Renormalization Group Analysis of Nonequilibrium Phase Transitions in Driven Disordered Systems [1st ed.] 978-981-13-6170-8, 978-981-13-6171-5

Table of contents :
Front Matter ....Pages i-xiii
Introduction (Taiki Haga)....Pages 1-28
Functional Renormalization Group of Disordered Systems (Taiki Haga)....Pages 29-46
Nonperturbative Renormalization Group (Taiki Haga)....Pages 47-77
Dimensional Reduction and its Breakdown in the Driven Random Field O(N) Model (Taiki Haga)....Pages 79-123
Nonequilibrium Kosterlitz-Thouless Transition in the Three-Dimensional Driven Random Field XY Model (Taiki Haga)....Pages 125-151
Summary and Future Perspectives (Taiki Haga)....Pages 153-156

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Springer Theses Recognizing Outstanding Ph.D. Research

Taiki Haga

Renormalization Group Analysis of Nonequilibrium Phase Transitions in Driven Disordered Systems

Springer Theses Recognizing Outstanding Ph.D. Research

Aims and Scope The series “Springer Theses” brings together a selection of the very best Ph.D. theses from around the world and across the physical sciences. Nominated and endorsed by two recognized specialists, each published volume has been selected for its scientific excellence and the high impact of its contents for the pertinent field of research. For greater accessibility to non-specialists, the published versions include an extended introduction, as well as a foreword by the student’s supervisor explaining the special relevance of the work for the field. As a whole, the series will provide a valuable resource both for newcomers to the research fields described, and for other scientists seeking detailed background information on special questions. Finally, it provides an accredited documentation of the valuable contributions made by today’s younger generation of scientists.

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Taiki Haga

Renormalization Group Analysis of Nonequilibrium Phase Transitions in Driven Disordered Systems Doctoral Thesis accepted by the Kyoto University, Kyoto, Japan

123

Author Dr. Taiki Haga Department of Physics Kyoto University Kyoto, Japan

Supervisor Prof. Shin-ichi Sasa Department of Physics Kyoto University Kyoto, Japan

ISSN 2190-5053 ISSN 2190-5061 (electronic) Springer Theses ISBN 978-981-13-6170-8 (hardcover) ISBN 978-981-13-6171-5 ISBN 978-981-13-6173-9 (softcover) https://doi.org/10.1007/978-981-13-6171-5

(eBook)

Library of Congress Control Number: 2018968379 © Springer Nature Singapore Pte Ltd. 2019 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 Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Supervisor’s Foreword

Phase transitions as a result of symmetry breaking occurs are classified into several classes (universality class) depending on the type of symmetry and the spatial dimension. This classification has been done for many systems in equilibrium, and various textbooks are published. Now, when a symmetry spontaneously breaks in systems out of equilibrium, there are not so many examples studied concretely. Under such circumstances, Springer Thesis by Taiki Haga provides us a new phase transition in a disordered system driven by an external force. The most striking result is that the quasi-long-range order appears in a three-dimensional system. This phenomenon is not clearly observed for equilibrium systems, and may be specific to systems out of equilibrium. The idea is simple. Some disorder destroys the phase order of the XY model in three dimensions, while an external driving restores the order. As a result, the system behavior is expected to be equivalent to that of pure XY model in two dimensions. This may be a sort of dimensional reduction for disordered nonequilibrium systems, which is his proposal. Of course, sufficient evidence is needed to assert qualitatively new properties of quasi-long-range order in three dimensions. He succeeded in raising its credibility by conducting numerical experiments and by performing the intensive renormalization group analysis. This theoretical analysis is quite tough. Indeed, for stochastic dynamics out of equilibrium, the renormalization group for the disordered system are formulated in a non-perturbative way. In order to complete it, it is necessary to deeply understand the contemporary development of the renormalization group, because the calculation requires advanced skills everywhere. Here, in this Springer Thesis, the renormalization group analysis for the disordered system and non-perturbative formulation are reviewed. These explanations and their concrete calculations are quite instructive. They also compile complicated calculations intelligently and give useful topics for future research.

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Supervisor’s Foreword

Readers of this Springer Thesis can learn the most advanced knowledge on the phase transition out of equilibrium. I am sure that graduate students in theoretical physics as well as researchers can enjoy reading this Springer Thesis. Kyoto, Japan December 2018

Prof. Shin-ichi Sasa

Parts of this thesis have been published in the following journal articles: 1. T. Haga, Nonequilibrium quasi-long-range order of a driven random-field O(N) model, Physical Review E 92, 062113 (2015). 2. T. Haga, Dimensional reduction and its breakdown in the driven random-field O(N) model, Physical Review B 96, 184202 (2017). 3. T. Haga, Nonequilibrium Kosterlitz-Thouless transition in a three-dimensional driven disordered system, Physical Review E 98, 032122 (2018).

vii

Acknowledgements

First of all, I would like to thank my supervisor, Shin-ichi Sasa, for his kind guidance and encouragement during my doctoral studies. I have learned from him not only physics, but also an attitude to scientific research. I am grateful to Gilles Tarjus and Matthieu Tissier in Université Pierre et Marie Curie. They provided me the opportunities to work in their laboratory when I visited in Paris. Many fruitful discussions with them helped me to proceed with my research. I also thank Bertrand Delamotte and Hugues Chaté for their stimulating discussions and remarks. I would like to express an appreciation to the members in Nonlinear laboratory in Kyoto University, Michikazu Kobayashi, Hiroki Ohta, Andreas Dechant, Masahiko Ueda, Yuki Minami, Tomokatsu Onaga, Hiroyoshi Takagi, Akito Inoue, Shuhei Taniguchi, Kazuki Fujita. They have helped me many times in my research, and in my daily life. Finally, I would like to thank my family for supporting me during my studies in the graduate course.

ix

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Physics of Phase Transitions . . . . . . . . . . . . . . . . . . . . 1.2 Phase Transitions in Disordered Systems . . . . . . . . . . . 1.2.1 Models of Disordered Systems . . . . . . . . . . . . . 1.2.2 Imry and Ma’s Argument . . . . . . . . . . . . . . . . . 1.2.3 Dimensional Reduction . . . . . . . . . . . . . . . . . . 1.2.4 Other Models of Disordered Systems . . . . . . . . 1.3 Disordered Systems Driven Out of Equilibrium . . . . . . 1.3.1 Collective Transports in Random Media . . . . . . 1.3.2 Phase Transitions in Driven Disordered Systems 1.4 Purpose of This Study . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Outline of This Thesis . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Functional Renormalization Group of Disordered Systems . . . . 2.1 Why is the Functional Renormalization Group Treatment Necessary? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 FRG of the Random Manifold Model . . . . . . . . . . . . . . . . . 2.2.1 RG Equation of the Disorder Correlator . . . . . . . . . . 2.2.2 Roughness Exponent Near Four Dimensions . . . . . . . 2.3 FRG of the Random Field and Random Anisotropy OðNÞ Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 RG Equation of the Disorder Correlator . . . . . . . . . . 2.3.2 Critical Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Fixed Point of the Random Field OðNÞ Model . . . . . 2.3.4 Fixed Point of the Random Anisotropy OðNÞ Model . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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47 47 47 50 52 52 53 58 60 61 65 67 69 69 71 76

4 Dimensional Reduction and its Breakdown in the Driven Random Field O(N) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Driven Random Field OðNÞ Model . . . . . . . . . . . . . . . . . . . . . 4.2 Dimensional Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 NP-FRG Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Scale-Dependent Effective Action . . . . . . . . . . . . . . . . . 4.3.2 Exact Flow Equation for the Effective Action . . . . . . . . 4.3.3 Derivative Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Dimensionless Quantities . . . . . . . . . . . . . . . . . . . . . . . 4.3.5 RG Equations Near the Lower Critical Dimensions . . . . 4.4 Critical Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Analytic Fixed Point . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Nonanalytic Fixed Point . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Fixed Line in the Case that N ¼ 2 and D ¼ 3 . . . . . . . . 4.4.4 Random Anisotropy Case . . . . . . . . . . . . . . . . . . . . . . . 4.5 Correlation Length in Three Dimensions . . . . . . . . . . . . . . . . . 4.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Propagators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Flow Equation for Fk ðqÞ . . . . . . . . . . . . . . . . . . . . . . . 4.6.3 Flow Equation for Dk ðˆ1 ; ˆ2 Þ . . . . . . . . . . . . . . . . . . . . 4.6.4 Flow Equations for Xk , vk , Zk , and Tk . . . . . . . . . . . . . . 4.6.5 Numerical Scheme to Calculate the Fixed Point . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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79 80 81 84 85 87 88 91 95 100 100 101 103 105 106 109 109 110 113 117 122 123

3 Nonperturbative Renormalization Group . . . . . . . . . . 3.1 General Formalism . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 NPRG of the OðNÞ Model . . . . . . . . . . . . . . . . . . 3.2.1 Derivative Expansion . . . . . . . . . . . . . . . . . 3.2.2 Flow Equations . . . . . . . . . . . . . . . . . . . . . 3.2.3 Fixed Point and Critical Exponents . . . . . . . 3.3 NP-FRG of Disordered Systems . . . . . . . . . . . . . . 3.3.1 General Formalism of the NP-FRG . . . . . . . 3.3.2 NP-FRG of the Random Manifold Model . . 3.3.3 NP-FRG of the Random Field OðNÞ Model 3.4 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Exact Flow Equation for Ck . . . . . . . . . . . . 3.4.2 Exact Flow Equations for Cp;k . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

5 Nonequilibrium Kosterlitz-Thouless Transition in the Three-Dimensional Driven Random Field XY Model 5.1 Spin-Wave Calculation . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Correlation Function . . . . . . . . . . . . . . . . . . . . . 5.2.2 Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 RG Analysis of the Spin-Wave Model . . . . . . . . . . . . . . 5.3.1 Exact Flow Equation for the Effective Action . . . 5.3.2 Flow Equations of the Disorder Cumulants . . . . . 5.3.3 RG Evolution of the Disorder Cumulants . . . . . . 5.4 Effect of Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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125 126 131 131 133 135 135 137 142 145 150

6 Summary and Future Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . 153 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

Chapter 1

Introduction

1.1 Physics of Phase Transitions Our naive intuition says that systems composed of simple constituents are also simple. In other words, the behavior of complex systems is trivially predictable from the property of their elements. This reasonable belief had been widely accepted by ancient people before the era of Descartes and Newton. Therefore, they relied on supernatural principles to explain the origin of “irreducible complexity” in our world, such as life. However, we today recognize that this intuition is incorrect and the whole system can have properties which its parts do not have. No matter how complicated living things seem to be, they are ultimately composed of atoms which obey a simple dynamical rule. This is one of the most remarkable and profound facts in the world. Then, we are naturally led to a question how rich macroscopic physics emerges from a simple microscopic principle, or more naively, why our universe is so complicated despite of the simpleness of the fundamental law. To answer this challenging question is the goal for those who are working in condensed matter physics, statistical physics, and biological physics. Unfortunately, our current understanding concerning this problem is very limited. As an elementary step toward clarifying the mechanism underlying the emergence of macroscopic physics, we restrict our attention to the simplest phenomena which are easy for mathematical modeling and analysis. It is phase transition [1, 2]. When one continuously changes external parameters, such as temperature and pressure, the macroscopic state of many-body systems can discontinuously change at some critical point. Examples include liquids to solids transitions in interacting particle systems, and para-to ferromagnetic transitions in magnetic systems. Although these phenomena are ubiquitous in our daily life, it is highly nontrivial task to understand their mechanism from a microscopic dynamical rule. Note that the property of the interactions among microscopic constituents, such as molecules or spins, do not change at the transition point. Therefore, phase transition is one of the simplest emergent phenomena that cannot be explained only from the nature of the individual parts composing the systems. © Springer Nature Singapore Pte Ltd. 2019 T. Haga, Renormalization Group Analysis of Nonequilibrium Phase Transitions in Driven Disordered Systems, Springer Theses, https://doi.org/10.1007/978-981-13-6171-5_1

1

2

1 Introduction

Phase transition is a consequence of the competition between interaction and fluctuation. Systems with large degrees of freedom tend to be “ordered” in the ground state, where the interaction energy is minimized. For example, interacting particles form into a crystal and spin systems settle in a completely ordered ferromagnetic or anti-ferromagnetic state at zero temperature. If the interactions between the constituents are simple, the ground states of the systems composed of them are also trivial. There is no interesting structure in the ground state. We now ask what happens when such interacting systems are stirred by fluctuations? There are two distinct types of fluctuations, time-dependent and time-independent fluctuations. The typical example of the former type is thermal fluctuation, which originates from the dynamics of microscopic degrees of freedom ignored in the effective description of the system. One can also consider nonthermal time-dependent fluctuation. For example, suppose that a macroscopic dissipative system, such as fluids and granular systems, is randomly agitated by an external agent. As a more subtle type of fluctuation, there is one resulting from the uncertainty principle in quantum systems, i.e., quantum fluctuations. In this section, let us concentrate on systems driven by thermal fluctuations. Note that fluctuations do not have any remarkable features themselves, and they are often described by Gaussian white noise, which is the simplest stochastic process. What is surprising in many-body physics is that the competition between simple interactions and random fluctuations yields a wide variety of nontrivial structures and complicated dynamics. The intuitive explanation of phase transitions is simple. When the fluctuation is quite small, the structure of the ground state is hardly affected and the system remains ordered. However, if the fluctuation prevail over the interaction, the order is destroyed and a disordered phase (high-temperature phase, paramagnetic phase) is realized. To clarify the mathematical structure underlying these phenomena, we introduce an analytically tractable model; an N -component spin system with ferromagnetic interaction. Let Si = (Si1 , . . . , SiN ) be a spin variable at site i in the D-dimensional hyper cubic lattice. The norm of each spin is fixed at unity: |Si |2 = 1. The Hamiltonian is given by   H({Si }) = −J Si · S j − h · Si , (1.1) i j

i

where i j indicates the nearest-neighbor sites and J is a positive constant. h is a uniform external field. The Hamiltonian is invariant with respect to the global rotation of the spins in the absence of the external field. Obviously, this Hamiltonian attains its minimum when all spins are completely ordered. It is convenient to introduce a continuous version of this spin model. Let φ(r) = (φ 1 (r), . . . , φ N (r)) be an N component real vector field. The simplest Hamiltonian with the O(N ) rotational symmetry is given by  H[φ] =

dDr

 1 α

2

 K |∇φ α (r)|2 + U (ρ(r)) − h · φ(r) ,

(1.2)

1.1 Physics of Phase Transitions

3

where K is a positive constant and ρ(r) = |φ(r)|2 /2 is the field amplitude. The choice of the local potential U (ρ) is arbitrary as long as it is lower bounded, for example, we have (1.3) U (ρ) = rρ + gρ 2 , with a positive constant g. The first term in Eq. (1.2) denotes an elastic energy, which favors uniform configurations, and the second term determines the amplitude of the field. If U (ρ) has its minimum at a nonzero value, the ground state of this model is a completely ordered ferromagnetic state. We call the model defined by Eq. (1.1) or (1.2) the O(N ) model. It is worth to note that the continuous model Eq. (1.2) can be considered as a long-distance description of the lattice model Eq. (1.1). Namely, the field φ(r) corresponds to a coarse-grained spin variable: φ(r) = l −D



Si ,

(1.4)

i∈Cr

where Cr is a hyper-cube with length l centered at r. The coarse-graining scale l is chosen such that it is much smaller than the correlation length, but Cr still contains many spins. For N = 1 this model is called the Ising model, for N = 2 the XY model and for N = 3 the Heisenberg model. The equilibrium dynamics are described by the following equation of motion: ∂t φ α = −

δH[φ] + ξ α. δφ α

(1.5)

The time-dependent Gaussian random noise ξ(r, t) satisfies ξ α (r, t) = 0, ξ α (r, t)ξ β (r  , t  ) = 2T δ αβ δ(r − r  )δ(t − t  ),

(1.6)

where T is a temperature. The physical meaning of Eq. (1.5) is as follows; the first term of the right-hand side represents the force that reduces the energy and the second term is the thermal fluctuation. At zero temperature, the system relaxes toward its ground state, while at finite temperature, it reaches a statistically steady state characterized by the Boltzmann-Gibbs distribution, PG [φ] =

e−βH[φ] , Z

(1.7)

where β = 1/T is the inverse temperature and  Z=

Dφe−βH[φ]

(1.8)

4

1 Introduction

is the partition function. The average of an arbitrary physical quantity A[φ] in the steady state is given by  (1.9) A[φ] = Dφ A[φ]PG [φ]. To calculate the average Eq. (1.9) is highly nontrivial and there are only a few exactly solvable examples. Various theoretical methods to approximately evaluate Eq. (1.9) have been developed, such as mean-field theory, high temperature expansion, spinwave approximation, and renormalization group (RG) theory. The ground state of the O(N ) model is a completely ordered state. When the spatial dimension of the system is higher than the so-called lower critical dimension Dlc , the ordered phase is stable against small thermal fluctuations. For the Ising model (N = 1), Dlc = 1, and for N ≥ 2, Dlc = 2. The important difference between the cases that N = 1 and N ≥ 2 is the absence or presence of massless modes. For N ≥ 2, there are infinitely many ground states which are continuously connected by the global rotation of the spins. Due to this degeneration, the system has infinitesimally low energy excitations, which are called massless or Goldstone modes. At two dimensions, such excitations destroy the ordered phase at any finite temperatures (Mermin-Wagner theorem). Above the lower critical dimension, there is a long-range ordered (LRO) phase at low temperatures, wherein the equal-time correlation function C(r  − r) = φ(r  ) · φ(r) attains a nonzero constant M 2 in the long-distance limit |r  − r| → ∞. The magnetization M decreases with temperature, and eventually it vanishes at some critical temperature Tc . In the high-temperature phase, the correlation function decays exponentially, C(r) ∼ e−|r |/ξc , where ξc is the correlation length. Near the critical point between two phases, the system exhibits anomalous behaviors, such as the divergence of the correlation length and the characteristic time-scale. These critical phenomena are characterized by the critical exponents and scaling functions, which are known to be independent of the microscopic detail of the system. This remarkable universality can be explained by the RG theory. Let us briefly sketch the concept of the RG theory [1–4]. The partition function Eq. (1.8) contains all fluctuations with the momentum (wavenumber) 0 ≤ |q| ≤ , where is an ultra-violet cutoff corresponding to the inverse of the lattice constant. In the RG theory, one follows the evolution of the effective Hamiltonian when high energy fluctuations are successively integrated out. We split the field φ into slowly and rapidly varying contributions, φ = φ L + φ S , where φ L contains long wavelength modes with |q| < k and φ S contains short wavelength modes with k < |q| < . The effective Hamiltonian H˜ k for the slowly varying component is defined by ˜

e−Hk [φ ] = L



Dφ S e−H[φ

L

+φ S ]

,

(1.10)

where the inverse temperature β is absorbed into the Hamiltonian. While in the original model the length is measured in units of the lattice constant −1 , in the coarse-grained model described by H˜ k the length should be measured in units of k −1 . This redefinition of the cutoff → k leads to the rescaling of the parameter

1.1 Physics of Phase Transitions

5

in H˜ k . When k is chosen to be close to , the successive application of the above transformation yields a continuous flow in the parameter space of the Hamiltonian. The thermodynamic phases (long-range ordered or disordered phase) and the critical point are characterized by a fixed point, at which the flow of the effective Hamiltonian vanishes. For example, the long-range ordered and disordered phases correspond to the zero and infinite temperature fixed points, respectively. Between these fixed points, there is a nontrivial fixed point that controls the critical behavior. The critical exponents and scaling functions can be obtained from the structure of the RG flow near the fixed point. The important point is that the nature of the fixed point is independent of the microscopic detail of the system. Therefore, the critical behaviors in a wide variety of systems can be classified into a few universality classes which depend only on symmetry and spatial dimension. In general, the RG transformation Eq. (1.10) cannot be performed exactly, thus we need some approximations. In many cases, the perturbative method would be useful, where the nonlinear term in the effective Hamiltonian is treated as a perturbation. It is justified when the coefficient of the nonlinear term evaluated at the fixed point is quite small. This condition is satisfied near the upper critical dimension, above which the nonlinear term becomes irrelevant. Although the perturbative RG approach has enjoyed considerable success, one should keep in mind that it fails for some types of systems. Interacting systems with quenched disorder are remarkable examples that the perturbative RG fails, as we will discuss later.

