A quantum parallel of this time-scale separation of the motions along the two principal directions of the classical phase space is the phenomenon of quantum localization: the quasi-energy eigenfunctions turn out localized in the angular momentum representation [1–3]. (We shall be concerned with the generic case of irrational values of a certain dimensionless version of Planck’s constant—for rational values extended eigenfunctions arise [4].) The difference between classical and quantum predictions for the rotator is nothing like a small quantum correction, even if states with large angular-momentum quantum numbers are involved. Classical behavior would be temporally unlimited diffusive growth of the kinetic energy, p2  ∝ t. Quantum localization must set an end to any such growth starting from a finite range of initially populated angularmomentum states, since only those Floquet states can be excited that overlap the initial angular momentum interval. Given localization of all quasi-energy eigenfunctions, it is clear that classical chaos cannot be accompanied by repulsion of all quasi-energies: Indeed, eigenfunctions without overlap in the angular momentum basis have no reason to keep their quasi-energies apart. On the other hand, if a level spacing distribution is established by including only eigenfunctions with overlapping supports one expects, and does indeed find, a tendency to level repulsion again [5]. From the point of view of random-matrix theory, the Floquet operator F of the kicked rotator appears as nongeneric in other respects as well. In the angular momentum representation, F takes the form of a narrowly banded matrix . In appreciating this statement, the reader should realize that the angular momentum © Springer Nature Switzerland AG 2018 F. Haake et al., Quantum Signatures of Chaos, Springer Series in Synergetics, https://doi.org/10.1007/978-3-319-97580-1_8

303

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8 Quantum Localization

representation is the eigenrepresentation of a natural observable of the kicked rotator, not merely a representation arrived at through an incomplete diagonalization. No wonder then that the sigma model for ensembles of banded matrices (see Sect. 8.8) turns out closely related to the ballistic sigma model for the kicked rotor( see Sect. 8.9). In the following we shall discuss some peculiarities of the dynamics of the kicked rotator related to localization. Particular emphasis will be placed on the analogy of the kicked rotator to Anderson’s hopping model in one dimension. Moreover, we shall contrast the kicked rotator with the kicked top. The latter system does not display localization except in a very particular limit in which the top actually reduces to the rotator. The absence of quantum localization from the generic top, it will be seen, is paralleled by the absence of any time-scale separation for its two classical phase-space coordinates. Correspondingly, there is no slow, nearly conserved variable. No justice can be done to the huge literature on localization, not even as regards spectral statistics of systems with localization. The interested reader can pick up threads not followed here in Refs. [6–8] and in Efetov’s and Imry’s recent books [9, 10]. Before diving into our theoretical discussions, it is appropriate to mention that quantum localization has been observed in rather different types of experiments. Microwave ionization of Rydberg states was investigated by Bayfield, Koch, and coworkers for atomic hydrogen [11–16] and by Walther and co-workers for rubidium, as reviewed in Ref. [17]. The degree of atomic excitation in these experiments was so high that one might expect and in some respects does find effectively classical behavior of the valence electron. Quasi-classical behavior would, roughly speaking, amount to diffusive growth of the energy toward the ionization threshold. What is actually found is a certain resistance to ionization relative to such classical expectation. Quantitative analysis reveals an analogy to the localization-imposed break of diffusion for the kicked rotator. For reviews on this topic, the reader is referred to Refs. [18, 19]. A second class of experiments realizing kicked-rotator behavior was suggested by Graham, Schlautmann and Zoller [20] and conducted by Raizen and co-workers, on cold (some micro-Kelvins) sodium atoms exposed to a standing-wave light field whose time dependence is monochromatic, apart from a periodic amplitude modulation [21–23]. The laser frequency is chosen sufficiently close to resonance with a single pair of atomic levels so that all other levels are irrelevant. The selected pair gives rise to a dipole interaction with the electric field of the standing laser wave. Thus, the translational motion of the atom takes place in a light-induced potential U (x, t) = −K cos(x−λ sin ωt) where ω is the modulation frequency, λ is a measure of the modulation amplitude, and K is proportional to the squared dipole matrix element. The periodically driven motion in that potential is again characterized by localization-limited diffusion of the momentum p. An account of the physics involved in the quantum optical experiments on localization is provided by Stöckmann’s recent book [24]. Experiments and the theory of localization in disordered electronic systems are reviewed in Ref. [25].

8.2 Localization in Anderson’s Hopping Model

305

8.2 Localization in Anderson’s Hopping Model In preparation for treating the quantum mechanics of the periodically kicked rotator, we now discuss Anderson’s model of a particle whose possible locations are the equidistant sites of a one-dimensional chain. At each site, a random potential Tm acts and hopping of the particle from one site to its rth neighbor is described by a hopping amplitude Wr . The probability amplitude um for finding the particle on the mth site obeys the Schrödinger equation [26–28] T m um +



Wr um+r = Eum .

(8.2.1)

r

If the potential Tm were periodic along the chain with a finite period q such that Tm+q = Tm , the solutions of (8.2.1) would, as is well known, be Bloch functions, and the energy eigenvalues would form a sequence of continuous bands. However, of prime interest with respect to the kicked rotator is the case in which the Tm are random numbers, uncorrelated from site to site and distributed with a density (Tm ). The hopping amplitudes Wr , in contrast, will be taken to be non-random and will decrease fast for hops of increasing length r. In the situation just described, the eigenstates of the Schrödinger equation (8.2.1) are exponentially localized. They may be labelled by their center ν such that uνm ∼ e−|ν−m|/ l for |ν − m| → ∞ .

(8.2.2)

The so-called localization length l is a function of the energy E and may also depend on other parameters of the model such as the hopping range and the density (Tm ). For the special case of Lloyd’s model, defined by a Lorentzian density (Tm ) and hoppings restricted to nearest-neighbor sites, an exact expression for l will be derived in Sect. 8.4. Typical numerical results for the eigenfunctions uνm are depicted in Fig. 8.1. Numerical results for the energy eigenvalues Eν indicate that in the limit of a long chain of sites (N  1), a smooth average level density arises. Moreover, the level spacings turn out to obey an exponential distribution just as if the levels were statistically independent. It is easy to see, in fact, that the model must display level clustering rather than level repulsion due to the exponential localization of the eigenfunctions: two eigenfunctions with their centers separated by more than a localization length l have an exponentially small matrix element of the Hamiltonian and thus no reason to keep their energies apart. A further aspect of localization is worthy of mention. The Hamiltonian matrix Hmn implicitly defined by (8.2.1) is a random matrix with independent random diagonal elements Hmm = Tm and with (noticeably) non-zero off-diagonal elements Hm,m+r = Wr only in a band around the diagonal; the “bandwidth” is small compared to the dimension N of the matrix. Upon numerically diagonalizing banded random matrices, one typically finds localized eigenvectors and eigenvalues

306

8 Quantum Localization 10 5 0 In Un –5 –10 –15 0

20

40 n

60

80

Fig. 8.1 Exponential localization of quasi-energy eigenfunctions of kicked rotator From [1]

with Poissonian statistics. Both the small bandwidth and the randomness of H are essential for that statement to hold true: As we have seen in Chap. 5, full random matrices whose elements are independent to within the constraints of symmetry typically display level repulsion. On the other hand, any Hermitian matrix with repelling eigenvalues can be unitarily transformed so as to take a banded and even diagonal form, but then the matrix elements must bear correlations. We shall return to banded random matrices in Sect. 8.8 and show there that localization takes place if the bandwidth b is small in the sense b2 /N < 1. An interesting argument for Anderson localization can be based on a theorem due to Furstenberg [29, 30]. This theorem deals with unimodular random matrices Mi (i.e., random matrices with determinants of unit modulus) and states that under rather general conditions lim

Q→∞

* + 1 ln Tr MQ MQ−1 . . . M2 M1 ≡ γ > 0 . Q

(8.2.3)

To apply (8.2.3) in the present context, we restrict the discussion to nearest neighbor hops (W±1 = W, Wr = 0 for |r| > 1) and rewrite the Schrödinger equation (8.2.1) in the form of a map for a two-component vector,        um um+1 um (E − Tm )/W −1 = ≡ Mm . 1 0 um um−1 um−1

(8.2.4)

The 2 × 2 matrix in this map is indeed unimodular, det Mn = 1, and random due to the randomness of the potential Tm . Starting from the values of the wave function

8.3 The Kicked Rotator as a Variant of Anderson’s Model

307

on two neighboring sites, say 0 and 1, one finds the wave function further to the left or to the right by appropriately iterating the map (8.2.4),     u1 um+1 = Mm Mm−1 . . . M2 M1 . (8.2.5) um u0 Far away from the starting sites, i.e., for |m| → ∞, Furstenberg’s theorem applies and means that the unimodular matrix Mm Mm−1 . . . M2 M1 has the two eigenvalues exp (±mγ ). It follows that almost all “initial” vectors u1 , u0 will give rise to wave functions growing exponentially both to the left and to the right. Thus, the parameter γ is appropriately called the Lyapunov exponent of the random map under consideration. By special choices of u0 and u1 , it is possible to generate wave functions growing only in one direction and decaying in the other. However, to enforce decay in both directions, special values of the energy E must be chosen. In other words, the eigenenergies of the one-dimensional Anderson model cannot form a continuum and in fact typically constitute a discrete spectrum. The corresponding eigenfunctions are exponentially localized and the Lyapunov exponent turns out to be the inverse localization length (we disregard the untypical exception of so-called singular continuous spectra). An interesting consequence arises for the temporal behavior of wave packets. An initially localized packet is described by an effectively finite number of the localized eigenstates. As a sum of an effectively finite number of harmonically oscillating terms, the wave packet must be quasi-periodic in time. At any site m, the occupation amplitude um (T ) will get close to its initial value again and again as time elapses. A particle initially placed at, say, m = 0 such that um (0) = δm,0 may, while hopping to the left and right in the random potential, display diffusive behavior (Δm)2  ∼ t. As soon as a range of the order of the localization length has been explored, however, the diffusive growth of the spread (Δm)2  must give way to manifestly quasi-periodic behavior.

8.3 The Kicked Rotator as a Variant of Anderson’s Model The periodically kicked rotator, one of the simplest and best investigated models capable of displaying classical chaos, has a single pair of phase-space variables, an angle Θ and an angular momentum p. As quantum operators, these variables obey the canonical commutation rule [p, Θ] =

h¯ . i

(8.3.1)

Their dynamics is generated by the Hamiltonian [31] H (t) =

+∞  I p2 + λ V (Θ) δ(t − nτ ) , 2I τ n = −∞

(8.3.2)

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8 Quantum Localization

where I is the moment of inertia, τ the kicking period, and λ a dimensionless kicking strength. The dimensionless potential V (Θ) must be 2π-periodic in Θ. We shall deal for the most part with the special case V (Θ) = cos Θ .

(8.3.3)

Due to the periodic kicking, a stroboscopic description is appropriate with the Floquet operator F = e−iλ(I /τ h¯ )V e−ip

2 τ/2I h

¯

(8.3.4)

which accounts for the evolution from immediately after one kick to immediately after the next. The wave vector at successive instants of this kind obeys the stroboscopic map Ψ (t + 1) = F Ψ (t) .

(8.3.5)

Equivalently, the Heisenberg-picture operators obey Xt +1 = F † Xt F .

(8.3.6)

In (8.3.5) and (8.3.6) and throughout the remainder of this chapter, the time variable t is a dimensionless integer counting the number of periods executed. Special cases of (8.3.6) are the discrete-time Heisenberg equations pt +1 = pt − λ τI V (Θt +1 ) , Θt +1 = Θt + τI pt ,

(8.3.7)

which follow from the commutator relation (8.3.1) (Problem 8.1). Due to the absence of products of p and Θ, the recursion relations (8.3.7) also hold classically as discrete-time Hamiltonian equations. A word about parameters and units is now in order. The parameter I /τ has the dimension of an angular momentum. Upon introducing a dimensionless angular momentum P = pτ/I , the stroboscopic equations of motion (8.3.7) simplify to reveal the dimensionless kicking strength λ as the only control parameter of classical dynamics. On the other hand, in the commutator relation [P , Θ] = h¯ τ/I i and also in the Floquet operator F, the dimensionless version of Planck’s constant h˜ = h¯ τ/I

(8.3.8)

appears as a second parameter controlling the effective strength of quantum fluctuations. With these remarks in mind, we shall sometimes (never, however, without explicit warning) simplify the notation and set τ and I equal to unity.

8.3 The Kicked Rotator as a Variant of Anderson’s Model

309

We now look at the classical behavior of the rotator. The period-to-period increment of angle Θ becomes greater, the larger the chosen kick strength λ. For sufficiently large λ, the angle may in fact jump by several 2π, without Θt +1 (mod 2π) showing any obvious preference for any part of the interval [0, 2π]. It follows that the momentum (which reacts only to Θt +1 (mod 2π)) tends to undergo a random motion for strong kicking. In a rough idealization, we may assume that a series of successive values of Θ uniformly fills the interval [0, 2π] and keeps no kick-to-kick memory; thus, the force −V (Θ) in (8.3.7) appears as noise with no kick-to-kick correlation. Then, the map (8.3.7) can be approximated by t  λI V (Θν ) τ

pt = p0 −

(8.3.9)

ν =1

and yields a constant mean momentum, t  λI pt = p0 − V (Θν ) = p0 , τ

(8.3.10)

ν =1

since the mean force V (Θ) =

1 2π



2π

dΘ V (Θ) = 0

(8.3.11)

0

vanishes for a periodic potential. Similarly, the mean-squared momentum, pt2

=

p02

t  λI 2  + V (Θμ )V (Θν ) τ μ,ν = 1

=

p02

+

   λI 2 τ

2π 0

 dΘ 2 V (Θ) t ≡ Dt , 2π

(8.3.12)

grows linearly with time inasmuch as the force displays no kick-to-kick correlations, V (Θμ )V (Θν ) = δμν V (Θ)2 .

(8.3.13)

Evidently, the crude idealization of the dynamics (8.3.7) for λ  1 by a random process suggests diffusive behavior of the (angular) momentum with the diffusion constant D=

 λI 2  τ

2π 0

dΘ V (Θ)2 . 2π

(8.3.14)

310

8 Quantum Localization

In particular, for V (Θ) = cos Θ, D=

1  λI 2 2 τ

(8.3.15)

From here on and throughout the remainder of this section we set I and τ equal to unity; the momentum becomes dimensionless and Planck’s constant must be read as the dimensionless representative defined in (8.3.8). In fact, by numerically iterating Chirikov’s standard map [31] pt +1 = pt + λ sin Θt +1 Θt +1 = (Θt + pt ) mod (2π)

(8.3.16)

for an ensemble of initial angles Θ0 and vanishing initial momentum, one does find the expected diffusive behavior if λ is chosen sufficiently large, λ  1.5. Figure 8.2 depicts this numerical evidence. The effectively diffusive behavior of the momentum is tantamount to chaos, as follows from the linearized version of the standard map that describes the fate of infinitesimally close phase-space points under iteration, 

δpt +1 δΘt +1



     δpt δpt 1 + λ cos Θt +1 λ cos Θt +1 = ≡ Mt . (8.3.17) 1 1 δΘt δΘt

Fig. 8.2 Classical (full curve) and quantum (dotted) mean kinetic energy of the periodically kicked rotator, determined by numerical iteration of the respective maps. The quantum mean follows the classical diffusion for times up to some “break” time and then begins to display quasiperiodic fluctuations. After 500 kicks, the direction of time was reversed; while the quantum mean accurately retraces its history, the classical mean reverts to diffusive growth thus revealing the extreme sensitivity of chaotic systems to tiny perturbations (here round-off errors). Courtesy of Dittrich and Graham [32]

8.3 The Kicked Rotator as a Variant of Anderson’s Model

311

Inasmuch as Θt is effectively random, Mt is a unimodular random matrix. Furstenberg’s theorem again applies (now in a classical context, in contrast to the application in Sect. 8.2) and secures a positive Lyapunov exponent. Exponential separation of classical trajectories, i.e., chaos, is manifest. Now, we turn to the quantum version of the kicked rotator. Slightly changing notation, we rewrite the quantum map (8.3.5) as |Ψ + (t + 1) = F |Ψ + (t) , t = 0, 1, 2, . . . .

(8.3.18)

Instead of looking at the wave vector |Ψ + (t) immediately after the tth kick, one may study the wave vector just before that kick |Ψ − (t) = e−iH0 /h¯ |Ψ + (t − 1) , H0 =

p2 . 2

(8.3.19)

To solve Schrödinger’s equation for either |Ψ + (t) or |Ψ − (t), a representation must now be chosen. The free motion in between kicks is most conveniently described in the basis constituted by the eigenvectors of p, p|n = nh¯ |n , n = 0, ±1, ±2, . . . ,

(8.3.20)

since the Hamiltonian is diagonal in that representation: H0 |n =

h¯ 2 2 n |n . 2

(8.3.21)

By expanding as |Ψ ± (t) =



Ψn± (t)|n ,

(8.3.22)

n

one finds that the free motion is described by Ψn− (t + 1) = e−ih¯ n

2 /2

Ψn+ (t) .

(8.3.23)

An individual kick, on the other hand, is most easily dealt with in the Θ representation, |Ψ ± (t) =



2π 0

dΘ Ψ ± (Θ, t)|Θ ,

(8.3.24)

312

8 Quantum Localization

since Ψ + (Θ, t) = e−iλV (Θ)/h¯ Ψ − (Θ, t) .

(8.3.25)

Combining the two steps in the p representation, one encounters the map +∞

2 Ψm+ (t + 1) = Jm−n e−ih¯ n /2 Ψn+ (t) , n = −∞ # 2π 1 i(m−n)Θ e−iλV (Θ) /h . Jm−n = 2π ¯ 0 dΘ e

(8.3.26)

For the special case V = cos Θ, the matrix element of the unitary kick operator becomes the Bessel function  π 1 Jn (z) = n dΘ eiz cos Θ cos nΘ . (8.3.27) πi 0 To establish a relation between the kicked rotator and Anderson’s model [1], we must consider the eigenvalue problem for the Floquet operator F, F |u+  = e−iφ |u+  ,

(8.3.28)

which, in the momentum representation, reads 

Jm−n e−ih¯ n

2 /2

−iφ + u+ um . n =e

(8.3.29)

n

The corresponding problem with the order of kick and free motion reversed will also play a role; the corresponding eigenvectors |u−  are most easily related to the |u+  in the Θ representation, u− (Θ) = eiλV (Θ)/h¯ u+ (Θ) ,

(8.3.30)

where u− (Θ) and u+ (Θ) pertain to the same eigenphase φ of e−iH0 /h¯ e−iλV /h¯ and F = e−iλV /h¯ e−iH0 /h¯ , respectively. With the help of (8.3.28), the relation (8.3.30) can be rewritten as u− (Θ) = ei(φ−H0 /h¯ ) u+ (Θ)

(8.3.31)

and thus in the p representation i(φ−h¯ m u− m =e

2 /2)

u+ m .

(8.3.32)

8.3 The Kicked Rotator as a Variant of Anderson’s Model

313

Now, we represent the unitary kick operator in terms of a Hermitian operator W , e−iλV /h¯ =

λV 1 + iW , W = −tan , 1 − iW 2h¯

(8.3.33)

and define the vector |u =

1 2



 |u+  + |u−  .

(8.3.34)

From (8.3.30), this vector is obtained as u− (Θ) u+ (Θ) = 1 + iW (Θ) 1 − iW (Θ)

u(Θ) =

(8.3.35)

and together with (8.3.32) yields an integral equation for the function u(Θ), [1 − iW (Θ)] u(Θ) = ei(φ−H0 /h¯ ) [1 + iW (Θ)] u(Θ) .

(8.3.36)

This equation assumes an algebraic appearance, however, when expressed in the p representation, T m um +



Wr um+r = Eum ,

(8.3.37)

r(=0)

where we have introduced Tm = i

2 /2)

1 − ei(φ−h¯ m 1 + ei(φ−h¯ m

2 /2)

 = tan

φ − h¯ m2 /2 2

 ,

(8.3.38)

E = −W0 . Now, the desired relationship between Anderson’s hopping model and the kicked rotator is established: the algebraic equation (8.3.37) for um = m|u does indeed have the form of the Schrödinger equation (8.2.1) for a particle on a lattice, where Tm is a single-site potential and Wr is a hopping amplitude. Interestingly, the quasienergy φ of the rotator has become a parameter in the potential Tm while the zeroth Fourier component W0 of the function W (Θ) has formally taken on the role of the energy of the hopping particle. A subtle difference between the kicked rotator and the Anderson model deserves discussion. The amplitudes Tm defined in (8.3.38) are not strictly random but only pseudorandom numbers. In fact, since the quantities (φ − hm ¯ 2 /2) enter Tm as the argument of a tangent, they become effective modulo π. According to a theorem of Weyl’s [33], the sequence (φ − h¯ m2 /2) mod π is ergodic in the interval [0, π] and

314

8 Quantum Localization

covers that interval with uniform density as ±m = 0, 1, 2, . . . . It follows that the Tm have a density W (T )dT = dφ/π. With dT /dφ = 1 + T 2 , one finds W (Tm ) =

1 , π(1 + Tm2 )

(8.3.39)

i.e., a Lorentzian distribution. Nevertheless, the Tm are certainly not strictly independent from one value of m to the next. Therefore, it may be appropriate to speak of the kicked rotator in terms of a pseudo-Anderson model or, actually, a pseudo-Lloyd model since the Anderson model with truly random Tm distributed according to a Lorentzian is known as Lloyd’s model [34]. There is another difference between the simplest version of the Anderson model and the simplest rotator. Rather than allowing for hops to nearest neighbor sites, the kicking potential V (Θ) = cos Θ gives rise to   λ cos Θ W (Θ) = −tan (8.3.40) 2h¯ with Fourier components 

2π

Wr = − 0

  dΘ irΘ λ cos Θ e tan . 2π 2h¯

(8.3.41)

These hopping amplitudes can be calculated in closed form for λ < π and, it may be shown, fall off exponentially with increasing r. Even though localization for the kicked rotator remains unproven, the pseudorandomness of the potential Tm and the finite range of the hopping amplitude Wr strongly suggest such behavior. Moreover, all numerical evidence available favors an effective equivalence of the Anderson model and the kicked rotator. It is also fair to say that the sigma model approach to be presented further below provides at least a good understanding.

8.4 Lloyd’s Model It may be well to digress a little and add some hard facts to the localization lore. Anderson’s Schrödinger equation allows an exact evaluation of the localization length l provided that the hopping amplitude is non-zero only for nearest neighbor hops and that the potential Tm is independent from site to site and distributed according to a Lorentzian [34], T m um +

κ (um−1 + um+1 ) = Eum , 2

(8.4.1)

1 . π(1 + Tm2 )

(8.4.2)

(Tm ) =

8.4 Lloyd’s Model

315

Now, we shall prove1 the following well-known result [28, 35] for the ensembleaveraged localization constant γ ≡ 1/ l, cosh γ =

) 1 ) (E − κ)2 + 1 + (E + κ)2 + 1 2κ

(8.4.3)

which implies γ > 0 for all values of the energy E. The starting point is the tridiagonal Hamiltonian matrix Hmn = Tm δmn +

 κ δm,n+1 + δm,n−1 2

inserted in the Green function   1 detnm (E − H ) 1 1 . Gmn = = (−1)m−n N E − H mn N det (E − H )

(8.4.4)

(8.4.5)

The determinant detmn (E − H ) is obtained from det (E − H ) by cancelling the nth row and the mth column. Assuming a finite number N of sites, the element G1N , i.e., the one referring to the beginning and end of the chain of sites, takes an especially simple form since  κ N−1 det1N (E − H ) = − . 2

(8.4.6)

It is an obvious result of the tridiagonality of Hmn that the element G1N reads simply G1N

1  κ N−1 + = N 2

@

N -

(E − Eν ) .

(8.4.7)

ν =1

On the other hand, by invoking the spectral representation of Green’s function, Gmn =

1  uνm uνn , N ν E − Eν

(8.4.8)

and comparing the residues of the pole at E = Eν in (8.4.7) and (8.4.8), one obtains uν1 uνN

 κ N−1 = + 2

@

-

(Eν − Eμ ) .

(8.4.9)

μ(=ν)

This latter expression lends itself to a determination of the localization length for the νth eigenvector. If that eigenvector were extended like a Bloch state, one

1 We

are indebted to H.J. Sommers for showing this proof to us.

316

8 Quantum Localization

would have uν1 uνN ∼ 1/N, i.e., |uν1 uνN |1/N ∼ (1/N)1/N ∼ exp (−(ln N)/N) → 1. For an exponentially localized state, however, uν1 uνN ∼ Ae−γν N and thus |uν1 uνN |1/N ∼ |A|1/N e−γν → e−γν < 1. Therefore, the localization length 1/γν can be obtained as γν =

 κ 1   ln Eν − Eμ  − ln . N 2

(8.4.10)

μ(=ν)

The limit N → ∞ is implicitly understood, so we may replace the sum in (8.4.10) by an integral with the level density (E) as a weight  γ (E) =

dx (x) ln |E − x| − ln

κ , 2

(8.4.11)

with the normalization of (x) chosen as that of a probability density, 

+∞ −∞

dx (x) = 1 .

(8.4.12)

More easily accessible than γ (E) itself is its derivative 



γ (E) = P

dx

(x) , E−x

(8.4.13)

since the principal-value integral occurring here is easily related to Green’s function. Indeed, the spectral representation (8.4.8) entails  1  1 (x) Tr {G(E)} = . → dx N ν E − Eν E−x

(8.4.14)

The continuum approximation endowed Green’s function with a cut along the real axis in the complex energy plane, so different limiting values of G arise immediately above and below the real energy axis2 * + Tr G(E ± i0+ ) = P

 dx

(x) ∓ iπ (E) . E−x

(8.4.15)

Usually, one is interested in the imaginary part of this equation which yields a strategy for calculating the level density (Sect. 5.7). Here, however, we compare the real part with (8.4.13) and conclude that * * ++ γ (E) = Re Tr G(E − i0+ ) .

2 Here

and in the following 0+ denotes an arbitrarily small positive number.

(8.4.16)

8.4 Lloyd’s Model

317

The latter identity holds for any configuration of the potential Tm . Averaging over all configurations {Tm } with the Lorentzian density (8.4.2) gives [35] * * ++ γ (E) = Re Tr G(E ± i0+ ) ,

(8.4.17)

a result often referred to as Thouless’ formula. Due to the assumed site-to-site independence, (...) =

  m

+∞

−∞

 1 dTm (...) . π(1 + Tm2 )

(8.4.18)

To carry out the average of G(E), it is convenient to represent Green’s function by a multiple Gaussian integral, 

N n  -

+∞

i = 1 α = 1 −∞

 dSiα

   + α α Sp1 Sq1 exp −i (E − H − i0 )ij Si Sj ij α



=

πN N Gpq (E − i0+ ) 2 det (E − H − i0+ )

n/2 .

(8.4.19)

An especially useful identity arises from (8.4.19) when the parameter n, after performing the integral, is elevated from integer to continuous real and then sent to zero,  +

NGpq (E − i0 ) = lim 2 n→0

N n  -

 dSiα

Sp1 Sq1

i =1α=1

   + α α × exp −i (E − H − i0 )ij Si Sj .

(8.4.20)

ij α

This representation of the inverse of the matrix (E − H − i0+ )pq by a multiple Gaussian integral with the subsequent manipulation of the parameter n is known in the theory of disordered spin systems as the “replica trick” [36]. Its virtue is that the random numbers Tm now appear in a product (of exponentials), each factor involving a single Tm . Therefore, the average (8.4.18) can be taken site by site according to N     exp [iTj Sjα Sjα ] exp i Tj Sjα Sjα = dTj π(1 + Tj2 j j =1    = exp − Sjα Sjα . j

(8.4.21)

318

8 Quantum Localization

The average Green function then takes the simple form  NGpq (E

− i0+ )

= lim

2

n→0

--

 dSiα

Sp1 Sq1

α

i

   + α α ˜ × exp −i (E − H − i0 )ij Si Sj

(8.4.22)

ij α

with the matrix  κ δp,q+1 + δp,q−1 . H˜ pq = iδpq + 2

(8.4.23)

It is remarkable that H˜ pq differs from the original Hamiltonian only by the replacement Tp → i. At any rate, one can now read the multiple integral (8.4.21) backwards as a representation of (E − H˜ − i0+ )−1 , i.e.,   −1 N −1 G

pq

= (E − i)δpq −

 κ δp,q+1 + δp,q−1 . 2

(8.4.24)

With the average over the random potential done, it remains to invert the tridiagonal matrix (8.4.23). To that end, we again employ (8.4.5) and write Tr {NG} =

 detmm (E − H˜ ) m

det (E − H˜ )

=

∂ ln det(E − H˜ ) ∂E

(8.4.25)

where in the last step we have exploited the fact that the energy E enters only the diagonal elements of det (E − H˜ ). Indeed, simplifying the notation for a moment in an obvious manner and expanding det (E − H˜ ) along the first row, ∂ det = det11 + (E − i)∂ det11 −

κ ∂ det12 = det11 + ∂1 det 2

(8.4.26)

where the differentiation ∂1 has to leave the first diagonal element untouched; similarly expanding along the second row gives ∂1 det = det22 + ∂12 det where ∂12 now has to leave the first two diagonal elements of the determinant undifferentiated; after N such steps, one arrives at ∂ det =

N 

detpp .

p=1

Finally, we must evaluate the N × N determinant DN ≡ det (E − H˜ ) .

(8.4.27)

8.4 Lloyd’s Model

319

That goal is most easily achieved recursively since D1 = E − i 2 D2 = (E − i)2 − κ4 Dn = (E − i)Dn−1 −

(8.4.28) κ2 4 Dn−2

.

These relations obviously allow the extension D0 = 1. Since the recursion relation in (8.4.28) has coefficients independent of n, it can be solved by the ansatz Dn ∼ x n where x is determined by the quadratic equation x 2 − (E − i)x +

κ2 =0. 4

(8.4.29)

The two solutions x± =

1 2

  ) E − i ± (E − i)2 − κ 2

(8.4.30)

together with the “initial” conditions D0 = 1, D1 = (E − i) give det (E − H˜ ) =

N+1 N+1 − x− x+ . x+ − x−

(8.4.31)

Combining (8.4.16), (8.4.25), (8.4.31), 4  N+1 N+1 x − x 1 ∂ + − . Re ln γ (E) = ∂E N x+ − x−

(8.4.32)

This result simplifies considerably in the limit N → ∞. Since |x+ | > |x− |, γ (E) =

∂ ln |x+ | . ∂E

(8.4.33)

Integrating and fixing the integration constant by comparing with (8.4.10) for E → ∞ gives   .    2 E − i  E − i  + γ (E) = ln  − 1 . κ  κ  A little algebra finally brings the localization constant to the form (8.4.3).

(8.4.34)

320

8 Quantum Localization

8.5 The Classical Diffusion Constant as the Quantum Localization Length It was argued in Sect. 8.3 that the classical map λI τ V (Θt +1 ) (Θt + τI pt ) mod (2π)

pt +1 = pt + Θt +1 =

(8.5.1)

may, for sufficiently strong kicking λ, be simplified to pt +1 = pt + noise ,

(8.5.2)

where the noise imparted to the momentum is uncorrelated from period to period. The ensuing diffusive behavior of the momentum, pt2  − p02  = Dt , pt  = p0  ,

(8.5.3)

with the diffusion constant [31]

(8.5.4) is in fact in good agreement with numerical data obtained by following the bundle of solutions of (8.5.1) originating from a cloud of many initial points; see Fig. 8.2. For the quantum rotator, however, diffusive growth of the squared momentum can prevail only until the spread of the wave packet along the momentum axis has reached the localization length l. Afterwards, pt2  must display quasi-periodicity in time. The transition from diffusive to manifestly quasi-periodic behavior, depicted in Fig. 8.2, will take place at a “break time” t ∗ , whose order of magnitude may be estimated by  (Δpt )2 = Dt ∗ = l 2 h¯ 2 .

(8.5.5)

An independent estimate for t ∗ or l is needed to complement (8.5.4) and (8.5.5) for a complete set of equations for D, t ∗ , and l. Chirikov et al. [37] have found a third relation that will be explained now. A quasi-periodic wave packet is spanned by about l basis states, for example, eigenstates of either the momentum p or the Floquet operator F. The corresponding quasi-energies, roughly l in number, all lie in the interval [0, 2π] and thus have a mean spacing of the order 2π/ l. The inverse of this spacing is the minimum time needed for the discreteness of the spectrum to manifest itself in the time dependence of the wave packet. Clearly, the time in

8.6 Absence of Localization for the Kicked Top

321

question may be identified with the break time t ∗ , whereupon one has the important order-of-magnitude result

(8.5.6) Shepelyansky [38] has numerically verified this result for the kicked rotator with V (Θ) = cos Θ and the kicking strength ranging in the interval 1.5 ≤ λ ≤ 29. While the above estimate of t ∗ certainly implies that pt2  must display quasiperiodicity for t > t ∗ , it does not explicitly suggest that pt2  precisely follows classical diffusion for 0 ≤ t  t ∗ . Clearly, for classical behavior to prevail at early times the dimensionless version of h¯ must be sufficiently small.

8.6 Absence of Localization for the Kicked Top It should be pointed out that not all periodically kicked systems display localization analogous to that of Anderson’s model. An interesting case is provided by kicked tops [39–42] which have already been alluded to several times and will be discussed now somewhat more systematically. Kicked tops have the three components of an angular momentum vector J as their only dynamic variables. In their Floquet operators F = e−iλV e−iH0 both H0 and V are polynomials in J such that the squared angular momentum is a constant of the motion,3 J2 = j (j + 1) , j = 12 , 1, 32 , . . . .

(8.6.1)

The classical limit is attained when the quantum number becomes large, j → ∞. The simplest model capable of chaotic motion in the classical limit is given by F = e−iλJz /2j e−iαJx . 2

(8.6.2)

The second factor in this F clearly describes a linear precession of J around the x-direction by the angle α. Similarly, the first factor may be said to correspond to a nonuniform “rotation” around the z axis; instead of being a constant c number the rotational angle is itself proportional to Jz . The dimensionless coupling constant λ might now be called a torsion strength. The quantum number j appears in V = Jz2 /2j to provide λV and H0 = αJz with the same asymptotic scaling with j when j → ∞ for finite constant λ and α.

3 While hτ/I for the kicked rotator naturally arises as a dimensionless measure of Planck’s ¯ constant, that role will be played by 1/j for the kicked top. Therefore, it is convenient to set h¯ = 1 in this section.

322

8 Quantum Localization

By using the angular momentum commutators [Ji , Jj ] = iεij k Jk ,

(8.6.3)

one finds the stroboscopic Heisenberg equations of motion Jt +1 = F † Jt F in the form   

 λ ˜ 1 ˜ − J˜y , sin . . . Jx,t +1 = Jx , cos Jz − j 2  i i + cos . . . , J˜y + sin . . . , J˜x , 2 2   

 λ ˜ 1 + J˜y , cos . . . Jy,t +1 = J˜x , sin Jz − j 2  i i cos . . . , J˜x , + sin . . . , J˜y − 2 2 Jz,t +1 = J˜z , J˜x = Jx,t ,

(8.6.4)

J˜y = Jy,t cos α − Jz,t sin α , J˜z = Jy,t sin α + Jz,t cos α . ˜ a linear precession by α around These clearly display the sequence Jt → J, the x-axis and J˜ → Jt +1 , a nonlinear precession around the z-direction. Since the latter rotation is by an angle involving the z component of the intermediate ˜ symmetrized products as well as commutators of noncommuting operators vector J, occur in the corresponding equations in (8.6.4), {A, B} = (AB + BA)/2. In the classical limit, j → ∞, the first two equations in (8.6.4) simplify inasmuch as the contribution λ/2j to the rotational angle is negligible and the symmetrized operator products become ordinary products of the corresponding c number observables while the commutators disappear. Formally, the classical limit can be achieved by first rescaling the operators J as X=

J j

(8.6.5)

whereupon the commutators (8.6.3) take the form [X, Y ] =

i Z. j

(8.6.6)

8.6 Absence of Localization for the Kicked Top

323

In the limit j → ∞, the vector X tends to a unit vector with commuting components, and the stroboscopic equations of motion take the form already described above, Xt +1 = X˜ cos λZ˜ − Y˜ sin λZ˜ Yt +1 = X˜ sin λZ˜ + Y˜ cos λZ˜ Zt +1 = Z˜ X˜ = Xt

(8.6.7)

Y˜ = Yt cos α − Zt sin α Z˜ = Yt sin α + Zt cos α . Due to the conservation law X2 = 1, the classical map (8.6.7) is two dimensional, expressible as two recursion relations for two angles defining the orientation of X, for example, X = sin Θ cos φ , Y = sin Θ sin φ , Z = cos Θ .

(8.6.8)

Actually, the surface of the unit sphere X2 = 1 is the phase space of the classical top with Z = cos Θ and φ as canonical variables. This can be seen most easily by replacing the quantum commutators (8.6.6) with the classical Poisson brackets {X, Y } = −Z and checking that the latter are equivalent to the canonical Poisson brackets {Z, φ} = 1. Indeed, from {Z, φ} = 1, one finds {f (Z), g(φ)} = f (Z)g (φ), and this immediately gives {X, Y } = −Z when the spherical representation (8.6.8) is used for X and Y. The classical trajectories generated by the map (8.6.7) depend in their character on the values of the torsion strength λ and the precession angle α. As shown in Fig. 8.3, the sphere X2 = 1 is dominated by regular motion for α = π/2, λ  2.5, whereas chaos prevails, at fixed α = π/2, for λ  3. It is important to realize that as soon as chaos has become global with increasing torsion strength, the typical trajectory traverses the spherical phase space rapidly in time: a single kick suffices for the phase-space point to hop all around the sphere in any direction. Such stormy exploration is in blatant contrast to the behavior of the kicked rotator in its cylindrical phase space: While the cylinder may be surrounded in the direction of the angular coordinate once or even several times between two subsequent kicks, the momentum is capable of only slow quasi-random motion along the cylinder; the distance covered in the p-direction in a finite time is only a vanishing fraction of the infinite length of the cylinder. Similarly striking is the difference in the quantum mechanical behavior of the top and of the rotator. The rotator displays localization along the one-dimensional angular momentum lattice, but the top, as we shall now proceed to show, does not. Rather, under conditions of classical chaos, wave packets pertaining to the top are in general limited in their spreads only by the finite size of the Hilbert space. To

324 Fig. 8.3 Classical phase space portraits for the kicked top (8.6.7) with α = π/2 and λ = 2 (a); λ = 2.5 (b); λ = 3 (c); and λ = 6 (d). Periodic orbits are marked by numerical labels

8 Quantum Localization

8.6 Absence of Localization for the Kicked Top

325

illustrate the typical behavior of the top, it is convenient to consider wave packets originating from coherent initial states. Coherent states of an angular momentum [43, 44] with the quantum number j assign a direction to the observables J that can be characterized by angles Θ and φ as Θφ|Jz |Θφ = j cos Θ   Θφ|Jx ± iJy |Θφ = j e±iφ sin Θ .

(8.6.9)

Clearly, the geometric meaning of these angles is the same as that of the angles used in (8.6.8) to specify the direction of the classical vector X. However, due to the noncommutativity of the components Ji , it is with a finite precision only that the coherent state |Θφ defines an orientation. To find that precision, one may observe that two of the states |Θφ have Θ = 0 and Θ = π, i.e., Jz  = ±j and therefore can be identified with the joint eigenstates |j m of J2 and Jz pertaining to Jz = m = ±j and J2 = j (j + 1). The relative variance of J with respect to these special coherent states is (J2  − J2 )/j 2 = 1/j. All other states |j m with m = ±j have larger variances of J and so do all of their linear combinations (for fixed j ) except the other coherent states |Θφ. In fact, all coherent states |Θφ can be generated from the “polar” state |Θ = 0, φ = |j, j  by a rotation,4 |Θφ = R(Θ, φ)|j, j  ,

(8.6.10)

   R(Θ, φ) = exp iΘ Jx sin φ − Jy cos φ ∗

= eγ J− e−Jz ln (1+γ γ ) e−γ J+ , where γ = eiφ tan (Θ/2) and J± = Jx ± iJy . The variance of J remains unchanged under the rotation in question, 

 1 Θφ|J2 |Θφ − Θφ|J|Θφ2 /j 2 = , j

(8.6.11)

and indeed constitutes the minimum uncertainty of the orientation of J permitted by the angular-momentum commutators. A solid angle ΔΩ = 1/j may be associated with the relative variance (8.6.11), and with respect to the classical unit sphere X2 = 1, a coherent state |Θφ may be represented by a spot of size ΔΩ located at the point (Θ, φ). Such spots of the minimal size allowed quantum mechanically are often called Planck cells. In the classical limit j → ∞, the spot in question shrinks to the classical phase-space point (Θ, φ).

4 The

reader will pardon the sloppiness of denoting the coherent state by |Θφ rather than |j Θφ.

326

8 Quantum Localization

Some further properties of the states |Θφ are worth mentioning for later reference. First, we should note the expansion in terms of the states |j m. It follows from (8.6.10) that |γ  ≡ |Θφ = (1 + γ γ ∗ )−j eγ J− |j, j 

(8.6.12)

and thus ∗ −j

j m|Θφ = (1 + γ γ )

γ

j −m

.

 2j . j −m

(8.6.13)

Therefore, the probability of finding Jz = m in the coherent state |Θφ is given by the binomial distribution   2j |j m|Θφ|2 = (1 + γ γ ∗ )−2j (γ γ ∗ )j −m , (8.6.14) j −m which we shall use presently. The expression (8.6.12) for the coherent state allows us to easily calculate expectation values like (8.6.9). For instance, obviously, γ |J− |γ  = (1 + γ ∗ γ )−2j

∂ 2j γ ∗ (1 + γ ∗ γ )2j = . ∂γ 1 + γ ∗γ

(8.6.15)

Next, by employing the easily checked identities e−γ J− Jz eγ J− = Jz − γ J− ,

e−γ J− J+ eγ J− = J+ + 2γ Jz − γ 2 J−

(8.6.16)

we get, similarly, 2j γ ∗ 1 − γ ∗γ , γ |J , |γ  = j z 1 + γ ∗γ 1 + γ ∗γ  2 2j γ |J+ J− |γ  = γ |J− |γ  + . (1 + γ ∗ γ )2 γ |J− |γ  =

(8.6.17)

A little trigonometry may finally be exercised to express these expectation values in terms of the angles Θ and φ and to thus recover (8.6.9). A coherent state |Θφ does not in general remain coherent under the time evolution generated by the Floquet operator F. Figure 8.4 depicts the variance of J with respect to a state F t |Θφ as the time grows. The behavior shown corresponds to the classical phase-space portrait of Fig. 8.3c, i.e., to α = π/2 and λ = 3 and to initial angles well outside the islands of regular motion. The angular-momentum quantum number was chosen as j = 100 to make the spot size ΔΩ = 1/j smaller than the solid angle range of the islands of regular motion. Several features of the

8.6 Absence of Localization for the Kicked Top

327

Fig. 8.4 Time-dependent variance (ΔJ/j )2  for the kicked top (8.6.2) (α = π/2, λ = 3, j = 100) for a coherent initial state localized well outside classical islands of regular motion in Fig. 8.3c. The dotted line refers to the classical variance based on a bundle of 1000 classical trajectories

full curve in Fig. 8.4 are worth noting. Figure 8.4 also displays the variance of the classical vector Xt for a bundle of phase-space trajectories originating from a cloud of 1000 initial points. The cloud was chosen to have uniform density and circular shape, and to be equal in location and size to the spot corresponding to the coherent initial state of the quantum top. Due to classical chaos, one would expect the classical and the quantum variance to become markedly different for times of the order ln j (a typical quantum uncertainty ∼ 1/j is amplified to order unity within such a time by exponentially separating chaotic trajectories); this expectation is consistent with the behavior displayed in Fig. 8.4. A further qualitative difference between the two curves in Fig. 8.4 becomes manifest for times exceeding the inverse mean level spacing (2j + 1)/2π : While the classical curve becomes smooth, the quantum curve displays quasi-periodicity. Incidentally, the recurrent events are quite erratic in their sequence. Most importantly, there appears to be no limit to the spread of either the quantum wave packet or the classical cloud of phase-space points other than the finite size, respectively, of the Hilbert space and the phase space. Further light may be shed on the problem of localization if coherent states |Θφ are represented as vectors with respect to the eigenstate of the Floquet operator F, |Θφ =

2j +1 

Cμ (Θ, φ)|μ ,

(8.6.18)

μ=1

taking the basis vectors |μ as normalized, 

|Cμ |2 = 1 .

(8.6.19)

μ

Useful information can be gained by ordering the components Cμ according to decreasing modulus and truncating the representation (8.6.18) so as to include only

328

8 Quantum Localization

the minimum number Nmin of basis vectors necessary to exhaust the normalization of |Θφ to within, say, 1 %. Numerical work [40] using α = π/2, λ = 3 indicates that Nmin scales quite differently with j , depending on whether the initial state lies in a region of classical chaos or in an island of classically regular motion, Nmin ∝ j x x=

1

chaotic

1 2

regular

(8.6.20)

The proportionality between Nmin and j for coherent states located in the classically chaotic region supports the conclusion that the Floquet eigenstates related to classical chaos are not confined in their angular spread to a region of solid angle smaller than the range of classical chaos on the unit sphere depicted in Fig. 8.3c. Moreover, Nmin ∼ j is precisely the result that random-matrix theory would suggest [45]. Conversely, the value 1/2 for the exponent x implies that Floquet eigenstates tend to be localized if they correspond to classically regular motion within islands around √ periodic orbits. Indeed, in the limit j → ∞, a vanishingly small fraction ∼ 1/ j of the full set of eigenstates suffices to build a coherent state located in such √ an island. The proportionality Nmin ∼ j is most easily understood when the island in question contains a fixed point of the classical map. A regular orbit surrounding such a fixed point must closely resemble linear precession of the vector X around the direction defined by the fixed point. Similarly, the eigenstates of the quantum dynamics must be well approximated by the states |j m provided the z-axis is oriented toward the classical fixed point in question. Then, the weight |j m|Θφ|2 of the approximate eigenstate |j m in the coherent state |Θφ is given by the binomial distribution (8.6.14) and that distribution is easily seen to have a width of the order √ 2j when j is large. Numerical evidence for the scaling (8.6.20) is presented in Fig. 8.5, which again corresponds to the classical situation of Fig. 8.3c. The three curves pertain to j = 200, 400, and 500. The drop of Nmin as the coherent state enters the island of classically regular motion is so pronounced that Nmin might even be taken as an approximate quantum measure of the classical Lyapunov exponent. The plots are consistent with Nmin ∝ j 1/2 in the classically regular region and support Nmin ∝ j quite convincingly whereas the coherent state ranges in the domain of classical chaos. The “regular localization” and “chaotic delocalization” of the Floquet eigenvectors for the top5 have interesting consequences for the time evolution of expectation

5 The

top does not, in general, display the phenomenon of quantum localization under conditions of classical chaos; from this fact it may be understood that the localization length is larger than 2j + 1; see, however, the subsequent section.

8.6 Absence of Localization for the Kicked Top

329

Fig. 8.5 Minimum number of eigenmodes of the kicked top (8.6.2) necessary to exhaust the normalization of coherent states to within 1% versus the polar angle Θy (with respect to positive y-axis) at which the coherent state is located; the azimuthal angle is fixed at φy = π/4. Coupling constants as in Fig. 8.3c. Classically, there is regular motion for 0.6  Θy  1.4 (the regular island in the lower right part of the y > 0 hemisphere in Fig. 8.3c). The curves follow the classical Lyapunov exponent better, the larger the j value (chosen here as 200, 400, and 500)

a 1.0

b

< Jy >/j 0.5

0. 0

500

t

1000

0

500

t

1000

Fig. 8.6 Quasi-periodic behavior in time of Jx  for the kicked top with the Floquet operator (8.6.2) for α = π/2, λ = 3, j = 100. Initial states are coherent, see (8.6.10), localized within a classically regular island around a classical fixed point (see Fig. 8.3c) for curve (a) and in a classically chaotic region for curve (b). Note the near-periodic alternation of collapse and revival in the “regular” case (a) and the erratic variety of quasi-periodicity in the “irregular” case (b)

values. In both cases quasi-periodicity must become manifest after a time of the order j, i.e., the inverse of the average quasi-energy spacing 2π/(2j + 1). However, as illustrated in Fig. 8.6, quasi-periodicity arises in two rather different varieties: a nearly periodic sequence of alternate collapses and revivals modulates oscillations of Jx ; those oscillations correspond to orbiting of the wave packet around the fixed point in one of the islands of Fig. 8.3c; this nearly periodic kind of quasi-periodicity accompanies Nmin ∝ j 1/2 and may be interpreted as a quantum beat phenomenon dominated by a very small number of excited eigenmodes. When Nmin ∝ j, on the

330

8 Quantum Localization

other hand, the wave packet has neither a fixed point to orbit around nor are there any regular modulations; instead, recurrent events form a seemingly erratic sequence; this type of behavior might be expected for broadband excitation of eigenmodes. Yet another quantum distinction of regular and chaotic motion follows from the different localization properties of the Floquet eigenvectors just discussed [46]. The “few” eigenvectors localized in an island of regular motion around a fixed point (as shown in Fig. 8.3c) should not vary appreciably when a control parameter, say λ, is altered a little, as long as the classical fixed point and the surrounding island of regular motion are not noticeably changed. On the other hand, an eigenvector that is spread out in a large classically chaotic region might be expected to react sensitively to a small change of λ, since it must respect orthogonality to the “many” other eigenfunctions spread out over the same part of the classical phase space. To check on the expected different degree of sensitivity to small changes of the dynamics, one may consider the time-dependent overlap of two wave vectors, both of which originate from one and the same coherent initial state but evolve according to slightly different values of λ, for example,  2    Θ, φ|F (λ , α)†t F (λ, α)t |Θφ  for t = 0, 1, 2, . . . .

(8.6.21)

Figure 8.7 displays that overlap for α = π/2, λ = 3, λ = 3(1 + 10−4 ). The classical phase-space portraits for these two sets of control parameters are hardly distinguishable and appear as in Fig. 8.3c. For the upper curve in Fig. 8.7, the initial state was chosen that it lies within one of the regular islands of Fig. 8.3c; in accordance with the above expectation, the overlap (8.6.21) remains close to unity for all times in this “regular” case. The lower curve in Fig. 8.7 pertains to an initial state lying well within the chaotic part of the classical phase space shown in Fig. 8.3c; the overlap (8.6.21), it is seen, falls exponentially from its initial value of unity down to a level of order 1/j. This highly interesting quantum criterion to distinguish regular and irregular dynamics, based on the overlap (8.6.21), was introduced by Peres [46]. Incidentally, the “regular” near-periodicity and “irregular” quasi-periodicity seen in Fig. 8.6 is again met in Fig. 8.7 on time scales of the order j. To summarize, the top and the rotator display quite different localization behavior. Under conditions of classical chaos, the eigenvectors of the rotator dynamics localize while those for the top do not. Somewhat different in character is the localization described above for eigenvectors confined to islands of regular motion; the latter behavior, well investigated for the top only, must certainly be expected for the rotator and other simple quantum systems as well.

8.6 Absence of Localization for the Kicked Top

331

Fig. 8.7 Overlap (8.6.18) of two wave vectors of the kicked top with the Floquet operator (8.6.2) for α = π/2, λ = 3, λ = 3(1 + 10−4 ), j = 1600. Initial coherent state as in Fig. 8.6a for upper curve (regular case) and as in Fig. 8.6b for lower curve. Note the extreme sensitivity to tiny changes of the dynamics in the irregular case. Courtesy of A. Peres [46]

332

8 Quantum Localization

8.7 The Rotator as a Limiting Case of the Top Now, we proceed to show that the top can be turned into the rotator by subjecting the torsion strength λ and the precession angle α to a special limit [47]. From a classical point of view, that limit must confine the phase-space trajectory of the top to a certain equatorial “waistband” of the spherical phase space to render the latter effectively indistinguishable from a cylinder. The corresponding quantum mechanical restriction makes part of the (2j + 1)-dimensional Hilbert space inaccessible to the wave vector of the top: with the axis of quantization suitably chosen, the orientational quantum number m must be barred from the neighborhoods of the extremal values ±j. To establish the limit in question, one may look at the Floquet operators of the top FT = e−iλT Jz /2j h¯ e−iαJx /h¯ 2

2

(8.7.1)

and the rotator FR = e−ip

¯ e−iλR (I /τ h¯ ) cos Θ

2 τ/2I h

.

(8.7.2)

To avoid possible confusion, all factors h, ¯ τ, and I are displayed here. The reader should note that the sequence of the unitary factors for free rotation and kick has been reversed with respect to the previous discussion of the rotator; the operator FT defined in (8.7.2) describes the time evolution of the rotator from immediately before one kick to immediately before the next. This shift of reference is a matter of convenience for the present purpose. By their very appearance, the two Floquet operators suggest the correspondence λT =

I 1 h¯ τ j , α = λR . I τ h¯ j

(8.7.3)

For j → ∞ with τ, I, and λR fixed, this clearly amounts to suppressing large excursions of J away from the “equatorial” region Jz /j ≈ 0. To reveal the rotatorlike behavior of the classical top in the limit (8.7.3), we call upon the classical recursion relations (8.6.7) of the top setting Jz = p, Jx = hj ¯ cos Θ, and Jy = h¯ j sin Θ and obtain p = p + λR τI sin Θ φ = φ + τI p .

(8.7.4)

These are indeed the equations of motion for the rotator. In fact, to bring (8.7.4) into the form (8.3.7), we must shift back the reference of time to let the free precession precede the nonlinear kick, Θ = Θt +1 , p = pt .

8.8 Banded Random Matrices

333

The equivalence of the quantum mechanical Floquet operators (8.7.1) and (8.7.2) in the limit (8.7.3) becomes obvious when their matrix representations are considered with respect to eigenstates of Jz and p, respectively, Jz |m = h¯ m|m , −j ≤ m ≤ +j p|m = h¯ m|m , m = 0, ±1, ±2, . . . .

(8.7.5)

Then, the left-hand factors in FT and FR coincide once λT is replaced by h¯ j τ/I according to (8.7.3). The exponents in the right-hand factors of FT and FR read  > =  α  1 )  (j − m)(j + m + 1)δm ,m+1 m  Jx  m = α 2 h¯  ) + (j + m)(j − m + 1)δm ,m−1 ,

(8.7.6)

 > =    + I I * 1 m λR cos Θ  m = λR δm ,m+1 + δm ,m−1 . τ h¯ 2 τ h¯ These matrices become identical when the correspondence (8.7.3) is inserted and the limit j → ∞ with m/j → 0 is taken, the latter condition constituting the quantum mechanical analogue of a narrow equatorial waist-band of the sphere. It is important to realize that quantum localization under conditions of classical chaos, impossible when α and λT remain finite for j → ∞, arises in the limit (8.7.3). According to (8.5.6) the localization length can be estimated as l ≈ (λI /h¯ τ )2 /2 and for localization to take place this length must be small compared to j, l j. The limit (8.7.3) clearly allows that condition to be met.

8.8 Banded Random Matrices 8.8.1 Banded Matrices Modelling Thick Wires The disordered systems with classical chaos and quantum localization considered in previous sections were strictly one-dimensional. That restriction will now be eased in favor of the quasi one dimensional behavior of a particle hopping from one lattice site to neighboring sites in a thick wire. We propose to return Anderson’s tightbinding Hamiltonian of (8.2.1) Hmn = Tm δmn + Wr δm,n+r ,

(8.8.1)

where the indices refer to sites, Tm is a single-site potential, and Wr a hopping amplitude.

334

8 Quantum Localization

Anderson’s model can account for an electron moving in a thick wire of length L and cross-section area S in the following way. We think of the wire as divided in L/ lel slices, whose length lel is the mean free path lel , i.e., the mean distance between scattering impurities. Each such slice provides b ≡ kF2 S transverse “channel states.” The moving electron can make intraslice transitions between different channel states and may also hop from one slice to one of the two neighboring sites. The site label in Anderson’s Hamiltonian may be chosen such that it increases by b within one slice before proceeding to doing the same for the neighboring slice to the right. Very schematically, we may associate r = ±b with an interslice hop and |r| < b with an intraslice transition. Thus, the total number of “sites” in the “lattice” is N = bL/ lel = kF2 SL/ lel , and the Hamiltonian matrix (8.8.1) has a band structure: Its elements Hmn are either strictly zero outside a band of width 2b around the diagonal, Hmn = 0 for |m − n| > b, or at least fall to zero sufficiently rapidly as the skewness |m − n|/b grows large. To introduce disorder, we take the elements Hmn as independent random numbers with the Gaussian joint density P (H ) = N

N i=1

 H2  |H |2  ij exp − ii exp − 2Jii 2Jij

(8.8.2)

i 0 for N → ∞. In fact, we shall find the semicircle law as a by-product of our discussion below. We now represent the IPR P with the help of two matrix elements of the resolvent μ Gnn (E) = n|(E − H )−1 |n = μ |ψn |2 (E − Eμ )−1 , one retarded (E = E + i0+ ) and the other advanced (E = E + i0− ), Nρ(E)P = lim ↓0

  Gnn (E − i)Gnn (E + i) π n

= lim

μ   |ψn |2 |ψnν |2 π n,μ,ν (E − Eμ − i)(E − Eν + i)

= lim

μ  μ   |ψn |4 = |ψn |4 δ(E − Eμ ) . 2 2 π n,μ (E − Eμ ) +  n,μ

↓0

↓0

(8.8.9)

The reader will appreciate that only diagonal terms in the foregoing double sum over eigenvectors survive the limit as  goes to zero. We should hurry to add that a full understanding of localization (and transport properties) of quasi one-dimensional wires cannot be given in terms of just our ensemble-averaged IPR. This is because finite samples are not self-averaging in their localization (and transport) behavior. That empirical fact is reflected in large fluctuations of the spectrally but not ensemble averaged IPR Pμ  given in Eq. (8.8.7) across the ensemble of banded random matrices. To capture such fluctuations one has to determine, beyond the ensemble mean P = Pμ , at least the ensemble variance (Pμ − Pμ )2 . Fyodorov and Mirlin have done even better, calculating the whole set of moments based on  μ

μ

μ

μ

|ψn1 |2q |ψn2 |2q . . . |ψnk |2q δ(E − Eμ )

(8.8.10)

8.8 Banded Random Matrices

337

with positive integer exponents q. Letting prevail pedagogical prudence we confine the following to the first moment P , not without stating that interested readers will not have problems digesting the full study of Ref. [48].

