Topology of strongly correlated systems : proceedings of the XVIII Lisbon Autumn School, Lisbon, Portugal, 8-13 October, 2000 9789812811455, 9812811451

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Proceedings of the XVIII Lisbon Autumn School

S

ogy of Strongly orrelated Systems '\i Pedro Bicudo J. Emflio Ribeiro • » edro Sacramento Joao Seixas Vitor Vieira

World Scientific

Topology of Strongly Correlated Systems

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Proceedinss of the XVIII Lisbon Autumn School

Topology of Strongly Correlated Systems Lisbon, Portugal

8-13 October, 2000

Pedro Bicudo J. Emflio Ribeiro Pedro Sacramento Joao Seixas Vitor Vleira Centro de Ffsica das interaccoes Fundamentals, Instituto Superior Tecnico, Portugal

V f e World Scientific wb

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TOPOLOGY OF STRONGLY CORRELATED SYSTEMS Proceedings of the XVIII Lisbon Autumn School Copyright © 2001 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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Preface

In October of 2000 took place in Lisbon, Portugal, the XVIII Autumn School of Centro de Fisica das Interacgoes Fundamentals on "Topology of Strongly Correlated Systems". This School continued a sequence of other schools that have been held on several topics since 1979. It was our purpose to bring together physicists from different areas ranging from QCD to Condensed Matter. We felt that this subject will be of ever growing importance in the coming years. Traditionally, in this series of schools there is a small group of invited speakers that give an introduction to the main ideas in the field. These are presented at a level that a graduate student may follow in a series of typically three lectures of one hour each, developing the topic to the present status of knowledge. Since the subject chosen for this school was of wide interest we also invited speakers to give a presentation of their research. Topics covered included vortices in superconductors and superfluids, Kosterlitz-Thouless transitions, effects of topology in fermionic systems, solitons, anomalies, regularization, non-trivial topology on the lattice and confinement (Wilson loops and strings, instantons, Abelian Higgs model, dual QCD). Also related topics presented by some participants are included in these Proceedings. It is a great pleasure to thank the invited speakers for the efforts made to present clear and pedagogical lectures and for the interest of the participants. Also it is a pleasure to thank all of our sponsors: Centro de Fisica das Interacc,6es Fundamentals, Caixa Geral de Depositos, Grupo Teorico de Altas Energias, Fundacao Calouste Gulbenkian, Fundacao para a Ciencia e a Tecnologia, Bank B.P.I., Fundagao Luso-Americana para o Desenvolvimento, French Embassy and Instituto Superior Tecnico. The organizers Pedro Bicudo J. Emilio Ribeiro Pedro D. Sacramento Joao Seixas Vitor Rocha Vieira

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Contents

Preface

v

Vortices in superfluids and superconductors, and topological defects in other materials D. J. Thouless

1

Interplay of real space and momentum space topologies in strongly correlated fermionic systems G. E. Volovik

30

Topological excitations and second order transitions in 3D O(N) models L. M. A. Bettencourt

50

Quantum numbers of solitons and theta-terms in non-linear sigma-models P. B. Wiegmann

62

Center vortices in continuum Yang-Mills theory H. Reinhardt and M. Engelhardt

74

Monopoles and confining strings in QCD M. N. Chernodub, F. V. Gubarev, M. I. Polikarpov and V. I. Zakharov

87

Effective string theory of vortices and Regge trajectories of hybrid mesons with zero mass quarks M. Baker and R. Steinke

129

The regularization problem and anomalies in quantum field theory J. Zinn-Justin

141

Regulated chiral gauge theory H. Neuberger

174

Probing the QCD vacuum N. Brambilla

184

Dynamics of topological excitations in low dimensional magnetic systems A. O. Caldeira and A. Villares Ferrer VII

196

VIM

Pointlike Hopf defects in Abelian projections F. Bruckmann

209

Topological symmetry breaking and the confinement of anyons P. W. Irwin and M. B. Paranjape