1.2 Phase Transitions in Disordered Systems In the previous section, we discussed phase transitions in interacting systems driven by time-dependent fluctuations. Next, let us consider what happens when timeindependent and spatially random fluctuations are exerted on the systems. This type of fluctuations is called quenched disorder. The precise meaning of “time-independent” is that the disorder does not change on the typical time scales in which we are interested. For example, suppose ferromagnetic materials with defects or impurities, which do not move on the time scale of the flip of individual spins. Obviously, the effect of the quenched disorder to destroy the ordered state is much stronger than that of the thermal fluctuations. Since all specimens in experiments inevitably contain impurities or defects, it is important from technical viewpoint to investigate the effect of the quenched disorder. In addition, it is also an intriguing problem from theoretical perspectives to consider what type of phase transitions and critical phenomena can emerge from the competition between the interaction and the quenched disorder. In this section, we introduce some well-studied models of disordered systems and discuss their properties. From a simple phenomenological argument, we first determine the lower critical dimensions for these models. We next show that the standard perturbative approach leads to a beautiful conclusion; dimensional reduction, which states that the critical behavior of disordered systems in spatial dimension D is the

6

1 Introduction

same as that of pure systems in D − 2 dimensions. However, it is known that this property breaks down in low enough dimensions due to the presence of multiple meta-stable states. Therefore, in disordered systems there is nontrivial physics that is inaccessible by the perturbative field theory.

1.2.1 Models of Disordered Systems The Hamiltonian of the O(N ) model is given by Eq. (1.1). We consider a class of disordered systems known as random field models, in which the external field h at each site is assumed to be a random variable. Thus, the Hamiltonian is given by HRF = −J

 i j

Si · S j −



h i · Si ,

(1.11)

i

where hi is independently chosen according to a mean-zero Gaussian probability distribution. The continuum representation of Eq. (1.11) is given by  HRF =

dDr

 1 α

2

 K |∇φ α (r)|2 + U (ρ(r)) − h(r) · φ(r) ,

(1.12)

where U (ρ) is given by Eq. (1.3) and h(r) obeys a mean-zero Gaussian distribution satisfying h α (r)h β (r  ) = h 20 δ αβ δ(r − r  ). (1.13) The over-bar in Eq. (1.13) represents the average over the disorder. This model is called the random field O(N ) model (RFO(N )M) and it is relevant to describe a variety of systems encountered in condensed matter physics. For instance, in diluted antiferromagnets, a uniform magnetic field leads to a random field like effect for the staggered magnetization [5]. Other examples that can be modeled by the random field Ising model include critical and binary fluids in porous media [6, 7], and absorbed mono-layers on impure substrates [8]. Note that the random field explicitly breaks the O(N ) symmetry. However, the O(N ) symmetry can be recovered after the disorder average is taken because the probability distribution function for the random field is invariant under the O(N ) rotation. We also introduce the O(N ) model with randomly distributed easy axis of magnetization:     1 K |∇φ α (r)|2 + U (ρ(r)) − τ αβ (r)φ α (r)φ β (r) , (1.14) HRA = d D r 2 α αβ where the random anisotropy tensor τ αβ (r) obeys a mean-zero Gaussian distribution satisfying

1.2 Phase Transitions in Disordered Systems

τ αβ (r)τ μν (r  ) =

7

 τ02  αμ βν δ δ + δ αν δ βμ δ(r − r  ). 2

(1.15)

This model is called the random anisotropy O(N ) model (RAO(N )M), which describes, for instance, amorphous magnets [9] and liquid crystals in porous media [10–13]. Note that for N = 1 the random anisotropy model is identical to the socalled random temperature Ising model (see Sect. 1.2.4). We further define another type of model; the random manifold model, which describes an elastic manifold in a random potential. The Hamiltonian is given by  HRM =

D

d r

 1 α

2

α



K |∇φ (r)| + V (r; φ(r)) , 2

(1.16)

where the random potential V (r; φ) satisfies V (r; φ)V (r  ; φ  ) = R(φ − φ  )δ(r − r  ).

(1.17)

There are two different types of elastic systems, interfaces and lattices. For interfaces, φ(r) is considered as the height of the surface at r and for lattices, it represents the displacement from the reference positions r (see Fig. 1.1). The former corresponds to domain walls in disordered magnets [14] and the contact line of a liquid wetting a rough substrate [15, 16]. The latter corresponds to vortex lattices in dirty type-II superconductors [17, 18]. The disorder correlator R(φ) depends on a physical situation under consideration. For interfaces, two sets of simple forms are usually employed, the short-range disorder, (1.18) R(φ) ∼ exp(−|φ|2 ), and the power-law correlations,

Fig. 1.1 Schematic pictures of elastic systems in a random potential. a Interfaces. b Lattices. The filled dots denote impurities

8

1 Introduction

R(φ) ∼

|φ|2(1−γ ) , γ −1

(1.19)

for large φ. For instance, in the case of domain walls in the random field Ising model, the energy resulting from the coupling to the random field h is given by φ  V (r; φ) ∼ dφ h(r, φ  ). Thus, we have [V (r; φ) − V (r; φ  )]2 ∼ |φ − φ  |, which leads to γ = 1/2 in Eq. (1.19). In the case of domain walls in the diluted Ising model, where each site is randomly occupied or empty by spins (see Sect. 1.2.4), the disorder correlator is described by the short-range form (1.18) because the global spin flip does not change the Hamiltonian and the total energy of the system is almost independent of the position of the domain wall. We define the random field and random bond correlators by R(φ) ∼ −|φ| and R(φ) ∼ exp(−|φ 2 ), respectively. For lattices in a random potential, R(φ) is assumed to be a periodic function, R(φ + eα ) = R(φ),

(1.20)

where {eα }α=1,...,N are the primitive translation vectors of the lattice. It is worth noting that the random field XY model (N = 2) without topological defects can be mapped into the random manifold model by introducing a phase parameter u ∈ (−∞, ∞) by φ/|φ| = (cos u, sin u). For a given realization of the quenched disorder, the equilibrium dynamics is given by Eq. (1.5). The thermal average of an aribitrary functional A[φ] is written as Eq. (1.9), which is a functional of the disorder. We denote the average over the disorder as  A[φ] = Dh PR [h]A[φ], (1.21) where PR [h] is the distribution of the disorder. Note that the disorder average is always calculated after the thermal average is taken. An important quantity measured in the random manifold models is the roughness exponent ζ , |φ(r) − φ(r  )|2  ∼ |r − r  |2ζ . (1.22) If the interface is flat, in other words, the left-hand side of Eq. (1.22) reaches a finite value at large distance, the roughness exponent is zero. Note that in the absence of the disorder, ζpure = (2 − D)/2 for D < 2 and ζpure = 0 for D > 2.

1.2.2 Imry and Ma’s Argument Let us consider the lower critical dimension of the RFO(N )M. There is a well-known phenomenological argument by Imry and Ma [19]. In the presence of the random field, the system can lower its energy by distorting the spin configuration according to the direction of the random field. For this spin distortion, we estimate the increase

1.2 Phase Transitions in Disordered Systems

9

Fig. 1.2 Schematic pictures of Imry and Ma’s argument. The arrows represent the spins, which are distorted over a region of linear size L. We consider whether the spin distortion is energetically favorable or not

of energy due to the ferromagnetic interaction between the spins and the decrease of energy due to the coupling to the random field. By comparing these energies, we can see whether or not an ordered phase is stable in the presence of the random field. For simplicity, we restrict our interest to the zero-temperature case. In fact, it is known that the thermal fluctuations are irrelevant at low temperatures. First, we consider the random field Ising model, which corresponds to the case that N = 1. Suppose that the system is prepared in a completely ordered state and the spins in a region of linear size L are flipped along the mean direction of the random field (see Fig. 1.2). Since the elastic energy E el is proportional to the surface area of the region, it scales as (1.23) E el ∼ σ L D−1 , where σ is the surface tension. On the other hand, the energy due to the coupling to the random field E R scales as E R ∼ h 0 L D/2 . (1.24) By comparing them, one finds that if D > 2 the elastic energy prevails the random field and the ordered phase is stable at weak disorder. Thus, the lower critical dimension of the random field Ising model is expected to be two. Indeed, it is proven by Aizenman and Wehr that for D ≤ 2 the random field Ising model does not exhibit any phase transition [20]. Furthermore, the three-dimensional random field Ising model is rigorously shown to exhibit LRO at weak disorder [21]. Next, we consider the case that N ≥ 2. The important difference from the case of the Ising model is that there is no domain wall with a definite thickness due to the presence of massless modes. As the spins in the region mentioned above are twisted along the random field, a gradual distortion of them yields a elastic energy density ∼ K L −2 , and E el is given by (1.25) E el ∼ K L D−2 . By comparing Eqs. (1.24) and (1.25), one finds that if D > 4 the ordered phase is stable at weak disorder. Thus, the lower critical dimension for the RFO(N )M with

10

1 Introduction

N ≥ 2 is expected to be four. In fact, it is proven by Aizenman and Wehr that for D ≤ 4 the model does not exhibit LRO [20].

1.2.3 Dimensional Reduction We show that a naive perturbation theory of the RFO(N )M leads to a remarkable result, which states that the critical exponents of the D-dimensional RFO(N )M is identical to those of the (D − 2)-dimensional pure O(N ) model [22, 23]. This property is called the dimensional reduction. Our starting point is the Hamiltonian (1.12). For simplicity, we consider the case of the Ising model (N = 1). For a given random field, we expand thermodynamic functions in a perturbation expansion in both g and h. In a diagrammatic representation, each four-point interaction gφ 4 is denoted by a vertex, and each random field hφ is denoted by an external line ending at a point with the variable h. The diagrams for the correlation function are given in Fig. 1.3. The upper diagrams (a) correspond to the correlation function for a given random field, φ(r)φ(r  ) − φ(r)φ(r  ), and the lower diagrams (b) corresponds to the disorder averaged correlation function, φ(r)φ(r  ) − φ(r)φ(r  ). A filled circle represents a four-point vertex and a cross symbol represents the random field. The combinatorial factors are omitted. Note that upon disorder averaging, each pair of the random field lines meet and creates an internal line with two propagators and with the disorder strength h 20 . The most divergent diagrams for the disorder averaged correlation function are obtained by adding cross symbols to those of the pure system as many as possible unless the diagrams become disconnected if one cuts the internal lines with a cross. In the resulting diagrams, each loop has one internal line with a cross symbol. For the pure model, a graph with l-loops and n internal lines yields an integral (pure)

ID

=

 n l d D ki (r + p 2j )−1 , D (2π ) i=1 j=1

(1.26)

(a) (b)

Fig. 1.3 Diagram representation for the correlation function. a Correlation function for a fixed random field. b Disorder averaged correlation function

1.2 Phase Transitions in Disordered Systems

11

where pj = sj +



η jm km ,

(1.27)

m

where s j is some function of the external momenta, η jm is equal to one if line j is contained in the loop m and is zero otherwise. The elastic constant K is set to unity for simplicity. For the random field model, the corresponding graph yields an integral I D(rand)

=

h 2l 0

n l  d D ki (r + p 2j )−λ j , D (2π ) C i=1 j=1

(1.28)

where λ j = 2 for lines with a cross and λ j = 1 without a cross. The summation C is over all arrangements of the crosses such that each loop has one internal line with a cross. Then, it can be shown that these integrals satisfy the following relation: I D(rand) =



h 20 4π

l

(pure)

I D−2 .

(1.29)

Thus, the diagram for the random field model is equal to the corresponding diagram for the pure model in two fewer dimensions. This leads to the dimensional reduction. The equivalence between the random field model and the lower dimensional pure model can be also understood from a hidden supersymmetry [24]. Let us consider the zero-temperature case because the thermal fluctuation is expected to be irrelevant. For a given random field h, the ground state satisfies − ∇ 2 φh + V  (φh ) − h = 0,

(1.30)

where V (φ) = (r/2)φ 2 + (g/4)φ 4 . Then, the (disconnected) correlation function is given by  Dh PR [h]φh (r)φh (r  ),

φ(r)φ(r  ) ∼

(1.31)

where the distribution function for the random field is    1 D 2 PR [h] ∼ exp − d rh(r) . 2

(1.32)

where the variance is set to unity. Using standard manipulations, we have  φ(r)φ(r  ) ∼ ∼ ∼

 

DhDφ PR [h]φ(r)φ(r  )δ(−∇ 2 φ + V  (φ) − h)|det(−∇ 2 + V  (φ))| ˆ

DhDφDφˆ PR [h]φ(r)φ(r  )e−i φ(−∇ ˆ

 −i φ(−∇ ˆ DφDφφ(r)φ(r )e

2

2

φ+V  (φ)−h)

ˆ 2 φ+V  (φ))+ 21 (i φ)

det(−∇ 2 + V  (φ))

det(−∇ 2 + V  (φ)), (1.33)

12

1 Introduction

where we have introduced an auxiliary field φˆ to take the average over the random field. Note that for the ground state, the Hessian det(−∇ 2 + V  (φ)) is positive. To represent the Hessian in terms of the exponential of some action, we introduce two anticommuting (Grassmann) fields ψ and ψ¯ [4], and we find  φ(r)φ(r  ) ∼

ˆ

¯

 −S[φ,φ,ψ,ψ] ˆ ¯ DφDφDψD ψφ(r)φ(r )e ,

(1.34)

where the action S is given by  S=

  1 2 2  2  ˆ ˆ ¯ d r − (i φ) + i φ(−∇ φ + V (φ)) + ψ(−∇ + V (φ))ψ . (1.35) 2 D

It is convenient to introduce the superspace characterized by the D-dimensional coordinate r and two anticommuting coordinates (θ, θ¯ ). If we define the superfield ¯ ˆ ¯ (r, θ ) = φ(r) + ψ(r)θ¯ + θ ψ(r) + i φ(r)θ θ,

(1.36)

Eq. (1.35) can be rewritten as  S=

  1 d D rd θ¯ dθ − ss  + V () , 2

(1.37)

where ss = ∇ 2 + ∂ 2 /∂ θ¯ ∂θ . When one constructs a perturbation theory based on the supersymmetric action Eq. (1.37), the resulting diagrams contain an integral of the form    D 2 D  2 ¯ ¯ d rd θ dθ f (r + θ θ ) = − d r f (r ) ∼ − dr r D−1 f  (r 2 )   = dr r D−3 f (r 2 ) ∼ d D−2 r f (r 2 ). (1.38) This implies that the Green’s functions calculated in the D-dimensional superspace are the same as those calculated in the (D − 2)-dimensional space. Thus, the dimensional reduction follows. Unfortunately, the dimensional reduction is known to break down in low enough dimensions. For instance, according to the dimensional reduction, the lower critical dimension of the random field Ising model is expected to be three because that of the pure Ising model is one. However, it contradicts the Imry-Ma’s argument, where the lower critical dimension is shown to be two. The failure of the above supersymmetric formalism is a consequence of the fact that there are a large number of stationary states (meta-stable states) satisfying Eq. (1.30). We denote the set of all stationary (χ) states as {φst (r)}χ=1,...,N , where N is the number of the stationary states. Note that (χ) φst (r) depends on the realization of the disorder. The correlation function is then given by

1.2 Phase Transitions in Disordered Systems

 C(r) =

Dh PR [h]

13 N  e−β Eχ χ=1

Z

(χ)

(χ)

φst (r)φst (0),

(1.39)

(χ) −β E χ where E χ = H[φst ] and Z = N . More rigorously, one should distinχ=1 e guish minima, maxima, and saddle-points of the Hamiltonian, and the summation in Eq. (1.39) should be taken over only local minima. However, at low temperatures, we expect that the contribution from the local minima dominates in this summation. Especially, at zero temperature, C(r) can be expressed in terms of the ground state. The calculation of Eq. (1.39) is highly nontrivial. Instead, we define a correlation (χ) function by averaging over {φst (r)} with the uniform weight,  Cuni (r) =

Dh PR [h]

N 1  (χ) (χ) sign(χ )φst (r)φst (0), N χ=1

(1.40)

where “sign(χ )” represents the sign of the Hessian det(δ 2 H/δφδφ) evaluated at (χ) φ = φst , for example, sign(χ ) = 1 for a local minimum of the Hamiltonian. Equation (1.40) is nothing but Eq. (1.34). Therefore, Cuni (r) is identical to the correlation function of the pure system in D − 2 dimensions. However, the large-scale behavior of Cuni (r) is not necessarily the same as that of the actual correlation function C(r). One can also derive the dimensional reduction for the random manifold models defined by Eq. (1.16). A naive perturbative expansion of any correlation function yields the same result that obtained from the Gaussian theory setting R(φ) = R(0) − |R  (0)|φ 2 /2. Especially, the two point correlation function reads to all orders, φ(q)φ(−q)DR =

|R  (0)| . |q|4

(1.41)

This dimensional reduction leads to a roughness exponent ζDR =

4− D , 2

(1.42)

which is known to be incorrect. For example, when the disorder cumulant is given by Eq. (1.19) with γ = 1/2, the roughness exponent is ζ = (4 − D)/3. How and when the dimensional reduction breaks down is one of the central issues in statistical mechanics of disordered systems and there are vast number of literature devoted to this topic. One of the theoretical approaches to overcome the dimensional reduction is the so-called functional renormalization group (FRG). We will review the formalism of the FRG treatment in the next chapter.