8.8.3 Sigma Model We now represent the product of two Green functions by the Gaussian superintegral (see Eq. (8.9.7)) # ∗ ∗ ψ+,B,n ψ−,B,n ψ−,B,n (8.8.11) Gnn (E − )Gnn (E + ) = (−1)N d(ψ, ψ ∗ )ψ+,B,n    × exp i 4α=1 Ψα† Lα H − E + iΛα  Ψα . The integral is over the 4N-component supervector ⎛ ⎞ ⎛ ⎞ ψ+,B,i Ψ1 ⎜ψ+,F,i⎟ ⎜Ψ2⎟ ⎟ ⎜ ⎟ Ψ =⎜ ⎝ψ−,B,i⎠ ≡ ⎝Ψ3⎠ ψ−,F,i

(8.8.12)

Ψ4

and its c.c. Ψ ∗ . Note that we have suppressed the QD index i = 1, 2, . . . , N in the definition of the components Ψα and that these latter are ordered in the advanced/retarded (AR) notation (retarded above advanced). The index α = 1, 2, 3, 4 runs over the spaces AR and BF. The four numbers Λα distinguish the retarded and advanced ‘sectors’ as +1, +1, −1, −1 and will eventually be written as the matrix Λ = diag(1, 1, −1, −1). Finally, the four numbers Lα ensure convergence of the Bosonic Gaussian integrals involved; they are conveniently arranged in the diagonal matrix L = 1, 1, −1, 1. The symbol  denotes a nonnegative infinitesimal which will eventually be sent to zero. A further remark on the Gaussian superintegral in (8.8.11) is in order. Were it ∗ ∗ not for the preexponential factor ψ+,B,n ψ+,B,n ψ−,B,n ψ−,B,n and were we to allow for four different energy arguments, E → Eα , the Asuperintegral would give the generating function Z = det(E3 − H ) det(E4 − H ) det(E1+ − H ) det(E2− − H ) which in turn yields the product  of two Green functions upon differentiating as G(E1+ )G(E2− ) = ∂ 2 Z/∂E1 ∂E2 E =E ,E =E . 1

3

2

4

338

8 Quantum Localization

Returning to our different goal and the pertinent more detailed expression (8.8.11) we proceed to the average over the Gaussian disorder ensemble (8.8.2) and (8.8.3) and obtain     1  exp i Hij uij = exp − Jij u∗ij uij , 2 ij

uij =



(8.8.13)

ij

∗ ψαi Lα ψαj .

(8.8.14)

α

It is precisely due to the dependence of the variance Jij on the site indices i, j that we must deal with the “local” 4 × 4 supermatrix ∗ Q˜ αβi = ψαi ψβi

(8.8.15)

and have exp(i ij Hij uij ) = exp(− 12 ij Jij StrQ˜ i LQ˜ j L). Our superintegral (8.8.11) thus takes the form  Gnn (E − )Gnn (E + ) = (−1)N

∗ ∗ ψ+,B,n ψ−,B,n ψ−,B,n dψ ∗ dψ ψ+,B,n

(8.8.16)



   1 ˜ ˜ ˜ ˜ × exp Str − Qi L −  Qi LΛ . Jij Qi LQj L − iE ij i i 2 With the disorder average out of the way, we ought to attack the integral over the supervector Ψ which enters the exponent quartically. A major difference to our treatment of the rotor in the previous section must be coped with here. There we had done a global spectral average which eventually allowed us to invoke the color-flavor transformation and thus ‘unfold’ the quartic terms to quadratic, at the expense of an integral over supermatrices. No such convenience is afforded by our present local spectral average in (8.8.7). The only known unfolding ‘quartic to quadratic’ in the present context is the superanalytic Hubbard-Stratonovich transformation (6.6.13) already used in calculating the two-point function of the GUE in Sect. 6.6. The variant of the HST needed here reads   1 exp Str − Jij Q˜ i LQ˜ j L (8.8.17) ij 2     1 N N −1 ˜ = (−1) d Q exp Str − (J )ij Qi LQj L − i Qi LQi L ij i 2 MN 4 where N auxiliary supermatrices Qi appear as integration variables; each Qi is to be integrated over its own manifold M4 (the set of Hermitian 4×4 supermatrices diagonalizable by pseudo-unitary matrices). As a little aside, we ought to convince ourselves of the validity of the foregoing identity. Really no more than the temporary introduction of an ordinary

8.8 Banded Random Matrices

339

orthogonal N × N matrix is required to diagonalize the real symmetric variance ˜ matrix, O T J O = diag(j 1 , j2 , . . . jN ). Thus, the quadratic

form in the Qi reads ˜ ˜ ˜ ˜ ij Jij StrQi LQj L = k jk Strq˜ k Lq˜ L with q˜k = i Oik Qi . But if Qi L is pseudounitarily diagonalizable,  6 so is q˜k#L such that   we may  use (6.6.13) N times, 

exp[− 12 k jk Str q˜k L)2 = k (−1)N dqk exp − Str 2j1k (qk L)2 − iqk Lq˜k L . Undoing the orthogonal transformation we confirm (8.8.17). With the help of this superanalytic Hubbard–Stratonovich transformation our averaged product of Green functions is brought to the form  Gnn (E − )Gnn (E + ) =



MN 4

dN Q

  ∗ ∗ dψ ∗ dψ ψ+,B,n ψ+,B,n ψ−,B,n ψ−,B,n

     1 × exp Str − (J −1 )ij Qi LQj L − Str Q˜ i L iE − Λ − iQi L , ij i 2 (8.8.18) where the ψ-integral has the desired Gaussian form. The superanalytic variant (6.4.17) of Wick’s theorem provides  Gnn (E − )Gnn (E + ) =

d N Q pn

(8.8.19)

MN 4

     1 × exp − (J −1 )ij Str Qi LQj L − Str ln L iE − Λ − iQi L ; ij i 2 and in particular the prefactor BB BB BB BB (n)gRR (n) + gAR (n)gRA (n) , pn = gAA

(8.8.20)

g(n) = (E − Qn + iΛ)−1 → (E − Qn )−1 ∗ which clearly stems from the previous pre-exponential factor ψ+,B,n ψ+,B,n ∗ ψ−,B,n ψ−,B,n “averaged over”; the imaginary infinitesimal will turn out to be dispensable in the 4 × 4 supermatrix g(n) since Qi will become effective with a finite imaginary part. Note that only Bosonic elements of g(n) arise in (8.8.19) due to the purely Bosonic character of the previous prefactor. Some embellishments are in order. We observe SdetL = −1, employ Qi L in place of Qi as integration variables and, finally, choose to expand the logarithm to first order in the positive infinitesimal . We thus get

Gnn

(E − )G

nn

(E + )

 =

( M4

  d N Q exp − A(Q)

L)N

×pn exp[−i

 i

Str (E − Qi )−1 Λ]

(8.8.21)

340

8 Quantum Localization

with the abbreviations A(Q) =

  Q2    1  −1 i (J )ij − J0−1 δij Str Qi Qj + Str + ln E − Qi 2 2J0 ij

i

and J0 =

N 

Jij =

j =1

λ2 1 + e−1/b ∼ λ2 . 2b 1 − e−1/b

(8.8.22)

Even though there is no explicit large factor decorating the “action” A(Q), the summation over the N sites suggests that A is proportional to N. The double sum in the first term also shares that property, as becomes obvious when we consider the exponential form of Jij given in (8.8.4) and its well known inverse [55]. We easily check  2bλ−2  (1 + e−2/b )δij − e−1/b (δi,j +1 + δi,j −1 ) , 1 − e−2/b  B 2δij − δi,j +1 − δi,j −1 , (J −1 )ij − J0−1 δij = (8.8.23) 2 (J −1 )ij =

B=

2b2 4bλ−2 e−1/b ∼ 2 . −2/b 1−e λ

At any rate, inasmuch as we are concerned with b  1, a saddle-point approximation appears in order for the Q-integrals. The saddle-point equation reads J0−1 Qsi − (E − Qsi )−1 +

  (J −1 )ij − J0−1 δij Qsj = 0

(8.8.24)

j

and obviously possesses homogeneous diagonal solutions determined by J0−1 Qs = (E − Qs )−1 .

(8.8.25)

For each diagonal element of Qs the forgoing quadratic equation allows for two roots, E/2 ± iπ(E)J0 where (E) = (2πJ0 )−1 (4J0 − E 2 )1/2 is the semicircular mean density of levels. Like in Sect. 6.6.3 there are sixteen saddle candidates to reckon with. Actually, only a single one qualifies, the one where the sequence of signs is given by Λ = diag(+1, +1, −1, −1). Moreover, that matrix Λ gives rise to a whole continuous manifold of saddles, Qsi = E/2 − iπ(E)J0 Ti ΛTi−1 ≡ E/2 − iπ(E)J0 Q(Ti )

(8.8.26)

8.8 Banded Random Matrices

341

at every site i, with Ti the site’s pseudo-unitary transformation. The continuous manifold of saddles Q(Ti ) = Ti ΛTi−1 obviously obeys Q(Ti )2 = 1

and

Str Q(Ti ) = 0

(8.8.27)

as is typical for the ‘Q-matrices’ in all sigma models. We can now harvest the fruits of the saddle-point approximation. Our average product of Green functions (8.8.21) simplifies considerably since the identities (8.8.27) entail Str(E −Qsi )Λ = Str Qsi Λ = Str Q(Ti )Λ and Str ln(E −Qsi ) = 0; it also follows that Qsi can be replaced by Q(Ti ) such that we obtain Gnn

(E − )G

nn

(E + )

=

 -

 dμ(Ti ) pn

(8.8.28)

i

     1  −1 × exp − i Str Q(Ti )Λ − (J )ij −J0−1δij Str Q(Ti )Q(Tj ) , J0 2 i

ij

The mean IPR (8.8.9) we further beautify with the help of the second of Eqs. (8.8.23),    dμ(Ti ) pn e−S (8.8.29) ↓0 πNρ(E) n i   2  S = πρ(E) Str Q(Ti )Λ + 12 πρ(E)J0 B Str Q(Ti )Q(Ti+1 ) .

P = lim

i

i

Notational differences apart, the exponential in the forgoing integrand is precisely the one that will be found in the next section for the kicked rotor, c. f. Eq. (8.9.24). The constituents of the prefactor pn are likewise simplified, g(n) = Qsn /J0 =

E − iπρ(E)Q(Tn ) . 2J0

(8.8.30)

8.8.4 Implementing the One Dimensional Sigma Model Efetov and Larkin first showed that the 1D sigma model describes thick disordered wires [56]. These authors also demonstrated the localization of eigenfunctions in the limit of long wires and found the localization length l = 2γ lel . Here, we keep following Ref. [48] where the full statistics of localization was treated.

342

8 Quantum Localization

Let us put our goal in view. The mean IPR is given by (8.8.9) and (8.8.29) as P = lim

→0

N  n=1

 Gnn (E − i)Gnn (E + i) Nπ

(8.8.31)

 N N  p   η n −S dμ(Ti ) e ≡ Pn , η→0 N (π)2 i n=1 n=1   S=η Str Q(Ti )Λ + 12 γ Str Q(Ti )Q(Ti+1 ) , = lim

i

i

2  and γ = π(E)J0 B ∝ 2b2 .

η = π

Before launching ourselves into the struggle of evaluation we should state that from this point on the treatments of the kicked rotor and banded random matrices run identically. We proceed to doing the superintegral in (8.8.31) recursively, working our way inwards site per site. We start at the edge, i = 1, and imagine that j − 1 ≥ 1 steps were already taken and yielded Yj (Tj ) ≡

j−1  i=1

where η = π

* γ + dμ(Ti ) e Str − 2 Q(Ti )Q(Ti+1 )−ηQ(Ti )Λ , 2  and γ = π(E)J0 B ∝ 2b 2 .

(8.8.32)

We can continue until after (n − 1) steps we arrive with Yn (Tn ) at the site whose contribution Pn to the inverse participation ratio P we focus on; that site requires special care since it attaches the extra factor pn to the integrand. Along the way we may rewrite the above integral as the recursion relation  Yj (Tj ) =

dμ(Tj −1 ) E(Tj , Tj −1 ) Yj −1 (Tj −1 ) ,

 γ E(T , T ) = exp − Str Q(T )Q(T ) − ηStr Q(T )Λ , 2 Y1 = 1 .

(8.8.33)

But we may equally well work our way inward starting from the other end, at i = N with YN = 1, and arrive after N − n steps with the contribution YN−n (Tn ) to the final integral over Tn . Combining all of the pieces, that last integral reads  Pn = lim η η→0

dμ(Tn ) Yn (Tn )YN−n (Tn )

pn −ηStr Q(Tn)Λ e . (π)2

(8.8.34)

8.8 Banded Random Matrices

343

Now, a parametrization of the matrix Tn must be invoked, together with the pertinent integration measure. The rational parametrization T =

 1 Z Z˜ 1

(8.8.35)

will again turn out convenient. Note that we have here omitted the site index n and will keep doing that until explicit notice to the contrary. We recall from Sects. 6.8.3 and 6.8.5 the singular-value decompositions A−1 ZD =

 √l

B

0

0  √ , −lF

with

˜ = D −1 ZA

lB ≥ 0

and

 √l

B

0

0  √ , − −lF

(8.8.36)

lF ≤ 0

in terms of supermatrices A and D, respectively unitary and pseudo-unitary, which also diagonalize the products The real eigenvalues lB , lF were given in (6.6.46) and the diagonalizing matrices A, D in (6.6.49). We note again the measure (6.6.44) where the eigenvalues mentioned together with two phases and four Grassmannians feature as ˜ = dμ(T ) = d(Z, Z)

dlB dlF dφB dφF dη∗ dηdτ ∗ dτ . 4π 2 (lB − lF )2

(8.8.37)

Since these eight parameters also make up the matrices A, D, Z, Z˜ they serve our purposes. We once more display the pertinent integration ranges (6.6.51), 0 < lB ≤ 1 ,

−∞ < lF < 0 ,

0 ≤ φB , φF ≤ 2π .

(8.8.38)

The recursion relation (8.8.33) then appears as no small hurdle to jump since the function Yj might depend on all eight independent variables entering Tj , i.e., the Grassmannians ηj∗ , ηj , τj∗ , τj , the phases φBj , φFj , and the eigenvalues lBj , lFj . A decisive step ahead of us is to show that the function Yj (Tj ) is in fact independent of the two phases and the four Grassmannians. Fearlessly launching ourselves into the fight for that simplification, we scrutinize the kernel E(T , T ) in (8.8.33). Simple to check is Str Q(T )Λ = 2

1 + l

B

1 − lB

−

1 + lF  ≡ 2(λB − λF ) . 1 − lF

(8.8.39)

The replacements lB,F → λB,F will presently turn out to be not just a momentary abbreviation but in fact an extremely helpful change of integration variables.

344

8 Quantum Localization

Slightly anticipating we right away note the pertinent measure, easily found from (8.8.37) as dμ(T ) =

dλB dλF dφB dφF dη∗ dηdτ ∗ dτ , 4π 2 (λB − λF )2

1 ≤ λB < ∞ ,

−1 ≤ λF ≤ 1 . (8.8.40)

The evaluation of Str Q(T )Q(T ) ≡ Str QQ is harder. To start we write out ⎛ Q = T ΛT −1 = ⎝

1+Z Z˜ 1−Z Z˜

˜ −1 −2Z(1 − ZZ) ˜ 1−ZZ

˜ − Z Z) ˜ −1 2Z(1

− 1+ZZ ˜

⎞ ⎠

(8.8.41)

and employ the singular-value decomposition √ √   2diag( l B , −lF )  −1 A 0  1+l 0  A − 1−l √1−l √ Q= 0 D −1 0 D 2diag( lB ,− −lF ) − 1+l 1−l

(8.8.42)

1−l

  with the diagonal 2 × 2 BF matrix l = diag lB , lF . At this point we wave good bye for good to the eigenvalues lB,F in favor of the variables λB,F according to  1 + l 1+l B 1 + lF = diag ≡ diag(λB , λF ) , 1−l 1 − lB 1 − lF  √  √ $  $ 2 l B 2 −lF , μ = diag = diag λ2B − 1 , 1 − λ2F lB − 1 lF − 1 λ=

(8.8.43)

μ˜ = diag(μB , −μF ) . Our readers will easily check  ηη∗ −η −η∗ ηη∗  ∗  ττ τ = −λ + (λB − λF ) τ∗ ττ∗

QRR = AλA−1 = λ + (λB − λF ) QAA = −DλD −1



(8.8.44) (8.8.45)

QRA = −AμD −1 (8.8.46)     1 + ηη∗ /2 ∗ (1 − τ τ /2)μB eB η −τ μB eB =− τ ∗ μF e F (1 + τ τ ∗ /2)μF eF η∗ 1 − ηη∗ /2   ∗ /2)(1−τ τ ∗ /2)μ e +ητ ∗ μ e −(1+ηη∗ /2)τ μB eB +η(1+τ τ ∗ /2)μF eF B B F F = − η(1+ηη ∗ (1−τ τ ∗ /2)μ e +(1−ηη∗ /2)τ ∗ μ e ∗ ∗ ∗ τ η μB eB +(1−ηη /2)(1+τ τ /2)μF eF B B F F

8.8 Banded Random Matrices

345

QAR = D μA ˜ −1 (8.8.47)    (1 + ηη∗ /2)μ e∗  1 − τ τ ∗ /2 τ −ημB eB∗ B B = ∗ ∗ η ∗ μF e F −(1 − ηη∗ /2)μF eF −τ ∗ 1 + τ τ ∗ /2   ∗ +τ η∗ μ e∗ ∗ −τ (1−ηη∗ /2)μ e∗ (1−τ τ ∗ /2)(1+ηη∗ /2)μB eB −(1−τ τ ∗ /2)ημB eB F F F F = −τ ∗ (1+ηη∗ /2)μ e∗ +(1+τ ∗ ∗ ∗ ∗ ∗ ∗ ∗ τ /2)η μ e −ητ μ e −(1+τ τ /2)(1−ηη /2)μ e B B

F F

B B

F F

with the unimodulars eB,F = e−iφB,F .

(8.8.48)

The bilocal expression we want to evaluate can now be written as   Str QQ = Str BF QRR Q RR + QRA Q AR + QAR Q RA + QAA Q AA .

(8.8.49)

Here, the first and the last of the four summands present no difficulty. Their contribution is immediately found as   Str BF QRR Q RR + QAA Q AA = 2(λB λ B − λF λ F )



+ (λB − λF )(λ B − λ F ) η˜ η˜ ∗ − τ˜ τ˜

(8.8.50)  ∗

with η˜ = η − η ,

τ˜ = τ − τ ,

η˜ ∗ = η∗ − η∗ ,

τ˜ ∗ = τ ∗ − τ ∗ . (8.8.51)

It is the remaining two terms in the supertrace (8.8.49) that require special thought. Among the plethora of additive terms arising quite a number involve products of six and even eight Grassmannians which do not cancel in a trivial way. Fortunately, these undesirables can all be absorbed in a phase shift iδφ =

1 ∗ (η η − ηη∗ + τ τ ∗ − τ τ ∗ ) 2

(8.8.52)

appearing only in φ˜ B = φB − φB − δφ

and

φ˜ F = φF − φF − δφ .

(8.8.53)

346

8 Quantum Localization

Having thus ridded ourselves of those undesirables we are left with Str Q(T )Q(T ) =

2(λB λ B

 −2 1 −

(8.8.54)

− λF λ F ) + (λB − λF )(λ B − λ F )(ρ˜ ρ˜ ∗ − σ˜ σ˜ ∗ )   ∗ 1 ˜ σ˜ ∗ μB μ B cos φ˜ B + μF μ F cos φ˜ F 4 ρ˜ ρ˜ σ

  −(ρ˜ ρ˜ ∗ − σ˜ σ˜ ∗ ) μB μ B cos φ˜ B − μF μ F cos φ˜ F  ˜ ˜  +iσ˜ ρ˜ ∗ μ B μF e−iφB + μ F μB eiφF e−i(φB −φF )  ˜ ˜  −ρi ˜ σ˜ ∗ μ B μF eiφB + μ F μB e−iφF ei(φB −φF ) .

We can now return to discussing the recursion relation (8.8.33). Therein, we had declared the primed variables η , η∗ , τ , τ ∗ , φB , φF , lB , lF as integration variables. , are now replaced with φ˜ The two phases among these, φB,F B,F ; that shift does not change the integration measure. Moreover, we change the Grassmannian integration variables to η˜ B , η˜B∗ , τ˜F , τ˜F∗ , again retaining the integration measure. Clearly then, independence of the local functions Yk of the Grassmannians is consistent; independence of the phases φB,F is not yet obvious since these phases still show up in (8.8.54). But let us try the ansatz Yk = Yk (λB , λF ) and write the recursion relation as 

∞

Yj (λB , λF ) = 1

dλ B



1

dλ F

−1

1 (λB − λ F )2



2π d φ˜ 0

B d φ˜ F (2π)2



˜ τ˜ ∗ d τ˜ d η˜ ∗ d ηd

  × exp −2η(λ B −λ F )−(γ /2)Str Q(T )Q(T ) Yj −1 (λ B , λ F ) .

(8.8.55)

The integration measure involves a pole at λ B = λ F = 1 which needs to be cancelled if the above integral is to exist. Searching for such cancellation, we inspect the phase-averaged exponential 

2π d φ˜ 0

B d φ˜ F (2π)2

  exp − (γ /2)Str Q(T0 )Q(T0 )

(8.8.56)

 = e−γ (λB λB −λF λF ) E0 + E1 η˜ η˜ ∗ − E1∗ τ˜ τ˜ ∗ + E2 (η˜ η˜ ∗ − τ˜ τ˜ ∗ ) + E3 η˜ η˜ ∗ τ˜ τ˜ ∗ where the four quantities Ei are functions of λB , λF , λ B , λ F as well as of the phases φB,F and read E0 = I0 (γ μ B μB )I0 (γ μ F μF ) ,  γ μB μF I1 (γ μ B μB )I0 (γ μ F μF ) + (B ↔ F ) e−i(φB −φF ) , E1 = 2 γ E2 = − (λB − λF )(λ B − λ F )E0 2

(8.8.57)

8.8 Banded Random Matrices

347

  + μB μ B I1 (γ μ B μB )I0 (γ μ F μF ) − (B ↔ F ) , E3 =

 γ2 (λB − λF )(λ B − λ F ) (λ B λB + λ F λF )I0 (γ μ B μB )I0 (γ μ F μF ) 2   − μB μ B I1 (γ μ B μB )I0 (γ μ F μF ) − (B ↔ F )

≡ (λB − λF )(λ B − λ F )L(λ B , λ F |λB , λF ) . Cancellation? Well, due to the definition of μB , μF in (8.8.43) and the smallargument behavior of the Bessel function I1 , the three functions E1 , E2 , E3 all approach zero linearly as λ B , λ F → 1 and thus give rise to converging integrals over λ B , λ F in (8.8.55). We can forget about the terms with E1 , E2 since these are annulled by the subsequent fourfold Grassmann integral; moreover, these two terms being the only ones involving the phases φB,F we conclude consistency of the phase independence of the functions Yk . On the other hand, the term with E3 does survive since it comes with the maximal Grassmann monomial which integrates # to unity, d ρ˜ ∗ d ρd ˜ σ˜ ∗ d σ˜ ρ˜ ρ˜ ∗ σ˜ σ˜ ∗ = 1. Finally, the term with E0 does not vanish at the boundary λ B = λ F = 1 and thus engenders a diverging Bosonic integral # multiplied by the vanishing Grassmann integral d ρ˜ ∗ dρdσ ∗ dσ = 0. The formal product ∞ × 0 is assigned the value exp[−γ (λB − λF )]Yj −1 (1, 1) by the Parisi– Sourlas–Efetov–Wegner theorem of Sect. 6.4.5. The worry about the pole in the integration measure is thus out of the way and the recursion relation takes the form Yj (λB , λF ) = e−γ (λB −λF ) Yj −1 (1, 1)  1  ∞ λB − λF dλ B dλ F + L(λB , λF |λB , λF ) λ − λ 1 −1 B F

(8.8.58)



× e−2η(λB −λF ) Yj −1 (λ B , λ F ) . Before trying to solve for Yj , it is well to pull the expression (8.8.34) for the inverse participation ratio Pn to the level reached for the functions Yj . Just inserting (8.8.39) in (8.8.34) we have the IPR  Pn = lim η η→0

dμ(Tn ) Yn (λnB , λnF )YN−n (λnB , λnF )

pn −2η(λB −λF ) e (π)2 (8.8.59)

Now, we recall the quantity pn from Eq. (8.8.20) and the saddle-point expression (8.8.26) for Qsn to write    E/2J0 − iπρQsBB pn = E/2J0 − iπρQsBB nAA nRR    sBB E/2J − iπρQ + E/2J0 − iπρQsBB 0 nAR nRA .

(8.8.60)

348

8 Quantum Localization

Herein, only terms involving the maximal Grassmann monomial ηn ηn∗ τn τn∗ will survive the remaining Grassmann integrals; among those, only the ones independent of the phases φnB , φnF withstand the remaining phase integrals. Inspecting the BoseBose entries in the BF matrices QRR , QAA , QRA , QAR given in (8.8.44) we easily see that all terms involving E/2J0 in the foregoing expression for pn go under and we are left with pn /(πρ)2 = −2ηn∗ ηn τn∗ τn λB (λB − λF ) + . . . ;

(8.8.61)

the dots refer to junk annulled by the said integrations. At this point it becomes clear that the mean density cancels from the mean IPR such that the only energy dependence left is in the parameter γ , cf. (8.8.31). The desired reformulation of (8.8.34) is Pn = lim 2η η→0

 ∞

1 dλ dλ λ B F B −2η(λB −λF )

−1

1

λB − λF

e

YN−n (λB , λF)Yn (λB , λF)

(8.8.62)

and suggests the next step ahead by its very appearance: The integral must be proportional to 1/η for the limit η → 0 to exist. Such singular behavior can be contributed only by large values of λB . But in that range, λB − λF → λB , and we are led to suspect that Yn (λB , λF ) becomes independent of λF . The ansatz Yn (λB , λF )|λB λF = yn (2ηλB ) ≡ yn (z) ,

y1 = 1

(8.8.63)

will in fact turn out consistent and immediately embellishes Pn to 

∞

dze−z yN−n (z)yn (z) .

Pn = 2

(8.8.64)

0

Now the recursion relation (8.8.58) must be updated to the ansatz (8.8.63) in the large-λB limit. As a first simplification, we can dispense with the exponentially small boundary term, and thus 

∞

yj (2ηλB ) = 1

dλ B



λB e−2ηλB λ B



1

dλ F L(λ B , λ F |λB , λF ) yj −1 (2ηλ B ) .

−1

(8.8.65) The integral over λ F becomes explicitly doable once we appreciate that μB μ B ≈ λ

λB λ B − 12 ( λBB +

λB ) λ B

 1 and replace the Bessel functions with large arguments by

their asymptotic forms expansions In (z) = (2πz)−1/2ez (1 −

4n2 −1 8z

. . .). Then, the

8.8 Banded Random Matrices

349

kernel L(λ B , λ F |λB , λF ) takes the asymptotic form  γ  λ γ 3/2 λB  B exp − + + γ λ F λF L(λ B , λ F |λB , λF ) = $ 2 λB λB 2 2πλB λ B  1  λ B 1 λB  + × λ F λF + + 2γ 2 λB λB ×I0 (γ μ F μF ) + μ F μF I1 (γ μ F μF )

(8.8.66) 

which we can subject to the λ F -integration required in (8.8.65). The integral identity (see Problem 8.12) √ ) sinh a 2 + b 2 2 dxe I0 (b 1 − x ) = 2 √ a 2 + b2 −1



1

ax

(8.8.67)

and its first derivatives w. r. t. the real parameters a, b yield  ∞ 7  ∞ z −z  z  yj (z)= dz y e L (z ) = duu−3/2e−uz Lγ (u)yj −1 (uz) , γ j −1 3 z z 0 0 7  γ sinh γ 1 1 γ − (u+ 1 ) u e 2 cosh γ − + u+ sinh γ . (8.8.68) Lγ (u)= 2π 2γ 2 u One arrow remains in our quiver and will be shot presently, the largeness of the parameter γ ∝ 2b2  1. The kernel Lγ (u) therefore has a sharp peak of width √ 1/ γ 1 at u = 1. Promising to demonstrate consistency later, we assume that the function yj (uz) varies slowly in u across the width of the peak of Lγ (u), whatever the value of z may be. Then, it is advisable to shift and rescale the integration variable as ξ u=1+ √ γ

(8.8.69)

√ such that the following expansions in powers of 1/ γ become available: 7   1 1 1 1 γ γ − γ (u+ 1 ) 2 u e + u+ Lγ (u) = 1− 2 2π 2γ 2 u 7   2 3 2 ξ ξ ξ4 ξ6 γ −ξ 1 −3/2 2 e + + √ − + + O(γ = ) , 1− 2π 4γ 4γ 2 γ 2γ 8γ 15ξ 2 3ξ + O(γ −3/2 ) , u−3/2 = 1 − √ + 2 γ 8γ

350

8 Quantum Localization

e

−zu

=e

−z



 z2 ξ 2 zξ −3/2 + O(γ ) , 1− √ + γ 2γ

zξ z2 ξ 2 y (z) + O(γ −3/2 ) . yj −1 (uz) = yj −1 (z) + √ yj −1 (z) + γ 2γ j −1

(8.8.70)

Upon doing the ξ -integrals, we transform the recursive integral equation (8.8.68) into the recursive differential equation6 ez yj (z) = yj −1 (z) +

+ z2 * yj −1 (z) − 2yj −1 (z) + yj −1 (z) . 2γ

(8.8.71)

We should appreciate that the unit steps of the site index j are minute compared to the correlation length γ ; thus, a continuum approximation according to j → 2γ τ is indicated, with τ ranging in the interval [0, N/(2γ ) ≡ x]. Finally, we anticipate that yj (z) will decay with growing z on a scale ∝ 1/γ and introduce the rescaled independent variable y = 2γ z, yj (z) = y2γ τ (y/2γ ) ≡ Y (y, τ ) with

y ∈ [0, ∞) ,

τ ∈ [0, x =

N ]. 2γ (8.8.72)

Such rescalings yield e−z = e−y/2γ → 1, reveal the first two terms in the curly bracket in (8.8.71) as negligible, and turn the recursion relation into the partial differential equation ∂2 ∂ ˆ (y, τ ) Y (y, τ ) = (y 2 2 − y)Y (y, τ ) ≡ LY ∂τ ∂y

(8.8.73)

and the “initial” condition in (8.8.33) and (8.8.63) into Y (y, τ = 0) = 1 .

(8.8.74)

The very absence of all parameters from the foregoing initial-value problem demonstrates the consistency of the scaling assumptions made above, provided, of course, that a solution exists. The (local) inverse participation ratio Pn last given in (8.8.64) can also be expressed in terms of the function Y (y, τ ), Pn →

6 Equation

1 γ



∞

dy Y (y, x − τ )Y (y, τ ) .

0

72 on p. 3818 of Ref. [48] contains typos and should be read as our Eq. (8.8.71).

8.8 Banded Random Matrices

351

Upon averaging over sites according to (8.8.31), we express the mean participation ratio as  x  N 1  2 ∞ P = Pn = dy dτ Y (y, x − τ )Y (y, τ ) . N N 0 0

(8.8.75)

n=1

This is already an important asymptotic result, worthy of highlighting. We may read the prefactor PGUE ≡ 2/N of the foregoing integral as the inverse participation ratio of the GUE and refer our P to that unit. The result, P PGUE





∞

=

x

dy 0

dτ Y (y, x − τ )Y (y, τ ) ≡ β(x) ,

(8.8.76)

0

depends only on the scaling parameter x = N/2γ ∝ N/b 2 .

(8.8.77)

As already noted above, Casati, Molinari, and Izrailev [49] established such scaling through numerical work and even conjectured that β(x) = 1 + x/3. Such surprisingly simple behavior was indeed borne out by Fyodorov’s and Mirlin’s analysis of Ref. [48] which we are spreading out here. It will be convenient to go for the final goal via the Laplace transform ˜ β(p) =



∞

dx e

−px



∞

β(x) =

0

dy Y˜ (y, p)2 .

(8.8.78)

0

Solutions of the above partial differential equation may be sought as composid2 tions of eigenfunctions of the operator Lˆ = y 2 dy 2 − y. Such eigenfunctions which decay to zero for y → ∞, are related to the modified Bessel functions Kr (t) as √ √ fr (y) = 2 yKr (2 y) and come with the eigenvalues (r 2 − 1)/4. For imaginary indices r = iν, ν ∈ (0, ∞), the functions fiν (y) are mutually orthogonal and can be used as a basis in the sense of the Lebedev–Kontorovich transformation [57]. That transformation relates a function F (x) to its transform F˜ (ν) as 

∞

F (x) =

dν Kiν (x)F˜ (ν) ,

0

2 F˜ (ν) = ν sinh πν π



∞ dx

x

0

(8.8.79)

Kiν (x)F (x) ,

provided (1) that F (x) is piecewise differentiable in (0, ∞) and (2) that there is some positive number  such that 

 0

dxx −1 |F (x) ln x| < ∞



∞

and 

dxx −1/2|F (x)| < ∞ .

(8.8.80)

352

8 Quantum Localization

A little thought shows that we cannot naively invoke that transformation to represent ν 2 +1 √ √ Y (y, τ ) as a superposition of eigenfunctions ei 4 τ 2 yKiν (2 y) since the initial condition Y (y, 0) = 1 would yield the function F (x) = 1/x which does not qualify as the member of a Lebedev–Kontorovich pair. There is a way out found by Fyodorov and Mirlin and easily followed. The differential operator Lˆ has one additional eigenfunction which decays to zero for y → ∞ and pertains to the ˆ 1 (y) = 0 where f1 = 2√yK1 (2√y). By simply including eigenvalue 0; indeed, Lf that eigenfunction, we represent the function Y (y, τ ) in search as √ √ Y (y, τ ) = b1 2 yK1 (2 y) +



∞

dν b(ν) e−

ν 2 +1 4 τ

√ √ 2 yKiν (2 y)

(8.8.81)

0

and determine the expansion coefficients b1 and b(ν) from the initial condition Y (y, 0) = 1. That condition reads  ∞ √ √ 1 − b1 2 yK1 (2 y) √ √ = dν b(ν) 2 yKiν (2 y) √ 2 y 0

(8.8.82)

and reveals b(ν) as the would-be Lebedev–Kontorovich transform of the function F (x) = x1 − b1 K1 (x). Due to the small-argument behavior of the Bessel function, K1 (x) → x1 + x2 ln x2 for x → 0, F (x) → (1 − b1 ) x1 ) − b1 x2 ln x2 , and we conclude that the coefficient b1 is uniquely determined as b1 = 1. No other choice would meet the first of the conditions (8.8.80); the second of these is also met since for b1 = 1, the large-x behavior F (x) → x1 . Therefore, our function Y (y, τ ) is determined up to quadratures, namely, b(ν) =

2 ν sinh πν π



∞ dx

x

0

Kiν (x)

1

2 ν sinh πν 2 − K1 (x) = x π 1 + ν2

(8.8.83)

and the composition (8.8.81)    2 2 ∞ ν sinh πν √ √ √ τ 2 − 1+ν dν e 4 Kiν (2 y) . Y (y, τ ) = 2 y K1 (2 y) + π 0 1 + ν2 (8.8.84) The final expression for the expansion coefficient in (8.8.81) as well as the following Laplace transform 1 Y˜ (y, p) = p ≡ μ=

1 p )



∞

du u 

0 ∞

(μ + 1) + u2 (μ − 1) √ Jμ−1 (2 yu) 2 2 (1 + u )

√ du F (u) Jμ−1 (2 yu) ,

0

4p + 1

(8.8.85)

8.9 Sigma Model for the Kicked Rotor

353

were found in Ref. [48] by some ingenious juggling with integrals involving trigonometric, hyperbolic, and Bessel functions. We skip these interesting calculations here. ˜ Inserting the Laplace transform Y˜ (y, p) into the scaling function β(p) of (8.8.77), we obtain ˜ p β(p) =



2



∞

∞

dy 0

√ du F (u) Jμ−1 (2 yu)

2 .

(8.8.86)

0

√ A simple change of the integration variable y to t = 2 y and the orthogonality #∞ 1 0 dt tJν (tu)Jν (tv) = u δ(u − v) of the Bessel functions produce ˜ p2 β(p) =

1 2



∞

du u−1 F (u)2 = p +

0

1 . 3

(8.8.87)

Reverting to the Laplace transform gives the celebrated scaling function β(x) = 1 + x/3 .

(8.8.88)

Remembering the scaling parameter x = N/2γ and the GUE value PGUE = find the mean IPR as a function of N and γ P (γ , N) =

1 2 + . N 3γ

2 N,

we

(8.8.89)

Clearly, then, there is no critical value of x for the onset of localization but rather a continuous transition from the delocalization typical of GUE matrices to well pronounced localization, as the scaling parameter x grows from O( N1 ) to O(N).

8.9 Sigma Model for the Kicked Rotor 8.9.1 A Rotor Without Time Reversal Invariance The sigma model treatment of the kicked rotor was initiated in Ref. [58]. Strong support for the originally controversial approach came with the recent extension to everything previously known about the rotor, including the so called quantum resonances and even higher dimensional versions [59, 60]. The principal difference between the rotor (or Anderson’s hopping model) and fully chaotic dynamics with universal spectral fluctuations á la RMT lies in the fact already alluded to several times in preceding sections that the rotor allows for arbitrarily slow modes of classical behavior (diffusion in angular momentum space). As a consequence, there cannot be a finite gap in the Frobenius-Perron spectrum of

354

8 Quantum Localization

resonances, even if there are no noticeable islands of regular motion in phase space. It is to that case that we shall confine the following. For simplicity, we consider a rotor without time reversal invariance with the Floquet operator a variant of (8.3.4), U = e−

ih˜ nˆ 2 2

e

− iλ˜ cos(θ+Φ) h

;

(8.9.1)

here θ and n = −i∂θ are dimensionless operators for the angular coordinate and angular momentum, obeying [θ, n] = i. Planck’s constant appears in the dimensionless form (8.3.8). The important difference to the previous Floquet operator (8.3.4) is the appearance of a phase Φ which, if non-zero, breaks invariance under the time reversal transformation t, θ, nˆ → −t, −θ, n. ˆ We shall need the matrix elements of the Floquet operator in the angular momentum representation, Unm

= n|Uˆ |m = e−

ihn ¯˜ 2 2

 0

2π

dθ −i λ˜ cos(θ+Φ)+i(n−m)θ e h¯ . 2π

(8.9.2)

We look at λ  1 since that limit produces strong chaos. To exclude the ˜ resonances mentioned before we can require h/4π to be an irrational number. Equally acceptable and in a sense preferable is ‘near irrationality’, i.e., h˜ = 4π M N with M, N both large primes. That latter case comes with the conservation law [TN , U ] = 0 where TN is a translation in angular momentum, TN |n = |n+N with n|n ˆ = n|n and integer n. We can then work in a Hilbert space of dimension N (the first Brillouin zone, in solid state language, see [59]). The physics of localization will be unchanged for N large compared to the localization length. In developing the sigma model for the rotor we shall closely follow Ref. [61].

8.9.2 Inverse Participation Ratio As for banded matrices above we employ the so called inverse participation ratio (IPR) as an indicator of localization. For an individual eigenvector |μ of U the inverse participation ratio is defined 4 as Pμ = N n=1 |n|μ| where {|n, n = 0, 1, 2, . . . N} are the angular momentum eigenstates forming the basis wherein we look for localization. In order to characterize the whole Floquet matrix rather than a single eigenvector, we average over all eigenvectors P =

N N N 1  1  Pμ = |n|μ|4 . N N μ=1

μ=1 n=1

(8.9.3)

8.9 Sigma Model for the Kicked Rotor

355

In order to reveal the two-point nature of the IPR thus defined we employ the 1 matrix elements G± nn (a, φ) = n| 1−ae±iφ U ±1 |n of the retarded/advanced Green − function with a = e at small positive . We propose to scrutinize the center-phase average of the product of these two matrix elements multiplied with N , 2 N



2π

0

dφ + G (a, φ)G− nn (a, φ) , 2π nn

a = e− ,

(8.9.4)

in the limit as  approaches 0. As long as  > 0 we can expand both Green functions in geometric series and afterwards do the φ-integral. Resumming the resulting new geometric series we get 2 N



2π 0

N dφ + 2 1  Gnn (a, φ)G− (a, φ) = |n|μ|2 |n|ν|2 . nn i(φ 2π N 1 − e ν −φμ )−2 μ,ν=1

(8.9.5) Only the diagonal terms of the double sum over Floquet eigenvectors survive the limit  ↓ 0 and we arrive at the spectrally averaged IPR after summing over n, P = lim ↓0

 N  2 n=1

N

2π 0

dφ + − − G (e , φ)G− nn (e , φ) . 2π nn

We may simply define a localization length as the inverse of the IPR, l =

(8.9.6) 1 P

.

8.9.3 Sigma Model − As in (8.8.11), the product G+ nn Gnn can be represented as a Gaussian superintegral in the fashion of (7.2.3) and the superanalytic version (6.4.17) of Wick’s theorem, − G+ nn (a, φ)Gnn (a, φ) =

× exp

N   k,l=1



∗ ∗ d(ψ, ψ ∗ )ψ+,B,n ψ+,B,n ψ−,B,n ψ−,B,n

(8.9.7)

∗ ∗ (δkl − aeiφ Ukl )ψ+,B,l − ψ−,B,k (δkl − ae−iφ Ulk∗ )ψ−,B,l − ψ+B,k

 ∗ ∗ − ψ+,F,k (δkl − aeiφ Ukl )ψ+,F,l − ψ−,F,k (δkl − ae−iφ Ulk∗ )ψ−,F,l .

Note that we could simplify, relative to (7.2.3), by setting a = b = c = d = e− . Indeed, then, the Gaussian integral gives just the l.h.s. of the foregoing equation.

356

8 Quantum Localization

We now do the center-phase average7 using the color-flavor transformation (6.8.2). Like in Sect. 7.2 we are dealing with a single dynamical system. The color-flavor transformation involves a single color, C = 1 (since the phase integral covers U(1)). The number of flavors is the QD dimension F = N. The − phase averaged product G+ nn Gnn then takes the form 

2π 0



=

dφ + G (a, φ)G− nn (a, φ) 2π nn   ∗ ∗ ∗T ∗T ψ+,B,n ψ−,B,n ψ−,B,n exp − ψ+ ψ+ − ψ− ψ− d(ψ, ψ ∗ )ψ+,B,n  ×

 =

(8.9.8)

 ∗T  ∗T † ˜ sdet(1 − Z Z) ˜ exp ψ+ ˜ − + ψ− d(Z, Z) U eˆ− Z eˆ+ U ψ+ Zψ

˜ sdet(1 − Z Z) ˜ d(Z, Z)



∗ ∗ ψ+,B,n ψ−,B,n ψ−,B,n e−ψ d(ψ, ψ ∗ )ψ+,B,n

† Mψ

with the supermatrix  M=

˜ † 1 −U ZU −Ze−2 1

(8.9.9)

already given in (7.2.7) albeit there with general a, b, c, d; again, M lives in QD⊗BF⊗AR but only the AR structure is explicit in the above expression. The integration variables Z and Z˜ are 2N × 2N supermatrices like in the superintegral representation (7.2.5) for the generating function Z. Doing the integral over ψ, ψ ∗ we arrive at 

2π 0

dφ + G (a, φ)G− nn (a, φ) = 2π nn



˜ pn e−S d(Z, Z)

(8.9.10)

where the action     ˜ †Z . ˜ = − str ln 1 − ZZ ˜ S(Z, Z) + str ln 1 − e−2 U ZU

(8.9.11)

from (7.2.9) reappears, but now with a = b = c = d = e− ; the prefactor pn is determined through Wick’s theorem (6.4.17) by the elements of the inverse of the the supermatrix M as −1 −1 −1 pn = M −1 +Bn,+Bn M −Bn,−Bn + M +Bn,−Bn M −Bn,+Bn .

(8.9.12)

7 The “global” spectral average possible here as it was in Sect. 7.2 saves us from having to use the superanalytic Hubbard-Stratonovich transformation.

8.9 Sigma Model for the Kicked Rotor

357

8.9.4 Slow Modes We propose to simplify the action S, invoking the slow diffusive motion of the angular momentum in the classical limit (slow on the time scale given by the kicking period). It is these slow modes that preclude universal fluctuations in the quantum spectrum. In order to isolate the corresponding ‘soft’ quantum fluctuations we start with the action (8.9.11) which now depends only the single source variable a = e− = 1 −  + . . .. Expanding in powers of  we have ˜ + str ln(1 − U ZU ˜ † Z) +  str S = −str ln(1 − ZZ)

˜ †Z U ZU + O( 2 ) ˜ †Z 1 − U ZU (8.9.13)

˜ Z), or, momentarily restricting our attention to the lowest order in (Z,   ˜ †Z + . . . . ˜ − U ZU ˜ † Z +  str U ZU S = str ZZ

(8.9.14)

In the end a symmetry argument will allow us to shed the restriction. Next, we look at the matrix elements Znm (and Z˜ nm ) in the angular momentum basis. In our search for slow modes we try to neglect off-diagonal terms, Znm ∼ δnm Z(n), for the following reason. Off-diagonal elements of Z carry information about the direction of propagation in n-space. A momentarily prevailing direction will be forgotten after a few kicks, simply since the angular momentum behaves diffusively. As the diffusion proceeds, off-diagonal elements must thus die out. (The situation is reminiscent of the unbiased diffusion of an electron on a 1D lattice which affords description by a density matrix diagonal w.r.t. the site label, on time scales larger than the scattering time.) Moreover, the diagonal elements Z(n) should vary slowly with n; this is because n-independent diagonal matrices Z and Z˜ commute with U and are thus strictly stationary; slowly varying diagonal elements must thus make up the slow modes. To formalize the argument we represent the slow modes by a truncated Fourier integral,  φ0 dφ Znm = δnm (8.9.15) Z(φ)e−inφ 2π 0 with a cut-off φ0 to be specified presently, certainly with φ0 1. We may of course imagine the cut-off worked into Z(φ) such that Z(φ) → 0 as φ becomes larger than φ0 . With that understanding we extend the φ-integral in (8.9.15) and in what follows to the upper limit 2π. Such prepared we write (the contribution of the slow modes to) the action as    S= (8.9.16) strBF Zn Z˜ m δnm − (1 − )|Unm |2 nm

 =

dφ 2π



dφ ˜ ) k(φ, φ ) strBF Z(φ)Z(φ 2π

358

8 Quantum Localization

with the kernel k(φ, φ ) =

  δnm − (1 − )|Unm |2 e−i(nφ+mφ ) .

(8.9.17)

nm

We here insert the matrix elements (8.9.2) of the Floquet operator and use the identity n einφ = 2πδ(φ) to get







k(φ, φ ) = 2πδ(φ + φ ) 1 − (1 − )

2π

0

dθ −i λ˜ [cos(θ+Φ)−cos(θ+Φ−φ)]  e h¯ . 2π (8.9.18)

The phase Φ breaking time reversal invariance is here seen to disappear, simply by changing the integration variable θ ; the IPR of the rotor is thus independent of Φ. The restriction 0 ≤ φ < φ0 1 allows to expand cos θ − cos(θ − φ) = −φ sin θ + . . .. Further restricting the cut-off as φ0

max(1, h¯˜ /λ)

iλ

we can expand the exponential e h¯ φ sin θ in powers of the small quantity   k(φ, φ ) * 2πδ(φ + φ )  +

λ 2h¯˜

2

φ2

  1 + O(, φ 2 , (λφ/h¯˜ )2 ) .

(8.9.19) λφ , h˜

(8.9.20)

#

Going back to the angular-momentum basis with n * dn (recall slow fields are # smooth), we have Z(φ) * dnZ(n)einφ and get the slow-mode action  S=

  dn strBF Zn Z˜ n +

λ 2h˜

2

 ˜ ∂n Zn ∂n Zn .

(8.9.21)

Here, we had to preserve consistency and recall that the restriction of Z(φ) to small φ allows to do the φ-integral over the interval [0, 2π] such that the orthogonality # 2π dφ i(n−m)φ = δnm arises. The slow-mode action can be said to be small, in the 0 2π e following sense: the first term is proportional to the infinitesimal  while the second cannot get large due to the smoothness condition (8.9.19). We now invoke the promised symmetry argument to shed the restrictionto the 0 quadratic action (8.9.14). Employing the matrix Q = T ΛT −1 with Λ = 10 −1  1 Z and T = Z˜ 1 as defined in (6.8.12) or (7.2.10) we first note that with Z and Z˜ diagonal in QD (the angular momentum space in our case) so are T and Q, i.e., Tnm = δnm Tn and similarly for Q. Moreover, transformations T close to unity entail

8.9 Sigma Model for the Kicked Rotor

359

˜ and Str QΛ ∼ 4 Str BF Z Z˜ and thus Str (∂n Q)2 ∼ −8 Str BF (∂n Z)(∂n Z) 1 S= 4 =



  dn Str Qn Λ − 12

1 Str 4 n

1 = Str 4 n

 Qn Λ − 

1 2



λ 2h˜

λ 2h˜

2  2  ∂n Qn

2 

Qn+1 − Qn

(8.9.22) 2 

  λ 2 Qn Λ + Qn+1 Qn 2h˜

(8.9.23)

with the supertrace now over AR and BF. At this point we can drop the restriction ˜ arguing as follows. The original matrix Q = T ΛT −1 has Z˜ BB = to small Z, Z, † † ˜ ZBB , ZF F = −ZF† F and |ZBB ZBB | < 1 as the only restrictions. In the present −1 context where each ‘site’ n has its separate  Qn and we can demand Qn = Tn ΛTn 1 Z

with single-site transformations Tn = Z˜ 1n which need not be close to unity. n Nor need their constituents Zn and Z˜ n be small. We have arrived at what is called the (action of the) diffusive 1D sigma model. In the previous Sect. 8.8 precisely this sigma model has appeared in a different context (banded random matrices). A critical remark on our somewhat cavalier construction of the slow-mode action is in order. It is to be complemented by checking stability against configurations outside the slow-mode ‘sector’. We shall forego that analysis which has been performed in Ref. [59]. The ‘massive’ modes we have neglected here turn out to renormalize the diffusion constant to a weak further dependence on h˜ and λ (see also [62, 63]). For the IPR we are left, due to the definition (8.9.6) and (8.9.10), with P = lim

 N  2  -

 dμ(Ti ) pn e−S

N i     2 S= Str (/4)Qn Λ + λ/2h˜ Qn+1 Qn ↓0

(8.9.24)

n=1

n

and the obligation to do the superintegral over the transformations Tn such that the full manifold for Qn is covered at each site n. It remains to scrutinize the prefactor pn of e−S in (8.9.10). It will turn out later that the overall factor  in the IPR cannot be compensated by the prefactor pn . Anticipating that right away we put a = e− → 1 in pn . Moreover, the slow modes are negligible there since in contrast to their role in the action they are but small 2  corrections of relative order λφ0 /h˜ in pn . So we can simply put U → 1. The thus simplified matrix M −1 reads   M −1 

→0 ,U →1

≡ M0−1 =



˜ −1 Z(1 ˜ − Z Z) ˜ −1  (1 − ZZ) , ˜ −1 (1 − Z Z) ˜ −1 Z(1 − ZZ)

(8.9.25)

360

8 Quantum Localization

with Z and Z˜ again both diagonal in QD and ranging over the Qn -manifolds for n = 1, 2, . . . , N. Preparing for the final steps we note the prefactor in the form 

pn =

BB  BB  BB  BB 1 1 1 1 + Z˜ n Zn , 1 − Z˜ n Zn 1 − Zn Z˜ n 1 − Zn Z˜ n 1 − Z˜ n Zn

as collected from (8.9.12), and (8.9.25). It differs from its analogue (8.8.60) for banded matrices. But the difference turns out less dramatic than one might fear. Like for banded matrices we determine the above superintegral recursively. The ansatz of functions Yi (λiB , λiF ) independent of phases and Grassmannians works again. Imagining the forgoing superintegral done for all but the nth site we arrive like for banded matrices at the ‘last-site’ integral (8.8.59), now with pn /(π)2 replaced by 8pn ,  Pn = lim η η→0

dμ(Tn ) Yn (λnB , λnF )YN−n (λnB , λnF ) 8pn e−2η(λnB −λnF ) . (8.9.26)

As in our previous processing of (8.8.59) only the phase independent terms involving the maximal Grassmann monomial ηn , ηn∗ , τn , τn∗ in pn survive the remaining phase and Grassmann integrals. Employing the singular-value decompositions of the supermatrices in the forgoing expression for the prefactor pn we find 1 pn = − ηn∗ ηn τn∗ τn λB (λB − λF ) + . . . ; 2

(8.9.27)

such that the ‘last-site’ contribution Pn to the mean IPR becomes Pn = lim 4η η→0

 ∞ 1

1 dλ dλ λ B F B −2η(λB −λF ) −1

λB − λF

e

YN−n (λB , λF)Yn (λB , λF) ,

(8.9.28)

twice the result (8.9.28) found for banded matrices. That factor carries through to the  2 final result (8.8.89). Inserting the above γ = 12 λ˜ and dropping the now negligible PGUE =

2 N

h

we find the result P =

4  h˜ 2 ; 3 λ

(8.9.29)

regarding the numerical prefactor this result should be considered as an improvement over the previous estimate (8.5.6) which involved order-of-magnitude arguments.

8.10 Problems

361

8.10 Problems 8.1 Derive the discrete-time Heisenberg equations for the kicked rotator. 8.2 The random-phase approximation of Sect. 8.3 does not assign Gaussian behavior to the force ξ = λV (Θ). Verify this statement by calculating ξ n  for V = cos Θ and uniformly distributed phases Θ. 8.3 The result of Problem 8.3 notwithstanding, the moments pt2n  tend to display Gaussian behavior for t → ∞. Show this with the help of the random-phase approximation. 8.4 Rewrite the quantum map (8.3.18) in the Θ-representation. 8.5 Show that V (Θ) = −2 arctan (κ cos Θ −E) for the kicked rotator corresponds to nearest-neighbor hops in the equivalent pseudo-Anderson model. 8.6 Discuss the analyticity properties of Green’s function G(E) = (E − H˜ )−1 with H˜ given by (8.3.24). 8.7 Generalize the simple Gaussian integrals 

+∞ −∞



+∞ −∞

ds e

2 −αs 2

ds s e

−αs 2

7 =

∂ =− ∂α

7

π α

1 π = α 2α

7

π α

to the multiple integrals 

+∞ −∞



+∞ −∞

N

d N s e−

d S S1 S2 e

−

.

i,j

αij Si Sj

= 

i,j

αij Si Sj

=

πN det α 1 2α

 1,2

7

πν det α

for a positive symmetric matrix α. Use the eigenvectors and eigenvalues of α. 8.8 Diagonalize the matrix Vij = κ(δi,j +1 + δi,j −1 )/2. 8.9 Use (8.4.29), (8.4.30) to determine the eigenvalues of the matrix Vij = κ(δi,j +1 + δi,j −1 )/2. Locate the poles of Green’s function given in (8.4.24). What happens to these poles in the limit N → ∞? 8.10 Verify the stroboscopic Heisenberg equation (8.6.4) for the kicked top with the Floquet operator (8.6.2).

362

8 Quantum Localization

8.11 Determine the amplitude of the temporal fluctuations of var (J)/j 2 from the point of view of random-matrix theory. 8.12 Even though possibly not aware of it, you know the integral identity (8.8.67). Consider the#Fourier transform of # ∞a spherically symmetric function F (|x|) in three dimensions, d 3 xeik·x F (|x|) = 0 drr 2 F (r)f (kr) where 



2π

f (ρ) =

π

dϕ 0

0

dθ sin θ eiρ cos θ = 4π

sin ρ ρ

To save labor the polar axis was chosen parallel to the vector k here. Now choose axes less wisely and enjoy 

1 −1

dx e

iax

√ ) sin a 2 + b 2 2 J0 (b 1 − x ) = 2 √ . a 2 + b2

as the fruit of lesser wisdom. Analytic continuation to imaginary a, b produces (8.8.67).