213

On the topological susceptibility in abelian-projected SU(2) Gluodynamics S. Kato Strong hadronic decays in QCD2 Yu. S. Kalashnikova and A. V. Nefediev

220 224

Universal properties in low dimensional fermionic systems and bosonization L. E. Oxman, D. G. Bard and E. R. Mucciolo

228

Quantum fluctuations of the Chern-Simons theory and the Sutherland model /. Andric, V. Bardek and L. Jonke

232

Antiferromagnetism and dx2_y2-wave pairing in the colored Hubbard model T. Baier and E. Bick

236

Quantum Friction A. Melikidze

240

List of Participants

247

VORTICES IN S U P E R F L U I D S A N D S U P E R C O N D U C T O R S , A N D TOPOLOGICAL D E F E C T S IN O T H E R MATERIALS

D. J. T H O U L E S S Dept. of Physics, Box 351560 University of Washington Seattle, WA 98195 U. S. A. E-mail: [email protected] Vortices in superfluids and flux lines in superconductors are described as examples of topological defects in materials, with which are associated the topological quantum numbers, superfluid circulation and quantized magnetic flux. A brief introduction is given to the mathematical description of such defects in terms of homotopy groups, and some comparisons are made between the situations in finite condensed matter systems and in a field theory in an unbounded space. The high precision of measurements of flux quantization through the Josephson effects and the lower precision of measurements of circulation in a neutral superfluid are discussed. There is a brief account of the homotopy classification of defects in superfluid 3 He and in liquid crystals. Finally, the application of these ideas to the quantum Hall effect is described, where it has been known since the first publication that the precision of quantization is high.

1

Introduction

My own interest in the topological aspects of condensed matter physics goes back for many years, since I first started reading Feynman's work on superfluidity, and talking to W. F. Vinen and P. W. Anderson on these topics. It is the topological nature of the various low-temperature phases that determines the answer to the following questions: 1. Why does current flow without loss in a superconductor, although the current-carrying state does not have low free energy? 2. Why is a solid rigid? 3. Why can a poorly defined Josephson junction provide the world's best voltage measurements? 4. Why can the quantum Hall effect provide the best standards of electrical resistance? 5. How many different types of line defects are there in various liquid crystal phases? 1

2

Recently I wrote a book summarizing the answers to such problems, with a collection of my favorite papers on the subject. 1 These lectures cover similar ground much more briefly. For further details I recommend you to look at some of the papers reprinted in my book. 2

Topological quantum numbers in superfluids and superconductors

The earliest example of a topological argument in quantum theory is probably Dirac's famous argument for the quantization of electric charge which was published in 1931,2 so that electric charge can reasonably be regarded as a topological quantum number. 3 In 1931 it was already known that charge is quantized with very high precision, 4,5 but we still lack experimental evidence for the existence of magnetic monopoles, which was a vital part of Dirac's argument. Although the experimental evidence for the high precision of charge quantization comes largely from macroscopic physics and chemistry, the theory is undoubtedly part of elementary particle theory. I will be discussing examples of topological aspects of condensed matter physics where the very existence of a topological quantum number depends on the fact that we are dealing with bulk matter. One of the simplest cases was discussed in 1949 by Onsager,6 in a published discussion remark at a conference. In this communication he made the following remarks about the condensate wave function in a Bose condensed system: 1. The phase may be multiple valued; but its increment over any closed path must be a multiple of 27r, so that the wave-function will be singlevalued. Thus the well-known invariant called hydrodynamic circulation is quantized, the quantum of circulation is h/m ... 2. Now we observe that a torus can be converted into a simply-connected space by shrinking the hole. If a circulating superfluid is subjected to such a deformation of its container, it must retain a quantized vortex in its interior. 3. If we admit the existence of quantized vortices, then a superfluid is able to rotate, but the distribution of vorticity is discrete rather than continuous. 4. The critical rate of creep (h/m) may be identical with the minimum rate of flow required before the liquid film can lose energy by the development of a vortex.