14

1 Introduction

Fig. 1.4 Schematic pictures of the diluted spin model, random field model, and spin glass model. In the diluted spin model, each site is randomly occupied or empty. In the random field model, there exists spatially inhomogeneous external field, which is depicted by the gray arrows. In the spin glass model, the interactions between spins are randomly distributed

1.2.4 Other Models of Disordered Systems Although in the following sections we consider the random field models, there are other important classes of disordered systems, spin glass models and diluted spin models. The schematic pictures for these models are shown in Fig. 1.4. We provide a brief review for these models below. Spin glass model In the spin glass models, the couplings J between spins are randomly distributed: HSG = −



Ji j Si · S j ,

(1.43)

i j

where Ji j are mean-zero Gaussian random variables. The spin glass models describe magnetic alloys in which magnetic impurities are randomly placed in magnetically inert host [14, 25, 26]. Since the interactions between the impurities depend on the distance with oscillation, they take random values that changes in sign. The random coupling creates frustration, which means that it is impossible to satisfy all couplings at the same time, as it could be in ferromagnetic system. Frustration yields multiple meta-stable states. For simplicity, we consider the Ising spin glass model, wherein each spin variable Si has two values ±1. The mean-field model of Eq. (1.43), in which the sum of the interaction is extended over all pairs of spins, has been extensively investigated [27–31]. This model is in paramagnetic phase at high temperatures, while at low temperatures it exhibits the so-called spin glass phase. In this phase, the total magnetization remains zero, and the local spin configurations are effectively frozen in time. To define the spin glass phase, we introduce the overlap between two configurations σ and τ , qσ τ

Ns 1  = σi τi , Ns i=1

(1.44)

1.2 Phase Transitions in Disordered Systems

(a)

15

(b)

Fig. 1.5 Schematic pictures of the disorder averaged overlap distribution. a Paramagnetic phase. b Spin glass phase. qEA denotes the Edwards-Anderson parameter

where Ns is the number of the spins. If σ and τ almost coincide, qσ τ has 1, and if they are totally uncorrelated, qσ τ is zero. By considering two systems with the same disorder, we define the overlap distribution, P(q) =

1 Z2



Dσ Dτ e−βH[σ ] e−βH[τ ] δ(q − qσ τ ).

(1.45)

For instance, in the pure Ising model, P(q) has a single peak at q = 0 in paramagnetic phase, and it has two peaks at ±M 2 , where M is the spontaneous magnetization, in ferromagnetic phase. In the spin glass phase, the phase space is decomposed into many sub-components, which are separated by infinitely high energy barriers, and the system explores only one of such sub-parts of the phase space. This ergodicity breaking leads to a broad distribution of the overlap. Figure 1.5 shows schematic pictures of the disorder averaged overlap distribution P(q) for the spin glass model. The left and right panels of Fig. 1.5 represent P(q) for paramagnetic phase (high temperature phase) and spin glass phase (low temperature phase), respectively. In the paramagnetic phase, P(q) has a single peak at q = 0, as in the pure Ising model. In the spin glass phase, it has a nontrivial broad structure. The edge point of the support of P(q) is known as the Edwards-Anderson parameter qEA , which corresponds to the overlap between states belonging to the same basin of the energy landscape. The transition where the overlap distribution develops such a nontrivial structure is called replica symmetry breaking (RSB). It is known that the random manifold model (1.16) also exhibits RSB in the large N limit. This RSB is directly related to the breakdown of the dimensional reduction [32, 33]. More precisely, its replica symmetric solution always leads to the roughness exponent predicted from the dimensional reduction (1.42), while the full RSB solution results in the correct roughness exponent. Diluted spin model In the diluted spin model, each site of the lattice is occupied by a spin with the probability p or empty with the probability 1 − p, where p is a fixed number between zero and one. The occupation of each site is independent of the occupation of other sites. Let m i be a random variable that has the value 1 if site i is occupied and 0 if

16

1 Introduction

site i is vacant. The Hamiltonian of the dilute spin model without an external field is given by  HDS = −J m i m j Si · S j . (1.46) i j

The minimal continuum representation of Eq. (1.46) is given by  HDS =

dDr

 α

  1 1 g K |∇φ α (r)|2 + r + δτ (r) |φ(r)|2 + |φ(r)|4 , (1.47) 2 2 4

where δτ (r) is a random variable that obeys a mean-zero Gaussian distribution satisfying δτ (r)δτ (r  ) = τ02 δ(r − r  ). (1.48) Equation (1.47) is also called the random temperature model because r + δτ (r) corresponds to an inhomogeneous temperature. The equivalence of Eqs. (1.46) and (1.47) can be shown by averaging the disorder with the aid of the replica method [34]. One of the important features of the diluted spin model is that the disorder does not break the O(N ) symmetry. In fact, the Hamiltonian Eq. (1.47) is invariant under the global rotation of φ. This nature leads to the fact that the lower critical dimension of this model is the same as the corresponding pure model. Thus, the diluted Ising model and XY model exhibit LRO in two and three dimensions, respectively. Next, it is natural to ask whether or not the disorder alters the critical behavior near the transition point. A simple criterion is known, the so-called Harris criterion [35], which enables us to predict the effect of the disorder from the critical exponents of the pure system. This criterion states that the disorder changes the critical exponents if Dνpure < 2 or αpure = 2 − Dνpure > 0, where νpure and αpure are the exponents of correlation length and specific heat, respectively, for the corresponding pure model. In contrast, when Dνpure > 2, the disorder appears to be irrelevant. For example, since αpure for the Ising model (N = 1) is 0.11, the diluted Ising model belongs to a different universal class from the pure one, while for the Heisenberg model (N = 3), the disorder does not affect the critical behavior because αpure −0.1. If the disorder changes the critical exponents, the second question is whether they depend on the disorder strength or not. At this time, it is widely believed that the critical behavior of the diluted spin model is universal, in other words, the critical exponents are independent of the disorder strength. As the disorder strength becomes weaker, the vicinity of the transition temperature, in which the critical behavior turns out to be different from that of the pure system, becomes narrower. The random temperature does not modify the system so strongly compared to the random field and random interaction. Thus, the theoretical analysis for the diluted spin model is relatively easier than those for the random field and spin glass models. However, there are some unsettled problems concerning this model. For the diluted Ising model, it has been theoretically suggested that a spin glass phase exists (do not

1.2 Phase Transitions in Disordered Systems

17

confuse with the spin glass model, which is just some class of disordered systems) between paramagnetic and ferromagnetic phases [36–38]. The presence of the spin glass phase implies that, near the phase boundary between paramagnetic and ferromagnetic states, there exist a large number of spanning (percolating) ferromagnetic clusters, whose magnetization remains random from cluster to cluster. However, a definite evidence of the spin glass state in experiments or numerical simulations is not reported yet.

1.3 Disordered Systems Driven Out of Equilibrium Suppose that a driving force is applied to a many-body system in a random environment. We ask what types of phenomena emerge from the interplay between interaction, disorder, and driving. This is, of course, a considerably broad subject. Thus, we restrict our attension to the following two subtopics, collective transports in random media and phase transitions in driven disordered systems. In the first topic, we consider transport properties of disordered systems driven out of equilibrium. Here, main object we are interested in is current-force characteristics, which measure how much current is induced by the applied external force. In the second topic, we discuss phase transitions and critical phenomena in driven disordered systems. Our problem is how the competition between disorder and driving leads to a novel type of phase transitions that cannot be observed in thermal equilibrium.

1.3.1 Collective Transports in Random Media Let us consider an elastic system in a random substrate, whose Hamiltonian is given by Eq. (1.16). For simplicity, suppose the case of the one-component field (N = 1). In the presence of a driving force f , the equation of motion is given by ∂t φ(r, t) = K ∇ 2 φ(r, t) + FR (r; φ(r, t)) + f + ξ(r, t).

(1.49)

The random force FR (r; φ) = −∂φ V (r; φ) satisfies FR (r; φ)FR (r  ; φ  ) = (φ − φ  )δ(r − r  ),

(1.50)

where (φ) = −∂φ2 R(φ). The thermal noise ξ(r, t) satisfies ξ(r, t)ξ(r  , t  ) = 2T δ(r − r  )δ(t − t  ).

(1.51)

First, we consider the zero-temperature case. When the driving force f , which tries to make the system move, is applied, this will be resisted by the random force exerted by the substrate. If f is smaller than a certain critical value f c , the system is

18

1 Introduction

pinned by the substrate and does not move. As f becomes large enough to overcome the resistance of the pinning forces, the system starts to move and eventually attains some steady velocity v. This transition from a pinned state to a moving state is known as depinning transition, in which the average velocity v corresponds to an order parameter. As in second-order transitions in equilibrium systems, the depinning transition can be described in the framework of standard critical phenomena. In fact, as f approaches to f c from above f → f c+ , the correlation function in the comoving frame behaves as  t 2ζ 2 , (1.52) (φ(r, t) − φ(0, 0))  ∼ r C rz where C(x) → const. for x → 0 and C(x) → x 2ζ /z for x → ∞. The dynamical roughening exponent ζ can be different from its equilibrium value ζeq . Several related exponents can be also introduced by v ∼ ( f − f c )β ,

(1.53)

ξ ∼ ( f − f c )−ν ,

(1.54)

where ξ is a correlation length. The exponents β and ν are connected with ζ and z by scaling relations. There are a large number of renormalization group [39–43] and numerical [44–47] studies devoted to determine these exponents. The transport properties at finite temperature is a more subtle problem. According to the description of a thermally assisted flow, the system overcome barriers via thermal activation and the velocity at small force is expected to behave as v ∼ e−U/T f , where U is typical energy scale of the barriers. For a single particle driven in a random potential, the barrier height U is obviously independent of the driving force. However, a remarkable fact is that in the presence of the elastic interaction, the motion is dominated by barriers which diverges as the driving force goes to zero. Indeed, well below the threshold critical force, the velocity-force characteristics is described by a stretched exponential form [42],

U v ∼ exp − T



f fc

−μ 

.

(1.55)

The exponent μ is universal in the sense that it is independent of microscopic details of the system. Such an extremely slow motion at low temperature is called “creep”. The typical velocity-force characteristics at zero and finite temperature are depicted in Fig. 1.6. There are many systems which exhibit a depinning transition and creep motion. Examples include domain walls in disordered magnets [14], charge density waves in solids [48], vortex lattices in type-II superconductor [17], and the contact line of a liquid on a rough substrate [16]. Especially, the depinning transition of the vortex lattices in dirty superconductors has a significant relevance to many industrial

1.3 Disordered Systems Driven Out of Equilibrium

19

Fig. 1.6 Typical v- f characteristics of an elastic system driven in a random medium. At zero temperature, the manifold is pinned by the disorder below the critical threshold force f c and starts to move above f c . At finite temperature, the manifold exhibits a creep motion well below f c

applications. In type-II superconductors, a vortex lattice is formed if a magnetic field between Hc1 and Hc2 , which are the lower and upper critical fields, respectively, is applied. An electric current exerts the Lorentz force on this vortex lattice. If the vortex lattice remains pinned, the current flows without dissipation. However, if the lattice moves, energy is dissipated and a finite resistance occurs. Therefore, the critical current, which is the maximum current that the superconductor can carry without loss, is closely related to the critical threshold force in the depinning transition. Not only the critical forces, but also the critical exponents are determined in many experiments. These measured critical exponents are not fully incompatible with theoretical predictions, however the origin of the deviation between them is still controversial [43]. Another remarkable property of the driven elastic system is avalanche dynamics [49]. Let us consider an elastic manifold driven by a harmonic potential over a random substrate, ∂t φ(r, t) = K ∇ 2 φ(r, t) + FR (r; φ(r, t)) − k(φ(r, t) − w(t)) + ξ(r, t), (1.56) where the position of the harmonic potential w(t) moves with a constant velocity w(t) ˙ = v. In the limit v → 0, the local velocity of the manifold is not smooth, but discontinuous. Most of time the manifold is pinned in a meta-stable state, but sometimes it performs large jumps (see Fig. 1.7). The distributions of their sizes S and durations T exhibit power-law behaviors, P(S) ∼ S −τ ,

P(T ) ∼ T −α ,

(1.57)

for small S and T . The exponents τ and α are believed to be universal. One of the well-studied examples of avalanche dynamics in disordered systems is the so-called Barkhausen noise [50, 51], which is the crackling signal emitted from a ferromagnet when an external field is slowly varied. The origin of the Barkhausen noise is discontinuous jumps of domain walls pinned by defects. Another example of avalanche phenomena is earthquake. It has been known for long time that the

20

1 Introduction

Fig. 1.7 Schematic picture ˙ of of the local velocity φ(t) the manifold. It exhibits discontinuous jumps (avalanches). S and T denote the size and duration of avalanches

distribution of the size of earthquakes exhibits power-law behavior. This is called the Gutenberg-Richter law [52].

1.3.2 Phase Transitions in Driven Disordered Systems Interacting many-particle systems in three dimensions exhibit a first-order phase transition from crystal to liquid as temperature increases. In the presence of quenched disorder, this transition is smeared and disappears. Crystalline phase is unstable with respect to an infinitesimally small disorder below four dimensions. This fact can be shown from the Imry and Ma’s argument in Sect. 1.2.2. The question is what type of phase transitions emerges when the interacting particle system is driven over a random potential by an external force. The equation of motion is simply given by  d ri =− ∇Vint (r i − r j ) − ∇VR (r i ) + f + ξ i (t), dt j ( =i)

(1.58)

where r i is the position of the i-th particle (i = 1, 2, ..., N ) and Vint is a interaction potential. The random potential VR (r) obeys a mean-zero Gaussian distribution satisfying VR (r)VR (r  ) = CR (|r − r  |), (1.59) where CR (r ) depends on a situation under consideration. The thermal noise ξ i (t) satisfies β (1.60) ξiα (t)ξ j (t  ) = 2T δi j δ αβ δ(t − t  ). Suppose that the particle density is high enough such that a crystalline phase is realized in the absence of disorder. The introduction of the random potential destroys this LRO. We now apply an external force f . Below the critical threshold force f c , the system is pinned and the average velocity is zero. Above f c , the system starts to move with a steady velocity v. This depinning behavior can be described by the driven random manifold model in Sect. 1.3.1. As v increases, the random forces that

1.3 Disordered Systems Driven Out of Equilibrium

21

each particle experiences vary rapidly and the inhomogeneity of the random substrate is effectively reduced. This implies that the system at large velocities become more ordered than that for small velocities. Therefore, one expects that at a critical velocity, a phase transition from a disordered phase to an ordered phase takes place. This nonequilibrium phase transition is called dynamical reordering transition and a large number of numerical studies are devoted to elucidate this phenomenon [53–57]. Since the random potential is expected to behave as an effective thermal noise in the moving frame, one might consider that an ordered moving phase is crystal. However, it is not true because some components of the disorder remains static and prevent the formation of a true LRO. In three dimensions, the ordered moving phase is not crystal, but a quasi-crystal in which the mean square displacement between two points grows logarithmically, (φ(r) − φ(0))2  ∼ C ln r,

(1.61)

where φ(r) denotes the displacement of the particle at r. Such quasi-long-range order phase is called a moving glass phase [58–60]. In the moving glass phase, the flow consists of the particles moving in coupled channels and the system exhibits a transverse pinning. This means that, if an additional force is applied perpendicular to the original driving force, a finite force is required to depin the system to transverse direction. The schematic phase diagram with respect to the disorder strength  and the driving force f is shown in Fig. 1.8. The red line denotes the depinning threshold f c (). In the plastic flow regime, there is no order and the structure factor, S(k) =

N 2 1   exp(−ik · r j ) ,  N j=1

(1.62)

has a ring-like structure. For the strong driving force, the moving glass phase is realized and the structure factor shows Bragg peaks. Since the moving glass is not a perfect crystal, the Bragg peak is not the delta function, but exhibits power-law divergence. There may be an intermediate phase, which is called a moving smectic, between the plastic regime and the moving glass phase. In the moving smectic,

Fig. 1.8 Typical phase diagram with respect to the disorder strength  and the driving force f . The insets show schematic pictures of the smectic and moving glass phases

22

1 Introduction

uncoupled channels of the flow are periodically aligned to the driving direction. Thus, a time-averaged density field is periodic for the transverse direction and uniform for the driving direction. The existence of the moving smectic phase depends on the explicit form of the interaction between the particles Vint and the disorder correlator CR . However, it is unclear whether there is a true transition between the plastic regime and the moving smectic phase. A typical experimental realization of the dynamical reordering transition is the driven vortex lattices in dirty type-II superconductors. Neutron scattering experiments show that the correlation length of the vortex lattice increases substantially as it is depinned and moving, eventually a transition to an ordered phase occurs [61, 62]. Furthermore, a transition from a moving smectic phase to a more ordered moving glass phase can be also observed [63].