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22. J.C. Robinson, C. Bharucha, F.L. Moore, R. Jahnke, G.A. Georgakis, Q. Niu, M.G. Raizen, B Sundaram, Phys. Rev. Lett. 74, 3963 (1995) 23. F.L. Moore, J.C. Robinson, C. Bharucha, B. Sundaram, M.G. Raizen, Phys. Rev. Lett. 75, 4598 (1995) 24. H.-J. Stöckmann, Quantum Chaos, An Introduction (Cambridge University Press, Cambridge, 1999) 25. B. Kramer, A. MacKinnon, Rep. Prog. Phys. 56, 1469 (1993) 26. P.W. Anderson, Phys. Rev. 109, 1492 (1958); Rev. Mod. Phys. 50, 191 (1978) 27. D.J. Thouless, in Session XXXI, 1979, Ill-Condensed Matter, ed. by R. Balian, R. Maynard, G. Thoulouse Les Houches (North-Holland, Amsterdam, 1979) 28. K. Ishii, Prog. Th. Phys. Suppl. 53, 77 (1973) 29. H. Furstenberg, Trans. Ann. Math. Soc. 108, 377 (1963) 30. A. Crisanti, G. Paladin, A. Vulpiani, Products of Random Matrices (Springer, Berlin, 1993) 31. B.V. Chirikov, preprint no. 367, Inst. Nucl. Phys. Novosibirsk (1969); Phys. Rep. 52, 263 (1979) 32. T. Dittrich, R. Graham, Ann. Phys. 200, 363 (1990) 33. H. Weyl, Math. Ann. 77, 313 (1916) 34. P. Lloyd, J. Phys. C2, 1717 (1969) 35. D.J. Thouless, J. Phys. C5, 77 (1972) 36. S.F. Edwards, P.W. Anderson, J. Phys. F5, 965 (1975) 37. B.V. Chirikov, F.M. Izrailev, D.L. Shepelyansky, Sov. Sci. Rev. 2C, 209 (1981) 38. D.L. Shepelyansky, Phys. Rev. Lett. 56, 677 (1986) 39. F. Haake, M. Ku´s, J. Mostowski, R. Scharf, in Coherence, Cooperation, and Fluctuations, ed. by F. Haake, L.M. Narducci, D.F. Walls (Cambridge University Press, Cambridge, 1986) 40. F. Haake, M. Ku´s, R. Scharf, Z. Phys. B65, 381 (1987) 41. M. Ku´s, R. Scharf, F. Haake, Z. Phys. B66, 129 (1987) 42. R. Scharf, B. Dietz, M. Ku´s, F. Haake, M.V. Berry, Europhys. Lett. 5, 383 (1988) 43. F.T. Arecchi, E. Courtens, G. Gilmore, H. Thomas, Phys. Rev. A6, 2211 (1972) 44. R. Glauber, F. Haake, Phys. Rev. A13, 357 (1976) 45. K. Zyczkowski, J. Phys. A23, 4427 (1990) 46. A. Peres, Quantum Theory: Concepts and Methods (Kluwer Academic, New York, 1995) 47. F. Haake, D.L. Shepelyansky, Europhys. Lett. 5, 671 (1988) 48. Y.V. Fyodorov, A.D. Mirlin, Int. J. Mod. Phys. B8, 3795 (1994) 49. G. Casati, L. Molinari, F. Izrailev, Phys. Rev. Lett. 64, 16 (1990); J. Phys. A24, 4755 (1991) 50. S.N. Evangelou, E.N. Economou: Phys. Lett. A151, 345 (1991) 51. M. Ku´s, M. Lewenstein, F. Haake, Phys. Rev. A44, 2800 (1991) 52. L. Bogachev, S. Molchanov, L.A. Pastur, Mat. Zametki 50, 31 (1991); S. Molchanov, L.A. Pastur, A. Khorunzhy, Theor. Math. Phys. 73, 1094 (1992) 53. M. Feingold, Europhys. Lett. 71, 97 (1992) 54. G. Casati, V. Girko, Random Oper. Stoch. Equ. 1, 1 (1993) 55. M. Kac, in Statistical Physics: Phase Transitions and Superfluidity, vol. 1 (Gordon and Breach, New York, 1991) 56. K.B. Efetov, A.I. Larkin, Sov. Phys. JETP 58, 444 (1983) 57. O.I. Marichev, Handbook on Integral Transforms of Higher Transcendental Functions (Ellis Horwood, New York, 1983) 58. A. Altland, M.R. Zirnbauer, Phys. Rev. Lett. 77 4536 (1996) 59. C. Tian, A. Altland, New J. Phys. 12, 043043 (2010) 60. C. Tian, A. Altland, M. Garst, Phys. Rev. Lett. 107, 074101 (2011) 61. A. Altland, S. Gnutzmann, F. Haake, T. Micklitz, Rep. Prog. Phys. 78, 086001 (2015) 62. D.L. Shepelyansky, Physica D28, 103 (1987) 63. F.M. Izrailev, Phys. Rep. 196, 299 (1990)

Chapter 9

Classical Hamiltonian Chaos

9.1 Preliminaries This chapter will present classical Hamiltonian mechanics to the extent needed for the semiclassical endeavors to follow. We can confine ourselves to the bare minimum since many excellent texts on classical chaos are available [1–5]. Readers with a good command of nonlinear dynamics might want to right away start with Sect. 9.14 where we begin expounding the fact that long periodic orbits of hyperbolic systems are not independent individuals but rather come in closely packed bunches. Different orbits in each bunch are topologically distinct but can be nearly indistinguishable in action, and for that reason have strong quantum signatures. The origin of orbit bunching can be seen in close self-encounters which a long orbit is bound to undergo. The orbit stretches involved in close self-encounters can be “switched” to form “partner orbits”. The classical bunching phenomenon was discovered in the course of semiclassical attempts to explain universal fluctuations in quantum energy spectra, the theme of the next chapter. It has turned out to be the key to a semiclassical understanding of universal features in quantum transport and of dynamical localization as well. Certain subtle issues related to extremal properties of the action, the so called focal points and Lagrangian manifolds could have been made part of the present chapter. They are, however, deferred to the semiclassical considerations of the next chapter since that context will provide a good motivation.

9.2 Phase Space, Hamilton’s Equations and All That At issue are dynamics with f degrees of freedom whose phase space is spanned by f coordinates q1 , . . . , qf and as many momenta p1 , . . . , pf . Configuration space is spanned by the f coordinates q. A Hamiltonian function H (q, p) generates the © Springer Nature Switzerland AG 2018 F. Haake et al., Quantum Signatures of Chaos, Springer Series in Synergetics, https://doi.org/10.1007/978-3-319-97580-1_9

365

366

9 Classical Hamiltonian Chaos

time evolution through Hamilton’s equations q˙ =

∂H , ∂p

p˙ = −

∂H ∂q

(9.2.1)

or, more compactly,1 X˙ = Σ∂X H = F (X),

 Σ=

0 1 −1 0

 .

(9.2.2)

The matrix Σ is antisymmetric and squares to minus the 2f × 2f unit matrix, Σ˜ = −Σ ,

Σ 2 = −1 .

(9.2.3)

When sets of phase space points are imagined transported à la Hamilton the picture of a flow emerges. That flow is free of sources, divF (X) = ∂? X Σ∂X H = 0 .

(9.2.4)

A solution of Hamilton’s equations yields the current point Xt as a function of the initial point X0 and may be written as Xt = Φt (X0 ) .

(9.2.5)

It will sometimes be convenient to represent the flow by a nonlinear operator Φt and write Xt = Φt X0 . Hamiltonian time evolution has an underlying group structure in the sense Φt +τ = Φt Φτ and Φ−t Φt = 1. For two phase-space functions f (q, p), g(q, p) one defines the Poisson bracket {f, g} =

∂g ∂f ∂f ∂g − ∂p ∂q ∂p ∂q

(9.2.6)

which might be written as {f, g} = (∂? X g)Σ∂X f . Coordinates and momenta are said to form canonical pairs in the sense {pi , qj } = δij ,

{pi , pj } = {qi , qj } = 0.

(9.2.7)

Any reparametrization of phase space, q, p → Q, P , for which the P , Q are again canonical pairs such that {Pi (q, p), Qj (q, p)} = δij , {Pi (q, p), Pj (q, p)} = {Qi (q, p), Qj (q, p)} = 0 is called a canonical transformation. Time evolution can be looked upon as a canonical transformation inasmuch as the basic brackets (9.2.7) hold for the time dependent coordinates and momenta at any time, {pi (t), qj (t)} = 1 Here

and below, it is best to read X =

q  p

, ∂X =

 ∂q  , and ∂? X = (∂q , ∂p ). ∂p

9.3 Action as a Generating Function

367

δij (while in general, of course, {pi (t), qj (t )} = δij for t = t ). Hamilton’s equations (9.2.1) are form invariant under canonical transformations. A phase space function f (q, p, t) with an explicit time dependence (beyond the implicit one through the time dependent coordinates and momenta) obeys the evolution equation df ∂f = {H, f } + , dt ∂t

(9.2.8)

of which Hamilton’s equations (9.2.1) are special cases. A further special case is dH /dt = ∂H /∂t; for autonomous dynamics the latter identity implies energy conservation. Unless explicitly noted otherwise we shall confine the subsequent discussion to that case. The non-integrable “chaotic” dynamics at issue here do not allow for solutions of Hamilton’s equations in terms of “quadratures” (i.e., explicit integrals). They have no or not enough independent constants of the motion. Chaotic behavior of autonomous dynamics requires at least two freedoms since energy is conserved.

9.3 Action as a Generating Function Any canonical transformation can be induced by a generating function. In particular, the transition along the classical trajectory from the initial coordinate q to the final coordinate q during the time span t is generated by a function S(q, q , t) which is the action, i.e. the time integral of the Lagrangian evaluated along the classical trajectory in question, S(q, q , t) =



t

dτ L(q(τ ), q(τ ˙ )) .

(9.3.1)

0

To show this and to get the final and initial momenta p, p we look at infinitesimal deflections in configuration space and the pertinent action S(q + δq, q + δq, t) =



t

dτ L(q(τ ) + δq(τ ), q(τ ˙ ) + δ q(τ ˙ ))

0





t

= S(q, q , t) +

dτ 0

(9.3.2)

 ∂L ∂L δq(τ ) + δ q(τ ˙ )) , ∂q ∂ q˙

higher order terms apart. Integrating by parts the second term in the integral and d ∂L ∂L realizing that (i) Lagrange’s equations dt ∂ q˙ − ∂q = 0 hold along the classical trajectory and (ii) the momentum is given by p = ∂L/∂ q˙ we get t  S(q + δq, q + δq , t) − S(q, q , t) = p(τ )δq(τ ) = pδq − p δq 0

(9.3.3)

368

9 Classical Hamiltonian Chaos

and thus the final and initial momenta as p=

∂S(q, q , t) , ∂q

p = −

∂S(q, q , t) . ∂q

(9.3.4)

The partial derivative ∂S/∂t is obtained when the action S(q, q , t+δt) is considered for the classical trajectory from q to q during the infinitesimally longer time span t +δt. That trajectory must run close to the original one and we may expand similarly as above, 



S(q, q , t + δt) =

t +δt

dτ L(q(τ ) + δq(τ ), q(τ ˙ ) + δ q(τ ˙ ))

(9.3.5)

0

t  = S(q, q , t) + δtL(q(t), q(t)) ˙ + p(τ )δq(τ ) . 0

But this time we have δq(0) = δq(t + δt) = 0 and thus δq(t) = −q(t)δt ˙ such that we can read off the time derivative of the action as ∂S(q, q , t) = L(q(t), q(t)) ˙ − pq˙ = −H (q, p) = −E . ∂t

(9.3.6)

Instead of fixing the time span t we may consider the transition q → q with prescribed energy E. That transition is generated by the Legendre-transformed action (to be called energy dependent action even though the standard name for the time dependent action is ‘Hamilton’s principal function’ and for the energy dependent variant ‘Hamilton’s characteristic function’) S0 (q, q , E) = S(q, q , t) + Et , p=

∂S0 (q, q , E) , ∂q

(9.3.7) p = −

∂S0 (q, q , E) . ∂q

In the foregoing definition of the energy dependent action S0 (q, q , E) the time argument of S(q, q , t) must be fixed according to ∂S0 (q, q , E) = t (q, q , E); ∂E

(9.3.8)

∂t ∂t the latter identity follows from (9.3.6) through ∂S ∂t ∂E = −E ∂E . The action of a periodic orbit with period T will repeatedly play a role in what follows. For autonomous flows, it is natural to consider the energy dependent action 0 S0 (q, q , E). Starting at an arbitrary point q , p = − ∂S ∂q on the orbit we sum up all differential increments dS0 (q, q , E) = pdq along the orbit, noting that q and E remain fixed and that p becomes a unique function of q. We then find

, S0 =

pdq .

(9.3.9)

9.4 Linearized Flow and Its Jacobian Matrix

369

9.4 Linearized Flow and Its Jacobian Matrix To study what happens near a given trajectory Xt = Φt (X0 ) it is appropriate to linearize the flow as δXt =

∂Φt (X0 ) δX0 = Mt (X0 )δX0 . ∂X0

(9.4.1)

An initial point deflected by δX0 from X0 ends up with a deflection δXt from Xt at time t. A deflection δX from X lives in a linear vector space called the tangent space at X. The 2f × 2f matrix Mt (X0 ) can be looked upon as the Jacobian matrix of the canonical transformation X0 → Xt . We may equivalently linearize the equations of motion δ X˙ =

 ∂F (X)  δX ∂X X=Φt X0

(9.4.2)

and are led to the Jacobian matrix as a time-ordered exponential 



t

Mt (X0 ) = exp 0

  ∂F  dτ . ∂X X=Φτ X0 )

(9.4.3)

+

The latter representation entails the group-type property Mt +τ (X) = Mt (Φτ X)Mτ (X)

(9.4.4)

as well as the evolution equation  ∂F  ˙ Mt (X0 ) . Mt (X0 ) = ∂X X=Φt X0

(9.4.5)

The Jacobian determinant of a Hamiltonian flow equals unity due to the absence of sources, see (9.2.4), as is clear from the following little calculation. The foregoing d Tr ln Mt and thus evolution equation (9.4.5) entails divF = TrM˙ t Mt−1 = dt #t Tr ln Mt = 0 dτ divF . An arbitrary flow therefore has det Mt = exp Tr ln Mt = #t exp 0 dτ divF while for a Hamiltonian one with divF = 0 we indeed infer det Mt (X) = 1 .

(9.4.6)

The determinant det Mt is the Jacobian of the canonical transformation X0 → Xt . Its constant value unity therefore means that the infinitesimal volume element is constant in time, d 2f X0 = d 2f Xt .

(9.4.7)

370

9 Classical Hamiltonian Chaos

In fact, the volume element is invariant under any canonical transformation, as can be shown with the help of the generating function S(q, q ); see Problem 9.1. It is worth noting that the linearized flow inherits Hamiltonian character from the original non-linear one.2 The transition δX0 → δXt may thus be seen as a canonical transformation, and that observation will be important in Sect. 9.7.

9.5 Liouville Picture The flow of sets of phase-space points mentioned above is worthy of further comments. Each point X fully specifies the state of the system under study. A set of points can be imagined to represent an ensemble of replicas of the system. A density ρ(X, t) in phase space reflects the dynamics of the ensemble as time evolves.3 The integral of ρ(X, t) over all of phase space remains constant in time since time evolution neither adds nor takes away any system from the ensemble,  d 2fX ρ(X) = const ;

(9.5.1)

if the constant value of the integral is chosen as unity ρ becomes a probability density, but other normalizations can be useful as well. The evolution equation of the density ρ is most easily found by momentarily assuming an ensemble of N replicas at the points Xμ (t), μ = 1 . . . N such that ρ(X, t) =

N 

δ(X − Xμ (t))

(9.5.2)

μ=1

becomes a number density of replicas in phase space. Taking the time derivative and using Hamilton’s equations one gets Liouville’s theorem, the evolution equation in search, ρ˙ + {H, ρ} = ρ˙ + ∂? X Fρ = 0 ,

(9.5.3)

which has the form of a continuity equation for the flow. Since the “velocity field" F (X) is devoid of sources (see (9.2.4)) the set of replicas behaves like an incompressible gas. Indeed, divF = 0 allows to write Liouville’s theorem in the form ρ˙ + F˜ ∂X ρ = 0 which reveals time independence of ρ in a locally co-moving frame of coordinates, X = X − F t, t = t. 2 The time dependent Hamiltonian for the linearized flow is obtained from the original H by expanding around the chosen trajectory and dropping terms of higher than second order in ΔX. 3 The notion of ensembles is of particular importance for the statistical treatment of many-particle systems.

9.6 Symplectic Structure

371

An important conclusion from Liouville’s theorem is a generalization of the constancy in time of the normalization integral (9.5.1) to any 2f dimensional subvolume V of phase space, irrespective of the shape,  V (t )

d 2fX ρ(X, t) = const ;

(9.5.4)

the constancy of the density ρ(X, t) in locally co-moving coordinates is at the basis of that invariance, together with the constancy of each differential volume element d 2f X, see (9.4.7). An interesting special case arises when the initial density ρ(X, 0) is chosen as the “characteristic function” of V(0) which vanishes outside and equals unity everywhere inside. Time evolution turns the density into the characteristic function of V(t) such that  d 2fX = const . (9.5.5) V (t )

The foregoing integral is known as one of Poincaré’s integral invariants. The reader is kindly# invited to slightly modify of the argument just given to show that the integral V d 2f X is in fact invariant under any canonical transformation.

9.6 Symplectic Structure Three infinitesimally close phase-space points X, X + δX, and X + δX define a parallelogram in the tangent space at X. The quantity ? δXΣδX = δqδp − δpδq

(9.6.1)

(which for f freedoms can be read as i [δqi δpi −δpi δqi ] ) is called symplectic area element.4 Alluding to area in that name is appropriate since in the ith phase plane the contribution δqi δpi − δpi δqi arises  which equals the area of the parallelogram spanned by the vectors

δqi δpi

and

δqi δpi

.

d The symplectic area element is conserved (Fig. 9.1). The column vector dt δX = ? ? F (X + δX) − F (X) = (δX · ∂X )F (X) = (δX · ∂X )Σ∂X H and its transpose are involved in the proof of that conservation law. The antisymmetry (9.2.3) of the “fundamental matrix Σ of the symplectic structure” yields the “row vector” d ? 2 ? ? dt δX = −(δX · ∂X )∂X ΣH . Due to Σ = −1 the announced conservation law then

4 The

symplectic area element can also be written as the antisymmetric wedge product δX ∧ δX .

372

9 Classical Hamiltonian Chaos p δp δq δp  δq q

Fig. 9.1 Conserved symplectic area element spanned by two vectors in tangent space to phase space

indeed results, d ? B · ∂X )∂X H = 0 . ? δX ? · ∂X )(∂? δXΣδX = (δX X H · δX ) − δX( dt

(9.6.2)

A (second, beyond (9.5.5)) Poincaré integral invariant arises when the symplectic area element is integrated over a surface S with a closed boundary γ . The resulting area   ? = (dqi dpi − dpi dqi ) (9.6.3) δXΣδX S

i

Si

receives an additive contribution from each phase plane wherein the projections Si and γi appear. Instead of the parallelogram-shaped symplectic area elements dqi dpi −dpi dqi , the rectangular elements dqi dpi may be collected. The symplectic area on S then takes the form   , ? δXΣδX = dqi dpi = dqi pi ; (9.6.4) S

i

Si

i

γi

the last member of the foregoing chain of equations is obtained through Stoke’s theorem, with the integrand pi (qi ) representing the closed curve γi . Now, when all points on the surface S and thus all points on the contour γ move with the Hamiltonian flow the integral remains invariant, due to the conservation of the differential symplectic area element. The integral invariant (9.6.4) is not only invariant under time evolution but in fact under general canonical transformations. It plays an important role in advanced presentations of analytical mechanics [6]. It will come up in the action difference for pairs of orbits differing in close encounters, (see Sect. 9.15.2).

9.7 Lyapunov Exponents

373

9.7 Lyapunov Exponents Initially close-by phase-space trajectories of chaotic dynamics in general separate exponentially as time elapses. In other words, the pertinent initial-value problem (i.v.p.) can be said to be exponentially unstable. That property is in fact often taken as the paradigm of chaos. The present section is devoted to a quantitative discussion. The Jacobian matrix Mt (X) allows to associate exponential growth rates alias Lyapunov exponents with 2f different directions e in the tangent space at every phase-space point X, λ(X, e) = lim

|t |→∞

1 ln ,Mt (X)e, . t

(9.7.1)

Inasmuch as the Hamiltonian flow leaves the generator dF (X)/dX of the linearized flow bounded the foregoing limit must exist, with λ < ∞. We label the 2f distinguished directions and the pertinent Lyaplounov exponents by an index as ei , λi (X) = λ(X, ei ); several directions may yield the same λ. To identify the direction e1 pertaining to the largest Lyapunov exponent λ1 one must vary a trial direction e until the rate λ1 is obtained both for large positive and negative times. By further variations of e another direction e2 must be found where either λ1 is again in effect for t → ±∞ (then λ1 would be degenerate) or the next largest exponent λ2 is. So proceeding stepwise one can establish all λ’s and the 2f distinguished directions. It is well to note that the different directions are in general not mutually orthogonal. Further insight into good strategies for finding the λ’s and e’s will arise below. Writing out the norm in the above definition (9.7.1) as ?t (X)Mt (X)e ,Mt (X)e,2 = e˜M

(9.7.2)

we see that the Lyapunov exponents may also be obtained through the eigenvalues ?t (X)Mt (X) as σi (t, X) of the non-negative real symmetric matrix M λi (X) = lim

t →∞

1 ln σi (t, X) . 2t

(9.7.3)

?t Mt must not be confused with the vector The mutually orthogonal eigenvectors of M fields ei (X) distinguished by the different Lyapunov exponents. In the direction e, (X) ∝ X˙ = Σ∂X H along a trajectory, the Lyapunov exponent vanishes. To see that, we consider a “reference” trajectory Xt with the initial point X0 and a “neighboring” one Xt +τ = Xt + X˙ t τ ≡ Xt + δXt which starts at Xτ = X0 + X˙ 0 τ ≡ X0 + δX0 and otherwise retraces the reference trajectory. According to the definition (9.4.1) of the linearized flow we have τ X˙ t = Mt (X0 )τ X˙ 0 . Recognizing that X˙ 0 points in the direction e, (X0 ) we conclude

374

9 Classical Hamiltonian Chaos

X˙ t /||X˙ 0 || = Mt (X0 )e, (X0 ) and therefore5 1 ,X˙ t , 1 ,F (Xt ), ln = 0. = lim ln t →∞ t ,F (X0 ), ,X˙ 0 , t →∞ t

λ(X0 , e, (X0 )) = lim

(9.7.4)

A second vanishing Lyapunov exponent arises in the direction perpendicular to the energy shell, eE ∝ ∂X H . Continuous symmetries and the pertinent conservation laws would entail further vanishing λ’s. In analogy with the conserved energy two neutral directions come with a conserved quantity C, and these can be shown to be ∂X C and Σ∂X C [2]. In particular, for integrable dynamics where trajectories wind around f dimensional tori such that initially close points stay close forever, no exponential divergence is possible and all λ’s vanish. For chaotic Hamiltonian flows the Lyapunov exponents arise in pairs ±λ. This important fact follows from the Hamiltonian character of the linearized flow (9.4.1). Consequently, the time evolution is again a canonical transformation. Parametrizing the deflections δXt by canonical pairs we have Poisson brackets {δpi (t), δqj (t)} = δij , {δpi (t), δpj (t)} = 0, {δqi (t), δqj (t)} = 0, valid at all times. The latter equations can be written compactly as the matrix identity ˜ MΣM =Σ

or

MΣ M˜ = Σ ;

(9.7.5)

˜ ˜ an iterate reads MMΣ MM = Σ and implies that an eigenvalue σi is accompanied with its inverse 1/σi as another one, with eigenvectors ui and Σui . The associated Lyapunov exponents therefore are ±λi . It follows that the number of vanishing

2f Lyapunov exponents is even. We further conclude i=1 λi = 0, and that property of Hamiltonian flows may be seen as a manifestation of Liouville’s theorem (preservation of phase-space volumes, see Sect. 9.2). To find further properties of the Lyapunov exponents we must dig a little deeper into nonlinear dynamics.

9.8 Stretching Factors and Local Stretching Rates By its definition (9.4.1) the Jacobian Mt (X0 ) transports a deflection δX0 in the tangent space at X0 to the deflection δXt in the tangent space at Xt = Φt X0 . Writing ei (X0 ) for the initial deflection and Λi ei (Xt ) for the one at time t we may note Mt (X0 )ei (X0 ) = Λi (t, X0 )ei (Xt ) ;

(9.8.1)

5 For the Hamiltonian dynamics under exclusive consideration here, no trajectory can end in, start out from, or pass through a stationary point such that neither F (X0 ) = 0 nor F (Xt ) = 0 make for worry; isolated moments of time with F (Xt ) = ∞ arise for elastic collisions of the hard-wall type but do not invalidate the conclusion.

9.8 Stretching Factors and Local Stretching Rates

375

if we were to normalize as ,ei (X0 ), = ,ei (Xt ), = 1 the “stretching factor” Λi (t, X0 ) would reflect all the stretching or shrinking incurred under the linearized flow. It is actually convenient to avoid the condition of unit length and stipulate the weaker condition that ,ei (Xt ), does not grow exponentially in time, 1 ln ,ei (Xt ), = 0 . t →∞ t lim

(9.8.2)

The stretching factor Λi (t, X0 ) then picks up all exponential growth associated with the direction ei , if there is any. We may choose 2f different directions ei (X0 ), i = 1 . . . 2f , in general not orthogonal, to get the Lyapunov exponents as λi (X) = λ(X, ei ) = lim

t →∞

1 ln ,Λi (t, X), . t

(9.8.3)

A second set of 2f vectors fi (X0 ) then exists such that the e’s and f ’s form biorthogonal pairs at each phase-space point X f˜i (X)ej (X) = δij

2f 

ei (X)f˜i (X) = 1 .

(9.8.4)

i=1

A “cocycle decomposition” of Mt (X) can now be written as Mt (X) =

2f 

ei (Φt X)Λi (t, X)f˜i (X) .

(9.8.5)

i=1

The group-type property (9.4.4) of the Jacobian entails one for the stretching factors, Λi (t + τ, X) = Λi (t, Φτ X)Λi (τ, X) .

(9.8.6)

An important conclusion from the foregoing group-type property and the relation (9.8.3) between the Lyapunov exponents and the stretching factors is λi (X) = λi (Φτ X), i.e. the constancy of all Lyapunov exponents along each trajectory. Different trajectories may have different λ’s, though. To reveal the final property of the Lyapunov exponents of relevance here we differentiate the definition (9.8.1) of the stretching factor w.r.t. time to get the evolution equation Λ˙ i (t, X) = χi (Φt X)Λi (t, X)

(9.8.7)

with the “local stretching rate”  ∂F (X) ∂ei (X)  χi (X) = f˜i (X) ei (X) − F (X) . ∂X ∂X

(9.8.8)

376

9 Classical Hamiltonian Chaos

In contrast to the Lyapunov exponents the local stretching rate in general varies along a trajectory. Upon formally # t integrating the evolution equation of the stretching factor we have Λi (t, X) = exp 0 dτ χi (Φτ X) and thus find the Lyapunov exponent as a time average of the local stretching rate along a trajectory, 1 t →∞ t



λi (X) = lim

t

dτ χi (Φτ X) .

(9.8.9)

0

For ergodic dynamics time averages equal ensemble averages (over the energy shell), and we can conclude that in that case all endless trajectories have the same Lyapunov exponents. Periodic orbits may still retain their individual λ’s; for growing periods, however, these λ’s can be expected to approach those of infinite trajectories. We shall come back to the effective ergodicity of long periodic orbits in Sect. 9.12 below. A final comment is in order on the 2f vector fields ei (X) giving rise to the various Lyapunov exponents. The ei (X) can be grouped into “stable directions” (λi < 0), “unstable directions” (λi > 0) and “neutral directions” (λi = 0). The three groups span the stable, unstable, and neutral subspaces of the tangent space at each phase-space point X.

9.9 Poincaré Map Laying the ground for modern nonlinear dynamics Henri Poincaré introduced a description of Hamiltonian flows in terms of a discrete map in a subspace of phase space which we propose to first consider for autonomous flows on the (2f − 1) dimensional energy shell H = E. A second (2f − 1) dimensional hypersurface intersecting the energy shell in a 2f − 2 dimensional manifold is to be chosen such that it is pierced by all trajectories of interest. That latter “Poincaré surface of section” P may be parametrized by (f −1) canonical pairs {qi , pi ; i = 1, . . . , f −1} which we summarily denote by x (Fig. 9.2). For example, for f = 2 and a Hamiltonian of the form H = p12 /2m + 2 p2 /2m + V (q1 , q2 ) the energy shell may be spanned by q1 , p1 , q2 with the missing $ momentum given by p2 = ± 2m(E − V (q1 , q2 )) − p12 . Fixing, say, the plane q2 = 0 to accompany the surface H = E we have the two dimensional Poincaré section pair q1 , p1 . For f > 2 we may similarly set qf = 0, pf = $ spanned by the f −1 ± 2m(E − V (q)) − i=1 pi2 . It is convenient to place P transversal to the flow such that it does not contain the direction e, along the flow. If there are no constants of the motion other than the energy, P then contains no neutral directions at all. In case there are further constants of the motion it is convenient to also exclude the pertinent pairs of neutral directions from P [2].

9.9 Poincaré Map

377

Fig. 9.2 Poincaré section in three-dimensional energy shell, pierced by periodic orbit

An endless trajectory pierces through the Poincaré section over and over again, each time in a different point x. For periodic orbits, of course, only a finite number of distinct piercing points arise. Of exclusive interest for us are piercings with the same sign of pf , say the positive one, since with that restriction a point x in P is one-to-one with a phase-space point and thus defines a phase-space trajectory. The n-th such piercing of a trajectory through P, denoted by xn , uniquely fixes the subsequent one, xn+1 , as well as the “first-return time” T (xn ). The points and times of piercings are related by the Poincaré map xn+1 = φ(xn ) ,

tn+1 = tn + T (xn ) .

(9.9.1)

The Jacobian matrix of the Poincaré map, m(x) =

∂φ(x) ∂x

(9.9.2)

is 2(f − 1) × 2(f − 1) if the energy is the only conserved quantity. Inasmuch as P is parametrized by canonical pairs of variables the Poincaré map amounts to a canonical transformation and therefore is “area” preserving, det m = 1 .

(9.9.3)

The linearized Poincaré map, δxn+1 = m(xn )δxn , allows to determine 2(f − 1) Lyapunov exponents of endless trajectories through λi (x0 ) = lim

1

n→∞ tn

ln ,m(xn )ei , .

(9.9.4)

378

9 Classical Hamiltonian Chaos

a

b

p2

q1 p1

Fig. 9.3 Piercings of regular (a) and chaotic (b) orbit through Poincaré section

For periodic orbits the Jacobian m is customarily called monodromy matrix and provides a convenient starting point for getting the Lyapunov exponents. If x ∈ P is a point of an orbit γ with the primitive period Tγ the eigenvalues Λi of m(x) yield the λ’s as λi = (1/Tγ ) ln Λi , without any limiting procedure. It may be well to summarize at this point the important properties of the eigenvalues Λi of the monodromy matrix m: (1) They are real or come in pairs of mutual complex conjugates (since m is a real matrix.) (2) They come in pairs of mutual inverses. (3) Inasmuch as the Poincaré section contains no neutral direction unity does not qualify as an eigenvalue. (4) Complex eigenvalues are unimodular and refer to stable subspaces; for f = 2 the underlying orbit is stable, and such behavior is forbidden for hyperbolic dynamics. The Poincaré surface of section allows to numerically test for integrability vs chaos in the case f = 2 where P is two dimensional. A constant of the motion independent of the energy would intersect P in a one dimensional curve. A sequence of piercings xn would “fill” that curve. In the non-integrable case where no such constant of the motion exists the xn will not be confined to a curve but rather explore a whole area in P. In generic systems one meets with islands of regular motion surrounded by a chaotic “sea” and in such cases P can accommodate sequences xn of both types, curve filling and area exploring (Fig. 9.3).

9.10 Stroboscopic Maps of Periodically Driven Systems Periodic driving makes for explicitly time dependent Hamiltonians, H (t) = H (t + T ) ,

(9.10.1)

9.11 Varieties of Chaos; Mixing and Effective Equilibration

379

where T is the period of the driving. A stroboscopic description with the strobe period T is then indicated and amounts to looking at the continuous flow Xt = Φt (X0 ) ≡ Φt X0 only at the discrete times t = nT , n = 0, 1, 2, . . .. The map XnT = ΦT X(n−1)T = ΦnT X0

(9.10.2)

still defines a group since Φ(n+m)T = ΦnT ΦmT and ΦnT Φ−nT = 1 and again is a canonical transformation. The Jacobian matrix MnT of the strobe map defines a linearized discrete map, δXnT = MnT (X0 )δX0 ,

MnT (X0 ) =

∂ΦnT (X0 ) , ∂X0

(9.10.3)

and again satisfies the group-type relation M(m+n)T (X0 ) = MnT (XmT )MmT (X0 ) .

(9.10.4)

as well as area preservation, det MnT = 1. Everything said above about Lyapunov exponents can be transcribed. Important differences lies in the absence of energy conservation and of the two neutral directions eE , e, . As a consequence, periodic driving enables single-freedom systems to behave chaotically. Should there be other constants of the motion the stroboscopic map can be reduced to the corresponding subspace of the phase space, as for flows. A popular

special case is periodic kicking involving Dirac deltas as H (t) = H0 + H1 +∞ n=−∞ δ(t − nT ). Examples are the kicked rotator and the kicked top. When dealing with maps below we shall suppress the strobe period T as XnT → Xn , ΦnT → Φn , MnT → Mn ; in brief, the dimensionless integer time n will be employed as well as dimensionless Lyapunov exponents, λT → λ. We should mention that there are area preserving maps which cannot be understood as stroboscopic descriptions of Hamiltonian flows. The baker map and the cat map are well known examples, see, e.g. Ref. [3]. Their Jacobian matrices still have the properties just mentioned for stroboscopic maps.

9.11 Varieties of Chaos; Mixing and Effective Equilibration Generic systems have mixed phase spaces where islands of regular motion are surrounded by “chaotic seas”. Upon changing a suitable control parameter periodic orbits “are born” or “die out” at bifurcations. For ergodic dynamics time averages along endless trajectories equal ensemble averages with suitable invariant measures. Phase-space densities of mixing dynamics effectively equilibrate, on the energy shell or, if more conserved quantities exist, the correspondingly reduced subman-

380

9 Classical Hamiltonian Chaos

ifold: thereon, cloud of points spreads out uniformly. Correlation functions of observables tend to factorize as lim|t −t |→∞ A(t)B(t ) → AB with stationary means A, B. Mixing implies ergodicity but not vice versa. We are mostly concerned with hyperbolic systems which have non-zero Lyapunov exponents everywhere or at least almost everywhere. Ergodicity and mixing are then granted (unless the phase space falls into disjoint regions). The “effective equilibration” for mixing dynamics deserves some more remarks, especially since one might suspect a contradiction between equilibration and Hamiltonian dynamics. For convenience, we restrict the discussion to autonomous hyperbolic systems with two freedoms. The energy shell is three dimensional and we assume the latter compact (volume V ). Each phase-space point has two neutral directions attached, one along the flow and the other transverse to the energy shell; two more directions e± are associated with a pair of Lyapounov exponents ±λ. Now imagine a “ball” extending circularly in the local plane spanned by the directions e± and similarly in the direction of the flow (volume v). As the dynamics starts to move the set of points of the ball the latter will be stretched (squeezed) in the unstable (stable) direction. The assumed compactness sooner or later forces the deformed ball to curl and fold but the total volume covered will remain constant, due to Liouville’s theorem. The lengthening, thinning, curling, and folding never ends. The “snake” thus arising will creep everywhere in the energy shell. The original phase space density always jumps back and forth between two values, zero (outside of) and the initial density inside the “snake” and forms an ever more fissured landscape. For an observer equipped to resolve phase-space volumina no smaller than w the phase-space density will begin to look homogeneous with the average value v/V at times of the order λ1 ln(w/v).6 In other words, the effective equilibration leads any observer to eventually seeing the microcanonical ensemble.

9.12 The Sum Rule of Hannay and Ozorio de Almeida Ergodicity implies that almost all endless trajectories visit everywhere in the accessible part of phase space. For autonomous flows with only the energy conserved the energy surface is then covered uniformly by a typical trajectory as 1 T →∞ T



T

dt δ(X − Φt X0 ) =

lim

0

1 δ(H (X) − H (X0 )) Ω

where Ω denotes the volume of the energy shell, Ω =

#

(9.12.1)

d 2f X δ(H (X) − H (X0 )).

6 Quantum mechanics of course does not allow for classical nonsense like infinitely fine phasespace structures. Quantum effects smoothen out the classical infinitely fissured landscape just mentioned [7]. In other words, “quantum diffusion” reinforces “Hamiltonian equilibration”.

9.12 The Sum Rule of Hannay and Ozorio de Almeida

381

For stroboscopic (or other area preserving) maps without any conserved quantities ergodicity means similarly N 1  1 δ(X − Φn X0 ) = N→∞ N Ω

lim

(9.12.2)

n=0

but now Ω is the full phase space volume which is assumed finite. As nonadmissible appear initial points X0 on periodic orbits since a finite-period orbit can certainly not visit everywhere. However, the set of all periodic orbits with periods in a window [T , T + ΔT ] might be expected to cover the available space uniformly as T → ∞. The sum rule of Hannay and Ozorio de Almeida [8, 9] substantiates that expectation in a most useful way, by identifying the appropriate weighting of each orbit. The following derivation of the HOdA sum rule assumes isolated periodic orbits. For pertinent mathematical work see Refs. [10, 11].

9.12.1 Maps Inasmuch as the ergodic property (9.12.2) holds for almost all initial points (excluded should remain short periodic orbits) and for all points X in phase space we may take the freedom to set X = X0 and to let the sum over n start at some value N0 overwhelmingly larger than all characteristic times of the dynamics at hand. Indeed then, the delta function δ(X0 −Φn X0 ) peaks at the periodic points on period-n orbits or on orbits whose primitive periods n/r fit an integer number r of times in n. With the promise to presently deliver justification we discard such repeated shorter orbits and write δ(X0 − Φn X0 ) =

(period n) n   p

i=1

1 p δ(X0 − Xi ) ; | det(Mp − 1)|

(9.12.3)

p

n here Mp = ∂X ∂X0 is the 2f × 2f Jacobian matrix of the period-n orbit p, customarily p called monodromy matrix, which is the same on all n points Xi on that orbit. On inserting the foregoing identity in the ergodicity property (9.12.2) and integrating over phase space we get

N 1  lim N→∞ N

n=N0



(period n) p

n = 1. | det(Mp − 1)|

(9.12.4)

Apart from fluctuations the quantity p | det(Mnp −1)| must itself approach unity as the period n grows. Doing away with such fluctuations by an average over a

382

9 Classical Hamiltonian Chaos

sufficiently large window Δn of periods with 1 3(period n)  p

n |Mp − 1|

5 =

1 Δn

Δn

Δn

(periods ∈ [n,n+Δn])

 p

n we get n ∼1 | det(Mp − 1)|

and, by finally dividing out n, the sum rule of Hannay and Ozorio de Almeida, 1 Δn

(periods ∈ [n,n+Δn])

 p

1 1 ∼ . | det(Mp − 1)| n

(9.12.5)

An immediate consequence of the HOdA sum rule is exponential proliferation of periodic orbits with growing period. Indeed, for large periods n the monodromy matrix is dominated by an exponential as | det(Mp − 1)| ∼ enλ where λ stands for the sum of all positive Lyapunov exponents. The sum rule thus implies #{period-n orbits} ∼

enλ . n

(9.12.6)

In fact, that exponential proliferation yields the selfconsistent justification of dropping r-fold repetitions of orbits of primitive period n/r with r = 2, 3, . . . from the above sums: these shorter orbits are exponentially outnumbered by the orbits with primitive period n while the determinant det(Mp − 1) keeps growing like eλn for all period-n orbits, primitive or not.

9.12.2 Flows For Hamiltonian flows, the HOdA sum rule looks rather the same as for maps, 1 ΔT

(periods ∈ [T ,T +ΔT ])

 p

1 1 = | det(mp − 1)| T

for ΔT

T,

(9.12.7)

but the monodromy matrix mp of the orbit p is now 2(f − 1) × 2(f − 1) and refers to a Poincaré map on the energy shell, see (9.9.1), (9.9.2). The derivation is a little trickier but proceeds in the same spirit as for maps. It is well to realize that the phase-space point Φt X0 never leaves the energy shell in which the initial point X0 lies. The ergodicity property (9.12.1) can thus be reformulated for the on-shell flow. To that end we choose the energy E as one phase-space coordinate and write X = E, x with suitable 2f − 1 coordinates x on the energy shell. With the on-shell flow denoted by xt = ΦtE x0 , ergodicity according

9.12 The Sum Rule of Hannay and Ozorio de Almeida

383

to (9.12.1) implies 1 lim T →∞ T



T 0

dt δ(x − ΦtE x0 ) =

1 , Ω

(9.12.8)

where E = H (X0 ) is understood. Now proceeding as for maps we set x = x0 and integrate over the energy shell. On the r.h.s. of (9.12.8) that integral yields

unity while on the l.h.s. a sum of contributions of periodic orbits arises, 1 = p Ip . To pick up the contribution Ip of an orbit p the energy-shell integral may be confined to a tube τp enclosing p; due to the delta function in the integrand the tube may be chosen so thin that no other orbit can squeeze in. Each tube can be parametrized by 2(f − 1) transverse coordinates x⊥ (with x⊥ = 0 on the orbit) and a longitudinal one along the flow. The latter coordinate must be a time-like one7 and will be called t, ; it runs from zero up to the period Tp . Within the tube τp the flow ΦtE shifts the longitudinal coordinate t, to (t, + t)modTp while the action on x⊥ can be approximated linearly,   ∂x⊥t  ΦtE (t, , x⊥ ) = (t, + t)modTp , x⊥ .  ∂x⊥ x⊥ =0

(9.12.9)

The 2f − 1 dimensional delta function in the integral over the tube τp thus includes a Tp -periodic temporal delta function, δ(x − ΦtE x) =

∞ 

δ(t, − t, − t + rTp )δ(x⊥ −

r=1

=

∞ 

∂x⊥t  x⊥ )  ∂x⊥ x⊥ =0

δ(t − rTp ) δ(x⊥ − mrp x⊥ ) ;

(9.12.10)

r=1



here, the Jacobian matrix

∂x⊥t  ∂x⊥ x =0 ⊥

is evaluated at successive completions of the

orbit p where it becomes a power of the monodromy matrix mp . As for maps we discard repetitions and keep δ(t − Tp ). The tube integral now results as 

Tp

Ip = δ(t − Tp ) 0

dt,



d 2f −2 x⊥ δ(x⊥ − mp x⊥ ) =

Tp δ(t − Tp ) . | det(mp − 1)|

(9.12.11)

The time average demanded on the l.h.s. of (9.12.1) remains to be done. By there setting the lower limit of the time integral to some value T0 much in excess of all

7 Imagine phase

space parametrized by a canonical pair p, , q, for momentum and coordinate along ∂E the flow, together with f −1 further pairs making up x⊥ ; then replace p, by E and realize ∂p = q˙, , to conclude dp, dq, = dEdt, ; here dt, equals the time differential dt, but it is well to distinguish the phase-space coordinate confined as 0 ≤ t, < Tp from the time t which never ends.

384

9 Classical Hamiltonian Chaos

characteristic times of the dynamics we again exclude short periodic orbits, 1 lim T →∞ T



T

dt T0

 Tp δ(t − Tp ) = 1. | det(mp − 1)| p

(9.12.12)

As for maps we now argue that the integrand must approach unity for growing time t, fluctuations apart. To get rid of those latter it suffices to do a time average over some finite interval ΔT , =

> Tp δ(t − Tp ) 1 = | det(mp − 1)| ΔT

(T 0. They may be imagined ordered as 0 < Re γ1 < Re γ2 < . . .. The leading one determines the fraction of the volume of a subspace σ of the energy shell that has not escaped by the time t as  d

2f −1

xd

2f −1

 x0 δ(x

σ

− ΦtE x0 )

−1 dx

∼ e−γ1 t .

(9.13.4)

σ

We follow Cvitanovi´c and Eckhardt [5, 13] to express the trace in terms of periodic orbits. To pick up the contribution of the (r-fold traversal of the) pth primitive orbit, we do as in the previous section and restrict the integration range to a narrow toroidal tube around that orbit. The tube integral has already been calculated in the previous section and now yields the trace 

d 2f −1 x P (x, t|x) =



Tp

p

i = 2π

∞  r=1



δ(t − rTp ) | det(Mpr − 1)|

∞

dk e

ikt

−∞

(9.13.5)

∞ ∂  e−ikrTp ; ∂k p r| det(Mpr − 1)| r=1

here we have taken the freedom to rename the monodromy matrix as mp → Mp , following common practice. The shorthand 8 Z(s) = exp −

∞  p r=1

esrTp r| det(Mpr − 1)|

9 (9.13.6)

offers itself now. It is known as the dynamical zeta function of the flow and allows us to write the trace under study as 

d 2f −1 x P (x, t|x) =

i 2π



+i∞ −i∞

ds e−st

Z (s)  = mi e−γi t . Z(s)

(9.13.7)

i

8 There is no contradiction between the propagator being a Dirac delta at all times and equilibration for t → ∞; note that the definition of Dirac deltas requires protection by integrals against smooth phase-space densities. See also the reasoning in Sect. 9.11.

386

9 Classical Hamiltonian Chaos

But the foregoing identity means that the resonances γi can be determined from the zeros of Z(s), just as if the latter were something like a secular determinant for the classical propagator; this is why Z(s) is called a zeta function. To facilitate comparison with the quantum zeta function to be met in Sect. 10.5 we reexpress the classical zeta function as a product over periodic orbits. For maximal convenience we do that for f = 2. Readers interested in arbitrary f will find pleasure in generalizing themselves or may consult [5, 14]. We must recall that for Hamiltonian flows of two-freedom systems, the stability matrix has two eigenvalues that are mutual inverses. For hyperbolic dynamics with no stable orbits, these eigenvalues are real and may be denoted as Λp and 1/Λp , where |Λp | > 1. The inverse-determinant weight of the pth orbit may be expanded as −1 −2 = |Λp |−r (1 − Λ−r | det(Mpr − 1)|−1 = |(1 − Λrp )(1 − Λ−r p )| p )

=

∞ ∞  

−(j +k)r

|Λp |−r Λp

.

(9.13.8)

j =0 k=0

Once this expansion is inserted in the zeta function (9.13.6), the sum over repetitions of the pth orbit yields a logarithm,   ∞ ∞   esTp ln Z(s) = ln 1 − . (9.13.9) j +k |Λp |Λp p j =0 k=0 Choosing l ≡ k + j and k as summation variables we can do the sum over k as well, and upon reexponentiating get the desired product form of the zeta function  l+1 ∞ -esTp 1− . (9.13.10) Z(s) = |Λp |Λlp p l=0

Now, it is easy to see that the zeta function has a simple zero at s = 0. Since the small-s behavior must be tied up with long orbits, we may, looking at ln Z(s) as sTp |Λ |−1 Λ−l and then given by (9.13.10), expand ln(1 − esTp |Λp |−1 Λ−l p p ) ≈ −e p keep only the leading term of the l-sum to get  ∞  esTp esT s→0 ≈− ≈ ln(sT0 ) ; ln Z(s) −→ − dT (9.13.11) |Λp | T T0 p here we have invoked the HOdA sum rule in the form (9.12.15) to recast the sum over long orbits into an integral, with T0 some reference time. We may imagine that T0 is the period upward of which the HOdA sum rule begins to apply and thus T0 TH ; the precise value does not matter. At any rate, the claim Z(s) ∝ s for s → 0 is borne out. Of course, the simple zero of Z(s) at s = 0 reflects the simple unit eigenvalue of the propagator corresponding to the stationary eigenfunction.

9.14 Exponential Stability of the Boundary Value Problem

387

In Sect. 10.5 we shall discuss a quantum zeta function related to the spectral determinant of a quantum Hamiltonian. The classical Z(s) as the exponentiated periodic-orbit sum (9.13.6) will find the quantum counterpart (see (10.5.10)) ∞   e ir Sp (s)/h¯ $ , ζ (s) ∝ exp − r p r=1 r | det(Mp − 1)|

(9.13.12)

where Sp is the action of the pth orbit with the so called Maslov phase included. Even the infinite-product representation (9.13.10) will have a quantum analogue. Both the classical and quantum zeta functions relate the spectrum of an evolution operator to periodic orbits. The differences between the two functions reflect the fact that probabilities are added in classical mechanics while in quantum mechanics probability amplitudes are superimposed, hence the square root of the inverse determinant and the action dependent phase factor in the quantum zeta function. It is also worth noting that the periodic-orbit form of the classical zeta is exact while the quantum zeta in general is not. Special cases (like geodesic flows on surfaces of constant negative curvature [15]) apart, the quantum zeta is a semiclassical approximation with corrections of higher orders in Planck’s constant.

9.14 Exponential Stability of the Boundary Value Problem The exponential instability of the initial-value problem of hyperbolic dynamics dwelled on above may be considered the paradigm of chaos. We turn to a consequence which on first sight might look like a contradiction in terms. A boundary-value problem à la Hamilton (b.v.p.) is defined by specifying initial and final positions q0 and qt (but no momentum) and asking for the connecting trajectory piece during a prescribed time span t. No solution need exist, and if one exists it need not be the only one. However, hyperbolicity forces a solution to be locally unique, unless q0 and qt happen to be conjugate points.9 Of foremost interest are time spans long compared to the inverse of the smallest positive Lyapunov exponent, for short t  1/λ (The present discussion need not be burdened by distinguishing between Lyapunov exponents and local stretching rates). As will be shown in a moment, slightly shifted boundary points yield a new trajectory piece, with the help of the linearized flow (9.4.1); the new trajectory piece approaches the original one within intervals of duration ∼ 1/λ in the beginning and at the end, like ; towards the “inside” the distance between the perturbed and the original trajectory decays exponentially. That fact is most easily comprehended

9 A focal or conjugate point in configuration space allows for a family of trajectories to fan out, each with a different momentum, which all reunite in another such point; we shall have to deal with conjugate points in the next chapter, see Sect. 10.2.2.

388

9 Classical Hamiltonian Chaos

Fig. 9.4 Search for a periodic orbit near a closed loop

by arguing in reverse: only an exponentially small transverse shift of position and momentum at some point “deep inside” the original trajectory piece can, if taken as initial data, result in but slightly shifted end points. Indeed, then, we may speak of exponential stability of the boundary-value problem as a consequence of the exponential instability of the initial-value problem. A wealth of insights is opened by that stability of the b.v.p. For instance, when beginning and end points for the b.v.p. are merged, qt = q0 with fixed t, each solution in general has a cusp (i.e. different initial and final momenta) there. If the cusp angle is close to π one finds, by a small shift of the common beginning/end, a close-by periodic orbit smoothing out the cusp and otherwise hardly distinguishable from the cusped loop, as shown schematically in Fig. 9.4. Similarly, any piece of an infinite trajectory is closely approached by periodic orbits. To find one such orbit, one may proceed in three steps. First, a b.v.p. with any two points close to the ends of the trajectory piece and with equal duration yields a nearby trajectory, as shown above. Next, the two new boundary points may be varied until a large-angle selfcrossing of the resulting trajectory is found somewhere. Finally, the cusp can be smoothed away as explained above. More than a century ago, Henri Poincaré came very close to such ideas when writing about periodic orbits (see the pertinent remark in the Introduction and [16]). The exponential stability of the boundary-value problem has become the clue to a world of phenomena related to close self-encounters of trajectories and long periodic orbits. As we shall see in the pages to follow, long periodic orbits are not mutually independent individuals but rather come in closely packed bunches.

9.15 Sieber-Richter Self-Encounter and Partner Orbit 9.15.1 Non-technical Discussion Figure 9.5 depicts a long periodic orbit in the so called cardioid billiard. That orbit bounces against the boundary in some dozens of distinct points. It appears to behave ergodically, i.e. to fill the area of the billiard densely and uniformly. Moreover, the

9.15 Sieber-Richter Self-Encounter and Partner Orbit

389

Fig. 9.5 Cardioid billiard with near ergodic periodic orbit

Fig. 9.6 Cartoon of Sieber-Richter pair of orbits. One orbit has small-angle crossing which the other narrowly avoids. Difference between orbits grossly exaggerated. Time reversal invariance required

depicted orbit crosses itself many times, in the two dimensional configuration space. The smaller the crossing angle the longer the two crossing orbit stretches remain close; if the closeness of those two stretches persists through many bounces we speak of a narrow 2-encounter, the “2” standing for two orbit stretches mutually close in configuration space. A periodic orbit with a small-angle self-crossing (a narrow 2-encounter with two encounter stretches connected by two “links”) is sketched in Fig. 9.6. In drawing such a configuration-space cartoon some artist’s licence is taken, inasmuch as all other self-crossings and all bounces against the boundary are dispensed with. The cartoon first appeared in a paper by Aleiner and Larkin [17] on weak localization in disordered electronic systems. It was fully exploited for chaotic

390

9 Classical Hamiltonian Chaos

dynamics by Sieber and Richter [18] who proved the following facts for the Hadamard-Gutzwiller model (geodesic flow on a compact surface of constant negative curvature of genus 2, a hyperbolic system with time reversal invariance): (1) The equations of motion allowing for the self-crossing orbit also allow for a partner orbit which has the crossing replaced by a narrowly avoided crossing, as shown by the dashed line in Fig. 9.6. Away from the crossing or avoided crossing, the partner orbit is exponentially close to the original one; (2) the action difference for the two orbits vanishes in proportion to the squared crossing angle, ΔS ∝  2 ; (3) In the limit of large periods T , the number of partners of an orbit related to small-angle crossings/avoided crossings has a principal term easily accessible through ergodicity (∝ T 2 sin ), with a relative correction (λT )−1 ln  2 ; (4) the orbit pairs in question yield the key to the semiclassical explanation of universal spectral fluctuations, and therefore the phenomenon of partner formation in close self-encounters deserves thorough discussion here. The existence of the partner avoiding the crossing follows for general hyperbolic dynamics from the exponential stability of the boundary value problem mentioned above: We may slightly shift apart beginning and end for each loop while retaining two junctions, as → , with nearly no change for those loops away from the junctions; by tuning the shifts we can smooth out the cusps in the junctions, → , and thus arrive at the partner orbit with an avoided crossing and reversed sense of traversal of one loop. The existence of the partner orbit may also be seen as a consequence of the shadowing theorem [19]. The Sieber-Richter pair of orbits of Fig. 9.6 exists only for time reversal invariant dynamics: The arrows indicate opposite sense of traversal of the left loop which is practically identical otherwise for the two orbits, away from the self-encounter. For a very close 2-encounter, neither orbit in the Sieber-Richter pair can exist without time reversal invariance, because each is required to have two long encounter stretches running mutually close, but with opposite senses of traversal; one may then say that the two encounter stretches are nearly “antiparallel”; in phase space, they are “nearly identical up to time reversal”. A long orbit can (and does) also display “parallel” 2-encounters and these again give rise to partner orbits, even for dynamics without time reversal invariance. Figure 9.7 depicts the interesting variant of partner formation then arising. The orbit with two crossing stretches (full line in the figure) may be said to consist of two loops, each of which has a cusp angle close to π. The partner (dashed in the figure) therefore decomposes into two shorter periodic orbits each of which smoothes out the cusp of the close-by loop of the original orbit. Such composite orbits (which together closely “shadow” a periodic orbit) are called pseudo-orbits in the physics literature. The existence of the pseudo-orbit as a partner of the self-crossing orbit in Fig. 9.7 follows from the above discussion of the exponential stability of the boundary-value problem. Intuitive as the phenomenon of partner formation in close self-encounters may now appear, the way towards that insight has been long and difficult. Action correlations between orbits have long been hunted for, mostly in the quest for a semiclassical explanation of universal spectral fluctuations in quantum chaos

9.15 Sieber-Richter Self-Encounter and Partner Orbit

391

Fig. 9.7 Cartoon of simplest pseudo-orbit, partner of orbit with parallel crossing. Time reversal invariance not required

[20, 21]. Only in 2001 Sieber and Richter found their “figure-eight pair” as the first manifestation of close self-encounters providing the general mechanism of creating correlated orbits, for the Hadamard-Gutzwiller model. Their reasoning was rapidly extended to general chaotic two-freedom systems [22] and eventually to more than two degrees of freedom [23, 24]. Self-encounters with more than two stretches close and the related orbit bunches (first pairs, later quadruplets) were identified in Refs. [24–29]; see Sect. 9.16 below. Extensions to encounters of endless trajectories were investigated in the context of (universal quantum features in) chaotic transport processes [30–36]. A general discussion of orbit bunches can be found in [37]. No genuinely classical “applications” of bunches of trajectories or orbits are known as yet but such might come with time. Readers wishing to think further in that direction are invited to look at Problem 9.5.