3 5. Finally we can have vortex rings in the liquid, and the thermal excitation of Helium II, apart from the phonons, is presumably due to vortex rings of molecular size. 6. As a possible interpretation of the A-point, we can understand t h a t when the concentration of vortices reaches the point where they form a connected tangle through the liquid, then the liquid becomes normal. T h e fifth of these s t a t e m e n t s is probably the only suggestion in his Comment t h a t is not generally accepted today. T h e last remark is the basis of the theory of defect driven phase transitions in two dimensions developed by Berezinskii 7 and by Kosterlitz and myself. 8 ' 9 Its relation to the transition in bulk superfluids is discussed in the lectures by Bettencourt and Tesanovic in this School. All this physical insight is compressed into 31 lines of text, so it is not surprising t h a t most of us acquired these ideas by reading Feynman, 1 0 rather t h a n by reading this p a r a g r a p h by Onsager. It is an i m p o r t a n t feature of this argument t h a t bulk superfluid helium or helium films can be characterized by a single-particle wave function, shared by the whole system; this is called the condensate wave function. This is particularly easy to understand for the dilute atomic systems which have been available since 1995, 1 1 where this condensate wave function is just the wave function into which the majority of the atoms condense at low temperatures. For liquid helium or other strongly interacting systems its meaning is not so obvious, since it is estimated t h a t only about 7% of the atoms are in the condensate wave function at low t e m p e r a t u r e s , 1 2 but Penrose and Onsager 1 3 developed the idea t h a t the condensate wave function could be uniquely identified as the eigenvector of the one-particle Dirac density m a t r i x corresponding to its macroscopic eigenvalue — the single eigenvalue whose m a g n i t u d e is of the order of the number of particles in the system. For superfluid 3 H e and superconductors, m a d e u p of fermions, the one-particle density m a t r i x can have no eigenvalues greater t h a n unity, but the two-particle density m a t r i x can have a macroscopic eigenvalue, and the corresponding eigenvector is the condensate wave function. This generalization was m a d e by Yang. 1 4 T h e condensate wave function ip(r) can be written in the form

i> = \4>\eis^

,

(2.1)

where the phase S can change by a multiple of 2ir round a loop. T h e velocity is given by v(r) = — V S ,

(2.2)

4

where m is the mass of a helium atom, or whatever other boson we are concerned with. T h e circulation is ,

/

h f

2TTnh

v , • dr = —

op of the order parameter at the points on the surface, this topological q u a n t u m number can be written as the integral of the Jacobian of the m a p p i n g over the surface, Nw = i - / % , 4TT J0

f d JO

M

m F

^

l ^ d(6s,4>s)

•

(3.1)

This invariant has the value + 1 if the order parameter points outwards over all the surface, and this defect is widely known as a hedgehog, in honor of

9 the animal which has quills sticking out from its surface when it rolls u p in a ball. T h e number Nw is -1 if the order parameter points inwards everywhere, which is more like a pin-cushion t h a n a hedgehog. This number can have any integer value. This homotopy group 7T2 is always either equivalent to the group of integers, or it is trivial, with all mappings equivalent to the identity element. Not all topological defects have a singular core, as the vortex in a simple Bose condensate or the hedgehog for the Heisenberg ferromagnet have. In some cases the order parameter changes slowly, and the topological charge is spread out over a region. Such an extended topological defect is known as a "texture". An example of a texture is a "skyrmion" in a two-dimensional Heisenberg magnet. In such a system the spins are aligned in a common direction except in one particular neighborhood, in which the spins rotate smoothly to form a defect with a topological q u a n t u m number analogous to Nw of eq. (3.1). A specific example centered at the origin could have an order parameter t h a t varied in space as 9op = f(rs/L), (j>op = a, where / is a smooth function equal to zero for r/L » 1, and equal to n at the origin. T h e topological charge Nw = 1-

j ^ d