1.4 Purpose of This Study The dynamical reordering transitions in interacting many-particle systems driven over random substrates have been extensively investigated numerically and experimentally. However, from theoretical viewpoints, our understanding for these phenomena remains limited in contrast to the situation of the depinning transitions, where RG approaches are successfully applied. There are two reasons that make the dynamical reordering transitions more complicated than the depinning transitions. First, the depinning transitions are quasi-static phenomena where the zero limit of the driving velocity is taken v → 0+ , while the dynamical reordering transitions are truly nonequilibrium phenomena where the system is driven at a finite driving velocity. The second reason why the depinning transitions are analytically tractable is that, if the disorder is not strong and the crystalline lattice keeps its local periodicity, the long-distance physics of the depinning transitions can be described by the elastic theory, Eq. (1.49). However, at the dynamical reordering transitions, the elastic theory does not work because topological defects plays a crucial role. In this study, to improve our theoretical understanding of the dynamical reordering transitions, we attempt to develop a RG theory for disordered systems which is driven at a finite velocity and contain topological defects. This study consists of the following steps: 1. We introduce simple models of driven disordered systems that exhibits a dynamical reordering transition. The interacting many-particle system discussed in Sect. 1.3.2 is too complicated for theoretical analysis. Therefore, we consider toy models which are suitable for a field theoretical formulation. 2. The dimensional reduction property discussed in Sect. 1.2.3 has been a fundamental starting point in the study of disordered systems in equilibrium. Therefore, we attempt to derive a novel type of dimensional reduction for driven disordered systems. It states that the dynamical reordering transitions in driven

1.4 Purpose of This Study

23

disordered systems are identical to equilibrium phase transitions in lower dimensional pure systems. 3. As in equilibrium, such a dimensional reduction is expected to break down in low enough dimensions due to a nonperturbative effect. We develop a RG theory for our models and elucidate the condition that the dimensional reduction fails. In the rest of this section, we define toy models that will be investigated in this thesis. The Hamiltonians of the random field, random anisotropy, and random manifold models are given by Eqs. (1.12), (1.14), and (1.16), respectively. The equilibrium dynamics of these models are described by the following equation of motion: ∂t φ α = −

δH[φ] + ξ α, δφ α

(1.63)

where the thermal noise ξ satisfies ξ α (r, t)ξ β (r  , t  ) = 2T δ αβ δ(r − r  )δ(t − t  ).

(1.64)

The distribution function of the steady state is given by the Boltzmann-Gibbs distribution Eq. (1.7). We now add a driving term [φ] to the equation of motion, ∂t φ α + [φ] = −

δH[φ] + ξ α. δφ α

(1.65)

We assume that the driving term [φ] satisfies the following conditions: • Nonpotentiality: [φ] cannot be expressed as a functional derivative of any potential, i.e., there is no functional V [φ] such that [φ] =

δV [φ] . δφ

This condition means that the driving force is not conservative and the system never reaches equilibrium. • Locality: [φ] is a local function. In other words, [φ](r) is a function of φ(r) and its spatial derivatives ∇ n φ(r) at the same coordinate r. • Uniformity: [φ] is translational invariant, ˜ t), [φ](r + r 0 , t + t0 ) = [φ](r, ˜ where φ(r, t) = φ(r + r 0 , t + t0 ). • Rotational invariance: [φ] does not break the O(N ) symmetry. In other words, [φ] is transformed in the same way as φ by the rotational transformation. • Linearity: For simplicity, [φ] is assumed to be linear with respect to φ. The simplest choice of the driving term [φ] satisfying the above conditions is

24

1 Introduction

[φ] = (v · ∇)φ,

(1.66)

where v is a constant vector. This is nothing but the streaming term, which describes an advection of φ with a uniform and steady velocity v. Equations (1.65) and (1.66) define the models that will be studied in this thesis. It is worthy to note the difference between our models and various driven systems discussed in previous studies. In Sect. 1.3.1, we consider the random manifold model driven by an external force f , ∂t φ = K ∇ 2 φ + FR (r; φ) + f + ξ.

(1.67)

In contrast, from Eqs. (1.65) and (1.66), we can also introduce another driven random manifold model, (1.68) ∂t φ + (v · ∇)φ = K ∇ 2 φ + FR (r; φ) + ξ. The crucial difference between these models is whether or not the driving term breaks the O(N ) symmetry. In Eq. (1.67), the external force f explicitly breaks the O(N ) symmetry, while in Eq. (1.68), the driving term (v · ∇)φ respects the symmetry. We call a type of the driving such as Eq. (1.68) transverse driving, because φ corresponds to the displacement field for the perpendicular direction to the driving. On the other hand, we call a type of the driving such as Eq. (1.67) longitudinal driving. Figure 1.9 shows schematic pictures of the random manifold models with the longitudinal and transverse driving. Note that the transverse driving makes the system anisotropic, but the longitudinal driving does not. In the longitudinal driving case Eq. (1.67), if the driving force f exceeds the depinning threshold, the displacement field φ grows linearly in time. When φ is trapped by a harmonic potential around φ = 0, the longitudinal driving force f just leads to a biased equilibrium state, not a nonequilibrium steady state. In contrast, the transversely driven random manifold Eq. (1.68) never reaches equilibrium state even if there is a trapping potential. In

Fig. 1.9 Schematic pictures of the random manifold models with a the longitudinal and b transverse driving. The arrows represent the driving direction

1.4 Purpose of This Study

25

this thesis, we deal with only the models with the transverse driving. We call the models defined by Eqs. (1.65) and (1.66) supplemented by the Hamiltonian HRF (random field), HRA (random anisotropy), and HRM (random manifold) as driven random field O(N ) model (DRFO(N )M), driven random anisotropy O(N ) model (DRAO(N )M), and driven random manifold model (DRM), respectively. The DRAO(N )M can be considered to describe the dynamics of liquid crystals flowing in a porous medium. Recently, the dynamics of liquid crystals confined in a complex geometry has attracted considerable attention, due to not only the fundamental scientific interest, but also the industrial applications [64, 65]. For liquid crystals in a porous medium, the irregular surface structure of the solid substrate results in a symmetry breaking random anchoring, which is similar to the random anisotropy in the O(N ) model.

1.5 Outline of This Thesis In Chap. 2, we review the so-called functional renormalization group (FRG) treatment, in which the evolution of the whole functional form of the disorder correlator is considered. We explain that the renormalized disorder correlator can develop a cusp at a finite renormalization scale and such a nonanalytic behavior leads to the breakdown of the dimensional reduction. As examples, we consider the random manifold and random field O(N ) models. The flow equations of the renormalized disorder correlator are derived by a perturbative approach and the critical exponents are calculated near the critical dimension. In Chap. 3, we introduce the nonperturbative renormalization group (NPRG) formalism. This approach is based on an exact evolution equation of an effective action and it enables us to access fixed points which are inaccessible by perturbative treatments. By combining the FRG and NPRG, the so-called nonperturbative functional renormalization group (NP-FRG) formalism is established. With the aid of this approach, we can go beyond the perturbative FRG in Chap. 2. As simple applications, we review the NPRG analysis of the pure O(N ) and random field O(N ) models. In Chap. 4, we investigate the dynamical reordering transition in the DRFO(N )M. First, from intuitive arguments, we introduce a new dimensional reduction property which predicts that the critical exponents of the D-dimensional DRFO(N )M at zero temperature are equal to those of the (D − 1)-dimensional pure O(N ) model. This dimensional reduction is expected to break down in low enough dimensions due to a nonperturbative effect resulting from the presence of multiple “meta-stable states”. Next, by extending the NP-FRG formalism to nonequilibrium cases, we derive the RG equations for the disorder correlators. The critical exponents of the DRFO(N )M are calculated in the first order of  = D − 3. We determine the range of N in which the dimensional reduction breaks down. Furthermore, by using the RG equations, we calculate the correlation length of the DRFO(N )M in three-dimensions.

26

1 Introduction

In Chap. 5, we especially consider the case that N = 2 and D = 3 in the DRFO(N )M, the driven random field XY model (DRFXYM). The dimensional reduction which will be discussed in Chap. 4 predicts that the 3D-DRFXYM exhibits a quasi-long-range order and the Kosterlitz-Thouless (KT) transition because it is mapped into the 2D pure XY model. We show that this expectation is certainly correct, in other words, the dimensional reduction holds for N = 2. In the first part of this chapter, we show several results of the numerical simulations of the DRFXYM. These results suggest that there is a transition between a quasi-long-range order phase at weak disorder and a disordered phase at strong disorder. The rest of this chapter is devoted to the RG analysis of this model. We introduce a spin-wave model of the DRFXYM, which is valid only when the field varies slowly in space. By applying the NP-FRG treatment, the spin-wave model is shown to exhibit a quasi-long-range order at weak disorder. However, the spin-wave model is invalid near a transition point because it is incapable of describing vortices. Next, we construct a phenomenological theory of the KT transition by taking into account the effect of the vortices. The main idea of this theory is to introduce an effective elastic constant, which is assumed to flow according to the RG equation similar to that of the 2D XY model. The structural change of vortices at the KT transition is also discussed. In Chap. 6, we give the summary of this thesis and an outlook to some open problems.

References 1. Ma S-K (1976) Modern theory of critical phenomena. W. A. Benjamin, Inc 2. Goldenfeld N (1992) Lectures on phase transitions and the renormalization group. Westview Press 3. Wilson KG, Kogut J (1974) The renormalization group and the  expansion. Phys Rep 12:75 4. Zinn-Justin J (1989) Quantum field theory and critical phenomena. Clarendon Press, Oxford 5. Fishman S, Aharony A (1979) Random field effects in disordered anisotropic antiferromagnets. J Phys C 12:L729 6. Gennes PG (1984) Liquid-liquid demixing inside a rigid network. Qualitative features. J Phys Chem 88:6469 7. Pitard E, Rosinberg ML, Stell G, Tarjus G (1995) Critical behavior of a fluid in a disordered porous matrix: an ornstein-zernike approach. Phys Rev Lett 74:4361 8. Villain J (1982) Commensurate-incommensurate transition with frozen impurities. J Physique Lett (France) 43:551 9. Harris R, Plischke M, Zuckermann MJ (1973) New model for amorphous magnetism. Phys Rev Lett 31:160 10. Bellini T, Buscaglia M, Chiccoli C, Mantegazza F, Pasini P, Zannoni C (2000) Nematics with quenched disorder: what is left when long range order is disrupted? Phys Rev Lett 85:1008 11. Bellini T, Buscaglia M, Chiccoli C, Mantegazza F, Pasini P, Zannoni C (2002) Nematics with quenched disorder: how long will it take to heal? Phys Rev Lett 88:245506 12. Rotunno M, Buscaglia M, Chiccoli C, Mantegazza F, Pasini P, Bellini T, Zannoni C (2005) Nematics with quenched disorder: pinning out the origin of memory. Phys Rev Lett 94:097802 13. Petridis L, Terentjev EM (2006) Nematic-isotropic transition with quenched disorder. Phys Rev E 74:051707

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14. Nattermann T (1997) Spin glasses and random fields. In: Young AP (ed). World Scientific, Singapore 15. Ertas D, Kardar M (1994) Critical dynamics of contact line depinning. Phys Rev E 49:R2532 16. Prevost A, Rolley E, Guthmann C (2002) Dynamics of a helium-4 meniscus on a strongly disordered cesium substrate. Phys Rev B 65:064517 17. Blatter G, Feigel’man MV, Geshkenbein VB, Larkin AI, Vinokur VM (1994) Vortices in hightemperature superconductors. Rev Mod Phys 66:1125 18. Nattermann T, Scheidl S (2000) Vortex-glass phases in type-II superconductors. Adv Phys 49:607 19. Imry Y, Ma S-K (1975) Random-field instability of the ordered state of continuous symmetry. Phys Rev Lett 35:1399 20. Aizenman M, Wehr J (1989) Rounding of first-order phase transitions in systems with quenched disorder. Phys Rev Lett 62:2503 21. Bricmont J, Kupiainen A (1987) Lower critical dimension for the random-field using model. Phys Rev Lett 59:1829 22. Aharony A, Imry Y, Ma S-K (1976) Lowering of dimensionality in phase transitions with random fields. Phys Rev Lett 37:1364 23. Young AP (1977) On the lowering of dimensionality in phase transitions with random fields. J Phys C 10:L257 24. Parisi G, Sourlas N (1979) Random magnetic fields, supersymmetry, and negative dimensions. Phys Rev Lett 43:744 25. Binder K, Young AP (1986) Spin glasses: experimental facts, theoretical concepts, and open questions. Rev Mod Phys 58:801 26. Mézard M, Parisi G, Virasoro MA (1987) Spin-glasses and beyond. World Scientific, Singapore 27. Sherrington D, Kirkpatrick S (1975) Solvable model of a spin-glass. Phys Rev Lett 35:1792 28. De Almeida JRL, Thouless DJ (1978) Stability of the sherrington-kirkpatrick solution of a spin glass model. J Phys A 11:983 29. Parisi G (1980) The order parameter for spin glasses: a function on the interval 0–1. J Phys A 13:1101 30. Parisi G (1983) Order parameter for spin-glasses. Phys Rev Lett 50:1946 31. De Dominicis C, Young P (1983) Weighted averages and order parameters for the infinite range Ising spin glass. J Phys A 16:2063 32. Mézard M, Parisi G (1991) Replica field theory for random manifolds. J Phys I 1:809 33. Le Doussal P, Wiese KJ (2003) Functional renormalization group at large N for disordered elastic systems, and relation to replica symmetry breaking. Phys Rev B 68:174202 34. Grinstein G, Luther A (1976) Application of the renormalization group to phase transitions in disordered systems. Phys Rev B 13:1329 35. Harris AB (1974) Effect of random defects on the critical behaviour of using models. J Phys C Solid State Phys 7:1671 36. Ma S-k, Rudnick J (1978) Time-dependent Ginzburg-landau model of the spin-glass phase. Phys Rev Lett 40:589 37. Dotsenko V, Harris AB, Sherrington D, Stinchcombe RB (1995) Replica symmetry breaking in the critical behavior of the random ferromagnet. J Phys Math Gen 28:3093 38. Tarjus G, Dotsenko V (2002) Is there a spin-glass phase in the random temperature Ising ferromagnet? J Phys Math Gen 35:1627 39. Fisher DS (1985) Sliding charge-density waves as a dynamical critical phenomenon. Phys Rev B 31:1396 40. Narayan O, Fisher DS (1992) Critical behavior of sliding charge-density waves in 4- dimensions. Phys Rev B 46:11520 41. Narayan O, Fisher DS (1993) Threshold critical dynamics of driven interfaces in random media. Phys Rev B 48:7030 42. Chauve P, Giamarchi T, Le Doussal P (2000) Creep and depinning in disordered media. Phys Rev B 62:6241

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43. Le Doussal P, Wiese KJ, Chauve P (2002) Two-loop functional renormalization group theory of the depinning transition. Phys Rev B 66:174201 44. Tanguy A, Gounelle M, Roux S (1998) From individual to collective pinning: effect of longrange elastic interactions. Phys Rev E 58:1577 45. Roters L, Hucht A, Lubeck S, Nowak U, Usadel KD (1999) Depinning transition and thermal fluctuations in the random-field Ising model. Phys Rev E 60:5202 46. Rosso A, Krauth W (2001) Origin of the roughness exponent in elastic strings at the depinning threshold. Phys Rev Lett 87:187002 47. Rosso A, Krauth W (2002) Roughness at the depinning threshold for a long-range elastic string. Phys Rev E 65:025101(R) 48. Grüner G (1988) The dynamics of charge-density waves. Rev Mod Phys 60:1129 49. Fisher DS (1998) Collective transport in random media: from, superconductor to earthquakes. Phys Rep 301:113 50. Urbach JS, Madison RC, Markert JT (1995) Interface depinning, self-organized criticality, and the barkhausen effect. Phys Rev Lett 75:276 51. Zapperi S, Cizeau P, Durin G, Stanley HE (1998) Dynamics of a ferromagnetic domain wall: avalanches, depinning transition, and the barkhausen effect. Phys Rev B 58:6353 52. Kagan YY (2002) Seismic moment distribution revisited: I. Statistical results. Geophys J Int 148:520 53. Reichhardt C, Olson Reichhardt CJ (2017) Depinning and nonequilibrium dynamic phases of particle assemblies driven over random and ordered substrates: a review. Rep Prog Phys 80:026501 54. Koshelev AE, Vinokur VM (1994) Dynamic melting of the vortex lattice. Phys Rev Lett 73:3580 55. Moon K, Scalettar RT, Zimanyi GT (1996) Dynamical phases of driven vortex systems. Phys Rev Lett 77:2778 56. Ryu S, Hellerqvist M, Doniach S, Kapitulnik A, Stroud D (1996) Dynamical phase transition in a driven disordered vortex lattice. Phys Rev Lett 77:5114 57. Dominguez D, Gronbech-Jensen N, Bishop AR (1997) First-order melting of a moving vortex lattice: effects of disorder. Phys Rev Lett 78:2644 58. Giamarchi T, Le Doussal P (1996) Moving glass phase of driven lattices. Phys Rev Lett 76:3408 59. Le Doussal P, Giamarchi T (1998) Moving glass theory of driven lattices with disorder. Phys Rev B 57:11356 60. Balents L, Marchetti MC, Radzihovsky L (1998) Nonequilibrium steady states of driven periodic media. Phys Rev B 57:7705 61. Yaron U, Gammel PL, Huse DA, Kleiman RN, Oglesby CS, Bucher E, Batlogg B, Bishop DJ, Mortensen K, Clausen K, Bolle CA, De La Cruz F (1994) Neutron diffraction studies of flowing and pinned magnetic flux lattices in 2H −NbSe2 . Phys Rev Lett 73:2748 62. Yaron U, Gammel PL, Huse DA, Kleiman RN, Oglesby CS, Bucher E, Batlogg B, Bishop DJ, Mortensen K, Clausen KN (1995) Structural evidence for a two-step process in the depinning of the superconducting flux-line lattice. Nature 376:753 63. Pardo F, De La Cruz F, Gammel PL, Bucher E, Bishop DJ (1998) Observation of smetic and moving-bragg-glass phases in flowing vortex lattices. Nature 396:348 64. Araki T (2012) Dynamic coupling between a multistable defect pattern and flow in nematic liquid crystals confined in a porous medium. Phys Rev Lett 109:257801 65. Sengupta A, Tkalec U, Ravnik M, Yeomans JM, Bahr C, Herminghaus S (2013) Liquid crystal microfluidics for tunable flow shaping. Phys Rev Lett 110:048303

Chapter 2

Functional Renormalization Group of Disordered Systems

2.1 Why is the Functional Renormalization Group Treatment Necessary? Renormalization group (RG) approaches have been successfully applied to the critical phenomena in pure systems. The Hamiltonian is parameterized by a few parameters and one follows the evolution of these when the high energy fluctuations are successively integrated out. The critical exponents can be obtained by linearizing the flow near a fixed point. In a standard perturbavie RG, the flow equation is expanded below the upper critical dimension above which all nonlinear terms become irrelevant. For example, in the case of the pure Ising model, the Hamiltonian reads 

 HIsing =

dDr

 1 |∇φ|2 + U (φ) , 2

(2.1)

where the interaction potential U is expanded as U (φ) =

1 1 1 g2 φ 2 + g4 φ 4 + g6 φ 6 + · · · 2 4! 6!