9.15.2 Quantitative Discussion of 2-Encounters Poincaré Section, Encounter Stretches, and Links For theoretical purposes it is best to characterize close self-encounters in phase space or rather the energy shell, for autonomous flows [38–40]. For simplicity we shall assume two degrees of freedom, but all ideas hold for f > 2 as well [23]. For each phase-space point X a Poincaré section P can be defined transverse to the orbit. Given hyperbolicity, P is spanned by one stable direction es and one unstable direction eu , and the normalization ||eu ∧ es || ≡ ||C eu Σes || = 1

(9.15.1)

can be adopted. Considering a neighboring trajectory whose piercing through P is, reckoned from X, δX = ses + ueu

(9.15.2)

392

9 Classical Hamiltonian Chaos

and imagining P moving with the flow, we see the components s, u change as u(t) = Λ(t, X)u(0) ,

s(t) = s(0)/Λ(t, X) ,

Λ(t, X) ∼ eλt

(9.15.3)

where Λ(t, X) > 1 is the local stretching factor; as indicated, for large times that factor grows exponentially with the Lyapunov exponent as the rate. To parametrize a 2-encounter within an orbit γ we put P at some point X1 on one of the two encounter stretches, and choose the moment of that “first” passage as time zero, t1 = 0. The second encounter stretch pierces through P at the time t2 in a point X2 . For a parallel 2-encounter X2 is very close to X1 while for an antiparallel one the time reverse T X2 is close to X1 ; in a unified notation we shall write Y2 . The small difference Y2 − X1 can be decomposed in terms of the stable and unstable directions at X1 , Y2 − X1 = (u2 − u1 )eu (X1 ) + (s2 − s1 )es (X1 ) , Y2 =

X2

parallel encounter

T X2

antiparallel encounter .

(9.15.4) (9.15.5)

When P is moved through the encounter along with the flow the components s2 − s1 , u2 −u1 change according to (9.15.3). However, since Hamiltonian time evolution amounts to a canonical transformation the symplectic surface element ΔS = (u2 − u1 )eu (X1 ) ∧ (s2 − s1 )es (X1 ) = (u2 − u1 )(s2 − s1 )

(9.15.6)

remains constant and therefore the precise location of P within the encounter is immaterial. The invariant surface element ΔS will turn out to determine all properties of the encounter of interest. To simplify the notation we shall mostly elevate the first piercing point X1 to the origin of unstable and stable axes in P such that the invariant surface element takes the form ΔS = u2 s2

if

u1 = s1 = 0 .

(9.15.7)

The definition of the self-encounter can now be completed by setting a bound c > 0 to the components s, u as P is moved, |u2 |, |s2 | ≤ c .

(9.15.8)

The encounter thus begins when |s2 | falls below the bound and ends when |u2 | rises above (see Fig. 9.8). The bound must be small enough for the encounter stretches to allow for the mutually linearized treatment of (9.15.3) and should roughly exhaust the range of linearizability, but otherwise the precise value is of no importance. Which of the two partner orbits is called γ and then used to define beginning and end of the encounter is, of course, also immaterial.

9.15 Sieber-Richter Self-Encounter and Partner Orbit

393

s

s

s

u

u

u

Fig. 9.8 Beginning and end of a 2-encounter

The bound c and the surface element ΔS = u2 s2 are two invariant parameters characterising the 2-encounter under consideration. The encounter becomes closer as ΔS/c2 decreases, and very close encounters enjoy the inequality ΔS c2 . The orbit pieces in between encounter stretches will be referred to as links. Encounter Duration The duration tenc of an encounter can be estimated due to the asymptotic exponential growth of the stretching factor: The time ts = λ−1 ln(c/|s2 |) passes in between the beginning of the encounter and the piercing through P and the time tu = (1/λ) ln(c/|u2 |) between the piercing and the encounter end, such that the encounter duration tenc = tu + ts =

c2 1 ln λ |u2 s2 |

(9.15.9)

is determined by the symplectic surface element ΔS. Partner Orbit The partner orbit γ coming with the 2-encounter is also uniquely fixed by the stable and unstable components s2 , u2 . Figure 9.9 again depicts the orbits γ , γ in configuration space (part a), as well as the Poincaré section (part b). In (a), the configuration-space piercing points are decorated with arrows indicating momenta for the quadruple of points mutually close in phase space, X1 , T X2 for one orbit and X1 , T X2 for the other. In (b), the Poincaré section P is shown with the latter four points X1 =

  0 , 0

T X2 =

  u2 , s2

X1 =

  u2 , 0

T X2 =

  0 . s2

(9.15.10)

Putting the origin of the stable and unstable axes at X1 is a convenient choice. Now, X1 and X1 must have practically coinciding stable components s1 = 0 ≈ s1 since when going backwards in time from those piercings we see the two orbits approach one another. Conversely, starting from X2 and X2 we see mutual approach of the two trajectories when going forward in time and conclude that those piercings must have very nearly the same unstable components; the time reverses then again share the same stable components s2 ≈ s2 . Similarly, X1 and T X2 must practically coincide

394

9 Classical Hamiltonian Chaos a) left port 1

right port 1 X1

Y2’ Y1’

Y2 left port 2

b)

right port 2

stable

s2

Y2’

Y2

X1

Y1’

unstable

u2 Fig. 9.9 Piercings X1 , Y2 of γ (full line) and piercings Y1 , Y2 of γ (dashed line) for a Sieber/Richter pair, depicted (a) in configuration space, with arrows indicating the momentum of the above phase-space points, and (b) in the Poincaré section P parametrized by stable and unstable coordinates. The symplectic area of the rectangle is the action difference ΔS

in their unstable components u 1 ≈ u2 . Finally, the unstable component of T X2 practically agrees with the vanishing one of X1 . The approximate equalities just invoked cannot be strict equalities since the mutual approaches of the two orbits invoked in each of the four cases eventually end somewhere within the links, to give way to divergence towards reentering the encounter; however, the error made for X1 , T X2 when taking ≈→= is exponentially small, of the order exp − λ(t2 −   t1 ) = exp − λt2 . Action Difference of Partner Orbits Next, we propose to show that the action difference of the two partner orbits γ , γ equals the invariant surface element ΔS = D u2 s2 . That surface element arises as the line integral dq p along the four edges of the parallelogram of Fig. 9.9b, say X1 → X1 → T X2 → T X2 within P. The action difference Sγ − Sγ can be seen as the sum of four pieces. To characterize them, a pair of exponentially close configuration-space points, ql on γ and ql on γ , is chosen somewhere in the middle of the left link and similarly a

9.15 Sieber-Richter Self-Encounter and Partner Orbit

395

pair qr , qr in the right link, see Fig. 9.9. The actions Sγ and Sγ then read  Sγ =

ql

dq p(q, ql ) +



dq p(qr , q) +

ql

dq p(q, qr ) +

qr

qr q1



q2

dq p(qr , q) +

q1

q1 ql



qr

dq p(q, ql ) +

 Sγ =



q1



q2 qr

dq p(ql , q) q2

dq p(q, qr ) +



ql q2

dq p(ql , q) .

In each of the foregoing integrals the path of integration is uniquely fixed (as a piece of γ or γ ) by the lower and upper limit; the integrand is a unique function of the integration variable q, given the fixed second argument (ql or qr on γ , ql or qr on γ ). One of the four pieces of the action difference arises from the first terms in Sγ and Sγ , and there we identify the points ql and ql , accepting an exponentially small error,  ΔS

(1)

≈



q1



q1

dq p(q, ql ) −

dq p(q, ql ) =

ql

q1

q1

ql

dq p(q, ql ) .

(9.15.11)

A contour integral along an arbitrary path from q1 to q1 results in the second of the foregoing equalities since the integrand p(q, ql ) is a unique function of q; that ∂ function is in fact the gradient p(q, ql ) = ∂q S(q, ql ) of the generating function for the canonical transformation {ql , pl } → {q, p} to which the time evolution along the trajectory starting at Xl amounts. The path of integration may be chosen to run in P since the end points X1 = {q1 , p1 } and X1 = {q1 , p1 } lie therein; as Fig. 9.9 reveals the path then runs along the unstable axis. For the next piece of the action difference we combine the second term in Sγ with the third in Sγ and set qr = qr , again tolerating an exponentially small error,  ΔS (2) ≈



qr

dq p(qr , q) −

 =

q1

dq p(qr , q) −

q1

 =

q2

dq p(q, qr )



qr

q2 qr qr q2

dq p(qr , q)

dq p(qr , q)

q1

where for the second line the sense of traversal was reverted in the integral between qr and q2 , by interchanging the integration limits and replacing p(q, qr ) → −p(qr , q). The contour integral in the last line may be evaluated along any path from q1 to q2 since p(qr , q) is a unique function of q. Viewed in phase space, the path starts at X1 and leads to T X2 , two points on P; the path may thus be chosen to remain within P and then runs along the stable axis as shown in part b of Fig. 9.9.

396

9 Classical Hamiltonian Chaos

The reader is kindly invited to reason similarly to get the remaining two pieces  ΔS

(3)

≈



q2

dq p(q, qr ) −

qr



qr q1

dq p(qr , q) =

q1

dq p(qr , q)

q2

with the final path in P from T X2 to X1 and  ΔS (4) ≈

ql



dq p(ql , q) −

q2



ql q2

dq p(ql , q) =

q2 q2

dq p(q, ql )

(9.15.12)

with the final path in P from T X2 to T X2 . The sum of the four pieces now gives the action difference as the closed-contour integral in P , Sγ − Sγ =

dq p

(9.15.13)

around the parallelogram spanned by the sequence of points X1 , X1 , T X2 , T X2 shown in part b of Fig. 9.9. That integral is a canonically invariant quantity; it is the Poincaré integral invariant discussed in Sect. 9.6. The evaluation in the simplest case of two degrees of freedom is an elementary task since the coordinates u, s form a canonical pair whereupon the line integral just gives the enclosed area. Moreover, the parallelogram is appropriately depicted as a rectangle in Fig. 9.9 since the normalization (9.15.1) absorbs the sine of the angle enclosed by eu and es . Indeed, then, the action difference in search is given by the symplectic surface element characterizing the 2-encounter, Sγ − Sγ = ΔS = u2 s2 .

(9.15.14)

That result must remain correct for f > 2, with u2 s2 to be interpreted as a sum of f − 1 terms, simply because the Poincaré invariant (9.15.13) is such D

f −1 D dqi pi . In view of generalizations to follow the a sum, dq p = i=1 following remark is in order. Without the identification of X1 with the origin in P the action difference would have come out in the form (9.15.6), i.e. as Sγ − Sγ = ΔS = (u2 − u1 )(s2 − s1 ). Phase-Space Densities we finally go for the expected number of close 2encounters within a periodic orbit, given hyperbolic dynamics. Of interest is the limit of very large periods T , larger by far than any other characteristic time of the classical dynamics. In particular, the Lyapunov time 1/λ, typical encounter durations tenc , and the time scale tmix on which ergodicity and mixing emerge will be assumed dwarfed by T . The term “expected” means an average over the set of all period-T orbits; within that set, the number of 2-encounters fluctuates. Moreover,

9.15 Sieber-Richter Self-Encounter and Partner Orbit

397

we shall focus on very narrow 2-encounters for which the encounter time obeys 1/λ, tmix

tenc

T.

(9.15.15)

More detailed and important information is contained in the expected number nT (ΔS)dΔS of 2-encounters with symplectic area elements in the differential interval [ΔS, ΔS + dΔS]. That quantity is in fact the most important statistical characteristic of 2-encounters since it equals the expected number of partner orbits of a given orbit, the partners “generated” in 2-encounters and differing in action from the given orbit by ΔS. We start from the probability for a Poincaré section P pierced by an orbit γ in a point X1 at time zero to contain a second piercing X2 of γ such that T X2 − X1 (for an antiparallel self-encounter) or X2 − X1 (for a parallel self-encounter) appears with stable and unstable coordinates in the differential box [u, u + du] × [s, s + ds] (with s, u inside the rectangle |s| ≤ c, |u| ≤ c defining the encounter) and a time delay t2 − t1 = t2 in the interval [t, t + dt]. The encounter duration is then fixed near tenc = λ−1 ln(c2 /|us|). Due to the assumption (9.15.15) the second piercing is statistically independent of the first, and the probability in question is given by the Liouville measure on the energy shell10 dudsdt , Ω  = [u, u + du] × [s, s + ds] ,

prob{Y2 − X1 ∈ , t2 ∈ [t, t + dt]} =

Y2 =

X2

parallel enc.

T X2

antiparallel enc.

(9.15.16)

where Ω is the volume of the energy shell. We shall show presently that the delay t2 can range in an interval of length T − 2tenc ; anticipating that restriction we proceed to the probability of the differential rectangle containing the second piercing after whatever delay, prob{T X2 − X1 ∈ } =

duds(T − 2tenc ) . Ω

(9.15.17)

The probability for the randomly chosen Poincaré section P to intersect a 2encounter characterized by the invariant symplectic surface element ΔS = us is now obtained by integrating as prob{ΔS ≤ us ≤ ΔS + dΔS} =

1 dΔS 2



c

du

−c



c

ds δ(ΔS − us)

−c

(T − 2tenc ) . Ω (9.15.18)

10 To check normalization to unity by integrating over the energy shell would require an extension of the coordinates u, s beyond P .

398

9 Classical Hamiltonian Chaos

The factor 12 prevents an overcounting of each 2-encounter, necessary since each of the two encounter stretches can be chosen as the “first” and yields a separate parametrization of the same encounter. The double integral in the foregoing expression is easily evaluated by changing the integration variables to the product us and, say, u; the result reads prob{ΔS ≤ us ≤ ΔS + dΔS} =

λtenc (T − 2tenc ) dΔS Ω

(9.15.19)

where the encounter duration must be read as a function of the invariant surface element, tenc = (1/λ) ln(c2 /|ΔS|). On the other hand, that latter probability can be expressed in terms of the expected number of 2-encounters in γ with fixed symplectic surface element, NT (ΔS)dΔS, as the ratio of the expected cumulative duration of all such 2encounters to the orbit period T , prob{ΔS ≤ us ≤ ΔS + dΔS} =

2tenc NT (ΔS)dΔS ; 2T

(9.15.20)

here, as in (9.15.18), a factor 12 had to be worked in order to count each encounter once. Comparison of the two expressions yields the number density of 2-encounters with fixed ΔS in a period-T orbit, NT (ΔS) =

λT (T − 2tenc ) . Ω

(9.15.21)

As already announced in the beginning of this subsection, NT (ΔS)dΔS also has the meaning of the number of partner orbits γ of a given period-T orbit γ with the action difference Sγ − Sγ in the interval [ΔS, ΔS + dΔS]. The overall number of partners generated in 2-encounters with any action difference in the permissible interval [−c2 , c2 ] is obtained by integrating as 

c2

−c2

dΔS NT (ΔS) =

1  λT 2 c2  1− . Ω λT

(9.15.22)

In the quantum applications to be discussed in the next chapter it will be technically convenient to work with an auxiliary ## c density wT (s, u) of which the above NT (ΔS) is the marginal NT (ΔS) = 12 −c duds wT (u, s)δ(ΔS − us), wT (u, s) =

T (T − 2tenc ) . tenc Ω

(9.15.23)

Note that to conform with established usage in the literature we have left out from wT the factor 12 which avoids overcounting. Somewhat loosely speaking one may interpret the latter bivariate quantity as twice the density of 2-encounters.

9.16 l-Encounters and Orbit Bunches

399

Both densities, NT (ΔS) and wT (u, s), have a leading term ∝ T 2 and a minute correction of relative order tenc /T . The latter will turn out of decisive importance for the semiclassical explanation of universal spectral fluctuations below. We finally fill the promise to show that the delay t2 of the second piercing ranges in an interval of length T − 2tenc . For the antiparallel encounter à la Sieber and Richter we need to inspect Fig. 9.9a. Since the first encounter stretch is taken as oriented towards the right, the duration of the right link can be read off as t2 − 2tu and that of the left link as T − t2 − 2ts . These link durations must be non-negative, and therefore t2 is confined to the interval 2tu ≤ t2 ≤ T − 2ts ; since tu + ts = tenc the length of that interval comes out as anticipated. For the parallel 2-encounter of Fig. 9.7 the same requirement of non-negative link durations is easily seen to imply tenc ≤ t2 ≤ T − tenc and the same interval length arises.

9.16 l-Encounters and Orbit Bunches Qualitative Discussion The 2-encounters of an orbit just discussed are but a special case of self-encounters. Any number of stretches can run mutually close and when that number is l it is appropriate to speak of an l-encounter [24–29]. Outside an l-encounter, an orbit has l links. Of interest are again orbits with very large periods T . With appeal to the exponential stability of the boundary value problem discussed in Sect. 9.14 we can again ascertain the existence of partner orbits γ of an orbit γ . The links of γ and γ are nearly identical in the following sense: Reckoned from each end, each link of γ approaches its correspondent of γ exponentially on the time scale 1/λ; the cartoon visualizes that behavior. To obtain a partner γ from γ the l encounter stretches are slightly deformed to differently connect links. The number of different connections is l! such that an l-encounter may be said to “generate” a bunch of altogether l! orbits; in other words, an orbit γ has l! − 1 partners. However, as is already obvious from Fig. 9.7 for l = 2, γ may be a decomposing pseudo-orbit. Moreover, some partners are formed without the active participation of all l encounter stretches, inactive being stretches which connect the same links as in γ . Equally contained in the number l! − 1 of partners are orbits for which the l-encounter under discussion acts like a collection of effectively independent encounters whose cumulative number of stretches is l. The interesting question as to how many of the l! − 1 partners are (i) genuine, i.e. non-decomposing orbits and (ii) generated such that all l encounter stretches are active without effectively forming a collection of distinct “sub-encounters” will be addressed below.11

11 In permutation-theory parlance, condition (ii) distinguishes l-encounters for which the permutation 1, 2 . . . , l → i1 , i2 , . . . , il of left-port labels to right-port labels is a single cycle, rather than

400

9 Classical Hamiltonian Chaos

Poincaré Section To parametrize an l-encounter a Poincaré section P transverse to the orbit at an arbitrary point X1 (passed at time t1 ) inside one of the encounter stretches. The other stretches pierce through P at times tj (j = 2, . . . , l) in points Xj . All l piercings are mutually close in configuration space while the momenta are either close to or nearly opposite to p1 ; in a unified notation, we shall write Yj for Xj in the first case and for T Xj in the second one, so as to have all Yj close to X1 . The small differences can be decomposed in terms of the stable and unstable directions at X1 ; with the choice u1 = s1 = 0 the differences read Yj − X1 = uj eu + sj es .

(9.16.1)

To define the encounter the components are restricted by |sj | , |uj | ≤ c .

(9.16.2)

Encounter Duration As in (9.15.9) we employ the time spans (1/λ) ln(c/|uj |) for the unstable components and (1/λ) ln(c/|sj |) for the stable components to reach c, reckoning from P towards the future and past, respectively. The encounter duration tenc = tu + ts thus generalizes to

tu = min j

c 1 ln λ |uj |

tenc = tu + ts =



,

ts = min j

c 1 ln λ |sj |



c2 1 ln ; λ maxi {|ui |} maxj {|sj |}

, (9.16.3)

Due to (9.15.3) the duration tenc again remains invariant as P is shifted. Partner Formation The initial and final points of each encounter stretch will be called “entrance” and “exit” ports. If all encounter stretches are almost parallel, as in - the entrance ports lie all on the same side of the encounter and the exit ports on the other side. If, however, the encounter involves mutually time reversed stretches - , it is useful to call “left” the ports on the side where the first encounter like  stretch begins and “right” those on the other side. Obviously, that distinction between left/right and entrance/exit is a necessary one only for time reversal invariant dynamics; without T -invariance, entrance and left are synonymous, as are exit and right. The l stretches and their ports can be numbered in the order of their traversal by the orbit γ : the ith encounter stretch of γ connects the left port i to the right port i. A partner γ has the left port i connected to a different right port j . For γ and γ to be genuinely related by an l-encounter, the l reshufflings i → j must not separate in disjoint groups, or else we would face several distinct encounters at work in the formation of the partner, see Fig. 9.10 and the previous footnote.

a collection of separate cycles; the cycle structure of permutations will become an issue presently; see Fig. 9.10.

9.16 l-Encounters and Orbit Bunches

401

Fig. 9.10 (a) Connections between left and right ports in partner orbit γ related to γ in 6encounter; permutation 123456 → 264513 is single 6-cycle. (b) 6-encounter splits into three pieces; γ thus in effect related to γ in three 2-encounters; permutation 123456 → 214365 has three 2-cycles

The points of piercing of a partner γ through P are fixed by those of γ , as already seen for l = 2 in the previous section. It is convenient to number the stretches of γ by the labels of their left ports and to also employ that label for the Yj . With that choice taken, the ith stretches of γ and γ have (to exponential accuracy) the same stable components of Yi and Yi since upon continuing to the left we see exponential approach of γ and γ (or T γ ) far into the adjoining links, si = si

(common left port i) .

(9.16.4)

Similarly, if the ith stretch of γ goes to the j th right port we observe subsequent exponential approach of γ and γ (or T γ ) towards the right and conclude equality of the unstable components of Yi and Yj , u i = uj

(common right port j ) .

(9.16.5)

Figure 9.11 depicts the locations of the Yi in P for a 3-encounter. stable

s3

Y’3

Y3 Y2

s2

X1

Y’2

unstable

Y’1

u2

u3

Fig. 9.11 Piercing points of γ ,γ differing in a 3-encounter; inside γ , left ports 1,2,3 are connected to right ports 2,3,1, respectively

402

9 Classical Hamiltonian Chaos

Action Difference The action difference Sγ − Sγ is uniquely determined by the ui , si , i = 2, . . . , l. The above reasoning for l = 2 can be taken over in l − 1 successive steps. Each step interchanges the right ports of two encounter stretches, as in a 2-encounter. Only the last, (l − 1)th, step produces the partner γ while each previous step introduces an intermediate “step”-partner. Each step contributes to the action difference an amount of the type (9.15.14) or rather (9.15.6). It is again convenient to label stretches and piercing points according to the left port. The only difference to the above treatment of 2-encounters is that the j th step has the j th piercing point uj , sj of γ as the reference point whose components vanish only for j = 1; the action difference according to (9.15.6) must therefore be employed with appropriate labels. The stepwise procedure is illustrated in Fig. 9.12. The port-to-port stretches of an orbit γ within a 4-encounter are depicted in part (a) of the figure and the reconnections for a partner γ in part (d). Three steps lead from (a) to (d). In the j th step, the left ports j and j + 1 are respectively connected to the right ports j + 1 and 1. Recalling that the stable and unstable coordinates of piercing points are respectively determined by the left and right ports, we get the separations between the intermediate piercings uj +1 − u1 , sj +1 − sj and thus the contribution of the j th step t the action difference as the product (uj +1 − u1 )(sj +1 − sj ). The overall action difference for the example under consideration is the sum Sγ − Sγ =

3 

(uj +1 − u1 )(sj +1 − sj ) .

(9.16.6)

j =1

All reconnections for l = 4, and in fact for any l can be treated analogously. A coordinate transformation is useful, {ui , si , i = 2, . . . , l} → {u˜ i , s˜i , i = 1, . . . , l − 1, such that the ith pair u˜ i , s˜i relates to the ith step and the product u˜ i s˜i becomes the contribution of that step to the action difference Sγ − Sγ =

l−1 

u˜ i s˜i .

(9.16.7)

i=1

For the example considered the transformation reads u˜ j = uj +1 − u1 , s˜j = sj +1 − sj ). In all cases, the transformation is linear and volume preserving. Due to the simple form (9.16.7) of the general overall action difference it is convenient to redefine the encounter region by bounding the new parameters rather than the old ones as |u˜ i |, |˜si | ≤ c. Phase-Space Densities We again invoke the Liouville measure for the probability to find the l − 1 points Yj for the piercings j = 2, . . . , l relative to an arbitrarily chosen reference one, X1 at t1 = 0, in a differential box j = [sj , sj + dsj ] × [uj , uj + duj ]

(9.16.8)

9.16 l-Encounters and Orbit Bunches

403

Fig. 9.12 Stepwise transition from γ (depicted in a) to γ (shown in d) within 4-encounter; each step interchanges right ports of two encounter stretches

and the respective delays in [tj , tj + dtj ], l  + * duj dsj dtj  . prob Yj − X1 ∈ j , j th delay ∈ [tj , tj + dtj ] j = 2, . . . , l = Ω j =2

(9.16.9) The piercing times are ordered as t1 < t2 < . . . < tl < t1 +T where T again denotes the orbit period. Moreover, the links in between the l encounter stretches must have positive lengths. With those restrictions we need to integrate over the l − 1 delays in order to get the piercing probability accumulated over all permissible delays, l  + * duj dsj  prob Yj − X1 ∈ j j = 2, . . . , l = Ω

 d l−1 t

(9.16.10)

j =2

To do the (l−1)-fold time integral it is convenient to shift the integration variables so as to have zero as the common lower limit of the integration ranges. The restrictions of the range then take the simple form 0 ≤ t2 ≤ t3 ≤ . . . tl ≤ T − ltenc . Since l−1 enc ) the integrand is time independent the integral equals (T −lt (l−1)! , the volume of the (l − 1) dimensional simplex with edge length T − ltenc . The piercing probability accumulated over all permissible delays thus reads l  + (T − ltenc )l−1 * prob Yj − X1 ∈ j j = 2, . . . , l = duj dsj . (l − 1)!Ω l−1

(9.16.11)

j =2

The reader will recall the next step in the above reasoning for 2-encounters, where the probability to find a randomly chosen P intersecting a 2-encounter with the symplectic area element ΔS was constructed, irrespective of the position of P inside the encounter and of which encounter stretch is chosen as the reference stretch. The appropriate generalization to an l-encounter requires the separation of the 2(l − 1) parameters s, u into 2l − 3 new variables (Δ1 , Δ2 , . . . , Δ2l−3 ) which describe the internal configuration of the encounter independent of the location of P and the choice of the reference stretch, and a single variable fixing the location

404

9 Classical Hamiltonian Chaos

of P along the orbit; clearly, for l = 2 there is just a single “internal” variable Δ1 ≡ ΔS. It is useful to at least momentarily think in terms of such Δ’s now and realize that the variables u, s uniquely determine the Δ’s, say as Δ = σ (u, s). A probability density p(Δ) can then be defined by integrating over u, s such that the Δ’s are confined to a small interval,  p(Δ)d 2l−3 Δ =

d l−1 ud l−1 s δ(Δ − σ (u, s)) Δ 1 as functions of q1α , q0 and taking the partial derivative with respect to q1α (at constant q0 ) in the equation of motion (10.2.7). We conclude that Dnα = det G(n,α) =

  ∂qn  m ∂qn  = ; ∂q1α q0 τ ∂p0α q0

(10.2.17)

2 Formally, the contribution of the αth classical path can be seen as arising from a Gaussian integral of the type (5.13.22) after analytic continuation to purely imaginary exponents; that continuation, achieved by the Fresnel integral (10.2.12), brings about the modulus operation and the ν α phase factors e−iπ/2 in (10.2.13).

414

10 Semiclassical Roles for Classical Orbits

the last member of the foregoing chain results from p0α = (m/τ )(q1α − q0). With the help of (10.2.5), we may express the inverse determinant by a second derivative of the action as ∂ 2 S (n,α) m ∂ m =− (q1α − q0 ) = − . ∂qn ∂q0 τ ∂qn τ Dnα

(10.2.18)

Replacing the determinant Dnα with the mixed second derivative of S (n,α) in the semiclassical propagator (10.2.13), (10.2.14) we get the latter in the Van Vleck form3 .   * +  1  ∂ 2 S (n,α)  i S (n,α) (q, q )/h¯ − ν α π/2 n , (10.2.19) q|U |q  ∼ e i2π h¯  ∂q∂q  α first spelled out for quantum maps by Tabor [18]. A simple prescription for determining the Morse index ν α derives from the α recursion relation (10.2.16) for the determinants Diα . A change of sign of Di+1 α relative to Di signals that the number of negative eigenvalues is larger by one for G(i+1,α) than for G(i,α) . This follows from the fact that G(i,α) is obtained from G(i+1,α) by discarding the ith row and column. A well-known corollary of Courant’s minimax theorem [19] then says4 that the i eigenvalues of G(i+1,α) form an alternating sequence with the i − 1 eigenvalues of G(i,α) . Therefore, the number ν α of negative eigenvalues of G(n,α) must therefore equal the number of sign changes in the sequence of determinants Diα from D1α to Dnα . Clearly, the prescription just explained relies on the kinetic energy to be a positive quadratic form in the momenta and distinguishes the coordinate representation; for more general situations see Ref. [20]. We leave to the reader as Problem 10.2 the generalization of the Van Vleck formula to f degrees of freedom. Actually, most applications of maps studied thus far are confined to f = 1, and reasonably so since periodically driven singlefreedom systems can display chaos. It is only for autonomous dynamics (alias flows), to be looked at in Sect. 10.3.2, that nonintegrability requires f > 1. The very appearance (10.2.19) of the Van Vleck propagator suggests validity beyond the limitations of its derivation given here. It is indeed not difficult to check that the simple product structure (10.2.2) of the Floquet#operator can be abandoned τ in favor of arbitrary periodic driving, U → (exp{−i 0 dtH (t)/h¯ })+ where the

3 This is equally valid in continuous time; if we let n → ∞, τ → 0 with nτ = t fixed, S (n,α) (q, q ) → S α (q, q , t). 4 When the eigenvalues of G(i+1,α) are ordered as g (i+1) ≤ g (i+1) ≤ . . . ≤ g (i+1) and those 1 2 i (i) of G(i,α) as g1(i) ≤ g2(i) ≤ . . . ≤ gi−1 the corollary to the minimax theorem in question yields (i) g1(i+1) ≤ g1(i) ≤ g2(i+1) ≤ g2(i) ≤ . . . gi−1 ≤ gi(i+1) .

10.2 Van Vleck Propagator

415

time-dependent Hamiltonian need not even be the sum of a kinetic and a potential term. According to its property (10.2.17), the determinant Dnα characterizes the stability properties of the classical path {qiα }i=0,1,...,n through δqn ≈ Dnα δq1 . We may define a Lyapunov exponent for the αth path between q0 and qn by 1 ln |Dnα | . n→∞ n

λα = lim

(10.2.20)

A path hopping about in a chaotic region will have positive λα such that the determinant in question will grow exponentially, Dnα ∼ eλα n ; a regular path, on the other hand, will have a vanishing Lyapunov exponent; its Dnα is uncapable of exponential growth but might oscillate or grow like a power of n. For a thorough discussion of such different behaviors the reader may consult any textbook on nonlinear dynamics; here, we must confine ourselves to the few hints to follow. For a rough qualitative characterization, we may replace all curvatures V (qn ) along the path by a suitable average V (qnα ) = mωα2 . Within this “harmonic” approximation dnα = 2 − ωα2 τ 2 = d is independent of n, whereupon the recursion relation (10.2.16) allows for solution by exponentials a n , whose bases 2 2 are ) determined by the quadratic equation a = d − 1/a as a± = 1 − ωα τ /2 ± 2 2 2 2 ωα τ (ωα τ − 4)/4). Accounting for D0 = 0, D1 = 1, we get the solution Dnα =

n − an a+ − . a+ − a−

(10.2.21)

Three cases arise: (1) For 0 < ωα2 τ 2 < 4, the determinant oscillates as Dnα =

sin nφα sin φα

with

sin

φα 1 = |ωα τ |, 2 2

0 ≤ φα ≤ π . (10.2.22)

The Morse index ν α is the integer part of nφα /π, the number of sign changes iφα in the sequence sin i = 1, 2, . . . , n. The boundary cases φα = 0 or π can sin φα , be included here as limiting ones and yield the subexponential growth Dnα = n with ν α = 0. Paths of this type are stable and are called elliptic. They exist where the curvature of the potential V (q) is positive and small. (2) For ωα2 τ 2 < 0, the determinant grows exponentially, Dnα =

sinh nλ˜ α sinh λ˜ α

with

sinh

1 λ˜ α = |ωα τ |, 2 2

λ˜ α > 0 .

(10.2.23)

The Morse index vanishes since Dnα never changes sign. One speaks here of hyperbolic paths. They experience a potential with negative curvature.

416

10 Semiclassical Roles for Classical Orbits

(3) Finally, for ωα2 τ 2 > 4, the determinant grows exponentially in magnitude but alternates in sign from one n to the next, Dnα = (−1)(n−1)

sinh nλ˜ α sinh λ˜ α

with

cosh

1 λ˜ α = |ωα τ |, 2 2

λ˜ α > 0 . (10.2.24)

The Morse index is read off as ν α = n − 1. Such inverse hyperbolic paths lie in regions where the potential V (q) has a large positive curvature. In the two unstable cases the parameter λ˜ α approximates the Lyapunov exponent. The exponential growth of |Dnα | with the stroboscopic time n entails the amplitude Aαn in (10.2.13) and (10.2.14) to decay like e−λα n/2 for unstable paths, while stable paths do not suffer such suppression. It would be wrong to conclude, though, that unstable paths have a lesser influence on the propagator than stable ones since the latter are outnumbered by the former, roughly in the ratio eλn where λ is a Lyapunov exponent. The reader will recall the discussion of exponential proliferation in Chap. 9.12. In the present context, that proliferation threatens the semiclassical propagator (10.2.13) with divergence as n → ∞.

10.2.2 Flows As already mentioned, the Van Vleck propagator (10.2.19) also applies to the continuous-time evolution generated by the Hamiltonian H = T + V : By taking the limit n → ∞, τ → 0 with the product t = nτ fixed, we get the result anticipated in (10.2.1), q|e

−iH t /h¯



|q  ∼

 α

.

    1  ∂ 2 S α (q, q , t)  i S α (q, q , t)/h¯ − ν α π/2 . e  i2π h¯  ∂q∂q (10.2.25)

Here, the discrete time index n was replaced with the continuous time t as an independent argument of the action of the αth classical path going from q to q during the time span t. As before, the action S α (q, q , t) is the generating function for the classical transition in question. To see what happens to the Morse index ν α in the limit mentioned (nothing, really, except for a beautiful new interpretation as a certain classical property of the αth orbit), a little excursion back to the derivation of the Van Vleck propagator of Sect. 10.1 is necessary. To avoid any rewriting, we shall leave the time discrete, ti = iτ with i = 1, 2, . . . , n and tn = t, and imagine n so large and τ so small that the discrete classical path from q0 = q to qn = q approximates the continuous path

10.2 Van Vleck Propagator

417

accurately. Recall that the Morse index was obtained as the number of negative eigenvalues of the matrix G(n,α) in the second variation of the action about the classical path, δ 2 S (n,α) ({φ}) =

n−1 m  (n,α) Gij φi φj . τ

(10.2.26)

i,j =1

Hamilton’s principle characterizes the classically allowed paths as extremalizing the action. An interesting question, albeit quite irrelevant for the derivation of the equations of motion, concerns the character of the extremum. The interest in the present context hinges on its relevance for the value of the Morse index. Clearly, the action is minimal if the second variation δ 2 S (n,α) is positive, i.e., if the eigenvalues (n,α) λ of the matrix Gij are all positive; in that case the Morse index vanishes. If we stick to Hamiltonians of the form H = T + V and to sufficiently short time spans, minimality of the action does indeed hold, simply because locally any classical path looks like a straight line and resembles free motion. But for free motion, the eigenvalue problem in question reads 2Φi − Φi−1 − Φi+1 = λΦi for i = 1, 2, . . . , n − 1 where the boundary conditions are Φ0 = Φn = 0. The eigenvectors Φ can be constructed through an exponential ansatz for their components, Φi = a i , which yields the quadratic equation a + λ − 2 + 1/a = 0; the two roots a± obey a+ a− = 1; thus, each eigenvector is the sum of two exponentials, i i Φi = c+ a+ + c− a − with coefficients c± restricted by the boundary conditions n n n n c+ +c− = c+ a+ +c− a− = 0; nontrivial solutions c± = 0 require a+ = a− and thus 2n 2n a+ = a− = 1; now, both bases a± are revealed as unimodular, i.e., determined by a phase χ through a± = e±iχ ; then, the foregoing quadratic equation for the bases yields the eigenvalue λ = 2(1 − cos χ) as bounded from below by zero; but zero is not admissible since a vanishing eigenvalue would entail a vanishing eigenvector; indeed, the eigenvectors are Φk ∝ eikχ − e−ikχ ∝ sin kχ and thus Φk ≡ 0 for χ = 0. To ease possible worry about the limit χ → 0 in the last conclusion, one should check that the normalization constant is independent of χ; one easily finds Φk = (2/n)1/2 sin kχ. As required for minimality of the action for free motion, all eigenvalues λ of G turn out positive. But back to general classical paths! As we let the time t grow (always keeping a sufficiently fine gridding of the interval [0, t]), the action will sooner or later lose minimality (even though not extremality, of course). When this happens first, say for time tn = t c , an eigenvalue of G reaches zero and subsequently turns negative. The corresponding point q(t c ) on the classical path is called “conjugate” to the initial one, q(0) = q . More such conjugate points may and in general will follow later. At any conjugate point the number of negative eigenvalues of G will change by one. A theorem of Morse’s [21] makes the stronger statement that the number of negative eigenvalues of G for Hamiltonians of the structure H = T + V actually always keeps increasing by one as the configuration space path traverses a point conjugate to the initial one. Therefore, the Morse index equals the number of conjugate points passed.

418

10 Semiclassical Roles for Classical Orbits

Fig. 10.1 Two consecutive focal (alias conjugate) points in configuration space: a bundle of trajectories fanning from the first reunites in the second

To locate a conjugate point, one must look for a vanishing eigenvalue, i.e., solve j Gij Φj = 0 which reads explicitly m (Φi+1 − 2Φi + Φi−1 ) = −V (qi )Φi . τ2

(10.2.27)

This so-called Jacobi equation may be read as the Newtonian equation of motion (10.2.7) linearized around a classical path q(ti ) = qi from q0 = q to qn = q in discretized time. Here, of course, we must look for a solution

Φi that satisfies the boundary conditions Φ0 = Φn = 0 and is normalizable as i Φi2 = 1. If such a solution exists, the “final” time tn = t is a t c , and the point q is conjugate to q . A geometrical meaning of conjugate points is worth a look. It requires considering the family of classical paths qi (p ) = q(p , ti ) with i = 1, 2, . . . originating from one and the same initial q0 ≡ q with different momenta p (see Fig. 10.1). i (p ) Two such paths move apart as qi (p + δp ) − qi (p ) = ∂q∂p δp . The “response function” J (p , ti ) =

∂qi (p ) ∂p

(10.2.28)

measuring that divergence vanishes initially since the initial coordinate is fixed as independent of p . At the final time tn = t, however, the response function obeys −1  2 ∂q(p ) ∂ S(q, q , t) =− J (p , t) = ∂p ∂q∂q

(10.2.29)

where the last member is given by (10.2.18) since p = m τ (q1 − q0 ); it reveals that the response function determines the preexponential factor in the Van Vleck propagator (10.2.25). Each path of the family considered obeys the equation of motion (10.2.7) which upon differentiation with respect to p yields m [J (p , ti+1 ) − 2J (p , ti ) + J (p , ti−1 )] = −V (qi )J (p , ti ) . τ2 The Jacobi equation (10.2.27) is met once more, and this time truly meant as a linearized equation of motion. If one wants to determine J (p , ti ) from here,

10.2 Van Vleck Propagator

419

one needs a second initial condition that follows from the initial momentum as τ q1 = q0 + m p . Should the response function J (p , ti ) vanish again at the final time tn = t, all member paths in the family would reunite in one and the same final point q(p , tn ) = q; moreover, that final point must be conjugate to q since, like the eigenvector of the stability matrix G with vanishing eigenvalue, the response function J (p , ti ) obeys the Jacobi equation and vanishes both initially and in the end. To underscore the geometrical image of the family fanning out from q and reconverging in q c , conjugate points are also called focal points . One more time anticipating material to be presented in the next section, we would like to mention yet another meaning of conjugate points. The propagator q|e−iH t /h¯ |q  sharply specifies the initial coordinate as q but puts no restriction on the initial momentum. From the classical point of view, a straight line L (for systems with a single degree of freedom) in phase space that runs parallel to the momentum axis is thus determined. If every point on that line is dispatched along the classical trajectory generated by the classical Hamiltonian function H (q, p), a time-dependent image L of the initial L will arise. For very short times, L will still be straight but will appear tilted against L (see Fig. 10.2). At some finite time t c , when the potential energy has become effective, the image may and in general will develop a first caustic above the q-axis, i.e., a point with vertical slope, ∂p/∂q = ∞ ⇔ ∂q/∂p = 0; but inasmuch as all points on L can still be uniquely labelled by the initial momentum p on L , these caustics can also be characterized

Fig. 10.2 Initial Lagrangian manifold L at q = 1, t = 0 and its time-evolved images L at t = 0.2 (left) and t = 2.0 (right) for the one-dimensional double-well oscillator with the potential V (q) = q 4 − q 2 . The directed thin lines depict orbits. The early-time L appears tilted clockwise against L and still rather straight, as is typical for systems with Hamiltonians H = p2 /2m+V (q). At the larger time t = 2.0, L has developed “whorls” and “tendrils” due to the anharmonicity of V (q); moreover, four configuration-space caustics appear. This type of Lagrangian manifold is associated with the time-dependent propagator; it is transverse to the trajectories which carry it along as time evolves. Courtesy of Littlejohn [22]

420

10 Semiclassical Roles for Classical Orbits

by the conjugate-point condition ∂q/∂p = 0. Therefore, a point in configuration space conjugate to the initial q yields a caustic of L. Subsequent to the appearance of the first such caustic on L, the Van Vleck propagator can no longer consist of a single WKB branch in (10.2.25) but must temporarily comprise two such since there are two classical paths leading from the initial configuration-space point q to the final q within the time span t (see Fig. 10.2). Eventually, when further caustics have appeared on L, more WKB branches arise in the propagator, as labelled by the index α and summed over in (10.2.25). No essential difficulty is added for f degrees of freedom. Then the use of nonCartesian coordinates and replacement of the Newtonian form of the equation of motion by the Lagrangian or Hamiltonian may be indicated. The fact is worth mentioning that conjugate points may and often do acquire multiplicities inasmuch as (at most f ) eigenvalues λ may vanish simultaneously. Then, the Morse index counts the conjugate points with their multiplicities.

10.3 Gutzwiller’s Trace Formula Here, we shall derive the Gutzwiller type traces of quantum propagators. These are the principal ingredients for semiclassical approximations of quantum (quasi) energy spectra and of measures of spectral fluctuations. For maps, the relevant propagator is the time-dependent one, q|U n |q , while for autonomous flows, the tradition founded by Gutzwiller demands looking at the energy-dependent propagator alias resolvent, (E − H )−1 . The traces to be established will be semiclassical, i.e., valid only up to corrections of relative order 1/h¯ and under additional restrictions to be revealed as we progress. The starting point is the Van Vleck propagator which itself is a sum of contributions from classical orbits. The latter structure is inherited by the traces, and the contributing orbits are constrained to be periodic: Closure in configuration space is an immediate consequence of the traces being sums (or integrals) of diagonal elements q|U n |q or q|(E − H )−1|q; then, periodicity in phase space results from a stationary-phase approximation in doing the integral which yields the condition p = p .

10.3.1 Maps As already explained in Chap. 5, the traces  tn = ∼

+∞

−∞

 α

dx x|U n |x .  * +  ∂ 2 S (n,α)  dx i S (n,α) (x,x)/h¯ −ν α π/2   e √ i2π h¯  ∂x∂x x=x

(10.3.1)

10.3 Trace Formulae and the Density of Levels à la Gutzwiller

421

are the Fourier coefficients of the density of levels (see (5.14.1), reproduced as (10.3.8) below) and building blocks for the secular coefficients through Newton’s formulae(see Sect. 5.15). Thus, their (h¯ → 0)-approximants are the clues to a semiclassical discussion of quasi-energy spectra. Having given the semiclassical propagator (10.2.19), we just need to do the single x-integral in (10.3.1) and there once more employ the stationary-phase approximation. The stationary-phase condition, dS (n,α) (q, q)/dq = [(∂/∂q + ∂/∂q )S (n,α) (q, q )]q=q = p − p = 0, restricts the contributing paths to periodic orbits, for which the initial phasespace point q , p and its nth classical iterate q, p coincide. The period n may be “primitive” or an integral multiple of a shorter primitive with period n0 = n/r and the number r of traversals a divisor of n. All n0 distinct points along any one period-n orbit make the same (yes, the same, see below) additive contribution to the integral which may again be found from the Gaussian approximation exp{iS (n,α) (x, x)/h¯ } ∼ exp {(i/h¯ )[S (n,α) (q, q) + 12 (S (n,α) (q, q)) (x − q)2 ]} and the Fresnel integral (10.2.12). So, we obtain Gutzwiller’s trace formula tn ∼

 α

. n0

  2 (n,α) * (n,α) + ∂ S 1 (q, q )  (n,α)  e i S (q,q)/h¯ −μ π/2   (n,α) ∂q∂q |(S (q, q)) | q=q (10.3.2)

where the sum is over all period-n orbits, q is the coordinate of any of the n points on the αth such orbit, and μ(n,α) is the Maslov index5 1 μ(n,α) = ν α (q, q) + {1 − sign((S (n,α) (q, q)) )} ≡ μαprop + μαtrace . 2

(10.3.3)

A word is in order on the Morse index ν α (q, q) ≡ μαprop in the integrand in (10.3.1). we already pointed out in the previous subsection that ν α (q, q ) depends on q, q and the connecting path. Upon equating q and q to the integration variable x, one must reckon with ν α (x, x) as an x-dependent quantity. On the other hand, ν α (x, x) is an integer. The n-step path under consideration will change continuously with x unless it gets lost as a classically allowed path; concurrent with continuous changes of x, its ν α (x, x) may jump from one integer value to another at some x, ˆ but the asymptotic h¯ → 0 contribution to the integral won’t take notice unless xˆ happened to be a stationary point of the action S (n,α) (x, x), i.e., unless the n-step path were a period-n orbit in phase space rather than only a closed orbit in configuration space. There is no reason to expect a point xˆ of jumping ν α (x, x) in coincidence with a point of stationary phase: Stationarity is a requirement for the first derivatives of the action while jumps of the Morse index take place when an eigenvalue of the matrix G(n,α) defining the second variation of the action around the path according ij

5 Somewhat arbitrarily but consistently, we shall speak of Morse indices in propagators and Maslov indices in trace formulae.

422

10 Semiclassical Roles for Classical Orbits

to (10.2.11) and (10.2.15) passes through zero.6 Thus, the contribution of a point q on the αth period-n orbit involves the Morse index ν α (q, q) which characterizes a whole family of closed-in-configuration-space n-step paths near the period-n orbit in question. The question remains whether all n points on the period-n orbit have one and the same Morse index; they do, but we beg the reader’s patience for the proof of that fact until further below. An alternative form of the prefactor of the exponential in the semiclassical trace (10.3.2) explicitly displays the stability properties of the αth period-n orbit by involving its stability matrix (called monodromy matrix by many authors) ⎡    ⎤ ⎛ S ∂q ∂q − Sq ,q ⎢ ∂q p ∂p q ⎥ ⎜ q,q ⎥ ⎜ M =⎢ ⎣ ∂p   ∂p  ⎦ = ⎝ (Sq,q )2 −Sq ,q Sq ,q ∂q

p

∂p

q

Sq,q

⎞ −S 1 ⎟ q,q ⎟ Sq,q ⎠ −S

(10.3.4)

q,q

which latter defines the n-step phase-space map as linearized about the initial point q , p . The second of the above equations follows from the generating-function property (10.2.9) of the action; to save on war paint, subscripts on the action denote partial derivatives while the superscripts (n, α) have been and will from here on mostly remain dropped. Due to area conservation, det M = 1, as is indeed clear from the last member in (10.3.4). Therefore, the two eigenvalues of M are reciprocal to one another; if they are also mutual complex conjugates, we confront a stable orbit; positive eigenvalues signal hyperbolic behavior and negative ones signal inverse hyperbolicity. According to (10.3.4), the trace of √ M is related√to derivatives of the√action such that the prefactor takes the form · · · = 1/ | det(M − 1)| = 1/ |2 − TrM|. Hence we may write tn as [17, 18, 23]7 tn ∼



n0 √ e i{S(q,q)/h¯ −μπ/2} . | det(M − 1)|

(10.3.5)

It is quite remarkable that the trace of the n-step propagator is thus expressed in terms of canonically invariant properties of classical period-n orbits. To check on the invariance of S (n,α) (q, q), first, think of a general periodically driven Hamiltonian system with H (p, q, t + τ ) = H (p, q, t), and let a canonical transformation p, q → P , Q be achieved by the generating function F (Q, q) according to P = ∂F (Q, q)/∂Q ≡ FQ and p = −∂F (Q, q)/∂q ≡ −Fq . The action accumulated along a classical path from q to q (equivalently, from Q to Q) during a time span t may be calculated with either set of coordinates, and the most basic #q property of the generating function yields the relation q [pd ˜ q˜ − H (q, ˜ p, ˜ t˜)d t˜] =

6 Should such nongeneric a disaster happen, there is a way out shown by Maslov: one changes to, say, the momentum representation; see next section. 7 Remarkably, the semiclassically approximate equality in (10.3.5) becomes a rigorous equality for the cat map, as was shown by Keating in Ref. [23].

10.3 Trace Formulae and the Density of Levels à la Gutzwiller

423

#Q

˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ Q {P d Q − H [q(Q, P ), p(Q, P ), t ]d t } + F (Q, q) − F (Q , q ); but for a periodic path with period nτ from q to q = q and similarly Q = Q , the difference

F (Q, q) − F (Q , q ) vanishes such that the two actions in question turn out to be equal. A little more labor is needed to verify the canonical invariance of the trace of the ∂Q ∂P stability matrix. Starting from ( ∂P )Q + ( ∂Q )P one thinks of the new phase-space coordinates P , Q as functions of the old ones and uses the chain rule to write  ∂Q  ∂Q

P

 ∂Q   ∂q   ∂Q   ∂p  + ∂q p ∂Q P ∂p q ∂Q P   ∂q   ∂p    ∂Q   ∂q   ∂q  + = ∂q p ∂q p ∂Q P ∂p q ∂Q P   ∂p   ∂p    ∂Q   ∂p   ∂q  + + ∂p q ∂q p ∂Q P ∂p q ∂Q P

=

(10.3.6)

and similarly for the second term in the trace. Upon employing the generating function F (Q, q, t), we get dP = FQQ dQ + FQq dq, dp = −FqQ dQ − Fqq dq and thus the Jacobian matrices (in analogy to (10.3.4)) ⎛ ⎛ ⎞ ⎞ Fqq F 1 1 − FqQ − FQQ − FqQ FQq ∂(Q, P ) ⎝ ∂(q, p) Qq ⎠, ⎠. = F F −F F = ⎝ F F −F F Fqq FQQ QQ qq qQ Qq qQ Qq QQ qq ∂(q, p) ∂(Q, P ) − − FqQ FqQ FQq FQq With these Jacobians inserted in (10.3.6) and the corresponding ∂P /∂P and realizing that for a periodic point q = q , p = p as well as Q = Q , P = P we immediately conclude that  ∂Q  ∂Q P

+

 ∂P  ∂P Q

=

 ∂q  ∂q p

+

 ∂p  ∂p

q

,

(10.3.7)

i.e., the asserted canonical invariance of the trace of the stability matrix of a periodic orbit. The more difficult proof of the canonical invariance of the Maslov index will be discussed in Sect. 10.4. At this point we can partially check, that in the trace formulae (10.3.2), (10.3.5), the n0 distinct points along an orbit with primitive period n0 contribute identically such that when summing over periodic orbits rather than periodic points we encounter the primitive period as a factor: According to (10.2.8) the function S (n,α) (qi , qi ) differs for the n0 points qi only in the irrelevant order of terms in the sum over single steps. The square root in the prefactor must also be the same for all points on the periodic orbit, simply since the stroboscopic time evolution may itself be seen as a canonical transformation and since the trace of the stability matrix is a canonical invariant, as shown right above. Similarly, once we have shown that the

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10 Semiclassical Roles for Classical Orbits

Maslov index is a canonical invariant, we can be sure of its independence of the periodic points along a periodic orbit. Inspection of (10.2.8) also reveals that the contribution of a period-n orbit with shorter primitive period n0 = n/r may be rewritten as S (n,α) = rS (n0 ,α) . As regards the prefactor for a repeated orbit of primitive period n0 = n/r, we may invoke the meaning of M as the map linearized about the orbit to conclude the multiplicativity M (n) = (M (n0) )r . Once again, the Maslov index is more obstinate than the other ingredients of the trace formula; its additivity for repeated traversals, μ(n,α) = rμ(n0 ,α) ≡ rμα , which unfortunately holds only for unstable orbits, will be shown in Sect. 10.4. The Maslov index for stable orbits, as appear for systems with mixed phase spaces, need not be additive with respect to repeated traversals. Now, the quasi-energy density is accessible through the Fourier transform (φ) =

∞ ∞    1 1  N + 2 Re tn einφ = tn einφ , 2πN n=−∞ 2πN

(10.3.8)

n=1

where a finite dimension N is assumed for the Hilbert space. We immediately proceed to a spectral average of the product of two densities, (φ +



2π

2π dφ (φ + e)(φ) 2π N 0  2  ∞ 1 2π = |tn |2 exp{i n e} 2πN n=−∞ N

2π e)(φ) = N

 =

1 2πN

2 

N +2 2

∞  n=1

|tn |2 cos (n

(10.3.9) 2π e) . N

This can be changed into the two-point cluster function in the usual way.8 It may be well to emphasize that a smooth two-point function results only after one more integration since the densities provide products of two delta functions. The form factor |tn |2 is such an integral and thus free of delta spikes (but still displays wild fluctuations in its n-dependence (see Sect. 5.20)). A cheap way of smoothing the above two-point function is to truncate the sum over n at some finite nmax . Already in the quasi-energy density (10.3.8), that truncation will “regularize”, roughly as δ(φ − φi ) → sin[(nmax + 12 )(φ − φi )]/ sin[ 12 (φ − φi )]. To fully define the spectrum within the semiclassical treatment we are pursuing we would need all periodic orbits with periods between 1 and N/2 since the traces tn with larger n can be determined with the help of self-inversiveness, Newton’s formulae, and the Hamilton Cayley theorem, as discussed in Sect. 5.15.

8 Subtract

and then divide by the product (1/2π)2 of two mean densities, take out the delta function provided by the self-correlation terms, and change the overall sign.

10.3 Trace Formulae and the Density of Levels à la Gutzwiller

425

The most important information about the specifics of a given dynamic system with global classical chaos is expected to be encoded in short periodic orbits, since as the period n grows to infinity, typical orbits will follow the universal trend to ergodic exploration of the phase space. Such classical universality corresponds to the quantum universality of the spectral fluctuations on the (quasi)energy scale of the mean nearest-neighbor spacing. The expectations just formulated constitute a strong motivation to seek semiclassical justification of the success of randommatrix theory. Now, the frame is set for a discussion of such efforts.

10.3.2 Flows Gutzwiller suggests [3] proceeding toward the energy spectrum through the energydependent propagator defined by the one-sided Fourier transform G(q, q , E) =

1 ih¯



∞

dt eiEt /h¯ q|e−iH t /h¯ |q  = q|

0

1 |q  . E−H

(10.3.10)

To ensure convergence, the energy variable E here must be endowed with a positive imaginary part. Indeed, the trace of the imaginary part of G(q, q , E) yields the density of levels as ImE → 0+ . After inserting the Van Vleck propagator (10.2.25), it is natural again to invoke the semiclassical limit and evaluate the time integral by a stationary-phase approximation. The new stationarity condition is E+

∂S α (q, q , t) = E − H (q, q , t) = 0 . ∂t

(10.3.11)

Note that the Hamiltonian function here does not appear with its natural vari ables therefore is time-dependent through H (q, p) = H q, p(q, q , t) =  and  H q , p (q, q , t) = H (q, q , t). The phase of the exponential in our time integral is stationary for the set of times solving (10.3.11). A new selection of classical paths is so taken since besides q and q , it is now the energy E rather than the time span t that is prescribed. Just as we could have labelled the paths contributing to the time-dependent propagator by their energy, we could so employ their duration now. Actually, one usually does neither but rather puts some innocent looking label on all quantities characterizing a contributing path. Calling t α (q, q , E) the roots of the new stationary-phase condition (10.3.11) and confining ourselves to the usual quadratic approximation in the expansion of the phase in powers of t − t α , we get .     ∂ 2 S α (q, q , t α )   i 1  exp [Et α + S α (q, q , t α )] − iν α π Gosc (q, q , E) =   h¯ ih¯ α ∂q∂q 2  ∞  i dt exp × (10.3.12) S¨ α (q, q , t α )(t − t α )2 √ 2h¯ i2π h¯ 0

426

10 Semiclassical Roles for Classical Orbits

.  1 ∂ 2 S α (q, q , t α )  1   =  S¨ α (q, q , t α )  ih¯ α ∂q∂q i π × exp [Et α + S α (q, q , t α )] − iκ α 2 h¯ where the double dot on S¨ α (q, q , t) requires two partial differentiations with respect to time at constant q and q . The “osc” attached to Gosc (q, q , E) signals that the Gaussian approximation about the times t α of stationary phase cannot do justice to very short paths where t α is very close to the lower limit 0 of the time integral; their contributions will be added below under the name G such that eventually G ∼ G + Gosc . The Morse index ν α counts the number of conjugate points along the αth path up to time t α , while for the successor κ α in the last member of the foregoing chain, by appeal to the Fresnel integral (10.2.12), κ α = να +

 + 1* 1 − sign S¨ α (q, q , tα ) . 2

(10.3.13)

We shall come back to that index in the next section and refer to it as the Morse index of the energy-dependent propagator; we shall see that it is related to the number of turning points (points where the velocity q(t) ˙ changes sign) passed along the orbit. Now it is more urgent to point to another feature of the exponential, the appearance of the time-independent action S0 (q, q , E) = S(q, q , t) + Et

(10.3.14)

which may be seen as related to the original action by a Legendre transformation. It is natural to express the preexponential factor in the last member of (10.3.12) in terms of S0 and its derivatives with respect to its “natural variables” as well. One obtains9 i π 1  α , A (q, q , E) exp S0α (q, q , E) − iκ α ih¯ α 2 h¯ G ⎛ ⎞ H ∂ 2 S0α ∂ 2 S0α  H H ⎜∂q∂q ∂q∂E⎟ (10.3.15) Aα (q, q , E) = H Idet⎝∂ 2 S α ∂ 2 S α⎠ 0 0   ∂E∂q ∂E 2

Gosc (q, q , E) =

by the usual prestidigitation of changing variables; here, the transformation t α → E according to a solution t α (q, q , E) of the stationary-phase condition; to avoid

9 The

structures of (10.2.25) and (10.3.15) are similar; one might see the difference as the result of an extension of the phase space by inclusion of the pair E, t.