(2.2)

Power counting shows that at D = 4, the fourth order term g4 is marginal and the higher order terms gn (n ≥ 6) are irrelevant. Therefore, it is suffice to retain only the fourth order term and all higher-order terms can be omitted near four dimensions. At D = 4 − , the value of the coefficient g4 near the fixed point is expected to be O(). This implies that the fourth order term can be treated as a perturbation in the derivation of the RG flow equation. The critical exponent is then calculated in the form of a polynomial of  [1]. For disordered systems, the situation is more complicated. In the case of the random manifold model defined by Eqs. (1.16) and (1.17), the correlation function of the random potential R(φ) is expanded as © Springer Nature Singapore Pte Ltd. 2019 T. Haga, Renormalization Group Analysis of Nonequilibrium Phase Transitions in Driven Disordered Systems, Springer Theses, https://doi.org/10.1007/978-981-13-6171-5_2

29

30

2 Functional Renormalization Group of Disordered Systems

Fig. 2.1 Schematic RG evolution of −R  (φ). It develops a linear cusp at the origin after a finite renormalization

R(φ) = R (0) +

1 (2) 2 1 R φ + R (4) φ 4 + · · · 2 4!

(2.3)

Then, the remarkable fact is that, as shown in the next section, all order terms R (n) are marginal at D = 4 in contrast to U (φ) of the pure Ising model. This means that we cannot truncate the above expansion at a finite order, but we have to consider the RG evolution of the whole function R(φ). The RG equation of R(φ) is given by a nonlinear partial differential equation. This approach is called functional renormalization group (FRG). The advantage of the FRG treatment is that one can deal with a nonanalytic behavior of R(φ). If we assume that R(φ) is analytic, in other words, the expansion Eq. (2.3) is possible, we are led to the dimensional reduction. Therefore, to overcome the dimensional reduction, we have to take into account a nonanalytic nature of R(φ). In fact, as we will explain in the next section, the second derivative R  (φ) develops a linear cusp at φ = 0, R  (φ)  R  (0) + R (3) (0)|φ|, after a finite renormalization. This implies that the fourth derivative at φ = 0, R (4) (0), diverges. Figure 2.1 shows the schematic evolution of −R  (φ). The generation of the cusp is a consequence of the presence of multiple local minima (meta-stable states) in the energy landscape and it leads to the breakdown of the dimensional reduction. In the following sections, we review the FRG treatment of the random manifold and random field O(N ) models.

2.2 FRG of the Random Manifold Model 2.2.1 RG Equation of the Disorder Correlator Let us consider the random manifold model defined by Eqs. (1.16) and (1.17) with N = 1. To take the average over the disorder, we introduce an n times replicated system with the same disorder [2], H[φa ] =

1 T



 dDr

 1 |∇φa (r)|2 + V (r; φa (r)) , 2

(2.4)

2.2 FRG of the Random Manifold Model

31

where the subscript a = 1, . . . , n is the replica index and temperature T is included in the Hamiltonian. The disorder average is given by    n  exp − H[φa ] = exp −H[{φa }] ,

(2.5)

a=1

where a replicated Hamiltonian H[{φa }] is written as H[{φa }] = H0 [{φa }] + H1 [{φa }]   1  1  d D r|∇φa |2 , H1 [{φa }] = − 2 d D r R(φa − φb ). 2T a 2T a,b (2.6) By noting that the field scales as φ ∼ r ζ , where ζ is the roughness exponent, one finds that the scaling dimensions of T and R(φ) are 2 − D − 2ζ and 4 − D − 4ζ , respectively. Since ζ ∝ 4 − D, all order terms in the expansion Eq. (2.3) are shown to be marginal at D = 4. We construct a renormalized Hamiltonian at D = 4 −  by eliminating high energy fluctuations. We split φ into slowly and rapidly varying contributions, φa = φaL + φaS , where φaL contains long wavelength modes with |q| < e−l  and φaS contains short wavelength modes with e−l  < |q| < . Substitution of φa = φaL + φaS into Eq. (2.6) yields H0 [{φa }] =

H[{φaL + φaS }] = H0 [{φaL }] + H1 [{φaL }] + H0 [{φaS }] + H[{φaL , φaS }], 

1  D r R  (φaL − φbL )(φaS − φbS ) d 2T 2 a,b  1 + R  (φaL − φbL )(φaS − φbS )2 + · · · . 2

(2.7)

H[{φaL , φaS }] = −

(2.8)

˜ aL }] is given by The renormalized Hamiltonian H[{φ ˜ aL }]) = exp(−H[{φ



DφaS exp(−H[{φaL + φaS }]),

(2.9)

a

thus we have a cumulant expansion, ˜ aL }] = H0 [{φaL }] + H1 [{φaL }] + H[{φaL , φaS }] 0 H[{φ 1 − H[{φaL , φaS }]2 0,c + · · · , 2

(2.10)

32

2 Functional Renormalization Group of Disordered Systems

where ... 0 denotes the average with respect to the distribution exp(−H0 [{φaS }]). We consider the correction to R(φ) in the zero temperature limit T → 0. In the following calculation, φaL are assumed to be uniform. By noting φaS (q)φbS (−q) 0 =

T δab , |q|2

(2.11)

one finds that the first order term H 0 does not contribute to R(φ) in the limit T → 0. Thus, let us consider the second order term, H2 0 =

1  1  L R (φa − φbL )R  (φcL − φdL ) 4T 4 a,b,c,d 4  × d D rd D r  (φaS (r) − φbS (r))2 (φcS (r  ) − φdS (r  ))2 0,c , (2.12)

where higher order derivatives in Eq. (2.8) do not contribute because they contain higher powers of T . The integral is calculated as 

d D rd D r  (φaS (r) − φbS (r))2 (φcS (r  ) − φdS (r  ))2 0,c  = d D rd D r  2(φaS (r) − φbS (r))(φcS (r  ) − φdS (r  )) 20 > = 4(δac δbd + δbc δad − δac δbc − δac δad − δbc δbd − δad δbd ) q

where

> q

=

e−l  T 2 2Cab,i j Cab,i j , |q|4

(2.30)

q

 where we have ignored terms which contain a sum of three replica indices a,b,c because they do not contribute to the second cumulant R(z). By using a relation  (a · e )(b · ei ) = a · b − (a · n L )(b · n L ), we have i i 

Aab Caa,ii = −(N − 1)z R  (1)R  (z),

i





Bab,i j Caa,i j = (1 − z 2 )R  (1)R  (z),

ij

Cab,i j Cab,i j = (N − 2 + z )R (z) − 2z(1 − z 2 )R  (z)R  (z) + (1 − z 2 )2 R  (z)2 , 2



2

ij

where z = naL · nbL . Therefore, at D = 4 we have 1 1 

H12 0,c = −(N − 1)z R  (1)R  (z) + (1 − z 2 )R  (1)R  (z) 2 2T 2 a,b 1 + (N − 2 + z 2 )R  (z)2 − z(1 − z 2 )R  (z)R  (z) 2  l 1 . + (1 − z 2 )2 R  (z)2 2 8π 2

(2.31)

Next, we calculate H0 H1 0,c , which leads to the renormalization of T . To determine the correction to the gradient term, we assume that naL depend on one spatial coordinate x only and lie in the same plane, ∂x ea,1 = −(∂x naL · ea,1 )naL , ∂x ea,i = 0, (i = 2, ..., N − 1).

(2.32)

Then, H0 can be rewritten as H0 [{naL , ψa,i }]

 N −1   1  =− |∇n aL ,α |2 (ψa,i )2 . dDr 2T a α i=2

By retaining terms which contain a sum of one replica index, we have

(2.33)

2.3 FRG of the Random Field and Random Anisotropy O(N ) Models

37

 N −1  1 1  L ,α 2 = |∇n a | Caa,ii 4 2T a α q i=2 >

H0 H1 0,c

q

1  l = |∇n aL ,α |2 (N − 2)R  (1) 2 . 2T a α 8π

(2.34)

Therefore, temperature renormalization T → T˜ is given by 1 1 l  1 − (N − 2)R  (1) 2 . = T 8π T˜

(2.35)

By combining Eqs. (2.31) and (2.35), we finally obtain the RG equation for R(z), ∂l R(z) = (4 − D)R(z) + 2(N − 2)R  (1)R(z) − (N − 1)z R  (1)R  (z) 1 + (1 − z 2 )R  (1)R  (z) + (N − 2 + z 2 )R  (z)2 2 1 2   − z(1 − z )R (z)R (z) + (1 − z 2 )2 R  (z)2 , (2.36) 2 where the factor 1/8π 2 is absorbed into R(z), and the first term of the right-hand side comes from the rescaling of R(z). The critical behavior can be obtained from a fixed point of Eq. (2.36). For N = 2 (random field XY model), if we introduce a parameter u by z = cos u, Eq. (2.36) can be reduced to ∂l R(u) = (4 − D)R(u) +

1  2 R (u) − R  (u)R  (0), 2

(2.37)

which is nothing but Eq. (2.16) with ζ = 0. This is obvious from the fact that, when the disorder is weak, the random field XY model can be mapped into the random manifold model with a periodic potential through a change of variable n = (cos u, sin u), where u is a phase variable.

2.3.2 Critical Exponents In disordered systems, the connected and disconnected Green’s functions are defined by, G c (r) = n(r) · n(0) − n(r) · n(0) , G d (r) = n(r) · n(0) − n(r) · n(0) ,

(2.38)

where the bracket and over-bar denote the thermal and disorder averages, respectively. From the fluctuation-dissipation theorem, the connected Green’s function is

38

2 Functional Renormalization Group of Disordered Systems

proportional to the response function. At the critical point, they behave as G c (r) ∼ |r|−(D−2+η) , ¯ G d (r) ∼ |r|−(D−4+η) ,

G c (q) ∼ |q|−2+η , G d (q) ∼ |q|−4+η¯ ,

(2.39)

where η and η¯ are anomalous dimensions corresponding to the connected and disconnected Green’s functions, respectively. We represent η and η¯ in terms of the fixed point function R∗ (z). First, note that the elimination of high-energy modes with momenta b < |q| <  and the rescaling q → b−1 q leads to    n(r) · n(0) = n(br) · n(0) 1 − (ψi )2 ,

(2.40)

i

where we have used Eq. (2.24) and n L (r) ∼ n(br). Note that in the above equation, we have assumed that the correlation between ψ at different points can be negligible because it varies rapidly in space. To calculate the fluctuation (ψi )2 due to the random field, we assume that n L is completely ordered, n L = (1, 0, ..., 0). Then, within the second order of ψ, the original Hamiltonian Eq. (2.22) can be rewritten as Hψ =

1 T

 dDr

 1 2

i

 |∇ψi |2 − h i ψi ,

(2.41)

where h i is a ei -direction component of the random field. At zero temperature, ψ is given by the stationary condition δHψ = −∇ 2 ψi − h i = 0, δψi

(2.42)

thus we have > (ψi

)2

= q

h 20 . |q|4

(2.43)

Since at the fixed point the disorder strength h 20 scales as |q|4−D R∗ (1), we have (ψi )2 = R∗ (1) ln b−1 , where the factor 1/8π 2 is absorbed into R. Equation (2.40) is rewritten as  n(r) · n(0) = 1 − (N − 1)R∗ (1) ln b−1 n(br) · n(0) 

−1

 e−(N −1)R∗ (1) ln b n(br) · n(0) 

= b(N −1)R∗ (1) n(br) · n(0), 

which leads to G d (r) = r −(N −1)R∗ (1) . Therefore, we have

(2.44)

2.3 FRG of the Random Field and Random Anisotropy O(N ) Models

η¯ = 4 − D + (N − 1)R∗ (1).

39

(2.45)

From Eqs. (2.35) and (2.44), the scaling dimensions of T and (ψ)2 are (N − 2)R∗ (1) and (N − 1)R∗ (1), respectively. Therefore, the gradient term of  Eq. (2.41) scales as |q|2−R∗ (1) , and we have η = R∗ (1).

(2.46)

2.3.3 Fixed Point of the Random Field O(N) Model We calculate the fixed point of the RG equation (2.36) at D = 4 + . First, we assume that the second derivative R  (z) is continuous around z = 1. The RG equations of R  (1) and R  (1) read ∂l R  (1) = − R  (1) + (N − 2)R  (1)2 , ∂l R  (1) = − R  (1) + R  (1)2 + 6R  (1)R  (1) + (N + 7)R  (1)2 .

(2.47) (2.48)

From Eq. (2.47), we have a nontrivial fixed point R∗ (1) =

 , N −2

(2.49)

which leads to the dimensional reduction value of the critical exponents, ηDR = η¯ DR =

 . N −2

(2.50)

Recall that the critical exponent of the pure O(N ) model is given by η = (D − 2)/(N − 2) near two dimensions [6]. Equation (2.48) combined with (2.49) has a stable fixed point, R∗ (1)

√ (N − 8) − (N − 2)(N − 18) , = 2(N − 2)(N + 7)

(2.51)

which exists only when N ≥ 18. For N < 18, there is no fixed point for Eq. (2.48) and instead R  (1) diverges at a finite renormalization scale, which corresponds to the Larkin length. Therefore, the fixed point controlling the critical behavior is nonanalytic around z = 1 when N < 18. More precisely, R  (z) has a cusp of the form √ 1 − z. In such case, Eq. (2.47) is modified as

40

2 Functional Renormalization Group of Disordered Systems

Fig. 2.3 Schematic pictures of the RG flow of Rk (1) = ηk for the RFO(N )M. The horizontal axis represents the spatial dimension D. The blue (red) dashed (dotted) line represents stable (unstable) fixed points. “D”, “LRO”, and “QLRO” denote the disordered, long-range ordered, and quasi-longrange ordered phases, respectively. Reprinted figure with permission from Ref. [7]. Copyright (2017) by the American Physical Society. https://doi.org/10.1103/PhysRevB.96.184202

∂l R  (1) = − R  (1) + (N − 2)R  (1)2 + lim {[(N + 1)R  (z) − 2(1 − z)R (3) (z)][R  (z) − R  (1)] z→1

+ 2(1 − z)R  (z)[2(1 − z)R (3) (z) − 3R  (z)]},

(2.52)

√ where the term lim z→1 {...} is nonzero when R  (z) ∼ 1 − z. Therefore, the fixed point R∗ (1) differs from the dimensional reduction value /(N − 2). We denote the number of the field component N below which the dimensional reduction breaks down as NDR . From the above argument, NDR = 18 for the RFO(N )M. Let us consider the detailed property of the nonanalytic fixed point. The schematic picture of the RG flow of Rk (1) = ηk is shown in Fig. 2.3. For N > Nc  2.83, a nonzero fixed point does not exist below D = 4 ( < 0), while above D = 4 ( > 0), an unstable fixed point exists, which corresponds to the transition between a long-range order (LRO) phase and a disordered phase. The critical exponents for several values of N are calculated as [8] η(N = 3) = 5.5||, η(N = 4) = 0.78||, η(N = 5) = 0.42||.