10.3 Trace Formulae and the Density of Levels à la Gutzwiller

427

notational hardship, we wave good-bye to the index α, write S 0 for S0α , and denote partial derivatives of S(q, q , t), S 0 (q, q , E), t (q, q , E), and E(q, q , t) by suffixes. Starting with the stationary-phase condition (10.3.11), we note the first derivatives of the two action functions, St = −E SE0 = St tE + t + EtE = t

(10.3.16)

Sq0 = Sq + St tq + Etq = Sq = p ,

Sq0 = Sq = −p .

The first of the foregoing identities, together with (10.3.14), constitutes the Legendre transformation from the time-dependent to the energy-dependent action, inasmuch as it yields the time t (q, q , E) taken by a classical phase-space trajectory from q to q on the energy shell; the third line in (10.3.16) reveals that the energy-dependent action is the generating function for the classical transition on the energy shell, p = ∂S 0 (q, q , E)/∂q and similarly p = −∂S 0 (q, q , E)/∂q . Now, we fearlessly 0 0 proceed to the second derivatives, starting with Sq,q = Sq,q + Sq,E Eq ; but here Eq can be expressed in terms of other second derivatives of S 0 since dt = tq dq + 0 /S 0 such that tq dq + tE dE at constant q and t yields Eq = −tq /tE = −SEq EE indeed Sqq is expressed through derivatives of S 0 as 0 Sqq = Sqq −

0 S0 SqE Eq 0 SEE

.

(10.3.17)

0 Finally, St t = −Et = −1/tE = −1/SEE , whereupon the radicand under study,

Sqq 0 0 0 0 = SqE SEq − SEE Sqq , St t

(10.3.18)

assumes the asserted determinantal form. Inasmuch as dynamics with nonintegrable classical limits are to be included we must generalize to f degrees of freedom, and f ≥ 2. The change is but little for G(q, q E): The determinant in the prefactor A becomes (f + 1) × (f + 1), with ∂ 2 S/∂q∂q an f × f matrix bordered by an f -component column ∂ 2 S/∂q∂E to the √ f −1 arises, as right and a row ∂ 2 S/∂E∂q from below.10 An overall factor 1/i2π h¯ is easy to check by going back to (10.2.3) where the free-particle propagator in √ f f dimensions comes with a prefactor m/i2π h¯ τ ; of that, the part m/τ is eaten up for good by the f -dimensional generalization of (10.2.19) such that the fate of √ f 1/i2π h¯ must be followed; the time integral in (10.3.12) takes away one of the

10 While this generalization is suggested by the very appearance of the prefactor in (10.3.15) and the previous footnote, and thus easy to guess, serious work is needed to actually verify the result.

428

10 Semiclassical Roles for Classical Orbits

√ f factors 1/ i2π h¯ ; these remarks should also prepare the reader for the further changes in the prefactor toward the final trace formula. we shall not bother with arbitrary f but rather specialize to f = 2 from here on since this is enough for most applications known and also simplifies the work to be done. Now, we must attack the trace   1 0 2 (10.3.19) d x Gosc (x, x, E) = √ d 2x A ei{S /h¯ −κπ/2} ih¯ i2π h¯ α by doing the twofold configuration-space integral, surely invoking the stationary∂ phase approximation. The stationary-phase condition, ( ∂x + ∂x∂ )S 0 (x, x )|x=x = p − p = 0, again restricts the contributing paths to periodic ones. But in contrast to the period-n orbits encountered in the previous subsection, which were a sequence of n points (qi , pi ) with contributions to be summed over, now we encounter a continuous sequence of periodic points along each periodic orbit. The sum over their contributions takes the form of a single integral along the orbit, and that integral cannot be simplified by any Gaussian approximation of the integrand. It is only the integration transverse to a periodic orbit for which the stationary-phase approximation invites a meaningful Gaussian approximation. We shall specify the two configuration-space coordinates summarily denoted by x above as one along the orbit, x, , and a transverse one, x⊥ ; the transverse coordinate is kin to the one and only one incurred for maps with f = 1 in the preceding subsection, cf (10.3.1). The two coordinates x, , x⊥ are assumed orthogonal such that the corresponding axes necessarily change along the orbit and d 2 x = dx, dx⊥ ; moreover, the x⊥ axis is taken as centered at the orbit, i.e., x⊥ = 0 and x˙⊥ = 0 thereon. Such a choice helps with the twofold integral over x, and x⊥ , first by giving a simple form to the 3 × 3 determinant in the prefactor A. To see this, we take various derivatives of the conservation law H (q, p) = E, H (q , p ) = E, considering the canonical momenta as well as the action S 0 (q, q , E) as functions of q, q , and E; also observing q = (x, , x⊥ ), p = (p, , p⊥ ), and ∂H /∂p⊥ = x˙⊥ = 0 (and similarly for the primed quantities), we get  ∂H (q, p) ∂E q   ∂H (q , p ) ∂E q   ∂H (q , p ) ∂x⊥ q ,E   ∂H (q , p ) ∂x, q ,E   ∂H (q, p) ∂x⊥ q,E 

=1 =

∂p, ∂H (q, p) ∂p, = x˙ , = x˙, Sx0, E , ∂p, ∂E ∂E

= 1 = x˙, = 0 = x˙, = 0 = x˙, = 0 = x˙,

∂p, ∂E ∂p, ∂x⊥ ∂p, ∂x,

= −x˙, Sx0 E , ,

= −x˙, Sx0 x , , ⊥

= −x˙, Sx0 x , , ,

∂p, 0 , = x˙ , Sx, x⊥ ∂x⊥

(10.3.20)

10.3 Trace Formulae and the Density of Levels à la Gutzwiller

429

where on the left-hand sides the suffixes attached to the closing brackets display the variables to be held constant while taking derivatives. Thus, the prefactor A from (10.3.15) becomes G ⎛ ⎞ G H  0 0 1/x˙,  H H 2S 0  H ⎜ H 1 ∂ 0 0  ⎟  0 Sx x Sx⊥ E⎠ = I A(q, q , E) = H I det⎝ ⊥ ⊥ ∂x ∂x  ,   x ˙ x ˙ , , ⊥ ⊥ 0 0   −1/x˙, SEx SEE

(10.3.21)

⊥

whereupon the trace (10.3.19) appears in the form  d 2x Gosc (x, x, E) = ,

dx, |x˙, |



dx⊥ √ i2π h¯

1  ih¯ α

.   ∂ 2 S 0 (x⊥ ,x, ,x⊥ ,x, ,E)      ∂x⊥ ∂x⊥

(10.3.22)

x⊥ =x⊥

0 e i{S (x, x, E)/h¯ − κπ/2} .

Integrating over x⊥ with the usual quadratic approximation in the exponent, )2 S(x , x , x , x , E)] 2 =0 x /2, we S 0 (x, x, E) = S(E) + [(∂/∂x⊥ + ∂/∂x⊥ ⊥ , ⊥ , x⊥=x⊥ ⊥ get 

.   ∂ 2 S(x⊥ ,x, ,x⊥ ,x, ,E)  dx⊥   √  ∂x⊥ ∂x⊥ i2π h¯  x⊥ =x⊥ G  H 2 H ∂ S(x⊥ ,x, ,x⊥ ,x, ,E)/∂x⊥ ∂x⊥  ∼ I ∂   ( ∂x⊥ + ∂x∂ )2 S(x⊥ ,x, ,x⊥ ,x, ,E)  ⊥

e iS(x, x, E)/h¯

(10.3.23)

e i{S(E)/h¯ − μtraceπ/2}

=0 x⊥ =x⊥

with the index ν according to the Fresnel integral (10.2.12), μtrace =

1 2 (1 − sign{[(∂/∂x⊥ + ∂/∂x⊥ ) S(x, x, , x , x, , E)]x=x =0 }) . 2

(10.3.24)

Note that we have somewhat sloppily written S(E) for S 0 (0, x, , 0, x, , E) thus expressing the fact that the action is independent of the coordinate x, along the periodic orbit and judging redundant the superscript “0” once writing the energy as an argument of the action. The index μtrace resulting from the trace operation combines with the Maslov index κ from the energy-dependent propagator to the full Maslov index of the periodic orbit μ = κ + μtrace ≡ μprop + μtrace .

(10.3.25)

In further implementing the restriction to periodic orbits, it is well to realize that ) of the action taken at x = x = 0 the transverse derivatives (w.r.t. x⊥ and x⊥ ⊥ ⊥

430

10 Semiclassical Roles for Classical Orbits

are also independent of x, . If we take for granted, subject to later Dproof, that the full Maslov index μ is x, -independent as well, the final x, -integral dx, /x˙, = T yields the primitive period of the orbit; it is the primitive period since the original configuration-space integral “sees” the orbit as a geometrical object without noticing repeated traversals. Thus, the trace we pursue becomes  d 2x Gosc (x, x, E) = G  H 1  H ∂ 2 S(x⊥ ,x, ,x⊥ ,x, ,E)/∂x⊥ ∂x⊥  T I ∂   ( ∂x⊥ + ∂x∂ )2 S(x⊥ ,x, ,x⊥ ,x, ,E)  ih¯ α ⊥

(10.3.26)

e i{S(E)/h¯ − μπ/2} . =0 x⊥ =x⊥

This trace formula looks remarkably similar to that obtained for maps with f = 1; see (10.3.2). In particular, the radicands in the prefactors are in complete correspondence such that by simply repeating the arguments of the previous subsection we express the present radicand in terms of the 2 × 2 stability matrix11 for a loop around the αth periodic orbit which should be envisaged as giving a map in a Poincaré section spanned by x⊥ , p⊥ . The final result for our trace reads, in analogy to (10.3.5),  d 2x Gosc (x, x, E) =

T 1  √ ih¯ α | det(M − 1)|

e iS(E)/h¯ ≡ gosc(E)

(10.3.27)

where for notational convenience the “action” S(E) ≡ S(E) − h¯ μπ/2 is defined to include the Maslov phase. Upon taking the imaginary part, we arrive at the oscillatory part of the density of levels12 osc (E) =

1  T cos (S/h¯ ) . √ π h¯ |det(M − 1)|

(10.3.28)

It is appropriate to emphasize that orbits which are r-fold traversals of shorter primitive ones contribute with their primitive period T while the stability matrix M is the rth power of the one pertaining to the primitive orbit, just as for maps. It is well to note that the above final results for the oscillatory parts gosc (E) of the Green function and osc of the level density also hold for more than two degrees of freedom [2].

11 Note that in contrast to the preceding chapter we here use the same symbol M for the stability matrix of maps and flows. 12 Departing from the convention mostly adhered to in this book, here we do not restrict ourselves to Hilbert spaces with the finite dimension N and thus do not write a factor 1/N into the density of levels; the reader is always well advised to check from the context on the normalization of (E).

10.3 Trace Formulae and the Density of Levels à la Gutzwiller

431

10.3.3 Weyl’s Law #∞ An integral of the form τ dtA(t)eiνΦ(t ) with large ν draws its leading contributions not only from the “points” tα of stationary phase but also from near the boundary τ of the integration range. The pertinent asymptotic result is [24] 

∞

dtA(t)eiνΦ(t ) ∼

τ

iA(τ )eiνΦ(τ ) + {stationary-phase contributions} . ˙ ) ν Φ(τ (10.3.29)

The one-sided Fourier transform defining the energy-dependent propagator in (10.3.10) is such an integral with 1/h¯ as ν and τ → 0. While the stationaryphase contributions were evaluated in the preceding subsection as Gosc (E), now we turn to the boundary contribution G(E). Employing the Van Vleck form of the time-dependent propagator, we run into the phase Φ(t) = Et + S(q, q , t) and its ˙ derivative Φ(t) = E − H (q, q , t). Thus, the boundary term reads 

 d fq G(q, q, E) ∼ lim

τ →0

d fqd fq

δ(q − q ) ˆ q|e−iH τ/h¯ |q  E − H (q, q , τ ) (10.3.30)

ˆ

where for brevity q|e−iH τ/h¯ |q  stands for the Van Vleck propagator; the reader will recognize Hˆ as the Hamiltonian operator and distinguish it from the Hamiltonian function H (q, q , τ ) appearing in the denominator. To see that the foregoing (τ → 0)-limit is well defined (rather than a bewildering 0#× ∞), we represent ∞ the delta function by a Fourier integral, δ(q) = (2π h¯ )−f −∞ d fp e−ipq/h¯ , and think of the q -integral as done by stationary phase. The stationary-phase condition ∂ ∂q (S(q, q , τ ) + pq ) = p − p (q, q , τ ) = 0 restores the natural variables q, p to the Hamiltonian function in the denominator, such that we may write 

1 τ →0 hf



d fq G(q, q, E) ∼ lim

d fqd fq d fp



e−ip(q−q )/h¯ ˆ q|e−iH τ/h¯ |q  . E − H (q, p) (10.3.31)

Note that while taking advantage of the benefit H (q, q , t) → H (q, p) of the aforementioned stationary-phase approximation, we have otherwise still refrained from integrating over q . Such momentary hesitation is not necessary but convenient ˆ since after first taking the limit τ → 0 to get q|e−iH τ/h¯ |q  → δ(q − q ), the q integral becomes trivial. Thus, the desired result  d fq G(q, q, E) ∼

1 hf

 d fqd fp

1 E − H (q, p)

(10.3.32)

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10 Semiclassical Roles for Classical Orbits

is reached. Providing the energy E with a vanishingly small imaginary part and using the familiar identity Im{1/(x − i0+ )} = πδ(x), we arrive at Weyl’s law, (E) ∼

1 hf

 d fqd fp δ(E − H ) ,

(10.3.33)

which expresses the nonoscillatory alias average density of states as the number of Planck cells contained in the classical energy shell. Following common usage, we refer to (E) also as the Thomas–Fermi density. Variants of and corrections to Weyl’s law can be found in Refs. [25–28]. Such corrections are not only interesting in their own right but become indispensable ingredients for semiclassical determinations of spectra for systems with more than two freedoms [29].

10.3.4 Limits of Validity and Outlook Most obvious is the limitation to the lowest order in h¯ , due to the neglect of higher orders when approximating sums and integrals by stationary phase. Readers interested in corrections of higher order in h¯ may consult Gaspard’s work [30] as well as the treatments of three-dimensional billiards by Primack and Smilansky [29] and Prosen [31].Interestingly, there are dynamical systems for which the trace formulae become rigorous rather than lowest-order approximations. Prominent among these are the cat map [23], billiards on surfaces of constant negative curvature [32], and graphs [33]. Inasmuch as all periodic orbits are treated to make independent additive contributions to the trace, one implicitly assumes that the orbits are isolated. Strictly speaking, one is thus reduced to hyperbolic dynamics with only unstable orbits. In practice, systems with stable orbits and mixed phase spaces can be admitted, provided one stays clear of bifurcations. Near a bifurcation, the periodic orbits involved, about to disappear or just arisen, are nearly degenerate in action and must be treated as clusters; that necessity is signalled by the divergence of the prefactor A in the single-orbit contribution AeiS/h¯ , due to the appearance of eigenvalues unity in the stability matrix. Some more remarks on mixed-phase spaces will be presented in Sect. 10.12. The greatest worry is caused by phase-space structures finer than a Planck cell, dragged into the trace formulae by orbits with long periods. Imagine a stroboscopic map for a system with a compact phase space with total volume Ω, a Hilbert space of finite dimension N, and a Planck cell of size hf ≈ Ω/N. Since the number of periodic orbits with periods up to n grows roughly like eλn , a Planck cell, on average, begins to contain more than one periodic point once the period grows larger than the Ehrenfest time, nE = λ−1 ln (Ωh−f ), and is crowded by exponentially many periodic points for longer periods. For maps with finite N, one escapes the worst consequences of that exponential proliferation by taking the periodic-orbit version of the tn only for 1 ≤ n ≤ N/2 and using unitarity, Newton’s formulae, and the

10.3 Trace Formulae and the Density of Levels à la Gutzwiller

433

Hamilton–Cayley theorem to express all other traces in terms of the first N/2 traces, as explained in Sect. 5.17.2. In particular, the N eigenphases can be obtained once the first N/2 traces tn are known. The proliferation seems more serious for the trace formulae (10.3.27), (10.3.28) for flows since orbits of all periods, even infinitely long ones, are involved. Much relief was afforded by a series of papers by Berry and Keating that culminated [34– 36] in a resummation, relating the contributions of long orbits to those of short ones in a certain functional equation to be presented in Sect. 10.5. It is interesting to see that convergence may be enforced for the trace formula (10.3.27) by letting the energy E become complex, E = E + i with [37],  = Im E > c ≡ h¯ λ/2 ,

(10.3.34)

where λ is the Lyapunov exponent that roughly gives the eigenvalues of the stability matrix M as e±λT . Indeed, assuming all orbits unstable and recalling that exponential proliferation implies that the number of orbits with periods near T is roughly eλT , we conclude that √ the contributions of long orbits run away as {# of p.o.’s with periods near T}/ det M ≈ eλT /2 ; but the addition of a classically small imaginary part to the energy ( of order h) ¯ provides an exponential attenuation exp ImS(E + i)/h¯ = e−T /h¯ sufficient to overwhelm the exponential runaway if  > c . Deplorably, an imaginary part Im E = O(h¯ ) wipes out all structure in the density of levels on the scale of a mean level spacing since the latter is, according to Weyl’s law, of the order h¯ f , i.e., already smaller than the convergence-enforcing c for the smallest dimension of interest, f = 2 for autonomous systems. An interesting attempt at unifying the treatment of maps and autonomous flows and at fighting the divergence due to long orbits was initiated by Bogomolny [38]. It is based on semiclassically quantizing the Poincaré map for flows. Similar in spirit and as fruitful is the scattering approach of Smilansky and coworkers [39]; these authors exploit a duality between bound states inside and scattering states outside billiards. Respectful reference to the original papers must suffice here, to keep the promise of steering a short course through semiclassical terrain. As yet insufficient knowledge prevents expounding yet another promising strategy of avoiding divergences for flows in compact phase spaces which would describe flows through stroboscopic maps. One would choose some strobe period T and aim at the traces tn = Tre−inT H/h¯ with integer n up to half the Heisenberg time. Thus, the unimodular eigenvalues of e−iH T /h¯ are accessible, and the periods of the classical orbits involved not exceeding half the Heisenberg time. The strobe period T would have to be chosen sufficiently small such that the energy levels yield phases Ei T /h¯ fitting into a single (2π)-interval. To efficiently determine spectra from semiclassical theory, we must learn how to make do with periodic orbits much shorter than half the Heisenberg time. Interesting steps in that direction have been taken by Vergini [40].

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10 Semiclassical Roles for Classical Orbits

10.4 Lagrangian Manifolds and Maslov Theory 10.4.1 Lagrangian Manifolds To bridge some of the gaps left in the proceeding two sections and to get a fuller understanding of semiclassical approximations, we pick up a new geometrical tool, Lagrangian manifolds [22]. These are f -dimensional submanifolds of a (2f )dimensional phase space to whose definition we propose to gently lead the willing here. We shall eventually see that these hypersurfaces have “generating functions” that are just the actions appearing in semiclassical wave functions or propagators of the structure AeiS/h¯ . Therefore, Lagrangian manifolds provide the caustics at which semiclassical wavefunctions diverge. Above its q-space caustics, one may climb up or down a Lagrangian manifold while appearing to stay put in q-space. A simple such manifold was encountered in Fig. 10.2 for f = 1 as the set of all points with a single fixed value of the coordinate and arbitrary momentum; it naturally arose as the manifold specified by the initial condition of the propagator, q|e−iH t |q t =0 = δ(q − q ). Inasmuch as all points on that manifold project onto a single point in q-space, the latter point is a highly degenerate caustic. The image of that most elementary Lagrangian manifold under classical Hamiltonian evolution at some later time t also qualifies as Lagrangian. For f ≥ 1, the foregoing examples are immediately generalized to all of momentum space above a single point in configuration space and the images thereof under time evolution. These may be seen as f -component vector fields p(q) defined on the f -dimensional configuration space, but not all vector fields qualify as momentum fields for classical Hamiltonian dynamics. Inasmuch as phase space allows for reparametrization by canonical transformations, we may think of the admissible momentum fields as gradients p = ∂S/∂q of some scalar generating function S, and such vector fields are curl-free, ∂pi /∂qj = ∂pj /∂qi . Curl-free vector fields are Lagrangian manifolds albeit not the most general ones. It has already become clear in the preceding section that the initial manifold specified by the propagator may and in general does develop caustics under time evolution, i.e., points in configuration space where some of the derivatives ∂pi /∂qj become infinite and in some neighborhood of which the momentum field is necessarily multivalued (see Fig. 10.2 again). Therefore, the proper definition of Lagrangian manifolds avoids derivatives: An f -dimensional manifold in phase space is called Lagrangian if in any of its points any two of its (2f )-component tangent vectors, δz1 = (δq 1 , δp1 ) and δz2 = (δq 2 , δp2 ), have an antisymmetric product ω(δz1 , δz2 ) ≡ δp1 ·δq 2 −δp2 ·δq 1 which vanishes, ω(δz1 , δz2 ) ≡ δp1 · δq 2 − δp2 · δq 1 = 0 ;

(10.4.1)

here the dot between two f -component vectors means the usual scalar product, like

f δp1 · δq 2 = i=1 δpi1 δqi2 . We leave to the reader as Problem 10.7 to show that the

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antisymmetric product ω(δz1 , δz2 ) is invariant under canonical transformations, i.e., independent of the phase-space coordinates used to compute it. We can easily check that the definition (10.4.1) comprises all of the aforementioned examples. For f = 1, the antisymmetric product ω(δz1 , δz2 ) measures the area of the parallelogram spanned by the two vectors in phase space; requiring that area to vanish means restricting the two two-component vectors in question so as to be parallel to one another; indeed, then, we realize that all curves p(q) are Lagrangian manifolds for single-freedom systems since all tangent vectors in a given point are parallel. For f > 1, one may use the antisymmetric product ω(δz1 , δz2 ) to define phase-space area in two-dimensional subspaces of the (2f )dimensional phase space. Moreover, we can consider two tangent vectors such that δz1 has only a single nonvanishing q-ish component, say δqi , and likewise only a single nonvanishing p-ish one, δpj , while, similarly, δz2 has only δqj and δpi as nonvanishing entries. For the f -dimensional manifold in question to be Lagrangian, we must have δpj δqj = δpi δqi ; now we may divide that equation by the nonvanishing product δqi δqj and conclude that curl-free vector fields yield Lagrangian manifolds. Finally, we check that {all of p-space above a fixed point in q-space} (as distinguished by the initial condition of the propagator in the q-representation) as well as {all of q-space above a fixed point in p-space} (distinguished by the initial condition of the propagator in the p-representation) fit the definition (10.4.1). Then, the image of the latter manifold under time evolution is Lagrangian as well since time evolution is a canonical transformation. It may be well to furnish examples for concrete dynamical systems. The simplest is the harmonic oscillator with f = 1 where the propagator in the coordinate representation distinguishes the straight line q = q in the phase plane as the initial manifold L . The time-evolved image L arises through rotation about the origin by the angle ωt; once every period, at the times nπ/ω with n = 2, 4, 6, . . ., L coincides with L . Clearly, that initial Lagrangian manifold is a caustic and even highly degenerate. One more such caustic per period arises at q = −q , at the times nπ/ω with n = 1, 3, 5, . . .. None of these caustics corresponds to turning points unless the initial momentum is chosen as p = 0. The caustics at q = ±q define mutual conjugate points in coordinate space since all trajectories fanning out of them refocus there again (and again). A less trivial example is presented in Fig. 10.3 which refers to a kicked top with a two-dimensional spherical phase space and depicts an initial Lagrangian manifold spanned by all of configuration space at fixed momentum. A type of Lagrangian manifold not contained in the set of examples given above arises when dealing with the energy-dependent propagator (10.3.15); we shall denote such manifolds by LE . Within the (2f )-dimensional phase space is the (2f − 1)-dimensional energy shell, wherein upon picking a point q in coordinate space, we consider a sub-manifold of f − 1 dimensions as well as its time-evolved images for arbitrary positive times. In the course of time, these time-evolved images sweep out an f -dimensional manifold that we recognize as Lagrangian as follows. For the sake of simplicity, we spell out the reasoning for f = 2 for which

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10 Semiclassical Roles for Classical Orbits

Fig. 10.3 Temporal successors of the line p = 0 (a Lagrangian manifold, see Sect. 10.4) for a kicked top under conditions of near integrability (left column) and global chaos after, from top to bottom, n = 1, 2, 3, 4 iterations of the classical map. The intersections with the line p = 0 define period-n orbits. The beginning of the exponential proliferation of such orbits is illustrated by the right column. Points qc with dp/dq = ∞ give configuration-space caustics and, correspondingly, points pc with dq/dp = ∞ give momentum-space caustics

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Fig. 10.4 Lagrangian manifold LE associated with the energy-dependent propagator for f = 2, depicted as a surface in the three-dimensional energy shell. LE is swept out by the time-evolved images of the initial line of fixed q1 , q2 ; it starts out flat but in general tends to develop ripples and thus caustics. Note that the trajectories lie within LE . Courtesy of Creagh, Robbins, and Littlejohn [41]

Fig. 10.4 provides a visual aid; moreover, we assume a Hamiltonian of the form H = p2 /2m + V (q). The manifold LE to be revealed as Lagrangian originates by time evolution from a set of points along, say, the p1 -direction $ above the picked

point q in the (q1 q2 )-plane; in the p2 -direction, p2 = ± 2m(E − V (q)) − p12 determined by the energy through √ H = E; the latter also restricts √ the set of admitted values of p1 to the interval − 2m(E − V (q)) ≤ p1 ≤ + 2m(E − V (q)). The initial one-dimensional manifold so characterized sweeps out LE as every initial point is dispatched along the trajectory generated by the Hamiltonian H . Within the three-dimensional q1 q2 p1 -space illustrated in Fig. 10.4, LE appears as a sheet which starts out flat but may eventually develop ripples and thus caustics. By complementing the three coordinates with the uniquely determined p2 attached to each point on the sheet, one gets LE as a two-dimensional submanifold of the fourdimensional phase space. Two independent tangent vectors offer themselves naturally at each point on LE : One along the trajectory through the point, δz1 = {q(t; ˙ q , p1 ), p(t; ˙ q , p1 )}δt, and the other, δz2 = {(∂q(t; q , p1 )/∂p1 ), ∂p(t; q , p1 )/∂p1 )}δp1 , pointing toward a neighboring trajectory. Once more, by expressing the fourth components of these vectors in terms of the first three and invoking Hamilton’s equations, we immediately check that their antisymmetric product vanishes, ω(δz1 , δz2 ) = 0, and thus reveals LE as Lagrangian. To acquire familiarity with the technical background of this reasoning, the interested reader will want to write out the pertinent few lines of calculation. The two tangent vectors just mentioned may be seen as elements of a texture covering the Lagrangian manifold LE associated with the energy shell. In particular, the trajectories running along LE climb “vertically” in some momentum direction

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10 Semiclassical Roles for Classical Orbits

Fig. 10.5 While momentarily climbing vertically in LE , a trajectory goes through a caustic that is associated with a turning point. Courtesy of Creagh, Robbins, and Littlejohn [41]

as they pass through a caustic of LE . It follows that a caustic arising in the energydependent propagator corresponds to turning points of the trajectory involved (see Fig. 10.5). This is in contrast to the contributions of classical trajectories to the timedependent Van Vleck propagator for which we have seen that caustics are related to points conjugate to the initial one, rather than to turning points. The statement that the stable manifold Ls of any periodic orbit is Lagrangian is useful for our subsequent discussion of Maslov indices.13 Imagine a point (q0 , p0 ) on Ls and a tangent vector δz = (δq, δp) attached to it. As this tangent vector changes into δz(t) by transport along the trajectory through q0 , p0 , eventually, by the definition of a stable manifold, it must become parallel to the flow vector δzp.o. (t) = (∂H /∂p, −∂H /∂q)p.o. along the periodic orbit whose stable manifold is under consideration. Thus, the antisymmetric product ω(δz(t), δzp.o. (t)) vanishes for t → ∞; but since it is also unchanged in time by Hamiltonian evolution, it must vanish at all times. So Ls is indeed Lagrangian. This statement will turn out to be useful for the discussion of Maslov indices in Sect. 10.4.3 below. While the notion of a Lagrangian manifold is defined without reference to a particular set of coordinates in phase space, their caustics are very much coordinate-dependent phenomena, as is clear already for f = 1 from Fig. 10.6. There we see a “configuration-space” caustic, i.e., a point with diverging ∂p/∂q; the canonical transformation q → P , p → −Q turns the curve p(q) into one P (Q) for which the previous caustic becomes a point with ∂P /∂Q = 0

13 The stable and unstable manifolds of a periodic orbit are defined as the sets of points which the dynamics asymptotically carries toward the orbit as, respectively, t → +∞ and t → −∞; the reader is kindly asked to think for a moment about why Ls , as well as the unstable manifold Lu , are f -dimensional.

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Fig. 10.6 A configuration-space caustic (at qc ) looks level from momentum space, as does a momentum-space caustic (at pc ) from configuration space. Near a caustic a Lagrangian manifold is double-valued

such that no configuration-space caustic appears; instead, we incur what should be called a “momentum-space” caustic inasmuch as the point in question comes with ∂Q/∂P = ∞. Conversely, we could say that in the original coordinates the coordinate-space caustic is not a momentum-space caustic there since ∂q/∂p = 0. For f > 1, a little more care is indicated in explaining caustics. One may uniquely label points on a Lagrangian manifold L by f suitable coordinates ξ1 , ξ2 , . . . , ξf such that q and p become functions q(ξ ) and p, ) on L. Whereever the q(ξ ) are locally invertible functions, ξ = ξ(q), and we can deal with the momenta as a vector field p[ξ(q)]; such invertibility requires that the Jacobian det(∂q/∂ξ ) not vanish. Conversely, (configuration-space) caustics arise where det(∂q/∂ξ ) = 0 since in such points one finds tangent vectors δz = (δq, δp) to L such that δq = (∂q/∂ξ )δξ = 0, even though δξ = 0. For every independent such δξ , the Lagrangian manifold L appears perpendicular to coordinate space, unless it happens that δξ is also annihilated by ∂p/∂ξ . The number of such independent δξ is called the order of the caustic; the most common value of that order is 1, but orders up to f may arise, and the latter extreme case indeed arises for the L distinguished by the initial form of the propagator q|e−iH t /h¯ |q . Again, caustics are phenomena depending on the coordinates used; if one encounters a coordinate-space caustic, one can canonically transform, as qi → Pi , pi → −Qi , for one or more of the f components so as to ban the caustic from coordinate space to momentum space. When dealing with the semiclassical approximation of a propagator, such a change corresponds to switching from the qi -representation to the pi -representation. In a region devoid of configuration-space caustics, a Lagrangian manifold L is characterized by a curl-free momentum field p(q) which in turn can be regarded as the gradient ∂S/∂q of a scalar field S. Taking up Littlejohn’s charming manner of speaking [22], we shall call S the generating function of L; this name reminds us of the other, already familiar role S plays in semiclassical games, that of the generating function of classical time evolution as a canonical transformation. Indeed, in the short-time version of the Van Vleck propagator, S appears in both meanings. As the generating function of L, we may get S from a path-independent integral in

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10 Semiclassical Roles for Classical Orbits

configuration space

 S(q) =

q

(10.4.2)

dx p(x) .

The starting point for that integral is arbitrary in accordance with the fact that the generating function of L is determined only up to an additive constant. Even within the semiclassical form AeiS/h¯ of a wave function such as the short-time version of the Van Vleck propagator, the additive constant in question is of no interest inasmuch as a constant overall phase factor is an unobservable attribute of a wave function. However, if L has caustics such as shown in Figs. 10.6 and 10.7, one may divide it into caustic-free regions separated by the caustics, here each region possesses a different generating function S α (q) and momentum field pα (q) = ∂S α (q)/∂q. In other words, the momentum field is no longer single-valued when caustics are around. Then, semiclassical wave functions take the form of sums over several

α branches, ψ = α Aα eiS /h¯ . From a classical point of view, one might be content for each S α to have an undetermined additive constant. For a wave function only, an overall phase factor is acceptable as undetermined, however; relative phases α between the various Aα eiS /h¯ determine interference between them.Therefore, we can leave open only one additive constant for one of the S α ’s and must uniquely determine all the other S α ’s relative to an arbitrarily picked “first” one. With this remark, we are back to Maslov indices. Fig. 10.7 Two branches (sheets, rather) of a Lagrangian manifold which may be seen as divided by a caustic (a configurationspace caustic here, depicted as the dashed line). Courtesy of Littlejohn [22]

p

p

2

1

L B

A

q

2

q q

1

10.4 Lagrangian Manifolds and Maslov Theory

L' L

Momentum

Fig. 10.8 Trajectory c has reached a caustic at the final time t, while that incident is imminent for orbit 1 and has already passed for orbit 2. Note again that the Lagrangian manifold associated with the time-dependent propagator is transverse to the trajectories. Courtesy of Littlejohn [22]

441

2 c 1

q'

Coordinate

10.4.2 Elements of Maslov Theory In Sect. 4.3, we already mentioned Maslov’s idea to temporarily switch to the momentum representation when a configuration-space caustic is encountered. It is now about time at least to sketch the implementation of that idea. Let us first recall the time-dependent Van Vleck propagator (10.2.25) which in α α general consists of several “branches”, i.e., terms of the structure Aα ei(S /h¯ −ν π/2) . Each of these corresponds to a classical trajectory leading from some point on the initial Lagrangian manifold L at q to a point with coordinate q on the final manifold (See Fig. 10.8). The various such trajectories have different initial and final momenta, p α = −∂S α /∂q and pα = ∂S α /∂q, respectively. But as explained above, for Hamiltonians of the structure H = T + V and sufficiently small times t, only a single term of the indicated structure arises, corresponding to a Lagrangian manifold L still without configuration-space caustics. Now assuming t so large that L has developed caustics, we consider a region in configuration space around one of them. If dealing with f = 1, it suffices to imagine that there are only two branches of the curve p(q) making up L “above” the q-interval in question; these branches are joined together at the phase-space point whose projection onto configuration space is the caustic. As pointed out in the previous subsection, there is no momentum-space caustic around then. For f > 1, we analogously consider a part of L with a single configuration-space caustic and no momentum-space caustic. Then, it is advisable to follow Maslov into the momentum representation where Schrödinger’s equation reads  p2 ∂  ˙˜ ˜ ih¯ ψ(p, t) = + V (ih¯ ) ψ(p, t) . 2m ∂p

(10.4.3)

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10 Semiclassical Roles for Classical Orbits

A semiclassical solution may here be sought through the WKB ansatz ˜ ˜ ˜ ψ(p, t) = A(p, t)eiS(p,t )/h¯ .

(10.4.4)

Inasmuch as this is a single-branch function due to the assumed absence of momentum-space caustics, no problem with Maslov or Morse phases arises. In full analogy to the usual procedure in the q-representation (see Problem 10.1), ˜ we find that to leading order in h¯ the phase S(p, t)/h¯ obeys the Hamilton–Jacobi equation ˜  p2 ∂ S(p, t)  ˙˜ + V (− ) = 0. S(p, t) + 2m ∂p

(10.4.5)

and thus may be interpreted as a momentum-space action. The next-to-leading order yields a continuity equation of the form + ∂ ˜ ∂ * ˜ ˜ |A(p, t)|2 − |A(p, t)|2 V [−S(p, t)] = 0 ; ∂t ∂p

(10.4.6)

here, the force −V provides the velocity in momentum space that makes ˜ ˜ −|A(p, t)|2 V (−S(p, t)) a probability current density in momentum space. Coordinate-space and momentum-space wave functions are of course related by a Fourier transform. In our present semiclassical context, that Fourier transform is to be evaluated by stationary phase, in keeping with the leading order in h¯ to which we are already committed. Therefore, we may write  df p ˜ ψ(q, t) = eipq/h¯ ψ(p, t) (10.4.7) (2π h¯ )f/2 with the single-branch momentum-space wave function (10.4.4). Momentarily setting f = 1, we arrive at the stationary-phase condition ˜ q = −∂ S/∂p

(10.4.8)

which for the envisaged situation has two solutions pα (q) with α = 1, 2. Thus, ˜ the single-branch ψ(p) gives rise to a two-branch ψ(q) which with the help of the Fresnel integral (10.2.12) we find as ψ(q) =

2 

Aα (q)eiS

α (q)−iν α π/2

α=1

˜ α (q)) S (q) = pα (q)q + S(p α

(10.4.9)

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443

G H −1  d 2 S(p)  ˜  α H H   α A (q) = A˜ p (q) I   dp2 

p=p α

να =

.   dpα (q)   ˜ = A(p (q))  dq  α

˜  1 d 2 S(p) dpα (q)  1 1 − sign( ) . ) = 1 + sign( 2 2 dp 2 dq

The two branches of the coordinate-space wave function differ not only by having their own action and prefactor each but also by the relative Maslov alias Morse phase π/2.14 A word and one more inspection of Fig. 10.8 are in order about the classical trajectories associated with the two branches of the wave function ψ(q, t) in (10.4.9); each goes from a point on the initial Lagrangian manifold L to a point on the image L of L at time t. One of them, that labelled “1”, has at some previous moment passed the caustic shown, while the trajectory labelled “2” has not yet gone through the “cliff”; the critical moment tc has arrived for the trajectory designated “c”, when the final point q is conjugate to the initial q . If, as shown in the figure, the slope dp/dq of the Lagrangian manifold at the caustic changes from negative to positive (while passing through infinity), the Maslov alias Morse index of branch “1” is larger by one than that of branch “2”, according to the above expression for ν α . This is in accord with our previous interpretation of the Morse index: The slope dp/dq of L should be written as the partial derivative ∂p(q, q )/∂q when the initialvalue problem pertaining to the propagator is at issue; but when ∂p/∂q changes from negative to positive so does ∂q/∂p and also, since points on L may still be uniquely labelled by the initial momentum p , the derivative ∂q/∂p ; we had seen that ∂q/∂p = 0 is the conjugate-point condition and that the Morse index increases by one when a conjugate point is passed. A similar picture arises for the energy dependent-propagator. The only difference is that the relevant Lagrangian manifold LE has the trajectories lying within itself (rather than piercing, as is the case for the time-dependent propagator and its L). Even more intuitively, then, the trajectory which has climbed through the vertical cliff in LE with the slope changing from negative to positive yields a Maslov/Morse index larger by one than that of the trajectory not yet through the cliff. When attempting to solve the initial-value problem for ψ(q, t) by a WKB ansatz in the coordinate representation, one finds a single-branch solution as long as the Lagrangian manifold L originating from the initial L is free of caustics. The singlebranch ψ(q) diverges at precisely that moment when a caustic of L arrives at the configuration-space point q. A little later, a WKB form of ψ(q, t) arises again but now as a two-branch function. That sequence of events might leave a spectator

14 In this subsection, it would seem overly pedantic to insist on the name Morse phase or index for wave functions or propagators, reserving the name Maslov index for what appears in traces of propagators; as a compromise we speak here of Morse alias Maslov phases for multibranch wave functions or propagators.

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puzzled who is borné to the coordinate representation. Maslov’s excursion into momentum space opens a caustic-free perspective and thus avoids the catastrophic but only momentary breakdown of WKB. Conversely, when in the momentum representation a momentum-space caustic threatens failure of WKB, one may take refuge in coordinate space (representation). At any rate, Maslov’s reasoning explains quite naturally why the breakdown of the coordinate-space Van Vleck propagator at a coordinate-space caustic is only a momentary catastrophe. Incidentally, the divergence of the time-dependent propagator upon arrival of a caustic at q may but does not necessarily imply breakdown of the semiclassical approximation. The harmonic oscillator treated in Problem 10.5 provides an example: The divergence of the semiclassical Van Vleck propagator takes the form of a delta function, G(q, q , tc ) = δ(q + q ) for times tc = (2n + 1)T /2 , n = 1, 2, . . . and G(q, q , tc ) = δ(q − q ) for times tc = nT , n = 1, 2, . . ., i.e., whenever a point conjugate to the initial q is reached; but this behavior faithfully reproduces that of the exact propagator. Conjugacy means simultaneous reunification at q of a oneparameter family of orbits which originated from the common q with different initial momenta and that family of orbits interferes constructively in building the “critical” propagator G(q, q , tc ) = q|e−iH tc /h¯ |q . The occurrence of multibranch WKB functions may be understood as due to several effective caustics. Maslov’s procedure then naturally assigns to each branch its own action branch S α (q) and Maslov alias Morse phase −ν α π/2, leaving open only a single physically irrelevant overall phase factor. It is well to keep in mind that each branch is provided by a separate classical trajectory and that the Maslov index equals the net number of passages of the trajectory through vertical cliffs in the relevant Lagrangian manifold where the slope dp/dq changes from negative to positive; “net” means that a passage in the inverse direction decreases the index by one. To liven up our words a bit, we shall henceforth speak of clockwise rotation of the tangent to L or LE at the caustic when the slope dp/dq changes from negative to positive and of anticlockwise rotation when the slope becomes negative. Thus, the net number of clockwise passages of that tangent through the momentum axis along a single traversal of a periodic orbit gives the Morse alias Maslov index in the contribution of that orbit to the propagator. The above (f =1) result (10.4.9) can be generalized to more degrees of freedom [22, 42]. The essence of the change is to replace |dpα (q)/dq| by | det dpα (q)/dq| ˜ in the amplitude Aα (q). If only a single eigenvalue of the matrix ∂ 2 S(p)/∂p∂p vanishes at the coordinate space caustic, one may imagine axes chosen such that in the above expression for the Maslov index dp(q)/dq → ∂pl (q)/∂qm where the subscripts indicate the components of p and q with respect to which the coordinatespace caustic appears with ∂pl (q)/∂qm = ∞.

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10.4.3 Maslov Indices as Winding Numbers We can finally proceed to scrutinizing an unstable periodic orbit for its Maslov index μ, staying with autonomous systems of two degrees of freedom. The presentation will closely follow Creagh, Robbins, and Littlejohn [41] but Mather’s pioneering contribution [43] should at least be acknowledged. The generalization to arbitrary f can be found in Robbins’ paper [44]. Intuition will be furthered by focusing on the three-dimensional energy shell. Therein the stable (and the unstable) manifold of the periodic orbit is a two-dimensional surface. Now imagine a surface of section Σ, also two-dimensional, transverse to the orbit, and let that Σ move around the periodic orbit. As in deriving the trace formula in Sect. 10.3.2, we shall specify the location of Σ by the coordinate x, along the orbit and parametrize Σ(x, ) by the transverse coordinate x⊥ and its conjugate momentum p⊥ . The stable manifold of the periodic orbit intersects Σ(x, ) in a line σ (x, ) through the origin where the orbit pierces through Σ(x, ); near the origin, σ (x, ) will appear straight. As Σ(x, ) is pushed along the orbit, the stable manifold σ (x, ) will rotate, possibly clockwise at times and then anticlockwise, as may happen. But σ (x, ) must return to itself when Σ(x, ) has gone around the periodic orbit. The net number of clockwise windings of σ (x, ) about the periodic orbit during a single round trip of Σ(x, ) is obviously an intrinsic property of that orbit, independent of the phasespace coordinates employed. We shall see presently that twice the net number of clockwise windings of the stable manifold for one traversal of the periodic orbit in the sense of growing time is the Maslov index μ appearing in the trace formula (10.3.27). With that fact established, the constancy of μ along the periodic orbit (i.e., the independence of x, ) is obvious, as is the additivity with respect to repeated traversals: If μ refers to a primitive orbit, its r-fold repetition has the Maslov index rμ. Thus, the trace formula (10.3.28) for the density of levels may be rewritten by explicitly accounting for repetitions of primitive orbits, ∞ 1   π h¯

T cos r(S/h¯ − μπ/2) , √ |2 − TrM r | r=1 prim.orb.

osc (E) =

(10.4.10)

where T , M, S, and μ all refer to primitive orbits and complete “hyperbolicity” is assumed, i.e., absence of stable orbits. To work, then! We shall have to deal with mapping the “initial” surface of section Σ(x, ) to the “running” one, Σ(x, ), as linearized about the origin (i.e., about the orbit in question),  δx  ⊥

δp⊥

⎡



⎢ =⎢ ⎣



∂x⊥ ∂x⊥ p⊥

∂p⊥ ∂x⊥ p⊥



 ⎤

∂x⊥   ∂p⊥ x ⎥ δx⊥  ⊥⎥ ⎦ ∂p⊥ δp⊥ ∂p⊥ x⊥

=

 a b  δx  cd

⊥ δp⊥

= M(x, , x, )

 δx  ⊥

δp⊥

.

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10 Semiclassical Roles for Classical Orbits

The special case of the matrix M(x, , x, ) pertaining to a full traversal of the periodic orbit is the stability matrix M entering the trace formula (10.3.28); kin to the present M(x, , x, ) is that found for area-preserving maps in (10.3.4); indeed, the close analogy of area preserving stroboscopic maps to mappings between surfaces of section for Hamiltonian flows permeates all of this chapter. Now instead of immediately concentrating on the (section with Σ(x, ) of the) stable manifold σ (x, ), it is convenient first to look at the section λ(x, ) of Σ(x, ) with the Lagrangian manifold LE associated with the energy-dependent propagator. The reader will recall that LE is the two-dimensional sheet swept out by the set of all trajectories emanating from the set of all points within the energy shell projecting to one fixed initial point in configuration space. The line λ(x, ) arises as the image under the mapping M(x, , x, ) of the momentum axis in Σ(x, ) or rather that interval of the momentum axis accommodated in the energy shell. Like σ (x, ), the line λ(x, ) goes through the origin of the axes used in Σ(x, ); close to that origin λ(x, ) looks straight and can thus be characterized by the vector (b, d) in Σ(x, ); that vector even provides an orientation to λ when attached to the origin like a clock hand. we shall refer to the part of λ extending from the origin along the vector b, d as the half line λ+ . Now, recall from (10.3.25) that the full Maslov index of a periodic orbit, μ = μprop + μtrace , gets the contribution μprop from the energy-dependent propagator while the trace operation furnishes μtrace . The first of these has already been given a geometric interpretation in Sect. 10.4.2. The consideration there was worded for a two dimensional phase space and thus fits our present Σ(x, ), and the slope of the Lagrangian manifold here is given by d/b. The net number of clockwise rotations of the vector (b, d) through the momentum axis while one goes around the periodic orbit gives μprop. Only a subtlety regarding the initial Lagrangian manifold, i.e., the momentum axis (rather, the part thereof fitting in the energy shell), needs extra comment. That initial λ(x, ) is as a whole one caustic and one that is not climbed through. Whether or not the index μprop gets a contribution here depends on the sign of ∂p⊥ /∂x⊥ at infinitesimally small positive times, t = 0+ ; if positive, no contribution to the index arises, whereas for negative slope, the index suffers a decrement of 1; this simply follows from our discussions of the Van Vleck propagator. In fact, we already saw that Hamiltonians of the structure H = T + V always give rise to minimal (rather than only extremal) actions for small times, such that the initial caustic is immaterial for μprop there. To proceed toward μtrace , we must determine where the vector (b, d) ends up after one round trip through the periodic orbit. Since we are concerned with an unstable orbit, we can narrow down the final direction with the help of the stable and unstable manifolds whose tangent vectors at the origin are given by the two real eigenvectors es and eu of the 2 × 2 matrix M pertaining to the completed round trip; the respective eigenvalues will be called 1/τ and τ with |τ | > 1. As shown in Fig. 10.9, these directions divide Σ(x, ) into four quadrants, labelled clockwise H, I, J, and K. H is distinguished by containing the upper momentum axis, so the positive-px part of the initial λ(x, ) lies therein. The final vector (b, d) must lie in

10.4 Lagrangian Manifolds and Maslov Theory

447

Fig. 10.9 The surface of section Σ transverse to a periodic orbit is divided into four sectors H, I, J, and K by the (sections with Σ of the) stable and unstable manifolds which look locally straight. At some arbitrarily chosen initial moment, the section λ of the Lagrangian manifold with Σ starts out as the momentum axis; its positive part, the half line λ+ , rotates as the section Σ(x, ) is carried around the periodic orbit and ends up, after one full traversal, either in sector H (hyperbolic case) or in sector J (inverse hyperbolic case). If λ+ ends up in H+ or J+ , μtrace = 0 while arrival in H− or J− implies μtrace = 1. Courtesy of Creagh, Robbins, and Littlejohn [41]

that same quadrant if the orbit is hyperbolic (τ > 2); conversely, for hyperbolicity with reflection (τ < −2), the vector ends up in the opposite quadrant, J . The index μtrace depends on where the vector (b, d) and thus the half line λ+ end up within the quadrants H or J . The definition (10.3.24) implies that μtrace = 0 if the quantity 2 ) S(x⊥ , x, , x⊥ , x, , E)]x⊥ =x⊥ =0 } w ≡ {[(∂/∂x⊥ + ∂/∂x⊥

=

TrM − 2 τ + 1/τ − 2 a+d −2 = = b b b

(10.4.11)

is positive while μtrace = 1 if w < 0. But since |τ + 1/τ | > 2, the signs of τ + 1/τ − 2, τ + 1/τ , and τ all coincide. It follows that the numerator of w is positive when M is hyperbolic and negative if M is hyperbolic with reflection. The denominator b, on the other hand, is positive or negative when the vector (b, d) ends up, respectively, to the right or the left of the momentum axis. Therefore, it is indicated to subdivide the quadrants H and J into the sectors H+ , H− , J+ , and J− , as shown in Fig. 10.9. The negative subscripts denote sectors counterclockwise to the momentum axis while the sectors clockwise of it have positive subscripts. It

448

10 Semiclassical Roles for Classical Orbits

follows that w > 0 if the vector (b, d) ends up in either H+ or J+ since in that case, τ and b have the same sign; conversely, w < 0 if the said vector ends up in either H− or J− , τ and b differs in sign then. What finally matters is that μtrace = 0 if the half line λ+ ends up running through the sectors H+ and J+ while μtrace = 1 if it ends up running through H− and J− . With both μprop and μtrace geometrically interpreted, now we can tackle their sum, the full Maslov index. To that we extend the definitions of the quadrants H, I, J, and K to the intermediate surfaces of section Σ(x, ). We do that by introducing the vectors es∗ (x, ) and eu∗ (x, ) such that their components along the x-axis and the px -axis in Σ(x, ) remain the same as those of es and eu in the initial surface of section Σ(x, ). Note that these starred vectors are not the ones specifying the stable and unstable manifolds in Σ(x, ) through the eigenvectors ∗ of M(x , x ), i.e., e ∗ (x ) = e (x ). Rather, the starred vectors carry the es,u , , s,u , s,u , quadrants H, I, J, and K along in rigid connection with the x- and px -axes, as we go through the sequence of sections Σ(x, ) around the orbit. During that round trip, we may define the current value of μprop as the net number of clockwise passages of the vector (b, d)(x, ) through the momentum axis along the part of the voyage done. A little more creativity is called for in defining intermediate values of μtrace and thus the sum μ since the connection with the trace must be severed. It proves useful to define μtrace = 0 if λ(x, ) and thus the vector (b, d)(x, ) is clockwise of the momentum axis, i.e., in the sectors H+ or J+ and μtrace = 1 if λ(x, ) is anticlockwise of it in the sectors H− or J− . There is no need to define μtrace for the quadrants I and K since, as seen above, λ cannot end up therein. Now the full Maslov index μ behaves quite simply as we go through the orbit. When λ(x, ) sweeps through the quadrants H and J , no change of μ occurs, since whenever λ(x, ) rotates through the momentum axis, the changes of μprop and μtrace cancel in their sum. But μ does increase by one for every completed clockwise passage of the vector (b, d)(x, ) through either quadrant I or K. Finally, we hurry to formulate the result just obtained in terms of the stable manifold σ (x, ). This is necessary since in contrast to the Lagrangian manifold λ(x, ), the stable manifold σ (x, ) is obliged to return to itself after one round trip through the periodic orbit. Incidentally, if λ(x, ) does not coincide with σ (x, ) initially, it cannot do so later either since the map M(x, , x, ) is area-preserving and cannot bring about coincidence of originally distinct directions. This means that λ(x, ) shoves along σ (x, ), though not with a constant angle in between. But both lines must have the same net number of clockwise passages through the quadrants I and K. When σ (x, ) returns to itself after one traversal of the orbit it must have undergone an integer number of “half-rotations” in the surfaces of section Σ(x, ) ; half-rotation means a rotation by π about the origin. During a half-rotation σ (x, ) sweeps through I and K. The number of half-rotations during one round trip through the orbit is the full Maslov index; we might also say, that μ is twice the number of full 2π-rotations if we keep in mind that the number of full rotations is half-integer if the orbit is hyperbolic with reflection.

10.4 Lagrangian Manifolds and Maslov Theory

449

Fig. 10.10 An orbit momentarily coming to rest at a caustic. In this nongeneric case, the coordinates x, , p, are momentarily ill defined. Then, the projection of the orbit onto configuration space displays a cusp. Courtesy of Creagh, Robbins, and Littlejohn [41]

Now the previous claim that the Maslov index is a winding number are shown true. In addition, we conclude that μ is even for hyperbolic orbits and odd for hyperbolicity with reflection. It may be well to remark that odd need not, but may be taken to mean either 1 or 3, and even either 0 or 2, since in the trace formula, μ appears only in a phase factor e−iμπ/2 . Before concluding, a sin must be confessed, committed in employing the specific coordinates x, , p, , x⊥ , p⊥ . Helpful as they indeed are, they may be ill defined at certain moments during the round trip along the periodic orbit. Such instants occur if when arriving at a caustic, the orbit momentarily comes to rest (see Fig. 10.10), such that (not only x˙⊥ = 0 which, by definition, is always the case but also) x˙, = 0. In coordinate space, the orbit then traces out a cusp. In the tip of the cusp, the coordinates x, , x⊥ are indeed ill defined since both axes become inverted there. Our final result for μ holds true, even when such cusps arise as one may check by temporarily switching to the momentum representation à la Maslov where the caustic is absent entirely. While the considerations of the present section establish the canonical invariance of all ingredients of the Gutzwiller trace formula (10.4.10) for Hamiltonian flows of autonomous systems with two degrees of freedom, the question is still open for the case of maps with f = 1. To settle that question, we proceed to show the equivalence, both classical and quantum mechanical, of autonomous dynamics with f = 2 and periodically driven ones with f = 1.

450

10 Semiclassical Roles for Classical Orbits

10.5 Riemann-Siegel Look-Alike Inspired by the famous Riemann-Siegel formula for Riemann’s zeta function, Berry and Keating [34–36] devised a semiclassical representation of the spectral determinant det(E − H ) with Im E ≥ 0 which is convergent and becomes real when the energy argument does, Im E → 0. The essence of the argument lies in the restriction to periodic orbits whose periods are bounded from above by half the Heisenberg time. The resulting Riemann-Siegel look-alike will turn out useful for the demonstration of spectral universality in the later sections. Readers with the strong nerves of physicists are invited to adventure in mathematically unsafe territory, starting with the definition of a zeta function, Hˆ ζ (E) = det{A(E, Hˆ )(E − Hˆ )} =

-

{A(E, Ej )(E − Ej )}

j

= det A(E, Hˆ ) exp Tr ln(E − Hˆ ) .

(10.5.1)

To provide a notational distinction between definitely real and possibly complex energies, we shall denote the former by E and the latter by E. The function A is arbitrary, save for having no real zeros, being real for real E, and ensuring convergence should the Hilbert space dimension N not be finite; elementary examples for such “regularizers” are found in [34] and Problems 10.9, 10.10. Then, the zeta function has the eigenvalues Ej as the only real zeros. The logarithm appearing above can be expressed as the integral of the Green function g(E) = Tr

1

(10.5.2)

E − Hˆ

which yields the level density for E = E + i0+ as (E) = − π1 Im g(E + i0+ ) and thus the level staircase as 1 N (E) = − π



E

dE Im g(E + i0+ ) ;

(10.5.3)

0

for convenience we imagine that the spectrum of Hˆ begins immediately to the right of E = 0. Now, we replace the logarithm, Tr ln(E − Hˆ ) − Tr ln(−Hˆ ) =

 E

dE g(E )

0

=

 E 0

dE gosc (E ) +

(10.5.4)  E

dE g(E ) ,

0

splitting the Green function into an “oscillatory” and a smooth part; the latter is defined as usual by a local spectral average. The smooth part is again split into

10.5 Riemann-Siegel Look-Alike

451

#E #E two pieces such that 0 dE g(E ) = −iπN (E) + { 0 dE g(E ) + iπN (E)}; as indicated by a somewhat cavalier notation, the first of these is defined so as to become −iπN (E) for E = E + i0+ while the second becomes real in that limit; the local spectral average N (E) is the Weyl staircase, i.e., the number of Planck cells contained in the part of phase space with energy up to E, N (E) = (2π h¯ )

−f

 d fqd fp Θ[E − H (q, p)] .

(10.5.5)

Thus, the zeta function assumes the form  E   ζ (E) = B(E) exp −iπN (E) + dE {g(E ) − g(E )} ,

(10.5.6)

0

wherein the prefactor B(E) = det(−AHˆ ) exp

 E

 dE g(E ) + iπN (E)

(10.5.7)

0

becomes real and free of zeros for real E = E. All is now prepared for the intended jump into semiclassical terrain which amounts to invoking the periodic-orbit expansion (10.3.28) of the oscillatory part of the level density, gosc (E) =

∞ e irSp (E)/h¯ 1  Tp $ , ih¯ p | det(Mpr − 1| r=1

(10.5.8)

where r-fold repetitions of primitive periodic orbits p are taken care of explicitly. Since we actually need the integral of gosc we should recognize that periodic-orbit sum as nearly a total energy derivative due to ∂Sp /∂E = Tp (E), ∂ e irSp (E)/h¯ e irSp (E)/h¯ i $ = rTp $ +... ; h¯ ∂E | det(M r − 1| | det(Mpr − 1| p

(10.5.9)

the dots stand for a term involving the derivative of the denominator which is smaller by one order in h¯ than the first term and may be dropped. Upon inserting all of this periodic-orbit stuff into our zeta function from (10.5.6), we get the periodic-orbit approximation ∞     e irSp (E)/h¯  $ . ζ (E) ∼ B(E) exp −iπN (E) exp − r p r=1 r | det(Mp − 1)|

(10.5.10)

452

10 Semiclassical Roles for Classical Orbits

Of the three factors in the zeta function, the last two are of prime importance: the second one, e−iπN (E ), becomes a “Weyl type” phase factor for real energies E; the third one, the exponentiated periodic-orbit sum, contains all semiclassical information about spectral fluctuations. For convergence, the periodic-orbit sum needs to be restricted to complex energies further away from the real energy axis than the limit established at the end of Sect. 10.3.4, Im E > hλ/2. ¯ It is helpful to write the infinite product over periodic orbits and their repetitions in (10.5.10) as a series. To that end, we first do the sum over repetitions. Barring dynamics with stable periodic orbits, we must reckon with eigenvalues of the stability matrix Mp that read15 exp(±λp Tp )

Poincaré map hyperbolic

− exp(±λp Tp )

(10.5.11)

Poincaré map hyperbolic with reflection .

Focussing on the first case we write the determinant det (Mpr − 1) = 2 − Tr Mpr as  2  2 | det(Mpr − 1)| = erλp Tp 1 − e−rλp Tp = erλp Tp /2 − e−rλp Tp /2

(10.5.12)

and expand the stability prefactor in a geometric series, $

1 | det(Mpr

− 1)|

= e−rλp Tp /2

∞ 

e−krλp Tp .

(10.5.13)

k=0

1 1 The sum over repetitions then yields − ∞ ¯ − (k + 2 )λp Tp ] = r=1 r exp r[iSp /h ln (1 − exp [iSp /h¯ − (k + 12 )λp Tp ]). In the zeta function we have thus bargained one product against another, ζ (E) = B(E)e −iπN (E)

∞  --

  1 − exp iSp /h¯ − (k + 12 )λp Tp .

p k=0

(10.5.14) A sum is to come through Euler’s identity ∞ k=0

(1 − ax k ) = 1 +

∞  r=1

(−a)r x r(r−3)/4 (x −1/2 − x 1/2 )(x −1 − x) . . . (x −r/2 − x r/2)

(10.5.15)

15 For simplicity, we specialize to two-freedom systems from hereon; there is only a single Lyapunov exponent λp for the pth orbit then.

10.5 Riemann-Siegel Look-Alike

453

and reads ζ (E) = B(E)e −iπN (E)

4  1 ¯ − 4 r(r − 1)λp Tp r exp irSp /h (−1) 1+ . × 6r j 1/2 p j =1 | det(Mp − 1)| r=1

(10.5.16)

∞ 

The phase factors suggest interpreting the summation variable r as a new repetition number. Upon expanding the product over primitive orbits, we finally arrive at the desired series  ζ (E) = B(E)e −iπN (E) FP (E) e i SP (E)/h¯ . (10.5.17) P

Now, the summation is over “pseudo-orbits” in which each primitive orbit is repeated rp times (possibly rp = 0) such that P is a multiple summation variable, P = {rp } ,

rp = 0, 1, 2, . . . .

(10.5.18)

A pseudo-orbit has as its action and pseudo-period the pertinent sums over the contributing primitive orbits, SP =



rp Sp ,

p

TP =

 ∂SP = rp Tp . ∂E p

(10.5.19)

The pseudo-orbit series (10.5.17) can be thought ordered by increasing pseudoperiod TP . The weight FP (E) of the P th pseudo-orbit is FP (E) =

⎧ -⎨ p

⎫  1 ⎬ exp − r (r − 1)λ T p p p p (−1)rp 6rp 4 . j 1/2 ⎭ ⎩ j =1 | det(Mp − 1)|

(10.5.20)

We leave to the reader to show that primitive orbits that are hyperbolic with reflection show up like hyperbolic ones in the pseudo-orbit sum (10.5.17), except for the replacement (−1)rp → (−1)Int[(rp +1)/2] in the weight FP (E). Incidentally, multiple repetitions of a primitive orbit within a pseudo-orbit have rapidly decreasing weights as rp grows, due to the factor   exp − 14 rp (rp − 1)λp Tp −→ exp{− 12 rp2 λp Tp } . 6rp j 1/2 | det(M − 1)| p j =1

(10.5.21)

Thus, the most important long pseudo-orbits are those composed of singly traversed primitive orbits. The predominance of singly traversed primitive orbits in the pseudo-orbit sum (10.5.17) is also due to the exponential proliferation of orbits with

454

10 Semiclassical Roles for Classical Orbits

growing period that we have seen to follow from the sum rule of Hannay and Ozorio de Almeida in Sect. 9.12. Mustering a lot of courage we follow Berry and Keating to real energy in the pseudo-orbit sum for the zeta function. Convergence is surely lost; reality, while required by the definition (10.5.1), is no longer manifest and definitely destroyed by any finite truncation of the infinite sum. To make the best of the seemingly desperate situation we may impose reality,

e−iπN (E)

∞ 

i h¯ SP (E)

FP (E) e

=

e iπN (E)

∞ i  − S (E) FP (E) e h¯ P ,

P

P

(10.5.22) in the hope of thus forbidding the divergence to play the most evil of games with us. But no amount of conjuration will retrieve convergence. Not even refuge to complex energies is possible since the l.h.s. would require Im E > h¯ λ/2 (see Sect. 10.4.3) and the r.h.s. Im E < −h¯ λ/2, leaving no overlap of the domains of convergence. Keating modestly calls the foregoing reality condition the “formal functional equation” [35], before cunningly drawing most useful consequences. # E+α(h) A Fourier transform E−α(h¯¯) dE exp (iτ E /h¯ ) (. . .) with respect to a classically small but semiclassically large energy interval, α(h¯ ) → 0 ,

α(h¯ )/h¯ → ∞

for h¯ → 0 ,

(10.5.23)

is now done on the functional equation. Neither the Weyl staircase N (E) nor the action SP vary much over the small integration range and may thus both be expanded around E to first order such that the l.h.s. of (10.5.22) yields ei[−πN (E)+τ E/h¯ ]



+α(h¯ )

−α(h¯ )



dE eiE [−π(E)+τ/h¯ ]





FP (E) ei(SP (E)+E TP )/h¯

P

(10.5.24) or, after rescaling the integration variable as E /h¯ → ω and boldly interchanging the order of integration and summation, h¯ ei[−πN (E)+τ E/h¯ ]

 P

FP (E) eiSP /h¯



+α(h¯ )/h¯ −α(h¯ )/h¯

+TP ] ¯ dω eiω[−π h(E)+τ .

(10.5.25) But now the integration range is large, and the η-integral gives a fattened delta function 2πδh¯ /α(h¯ ) [−π h¯ (E) + τ + TP ] whose width h¯ /α(h¯ ) vanishes as h¯ → 0.

10.5 Riemann-Siegel Look-Alike

455

Exactly the same procedure brings the r.h.s. of the functional equation into to a form identical to (10.5.26) save for sign changes in front of N , , SP , and TP . Hence, the functional equation reappears as e−iπN (E)



  + τ + TP (E) (10.5.26) FP (E) eiSP (E)/h¯ δh¯ /α(h¯ ) − π h(E) ¯

P

= eiπN (E)



  FP (E) e−iSP (E)/h¯ δh¯ /α(h¯ ) π h(E) + τ − TP (E) . ¯

P

As a final step we integrate on both sides over τ from 0 to ∞ and recognize π h¯ (E) = TH (E)/2 as half the Heisenberg time, TP ≤

TH

2

TP >

FP exp i(SP /h¯ − πN ) =

P

TH

2

FP exp i(−SP /h¯ + πN ) .

P

(10.5.27) We have arrived at a sum rule connecting the manifestly finite contribution from pseudo-orbits with pseudo-periods below half the Heisenberg time to the ill-defined contribution of the infinitely many longer pseudo-orbits; the two contributions are mutual complex conjugates. If we import that sum rule into the zeta function (10.5.17), we get a finite semiclassical approximation, the celebrated Riemann–Siegel look-alike ζs.cl. (E) = 2B(E)

TP (E)≤T H (E)/2

  FP (E) cos SP (E)/h¯ − πN (E) .

P

(10.5.28) What a relief indeed, after all the chagrin about lost convergence! The zeros of this finite semiclassical zeta function [45, 46] give semiclassically accurate energy eigenvalues up to the energy E which one is free to choose, provided that the periodic orbits with periods up to TH (E)/2 are available. It is immaterial now whether the Hilbert space is finite- or infinite-dimensional. Some further remarks are in order in view of later applications. First, even though the special charm of the Riemann-Siegel lookalike lies in the reality of ζ(E) for real E, it is allowable to wander to E ± = E ± iγ and   + − 2 cos SP (E)/h¯ − iπN (E) → ei(SP (E )/h¯ −πN (E)) + e−i(SP (E )/h¯ −πN (E)) , (10.5.29)

456

10 Semiclassical Roles for Classical Orbits

with the aim to enhance convergence in the limit h¯ → 0 ⇔ TH ∝ h¯ −f +1 → ∞; of course, the quantities N (E) and B(E) can be kept at real E since they are the same for all pseudo-orbits; the stability factor FP (E) also does not need to be complexified since the difference F (E) − F (E ± ) is not referred to a quantum scale in the above sums. Second, if one wants to forsake orbit repetitions to begin with, the RiemannSiegel lookalike can be obtained somewhat faster, by dropping all terms with r > 1 from the exponentiated orbit sum (10.5.10) and expanding the exponential in powers of the sum over primitive orbits; see Problem 10.11. Third, we kindly invite the reader to think about the generalization of the reasoning of the present section to more than two degrees of freedom.

10.6 Spectral Two-Point Correlator The universality of spectral fluctuations under conditions of classical hyperbolicity and in the absence of any symmetries (beyond the ones distinguishing the WignerDyson universality classes) will be illustrated here, by a semiclassical determination of the two-point correlator of the density of levels. Due to a suitable energy average within a single energy spectrum, that correlator takes the form predicted by random-matrix theory. The four essential ingredients to be used are (1) a generating function involving four spectral determinants, (2) Gutzwiller sums for the spectral determinants, (3) the Riemann-Siegel lookalike, and (4) bunches of orbits differing in close self-encounters giving constructively interfering contributions to the fourfold Gutzwiller sum. Throughout the present section we shall closely follow Ref. [6]. Before diving into the announced periodic-orbit expansions it is well to throw a glance back to the times before the advent of the classical phenomenon of periodic-orbit bunching on the semiclassical scene. When nearly a decade had passed without semiclassical progress since Berry’s diagonal approximation (see below) a group of chaoticians (Argaman, Dittes, Doron, Keating, Kitaev, Sieber, and Smilansky) asked a most interesting question: Given universal behavior of the twopoint spectral correlator and Gutzwiller’s periodic-orbit expansion, what classical correlations between periodic orbits follow? They indeed found nontrivial ‘universal action correlations’ as a classical pendant to universal quantum spectral fluctuations [47]. Indeed, all known exceptional dynamics characterized by full classical chaos but non-universal quantum spectral fluctuations display very specific non-universal action correlations. Examples are the so called arithmetic billiards [48–50] and the cat map [23]. For an early precursor of Ref. [47] see [51].

10.6 Spectral Two-Point Correlator

457

10.6.1 Real and Complex Correlator The simplest indicator of spectral correlations is the connected correlator R(e) =

   1   e e  E+  E− −1 2 (E) 2π(E) 2π(E)

(10.6.1)

where the angular brackets (or an overbar) again denote a local average over the center energy E. The dimensionless variable e gives the energy offset in units of the mean level spacing 1/ ≡ 1/¯ (divided by 2π, for convenience). Dyson’s cluster function which was studied in the framework of random-matrix theory in Chap. 5 is related to the above correlator as Y (e) = δ(e) − R(πe). RMT yields ⎧ ⎨−s(e)2 = − 1−cos 2e unitary class 2e 2  R(e) − πδ(e) = 1 1 2 ⎩−s(e) + s (e) Si(e) − sgn(e) orthogonal class π 2 (10.6.2) #e with s(e) = sine e and Si(e) = 0 de s(e ). The correlation function R(e) is closely connected to the complex correlator of Green functions of the Hamiltonian H C(e+ ) =

1 1 1 1  Tr Tr − + + 2 2 e e 2π ¯ 2 E + 2π E − 2π ¯ − H ¯ − H

(10.6.3)

where the superscript + denotes a positive imaginary part +iη. The first Green function in (10.6.3) is a retarded one due to Im (E + e+ /2π ) ¯ > 0 while the second is an advanced one. The real part of C(e+ ) at real energy offset yields R(e) = Re lim C(e+ ) .

(10.6.4)

η→0

The RMT prediction for the complex correlator can be written as ⎧ + 1 e2ie ⎪ ⎪ ⎨ 2(ie+ )2 − 2(ie+ )2

∞ (n−3)!(n−1) 1 C( + ) ∼ + )2 + n=3 2(ie+ )n (ie ⎪ ⎪ ⎩ 2ie+ ∞ (n−3)!(n−3) +e n=4 2(ie+ )n

unitary class (10.6.5) orthogonal class ,

and the semiclassical confirmation of that prediction is the principal goal of the present section. For the unitary symmetry class, C(e+ ) is the sum of a monotonous term ∝ e12 and an oscillatory term ∝ e12 e2ie . That structure of C also arises for the orthogonal class, except that both the monotonous term and the cofactor of

458

10 Semiclassical Roles for Classical Orbits

the oscillating exponential e2ie are asymptotic series in 1e ; the series can be Borel summed and a closed from analogous to (10.6.2) be given. Before launching the semiclassical calculation of C(e+ ) it is well to pause for an interlude and prove the relation (10.6.4) between complex and real correlator.

10.6.2 Local Energy Average We start with N consecutive levels E1 < E2 < . . . EN , disregarding levels below E1 and above EN . The size ΔE = EN − E1 of the energy window so filled should be much larger than (1) the range allowing to define a smooth local mean density of levels and (2) the range over which level correlations extend. On the other hand, the window must not be too large; the local mean density must be constant therein and thus equal ¯ = ENN−E1 ; for the sake of concreteness we may imagine the window size ΔE to be the maximal one compatible with the internal constancy of . ¯ Any location of the window in the complete spectrum is possible. The included levels allow definition of “local”

for the 6N Green function and spectral determinant as 1 g(E) = N and Δ(E) = i=1 E−Ei i=1 (E − Ei ). With the goal of picking up spectral correlations somewhere in the spectrum, say at E, we place a window of the above kind symmetrically around E. we can now # E+δE/2 1 specify a local energy average as . . . = δE E−δE/2 dE (. . .), with the length δE of the averaging interval small compared to ΔE but larger than the correlation range and thus containing many levels. The symmetry of both the averaging interval and the larger energy window is a matter of convenience. In the semiclassical limit we have the hierarchy of energy scales 1¯ δE ΔE and can even afford the limit → 0. The “local” complex correlator can now be written as16

δE ΔE

N  1 1 C(e ) + = 2 2π 2 ¯ 2 δE +

e+ i,k=1 π ¯

1 + Ei − Ek

 ln

E − E +

e+ 2π ¯ e+ 2π ¯

− Ei E =+δE/2 − Ek

E =−δE/2

where for notational convenience we have chosen the reference energy E as the origin for the energy axis. With (x + iη)−1 ↔ P x −1 − iπδ (x) for η → 0+ and ln x = ln |x| + i arg x, the real part of C can be written as Re C(e+ ) +

1 = RI I + RRR 2

16 As already pointed out in the footnote accompanying the definition (5.20.3) of the two-point correlator, two averages are needed to produce at least a piecewise constant function with step heights, upwards or downwards, of order N1 . One of these is over the center energy as discussed here; the second, not worked into the formulae here in order to save space can be over a small window for the offset variable e.

10.6 Spectral Two-Point Correlator

459

where RI I (RRR ) stems from the product of the imaginary (real) parts of the fraction

and the square bracket in the above double sum ik . Starting with RI I , we take Im e+ /¯ = η/¯ → 0+ . In the (i, k)-th summand in the above double sum for C(e+ ), the phase of the expression in the square brackets can be (1) 2π (both Ei and Ek lie within the averaging interval [−δE/2, δE/2]); (2) π (only one eigenvalue within [−δE/2, δE/2]); (3) zero (both eigenvalues outside [−δE/2, δE/2]). Of exclusive interest are values of the energy offset e small compared with the size δE of the averaging interval and for these, in view of the factor δ( πe + Ei − Ek ), the possibility (2) can be discarded. Up to an additive constant, RI I must then coincide with the real correlation function,    1 e RI I = + Ei − Ek = R (e) + 1. δ δE π ¯ −δE/2 2 in one is split into an (l − 1)-encounter and a 2-encounter in the other, and the respective contributions cancel. In the absence of time reversal invariance only the diagonal contribution thus survives. For time reversal invariant dynamics from the orthogonal symmetry class, the foregoing cancellation mechanisms still “wipe out” all quadruplets with exclusively parallel encounters.

10.9 Semiclassical Construction of a Sigma Model, Unitary Symmetry Class Even though the nullity of off-diagonal corrections for the unitary symmetry class is now established there is more to learn about that class. we propose to encapsulate the whole pseudo-orbit series for Z (1) in a matrix integral which will turn out to coincide with the one known from the so-called sigma model in RMT.

10.9.1 Matrix Elements for Ports and Contraction Lines for Links Structures of pseudo-orbit quadruplets were defined by (1) fixing the set of encounters and numbering the encounter ports, (2) connecting the encounter ports by links and thereby defining the pre- and post-reconnection periodic orbits , and (3) dividing the pre-reconnection orbits between the pseudo-orbits A, C and the post-reconnection orbits between B and D. Now we encode each structure in (10.8.14) by a sequence of alternating symbols19 Bkj and B˜ j k respectively standing for the entrance and exit encounter ports. For each encounter starting with the first entrance and exit ports and following the port labels we pairwise list the symbols for all further ports. The indices of B, B˜ indicate the affiliation with the pseudo-orbits: the index j , the second on B ˜ determines whether before reconnection the port belongs to A and the first on B, (then j = 1) or to C (j = 2). Similarly, k shows whether the port belongs after

19 We must apologize for using the tilde on one of these two symbols which in contrast to everywhere else in this book does not mean transposition.

480

10 Semiclassical Roles for Classical Orbits

reconnection to B (if k = 1) or to D (k = 2). Due to the two possible values for both k and j , altogether four Bkj ’s and four B˜ j k ’s are brought into play. The port numbering introduced before will now prove well suited to the goal posed (Recall that each entrance port is connected by an encounter stretch to the exit port with the same number before reconnection, and the number obtained by backward cyclic shift, within each encounter, after reconnection). The ports connected by an encounter stretch necessarily enter the same orbit and pseudo-orbit. Consequently, neighboring indices of successive port symbols must coincide: Bkj is followed by B˜ j k and preceded by B˜ j k . In particular, a 2-encounter with two entrances and exits as in Fig. 10.13a is assigned the four-symbol sequence Bk1 ,j1 B˜ j1 ,k2 Bk2 ,j2 B˜ j2 ,k1 ,

(10.9.1)

such that the indices follow one another just like in a trace of a matrix product; however, until further notice no summations are implied; note that the sub-indices on the indices j, k signal encounter stretch labels within the encounter. For a structure with arbitrarily many encounters indexed by σ = 1, 2, . . . V and stretches inside each encounter indexed by a = 1, 2, . . . l(σ ) the sequence of ports gives rise to the product V   Bkσ 1 ,jσ 1 B˜ jσ 1 ,kσ 2 Bkσ 2 ,jσ 2 . . . B˜ jσ l(σ ) ,kσ 1 ;

(10.9.2)

σ =1

here, the indices j and k carry two subindices each, σ = 1, 2, . . . , V to label the encounter and i = 1, 2, . . . , l(σ ) to label the stretches within an encounter; such proliferation of address labels notwithstanding there are still only four different Bkj ’s and four B˜ j k ’s; whenever possible the subindices on k and  j will  be B11 B12 ˜ and suppressed. Taking Bkj , Bj k as elements of 2 × 2 matrices B = B21 B22

similarly for B˜ we can interpret the foregoing symbol sequence could be as one of  l(σ ) 6 the summands in the expansion of the trace product Vσ =1 tr B B˜ . The links of a structure can be indicated on the symbol sequence by drawing “contraction lines” between the respective exit and entrance ports. For instance, in the four-symbol sequence (10.9.1) contraction lines can be drawn as Bk1 j1 B˜ j1 k2 Bk2 j2 B˜ j2 k1

or

Bk1 j1 B˜ j1 k1 Bk1 j2 B˜ j2 k1 ,

(10.9.3)

in correspondence to the link configurations in Fig. 10.13b and c, respectively. The periodic orbit sets before and after reconnection can be read off from the ˜ B B-sequence with the link lines drawn. Namely, complementing the plot by the encounter stretches connecting each Bkj with the cyclically following (within each encounter) exit port B˜ j k we obtain a graph of pre-reconnection periodic orbits. Similarly drawing the encounter stretches in the opposite direction from each Bkj to

10.9 Semiclassical Construction of a Sigma Model, Unitary Symmetry Class

481

the cyclically preceding B˜ j k we obtain the post-reconnection orbits. In our above example of a four-symbol sequence all subscripts might be chosen as 1 (thereby ascribing all pre-reconnection orbits to A and all post-reconnection ones to B) we obtain two structures each depictable by two plots, B11 B˜ 11 B11 B˜ 11

and

B11 B˜ 11 B11 B˜ 11

(10.9.4)

B11 B˜ 11 B11 B˜ 11 ,

(10.9.5)

or B11 B˜ 11 B11 B˜ 11

and

where lower lines indicate encounter stretches. The upper pair represents the first structure in the list (10.8.3) and pertains to Fig. 10.13b while the lower pair is one of the eight structures pertaining to Fig. 10.13c. The previous diagrammatic rules for the contribution of a structure in (10.8.14) can be translated to the new language of contracted sequences of matrix elements and start with the link factors. The two ports connected by a link belong to the same pseudo-orbit, both before and after reconnection. Consequently, a contraction of Bkj to B˜ j k may exist only if k = k and j = j . Recalling the link factor in (10.8.14), [−i(A or C + B or D )]−1 , we can write the factor provided by a contraction between Bkj , B˜ j k as [−i(j + k )]−1 δkk δjj . Here j stands for A (j = 1) or C (j = 2); similarly k equals B for k = 1 and D for k = 2. The primed energy will always refer to the post-reconnection pseudo-orbits B or D. The indices in an encounter factor i (A or C + B or D ) in (10.8.14) are determined by the pseudo-orbits containing the first entrance port of the encounter. In (10.9.2) the first entrance port of the σ th encounter is denoted by Bkσ 1 ,jσ 1 such that the respective encounter factor can be rewritten as i(jσ 1 + k σ 1 ).

10.9.2 Wick’s Theorem and Link Summation Summation over all structures involves, in particular, summation over all possible links alias contractions. A useful tool for that summation is provided by Wick’s theorem. Consider the Gaussian integral 

∗

dbdb∗ eibb f (b, b ∗ )

(10.9.6)

over the complex plane with dbdb∗ ≡ dReb dImb/π, f (b, b∗) a product involving an equal number of b’s and b∗ ’s, and Im  > 0 to ensure convergence. That integral can be written as a sum over diagrams where each b in f (b, b∗ ) is connected to one

482

10 Semiclassical Roles for Classical Orbits

b∗ by a “contraction line”. These diagrams can be evaluated using the rule 

dbdb∗ b b∗ g(b, b∗ ) =

1 −i



d 2 b g(b, b∗ ) ;

(10.9.7)

step by step one can remove contraction lines and the associated pairs b, b∗ and in each step one obtains a factor − i1 . For instance, if g(b, b∗ ) = (bb∗)n one so obtains # ∗ n! dbdb∗eibb (bb∗)n = (−i) n+1 with the factorial giving the number of possibilities to draw contraction lines connecting each b with some b∗ . The same result holds if the integration variables are replaced by Grassmann variables20 η and η∗ . Imagining exp (iηη∗ ) Taylor-expanded we obtain 

∗ iηη∗

∗

dηdη η η e

 ≡

∗

∗ iηη∗

dηdη η η e

1 = −1 = i



∗

dηdη∗ eiηη . (10.9.8)

This is the analogue of (10.9.7) with g = 1 which is the only case of interest since all powers of the Grassmann variables higher than 1 vanish. We shall now employ Wick’s theorem to represent sums of all possibilities of contracting Bkj ’s and B˜ j k ’s as in (10.9.3), for general products of the form (10.9.2). To that end we declare the Bkj to be either complex Bosonic or Fermionic variables and stipulate the variables Bkj and B˜ j k to be connected by contraction lines to be ∗ ; the choice of the Bosonic mutual complex conjugates up to a sign, B˜ j k ≡ ±Bkj or Fermionic character of the variables and of the sign is reserved for later. Taking the general symbol sequence, we integrate with a Gaussian weight that has a term ∗ in the exponent. Wick’s theorem, written with an arbitrary proportional to Bkj Bkj multiplier i in the exponent, yields a factor − i1 for each contraction line. The ∗ in the Gaussian desired link factor [−i(ej + ek )]−1 arises if the multiplier of Bkj Bkj exponent is chosen as i(ej + ek ). By simply including the encounter factor i(ejσ 1 + ek σ 1 ) we are led to the integral  ±

˜ e± d[B, B]

j,k=1,2

∗ i(ej +ek )Bkj Bkj

(10.9.9)

4 V     ±i ejσ 1 + ekσ 1 Bkσ 1 ,jσ 1 B˜ jσ 1 ,kσ 2 Bkσ 2 ,jσ 2 . . . B˜ jσ l(σ ) ,kσ 1 × σ =1

where the integration measure involves the product of all independent differentials, 62 ∗ ˜ ∝ dB d[B, B] kj dBkj . All (convergent) expressions of this type will k,j =1 presently be seen to generate the right contributions for each structure. 20 Grassmann algebra and Grassmann integrals will accompany us in the remainder of this chapter. The reader may want to refresh versatility with these techniques by a glance at Sect. 5.13.

10.9 Semiclassical Construction of a Sigma Model, Unitary Symmetry Class

483

10.9.3 Signs The most delicate task is to get the sign factor (−1)nC +nD in the sum over structures (10.8.14) by appropriately choosing the Bosonic vs. Fermionic character of the integration variables and the signs in (10.9.9). The choices to be made are inspired the supersymmetric sigma model of random-matrix theory (see Chap. 6) but will here be justified on purely semiclassical grounds. We choose B11 , B22 Bosonic and B12 , B21 Fermionic, define the relation between the “supermatrices” B and B˜ as B˜ =



∗ B∗ B11 21 ∗ −B ∗ B12 22

 ,

(10.9.10)

and pick the signs in the exponent of the Gaussian in (10.9.9) as ±



∗ i(ej + ek )Bkj Bkj −→

(10.9.11)

j,k=1,2 ∗ ∗ ∗ ∗ i(e1 + e1 )B11 B11 − i(e1 + e2 )B21 B21 + i(e2 + e1 )B12 B12 + i(e2 + e2 )B22 B22 .

To compact the notation we employ the diagonal 2 × 2 matrices eˆ = diag(e1 , e2 ) ,

eˆ = diag(e1 , e2 )

(10.9.12)

and introduce the “supertrace” of 2 × 2 supermatrices with Bosonic diagonal entries and Fermionic off-diagonal entries as  Str

M11 M12 M21 M22

 = M11 − M22 .

(10.9.13)

Readers not familiar with superalgebra are kindly requested to spend a minute with checking that the foregoing definition of a supertrace entails the property of cyclic invariance, Str XY = Str Y X, precisely due to the Fermionic character of the offdiagonal elements M12 , M21 of our supermatrices. Profiting from such notational artifices we may now enjoy the slimmed version of the above quadratic form (10.9.11), ±



  ∗ ˜ . i(ej + ek )Bkj Bkj −→ Str eˆ B B˜ + eˆBB

(10.9.14)

j,k=1,2

The choices for the signs and the “superjargon” just established yield the Gaussian integral in (10.9.9) correspondingly compacted to  −

* +  ˜ ˜ exp i Str eˆ B B˜ + eˆBB . . . ≡ {. . .} ; d[B, B]

(10.9.15)

484

10 Semiclassical Roles for Classical Orbits

here the curly brackets {. . .} have the same content as in (10.9.9), namely the sequence of 2V alternating factors B and B˜ involved in the representation of a general structure; for future convenience we denote the integral with the (negative and non-normalized) Gaussian weight by double angular brackets . . .; the integration measure is finally fixed as ∗ ∗ ˜ = dRe B11 dIm B11 dRe B22 dIm B22 dB12 d[B, B] dB12 dB21 dB21 . π π

(10.9.16)

Repeated application of the contraction rule to all integrals of the type (10.9.15) ˜ until finally only the elementary integral successively removes all B’s and B’s  1 = − =

  ˜ d[B] exp i Str eˆ B B˜ + eˆBB

(e1 + e2 )(e2 + e1 ) (eA + eD )(eC + eB ) = (e1 + e1 )(e2 + e2 ) (eA + eB )(eC + eD )

(10.9.17)

(1)

is left. Remarkably, the diagonal part Zdiag of the generating function, up to the Weyl factor ei(eA +eB −eC −eD )/2 thus arises. A final specification of signs will presently prove beneficial: For each encounter, we insert a factor sk1 which equals 1 if k1 = 1 (after the reconnection the first exit port of the encounter belongs to B), and sk1 = −1 if k1 = 2 (the port belongs to D). Moreover, we install the weight 1/V ! of a structure in (10.8.14) and the Weyl factor. The resulting expression Z (1)(V , {l(σ ), jσ i , kσ i }) = 33 ×

V  σ =1

ei(eA +eB −eC −eD )/2 V!

(10.9.18)

skσ 1 i(ejσ 1 + ek σ 1 )Bkσ 1 ,jσ 1 B˜ jσ 1 ,kσ 2 Bkσ 2 ,jσ 2 . . . B˜ jσ l(σ ) ,kσ 1

55 

(1)  (1) ) of all structures with V 1 + Zoff is the cumulative contribution to Z (1) = Zdiag encounters, each having a fixed number of stretches l(σ ), σ = 1 . . . V and a fixed distribution of ports {jσ i , kσ i } among the pseudo-orbits; herein each allowed way of drawing contraction lines makes for one structure. The purely diagonal additive term is due to V = 0. Each factor in the product over encounters in (10.9.18) can now be written as a supertrace which automatically takes care of the factor skσ 1 . Still refraining from summing over port labels, we keep the pertinent indices fixed using projection

10.9 Semiclassical Construction of a Sigma Model, Unitary Symmetry Class

485

matrices P1 = diag(1, 0), P2 = diag(0, 1), to get Z (1) (V , {l(σ ), jσ i , kσ i }) = ei(eA +eB −eC −eD )/2 33 ×

V σ =1

i(ejσ 1 +

1 V!

˜ kσ 2 ek σ 1 ) Str Pkσ 1 BPjσ 1 BP

(10.9.19) 55 . . . Pjσ l(σ ) B˜

.

All the foregoing choices of signs indeed produce the factor (−1)nC +nD in (10.8.14). To see that we employ a more explicit form of the formal contraction rule (10.9.7). Contraction lines in Eq. (10.9.19) may be drawn either between ports within the same encounters (inside the same supertrace) or from different encounters (supertraces). For these two cases the formal rule can be written in terms of the matrices B and with the Gaussian of (10.9.15) accounted for, as == >>  δ δ  ˜ k ) . . . = − jj kk Str(Pk XPj Y ) . . . Str(Pk BPj Y )Str (XPj BP i(ej + ek )

(10.9.20)

for inter-encounter contractions and >> ==  δjj δkk  ˜ Str(Pj U )Str(Pk V ) . . . Str(Pk BPj U Pj BPk V ) . . . = − i(ej + ek ) (10.9.21) for intra-encounter contractions. The symbols X, Y, U, V represent matrix products ˜ X = B . . . B, U = B˜ . . . B, V = B . . . B. ˜ The as in (10.9.19) with Y = B˜ . . . B, Kronnecker deltas in the arising link factors −δjj δkk i(e 1+e ) reflect the fact that j

k

links connect ports associated to the same pre-reconnection pseudo-orbit (j = j ) and the same post-reconnection pseudo-orbit (k = k ). The “averages” on both sides of each contraction rule comprise all structures accessible by drawing further contraction lines; the set of periodic orbits so included by the right-hand-side average can be smaller than the set included in the left-hand-side average, since two ports and one link are taken out. we shall prove these contraction rules in the following subsection. The contraction rules can now be used to simplify our expressions for the structure contributions in steps, removing two matrices at a step. This corresponds to removing a link connecting two ports from a periodic-orbit structure, without changing other links. At first sight, the rules (10.9.20), (10.9.21) do not seem to yield any sign factors. But special attention must be payed to the steps where the contraction line to be removed pertains to a single-link orbit; such a remnant arises when previous steps have removed all other contraction lines from some multiple-link orbit. Then the B and B˜ connected by the last line represent the last two remaining ports of the orbit. Since the orbit is periodic, these ports must be connected not only by the link but ˜ also by an encounter stretch. According to our convention for ordering B’s and B’s

486

10 Semiclassical Roles for Classical Orbits

the matrices representing these ports must follow each other with just one projection matrix in between. If the orbit in question is an original one, the matrices follow each ˜ The contraction rule (10.9.21) then applies, with U = 1. Then the other like BPj B. first supertrace on the r.h.s. turns into Str Pj ; but that supertrace equals 1 for j = 1, i.e., if the orbit belongs to pseudo-orbit A, while for j = 2, i.e., if the orbit belongs to C we have Str P2 = −1. The desired sign factor −1 for each orbit included in C thus arises. An analogous result holds for partner orbits. When the last contraction line associated to a partner orbit is removed the ports connected by that line must also be connected by a partner encounter stretch. The matrix B representing the entrance ˜ k B. port and the matrix B˜ representing the exit port must follow each other like BP Rule (10.9.21) then applies with V = 1 and the second supertrace on the r.h.s. thus equals Str Pk , i.e. −1 if k = 2 and the orbit belongs to D. To conclude, by turning B and B˜ into supermatrices with Grassmannian offdiagonal entries, we have successfully incorporated the sign factor (−1)nC +nD .

10.9.4 Proof of Contraction Rules, Unitary Case An interlude is in order to derive the contraction rules (10.9.20) and (10.9.21) from the raw versions (10.9.7) and (10.9.8). The exponent (10.9.15) permits nonzero contractions only for mutually complex conjugate matrix elements. For Bkj and B˜ j k (the latter agreeing up to the sign with Bk∗ j ) we have ==

>> ˜ =− Bkj B˜ j k g(B, B)

 sj ˜ . δjj δkk g(B, B) i(j + k )

(10.9.22)

Here sj is a sign factor equal to 1 if j = 1 and to −1 if j = 2. To understand its origin we must appreciate the rule (10.9.7) for the Bosonic variables (j = k) yielding − i1 and (10.9.8) for the Fermionic variables (j = k) yielding i1 ; a additional factors −1 emerges when j = k = 2 due to the negative sign in ∗ and when j = 1, k = 2 because of minus at the term proportional B˜ 22 = −B22 to B21 B˜ 12 in the exponent of (10.9.15). Using Eq. (10.9.22) we obtain the rule (10.9.20) for contractions between B and B˜ in different supertraces (inter-encounter contractions) as ==

>> >> == ˜ ˜ Str (Pk BPj Y )Str (Pk XPj B) . . . = Str (Pj Y Pk B)Str (Pj BPk X) . . . == >> = sj Yj k Bkj sj B˜ j k Xk j . . . =−

 sj δjj δkk  sj Yj k sj Xkj . . . i(j + k )

10.9 Semiclassical Construction of a Sigma Model, Unitary Symmetry Class

=−

 δjj δkk  sj Yj k Xkj . . . i(j + k )

=−

 δjj δkk  Str (Pj Y Pk X) . . . . i(j + k )

487

Here we used the cyclic invariance of the supertrace, wrote out the supertrace in components, with the help of Str (Pk Z) = sk Zkk for any supermatrix Z, and finally used Eq. (10.9.22). Equation (10.9.21) for intra-encounter contractions follows similarly from ==

>> == >> ˜ k V ) . . . = sk Bkj Ujj B˜ j k Vk k . . . Str (Pk BPj U Pj BP == >> = sk Bkj B˜ j k Ujj Vk k . . . =−

 sj δjj δkk  sk Ujj Vkk . . . i(j + k )

=−

 δjj δkk  Str (Pj U )Str (Pk V ) . . . . i(j + k )

In the second line we could interchange Ujj and B˜ j k since a nonzero result arises only for j = j in which case Ujj is Bosonic and commutes with all other variables.

10.9.5 Emergence of a Sigma Model To determine Z (1) we sum over all structures of pseudo-orbit quadruplets. In Eq. (10.9.19) all ways of linking ports with fixed labels are already collected. Now all possibilities of assigning ports to pseudo-orbits must be picked up, i.e., the indices jσ i , kσ i in (10.9.19) be summed over. That sum just amounts to dropping the projection matrices. To keep the factors i(eσ 1 + eσ 1 ) connected to the indices in the supertrace we once more use the diagonal offset matrices e, ˆ eˆ and write Z (1) (V , {l(σ )}) =

ei(eA +eB −eC −eD )/2 V!

== V

>>  ˜ l(σ ) + eˆ (BB) ˜ l(σ ) . i Str e(B ˆ B)

σ =1

It remains to sum over the number V of encounters and over their sizes l(σ ), σ = 1, 2, . . . , V . Including the trivial term V = 0 which takes care of the purely diagonal

488

10 Semiclassical Roles for Classical Orbits

contribution, we have

Z

(1)

=e

=e

∞  1 V! V =0 33

i (eA +eB −eC −eD )/2

338 i Str

∞  

˜ ˜ eˆ(BB) + eˆ (B B)

l

l

9 55  V

l=2

55 ∞    l l ˜ ˜ e( ˆ BB) + eˆ (B B) . exp i Str

i (eA +eB −eC −eD )/2

l=2

Yet one more time we take advantage of the precaution of a large imaginary part of all energy offsets. Due to that restriction the B-integral draws dominant contributions from near the saddle of the integrand at B = 0. In fact, without changing the previous moments of the Bosonic Gaussians by more than corrections of the order e−η it is possible to restrict the integration ranges for the Bosonic variables as |B11 |, |B22 | < 1 and then sum the geometric series in the foregoing exponent, to get 9 ˜ ˜ B B BB ˜ exp i Str eˆ = −e d[B, B] + eˆ ˜ 1 − BB 1 − B B˜ 9  8  ˜ ˜ ˜ exp i Str eˆ 1 + BB + eˆ 1 + B B . (10.9.23) = − d[B, B] ˜ 2 1 − BB 1 − B B˜ 

Z

(1)

8



i (eA +eB −eC −eD )/2

The matrix integral arrived at compactly sums up the asymptotic 1e series. Due to the restriction Im e = η  1 of the present semiclassical derivation it yields only the large-η asypmtotics Z (1) . The integral can be done and the large-η asypmtotics exhibited. To that end, we propose to establish a transformation of the integration variables bringing the integral to a Gaussian form. ˜ are respectively diagonalized by U and V , The matrix products B B˜ and BB ˜ ˜ = V −1 BBV U −1 B BU = diag (lB , lF ),

(10.9.24)

with eigenvalues 0 ≤ lB < 1, lF ≤ 0. The diagonalizing matrices read21  U =  V =

η 1 + ηη∗ /2 ∗ η 1 − ηη∗ /2 1 − τ τ ∗ /2 τ −τ ∗ 1 + τ τ ∗ /2

 

e−iφB 0 0 e−iφF

 ,

(10.9.25)

= V † = σz V −1 σz .

The eight independent parameters in the matrices B, B˜ are thus expressed in terms of eight new variables four of which (lB , lF , φB , φF ), are Bosonic and four 21 The

matrices U, V have appeared in Sects. 6.6, 6.9, and 8.8 under the names A, D.

10.9 Semiclassical Construction of a Sigma Model, Unitary Symmetry Class

489

(η, η∗ , τ, τ ∗ ) Fermionic. It will be useful to know that the two eigenvalues lB , lF have different signs, lB = |B11 |2 + . . . ,

lF = −|B22 |2 + . . . ,

(10.9.26)

where the dots refer to “non-numerical” additions bilinear in the original Grass∗ , B , B ∗ . The rational functions of B B ˜ and BB ˜ appearing in mannians B12 , B12 21 21 the exponent of the matrix integral (10.9.23) are similarly diagonalized, U −1

B B˜ 1 − B B˜

U = V −1

  ˜ BB lF lB , ≡ diag(mB , mF ) , V = diag ˜ 1 − lB 1 − lF 1 − BB (10.9.27)

and the eigenvalues again have different signs (of their “numerical parts”), mB > 0, mF < 0, provided the restriction |B11 | < 1 met above is respected. Next, we can rejoice in realizing that the matrices U and V allow for a certain ˜ analogue of singular-value decompositions of B and B, ) ) B = U diag ( lB , −lF )V −1 ,

) ) B˜ = U diag ( lB , − −lF )V −1 . (10.9.28)

The matrices √ √ C = U diag ( mB , −mF )V −1 ,

√ √ C˜ = U diag ( mB , − −mF )V −1 (10.9.29)

thus have the products C C˜ =

B B˜ 1 − B B˜

,

˜ = CC

˜ BB . ˜ 1 − BB

(10.9.30)

Indeed then, the integrand of the matrix integral (10.9.23) becomes a Gaussian in ˜ The transformation B, B˜ → C, C˜ has the Jacobian unity such that the terms of C, C. ˜ = d[C, C] ˜ with d[C, C] ˜ structured like d[B, B] ˜ integration measures obey d[B, B] as given in (10.9.16). The integral in question thus takes the form     ˜ + eˆ C C˜ ˜ exp i Str eˆ CC Z (1) = −ei (eA+eB −eC −eD )/2 d[B, B] 2 = − e 2 (A +B −C −D )/2  2 d B11 d 2 B22 i{B11 B ∗ (A +B )+B22 B ∗ (C +D )} 11 22 e × π π  ∗ ∗ ∗ ∗ × dB12 dB12 dB21 dB21 ei{B12 B12 (B +C )+B21 B21 (A +D )} . i

(10.9.31) (10.9.32)

490

10 Semiclassical Roles for Classical Orbits

Admitting an error of the order e−η we can extend the Bosonic integrations over the whole complex planes. The elementary integrals then yield Z (1)(A , B , C , D ) = ei (A +B −C −D )/2

(A + D )(B + C ) , (A + B )(C + D )

η  1. (10.9.33)

The error being exponentially small, no powers in 1e can arise as corrections. Nothing prevents us from continuing the result to real offsets. But then the foregoing Z (1) cannot exhaust the high-energy asymptotics of the generating function Z since the exponentially suppressed Riemann-Siegel type complement remains suppressed. However, the Riemann-Siegel lookalike does yield the full high-energy asymptotics through (10.6.19) as Z(A , B , C , D ) ∼ e 2 (eA +eB −eC −eD ) i

−e 2 (eA +eB +eC +eD ) i

(eC + eB )(eD + eA ) (eA + eB )(eD + eC ) (eB − eD )(eA − eC ) , (eA + eB )(eC + eD )

(10.9.34) |e|  1 ,

like in the diagonal approximation (10.7.7). The high-energy asypmtotics of the correlator C(e) thus comes out as in the diagonal approximation (10.7.8), C(e) ∼ (1−e2ie)/2(ie)2 . Given the absence of any 1 e -corrections and assuming the exact correlator analytic in the upper half e-plane, we can be sure [61] that the high-energy asymptotics fixes the exact correlator and conclude C(e) = (1 − e2ie )/2(ie)2 . The RMT result for the unitary symmetry class is thus recovered semiclassically. ˜ The B, B-integral in (10.9.23) is well known from random-matrix theory where it arises as the so-called rational parametrization of the zero dimensional sigma model, there for real eC , eD and with fully specified integration ranges for the Bosonic variables B11 , B22 , and thus giving the exact generating function in full. We just mention here that the contribution Z (1) is associated with the stationary point B = B˜ = 0 (“standard saddle”) while the Riemann-Siegel complement comes from a second stationary point, the “Andreev-Altshuler saddle” B = B˜ = diag(0, ∞).

10.10 Semiclassical Construction of a Sigma Model, Orthogonal Symmetry Class When time reversal invariance reigns close self-encounters become possible where some encounter stretches are almost mutually time reversed rather than being all close in phase space. In configuration space, all stretches of such an encounter still run close to one another but with opposite senses of traversal. If connections inside such an encounter are changed, the links outside still look almost the same in

10.10 Semiclassical Construction of a Sigma Model, Orthogonal Symmetry Class

491

configuration space, but some of them acquire opposite directions. A prime example is the Sieber-Richter pair, see the uppermost left in Fig. 10.11. The new pseudo-orbit quadruplets with oppositely oriented encounter stretches and links now make for non-cancelling contributions to the generating function Z in all orders in 1e . The summation of the relevant quadruplets can again be done with the help of a sigma-model type matrix integral, much in parallel to the above treatment of the unitary symmetry class. The faithfulness of individual dynamical systems to the RMT prediction for the two-point correlator of the level density will thus again find its semiclassical explanation.

10.10.1 Structures Since mutually time-reversed stretches enter and leave the encounter from different sides, it becomes cumbersome to work with the notions of entrance and exit ports. Instead we arbitrarily refer to the ports on one side of each encounter as “left” ports and those on the opposite side as “right” ports. Encounter stretches may lead either from left to right or from right to left. The links may connect left and left, right and right, or left and right ports. The definition of structures must therefore be modified: (S1’) When numbering encounters one must make sure that in the partner pseudo-orbit the left port of the i-th stretch is always connected to the right port of the (i − 1)-st stretch. (S2’) When connecting ports by links arbitrary connections are permissible such that (2L)! possibilities arise. (S3’) Beyond partitioning n original orbits among the pseudoorbits (A, C) and n partner orbits among B, D, now every orbit can be assigned either of two senses of traversal; there are thus 4n+n structures for each of the choices made in (S1’,S2’). Again, different structures can describe one and the same quadruplet. In particular, the choice of the right and left side of each encounter is arbitrary, and for an overall number V of encounters there are 2V equivalent possibilities of labelling the sides. To avoid overcounting, that factor must be divided out. The starting point for the construction of the sigma model therefore becomes modified from (10.8.14) to (1)

Zoff =

6 (−1)nC +nD enc i(eA or C + eB or D ) 6 , VV! 2 links (−i(eA or C + eB or D )) structures 

(10.10.1)

(1) (1) with Z (1) = Zdiag (1 + Zoff ) still intact but the diagonal part given by (10.7.9) as

(1) Zdiag

=e

i 2 (eA +eB −eC −eD )



(eC + eB )(eD + eA ) (eA + eB )(eD + eC )

2 .

(10.10.2)

492

10 Semiclassical Roles for Classical Orbits

10.10.2 Leading-Order Contributions To illustrate the use of the foregoing master formula (10.10.1) we here consider the contributions responsible for the leading non-oscillatory and oscillatory terms in the correlator C(e). To get the first non-oscillatory correction δC(e) ∝ e13 we check the two diagrams with L − V = 1 in the gallery of Fig. 10.11. The one with a single parallel 2encounter does not contribute, as a reasoning completely parallel to the one in Sect. 10.8.3 reveals. So only the Sieber-Richter diagram with a single anti-parallel 2encounter remains, the uppermost left in Fig. 10.11. That diagram has one encounter (V = 1) and one orbit both before and after reconnection. we first allot these orbits to A and B such that C and D remain empty. There are four equivalent associated structures, in accordance with two possible senses of traversal of the initial and final (1) 2 orbit, and their joint yield for Zoff reads 4 2111! i(eA +eB ) 2 = i(eA +e . For the three B) [−i(eA +eB )] other allotments of orbits to pseudo-orbits different subscripts on the offset variables and the sign factor (−1)nC +nD arise. Altogether the Sieber-Richter contribution to (1) Zoff results as (1)

Zoff,SR = = =

  2 1 1 1 1 + − − i eA + eB eC + eD eA + eD eC + eB 2 (eA − eC )(eB − eD )(eA + eB + eC + eD ) . i (eA + eB )(eB + eC )(eA + eD )(eC + eD )

(10.10.3)

(1) (1) The addition to Z (1) is Zdiag Zoff,SR and the one to C(e) is given by (10.6.12),



∂2 CSR (e) = −2 Z (1) Z (1) ∂eA ∂eB diag off,SR

 . eA,B,C,D =e

(1) To save labor it is well to realize that the factor (eA − eC ) (eB − eD ) in Zoff,SR vanishes as the four offset variables are all equated. Both derivatives must therefore act on that factor while everywhere else the arguments can be set equal to e. In this way the result

CSR (e) = −2

  2 4e i = 3 4 4 i 2 e e

(10.10.4)

is readily reached; it indeed agrees with the third-order term of the RMT correlator given in (10.6.5); its Fourier transform −2τ 2 was first found by Sieber and Richter [60] as the leading correction to the diagonal approximation, 2τ , for the spectral form factor K(τ ) = 2τ − τ ln(1 + 2τ ), and that discovery opened the door to the complete semiclassical construction of the correlator described here.

10.10 Semiclassical Construction of a Sigma Model, Orthogonal Symmetry Class

493

For the leading oscillatory term, on the other hand, we consider Z (1) (eA , eB , −eD , −eC ) (10.10.5)  2 i (eA − eC ) (eB − eD ) (1) Zoff = e 2 (eA +eB +eC +eD ) (eA , eB , −eD , −eC ) , (eA + eB ) (eC + eD ) immediately resorting to real energy offsets. The factor [(eA − eC )(eB − eD )]2 vanishes for eA,B,C,D = e even after application of ∂ 2 /∂eA ∂eB . The oscillatory contribution in search must therefore come from the summands in (1) (eA , eB , −eD , −eC ) proportional to [(eA − eC )(eB − eD )]−1 . The respective Zoff (1) terms in Zoff (eA , eB , eC , eD ) behave like [(eA + eD )(eC + eB )]−1 . The contributing diagrams must (i) be specific for the orthogonal case and (ii) contain at least two pre- and two post-reconnection orbits (the denominator indicates that one link was in A before and D after reconnection; the second link was in C before and in B after reconnection). A glance at the gallery of Fig. 10.11 reveals that the only relevant diagram is the “double Sieber-Richter pair” (to be shorthanded as dSR below) in the third row of the column 2:2. The set of structures is further restricted by the demand that of the two pre-reconnection orbits one must be in A and the other in C; after reconnection one orbit must belong to B and the other to D. We arbitrarily name the pre-reconnection orbits γ1 , γ2 and their postreconnection partners γ¯1 , γ¯2 . Since each of these orbits contains one encounter and two links and each can have two senses of traversal, there are 22 × 22 = 16 structures for each allotment to pseudo-orbits. Four allotments are possible, 1. {γ1 ∈ A, γ2 ∈ C, γ¯1 ∈ D, γ¯2 ∈ B}, 2. {γ1 ∈ C, γ2 ∈ A, γ¯1 ∈ B, γ¯2 ∈ D}, 3. {γ1 ∈ A, γ2 ∈ C, γ¯1 ∈ B, γ¯2 ∈ D}, 4. {γ1 ∈ C, γ2 ∈ A, γ¯1 ∈ D, γ¯2 ∈ B}. Only the first two of these yield the desired net denominator, and they contribute identically to (1) Zoff,dSR (eA , eB , eC , eD ) = 2 × 16 ×

=−

[i (eA + eD )] [i (eC + eB )] 2 2 2! [−i (eA + eD )]2 [−i (eC + eB )]2

4 . (eA + eD ) (eC + eB )

(10.10.6)

The exchange eC ↔ −eD then gives (2) Zoff,dSR (eA , eB , eC , eD ) = −

4 + ... (eA − eC ) (eB − eD )

where the dots stand for terms present in the generating function but irrelevant for the oscillatory part of the correlator. Upon differentiating we finally arrive at the

494

10 Semiclassical Roles for Classical Orbits

leading oscillatory contribution  ∂2 (2) (2) Z Z CdSR (e) = −2 ∂eA ∂eB diag off,SR eA,B,C,D =e    4 ∂2 − = −2 ∂eA ∂eB (eA − eC ) (eB − eD ) 

e =

1 i2e e , 2e4

i 2 (eA +eB +eC +eD )

(eA − eC )2 (eB − eD )2 (eA + eB )2 (eC + eD )2

9 eA,B,C,D =e

(10.10.7)

in agreement with (10.6.5); it was first found in a daring foray by Bogomolny and Keating in 1996 [62] and then confirmed within the systematic approach described here in Ref. [56].

10.10.3 Symbols for Ports and Contraction Lines for Links Turning from the leading terms towards doing the sum over all structures we again propose to consider the sequence of ports starting with the first left port of the first encounter, continuing with the first right port, etc. The left ports will be denoted by Bνk,μj and the right ports by B˜ μj,νk . The first and the second pair of indices of B respectively refer to the properties of the port before and after reconnection, and ˜ In particular, similar to the unitary case the Latin index j = 1, 2 vice versa for B. indicates whether the port belongs to the original pseudo-orbit A (j = 1), or C (j = 2). The index k reveals whether after reconnection the port belongs to the partner pseudo-orbit B (k = 1) or D (k = 2). As a new element relative to the unitary class, Greek indices μ, ν account for the directions of motion through the port and the attached encounter stretches, both in the pertinent original and partner orbits: If μ = 1 the direction is from left to right in the original orbit; if μ = 2 it is from right to left. The index ν = 1, 2 indicates the same for the partner orbits. The symbols Bνk,μj , B˜ μj,νk can be considered as ˜ whenever convenient a capital Latin letter will be elements of 4 × 4 matrices B, B; employed for the composite index like J = (μ, j ). The order of alternating symbols is such that a B and the immediately following B˜ represent ports connected by an encounter stretch of an original orbit. The two ports are therefore traversed in the same direction and hence their indices J = (μ, j ) coincide. Similarly each B˜ and the immediately following B represent ports connected by a stretch of a partner orbit and the corresponding subscripts K = (ν, k) coincide. (When a B˜ represents the last right port in its encounter, its subscripts ν, k have to coincide with those of the first B.) At any rate, the indices J and K are

10.10 Semiclassical Construction of a Sigma Model, Orthogonal Symmetry Class

495

arranged like in a product of matrices, in fact the product already met in Eq. (10.9.2) but with j, k replaced by J, K. Next, the links can be built in and depicted by “contraction lines” above the symbol sequences; such lines can now connect not only B to B˜ but also B to B and ˜ Any two ports connected by a link must belong to the same pseudo-orbit, B˜ to B. before and after the reconnection and hence their Latin indices must coincide, as in the unitary case. The Greek direction indices require new reasoning. A contraction between a B˜ and a B stands for a link connecting a right port to a left one. In this case, if an orbit leaves one encounter at the right port it must enter the other encounter stretch at the left port. The direction of motion (left to right) is thus the same for both encounter stretches. The same applies for a direction of motion from right to left, and for original and partner orbits alike. Hence for contractions between B˜ and B the Greek subscripts indicating the directions of motion must coincide, μ1 = μ2 , ν1 = ν2 . On the other hand, in the contractions B . . . B and B˜ . . . B˜ the connected ports lie on the same side, and hence their directions of motion must be opposite. We shall use a bar over ν, μ to indicate that the port direction of the motion is flipped like 1  2 (such that μ¯ = 3 − μ). Again denoting the energy offsets as e1 = eA , e2 = eC , e1 = eB , e2 = eD we have the link contribution δ δ δμμ ¯ δν¯ ν × −i(ej − ek ) δμμ δνν kk

jj

for

⎧ ⎨B

νk,μj Bν k ,μ j

⎩

and B˜ μj,νk B˜ μ j ,ν k

Bνk,μj B˜ μ j ,ν k .

(10.10.8)

The contribution of an encounter is defined by its first left port and can be written as i(ejσ 1 + ek σ 1 ). And again, there will be a factor (−1)nC +nD depending on the numbers of orbits included in C and D.