(2.53)

Figure 2.4 shows η (solid red line) and η¯ (dashed green line) for N > Nc . They diverge ¯ DR , where ηDR is the as N approaches to Nc  2.83. The inset shows η/ηDR and η/η dimensional reduction value (2.50). Note that for N > NDR = 18, η = η¯ = ηDR . For N < Nc , a stable fixed point exists below D = 4 ( < 0). This fixed point corresponds to a quasi-long-range order (QLRO), in which the disconnected Green’s function exhibits power-law decay with a universal exponent  + η. ¯ For example, in the case of the random field XY model (N = 2), we can easily obtain the fixed point from Eq. (2.37),

2.3 FRG of the Random Field and Random Anisotropy O(N ) Models

41

Fig. 2.4 Critical exponents η and η¯ of the RFO(N )M for N > Nc . The red solid and green dashed lines represent η and η, ¯ respectively. The blue dotted line represents the dimensional reduction value ηDR = 1/(N − 2).  is set to unity. The inset shows η/ηDR and η/η ¯ DR

∗ (u) = −R∗ (u) =

|| π2 (u − π )2 − ||, (0 ≤ u < 2π ), 6 18

(2.54)

and the exponent is given by  + η¯ = ∗ (0) =

π2 ||  1.10||. 9

(2.55)

Note that ∗ (u) obviously satisfies the potentiality condition, 2π ∗ (u)du = 0.

(2.56)

0

This QLRO is known as the Bragg glass in the context of the vortex lattices in superconductors [9–12]. Above D = 4 ( > 0), any nonzero fixed point does not exist. It is difficult question whether the Bragg glass phase exists in three dimensions ( = −1). We cannot answer this question within the one-loop FRG equation (2.37). There are two questions concerning the existence of the Bragg glass. The first question is whether the random manifold model with a periodic correlator exhibits QLRO in three dimensions. Note that the mapping between the random field XY model and the random manifold model is valid only for weak disorder because the random manifold model cannot describe vortices, at which the phase variable u becomes singular. If the random manifold model is shown to exhibit the Bragg glass in three-dimensions, the second question is whether such a QLRO is stable with respect to the nucleation of vortices. Unfortunately, there is no clear answer even for the first question at this time. For example, if we literally believe a two-loop perturbative FRG equation of the random manifold model, the stable fixed point corresponding to the Bragg glass disappears below D  3.9 [13, 14]. The corresponding RG flow in three dimensions

42

2 Functional Renormalization Group of Disordered Systems

shows a signature of a “pseudo” fixed point, at which the beta function has a small but nonzero value. We cannot exclude a possibility that the Bragg-glass like behavior found in experiments and simulations [15] is a fake due to the presence of the pseudo fixed point. There are some phenomenological arguments which states that the formation of topological defects is energetically unfavorable [16], related to the second question mentioned above. However, the absence of defects does not necessarily imply the existence of QLRO, unless the answer of the first question is positive. Furthermore, it is worth noting that the absence of QLRO in the three-dimensional random field XY model is suggested from a nonperturbative functional RG analysis, which can take account into the effect of vortices [17] (The formalism of the nonperturbative functional RG will be reviewed in Chap. 3). Therefore, at this time, it is unclear whether the Bragg glass phase exists in three dimensions. It is interesting to consider the case of the four-dimensional (4D) random field XY model (N = 2 and D = 4). Recall that the lower critical dimension of the random field XY model is four. Since the pure XY model exhibits a QLRO and the KosterlitzThouless (KT) transition in two dimensions [18, 19], one might speculate that the 4D random field XY model also exhibits a QLRO and the KT transition. However, this is not true. Let us consider the FRG equation (2.37) with D = 4, ∂l (u) =  (u)[(0) − (u)] −  (u)2 ,

(2.57)

where (u) = −R  (u). This equation does not have any nonzero fixed point satisfying the potentiality condition (2.56). This can be understood as follows. By taking the limit u → 0+ , we have ∂l (0) = − (0+ )2 . At the fixed point, ∗ (0+ ) = 0, thus ∗ (u) does not have a cusp at the origin. Next, note that ∗ (u) should be nonnegative for all u because ∗ (0) − ∗ (u) and ∗ (u)2 are nonnegative. However, since ∗ (u) does not have a cusp, ∗ (u) is not positive near the origin. Therefore, the fixed point satisfying Eq. (2.56) is only ∗ (u) = 0. This implies that the 4D random field XY model does not exhibit a QLRO with continuously varying exponent, which results from a line of fixed points. Instead, we consider a solution of the form (u) =

˜ (u) . l + l0

(2.58)

Substituting this expression into Eq. (2.57) yields ˜ ˜  (u)[(0) ˜ ˜ ˜  (u)2 = 0, (u) + − (u)] −

(2.59)

˜ which is the original fixed point equation with  = −1. Thus, (u) is given by 2 ˜ Eq. (2.54), especially, (0) = π /9. From Eq. (2.44), the disconnected Green’s function satisfies r ∂r G d (r ) = −(0)G d (r ) = −

π2 (ln r )−1 G d (r ), 9

(2.60)

2.3 FRG of the Random Field and Random Anisotropy O(N ) Models

43

where we have used l ∼ ln r . This leads to G d (r ) ∼ (ln r )−π /9 . The above discussion is correct in the larger scale than the Larkin length ξL . Below the Larkin length, since (u) is analytic, η = (0) does not evolve along the RG flow. Thus, G d (r) exhibits power-law decay with η = h 20 /(8π 2 ), where h 0 is the bare disorder and the implicit factor 1/8π 2 is reintroduced. Finally, we have 2

 r −η , (r < ξL ), G d (r ) ∝ (ln r )−α , (r  ξL ),

(2.61)

where η = h 20 /(8π 2 ) and α = π 2 /9 [5, 20]. The Larkin length is given by ξL = exp(cη−1 ), where c is a universal constant. We call this phase, in which the correlation function behaves as ∼ (ln r )−α , a logarithmic-QLRO phase. Note that for N < Nc , we cannot obtain an unstable fixed point corresponding to the second-order transition between the (logarithmic-) QLRO and a disordered phase within the first order of . To obtain this fixed point, the beta function of R(z) to the higher order of  must be calculated [13].

2.3.4 Fixed Point of the Random Anisotropy O(N) Model We calculate the fixed point of the random anisotropy case. The bare disorder correlator is given by R(z) = τ02 z 2 , and the RG equation (2.36) preserves the symmetry R(z) = R(−z). Thus, we seek a fixed point satisfying this inversion symmetry. It is convenient to rewrite R(z) = 21 S(z 2 ). If we assume that S(z 2 ) is analytic at z = 1, we have the RG equations for S  (1) and S  (1), ∂l S  (1) = − S  (1) + (N − 2)S  (1)2 ,

(2.62)

∂l S  (1) = − S  (1) + 8S  (1)2 + 2(N + 10)S  (1)S  (1) + 2(N + 7)S  (1)2 . (2.63) Equation (2.62) leads to the dimensional reduction fixed point R∗ (1) = S∗ (1) = /(N − 2) and then Eq. (2.63) has solutions S∗ (1) =

√ −(N + 22) ± (N − 2)(N − 18) . 4(N − 2)(N + 7)

(2.64)

Note that, since these both solutions are negative, the flow of S  (1) started from a positive initial value always diverges for any N . This means that an analytic fixed point does not exist for all N , in contrast to the case of the random field in which it exists for N ≥ NDR = 18. In other words, for the random anisotropy case, the dimensional reduction always breaks down near four dimensions, NDR = ∞. The RG flow of R  (1) is qualitatively similar to that depicted in Fig. 2.3. For the random anisotropy, Nc  9.4. Figure 2.5 shows the critical exponent η corresponding

44

2 Functional Renormalization Group of Disordered Systems

Fig. 2.5 Critical exponent η of the random anisotropy O(N ) model for N > Nc above four dimensions.  = D − 4 is set to unity. The inset shows η/ηDR . The green dashed line represents the asymptotic value lim N →∞ η/ηDR = 3/2. The blue dotted line represents the dimensional reduction value η/ηDR = 1

Fig. 2.6 Critical exponent η characterizing the QLRO for N < Nc below four dimensions. η diverges as N approaches to Nc  9.4. || is set to unity

to the transition between the LRO and disordered phase above four dimensions. The inset shows η/ηDR . Note that η does not converge to the dimensional reduction value ηDR in the limit N → ∞, but we have lim

N →∞

η 3 = . ηDR 2

(2.65)

The 1/N correction of η is also analytically obtained in Ref. [14]. Figure 2.6 shows η corresponding to the QLRO phase for N < Nc below four dimensions. Especially, for the XY case (N = 2), the fixed function can be obtained by simply rescaling the solution Eq. (2.54) such that its periodicity becomes π , and we have η = ||π 2 /36  0.27||.

2.3 FRG of the Random Field and Random Anisotropy O(N ) Models

45

Finally, we remark about liquid crystals confined in random porous media. Let n(r) = (n 1 (r), n 2 (r), n 3 (r)) be the director of the liquid crystal, which satisfies |n(r)| = 1. The Hamiltonian of this system is given by 



1 1 K 1 ∂α n β (r)∂α n β (r) + K 2 ∂α n α (r)∂β n β (r) 2 2  1 α β γ γ αβ α β + K 3 n (r)n (r)∂α n (r)∂β n (r) − τ (r)n (r)n (r) , (2.66) 2

1 H[n] = T

D

d r

where the summation over the repeated indices α, β, γ is assumed. For the liquid crystal, there are three elastic constants, and Eq. (2.66) is reduced to the Hamiltonian of the RAO(3)M when K 2 = K 3 = 0. However, it is shown that K 2 and K 3 are irrelevant to the critical behavior [21]. Therefore, the liquid crystals in porous media belong to the universality class of the RAO(3)M.

References 1. Wilson KG, Kogut J (1974) The renormalization group and the  expansion. Phys Rep 12:75 2. Mézard M, Parisi G, Virasoro MA (1987) Spin-glasses and beyond. World Scientific, Singapore 3. Fisher DS (1985) Random fields, random anisotropies, nonlinear σ models, and dimensional reduction. Phys Rev B 31:7233 4. Fisher DS (1986) Interface fluctuations in disordered systems: 5 −  expansion and failure of dimensional reduction. Phys Rev Lett 56:1964 5. Feldman DE (2000) Quasi-long-range order in the random anisotropy Heisenberg model: functional renormalization group in 4 −  dimensions. Phys Rev B 61:382 6. Zinn-Justin J (1989) Quantum field theory and critical phenomena. Clarendon Press, Oxford 7. Haga T (2017) Dimensional reduction and its breakdown in the driven random-field O(N ) model. Phys Rev B 96:184202 8. Feldman DE (2002) Critical exponents of the random-field O(N ) model. Phys Rev Lett 88:177202 9. Nattermann T, Scheidl S (2000) Vortex-glass phases in type-II superconductors. Adv Phys 49:607 10. Giamarchi T, Le Doussal P (1994) Elastic theory of pinned flux lattices. Phys Rev Lett 72:1530 11. Giamarchi T, Le Doussal P (1995) Elastic theory of flux lattices in the presence of weak disorder. Phys Rev B 52:1242 12. Menon GI (2002) Phase behavior of type-II superconductors with quenched point pinning disorder. Phys Rev B 65:104527 13. Le Doussal P, Wiese KJ (2006) Random-field spin models beyond 1 loop. Phys Rev Lett 96:197202 14. Tissier M, Tarjus G (2006) Two-loop functional renormalization group of the random field and random anisotropy O(N ) models. Phys Rev B 74:214419 15. Gingras MJP, Huse DA (1996) Topological defects in the random-field XY model and the pinned vortex lattice to vortex glass transition in type-II superconductors. Phys Rev B 53:15193 16. Fisher DS (1997) Stability of elastic glass phases in random field XY magnets and vortex lattices in type-II superconductors. Phys Rev Lett 78:1964 17. Tissier M, Tarjus G (2006) Unified picture of ferromagnetism, quasi-long-range order, and criticality in random-field models. Phys Rev Lett 96:087202

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18. Kosterlitz JM, Thouless DJ (1973) Ordering, metastability and phase transitions in twodimensional systems. J Phys C 6:1181 19. Jose JV, Kadanoff LP, Kirkpatrick S, Nelson DR (1977) Renormalization, vortices, and symmetry-breaking perturbation in the two-dimensional planar model. Phys Rev B 16:1217 20. Chitra R, Giamarchi T, Le Doussal P (1999) Disordered periodic systems at the upper critical dimension. Phys Rev B 59:4058 21. Feldman DE (2000) Quasi-long-range order in nematics confined in random porous media. Phys Rev Lett 84:4886

Chapter 3

Nonperturbative Renormalization Group

3.1 General Formalism In this section, we review the so-called nonperturbative renormalization group (NPRG) formalism [1]. The main objective of this approach is to introduce a family of effective actions, or Gibbs free energies, which have different cutoff scales, and derive an exact flow equation of them. Since the NPRG treatment enables us to access a fixed point which is inaccessible by perturbative approaches, it has been applied to a wide variety of systems, such as frustrated magnets [2], strongly correlated quantum gases [3], reaction-diffusion systems [4], and the Kardar-Parisi-Zhang equation [5].

3.1.1 Statics We present a static (equilibrium) formalism of the NPRG. First, let us recall the definition of the effective action or Gibbs free energy. The partition function corresponding to the Hamiltonian H[φ] reads  Z [J ] =

   Dφ exp −H[φ] + J φ ,

(3.1)

r

where



=



d D r. The generating functional or Helmholtz free energy is given by

r

W [J ] = ln Z [J ].

(3.2)

The effective action [ψ] is defined by a Legendre transformation of W [J ],  J ψ,

[ψ] = −W [J ] +

(3.3)

r

© Springer Nature Singapore Pte Ltd. 2019 T. Haga, Renormalization Group Analysis of Nonequilibrium Phase Transitions in Driven Disordered Systems, Springer Theses, https://doi.org/10.1007/978-981-13-6171-5_3

47

48

3 Nonperturbative Renormalization Group

Fig. 3.1 Typical cutoff function Rk (q)

where J and ψ are related by ψ=

δW [J ] . δJ

(3.4)

The second-order derivative of [ψ] is the inverse of the connected Green’s function and the higher-order derivatives yield the irreducible vertex functions [6]. We introduce a one-parameter family of k , which includes only high-energy modes with momenta larger than the running scale k. As k goes from the cutoff  to zero, k interpolates between the bare Hamiltonian H and the full effective action Eq. (3.3). To suppress the contribution from the low-energy modes, a mass-like quadratic term is added to the bare Hamiltonian,   1 1 Hk [φ] = Rk (q)φq φ−q = Rk (r − r  )φr φr  , (3.5) 2 2 r,r 

q

where



=



d D q/(2π) D . The cutoff function Rk (q) satisfies the following proper-

q

ties: • When k = 0, Rk (q) = 0 for all q. • When k → ∞, Rk (q) → ∞ for all q. In usual, Rk (q) is chosen to be proportional to k 2 . • Rk (q) rapidly decreases to zero for |q| > k. Figure 3.1 shows a schematic picture of the cutoff function. The partition function and generating functional with the running scale k read     Z k [J ] = Dφ exp −H[φ] − Hk [φ] + J φ , (3.6) r

Wk [J ] = ln Z k [J ]. The scale-dependent effective action is defined by  k [ψ] = −Wk [J ] + J ψ − Hk [ψ], r

(3.7)

(3.8)

3.1 General Formalism

49

where J and ψ are related by ψ=

δWk [J ] . δJ

(3.9)

It is obvious from the definition that k =  when k = 0. Let us show that k = H in the limit k → ∞. When Rk is large, the dominant contribution to Wk comes from the saddle-point value of −H + Hk + J φ. Thus, Wk can be evaluated as r

 Wk [J ]  −H[φ J ] − Hk [φ J ] +

J φJ ,

(3.10)

r

where φ J is the solution of δ (−H[φ] − Hk [φ]) + J = 0. δφ

(3.11)

Then, by using δWk [J ] ψr = = δ Jr

  r

 δ δφ J,r  (−H[φ] − Hk [φ]) + Jr  + φ J,r = φ J,r , δφr  δ Jr (3.12)

we have k [ψ]  H[ψ].

(3.13)

The exact evolution of the effective action k is described by the so-called Wetterich equation [1], ∂k k [ψ] = =

1 2 1 2



−1  ∂k Rk (r − r  ) k(2) [ψ] + Rk  r,r

r,r 



−1  ∂k Rk (q) k(2) [ψ] + Rk , q,−q

(3.14)

q

where Rk in the second line implicitly contains the delta function (2π) D δ(q − q ) and k(2) [ψ]q,q =

δ 2 k [ψ] . δψ(q)δψ(q )

(3.15)

If ψ is a multi-component vector (ψ 1 , . . . , ψ N ), the trace over the component should be taken. The detailed derivation of Eq. (3.14) is given in Sect. 3.4.1. For a given bare Hamiltonian k= = H, by solving Eq. (3.14), we can obtain the full effective action k=0 = . However, to solve this equation, we have to introduce an approximation for the functional form of k . Since we are interested in the large-

50

3 Nonperturbative Renormalization Group

scale behaviors of the system, we expand the effective action in an increasing number of derivatives of the field and retain only a limited number of terms. This systematic truncation scheme is called “derivative expansion” [1]. In the next section, we review how this scheme is applied to the O(N ) model.

3.1.2 Dynamics We also present a dynamic (nonequilibrium) formalism of the NPRG [7]. Suppose that the dynamics of φ is described by a Langevin equation, ∂t φ(r, t) = F(φ; r) + ξ(r, t),

(3.16)

where the thermal noise satisfies ξ(r, t)ξ(r  , t  ) = 2T δ(r − r  )δ(t − t  ).