10.10.4 Gauss and Wick Trotting along the path laid out in the previous section we seek to represent additive terms in (10.10.1) by contractions we again stipulate any two symbols connectable by a link to be represented by a pair of mutually complex conjugate variables, up to a sign. Since rule (10.10.8) allows for contractions between two B’s differing in both their subscripts μ and ν we must set Bνk,μj = ±Bν∗¯k,μj ¯ .

(10.10.9)

The identities so arising for the various choices of the indices can be compactly written as a matrix identity if the Bνk,μj are assembled into a 4 × 4 matrix B.

496

10 Semiclassical Roles for Classical Orbits

Organized with the help of four 2 × 2 blocks that matrix reads  ?

B=

bp ba∗ ba bp∗

 ,

(10.10.10)

with the question mark on the equality a reminder of eight as yet unspecified signs. The index p on bp , bp∗ signals μ = ν, i.e., the ports associated with these subblocks are traversed in parallel directions by the original and the partner orbits. In contrast, the ports associated with ba , ba∗ have μ = ν and their traversal directions by original and partner orbits are antiparallel. On the other hand, the rule (10.10.8) allows contractions between matrix elements of B and B˜ with identical subscripts. Hence these matrix elements, too, must be chosen as mutually complex conjugate (up to a sign). Since the subscripts in B and B˜ are ordered differently, B˜ must coincide with the adjoint of B (again up to signs), i.e., ? B˜ = B † .

(10.10.11)

A Gaussian-weight integral of a product of mutually conjugate matrix elements will yield a sum over all permissible ways of drawing contraction lines. To obtain ∗ the correct contribution from each link the prefactor of Bμk,νj Bμk,νj in the exponent of the Gaussian again has to be chosen as i(ej + ek ). Furthermore, due to the ∗ internal structure of B the sum over all Bμk,νj Bμk,νj in the exponent includes each pair of complex conjugate elements twice, and hence that sum must be divided ≡ e (independent of the Greek part of by 2. With the help of eJ ≡ ej and eK k

)B ∗ J, K) the exponent in the Gaussian can be written as ± 2i J K (eJ + eK KJ BKJ . Incorporating also the factor i(eJσ 1 − eK ) for each encounter we have σ1  ± ×

˜ ±2 d[B, B]e i

∗ J,K (eJ +eK )BKJ BKJ

(10.10.12)

V   ˜ Jσ 1 ,Kσ 2 BKσ 2 ,Jσ 2 . . . B˜ Jσ l(σ ) ,Kσ 1 . ±i(eJσ 1 + eK B )B K ,J σ 1 σ 1 σ1

σ =1

Each integral of this type includes the correct encounter and link factors for all structures with V encounters involving l(1), l(2) . . . l(V ) encounter stretches.

10.10.5 Signs Towards re-enacting the sign factor (−1)nC +nD and the correct prefactor of the diagonal approximation, we finally specify the matrices B and B˜ in full: The offdiagonal entries in all four blocks of B are chosen Fermionic and the diagonal

10.10 Semiclassical Construction of a Sigma Model, Orthogonal Symmetry Class

497

entries Bosonic, and the signs in (10.10.11) are picked as  B=

bp −ba∗ ba −σz bp∗

 ,

B˜ = σz ⊗ B † ;

(10.10.13)

the tensor product σz ⊗ B † means that each 2 × 2 block of B † is multiplied with σz ; taking the adjoint of our 4 × 4 supermatrices involves, apart from complex conjugation and transposition in the sense of ordinary matrices, a sign flip in the lower left (Fermi-Bose) entries of all 2 × 2 blocks. Various signs in the Gaussian weight and in the matrix product must still be fixed, and we do that such that both quantities can be written in terms of supertraces. The supertrace is now defined as the sum of the two upper left (Bose-Bose) entries minus the sum of the lower right (Fermi-Fermi) entries of the two diagonal blocks; in other words, the supertrace includes a negative sign for all diagonal elements associated to a Latin index 2 (regardless of the Greek index). The integral (10.10.12) then acquires the form    ˜ exp i Str eˆ B B˜ + Str eˆBB ˜ d[B, B] (. . .) ≡ . . . , (10.10.14) 2 = e . with e, ˆ eˆ 4×4 diagonal matrices whose diagonal elements are eˆJ = ej , eˆK k After eliminating all contraction lines from any integral of this type one is left with the Gaussian integral

 1 = =

  (e + e )2 (e + e )2 1 2 2 1 ˜ exp i Str eˆ B B˜ + Str eˆBB ˜ d[B, B] = 2 2 (e1 + e1 ) (e2 + e1 )2

(eA + eD )2 (eC + eB )2 , (eA + eB )2 (eC + eD )2

(10.10.15)

as in the diagonal factor for time-reversal invariant systems. Upon adding to (10.10.14) the Weyl factor and the factor 1/(2V V !) from (10.10.1) we obtain the contribution to the generating function Z (1) (V , {l(σ ), Jσ i , Kσ i }) = ei(eA +eB −eC −eD )/2 33 ×

V -

σ =1

1 V !2V

˜ Kσ 2 . . . PJσ l(σ ) B˜ i(eJσ 1 + eK )Str PKσ 1 BPJσ 1 BP σ1

(10.10.16) 55 .

Here PJ is a 4 × 4 projection matrix within which the J -th diagonal element equals 1 while all other elements vanish, and PK is defined analogously. To show that the sign factor (−1)nC +nD is also obtained correctly, the contraction rules (10.9.20) and (10.9.21) must be invoked once more. In fact, two more such ˜ which rules are needed here, for the contractions between two B’s or two B’s

498

10 Semiclassical Roles for Classical Orbits

represent links connecting two left ports or two right ports, == >> Str (PK BPJ Y )Str (PK BPJ Z) . . . = == >> Str (PK BPJ Y PK BPJ Z) . . . =

 δKK ¯ δJ¯J ˜ ... Str (PJ Y PK Z) −i(eJ + eK )

 δKK ¯ δJ¯J ˜ P ¯ Z) . . . Str (P Y K J ) −i(eJ + eK (10.10.17)

˜ Here Y and Z are products of the type required in (10.10.12) and analogously for B. ˜ The matrix Z˜ differs from Z but has the order of starting and ending with B. matrices interchanged, and B replaced by B˜ and vice versa. The familiar link factor shows up, notwithstanding the more compact notation J = (μ, j ) ,

J¯ ≡ (μ, ¯ j) ,

K = (ν, k) ,

K¯ ≡ (¯ν , k) .

(10.10.18)

The Kronecker deltas imply that the connected ports must belong to the same pseudo-orbits and have opposite direction of motion. The four contraction rules can now be employed to stepwise remove the contraction lines corresponding to all links of a structure. Sign factors arise only in the “final” steps where single-link orbits are removed. A final link always connects a left to a right port, or else it could not form a periodic orbit together with one encounter stretch. Hence the rules (10.9.20) and (10.9.21) apply, the ones for removing links between left ports (denoted by B) and right ports (denoted by ˜ The same argument as in Sect. 10.9.3 shows that the final step yields a factor B). Str (PJ ) for each original orbit and a factor Str (PK ) for each partner orbit. The latter supertraces equal −1 if the corresponding lower-case indices are j = 2 (corresponding to the pseudo-orbit C) or k = 2 (corresponding to the pseudo-orbit D), and these factors combine to the desired term (−1)nC +nD .

10.10.6 Proof of Contraction Rules, Orthogonal Case The derivation of the rules (10.9.20) and (10.9.21) already established for the unitary symmetry class carries over directly. The only difference is that J, K are now double indices consisting of the Greek and Latin parts, J = (μ, j ), K = (ν, k); the sign factor sJ has the values 1 for j = 1 and −1 for j = 2 regardless of the Greek index, and sK is defined analogously. In addition, both B and B˜ now contain pairs of complex conjugate matrix ˜ become possible. elements, and therefore contractions between two B’s or two B’s The further contraction rules (10.10.17) for contractions between two B’s thus arose ˜ To derive these above, as well as analogous ones for contractions between two B’s.

10.10 Semiclassical Construction of a Sigma Model, Orthogonal Symmetry Class

499

rules we employ the symmetry of B implied by the definition (10.10.13),   0 σz ΣB t Σ t = −B˜ , Σ B˜ t Σ t = −B , Σ = , (10.10.19) 1 0 where B t is the transpose22 of B. Similarly conjugating the projection matrices PK , PJ we get ΣPK Σ t = PK¯ ,

ΣPJ Σ t = PJ¯ ;

(10.10.20)

Now we consider the matrix products Y and Z in (10.10.17) which involve ˜ and interspersed projection matrices. Under alternating sequences of B’s and B’s transposition and conjugation with Σ these products have (i) the order of matrices inverted (due to the transposition), (ii) B replaced by B˜ and vice versa (due to (10.10.19)), and (iii) the subscripts J and K of all projection matrices replaced ˜ are equal the sign by J¯ and K¯ (due to (10.10.20)). Since the numbers of B’s and B’s factors from (10.10.19) all mutually compensate. Thus equipped we attack the rule for inter-encounter contractions between two B’s in (10.10.17). By using the invariance of the supertrace both under transposition of its argument and under conjugation with Σ we have ==

>> Str (PK BPJ Y )Str (PK BPJ Z) . . . == >> = Str (PK BPJ Y )Str (ΣZ t PJ B t PK Σ t ) . . . == >> ˜ ¯ ) . . . ˜ ¯ BP = Str (PK BPJ Y )Str (ZP J K =−

 δJ¯,J δK,K ¯

) i(J + K

˜ ... . Str (PJ Y PK Z)

(10.10.21)

Here in the third line Z˜ denotes the product obtained from Z by steps (i)–(iii) above. The first of the rules in (10.10.17) is thus established. We proceed to the intra-encounter contractions Str (PK BPJ Y PK BPJ Z) . . .. The contracted matrix elements are BKJ and BK J , the latter coinciding up to a ∗ . Hence the only relevant case is J = J¯, K = K ¯ and we get sign with BK, ¯ J¯ ==

>> == >> Str (PK BPJ Y PK¯ BPJ¯ Z) . . . = sK BKJ (Y PK¯ B)J J¯ ZJ¯K . . . (10.10.22)

22 Transposition of a supermatrix involves interchanging the indices of the matrix elements and afterwards flipping the sign of the lower left elements in each 2 × 2 block.

500

10 Semiclassical Roles for Classical Orbits

with  t (Y PK¯ B)J J¯ = Σ t (ΣB t Σ t )(ΣPK¯ Σ t )(ΣY t Σ t )Σ J J¯   ˜ K Y˜ Σ = Σ t BP J¯J

=

ΣJt¯,J B˜ J,K Y˜K,J¯ ΣJ¯,J

= sJ B˜ J,K Y˜K,J¯ .

(10.10.23)

Here we first replaced the matrix sequence by itself doubly transposed; note that double transposition of a supermatrix (not the identity operation!) leaves its Bosonic elements like the one with the subscripts J J¯ unchanged. Then we invoked Eqs. (10.10.19), (10.10.20) and noted that for the matrix elements at hand transposition just amounts to interchanging the two subscripts. In the third line we exploited the fact that only those elements of Σ, Σ t are nonzero for which one subscript equals another one barred, i.e., the port directions of motion are opposite. Finally, we employed the identity ΣJt¯,J ΣJ¯,J = sJ . Altogether we get == >> Str (PK BPJ Y PK BPJ Z) . . . == = δJ¯,J δK,K ¯ =− =−

>> ˜ ˜ sK BKJ sJ BJ,K YK,J¯ ZJ¯,K . . .

 δJ¯,J δK,K ¯

sK Y˜K,J¯ ZJ¯,K . . .

 δJ¯,J δK,K ¯

Str (PK Y˜ PJ¯ Z) . . . ,

) i(J + K ) i(J + K

(10.10.24)

the intra-encounter contraction rule in (10.10.17); the intermediate steps rely ˜ on (10.9.22) and the quantity Y˜ defined in analogy to Z. ˜ as the Rules analogous to (10.10.17) also hold for contractions between two B’s, interested reader will easily check.

10.10.7 Sigma Model Summing over structures as in Sect. 10.9.5 and again using the assumed largeness of the imaginary part of all energy offsets we get the generating function Z (1) = ei (eA +eB −eC −eD )/2



9  ˜ ˜ BB B B i ˜ exp + eˆ , Str eˆ d[B, B] ˜ 2 1 − BB 1 − B B˜ 8

(10.10.25)

10.11 Outlook

501

equal in appearance as (10.9.23) for the unitary symmetry class, up to a factor 12 in the exponent. The matrix integral again sums up the contributions from all structures which in its raw form is an asymptotic expansion in inverse powers of energy offsets. The integration domain must include the stationary point B = B˜ = 0 and let the ˜ be positive; it needs no further specification eigenvalues of 1 = B B˜ and 1 = BB due to the underlying assumption η  1. To see that the prediction of the Gaussian orthogonal ensemble (GOE) of random-matrix theory is recovered we can argue like in the unitary case. The sigma model of the GOE yields the same matrix integral, two important differences apart. ˜ First, the integration domain is fully specified for all Bosonic elements of B, B; second, the offsets are unrestricted and can even be real. The large-e asymptotics of the integral is again dominated by two saddles of the exponent both of which need to be accounted for unless the offsets have large imaginary parts. The standard saddle at B = B˜ = 0 yields a contribution dominated by small B and identical to the above semiclassical Z (1) . The contribution of the Andreev-Altshuler saddle yields the Riemann-Siegel complement. Our semiclassical results thus recover the high-energy asymptotics of the generating function in agreement with RMT. The same must then be true for the correlator C(e). Moreover, an analytic function can, under rather general conditions which we dare to assume fulfilled, be restored from its asymptotic series by Borel summation [61]. That method involves the term-by-term Fourier transform of the asymptotic expansion leading to a converging series, followed by the inverse Fourier transform of the resulting analytic function. The first stage of the Borel summation applied to the two components of the correlation function gives the spectral form factor K(τ ) both for τ < 1 (from Z (1) ) and τ > 1 (from the Riemann-Siegel complement). The inverse Fourier transform recovers the closed RMT expression for the correlation function in Eq. (10.6.2), for all energies .

10.11 Outlook The interplay of orbit-based semiclassics and the nonlinear sigma model can be expected to become, and in fact has already become helpful in tackling further challenges, beyond the two-point correlator of the level density for the unitary and the orthogonal symmetry classes considered in this chapter. The form factor for the symplectic symmetry class was first determined for times up to the Heisenberg time TH [55, 63–65] and later also beyond TH [66]. Transitions between the symmetry classes are largely understood [67–71]. Higher-order correlations of the level density have thus far been treated only for the unitary class [72]. A somewhat more ambitious goal will be the density of level spacings. Arithmetic billiards [48] and their pseudo-arithmetic variants have recently found a semiclassical explanation of their different spectral statistics [49, 50].

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10 Semiclassical Roles for Classical Orbits

Periodic orbits with periods of the order of the Ehrenfest time [73, 74] or shorter can be investigated for their influence on system specific behavior without RMT counterpart. Transport through ballistic chaotic conductors [7–14] including the “full counting statistics” [75] and localization in long thin wires [15] have been treated successfully. An important step towards understanding the so called Andreev gap in mesoscopic normal/superconducting hybrid structures in terms of periodic orbits was recently taken in [76]. More semiclassical work on the new nonstandard symmetry classes will surely come forth.

10.12 Mixed Phase Space Up to this point, we have mostly taken for granted that the classical dynamics under consideration are hyperbolic, i.e., have no stable periodic orbits. Generic systems, however, are neither integrable nor fully chaotic in the sense mentioned but rather fill their “mixed” phase spaces with chaotic as well as stable regions. we shall not enter an extensive discussion of the rich behavior but confine ourselves to a few remarks and references. If a regular region is so small that it is not resolved by a Planck cell, it leaves but feeble signatures in quantum behavior. For differences to fully hyperbolic systems to become sizable, the islands of regular motion must contain a good fraction of the total number of Planck cells at all visited. Chaotic regions are populated by hyperbolic periodic orbits that allow for semiclassical treatment by Gutzwiller’s trace formula. Islands of regular motion, on the other hand, have central elliptic periodic orbits as their backbones; these are surrounded by chains of further elliptic and hyperbolic orbits. Under refined resolution, that phase space structure appears repeated to ever finer scales in a selfsimilar fashion: there is an infinite hierarchy of islands within islands [77]. Needless to say, quantum mechanics is immune to such excesses, and Planck’s constant sets the limit of resolution. Some spectral characteristics can, at least summarily, be understood by accounting for the separate chaotic and regular regions with their phase space volumes. According to a rather successful hypothesis by Berry and Robnik [78], for instance, the spacing distribution P (S) for the spectrum of a system with ω = Ωc /Ω and ω = 1 − ω = Ωr /Ω as phase-space fractions is approximated by superimposing two independent ladders of levels, one Poissonian and the other Wignerian, in the way explained in Sect. 4.6. The resulting spacing distribution reads, for β = 1, P (S) = ω2 e−ωS erfc

  √    π π π ωS + 2ωω + ω3 S exp −ωS − ω2 S 2 . 2 2 4 (10.12.1)

10.12 Mixed Phase Space

503

As a control parameter is varied to steer a dynamical system from regular toward increasingly chaotic behavior, one passes through bifurcations at which new periodic orbits are born. All periodic orbits, elliptic or hyperbolic, with periods up to half the Heisenberg time TH are relevant for a semiclassical description of spectral properties. The number of times a single Planck cell (located in a chaos dominated region) is visited by periodic orbits with periods up to some value T grows larger than unity once T exceeds the Ehrenfest time TE . At T ≈ TH /2, a typical Planck cell is crowded by exponentially many periodic orbits. We still do not really know how to deal with such a multitude of orbits which tends to give a quantum mechanically illegitimate structure to Planck cells. Worse yet, since all of these orbits must have arisen through bifurcations the cell in question must typically contain many orbits that have just arisen in bifurcations. Fortunately, some progress has been achieved lately in dealing with phase-space structures like chains of islands around central orbits. Such local structures are classically generated by collective actions that can be classified by so-called normal forms typical for the bifurcation at which the chain structure is born from the central orbit. Normal forms themselves are characterized by their codimension, i.e., the number of controllable parameters needed for their location: Codimension-one bifurcations are seen when a single parameter is varied; if one gets close to some bifurcation by changing a single parameter but does not quite hit, one may zero in by fine-tuning a second parameter and then has located a case of codimension two. Codimension-one bifurcations were discussed by Meyer [79] and codimension-two ones more recently by Schomerus [80]. Phase-space structures generated by a normal-form action collectively contribute to the trace of the quantum propagator through terms of the form AeiS/h¯ where S is the normal form in question and A is a suitable prefactor. Such collective contributions uniformly regularize the sum of single-orbit terms that diverge at the underlying bifurcation. Such uniform approximations were pioneered by Ozorio de Almeida and Hannay [81] and brought to recent fruition by Tomsovic, Grinberg, and Ullmo [82, 83] and Schomerus and Sieber [80, 84–87]. It is well to acknowledge that uniform approximations have a long history in optics and in catastrophe theory [88]. A most interesting phenomenon related to bifurcations is the semiclassical relevance of so-called ghost orbits. These are complex solutions of the nonlinear classical equations of motion which of course have no classical reality to them; they arise as saddle-point contributions to integral representations of quantum propagators, with equal formal right as the stationary-phase contributions of real periodic orbits, provided the original path of integration can legitimately be deformed so as to pass over the corresponding saddles; that condition always rules out ghost orbits whose complex actions have a positive real part such that eiS/h¯ would diverge for h¯ → 0. Contributions from ghost orbits are usually suppressed exponentially in the latter limit except in some neighborhood of a bifurcation where the competition of the two “limits” h¯ → 0 and ImS → 0 may render them important, sometimes even so far away from the bifurcation that the ghost can be accounted for with a separate term AeiS/h¯ rather than including it in a collective normal-form term [89].

504

10 Semiclassical Roles for Classical Orbits

If Gutzwiller’s trace formula is augmented by contributions from stable orbits, ghost orbits and clusters of orbits making up phase-space structures characteristic of bifurcations, reasonable semiclassical approximations for whole spectra become accessible for the kicked top (with not too large Hilbert spaces), and the mean error for a level is a small single-digit percentage of the mean spacing [90]. For general systems, and in particular for finite dimensional Hilbert spaces, no competitive semiclassical schemes for calculating spectra is available as yet. The traces tn = Tr F n of powers of the Floquet operator of periodically driven systems and thus the form factor display anomalously large variations in their dependence on control parameters as a bifurcation is passed. The same is true for fluctuations of the level staircase. Every type of bifurcation produces a peaked contribution, and the maximum amplitude follows a power law with a bifurcation specific exponent for its h¯ -dependence [91–94]. Interesting consequences of the coexistence of regular and chaotic motion can be seen in the time evolution of the classical phase-space density, which obeys the Liouville equation in the Hamiltonian case. The pertinent time-evolution operator, commonly called the Frobenius–Perron operator is unitary w.r.t. the Hilbert space of functions one may define on phase space. Thus, the spectrum of the FrobeniusPerron operator lies on the unit circle in the complex plane. However, the resolvent of the Frobenius–Perron operator may have poles inside the unit circle on a second Riemann sheet; these so-called Pollicott–Ruelle resonances [95] have a solid physical meaning as the rates of probability loss from the phase space regions supporting the corresponding eigenfunctions. These resonances can rather simply be detected by looking on phase space with limited resolution. Such blurring automatically arises when the infinite unitary Frobenius-Perron operator is cut to finite size, say N × N, in a basis whose functions are ordered by resolution [96, 97]. Of course, the N × N approximant of the Frobenius-Perron operator is non-unitary and has eigenvalues not exceeding unity in modulus. Those eigenvalues of the finite approximating matrix which are insensitive to variations in N once N is sufficiently large, turn out to be Pollicott-Ruelle resonances. The corresponding eigenfunctions are located on and immediately around elliptic periodic orbits for resonances approaching unity in modulus for large N; resonances smaller than unity in modulus, on the other hand, have eigenfunctions supported by the unstable manifolds of hyperbolic periodic orbits; at any rate, these eigenfunctions are strongly scarred [97]. Once a resonance is identified, it can be recovered through the so-called cycle expansion, i.e., a representation of the spectral determinant in the fashion explained in Sect. 10.5, where the periodic orbits included are those visible in the scars (and their repetitions). These classical findings have quantum analogues. Instead of the phase-space density, one must employ any of the so-called quasi-probability densities that are defined so as to have the phase space coordinates as independent variables and to represent the density operator; examples are the Wigner and Husimi or Glauber’s Q-function [98]. The Liouvillevon Neumann equation for the density operator may be written as an evolution equation for the quasi-probability used, say the Q-function. Thus, a Q propagator arises whose classical analogue is the Frobenius–Perron operator. In contrast to the

10.13 Problems

505

2 ×N 2 matrix whose classical Frobenius–Perron operator, the Q propagator is an NQ Q dimension is fixed through Weyl’s law by the number of Planck cells contained in the classically accessible phase-space volume Ω, i. e. NQ = hΩf . The spectrum ¯ of the Q propagator must lie on the unit circle due to the unitarity time evolution of the density operator. It turns out that classical and quantum dynamics become indistinguishable if looked upon with a resolution much coarser than a Planck cell: N × N approximants of the Frobenius–Perron operator and the Q propagator with N Nq yield the same resonances and scarred eigenfunctions [99], which is a rather intuitive result indeed. Of the host of observations of mixed phase spaces in real systems, at least some examples deserve mention. Conductivity measurements for two-dimensional antidot structures in semiconductors [100] have revealed classical [101] and semiclassical [102] manifestations of mixed phase spaces. Magneto-transport through chaotic quantum wells has recently attracted interest [103], as has the radiation pattern emerging from mesoscopic chaotic resonators [104]. The mass asymmetry in nuclear fission has found a semiclassical interpretation [105]. Atomic hydrogen in a strong magnetic field remains a prime challenge for atomic spectroscopy [106]. Finally, quasi-particle excitations of Bose–Einstein condensates trapped in anisotropic (even axially symmetric) parabolic potentials are nonintegrable if their energies are comparable to the chemical potential [107]; these latter phenomena make for a particularly nice enrichment of known chaotic waves, inasmuch as the underlying wave equation is the nonlinear Gross–Pitaevski equation.

10.13 Problems 10.1 Retrace your first steps into WKB terrain: Entering Schrödinger’s equation ih¯ ψ˙ = (p2 /2m+V (x))ψ with the ansatz ψ = A(x, t)eiS(x,t )/h¯ show that to leading order in h¯ the phase S/h¯ obeys the Hamilton–Jacobi equation S˙ + H (x, ∇S) = 0 while the next-to-leading order yields the continuity equation ˙ + divj = 0 for the probability density  = |A|2 and the probability current density j = ∇S/m. 10.2 Modify the Van Vleck propagator (10.2.19) and the trace formula (10.3.5) for f degrees of freedom. 10.3 The baker map operates on the unit square as phase space in two steps, like a baker on incompressible dough: Compression to half height in the q-direction and double breadth in p is followed by stacking the right half of the resulting rectangle on top of the left. Show graphically that the number of intersections of the nth iterate of a line of constant q has 2n intersections with any line of constant p. 10.4 Transcribing the reasoning of Sect. 10.3.2 from discrete to continuous time shows that free motion has minimal action. You will enjoy being led in this classical problem to the Schrödinger equation of a particle in a box.

506

10 Semiclassical Roles for Classical Orbits

mω 2 10.5 Show that the harmonic oscillator has (1) the action S(q, q , t) = 2 sin ωt [(q + q )2 cos ωt − 2qq ], (2) a Van Vleck propagator coinciding with the exact one and shrinking to a delta function every half period, and (3) a conjugate point every half period. After having studied Sect. 10.4, draw out the fate of the initial Lagrangian manifold.

10.6 The Kepler ellipse can be parametrically represented by r = a(1 − e cos x, ) . ma 3 (x, − e sin x, ) t= GM where the radial coordinate r is reckoned from a focus, a is the semimajor axis, e the excentricity, m the reduced and M the total ) mass, and G the gravitational constant. Using a = −GmM/2E and e = 1 + 2EL2 /m3 M 2 G2 with E the energy and L the angular momentum, find the configuration-space caustics and check that ∂ 2 S/∂t 2 = −∂E/∂t diverges thereon. Hint: Look at the turning points of the radial oscillation. Note that for f > 1, in contrast to oscillations of a single degree of freedom, the turning points of one coordinate are not characterized by vanishing kinetic energy. 10.7 Show that the symplectic form (10.4.1) is independent of the coordinates used to compute it. Hint: Use a generating function F (Q, q) to transform canonically from q, p to Q, P .

10.8 Use the definition (E) = i δ(E − Ei ) of the level density of autonomous systems to write the form factor as a sum of unimodular terms e−i(Ei −Ej )TH τ/h¯ . Before embarking on the few lines of calculation, think about what expression for the Heisenberg time TH you will come up with. 10.9 A particle in a one-dimensional box has the energy spectrum Ej = j 2 where for notational convenience h¯ 2 /2m is set equal to unity. Use the infinite-product representation of the sine, sin z = z

∞ j =1

(1 −

z2 ) π 2j 2

(10.13.1)

to show that a convergent zeta function ζ (E) =

∞ j =1

{Aj (E − j 2 )} =

√ sin π E √ π E

results from Aj = −1/Ej . Note that the energy levels are the zeros of ζ(E).

References

507

10.10 Consider the harmonic oscillator with the spectrum E = j − 1/2 for j = 1, 2, . . .. Use Euler’s infinite product for the Gamma function ∞

1 = zeγ z Γ (z)

j =1

   z −z/j e 1+ j

(10.13.2)

and the product (10.13.1) once more to show that the regularizer A(E, Ej ) = −

1 Ej +

1 2

exp

E+

1 2 Ej + 12

yields the zeta function ζ (E) =

1 1 γ (E+ 1 ) 1 2 Γ (E + e ) sin π(E + ) π 2 2

where γ is Euler’s constant. Check that ζ (E) has the eigenvalues as its zeros, i.e., that the additional zeros of the sine are cancelled by the poles of the Gamma function. 10.11 Rederive the Riemann-Siegel lookalike (10.5.28) in a simplified manner by neglecting orbit repetitions already in the exponentiated orbit sum (10.5.10) and expand the exponential in powers of the periodic-orbit sum. Does the stability amplitude FP now differ for hyperbolic orbits with and without reflection? 10.12 Recover the diagonal approximation (10.7.6) starting from the pseudo-orbit expansion (10.6.20) for Z (1) . Drop the restriction TTC ,TD λc . It follows that p oscillates back and forth within only a single allowed interval if λ < λc whatever value the energy takes, whereas for λ > λc , there is an energy range with two separate accessible intervals for p. The classically forbidden region in between separate intervals of oscillation becomes penetrable by quantum tunneling. Consequently, energy levels within the energy range allowing for classically separate motions but well away from the “top of the barrier” (the minimum of U+ (p) in the present example)#can come in close pairs with the small splitting proportional to the factor exp{−| barrier dϕp|/h¯ }; in other words, tunneling through a high and thick barrier is exponentially suppressed and thus very slow. The three parts of Fig. 11.5 refer to three different values of λ, one of which is subcritical so as to allow for only one classically accessible p-interval; quantum mechanically. no level spacing is made small by tunneling if λ varies in the subcritical range, as is obvious from the level dynamics in Fig. 11.4. The middle part of Fig. 11.5 pertains to a slightly supercritical value in the neighborhood of which near crossings appear in the level dynamics which persist onto the third part which has its λ closer to π/2. We should clarify why some and which fraction of the avoided crossings are strongly avoided. These distinguished encounters happen near a line E(λ), not drawn into the level dynamics of Fig. 11.4, which gives the energy of the minimum of the upper boundary U+ (p, λ) of the banana of Fig. 11.5, corresponding to the “top of the potential barrier.” Clearly, pairs of levels near that “top” cannot come very close to one another since here tunneling # processes have but a small and thin barrier to penetrate; thus the factor exp{−| barrier dϕp|/h¯ } determining the splitting is not small. A substantial fraction (finite even when N → ∞) of all levels will get close to the curve E(λ) such that the number of strongly avoided crossings is proportional to N, quite small a minority indeed compared to the unresolved crossings whose number, we have seen, is of the order N 2 . It may be well to note that for a given value of λ the energy E(λ) defines a contour line in phase space which is the separatrix between periodic orbits encircling either one or the other of the two stable fixed points (the latter corresponding to the two minima of U+ (p)) and those encircling both fixed stable points; the separatrix itself of course intersects itself at the hyperbolic fixed point. One final example of an integrable Hamiltonian is needed to convince us that strongly avoided crossings are exceptional events in the semiclassical limit N → ∞. In search of a Hamiltonian H (λ) whose N levels display roughly N 2 avoided crossings with minimal spacings a sizable fraction of the mean spacing, we might try a large-order polynomial in angular momentum components like a Chebyshev polynomial of the first kind, Tn (Jz /j ) with n  1, in   H = Tn (Jz /j ) + 2Jx /j cos λ + (Jz /j ) sin λ .

(11.5.8)

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11 Level Dynamics

It is easily checked that the ensuing level dynamics has the majority of crossings strongly avoided, if the dimension 2j + 1 of the Hilbert space is comparable to the order n; but as soon as j  n, the majority of crossings become barely resolvable as avoided. If for the sake of illustration we fix n = 10 and count the percentage of closest approach spacings whose ratio with the mean spacing is in excess of e−4 , we find that percentage as 98, 44, and 31, respectively, for j = 10, 20, 100. To conclude, integrable dynamics have level dynamics corresponding to near ideal PYG’s. The weak nonideality lies in the rare strongly avoided crossings and the exponential smallness of most closest-approach spacings.

11.6 Two-Body Collisions As a final prelude to the promised statistical analyses, it is useful to consider what can be learned from the close encounter of two particles of the Pechukas–Yukawa gas corresponding to a single multiplet in a system with global classical chaos. We shall recover the insight already obtained by almost degenerate perturbation theory in Sect. 3.4. When two of the N particles, to be labeled 1 and 2, approach each other so closely that their mutual force by far exceeds the force exerted on either by any other particle, the collision effectively decouples from the rest of the dynamics. Restricting the consideration to the Floquet case, from (11.2.13) and (11.2.15), φ¨1 = −φ¨2 = 14 |l12 |2 cos3 (φ12 /2) l˙12 = 0 ,

sin (φ12 /2)

(11.6.1)

the latter equation is due to the absence of small denominators sin2 (φ12 /2) in l˙12 given by Eq. (11.2.18). Obviously, the center of mass (φ1 + φ2 )/2 moves uniformly, while the relative coordinate φ = φ12 /2 obeys (setting l12 = l for short) cos φ 1 ∂V φ¨ = |l|2 3 = − 4 ∂φ sin φ

(11.6.2)

where the potential energy U (φ) =

1 |l|2 8 sin2 φ

(11.6.3)

is π-periodic in φ and confines the coordinate φ to [0, π] if φ is in that interval initially. By using energy conservation, one easily constructs the solution  √  φ(λ) = arccos cos φˆ cos 2E(λ − λˆ .

(11.6.4)

11.7 Ergodicity of Level Dynamics and Universality of Spectral Fluctuations

529

Here λˆ denotes one of the times of closest approach of the two particles, 2φˆ is the corresponding angular distance, and E=

|l|2

(11.6.5)

8 sin2 φˆ

is the energy of the nonlinear oscillation. Consistent with the neglect of all other particles, we may assume that φˆ 1 and study the close encounter at times λ ≈ λˆ by expanding all cosines as cos x = 1 − x 2 /2. The result is 8 φ = ± φˆ 2 + |l|2

ˆ 2 (λ − λ) 4φˆ 2

91/2 .

(11.6.6)

The discriminant in (11.6.6) is a sum of two or three nonnegative terms depending on whether the angular momentum l is real or complex. Thus, the codimension of a level crossing is found once more as n = 2 and n = 3 for systems with, respectively, O(N) and U (N) as their canonical groups. As could have been expected on intuitive grounds, isolating a pair of colliding particles from the remaining N − 2 particles is equivalent to the nearly degenerate perturbation theory applied to an avoided level crossing in Sect. 3.2. The case in which Sp(N) is the canonical group can be treated similarly, even though Kramers’ degeneracy looks slightly strange in the fictitious-particle picture: the particles have pairwise identical positions φ and an avoided crossing corresponds to a collision of two pairs of particles with φ1 = φ1¯ , φ2 = φ2¯ . Now, the equations of motion read (see Problem 11.6) φ¨1 = −φ¨2 =

 cos (φ /2) 1 12 |l12 |2 + |l12¯ |2 3 4 sin (φ12 /2)

(11.6.7)

with the coupling strength |l12 |2 + |l12¯ |2 again effectively constant. The solution φ(λ) given in (11.6.6) for the nondegenerate cases holds here, too, but with the replacement |l12 |2 → |l12 |2 + |l12¯ |2 , which immediately yields the correct codimension of a level crossing, n = 5.

11.7 Ergodicity of Level Dynamics and Universality of Spectral Fluctuations 11.7.1 Ergodicity Now, we can take up one of the most intriguing questions of the field. Why do dynamical systems with global classical chaos display universal spectral fluctuations

530

11 Level Dynamics

in fidelity to random-matrix theory, and for what reasons do exceptions occur? Our discussion will focus primarily on periodically driven systems and their quasienergy spectra. A motivating glance at Fig. 11.3 is recommended; there we see the quasi-energy spectrum of a kicked top (F = exp(−i 2jλ+1 Jz2 ) exp(−ipJy )) with the torsion strength λ varied from the integrable case λ = 0 to the range for which the classical motion is globally chaotic, λ  5. Clearly, equilibrium reigns in the chaotic range while for λ up to, say, 5 we encounter an equilibration process. We had already seen in Sect. 11.2 that the level dynamics of an N × N Floquet matrix F = e−iλV e−iH0 proceeds on an N-torus. The pertinent frequencies are the eigenvalues ωμ of V for which we assume the generic case of no degeneracy. Moreover, if the eigenvalues ωμ are incommensurate, the motion on the N-torus will be ergodic. Ergodicity means that we can equate “time” averages, i.e., λ averages of certain observables with ensemble averages. The appropriate ensemble to use is the generalized microcanonical ensemble nailing down the independent constants of the motion. In particular, it is not legitimate to work with the primitive microcanonical ensemble Z −1 δ(H − E) which fixes only the energy of the Pechukas–Yukawa gas, simply because the trajectory of level dynamics explores only an N-torus rather than the full energy shell. As to the observables for which ensemble average equals “time” average (and ultimately even typical instantaneous values), the same thoughts as for other many-body systems apply. Quantities like spectrally averaged products of the level density or the spacing distribution can be expected to be self-averaging and thus to qualify for statistical description. Others, like the not spectrally averaged products of the level density, the (neither spectrally nor temporally averaged) form factor, or the localization length we shall meet in Chap. 8, allow for weaker statements only: Averages over the “time” λ still equal ensemble averages but there are strong ensemble fluctuations. Here, we are concerned mostly with self-averaging quantities. Time averages for the Pechukas–Yukawa gas are control parameter averages over a family of original dynamical systems: A λ average over, say, the spacing −i(

λ

J 2 +αJ )

z distribution for F (λ) = e 2j+1 z e−iβJx involves not a single kicked top but a whole one-parameter family of such. How can we draw conclusions for the actually observed universality of spectral fluctuations of individual dynamic systems like a single kicked top? The clue to the answer lies in the further question over how long an interval Δλ must we calculate time averages before the ensemble means are approached. One often stipulates that the interval Δλ is large in the sense Δλ → ∞. Such an extreme request allows us to accommodate and render weightless equilibration processes starting from initial conditions far from equilibrium. For level dynamics, such equilibration would arise if F (λ)|λ=0 = e−iH0 were the quantized version of an integrable classical dynamics and the switch-on of e−iλV paralleled the expansion of chaos to (almost) global coverage of phase space. On the other hand, if we bar such initial situations and assume that level dynamics displays avoided crossings right away, the interval need not be much larger than the mean control parameter

11.7 Ergodicity of Level Dynamics and Universality of Spectral Fluctuations

531

distance between subsequent avoided crossings of a pair of neighboring levels. For the Pechukas–Yukawa gas, we could call that interval a collision time, i.e., the mean temporal distance between two subsequent collisions of a particle with its neighbors.

11.7.2 Collision Time The collision time just alluded to should scale with the number of particles (i.e., the number of levels N) like a power [23], λcoll ∝ N −ν with ν > 0 .

(11.7.1)

It is well to devote a little consideration to that power-law behavior and to indicate that the exponent ν cannot be expected to be universal, save for its positivity under conditions of chaos. First, let us consider a Hamiltonian of the structure H (λ) = H0 + λV and take H0 and V as independent random N × N matrices from, say, the GUE. Both should have zero mean and the same variance, H0ij = Vij = 0 ,

|H0ij |2 = |Vij |2 = 1/N .

(11.7.2)

According to Wigner’s semicircle law, see (5.7.2), (5.7.3), the mean level spacing is Δ ≡ Ei+1 − Ei ∼ N1 . The level velocity vanishes in the ensemble mean and has the mean square pi2 = ψi |V |ψi 2 ∼

1 . N

(11.7.3)

Thus, a typical level velocity is p ∼ N−2 1

(11.7.4)

and the simple estimate λcoll ≈ Δ/p ∼ N −1/2 yields the exponent ν = 1/2. The same value of the exponent results if we consider a Floquet operator F (λ) = e−iH0 e−iλV and again take H0 and V random as before. For the Floquet operator of the kicked top with V = Jz2 /(2j + 1), the following argument will reveal ν = 32 . Fixing the number of levels as N = 2j + 1 we get the

1 N2 1 mean level velocity v ≡ N1 N i Vii = N TrV = 12 (1 + O( N )); but this describes a drift common to all levels and is irrelevant A typical velocity

for collisions.

N relative 2 , where again 2 ≡ 1 ˜ p results from the variance p2 = N1 N (V − v) V i=1 ii i=1 ii N the matrix elements are meant in the eigenrepresentation of the Floquet operator F (λ); but in that representation the perturbation V˜ looks like a full matrix with

2 1 ˜2 1 N2 1 ˜2 ˜ ˜ i Vii ≈ 0 and i Vii ≈ N ij Vij = N TrV = 180 (1 + O( N )); we conclude

532

11 Level Dynamics

−2 p ∝ N 2 . The mean level spacing 2π and the N yields the collision time ∝ N 3 exponent ν = 2 ; the latter value was confirmed by numerically following level dynamics for 10 < j < 160 [23]. At any rate, the asymptotic disappearance of the collision time under conditions of chaos is in sharp contrast to the parametric change of levels in integrable systems. No collision time could at all sensibly be established for the latter, as is clear from our discussion of Sect. 11.5. 1

3

11.7.3 Universality But inasmuch as Weyl’s law implies N ∝ (2π h¯ )−f , the minimal control parameter window under discussion can be specified as Δλ ∼ h¯ f ν and thus vanishingly small from a classical perspective. Thus, all of the different quantum systems involved in the control parameter alias “time” average can be said to have the same classical limit. Such classically vanishing control parameter intervals have been suggested by Zirnbauer [24] in an attempt at demonstrating the universality of spectral fluctuations by the superanalytic method expounded in Chaps. 6 and 7. The assumption of a close-to-equilibrium initial condition for the Pechukas– Yukawa gas means roughly, as already indicated above, that the original dynamical system should have global chaos to begin with. That assumption is crucial since an equilibration of the gas corresponding to the classical transition from regular to dominantly chaotic behavior takes a time of classical character independent of N ∝ h¯ −f and thus much larger than λcoll . Back to the assumption of incommensurate eigenvalues ωμ of V which entails ergodicity on the N-torus. We can and should partially relax that assumption; we should since we often encounter commensurate eigenvalues, most notably perhaps −i

λ

J2

for the kicked top with V ∝ Jz2 . The unitary operator e 2j+1 z is periodic in λ with period 2π(2j + 1) if 2j + 1 is odd and 8π(2j + 1) if 2j + 1 is even. We may be permissive since, inasmuch as that period is O(N) while the collision time is O(N −1 ), the distinction of strict and approximate ergodicity on the torus is quite irrelevant. In such cases we can still equate time averages with ensemble averages for the Pechukas–Yukawa gas, choosing Δλ of the order of λcoll and the generalized microcanonical ensemble to describe equilibrium on the torus. To understand finally why spectral fluctuations for systems with global classical chaos are universal and faithful to random-matrix theory, we now have to take the most difficult step and show that equilibrium statistical mechanics for the Pechukas–Yukawa gas entails random-matrix type behavior for the relevant spectral characteristics. This will be the object of the next section. Before precipitating ourselves into that rather serious adventure, it is well to mention exceptional classes of dynamical systems that, although globally chaotic in their classical behavior, do not display level repulsion and therefore do not fall into the universality classes we are principally dealing with here. Prototypical for

11.8 Equilibrium Statistics

533

one exceptional class is the kicked rotor which we shall treat in some detail in Chap. 8. Classical chaos notwithstanding, the eigenfunctions of the Floquet operator are exponentially localized in the H0 basis. It follows that matrix elements of V with respect to different quasi-energy eigenstates are exponentially small if the states involved do not overlap; should the corresponding levels engage in an avoided crossing, the distance of closest approach would be exponentially small compared to the mean spacing. But since almost all pairs of states have that property in a Hilbert space whose dimension N is large compared to the (dimensionless) localization length l, almost all avoided crossings become visible only after a blow-up of the eigenphase scale by a factor of the order el  1; on the scale of a mean spacing such narrow anticrossings look like crossings and this is why quantum localization comes with Poissonian spectral fluctuations. The qualitative reasoning just presented about localization may be formalized in the fashion of our above treatment of two-body collisions. By appeal to conservation of the energy 12 ϕ˙ 2 + U (ϕ) according to (11.6.2) and (11.6.3), we can express the distance of closest approach 2ϕˆ in terms of an initial distance 2ϕ0 taken to equal the v2

ϕ0 2 mean distance 2π/N and an initial velocity −ϕ˙ 0 = v0 as ( sin ) = 1+ |V 0 |2 where sin ϕˆ 12 the matrix element V12 refers to the two Floquet eigenstates of the colliding levels at the initial “time”; that matrix element is exponentially small for nonoverlapping states and makes the avoided crossing look like a crossing. A second class of exceptions is constituted by some billiards on surfaces of constant negative curvature that display what is sometimes called arithmetic chaos. These have certain symmetries of the so-called Hecke type that produce effectively independent multiplets of quantum energy levels; though their classical effect is not to provide integrals of the motion which would make for integrability; they rather provide periodic orbits degenerate in action (length) whose degeneracy increases exponentially with the period [25]. The reason for such exceptional quantum behavior was identified in Refs. [26, 27]. Quantum symmetries without classical counterparts have also been found for cat maps [28, 29] and shown to cause spectral statistics not following the usual association of classical symmetries with quantum universality classes. In brief, level dynamics and the equilibrium statistics based thereon goes an important step towards explaining the universality of spectral fluctuations: full chaos in the classical limit is revealed as necessary. It does not yield, however, a simple criterion allowing us to understand why the exceptional systems mentioned do not qualify for universality.

11.8 Equilibrium Statistics In the limit of a large number of levels N, the fictitious N-particle system calls for a statistical description [30, 31]. As already mentioned, the phase-space trajectory ergodically fills an N-torus and the appropriate equilibrium phase-space density is

534

11 Level Dynamics

therefore the generalized microcanonical one 98 N 9 8 -   δ Cmn (φ, p, l) − C¯ mn δ(lmm ) . (φ, p, l) ∼ mn

(11.8.1)

m=1

Accounted for in the distribution function (11.8.1) are the N(N − 1) independent constants of the motion Cmn (φ, p, l) identified in Sect. 11.3 as well as the N conserved diagonal angular momenta lmm . Inasmuch as only the reduced distribution of the eigenphases,  P (φ) =

dp dl(φ, p, l) ,

(11.8.2)

will be of interest in the remainder of the present chapter we do not have to worry about the gauge phases; whether or not we fix N phases for the off-diagonal angular momenta to sharp values, P (φ) will not be affected. In order not to carry along unnecessary burden we propose to imagine the primitive gauge, θ˙m = 0, such that neither the Hamiltonian H nor the Hamiltonian equations of motion for the φ, p, l contain the gauge phases any longer. The φ dependence of P (φ) arises through the off-diagonal elements of the matrix v given in (11.2.7). As is obvious from that definition, for a fixed finite value of the angular momentum lmn , the associated vmn tends to infinity when the two coordinates φm and φn become equal; the phase-space functions Cμ (φ, p, l) involving v diverge then, too, and the coordinate distribution thus vanishes. To find out how P (φ) approaches zero as two particles suffer a close encounter, we may change the integration variables in (11.8.2) according to lmn → vmn =

lmn iφ i(e mn − 1)

, m = n .

The Jacobian of this transformation,     -   ∂l   J (φ) = det = e−iφm − e−iφn  ,  ∂v

(11.8.3)

(11.8.4)

(m,n)

is a function of the coordinates φ deserving special attention. The product in (11.8.4) is over all distinct pairs of particles, each pair counted with a multiplicity characteristic of the group of canonical transformations of the underlying Floquet operator. In the orthogonal case, the lmn = −lnm are real, and N(N − 1)/2 in number and the multiplicity in question is one. When U (N) applies, however, the multiplicity is two ∗ are complex and the original l integral is over N(N − 1)/2 since the lmn = −lnm complex planes. The symplectic case, finally, produces the multiplicity four, as can be seen from the following symmetry argument.

11.8 Equilibrium Statistics

535

The time reversal covariance T F T −1 = F † yields, on differentiation w.r.t. λ, T V T −1 = F † V F .

(11.8.5)

We assume that the eigenbasis of F is organized such that |m and T |m ≡ |m ¯ are the two eigenvectors pertaining to the quasi-energy φm . Taking matrix elements in (11.8.5) between the states pertaining to a pair of quasi-energies, one obtains Vmn = e−iφmn Vn¯ m¯ ,

Vmn¯ = −e−iφmn Vnm¯ .

(11.8.6)

Now, it is obvious that, of the four matrix elements of V associated with the pair of levels φm , φn with m < n, Vmn , Vmn ¯ , Vmn¯ , Vm ¯ n¯ , only two are independent. Thus, the integral in (11.8.2) is over the complex plane for both lmn and lmn¯ , and the Jacobian (11.8.4) acquires four factors |e−iφm − e−iφn | for each pair m < n. To summarize, the Jacobian in question can be written as J (φ) =

 n−1 n−1 -  -    2 sin φij  = e−iφi − e−iφj   2 

i −1/2 the intracluster positions Ei in the Coulomb-force denominator can be replaced by the center energy E of the cluster; after inserting the rescaling (11.14.12), (11.14.13) for the n-level correlation function, the local homogeneity (11.14.15), (11.14.16) can be invoked. Finally, the integral over the interval ΔE ∼ N ε can, using ε > −1/2 once more, be extended over the whole energy axis such that the hierarchy takes the form ∂ ˜ n = ∂ τ˜

1 ... n i

−

⎛ ∂ ⎝ − ∂ei

1 ... n i

∂ P ∂ei

1 ... n j (=i)

⎞ 1 1 ∂ ⎠ + ˜ n ei − ej 2 ∂ei

 den+1

˜ n+1 , n>1. ei − en+1

(11.14.21)

Now, the correlation function ˜ n+1 under the integral represents a local cluster of n + 1 levels. Remarkably enough, apart from the absence of the linear drift term from (11.14.21), the original hierarchy (11.14.4) and the rescaled one have identical structures.

11.15 Problems

585

Having invoked local homogeneity for the correlation functions ˜ n of local clusters in showing the “locality” of the integral term in (11.14.21), it is imperative to demonstrate the consistency of the homogeneity property (11.14.16) with the hierarchy (11.14.21). Indeed, by differentiating (11.14.21) and

integrating by parts in the last term, it is easy to check that the quantities Dn = 1i ... n (∂/∂ei )˜ n obey the same hierarchy of integrodifferential equations (11.14.21) as the correlation functions ˜ n . Consequently, if all Dn are of order 1/N initially, none of them can grow enough to exceed that order of magnitude. To appreciate this statement, the reader should realize that both the diffusion and the Coulomb repulsion tend to make ˜ n smooth and thus Dn small. The separation of energy and time scales for local fluctuations and global relaxation of the density must hold for any reasonable initial condition. A particularly interesting example is the transition from the GOE to the GUE treated in the preceding section. The transition from the Poissonian ensemble to the GUE, also briefly mentioned in Sect. 11.13, must abide by that separation as well. It follows that the latter transition cannot be considered the quantum parallel of classical crossovers from predominantly regular motion to global chaos, except when that classical transition is abrupt. In fact, we know of no classical system with a Hamiltonian H0 + λV or a Floquet operator e−iλV e−iH0 which, while integrable for λ = 0, is globally chaotic for all non-zero values of λ. Such abrupt classical crossovers can happen for billiards upon changes of the boundaries [66]. In that case, however, level dynamics does not apply: Even a small deformation of the shape of a billiard boundary amounts to strong perturbations for sufficiently high degrees of excitation. Neither the Gaussian ensembles of Hermitian random matrices nor their Brownian-motion dynamizations could yield good models of Hamiltonians of quantum systems with globally chaotic classical limits, were it not for the disparity of scales under discussion. In fact, concrete dynamic systems do not in general obey the semicircle law for their level densities; it is only the local fluctuations in the spectra which tend to be faithful to random-matrix theory. Were there not a complete decoupling of local fluctuations and global variations of the density, provided by the scale separation, the predictions of random-matrix theory would be as unreliable locally as they in fact are globally [64].

11.15 Problems 11.1 Determine the number of independent dynamic variables of the fictitious N-particle system when O(N), U (N), and Sp(N) are the groups of canonical transformations. 11.2 Show that the angular momentum matrix l fulfills l˜ = −l, l † = −l, and ˜ = −Zl when the Floquet operator has O(N), U (N), and Sp(N), respectively, lZ as its canonical group.

586

11 Level Dynamics

11.3 Having done this problem you will understand why the variables lmn are often called angular momenta and why their Poisson brackets define the Lie algebras o(N), u(N), and sp(N) in the orthogonal, unitary, and symplectic cases, respectively. Imagine an N-dimensional real configuration space with Cartesian coordinates x1 . . . xN . A 2N-dimensional phase space arises by associating N conjugate momenta p1 . . . pN . Rotations in the configuration space are generated by the angular momenta lmn = 12 (xm pn − xn pm ). The Poisson brackets Think about how to generalize to complex x, p to get the Poisson brackets (11.2.25) for the unitary case where the lmn generate rotations in an N-dimensional complex vector space. In what vector space do the lmn generate rotations for the symplectic case? 11.4 Verify x˙m = pm p˙m = −

 l(=m)

l˙mn =



2lml llm (xm − xl )3

 lml lln (xm − xl )−2 − (xl − xn )−2

l(=m,n)

as the level dynamics for time-independent Hamiltonians H = H0 + λV . 11.5 Due to the Jacobi identity (11.2.24) the Poisson bracket of two conserved quantities is conserved as well. As a check on the reasoning of Sect. 11.3 it might be interesting to verify by explicit calculation that the Poisson bracket of any two of the independent constants of the motion either vanishes identically or is a linear combination of the independent ones. Please do not try to do this for the general case {Cmn , Cm n }. However, an instructive and easy exercise is to check {lkk , Cmn } = 0 and {C0m , C0n } = 0. 11.6 Prove (11.6.7) by showing that l11¯ = l22¯ = 0 and |l12 |2 + |l12¯ |2 = const. Use the T -invariance of F with T 2 = −1 and the ensuing identity for T V T −1 . 11.7 Using the arguments of Sect. 5.7 show that the ensemble (11.13.12) implies Wigner’s semicircle law for the mean level density in the limit N → ∞. 11.8 Show that the free diffusion described by (11.13.2) with λ2 as a “time” implies a Brownian motion with Coulomb gas interaction for the eigenvalues of H. Discuss how this differs from (11.13.13) and (11.13.19). 11.9 Study a two-body collision in the Coulomb gas model (11.13.13) and (11.13.19).

11.15 Problems

587

11.10 Write the Fokker–Planck equation (11.13.13) and (11.13.19) in the compact

√ −1 form ˙ = L, L = ∂  ∂i GUE . Transform as  = GUE , ˜ and show i GUE i that the transformed generator L˜ is a Hermitian differential operator. Furthermore, show that L˜ contains pair interactions inversely proportional to the squared distance between the “particles”, as does the Hamiltonian level dynamics (11.2.29). Appre˜ ciate the conceptual difference between the latter and the motion generated by L. 11.11 Derive the Hamiltonian flows generated by the Hamiltonians (11.14.13) and (11.14.15) by inserting the transformation Em (τ ) = g(τ ) · xm (λ(τ )) into Pechukas’ dynamics (11.2.29). 11.12 Proceeding as in Sect. 11.3, find the Lax form of the Pechukas equations (11.2.29). Hint: The Pechukas equations follow from (11.2.13), (11.2.15), and (11.2.18) by linearizing as cos x → 1, sin x → x. 11.13 Find the Lax form of the modified Pechukas equations pertaining to the Hamiltonian function (11.14.15). Hint: Proceed as in Sect. 11.3 but use (11.14.6). 11.14 Establish the most general canonical ensemble for the modified Pechukas– Yukawa gas (with the confining potential) which rigorously gives the joint distribution of eigenvalues (5.4.2) of the Gaussian ensembles of random matrices. Use the results of Problem 11.13, and argue as in Sect. 11.5. Where does the Gaussian factor come from? 11.15 Find the spacing distribution P (S, λ) interpolating between the Gaussian orthogonal and unitary ensemble of 2 × 2 matrices. Use the method of Sect. 5.6, but start from the matrix density (11.13.12), taking the latter for N = 2. The result is [60] P (S, λ) =

$ 2 2 1 + λ2 /2Sφ(λ)2 e−S φ(λ) /2 erf (Sφ(λ)/λ)

where φ(λ) =

$

8  √ 9 ) λ 2 2 λ 1 + λ2 /2 π/2 1 − arctan √ − π 2 2 + λ2

and λ2 = e2τ − 1. 11.16 Repeat Problem 11.15 but for the transition of Poisson to Wigner. Start from the Gaussian defined by (11.13.28) to find 2 −S 2 ψ(λ)2 /4



∞

P (S, λ) = Sλψ(λ) e

dξ e−ξ

2 −2ξ λ

I0 [Sψ(λ)] .

0

By requiring S¯ = 1, the scale factor ψ(λ) can be determined and is related to the Kummer function [60, 61].

588

11 Level Dynamics

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

P. Pechukas, Phys. Rev. Lett. 51, 943 (1983) T. Yukawa, Phys. Rev. Lett. 54, 1883 (1985) T. Yukawa, Phys. Lett. 116A, 227 (1986) F. Haake, M. Ku´s, R. Scharf, Z. Phys. B65, 381 (1987) K. Nakamura, H.J. Mikeska, Phys. Rev. A35, 5294 (1987) H. Frahm, H.J. Mikeska, Z. Phys. B65, 249 (1986) M. Ku´s, Europhys. Lett. 5, 1 (1988) O. Bohigas, M.J. Giannoni, C. Schmit, Phys. Rev. Lett. 52, 1 (1984) F. Haake, M. Ku´s, Europhys. Lett. 6, 579 (1988) F. Dyson, J. Math. Phys. 3, 140 (1962) M.L. Mehta, Random Matrices (Academic, New York, 1967; 2nd edition 1991; 3rd edition Elsevier 2004) 12. M. Ku´s, F. Haake, D. Zaitsev, A. Huckleberry, J. Phys. A 30, 8635 (1997) 13. A. Huckleberry, D. Zaitsev, M. Ku´s, F. Haake, J. Geom. Phys. 37, 156 (2001) 14. M. Hardej, M. Ku´s, C. Gonera, P. Kosi´nski, J. Phys. A40, 423 (2007); for a more extended version see arXiv:nlin.cd/0608026 v1 11 Aug 2006 15. K. Mnich, Phys. Lett. A176, 189 (193) 16. P. Gaspard, S.A. Rice, H.J. Mikeska, K. Nakamura, Phys. Rev. A42, 4015 (1990) 17. F. Calogero, C. Marchioro, J. Math. Phys. 15, 1425 (1974) 18. B. Sutherland, Phys. Rev. A5, 1372 (1972) 19. J. Moser, Adv. Math. 16, 1 (1975) 20. S. Wojciechowski, Phys. Lett. 111A, 101 (1985) 21. P.D. Lax, Comm. Pure Appl. Math. 21, 467 (1968) 22. P.A. Braun, Rev. Mod. Phys. 65, 115 (1993) ˙ 23. P.A. Braun, S. Gnutzmann, F. Haake, M. Ku´s, K. Zyczkowski, Found. Phys. 31, 614 (2001) 24. M.R. Zirnbauer, in Supersymmetry and Trace Formulae; Chaos and Disorder, ed. by I.V. Lerner, J.P. Keating, D.E. Khmelnitskii (Kluwer Academic/Plenum, New York, 1999) 25. E.B. Bogomolny, B. Georgeot, M.-J. Giannoni, C. Schmit, Phys. Rep. 291, 219 (1997) 26. P. Braun, F. Haake, J. Phys. A43 262001 (2010); arXiv:1001.3339v2 [nlinCD] (2010) 27. P. Braun, arXiv:1508.02075v1 [nlin.CD] (2015) 28. J.P. Keating, Nonlinearity 4, 309 (1991) 29. J.P. Keating, F. Mezzadri, Nonlinearity 13, 747 (2000) 30. B. Dietz, Dissertation, Essen (1991) 31. B. Dietz, F. Haake, Europhys. Lett. 9, 1 (1989) 32. F. Haake, G. Lenz, Europhys. Lett. 13, 577 (1990) 33. D.J. Thouless, Phys. Rep. 13C, 93 (1974) 34. P. Gaspard, S.A. Rice, H.J. Mikeska, K. Nakamura, Phys. Rev. A42, 4015 (1990) 35. D. Saher, F. Haake, P. Gaspard, Phys. Rev. A44, 7841 (1991) 36. B.D. Simons, A. Hashimoto, M. Courtney, D. Kleppner, B.L. Altshuler, Phys. Rev. Lett. 71, 2899 (1993) 37. M. Kollmann, J. Stein, U. Stoffregen, H.-J. Stöckmann, B. Eckhardt, Phys. Rev. E49, R1 (1994) 38. D. Braun, E. Hofstetter, A. MacKinnon, G. Montambaux, Phys. Rev. B55, 7557 (1997) 39. D. Braun, G. Montambaux, Phys. Rev. B50, 7776 (1994) 40. J. Zakrzewski, D. Delande, Phys. Rev. E47, 1650 (1993); ibid. 1665 (1993); D. Delande, J. Zakrzewski, J. Phys. Soc. Jpn. 63, Suppl. A, 101 (1994) 41. F. von Oppen, Phys. Rev. Lett. 73,798 (1994); Phys. Rev. E51, 2647 (1995) 42. Y.V. Fyodorov, H.-J. Sommers, Phys. Rev. E51, R2719 (1995); Z. Phys. B 99, 123 (1995) 43. S. Iida, H.-J. Sommers, Phys. Rev. E49, 2513 (1994) 44. E. Akkermans, G. Montambaux, Phys. Rev. Lett. 68, 642 (1992) 45. Y.V. Fyodorov, A.D. Mirlin, Phys. Rev. B51, 13403 (1995) 46. M. Sieber, H. Primack, U. Smilanski, I. Ussishkin, H. Schanz, J. Phys. A28, 5041 (1995)

References

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47. H.-J. Stöckmann, Quantum Chaos, An Introduction (Cambridge University Press, Cambridge, 1999) 48. M. Barth, U. Kuhl, H.-J. Stöckmann, Phys. Rev. Lett. 82, 2026 (1999) 49. M.V. Berry, J. Phys. A10, 2083 (1977) 50. V.N. Prigodin, Phys. Rev. Lett. 75, 2392 (1995) 51. X. Yang, J. Burgdörfer, Phys. Rev. A46, 2295 (1992) 52. B.D. Simons, B.L. Altshuler, Phys. Rev. Lett. 70, 4063 (1993); Phys. Rev. B48, 5422 (1993) 53. J. Zakrzewski, Z. Phys. B98, 273 (1995) ˙ 54. I. Guarneri, K. Zyczkowski, J. Zakrzewski, L. Molinari, G. Casati, Phys. Rev. E52, 2220 (1995) 55. G. Lenz, F. Haake, Phys. Rev. Lett. 65, 2325 (1990) 56. G. Lenz, Dissertation (Essen, 1992) 57. H. Risken, The Fokker Planck Equation. Springer Series in Synergetics, vol. 18 (Springer, Berlin/Heidelberg, 1984) 58. A. Pandey, M.L. Mehta, Commun. Math. Phys. 87, 449 (1983) 59. M.L. Mehta, A. Pandey, J. Phys. A16, 2655 (1983) 60. G. Lenz, F. Haake, Phys. Rev. Lett. 67, 1 (1991) 61. E. Caurier, B. Grammaticos, A. Ramani, J. Phys. A23, 4903 (1990) 62. J.B. French, U.K.B. Kota, A. Pandey, S. Tomsovic, Ann. Phys. (N.Y.) 181, 198 (1988) 63. A. Pandey, Quantum Chaos and Statistical Nuclear Physics, ed. by T.H. Seligman, H. Nishioka (Springer, Berlin/Heidelberg, 1986) 64. F.J. Dyson, J. Math. Phys. 13, 90 (1972) 65. L.A. Pastur, Theor. Math. Phys. 10, 67 (1972) 66. L.A. Bunimovich, Funct. Anal. Appl. 8, 73 (1974)

Chapter 12

Dissipative Systems

12.1 Preliminaries Regular classical trajectories of dissipative systems eventually end up on limit cycles or settle on fixed points. Chaotic trajectories, on the other hand, approach so-called strange attractors whose geometry is determined by Cantor sets and their fractal dimension. In analogy with the Hamiltonian case, the two classical possibilities of simple and strange attractors are washed out by quantum fluctuations. Nevertheless, genuinely quantum mechanical distinctions between regular and irregular motion can be identified. The main goal of this chapter is the development of one such distinction, based on the generalization of energy levels to complex quantities whose imaginary parts are related to damping. An important time scale separation arising for dissipative quantum systems will also be expounded: Coherences between macroscopically distinct states tend to decay much more rapidly than quantities with well-defined classical limits. An example relevant in the present context is the dissipative destruction of quantum localization for the kicked rotator. For background information on dissipation the reader may consult the comprehensive treatise of U. Weiss [1]. More special issues related to quantum chaos are dealt with by D. Braun [2].

12.2 Hamiltonian Embeddings Any dissipative system S may be looked upon as part of a larger Hamiltonian system. In many cases of practical interest, the Hamiltonian embedding involves weak coupling to a heat bath R whose thermal equilibrium is not noticeably perturbed by S. Strictly speaking, as long as S + R is finite in its number of degrees of freedom, the Hamiltonian nature entails quasi-periodic rather than truly irreversible temporal behavior. However, for practical purposes, quasi-periodicity © Springer Nature Switzerland AG 2018 F. Haake et al., Quantum Signatures of Chaos, Springer Series in Synergetics, https://doi.org/10.1007/978-3-319-97580-1_12

591

592

12 Dissipative Systems

and true irreversibility cannot be distinguished in the “effectively” dissipative systems S in question. The equilibration time(s) imposed on S by the coupling to R are typically large compared to the time scales τR of all intrabath processes probed by the coupling; in such situations, to which discussions will be confined, the dissipative motion of S acquires a certain Markovian character. Then, the density operator (t) of S obeys a “master” equation of the form (t) ˙ = l(t)

(12.2.1)

with a suitable time-independent generator l of infinitesimal time translations. The dissipative motion described by (12.2.1) must, of course, preserve the Hermiticity, positivity, and normalization of . Now, we shall proceed to sketch how the master equation (12.2.1) for the density operator  of S can be derived from the microscopic Hamiltonian dynamics [3, 4] of S + R. The starting point is the Liouville–von Neumann equation for the density operator W (t) of S + R, i W˙ (t) = − [H, W (t)] ≡ LW (t) . h¯

(12.2.2)

Here H denotes the Hamiltonian which comprises “free” terms HS and HR for S and R, as well as an interaction part HSR , H = HS + HR + HSR .

(12.2.3)

A similar decomposition holds for the Liouvillian L. The formal integral of (12.2.2) reads W (t) = eLt W (0) .

(12.2.4)

To slightly simplify the algebra to follow, we assume stationarity with respect to HR and the absence of correlations between S and R for the initial density operator, W (0) = (0)R , [HR , R] = 0 .

(12.2.5)

Inasmuch as the bath is macroscopic, it is reasonable to require that R = ZR−1 e−βHR , TrR {R} = 1 .

(12.2.6)

The formal solution (12.2.4) of the initial-value problem (12.2.2), (12.2.5) suggests the definition of a formal time-evolution operator U (t) for S, (t) = U (t)(0) , U (t) = TrR {eLt R} .

(12.2.7)

12.2 Hamiltonian Embeddings

593

Due to the coupling of S and R, this U (t) is not unitary. Assuming that U (t) possesses an inverse, we arrive at an equation of motion for (t), (t) ˙ = l(t)(t) ,

(12.2.8)

l(t) = U˙ (t)U (t)−1 ,

in which l(t) may be interpreted as a generator of infinitesimal time translations. In general, l(t) will be explicitly time-dependent but will approach a stationary limit l = lim U˙ (t)U (t)−1

(12.2.9)

t →∞

on the time scale τR characteristic of the intrabath processes probed by the coupling. On the much larger time scales typical for S, the asymptotic operator l generates an effectively Markovian process. Only in very special cases can l be constructed rigorously [5, 6]. A perturbative evaluation with respect to the interaction Hamiltonian HSR is always feasible and in fact quite appropriate for weak coupling. Assuming for simplicity that TrR {HSR R} = 0 ,

(12.2.10)

one easily obtains the perturbation expansion of l as 

∞

l = LS +

  dt TrR LSR e(LR +LS )t LSR e−(LR +LS )t R

(12.2.11)

0

where third- and higher order terms are neglected (Born approximation). The few step derivation of (12.2.11) uses the expansion of eLt in powers of LSR , eLt = e(LR +LS )t +



t





dt e(LR +LS )t LSR e(LR +LS )(t −t ) + . . . ,

(12.2.12)

0

the identities TrR {LS ( . . . )} = LS TrR {( . . . )}, TrR {LR ( . . . )} = 0, and the assumed properties (12.2.5), (12.2.10) of R and HSR . Of special interest for the remainder of this chapter will be an application to spin relaxation. Therefore, we shall work out the generator l for that case. The spin or angular momentum is represented by the operators Jz , J± which obey [J+ , J− ] = 2Jz , [Jz , J± ] = ±J± .

(12.2.13)

The free spin dynamics may be generated by the Hamiltonian HS = h¯ ωJz

(12.2.14)

594

12 Dissipative Systems

and for the spin bath interaction, we take   HSR = h¯ J+ B + J− B †

(12.2.15)

where B and B † are a pair of Hermitian-conjugate bath operators. We shall speak of the observables entering the interaction Hamiltonian as coupling agents. The Hamiltonian of the free bath need not be specified. By inserting HSR from (12.2.14) in the second-order term of l given in (12.2.11), we obtain  ∞  l (2)  = dt B(t)B † eiωt [J− , J+ ] 0



+ 0

∞

 dt B † B(t) eiωt [J+ , J− ] + H.c. .

(12.2.16)

Interestingly, the c-number coefficients appearing here are the Laplace transforms of the bath correlation functions B(t)B †  and B † B(t), taken at the frequency ω of the free precession (apart from the imaginary unit i). The correlation functions refer to the reference state R of the bath which is assumed to be the thermal equilibrium state given in (12.2.6); their time dependence is generated by the free-bath Hamiltonian HR . For the sake of simplicity, we have assumed, in writing (12.2.16), that B(t)B(0) = B † (t)B † (0) = 0. The Laplace transforms of the bath correlation functions are related to their Fourier transforms by #∞

dt eiωt f (t) = 12 f (ω) + # +∞ f (ω) = −∞ dteiωt f (t) , 0

# +∞ f (ν) i 2π P −∞ dν ω−ν

,

(12.2.17)

where f (t) stands for either B(t)B †  or B † B(t) and P denotes the principal value. Furthermore, since one and the same bath Hamiltonian HR generates the time dependence of B(t) and determines the canonical equilibrium (12.2.6), the well-known fluctuation-dissipation theorem holds in the form [7] # +∞ iωt † −∞ dte B(t)B  = 2κ(ω) [1 + nth (ω)] (12.2.18) # +∞ iωt † −∞ dte B B(t) = 2κ(ω)nth (ω) where  2κ(ω) =

+∞

−∞

 dteiωt [B(t), B † ]

(12.2.19)

is a real Fourier-transformed bath response function and nth (ω) =

1 ¯ −1 eβ hω

(12.2.20)

12.2 Hamiltonian Embeddings

595

is the average number of quanta in an oscillator of frequency ω at temperature 1/β. Combining (12.2.11), (12.2.14)–(12.2.19) we obtain the master equation  ˙ = l = −i (ω + δ + δth )Jz − δJz2 ,  +κ(1 + nth ) {[J− , J+ ] + [J− , J+ ]} +κnth {[J+ , J− ] + [J+ , J− ]}

(12.2.21)

with the “frequency shifts” 1 δ= P π



+∞

−∞

κ(ν) 1 , δth = P dν ω−ν π



+∞ −∞

dν

nth (ν)κ(ν) . ω−ν

(12.2.22)

This master equation was first proposed to describe superfluorescence in Ref. [8] and put to experimental test in that context in Ref. [9]. The first commutator in (12.2.21) describes a reversible motion according to a Hamiltonian h¯ (ω + δ + δth )Jz − h¯ δJz2 ; the first term of this generates linear precession around the z-axis at a shifted frequency ω + δ + δth , whereas the second term yields a nonlinear precession around the z-axis. The remaining terms in the master equation are manifestly irreversible. In view of the temperature dependence of nth , the stationary solution ¯ of the master equation (12.2.21) must be separately stationary with respect to both the reversible and the irreversible terms. While the reversible terms admit any function of Jz , the irreversible ones single out ¯ = ZS−1 e−βHS ,

(12.2.23)

i.e., the unperturbed canonical operator with the temperature ∝ 1/β equalling that of the bath. The stationarity of this ¯ may be checked by a little calculation with the help of the identity J± exJz = e±x e−xJz J±

(12.2.24)

but in fact follows from very general arguments [10]: Due to the structure of (12.2.21), ¯ must be independent of δ and κ, i.e. of zeroth order in HSR . In the limit of vanishing coupling, however, the angular momentum and the bath become statistically independent and their total energy tends to the sum of the respective unperturbed parts. The only function of HS or, equivalently, Jz meeting these requirements is the exponential (12.2.23). The master equation obviously respects the conservation law for the squared angular momentum, as indeed it must since [HS + HSR + HR , J2 ] = 0, J2 = j (j + 1) = const .

(12.2.25)

596

12 Dissipative Systems

Once the quantum number j is fixed, the Hilbert space is restricted to 2j + 1 dimensions, and the density operator is representable by a (2j + 1) × (2j + 1) matrix. A convenient basis is provided by the joint eigenvectors |j m of J2 and Jz pertaining to the respective eigenvalues j (j + 1) and m. By employing J± |j m =

)

(j ∓ m)(j ± m + 1) |j, m ± 1 ,

(12.2.26)

the master equation (12.2.21) is easily rewritten as a set of coupled differential equations for the matrix elements mm (t). Since no phase is distinguished for the polarizations J± , there is a closed set of “rate equations” for the probabilities mm (t) ≡ m (t), ˙ m (t) = 2κ(1 + nth )[(j − m)(j + m + 1)m+1 − (j − m + 1)(j + m)m ] +2κnth [(j − m + 1)(j + m)m−1 − (j − m)(j + m + 1)m ] . (12.2.27) These actually form a master equation of the Pauli type with energy-lowering transition rates w(m + 1 → m) = 2κ(1 + nth)(j − m)(j + m + 1) and upward rates w(m → m + 1) = 2κnth(j − m)(j + m + 1). As revealed by the factor 1 + nth , downward transitions of the angular momentum are accompanied by spontaneous or induced emissions of quanta hω ¯ into the bath. Upward transitions, on the other hand, require the absorption of quanta h¯ ω from the bath and thus are proportional to nth . Of course, neither absorption nor induced emission can take place when the heat bath temperature is too low, i.e., when kB T hω. ¯ In this latter limit, the angular momentum keeps dissipating its energy, i.e., emitting quanta into the bath, until it settles into the ground state, m = −j . It is as well to note in passing that the master equation (12.2.21) or (12.2.27) is nothing but a slightly fancy version of Fermi’s golden rule. Indeed, the transition rates w(m + 1 → m) and w(m → m + 1) may be calculated directly with that rule; then, the constant κ(ω) ≡ κ appears as proportional to the number of bath modes capable, by resonance, of receiving quanta hω ¯ emitted by the precessing angular momentum. We should also add a word about the limit of validity of the master equation (12.2.21). Due to the perturbative derivation, we are confined to the case of small coupling, κ, δ

ω,

(12.2.28)

in which the free precession is only weakly perturbed. Moreover, the damping constant κ, in energy units, must be smaller than the bath temperature, hκ ¯

kB T .

(12.2.29)

12.3 Time-Scale Separation for Probabilities and Coherences

597

To explain this latter restriction, it is useful to consider (12.2.20) and to replace the free-precession frequency ω by a complex variable z. The function nth (z) has imaginary poles at β h¯ zμ = 2πiμ, μ = 0, ±1, ±2, . . . . Then, one may reconstruct the correlation functions B(t)B †  and B † B(t) by inverting the Fourier transforms (12.2.18). Closing the frequency integrals in the upper half of the z plane, as is necessary for t > 0, one encounters thermal transients eizμ t = e−|zμ |t ; these transients must decay much faster than e−κt or else the assumed time-scale separation for the bath and the damped subsystem is violated, and the asymptotic generator l becomes meaningless. A limit of special relevance for the remainder of this chapter is that of low temperatures, nth 1, and large angular momentum, j  1. In this quasi-classical low-temperature regime, classical trajectories become relevant as a reference behavior about which the quantum system displays fluctuations. Classical dynamics may be obtained from (12.2.21) by first extracting equations of motion for the mean values J(t) and then factorizing as Jz J±  = Jz J±  etc. In the limit of negligible nth and with the help of a spherical-coordinate representation, Jz  = j cos Θ , J±  = j sin Θe±iφ ,

(12.2.30)

classical dynamics turns out to be that of the overdamped pendulum, Θ˙ = Γ sin Θ , φ˙ = 0 , Γ = 2κj .

(12.2.31)

Evidently, the classical limit must be taken as j → ∞ with constant Γ . Then, the angle Θ relaxes toward the stable equilibrium Θ(∞) = π according to tan

Θ(0) Θ(t) = eΓ t tan . 2 2

(12.2.32)

12.3 Time-Scale Separation for Probabilities and Coherences Off-diagonal density matrix elements between states differing in energy by not too many quanta display lifetimes of the same order of magnitude, under the dissipative influence of heat baths, as diagonal elements. A typical example is provided by the damped angular momentum considered above, when the quantum number j is specified as j = 1/2. The damping rates for the off-diagonal element 1/2,−1/2 and the population probability 1/2,1/2 are easily found from (12.2.21), respectively, as γ⊥ = κ(1 + 2nth ) , γ, = 2κ(1 + 2nth ) , i.e., they are indeed of the same order of magnitude.

(12.3.1)

598

12 Dissipative Systems

A different situation arises for large quantum numbers. Coherences between mesoscopically or even macroscopically distinct states usually have lifetimes much shorter than those of occupation probabilities. We propose to illustrate that phenomenon of “accelerated decoherence” for an angular momentum subject to the low-temperature version (nth = 0) of the damping process (12.2.21); it suffices to consider the dissipative part of that process whose generator Λ is defined by * ˙ = Λ = κ J− , J+ ] + [J− , J+ ]} ;

(12.3.2)

the classical limit of that process was characterized in (12.2.31) and (12.2.32). For reasons to be explained presently, it is convenient to start with a superposition of two coherent states (8.6.12), |  = c|Θφ + c |Θ φ 

(12.3.3)

where the complex amplitudes c and c are normalized as |c|2 + |c |2 = 1. In the limit of large j , the two component states may indeed be said to be macroscopically distinguishable; the superposition is often called a Schrödinger cat state, reminiscent of Schrödinger’s bewilderment with the notorious absence of quantum interference with macroscopically distinct states from the classical world. In fact, the phenomenon of accelerated decoherence that we are about to reveal explains why such interferences are usually impossible to observe. The rigorous solution exp{Λt}|  | of the master equation (12.3.2) originating from the initial density operator |  | for the cat state (12.3.3) can be constructed without much difficulty; see Refs. [11–14]. However, we reach our goal more rapidly by considering the piece Θφ,Θ φ (t) = exp{Λt}|ΘφΘ φ |

(12.3.4)

and estimating its lifetime from the initial time rate of change of its norm NΘφ,Θ φ (t) = TrΘφ,Θ φ (t)† Θφ,Θ φ (t) .

(12.3.5)

By differentiating w.r.t. time, setting t = 0, inserting the generator Λ, and using (8.6.17), we immediately get  *  N˙ Θφ,Θ φ (t) = −(2κj ) j sin2 Θ + sin2 Θ − 2 cos(φ − φ ) sin Θ sin Θ  + + 12 (1 + cos Θ)2 + (1 + cos Θ )2 . (12.3.6) Now, the announced time-scale separation may be read off: The probabilities described by Θ = Θ , φ = φ have the classical rate of change Γ = 2κj ; in contrast, for sin Θ = sin Θ and/or φ = φ , i.e., coherences, the first square bracket in the foregoing rate does not vanish and carries the “acceleration factor” j .

12.3 Time-Scale Separation for Probabilities and Coherences

599

Inasmuch as j is large, we conclude that the cat state (12.3.3) rapidly (on the time scale 1/j Γ ) decoheres to a mixture,    c|Θφ + c |Θ φ  cΘφ| + c Θ φ | −→ |c|2 |ΘφΘφ| + |c |2 |Θ φ Θ φ | ,

(12.3.7)

while the weights of the component states begin to change only much later at times of the order 1/Γ . An exception of the general rule of accelerated decoherence is highly interesting: Coherences between coherent states which lie symmetrically to the “equator” (Θ = π − Θ) on one and the same “great circle” (φ = φ) are exempt from the rule, simply because sin Θ = sin(π − Θ). These give rise to long-lived Schrödinger cat states (12.3.3) [14]. A deeper reason for the immunity of these exceptional cat states will appear once we have clarified the distinction of the coherent states for the process under consideration. To that end, we must come back to the coupling agents J± entering the interaction Hamiltonian (12.2.15). These happen to have coherent states as approximate eigenstates in the semiclassical limit of large j , J± |Θφ ≈ j e±iφ sin Θ |Θφ

if

tan2 Θ  1/j.

(12.3.8)

That property is worth checking since the notion of an approximate eigenvector is not met all too often. But indeed, it is easily seen, again with the help of (8.6.17), that the two vectors J± |Θφ and j e±iφ sin Θ |Θφ have norms with a relative difference of order 1/j and √ comprise an angle (in Hilbert space) in between themselves which is of order 1/ j , provided that the coherent state in question does not appreciably overlap the polar ones at Θ = 0 and Θ = π. The approximate eigenvalues e±iφ sin Θ are doubly degenerate, and the two pertinent approximate eigenstates have the same azimuthal angle φ whereas the polar angles are symmetric about π/2. The Schrödinger cat state with equatorial symmetry cannot be rapidly decohered by the damping mechanism in consideration since its two component states are precisely such a doublet. To appreciate the foregoing reasoning better, we may think of the unitary motion generated by some Hamiltonian that has a degenerate eigenvalue E. An initial superposition of states from the degenerate subspace will subsequently acquire the overall prefactor exp(−iEt/h¯ ) but will not undergo any internal change. Or, to stay with dissipative motions, we may consider an interactive Hamiltonian HSR = XB with Hermitian coupling agents X and B for the system S and the reservoir R, respectively. Then, the Fermi golden rule type arguments of Sect. 12.2 yield a master equation with the dissipative generator * + Λ = (κ/x02 ) [X, X] + [X, X]

(12.3.9)

600

12 Dissipative Systems

where κ is a rate constant and x0 is a constant of the same dimension as the coupling agent X. Obviously now, occupation probabilities of eigenstates |x of the coupling agent X are not influenced at all by the damping: Λ|xx| = 0. 2  Coherences are affected, however, since Λ|xx | = κ (x − x )/x0 |xx |, and the proportionality of the decoherence rate to (x − x )2 /x02 signals accelerated decoherence. Acceleration would fail only for superpositions of eigenstates |x, α belonging to one degenerate eigenvalue x. In principle, such states could still be very different, even macroscopically, as suggested by the toy example X = Y 2 for which the eigenstates | ± y of Y form a doublet with common eigenvalue y 2 of X. The phenomenon considered here is universal. The unobservability of superpositions of different pointer positions in measurement devices finds its natural explanation here and so does the reduction of superpositions to mixtures in microscopic objects coupled to macroscopic measurement or preparative apparatuses [15–18]. Examples of current experimental interest include superconducting quantum interference devices (SQUIDS), Bloch oscillations in Josephson junctions, and optical bistability. The common goal of such investigations is to realize the largest possible distinction, hopefully mesoscopic, between two quantum states that still gives rise to detectable coherences in spite of weak dissipation. The first actual observations of accelerated decoherence were reported for photons in microwave cavities in Ref. [19]. A negative implication of the phenomenon in question also deserves mention. The classical alternatives of regular and chaotic motion can be reflected, as shown in Chap. 8, in different appearances of the sequence of quantum recurrences in the time evolution of certain expectation values. Quantum recurrence events of whatever appearance, however, are constructive interference phenomena requiring preservation of coherences between successive events. Inasmuch as recurrence events involve coherences between classically distinct states, rather feeble damping suffices to destroy the phase relations necessary for substantial constructive interference.

12.4 Dissipative Death of Quantum Recurrences We proceed to a quantitative analysis of recurrences in the presence of feeble damping [20–22]. Due to the finiteness of its Hilbert space, the kicked top will be a convenient example to work with again. The main conclusions, however, will be of quite general validity. The clue to the treatment lies in the fact that very weak dissipation suffices to suppress recurrence events. Thus, the damping may be considered a small perturbation of the unitary part of the dynamics. Furthermore, under conditions of classical chaos ideas borrowed from random-matrix theory allow a full implementation of first-order perturbation theory.

12.4 Dissipative Death of Quantum Recurrences

601

The kicked top to be employed here must be described now by a dissipative map for the density operator (t) after the tth kick, (t + 1) = eΛ eL (t)

(12.4.1)

where eL represents the unitary motion discussed in previous chapters, eL  = F F † F = e−ipJz e−iλJy /2j . 2

(12.4.2)

For the sake of convenience, the axis of the nonlinear precession is now taken to be the y-axis and that of the linear precession to be the z-axis. The nonunitary factor eΛ describes the dissipative part of the evolution. To keep the situation simple, we will assume low temperatures, nth ≈ 0, and neglect the frequency shift terms, δ = 0. Then, the generator Λ is obtained from (12.2.21) as Λ =

Γ ([J− , J+ ] + [J− , J+ ]) 2j

(12.4.3)

except that here the classical damping constant Γ = 2j κ is used to facilitate comparison with the classical behavior (12.2.31), (12.2.32). The free-precession term in (12.2.21) is not included in Λ but is written instead as a separate factor in the unitary evolution operator eL ; no approximation is involved in that separation since Λ[Jz , ] = [Jz , Λ]. Thus, the whole dissipative quantum map (12.4.1) may be said to allow linear precession around the z-axis with concurrent damping, whereas the nonlinear precession prevails during a separate phase of the driving period. Note that the time t = 0, 1, 2, . . . and the coupling constants Γ, λ, p are all taken as dimensionless and that h¯ = 1. In the weak-damping limit, the generator of the map can be approximated as eΛ eL = (1 + Λ + . . . )eL .

(12.4.4)

With the eigenstates of the unitary Floquet operator F denoted by |μ, the dyadic eigenvectors of eL read |μν| and the corresponding unimodular eigenvalues exp (−iφμ + iφν ). To first order in Λ, the perturbed eigenvalue, for μ = ν, is λμν = e−i(φμ −φν ) (1 − Λμν ) ≈ e−i(φμ −φν )−Λμν , μ = ν ,

(12.4.5)

where the real damping constant   Λμν = −Tr (|μν|)† Λ|μν| =

Γ (−2μ|J− |μν|J+ |ν + ν|J+ J− |ν + μ|J+ J− |μ) (12.4.6) 2j

602

12 Dissipative Systems

is the μν element of the dyad −Λ|μν|; the latter is the infinitesimal increment of |μν| generated by Λ during the time interval dt, divided by dt. With Λ expressed in the |j, m basis, the damping constant Λμν takes the form +j 

Λμν =



a(m, n) |μ|j, m|2 |ν|j, n|2

m,n = −j

−b(m, n)μ|j, mj, m + 1|μν|j, n + 1j, n|ν , a(m, n) =

Γ [(j + m)(j − m + 1) + (j + n)(j − n + 1)] , 2j

b(m, n) =

Γ [(j − m)(j + m + 1)(j − n)(j + n + 1)]1/2 . j

(12.4.7)

Once the eigenvectors |μ of the Floquet operator F are known, the sums in (12.4.7) can be evaluated. The classical dynamics implied by (12.4.1)–(12.4.3) in the limit j → ∞ is chaotic on the overwhelming part of the sphere (J/j )2 = 1 for λ ≈ 10, p ≈ π/2, Γ ≈ 10−4 , the case to be studied in the following. The strange attractor accommodating the chaotic trajectories covers the classical sphere almost uniformly. Of course, for dissipation to become manifest in the classical dynamics, a time of the order 1/Γ = 104 must elapse, and the strangeness of the attractor is revealed only on even larger time scales. In the situation of global classical chaos chosen here, the Floquet operator F may be considered a random matrix drawn from Dyson’s circular orthogonal ensemble. Then, the second term in Λμν , as given by (12.4.7), is, for j  1, unable to make a non-zero contribution to the sum over m, n since the individual eigenvector components are random in sign. The remaining term in Λμν may be simplified as

Λμν

+j   Γ  = (j + m)(j − m + 1) |μ|j, m|2 + |ν|j m|2 . 2j

(12.4.8)

m = −j

The damping constants Λμν now appear to be random numbers. Due to the sum over √ m, however, the relative root-mean-square deviation from the mean is ∼ 1/ j and to within this accuracy the Λμν are in fact all equal to one another

12.4 Dissipative Death of Quantum Recurrences

603

and to their ensemble mean (Problem 12.8). Therefore, they must also equal their arithmetic mean, Λμν =

2j +1   Γ (j + m)(j − m + 1) |μ|j, m|2 j (2j + 1) m μ=1

=

 Γ (j + m)(j − m + 1) j (2j + 1) m

= 23 Γ (j + 1) ≈ 23 Γ j , μ = ν .

(12.4.9)

In short, the random numbers Λμν are self-averaging in the limit of large j. The result just obtained is in fact quite remarkable. It is independent of the coupling constants λ and p and even comes with the claim of validity for all tops with global chaos in the classical limit which are weakly damped according to (12.4.3). Furthermore, since Λμν turns out to be independent of μ and ν for μ = ν, all off-diagonal dyadic eigenvectors |μν| of eL display the same attenuation. Finally, the proportionality of Λμν to j reflects the phenomenon of accelerated decoherence. The life time of the coherences Λμν is ∼ 1/Γ j while classically the damping is noticeable only on a time scale larger by a factor j. For perturbation theory to work well, the damping constants Λμν must be small. As to how small, we may argue that our first-order estimate should not be outweighed by second-order corrections. In the latter, small denominators of order 1/j 2 appear since 2j (2j + 1) nonvanishing zero-order eigenvalues φμ − φν lie in the accessible (2π)-interval. Thus, we are led to require that Γ

1 . j3

(12.4.10)

A word about the diagonal dyads |μμ| may be in order. They are all degenerate in the conservative limit, because they are eigenvectors of eL with eigenvalue unity. The fate of that eigenvalue for weak damping must be determined by degenerate perturbation theory, i.e., by diagonalizing Λ in the (2j + 1)-dimensional space spanned by the |μμ|, a task in fact easily accomplished. The elements of the (2j + 1) × (2j + 1) matrix in question read Tr {|μμ|(Λ|νν|)}  Γ  δμν μ|J+ J− |μ − |μ|J− |ν|2 =− j

 Γ |μ|m|2 (j + m)(j − m + 1) δμν =− j m  1/2 (j + m)(j − m + 1)(j + m )(j − m + 1) − mm

 × ν|j, m − 1j, m|μμ|j, m j, m − 1|ν .



(12.4.11)

604

12 Dissipative Systems

They are quite similar in structure to the damping constants Λμν given in (12.4.7). As was the case for the latter, random-matrix theory can be invoked in evaluating the various sums in (12.4.11), provided there is global chaos in the classical limit. Due to the sums over m and m , all matrix elements (12.4.11) are again self-averaging, i.e., have root-mean-square deviations √ from their ensemble mean that are smaller than the mean by a factor of order 1/ j . In the ensemble mean, only the terms with m = m in the double sum contribute. For the off-diagonal elements, μ = ν, the probabilities |j m|μ|2 and |j, m − 1|ν|2 may be considered independent, and therefore both ensembles average to 1/(2j + 1) ≈ 1/2j. For the diagonal elements, μ = ν, the contribution of the second sum in (12.4.11) need not be considered further since it is smaller than the first by a factor of order 1/j. Thus, the matrix to be diagonalized reads   2 1 − δμν μ |(Λ|νν|)| μ = − Γ j δμν − . 3 2j

(12.4.12)

Its diagonal elements precisely equal the eigenvalues −Λμν given in (12.4.9) while its off-diagonal elements are smaller by the factor 1/2j. Interestingly, in spite of their relative smallness, the off-diagonal elements may not be neglected since probability conservation requires that 2j +1 

μ |(Λ|νν|)| μ = 0 ;

(12.4.13)

μ=1

indeed, μ|(Λ|νν|)|μdt is the differential increment, due to the damping, in the probability of finding the state |μ a time span dt after the state |ν had probability one. The matrix (12.4.12) is easily diagonalized. It has the simple eigenvalue zero and the 2j -fold eigenvalue (−2Γ j/3). The corresponding eigenvalues of the map eΛ eL are λ = 1 simple , λ = e−2Γj/3 2j −fold .

(12.4.14)

Clearly, the simple eigenvalue unity pertains to the stationary solution of the quantum map (12.4.1). Special comments are in order concerning the equality of the 2j -fold eigenvalue in (12.4.14) and the modulus of those considered previously in (12.4.5), (12.4.9). Even though this equality implies one and the same time scale for the relaxation of the coherences μ|(Λ|μν|)|ν and the probabilities μ|(Λ|νν|)|μ, there is no contradiction with the time-scale separation discussed in Sect. 12.3; the latter refers to coherences j, m||j, m  and probabilities j, m||j, m. The eigenstates of J2 and Jz form a natural basis when the damping

12.4 Dissipative Death of Quantum Recurrences

605

under consideration is the only dynamics present, whereas the eigenstates |μ of J2 and the Floquet operator F are the natural basis vectors when the damping is weak and chaos prevails in the classical limit. The equality of all damping rates encountered here suggests an interpretation in which, under conditions of global classical chaos, each eigenstate |μ and thus also each probability μ|(Λ|νν|)|μ comprises coherences between angular momentum eigenstates |j, m and |j, m  where |m − m | takes values up to the order j. Such an interpretation is indeed tantamount to the applicability of random-matrix theory, which assigns, in the ensemble mean, one and the same value 1/(2j + 1) ≈ 1/2j to all probabilities for finding Jz = m in a state |μ. This may be viewed as a quantum manifestation of the assumed globality of classical chaos: The strange attractor covers the classical sphere (J/j )2 = 1 uniformly. Global uniform support is evident in the stationary solution of the quantum map (12.4.1): Indeed, it is easily verified that the eigenvector of the matrix (12.4.12) pertaining to the simple vanishing eigenvalue has all components equal to one another. Therefore, the unique density operator invariant under the map (12.4.1) is the equipartition mixture of all 2j + 1 Floquet eigenstates |μ, ¯ =

2j +1  1 |μμ| . 2j + 1

(12.4.15)

μ=1

These results for the eigenvalues of the map eΛ eL enable strong statements about the time dependence of observables. For instance, all observables with a vanishing stationary mean display the universal attenuation law O(t) = e−2Γj t /3 O(t)Γ = 0 .

(12.4.16)

The angular momentum components Ji provide examples of such observables. For instance, Jz (∞) =

2j +1  1 μ|Jz |μ 2j + 1 μ=1

=

2j +1  +j  1 m |m|μ|2 2j + 1 μ = 1 m = −j

+j  1 = m=0. 2j + 1

(12.4.17)

m = −j

Figure 12.1 compares the ratio Jz (t)/Jz (t)Γ = 0 , calculated by numerically iterating the map (12.4.1), for Γ = 1.25 × 10−4 and Γ = 0 with the random-matrix

606

12 Dissipative Systems

Fig. 12.1 Ratio of the time-dependent expectation value Jz (t) with and without damping for the kicked top (12.4.1), (12.4.2). Parameters were chosen as j = 40, p = 1.7, λ = 10. The coherent initial state was localized at Jx /j = 0.43, Jy /j = 0, Jz /j = 0.9. The smooth curve represents the prediction (12.4.16) of random-matrix theory. The optimal fit for the decay rate to the numerical data is 0.61j Γ whereas random-matrix theory predicts (2/3)j Γ

prediction (12.4.16). The agreement is quite impressive even though the numerical work was done for j as small as 40. An implication of the above concerns the fate of quantum recurrences under dissipation. Quasi-periodic reconstructions of mean values in the conservative limit must suffer the attenuation described by (12.4.16). The mean frequency of large fluctuations for Γ = 0 roughly equals the quasi-energy spacing 2π/(2j + 1). Therefore, the critical damping above which recurrences are noticeably suppressed can be estimated as π/j ≈ 2Γ j/3, i.e., Γcrit ≈

1 . j2

(12.4.18)

For Γ < Γcrit , the damping constants 2Γ j/3 are smaller than even a typical nearest neighbor spacing 2π/(2j + 1) of conservative quasi-energies. Then, the quasiperiodicity of the conservative limit must still be resolvable, of course. Figure 12.2 displays the fate of recurrences in Jz (t) for the map (12.4.1)– (12.4.3) under conditions of global classical chaos. The survival condition Γ < Γcrit for recurrences is clearly visible. The amplitude of the quasi-periodic temporal fluctuations of Jz (t) displayed in Fig. 12.2 calls for comment. The ordinates in

12.4 Dissipative Death of Quantum Recurrences

607

√ Fig. 12.2 Time-dependent expectation value Jz / j for the kicked top (12.4.1), (12.4.2) without −4 damping (left column) and with Γ = 5 × 10 for several values of j. Initial conditions and coupling constants p, λ are as in Fig. 12.1.√The influence of j on the lifetime of recurrences is clearly visible. The mean Jz  is referred √ to j in the expectation that the strength of the temporal fluctuations of Jz /j is of the order 1/ j ; see also Fig. 12.3

608

12 Dissipative Systems

σ .1

.05 10

20

50

100

j

Fig. 12.3 Variance of the temporal fluctuations of Jz  according to (12.4.19). Dynamics and parameters are as in Figs. √ 12.1 and 12.2. The straight line represents the prediction of randommatrix theory, σ ∼ 1/ j . The four points were obtained by numerically iterating the map (12.4.1), (12.4.2) and performing the time average indicated in the text; linear regression on the four points suggests an exponent −0.51

that figure are scaled in the expectation that the variance of the temporal fluctuations behaves as σ =

  2 1/2 1  1 Jz  − Jz  ∼ √ j j

(12.4.19)

#T where the bar denotes a time average, Jz  = T1 0 dtJz (t) with T  j. This behavior is indeed borne out in Fig. 12.2 and somewhat more explicitly in Fig. 12.3. The scaling (12.4.19) can be understood on the basis of random-matrix theory. Since the damping is taken care of by the factor exp (−2Γ j t/3) according to (12.4.16), the argument may be given for Γ = 0; it uses the scaling Jz ∼ j and the eigenvector statistics from Sect. 5.10. Since similar arguments have been expounded before, this particular one is left to the reader as Problem 12.11. As a further check of random-matrix theory the damping constants Λμν of (12.4.7) and the eigenvalues of the matrix (12.4.11) were evaluated numerically, using the numerically determined eigenstates |μ of the Floquet operator given in (12.4.2). The calculations were done for various pairs of p, λ, all corresponding to global classical chaos and for j = 5, 10, 20, 40, 80. The arithmetic mean of all (2j + 1)2 − (2j + 1) off-diagonal damping constants so determined is independent √ of p and λ and matches the random-matrix result to within less than 1/ j. The variation with j of the relative root-mean-square deviations from the mean is √ consistent with a proportionality to 1/ j ; see Fig. 12.4.

12.5 Complex Energies and Quasi-Energies

609

σ .1

5

10

20

50

100

j

Fig. 12.4 Check on the equality of all “off-diagonal” damping constants Λμν in the weakdamping limit Γ j 1 of the kicked top (12.4.1), (12.4.2) with p = 1.7, λ = 6. The plot shows the variation with j of the root-mean-square deviation of the Λμν from their arithmetic mean Λ¯ μν , 1/2  /Γ j . The five points, based on j = 5, 10, 20, 40, 80, yield, by linear σ = [(Λμν − Λ¯ μν )2 ] regression, the power law σ ∼ j −0.50 which is precisely the prediction of random-matrix theory (straight line)

12.5 Complex Energies and Quasi-Energies Dissipative quantum maps of the form (t + 1) = M(t)

(12.5.1)

have generators M that are neither Hermitian nor unitary. Their eigenvalues e−iφ are complex numbers that, for stable systems, are constrained to have moduli not exceeding unity, |e−iφ | ≤ 1 .

(12.5.2)

Indeed, an eigenvector of M with Im {φ} > 0 would grow indefinitely in weight with the number of iterations of the map. In the conservative limit, all eigenvalues actually lie on the circumference of the unit circle around the origin of the complex plane. Their phases φ are differences of eigenphases of the unitary Floquet operator, φμν = φμ − φν . If the Hilbert space is N dimensional, N of the conservative φμν vanish, and the remainder falls into N(N − 1)/2 pairs ±|φμ − φν |. Only N of these pairs refer to adjacent quasi-energy levels, i.e., give nearest neighbor spacings. As the damping is increased, the eigenvalues e−iφ tend to wander inward from the unit circle and eventually can no longer be associated with the pairs of conservative quasi-energies from which they originated; such an association is possible only within the range of applicability of a perturbative treatment of the damping, such as that given in the last section. In general, only one eigenvalue, that pertaining to the

610

12 Dissipative Systems

Fig. 12.5 Inward migration from the unit circle in the complex plane of the eigenvalues of the dissipative map (12.4.1), (12.6.1) for a kicked top with j = 6, p = 2, λ0 = 8, λ1 = 10 as the damping constant Γ is increased from 0 to 0.4 in steps of 0.005. A crossing of two eigenvalue “lines” in this picture does not in general correspond to a degeneracy since the crossing point may have different values of Γ on the two lines. The reflection symmetry about the real axis is a consequence of Hermiticity conservation for the density operator by the dissipative quantum map; see Sect. 10.5

stationary solution of the map (12.5.1), is excepted from the inward migration and actually rests at unity. Figure 12.5 depicts the inward motion of the e−iφ for a kicked top. We shall refer to the complex φ as complex or generalized quasi-energies. Given that, for conservative maps in N-dimensional Hilbert spaces with N  1, statistical analyses of the Floquet spectrum yield important characteristics and, especially, the possibility of distinguishing “regular” and “chaotic” dynamics, the question naturally arises whether statistical methods can be usefully applied to dissipative maps as well [23]. The density of eigenvalues is worthy of particular attention. In the conservative limit, the quasi-energies tend to be uniformly distributed along the circumference of the unit circle. A linear distribution of uniform density along a circle of smaller radius will still arise for weak damping, as long as first-order perturbation theory is still reliable and provided that random-matrix theory is applicable in zeroth order, i.e., provided chaos prevails in the classical limit. Indeed, as shown in the preceding section, the generalized quasi-energies φμν begin their inward √ journey with uniform radial “speed” ∂φ/∂Γ ; the relative spread of speed is ∼ 1/ j . However, for large damping, a two-dimensional surface distribution tends to arise in place of the onedimensional line distribution. Numerical results like those displayed in Fig. 12.5 suggest isotropy but radial nonuniformity for that surface distribution.

12.6 Different Degrees of Level Repulsion for Regular and Chaotic Motion

611

To render density fluctuations in different parts of the spectrum meaningfully comparable, the coordinate mesh within the unit circle must in general be rescaled to achieve a uniform mean density of points throughout. This is similar in spirit to the unfolding of real energy spectra discussed in Sect. 5.19. Now, fluctuations may be characterized by, e.g., the distribution of nearest neighbor spacings; when the spacing between two points in the plane is defined as their Euclidean separation, a nearest neighbor can be found for each point and a spacing distribution established. All of the above considerations require only slight modifications for damped systems under temporally constant external conditions. In such cases, the density operator obeys a master equation of the form (t) ˙ = l(t) , l = L + Λ ,

(12.5.3)

where the generator l of infinitesimal time translations generally has a conservative part L and a damping part Λ. The eigenvalues of l, which are more analogous to the exponents iφ than to the eigenvalues e−iφ of discrete-time maps, must have negative real parts for stable systems; distributed along the imaginary axis in the conservative limit, they tend to spread throughout the left half of the complex plane once the damping becomes strong. In view of the conservative limit and the analogy to periodically driven systems, it is convenient to denote the eigenvalues of l by −iE and to regard the E as complex or generalized energies. It should be kept in mind, though, that for vanishing damping, the eigenvalue in question becomes the difference of two eigenenergies of the Hamiltonian, Eμν = Eμ − Eν , and the corresponding eigen-dyad of l becomes |μν| with |μ and |ν energy eigenstates.

12.6 Different Degrees of Level Repulsion for Regular and Chaotic Motion Now, we consider strong damping outside the range of applicability of first-order perturbation theory. For greater flexibility, the kicked-top dynamics of Sect. 12.4 will be generalized slightly to allow two separate conservative kicks per period F = e−ipJz e−iλ0 Jz /2j e−iλ1 Jy /2j . 2

2

(12.6.1)

The damping generator Λ in the map (12.4.1) is kept unchanged, i.e., given by (12.4.3). Now, the map eΛ eL must be diagonalized numerically. As in the zero-damping case, the angular-momentum basis |j m proves convenient for that purpose, especially since the dissipative part eΛ of the map can be given in closed form as the tetrad   (eΛ )mn,pq = j m| eΛ |jpj q| |j n ,

(12.6.2)

612

12 Dissipative Systems

Fig. 12.6 Complex eigenvalues of the dissipative map (12.4.1), (12.6.1) for j = 15, Γ = 0.07, p = 2, λ0 = 11.7 under conditions of (a) classically regular motion (λ1 = 0) and (b) chaos (λ1 = 10). Note again the reflection symmetry about the real axis. The inhibition to close proximity is clearly lower in the regular than in the chaotic case

which represents the exact solution of the master equation ˙ = Λ at t = 1 originating from an arbitrary initial dyad |jpj q|. For the explicit form of the tetrad, the reader is referred to [8, 11]. Figure 12.6 displays the eigenvalues of the map eΛ eL which were obtained by numerical diagonalization for j = 15, Γ = 0.07. The coupling constants p, λ0 , λ1 in the Floquet operator were chosen to yield either regular classical motion (Fig. 12.6a; p = 2, λ0 = 11.7, λ1 = 0) or predominantly chaotic classical motion (Fig. 12.6b; p = 2, λ0 = 11.7, λ1 = 10). Mere inspection of the two annular clouds suggests a lesser inhibition with respect to close proximity under conditions of classically regular motion (Fig. 12.6a). To unfold the spectra, a local mean density was determined around each point as ¯ = n/πdn2 where dn is the distance to the nth nearest neighbor; after making sure that the precise value of n does not matter, the choice n = 10 was made as a compromise respecting both limits in 1 n (2j + 1)2 . Then, the distance to the first neighbor was rescaled as ) S = d1 ¯

(12.6.3)

in analogy to (5.19.11). (Note that in (5.19.11), ¯ is normalized as a probability density and refers to a distribution of points along a line.) Figure 12.7 depicts the spacing staircases I (S) for the two clouds of Fig. 12.6. Clearly, linear repulsion [i.e., a quadratic rise of I (S) for small S] is obtained when the classical dynamics is regular, whereas cubic repulsion [I (S) ∝ S 4 ] prevails under conditions of classical chaos. The distinction between linear and cubic repulsion of the complex levels and its correlation with the classical distinction between predominantly regular motion and global chaos does not seem to be a peculiarity of the dynamics chosen. At any rate,

12.6 Different Degrees of Level Repulsion for Regular and Chaotic Motion

613

Fig. 12.7 Integrated spacing distributions I (S) for two different universality classes. One curve in each of the two pairs refers to the map (12.4.1), (12.6.1) with j = 10, Γ = 0.1, p = 2, and 100 different values of λ0 from 10 ≤ λ0 ≤ 12. The “regular” case has λ1 = 0 and the “chaotic” one λ1 = 8. The second line in the regular pair represents the Poissonian process in the complex plane according to (12.7.4); in the “chaotic” pair, the spacing staircase of general non-Hermitian matrices of dimension 2j + 1 = 21 appears. The insert reveals linear and cubic repulsion

for the model considered, linear repulsion prevails not only in the strictly integrable case λ1 = 0 but also for all λ1 in the range 0 ≤ λ1 ≤ 0.2 which corresponds to the predominance of regular trajectories on the classical sphere; cubic repulsion, on the other hand, arises not only for λ1 = 6 but was also found for larger values of λ1 . These results certainly suggest universality of the two degrees of level repulsion, but they cannot be considered conclusive proof thereof. An interesting corroboration and, in fact, extension comes from the observation that the damped periodically driven noisy rotator displays linear and cubic repulsion of the complex Floquet eigenvalues of its Fokker-Planck equation, depending on whether the noiseless deterministic limit is, respectively, regular or chaotic [24]. The reader will have noticed that Fig. 12.7 depicts a pair of curves for each of the “regular” and “chaotic” cases. One curve in a pair pertains to the map in consideration while the other represents a tentative interpretation which we shall discuss presently. In intuitive generalizations of the conservative cases, the interpretations involve a Poissonian random process in the plane (rather than along a line) for the regular limit and a Gaussian ensemble of random matrices restricted neither by unitarity nor by Hermiticity in the chaotic limit. Figure 12.7 suggests that both interpretations work quite well. It would be nice to have an exactly solvable damped system with generic spectral fluctuations, i.e.; linear repulsion. A good candidate seems to be the map eΛ eL with F = e−ipJz e−iλJz /2j 2

Λ =

Γ 2j

([Jz , Jz ] + [Jz , Jz ]) +

γ j3

  2 [Jz , Jz2 ] + [Jz2 , Jz2 ] .

(12.6.4)

614

12 Dissipative Systems

Fig. 12.8 Integrated spacing distribution I (S) for the eigenvalues (12.6.5) of the classically integrable dissipative map (12.6.4) with j = 30, p = 17.3, λ = 24.9, Γ = 0.012, γ = 0.011. The staircase is hardly distinguishable from the prediction for the two dimensional Poissonian process, Sect. 10.7

1 I(S)

0.5

0

0

1

2

3

S

Its “eigen-dyads” |j, mj, m | are built by the eigenvectors of J2 and Jz . The eigenvalues can be read off immediately as 

 λ λmm = exp −i (m − m )p + (m2 − m 2 ) 2j  Γ γ −(m − m )2 − (m2 − m 2 )2 3 . (12.6.5) 2j j A peculiarity of the dynamics (12.6.4) is that the damping generator does not permit transitions between the conservative Floquet eigenstates |j, m but only phase relaxation in the coherences between such states. Consequently, the (2j + 1)fold degeneracy of the conservative eigenvalue unity, common to all diagonal dyads |j mj m|, is still present in the λmm for γ , Γ > 0. One might expect generic spacing fluctuations since both the real and the imaginary part of ln λmm depend on the two quantum numbers m ± m . Thus, the whole spectrum may be looked upon as a superposition of many effectively independent ones. This expectation is indeed borne out by the spacing staircase of Fig. 12.8.

12.7 Poissonian Random Process in the Plane Imagine that N points are thrown onto a circular disc of radius R, with uniform mean density and no correlation between throws. The mean area per point is 2 /N, and the mean separation s¯ between nearest neighbors is of the order πR√ R/ N. The distribution of nearest-neighbor spacings and the precise mean separation s¯ are also readily determined. The probability P (s)ds of finding the nearest neighbor of a given point at a distance between s and s + ds equals the probability that one of the (N − 1) points is located in the circular ring of thickness ds around the given point and that all (N − 2) remaining points lie beyond this: P (s)ds = (N − 1)

2πs ds πR 2

 N−2 πs 2 . 1− πR 2

(12.7.1)

12.7 Poissonian Random Process in the Plane

615

It is easy to check that this distribution is correctly normalized, The mean spacing reads    1 R √ N(N − 1)2 dt t 2 (1 − t 2 )N−2 . s¯ = √ N 0

#R 0

dsP (s) = 1.

(12.7.2)

√ After rescaling the integration variable as t = τ/ N , the square bracket in (12.7.2) is seen, as N → ∞, to approach the limit √ N

 2 0

√  N  ∞ π τ2 2 −τ 2 . dτ τ 1 − →2 dτ τ e = N 2 0 2

(12.7.3)

To find the asymptotic spacing distribution for large N, the spacing must be √ referred to its mean s¯ = R π/4N. After introducing the appropriately rescaled variable S = s/¯s and performing the limit as in (12.7.3), the distribution (12.7.1) is turned into P (S) = 12 πSe−πS

2 /4

.

(12.7.4)

Amazingly, (12.7.4) gives Wigner’s conjecture exactly for the spacing distribution in the GOE and COE discussed in Chap. 5. As was shown there, the Wigner distribution is rigorous for the GOE of 2 × 2 matrices but is only an approximation, albeit a rather good one, for N × N matrices with N > 2. In the present context, the Wigner distribution arises as a rigorous result in the limit N → ∞. According to Figs. 12.7 and 12.8, this distribution is reasonably faithful to the spacing distributions of the respective regular dynamics. The question arises why the Poissonian random process in the plane should so accurately reproduce the spacing fluctuations of generic integrable systems with damping. To achieve an intuitive understanding, the reader may recall from Chap. 4 the most naive argument supporting the analogous statement for Hamiltonian dynamics: the spectrum of an integrable Hamiltonian with f degrees of freedom may be approximated by torus quantization. Its levels are labelled by f quantum numbers, and the spectrum may be thought of as consisting of many independent subspectra. An analogous statement can be made about the spectrum of the map eΛ eL with the conservative Floquet operator F from (12.6.1) and the damping generator Λ from (12.4.3), provided λ1 = 0 in F to make the classical limit integrable. As is easily checked, this map has the symmetry 

Jz , eΛ eL  = eΛ eL [Jz , ]

(12.7.5)

616

12 Dissipative Systems

for arbitrary . It follows that the map in question does not couple dyads |j, m + kj, m| with different values of k. Indeed, by using |j, m+kj, m| for  in (12.7.5) and writing eΛ eL |j, m + kj, m| =

 m k

mk Cm k |j, m + k j, m | ,





m k = 0, i.e., C m k ∝ δ . one obtains from (12.7.5) the identity (k − k)Cmk kk mk Consequently, k is a good quantum number for the map in question and the cloud of eigenvalues depicted in Fig. 12.6a may be segregated into many subclouds, one for each value of k. When the map defined by (12.4.3) and (12.6.1) is turned into a classically nonintegrable map by allowing λ1 > 0, the symmetry (12.7.5) is broken. Consequently, now the previously segregated (λ1 = 0) subspectra begin to interact. It is still not counterintuitive that the spacing distribution remains Poissonian as long as λ1 is small (0 ≤ λ1 ≤ 0.2); experience with conservative level dynamics suggests that a large number of close encounters of levels must have taken place before strong level repulsion can arise throughout the spectrum.

12.8 Ginibre’s Ensemble of Random Matrices By dropping the requirement of Hermiticity, Ginibre [25] was led from the Gaussian unitary ensemble to a Gaussian ensemble of matrices with arbitrary complex eigenvalues z. The joint probability density for the N eigenvalues of such N × N matrices looks like the joint distribution of eigenvalues for the GOE,   N 1 ... N  N −1 2 2 , P (z1 , . . . , zN ) = N |zi − zj | exp − |zi | π i