(3.17)

The force F(φ; r) contains not only a function of φ, but also its spatial derivatives. We introduce a path integral representation of the Langevin equation (3.16). The average of a function of the field A[φ] over the thermal noise is written as  A[φ] =

 Dξ P[ξ]

Dφδ(φ − φ[ξ])A[φ],

(3.18)

where φ[ξ] is the solution of Eq. (3.16) for a realization of the noise ξ and    1 ξ(r, t)2 . P[ξ] ∼ exp − 4T

(3.19)

r,t

This average can be calculated as 





DφJ [φ]A[φ]δ ∂t φ − F(φ) − ξ ,      ˆ [φ]A[φ] exp − i φ{∂ ˆ t φ − F(φ) − ξ} = Dξ P[ξ] DφDφJ

A[φ] =

Dξ P[ξ]

 =

r,t

   ˆ t φ − F(φ) − T i φ} ˆ , ˆ [φ]A[φ] exp − i φ{∂ DφDφJ

(3.20)

r,t

where J [φ] is the Jacobian,   δ F(φ) J [φ] = det ∂t − , δφ

(3.21)

3.1 General Formalism

51

which is set to unity [7]. Therefore, we have  A[φ] =

ˆ ˆ DφDφA[φ] exp(−S[φ, i φ]),

(3.22)

ˆ = S[φ, i φ]

  ˆ t φ − F(φ)} − T (i φ) ˆ 2 . i φ{∂

(3.23)

r,t

The partition function and generating functional are defined by ˆ = Z [ j, j]



   ˆ ˆ ˆ ˆ DφDφ exp −S[φ, i φ] + (φ j + i φ j) ,

(3.24)

r

ˆ = ln Z [ j, j]. ˆ W [ j, j]

(3.25)

The effective action is given by ˆ = −W [ j, j] ˆ + [ψ, i ψ]



ˆ (ψ j + i ψˆ j),

(3.26)

r

ˆ are related by ˆ and ψ (ψ) where j ( j) ψ=

ˆ ˆ δW [ j, j] δW [ j, j] , i ψˆ = . ˆ δj δj

(3.27)

ˆ is similar to that The definition of the scale-dependent effective action k [ψ, i ψ] of the static case. We introduce the following quadratic term, ˆ = Sk [φ, i φ]



Rk (q)i φˆ q,ω φ−q,−ω =



Rk (r − r  )i φˆ r,t φr  ,t .

(3.28)

r,r  ,t

q,ω

Since adding this term to the bare action S is equivalent to adding a force −Rk φ to the left-hand side of Eq. (3.16), S suppresses the low-energy fluctuations. S can be rewritten as  t ˆ = 1 q,ω Rk (q)−q,−ω , (3.29) Sk [φ, i φ] 2 q,ω

ˆ and where  = t (φ, i φ)  Rk (q) =

 0 Rk (q) . Rk (q) 0

(3.30)

52

3 Nonperturbative Renormalization Group

The exact flow equation is given by ∂k k [] =

1 Tr 2



−1  ∂k Rk (q) k(2) [] + Rk

q,ω;−q,−ω

,

(3.31)

q,ω

ˆ where Tr denotes the trace over ψ and ψ.

3.2 NPRG of the O(N) Model 3.2.1 Derivative Expansion We review the NPRG analysis of the O(N ) model [8]. The Hamiltonian is given by  H=

dDr

 α

 1 |∇φα |2 + U (ρ) , 2

(3.32)

where ρ = |φ|2 /2 is the field amplitude. The potential U (ρ) is assumed to have a double-well form, 1 (3.33) U (ρ) = g(ρ − ρ0 )2 , 2 where g, ρ0 > 0. The derivative expansion of the effective action k reads  k =

 

1 1 α 2 2 4 Z k (ρ)|∇φ | + Yk (ρ)|∇ρ| + O(∂ ) , d r Uk (ρ) + 2 4 α D

(3.34)

where Uk (ρ) is the renormalized potential. The field renormalization factors Z k and Yk depend on the field amplitude ρ. Although Yk and other terms with higher-order derivatives do not exist in the bare model, they can be generated along the RG flow. One of the simplest choices of k is  k =

 

1 Z k |∇φα |2 , d D r Uk (ρ) + 2 α

(3.35)

where Yk = 0 and the ρ-dependence of Z k is ignored. Note that Z k= = 1 and Uk= (ρ) = U (ρ). In the following, we derive the flow equations for Z k and Uk (ρ) from Eq. (3.14).

3.2 NPRG of the O(N ) Model

53

3.2.2 Flow Equations First, we √ evaluate the both hand side of Eq. (3.14) for a uniform field configuration, ψ = ( 2ρ, 0, . . . , 0). Then, the left-hand side yields ∂k Uk (ρ). The second derivative of k in the right-hand side can be calculated as ψ(2)1 ψ1 (q, q ) = [Z k |q|2 + Uk (ρ) + 2ρUk (ρ)](2π) D δ(q + q ),

(3.36)

ψ(2)ν ψν (q, q ) = [Z k |q|2 + Uk (ρ)](2π) D δ(q + q ), (ν = 2, . . . , N ),

(3.37)

where the following notation has been introduced, δ 2 k [ψ] . δψ α (q)δψ β (q )

(3.38)

 1 N −1 + , ∂k Rk (q) M0 (q; ρ) M1 (q; ρ)

(3.39)

ψ(2)α ψβ (q, q ) = Therefore, we have 1 ∂k Uk (ρ) = 2

 q



where M0 (q; ρ) = Z k |q|2 + Rk (q) + Uk (ρ), M1 (q; ρ) = Z k |q|2 + Rk (q) + Uk (ρ) + 2ρUk (ρ).

(3.40)

This is the RG equation for Uk (ρ). Next, we derive the flow equation for Z k , which is given by Z k = ∂|p|2 ψ(2)2 ψ2 (p, −p)

p=0

.

(3.41)

In Eq. (3.41), we have defined Z k in terms of the derivative of the transverse mode ψ 2 because the transverse (massless) modes have dominant contribution to the fluctuations near the critical point, rather than the longitudinal (massive) mode. Within the simplest approximation Eq. (3.35), the replacement of ψ 2 in Eq. (3.41) with ψ 1 makes no difference in the definition of Z k , however, the resulting flow equations are different. This is because the field renormalization factors for the transverse and longitudinal modes flow in a different way. To take into account this effect, we have derivative in Eq. (3.41) to add a term Yk (ρ)|∇ρ|2 to Eq. (3.35). When the functional √ is evaluated for a uniform field configuration, ψ = ( 2ρ, 0, . . . , 0), the right-hand side still depends on ρ. We fix ρ such that Uk (ρ) attains its minimum, Uk (ρm,k ) = 0.

(3.42)

54

3 Nonperturbative Renormalization Group

We call ρm,k the scale-dependent spontaneous magnetization. Therefore, the flow equation of Z k is given by ∂k Z k =

 ∂|p|2 ∂k ψ(2)2 ψ2 (p, −p) p=0 ρ=ρm,k  (2) +(∂k ρm,k ) ∂ρ ∂|p|2 ψ2 ψ2 (p, −p)

p=0 ρ=ρm,k

.

(3.43)

Since Z k is independent of ρ in the approximation (3.35), the second term can be omitted. To derive the flow equation for Z k , the flow equation for  (2) is required. Taking the derivative of Eq. (3.14) leads to (2) = Tr ∂k p,−p



(3) (3) ∂k Rk (q)Pq p,q,−p−q Pp+q −p,p+q,−q Pq

q

1 − Tr 2



(4) ∂k Rk (q)Pq p,−p,q,−q Pq ,

(3.44)

q

where the propagator is defined by

−1 Pq =  (2) + Rk q,−q ,

(3.45)

which is an N × N matrix. The vertex functions  (3) and  (4) in Eq. (3.44) should be also considered as N × N matrices, whose elements are given by ψ(3)2 ψα ψβ and

ψ(4)2 ψ2 ψα ψβ , (α, β = 1, . . . , N ), respectively. Figure 3.2 shows a graphical representation of the right-hand side of Eq. (3.44). The rule for the graphical representation is as follows: • An inner line denotes the propagator Pq . • A filled circle represents a vertex function  (n) . • A cross symbol denotes ∂k Rk (q).

Fig. 3.2 Graphical representation for the flow equation of  (2)

3.2 NPRG of the O(N ) Model

55

From Eq. (3.35), the three-point vertex function  (3) can be calculated as ψ(3)2 ψ2 ψ1 (q1 , q2 , q3 ) =



2ρUk (ρ)(2π) D δ(q1 + q2 + q3 ),

(3.46)

where the functional derivative is evaluated for a uniform field configuration, ψ = √ ( 2ρ, 0, . . . , 0). The first term of Eq. (3.44) is given by (2) = 2ρUk (ρ)2 ∂k p,−p



∂k Rk (q)[M0 (q; ρ)−2 M1 (p + q; ρ)−1

q

+M1 (q; ρ)−2 M0 (p + q; ρ)−1 ].

(3.47)

Since the second term in Eq. (3.44) does not depend on the external momentum p, this term does not contribute to the flow equation of Z k . Thus, we omit the second term in Eq. (3.44). The |p|2 -derivative of Eq. (3.47) is calculated as (2) ∂|p|2 ∂k p,−p

p=0

   2 ∂k Rk (q) −2 Z k + Rk (|q|2 ) + |q|2 Rk (|q|2 ) D q  4 2  2 2 −1 −1 + |q| [Z k + Rk (|q| )] [M0 (q; ρ) + M1 (q; ρ) ] D

= 2ρUk (ρ)2



×M0 (q; ρ)−2 M1 (q; ρ)−2 ,

(3.48)

2 2 where Rk (|q|2 ) = ∂|q|2 Rk (|q|2 ) and Rk (|q|2 ) = ∂|q| 2 Rk (|q| ). Equations (3.43) and (3.48) reads the flow equation of Z k . In the original model, the length r is measured in units of −1 , where  is the cutoff wavenumber. In the coarse-grained model k , the length r should be measured in units of k −1 . Therefore, we rewrite the effective action in terms of the dimensionless length r˜ = rk. The dimensionless quantities corresponding to |q|2 , ρ, and Uk are defined by |q|2 (3.49) y= 2 , k

ρ˜ = Z k k 2−D ρ,

(3.50)

˜ = k −D Uk (ρ), u k (ρ)

(3.51)

respectively. In the following, we employ a renormalization scale l = − ln(k/), which moves from 0 to ∞ when k goes from  to 0. The cutoff function Rk (q) is assume to be written as Rk (q) = Z k |q|2 r (y) = Z k k 2 r˜ (y),

(3.52)

56

3 Nonperturbative Renormalization Group

where r˜ (y) = yr (y). The dimensionless cutoff function r˜ (y) has a finite value at y = 0 and rapidly decreases for y > 1. For example, r˜ (y) =

αy , −1

ey

(3.53)

or r˜ (y) = (1 − y)θ(1 − y),

(3.54)

where θ(x) is the step function; θ(x) = 1 for x > 0 and θ(x) = 0 for x < 0. We now remark about an ambiguity associated with the choice of the cutoff function. It is obvious that the full effective action, which is the endpoint of the flow described by Eq. (3.14), and the critical exponents should be independent of the cutoff function. However, the approximation scheme such as the derivative expansion leads to a spurious cutoff function dependence of them. Therefore, to obtain accurate values of the critical exponents, an appropriate optimization of the cutoff function is required. One of the simple criteria that determines the optimized cutoff is “minimum sensitivity condition”. By introducing a one-parameter family of cutoff function Rk(α) (q), for example Eq. (3.53), one obtains a critical exponent as a function of α. Then, an optimized parameter αopt is determined from the condition that the α-derivative of the critical exponent vanishes. This optimization scheme enables us to obtain the critical exponent with high accuracy. The cutoff function defined by Eq. (3.54) is called the optimized cutoff function [9]. It is believed that, in the leading order of the derivative expansion, the critical exponent attains its minimum for this optimized cutoff function [10]. At a critical point, the full two-point vertex function scales as  (2) (q) ∼ |q|2−η , where η is the anomalous dimension. Since the infrared regulator Eq. (3.5) suppresses the large-scale fluctuations, k(2) (q) is expected to behave as k(2) (q) ∼ |q|2 (|q|2 + ck 2 )−η/2 ,

(3.55)

where c is a constant. Note that this equation recovers |q|2−η as k goes to zero. Within this assumption, the field renormalization factor Z k is found to scale as k −η . Thus, we define a scale-dependent anomalous dimension, ηk = ∂l ln Z k ,

(3.56)

where ∂l = −k∂k . We rewrite the flow equation derived above in terms of these dimensionless quantities. ˜ is given by From Eq. (3.39), the flow equation for u k (ρ) ˜ = Du k (ρ) ˜ − (D − 2 + ηk )ρu ˜ k (ρ) ˜ − 2 A D [(N − 1)l0D (u k (ρ)) ˜ ∂l u k (ρ) D   ˜ + 2ρu ˜ k (ρ))], ˜ (3.57) +l0 (u k (ρ) where the function lnD (w) is given by

3.2 NPRG of the O(N ) Model

lnD (w)

57

n + δn,0 = 2

∞ dyy D/2 0

−2yr  (y) , [(1 + r (y))y + w]n+1

(3.58)

D+1 D/2 and A−1 π (D/2). The first and second terms in Eq. (3.57) come from D =2 the rescaling of Uk and ρ, respectively. We have ignored ηk coming from ∂t Rk (q) because it is negligible compared to other terms. It is convenient to approximate u k in a quadratic form 1 ˜ = λk (ρ˜ − κk )2 . (3.59) u k (ρ) 2

The flow equations of λk and κk can be obtained from Eq. (3.57), u k (κk ) = 0, and u k (κk ) = λk . They read ∂l κ = (D − 2 + ηk )κ − 2 A D [(N − 1)l1D (0) + 3l1D (2κλ)],

(3.60)

∂l λ = −(D − 4 + 2ηk )λ − 2 A D λ2 [(N − 1)l2D (0) + 9l2D (2κλ)].

(3.61)

The equation for ηk can be obtained from Eq. (3.48), ηk =

16A D 2 D κλ m 2,2 (2λκ, 0), D

(3.62)

D where the function m 2,2 (w) is given by

 1 + r + yr  2y(2r  + yr  ) [(1 + r)y + w1 ]n 1 [(1 + r)y + w2 ]n 2 0   n1 n2 −y 2 r  (1 + r + yr  ) + . (3.63) (1 + r)y + w1 (1 + r )y + w2 ∞

m nD1 ,n 2 (w1 , w2 ) =

dyy D/2

In the right-hand side of Eq. (3.62), we have ignored ηk coming from ∂t Rk (q). If we employ the optimized cutoff function (3.54), the functions lnD (w) and m nD1 ,n 2 (w1 , w2 ) can be calculated as lnD (w) =

2 1 (n + δn,0 ) , D (1 + w)n+1

m nD1 ,n 2 (w1 , w2 ) =

1 (1 + w1

)n 1 (1

+ w 2 )n 2

(3.64) .

(3.65)

Equations (3.60), (3.61), and (3.62) constitute the set of flow equations for the O(N ) model.

58

3 Nonperturbative Renormalization Group

3.2.3 Fixed Point and Critical Exponents Let κ∗ and λ∗ be fixed point values of Eqs. (3.60) and (3.61). When we start with a bare value κl=0 below κ∗ , κl flows to zero at a finite renormalization scale. This regime corresponds to a disordered phase without a spontaneous magnetization. When we start with κl=0 above κ∗ , κl increases exponentially. This regime corresponds to a long-range order (LRO) phase with a nonzero spontaneous magnetization. To calculate the critical exponents, we linearize Eq. (3.60) as ∂l (κ − κ∗ )  yκ (κ − κ∗ ),

(3.66)

near the fixed point. Then, the critical exponent for the correlation length ν can be written as 1 (3.67) ν= . yκ The other critical exponents α, β, γ, δ can be determined from ν and η by using the scaling relations [11]. One of the remarkable properties of Eqs. (3.60), (3.61), and (3.62) is that they recover the results of the perturbative RG at D = 2 +  and D = 4 − . Below, let us calculate ν and η in the leading order of . First, we consider the case of D = 2 + . Since D = 2 is the lower critical dimension of the O(N ) model, the fixed point κ∗ is expected to scale as ∼ 1/. In contrast, λ∗ = O(1). From Eqs. (3.60) and (3.62), we have the following equations in the leading order of : (3.68) ( + η)κ∗ − 2 A2 (N − 1) = 0, η=

2 A2 . κ∗

(3.69)

Therefore, we have the critical exponents, η=

 , N −2

(3.70)

1 . 

(3.71)

ν=

These exponents are equal to those obtained from the perturbative RG approach in the nonlinear sigma model [6]. Next, we consider the case of D = 4 − . Since D = 4 is the upper critical dimension, λ∗ = O(). In contrast, κ∗ = O(1), thus η = O(2 ). From Eqs. (3.60) and (3.61), we have the following equations in the leading order of : (2 − )κ∗ − A4 (N + 2 − 12κ∗ λ∗ ) = 0,

(3.72)

3.2 NPRG of the O(N ) Model

59

Fig. 3.3 η calculated from Eqs. (3.60), (3.61), and (3.62). The red solid line and the green dashed line represent η for N = 2 and N = 3, respectively

λ∗ − 2 A4 (N + 8)(λ∗ )2 = 0,

(3.73)

where we have used l14 (w)  (1 − 2w)/2. Therefore, we have the critical exponents, N +2 2  , 2(N + 8)2

(3.74)

N +2 1 + . 2 4(N + 8)

(3.75)

η= ν=

These exponents are equal to those obtained from the perturbative RG approach [11]. Figure 3.3 shows η as a function of D for N = 2 (XY) and 3 (Heisenberg). In three dimensions, Monte Carlo simulations predict η = 0.03 ∼ 0.04 for both XY and Heisenberg models [12]. Since η is seriously affected by the omission of the higher derivative terms in the effective action, in three dimensions it is poorly determined within the approximation Eq. (3.35). However, the NPRG approach provides a firstprinciple theory which interpolates the perturbative results at D = 2 +  and D = 4 − . Let us consider the case of the 2D XY model (N = 2 and D = 2). This model does not exhibit LRO at finite temperature (Mermin-Wagner theorem). However, at low temperatures T < TKT , this model exhibits quasi-long-range order (QLRO), in which the correlation function shows power-law decay, C(r ) ∼ r −η(T ) .

(3.76)

Thus, the correlation length is divergent for all temperatures below TKT . The exponent is given by T , (3.77) η(T ) = 2πK eff (T ) where K eff (T ) is a renormalized elastic constant, which is almost equal to unity for sufficiently low temperatures. At T = TKT , η(TKT ) = 1/4. At high temperatures T > TKT , this model is in a disordered phase, in which the correlation function shows

60

3 Nonperturbative Renormalization Group

Fig. 3.4 Schematic picture of η as a function of D. The red solid line and the green dashed line represent η for N = 2 and N > 2, respectively. The dotted line represents Eq. (3.70)

exponential decay. The transition between the QLRO phase to the disordered phase is known as the Kosterlitz-Thouless (KT) transition [13, 14]. The presence of the QLRO below TKT implies that the right-hand side of the flow equation (3.60) should identically vanish. In other words, there is a line of fixed points, “fixed line”. In fact, at low temperatures (or large κ), the leading order of the flow equation vanishes, (3.78) ∂l κ = O(κ−1 ), where we have used η = 2 A2 /κ + O(κ−2 ). However, there are finite contributions from higher-order terms. This insufficient vanishing of the flow equation is an artifact of the approximation. Thus, we observe a “pseudo fixed line”, in which the flow of κ is extremely slow. √ In Fig. 3.3, the curve for N = 2 has infinite slope at D = 2 and η behaves as ∼ (D − 2). This square root dependence comes from the remaining term in Eq. (3.78), which is an artifact of the approximation. Figure 3.4 shows the schematic picture of η as a function of D. For N > 2, the qualitative feature is similar to that shown in Fig. 3.3. For N = 2, there is a fixed line for 0 ≤ η ≤ 1/4, which cannot be captured by the approximation employed in this calculation.

3.3 NP-FRG of Disordered Systems Recently, the NPRG scheme is combined with the functional renormalization group of disordered systems, and this formalism is called nonperturbative functional renormalization group (NP-FRG) [15, 16]. The NP-FRG enables us to go beyond the perturbative FRG near the critical dimension, which has been discussed in Chap. 2. In this section, we present the general formalism of the NP-FRG and review some applications to the random manifold and random field O(N ) models (RFO(N )M). Especially, for the RFO(N )M, one obtains a phase diagram which shows the parameter region wherein the dimensional reduction fails.

3.3 NP-FRG of Disordered Systems

61

3.3.1 General Formalism of the NP-FRG We write the Hamiltonian of a disordered system as H[φ; h] = H1 [φ] + HR [φ; h],

(3.79)

where h denotes a disorder such as a random field or random potential. The partition function and generating functional (Helmholtz free energy) are given by  Z [J ; h] =

   Dφ exp −H[φ; h] + J φ ,

(3.80)

r

W [J ; h] = ln Z [J ; h].

(3.81)

We now introduce replicated fields {φa }a=1,...,n and sources {Ja }a=1,...,n . Then, we have     exp W [Ja ; h] = exp W [{Ja }] , (3.82) a

where the replicated generating functional W [{Ja }] is defined by W [{Ja }] = ln Z [{Ja }],  Z [{Ja }] =

(3.83)

 

 Dφ exp −H[{φa }] + Ja φa . a

(3.84)

r

The replicated Hamiltonian H[{φ}] is written as the following form:

1 1 H2 [φa , φb ] + H3 [φa , φb , φc ] − · · · , 2 a,b 3! a,b,c a (3.85) where H p is the p-th cumulant of HR , for example, H[{φa }] =

H1 [φa ] −

H2 [φ1 , φ2 ] = HR [φ1 ; h]HR [φ2 ; h] − HR [φ1 ; h] HR [φ2 ; h].

(3.86)

The replicated effective action [{ψa }] is defined by a Legendre transformation, [{ψa }] = −W [{Ja }] +

 a

Ja ψa ,

(3.87)

r

where Ja and ψa are related by ψa =

δW [{Ja }] . δ Ja

(3.88)

62

3 Nonperturbative Renormalization Group

Let us consider the physical meaning of [{ψa }]. Quantities in which we are interested are the averaged free energy and its cumulants, W1 [J1 ] = W [J1 ; h], W2 [J1 , J2 ] = W [J1 ; h]W [J2 ; h] − W [J1 ; h] W [J2 ; h].

(3.89)

In terms of W1 and W2 , the connected and disconnected Green’s functions are given by δ 2 W1 [J ] δ J (r)δ J (r  ) δ 2 W2 [J1 , J2 ] G d (r − r  ) ≡ φ(r)φ(r  ) − φ(r) φ(r  ) = . δ J1 (r)δ J2 (r  ) G c (r − r  ) ≡ φ(r)φ(r  ) − φ(r)φ(r  ) =

(3.90)

From Eq. (3.82), W [{Ja }] can be expanded by using these cumulants,

1 1 W2 [Ja , Jb ] + W3 [Ja , Jb , Jc ] + · · · . 2 a,b 3! a,b,c a (3.91) We also expand the effective action [{ψa }] in increasing number of free replica sum, W [{Ja }] =

W1 [Ja ] +

1 1 2 [ψa , ψb ] + 3 [ψa , ψb , ψc ] − · · · . 2 a,b 3! a,b,c a (3.92) From Eq. (3.87), one finds that 1 [ψ] is the Legendre transform of W1 [J ], [{ψa }] =

1 [ψa ] −

 1 [ψ] = −W1 [J ] +

J ψ,

(3.93)

r

with ψ=

δW1 [J ] . δJ

(3.94)

The second-order term is given by 2 [ψ1 , ψ2 ] = W2 [J1 , J2 ],

(3.95)

where ψ and J are related by Eq. (3.94). Although the higher-order terms  p , ( p ≥ 3) cannot be identical to W p directly, one can express  p in terms of such cumulants of order equal to or lower than p (see Ref. [15]). Various physical quantities can be calculated from 1 and 2 . For example, the connected and disconnected Green’s functions are calculated as

3.3 NP-FRG of Disordered Systems

63





G c (r − r ) =





G d (r − r ) = r1 ,r2

−1 r,r 

,

(3.96)

δψ1 (r 1 ) δ 2 2 [ψ1 , ψ2 ] δψ2 (r 2 ) δ J1 (r) δψ1 (r 1 )δψ2 (r 2 ) δ J2 (r  )



=

δ 2 1 [ψ] δψδψ

G c (r − r 1 ) r1 ,r2

δ 2 2 [ψ1 , ψ2 ] G c (r 2 − r  ), δψ1 (r 1 )δψ2 (r 2 )

(3.97)

where we have used Eqs. (3.90) and (3.94). According to the general formalism of the NPRG, we introduce the scaledependent effective action k [{ψa }]. We add the following mass-like term to the replicated Hamiltonian, Hk [{φa }] =

1 2 a

 Rk (q)φa (q)φa (−q),

(3.98)

q

where Rk (q) is the cutoff function. The scale-dependent partition function and generating functional can be defined as  Z k [{Ja }] =

 

 Dφ exp −H[{φa }] − Hk [{φa }] + Ja φa , a

(3.99)

r

Wk [{Ja }] = ln Z k [{Ja }].

(3.100)

The scale-dependent effective action k [{ψa }] is given by k [{ψa }] = −Wk [{Ja }] +

 a

Ja ψa − Hk [{ψa }],

(3.101)

r

where Ja and ψa are related by ψa =

δWk [{Ja }] . δ Ja

(3.102)

The exact flow equation of k [{ψa }] is given by ∂k k [{ψa }] =

1 2 a



 −1 ∂k Rk (q) k(2) [{ψa }] + Rk I

(q,a);(−q,a)

q

,

(3.103)

64

3 Nonperturbative Renormalization Group

where I is the unit matrix in the replica space. Note that the term [. . .]−1 in the right-hand side contains the inversion with respect to not only the momentum, but also the replica index. As in Eq. (3.92), we expand the effective action k [{ψa }] in increasing number of free replica sum,

1 1 2,k [ψa , ψb ] + 3,k [ψa , ψb , ψc ] − · · · . 2 a,b 3! a,b,c a (3.104) Substitution of Eq. (3.104) into (3.103) leads to the exact flow equations for  p,k . The detailed derivation of these equations is presented in Sect. 3.4.2. To express these in a compact form, we define the one-replica propagator with the infrared cutoff, k [{ψa }] =

1,k [ψa ] −

−1

(2) [ψ] + Rk . Pk [ψ] = 1,k

(3.105)

We also introduce the following notation for the functional derivatives: δ 2 2,k [ψ1 , ψ2 ] , δψ1 δψ2 δ 2 2,k [ψ1 , ψ2 ] (20) [ψ1 , ψ2 ] = . 2,k δψ1 δψ1

(11) [ψ1 , ψ2 ] = 2,k

(3.106)

The exact flow equations for 1,k [ψ] and 2,k [ψ1 , ψ2 ] read ∂k 1,k [ψ] =

1 2



 (11) ∂k Rk (q) Pk [ψ] + Pk [ψ]2,k [ψ, ψ]Pk [ψ]

q,−q

, (3.107)

q

∂k 2,k [ψ1 , ψ2 ] = −

1 2



  (20) (110) ∂k Rk (q) Pk [ψ1 ] 2,k [ψ1 , ψ2 ] − 3,k [ψ1 , ψ1 , ψ2 ]

q (11) (11) +2,k [ψ1 , ψ2 ]Pk [ψ2 ]2,k [ψ2 , ψ1 ] (20) (11) +2,k [ψ1 , ψ2 ]Pk [ψ1 ]2,k [ψ1 , ψ1 ]

 (11) (20) +2,k [ψ1 , ψ1 ]Pk [ψ1 ]2,k [ψ1 , ψ2 ] Pk [ψ1 ] +perm(ψ1 , ψ2 ) , q,−q

(3.108)

where perm(ψ1 , ψ2 ) denotes the permutation of ψ1 and ψ2 . For the multi-component field ψ = (ψ 1 , . . . , ψ N ), we have to take the trace with respect to the field component index in the right-hand side of Eqs. (3.107) and (3.108). Note that  p+1 appears on the right-hand side of the flow equation for  p , thus we have an infinite hierarchy of the coupled flow equations (see Sect. 3.4.2 for a graphical representation of the

3.3 NP-FRG of Disordered Systems

65

flow equations). To solve Eqs. (3.107) and (3.108), approximations for the functional forms of 1,k [ψ] and 2,k [ψ1 , ψ2 ] are required. In the following sections, we apply the general formalism developed above to the random manifold and random field O(N ) models.

3.3.2 NP-FRG of the Random Manifold Model The Hamiltonian of the one-component random manifold model is given by  1 |∇φ|2 , 2T r  1 V (r; φ), HR [φ; V ] = T H1 [φ] =

(3.109)

r

where V (r; φ) is a mean-zero Gaussian random potential satisfying V (r; φ)V (r  ; φ ) = RB (φ − φ )δ(r − r  ). The two-replica Hamiltonian in Eq. (3.85) reads  1 RB (φ1 − φ2 ). (3.110) H2 [φ1 , φ2 ] = 2 T r

The higher-order cumulants of HR vanish. We employ the following approximation for 1,k [ψ] and 2,k [ψ1 , ψ2 ], 1,k [ψ] =

1 2T

 Z k |∇ψ|2 , r

1 2,k [ψ1 , ψ2 ] = 2 T

 R(ψ1 − ψ2 ),

(3.111)

r

where R(ψ) is a renormalized disorder correlator (do not confuse with the cutoff function Rk (q)) and T is assumed to be constant. We ignore the higher-order cumulants, 3,k = 4,k = · · · = 0. It can be easily shown that the field renormalization factor Z k is not renormalized, ∂k Z k = 0, because the vertex functions for the onereplica effective action 1(3) and 1(4) vanish, which appear in the flow equation of 1(2) , thus we set Z k = 1 below. The one-replica propagator is given by Pk [ψ]q,q =

|q|2 /T

1 (2π) D δ(q + q ). + Rk (q)

The functional derivatives of 2 are calculated as

(3.112)

66

3 Nonperturbative Renormalization Group (20) 2;ψ (q, q ) = 1 ψ1

1  R (ψ1 − ψ2 )(2π) D δ(q + q ), T2

(11) 2;ψ (q, q ) = − 1 ψ2

1  R (ψ1 − ψ2 )(2π) D δ(q + q ), T2

(3.113) (3.114)

where the functional derivatives are evaluated for a uniform field. Equation (3.108) leads to the flow equation for R(ψ),  ∂l R(ψ) = −

 R  (ψ) R  (ψ)2 − 2R  (ψ)R  (0) , ∂l Rk (q) + M(q)2 T 2 M(q)3 

(3.115)

q

where l = − ln(k/) and M(q) = |q|2 /T + Rk (q). We introduce dimensionless field and disorder correlator, which are denoted with a tilde, (3.116) ψ˜ = k ζ ψ, ˜ = k D−4+4ζ R(ψ), ˜ ψ) R(

(3.117)

where ζ is the roughness exponent. The cutoff function is given by Rk (q) =

1 2 k r˜ (y), T

(3.118)

where y = |q|2 /k 2 . If we employ the optimized cutoff function r˜ (y) = (1 − y)θ(1 − y), the integrals in Eq. (3.115) can be calculated as  −

∂l Rk (q)M(q)−n = T n−1 k D−2n+2

8 AD, D

(3.119)

q D+1 D/2 where A−1 π (D/2). From Eqs. (3.116), and (3.117), D =2

˜ = ∂l [k D−4+4ζ R(k −ζ ψ)] ˜ ˜ ψ) ∂l R( ˜ + k D−4+4ζ ∂l R(ψ) −ζ ˜ , ˜ + ζ ψ˜ R˜  (ψ) ˜ ψ) = (4 − D − 4ζ) R( ψ=k ψ

(3.120)

˜ = k D−4+3ζ R  (ψ). By using Eqs. (3.115) and (3.119), we where we have used R˜  (ψ) ˜ ˜ ψ), obtain the flow equation for R( ˜ + 8 A D T k D−2+2ζ R˜  (ψ) ˜ ˜ = (4 − D − 4ζ) R( ˜ + ζ ψ˜ R˜  (ψ) ˜ ψ) ˜ ψ) ∂l R( D  1 ˜  ˜ 2 16 ˜ R˜  (0) . (3.121) + AD R (ψ) − R˜  (ψ) D 2

3.3 NP-FRG of Disordered Systems

67

Fig. 3.5 Schematic picture of (ψ) for zero and finite temperature. At finite temperature, the linear cusp is smoothed out over a width proportional to T

˜ and T by (16/D)A D R( ˜ → R( ˜ and (8/D)A D T k D−2+2ζ → ˜ ψ) ˜ ψ) ˜ ψ) We redefine R( ˜ ˜ Tl , and we omit the tilde in R(ψ). Finally, we have ∂l R(ψ) = (4 − D − 4ζ)R(ψ) + ζψ R  (ψ) + Tl R  (ψ) 1 + R  (ψ)2 − R  (ψ)R  (0), 2 dTl = (2 − D − 2ζ)Tl . dl

(3.122) (3.123)

Note that near D = 4 the “renormalized temperature” Tl flows to zero in the large scale, namely the temperature is irrelevant. If we consider the zero-temperature case Tl = 0, we have the same equation as that obtained from the perturbative approach, Eq. (2.16). Recall that at zero temperature, the cumulant for the random force (ψ) = −R  (ψ) exhibits a liner cusp, (ψ)  (0) +  (0+ )|ψ|. At finite temperature, the diffusion term Tl R  (ψ) in Eq. (3.122) acts to smooth out the cusp (see Fig. 3.5). This is because the thermal fluctuations lead to the local averaging of the disorder.

3.3.3 NP-FRG of the Random Field O(N) Model We next consider the NP-FRG treatment of the RFO(N )M. The detailed discussion is presented in Refs. [15, 16]. Here, we just explain the approximation of the effective action and briefly review the main result of the above papers. The Hamiltonian is given by 1 H1 [φ] = T

  1 r

α

2

 |∇φ | + U (ρ) , α 2

(3.124)

68

3 Nonperturbative Renormalization Group

HR [φ; h] = −

1 T

 h · φ,

(3.125)

r

where ρ = |φ|2 /2. The random field obeys a mean-zero Gaussian distribution satisfying h α (r)h β (r  ) = h 20 δ αβ δ(r − r  ). (3.126) The second cumulant of HR reads 1 H2 [φ1 , φ2 ] = 2 T

 h 20 φ1 · φ2 .

(3.127)

r

We employ the following approximation for the effective action, 1 1,k [ψ] = T

  1 r

α

2,k [ψ 1 , ψ 2 ] =

2 1 T2

α 2



Z k |∇ψ | + Uk (ρ) ,

(3.128)

 Vk (ψ 1 , ψ 2 ),

(3.129)

r

where Vk is the second cumulant of the renormalized random potential. Substitution Eqs. (3.128) and (3.129) into the exact flow equations (3.107) and (3.108) leads to the flow equation for Uk (ρ), Z k , and Vk (ψ 1 , ψ 2 ), which are given in Ref. [15]. It is worth noting that the approximation Eqs. (3.128) and (3.129) can recover the perturbative FRG result at D = 4 + . From the rotational symmetry, Vk (ψ 1 , ψ 2 ) is represented as (3.130) Vk (ψ 1 , ψ 2 ) = Vk (ρ1 , ρ2 , z), √ where ρ1 = |ψ1 |2 /2, ρ2 = |ψ2 |2 /2, and z = ψ 1 · ψ 2 / 4ρ1 ρ2 . We define R(z) =

Vk (ρm,k , ρm,k , z) , (2ρm,k )2

(3.131)

where ρm,k is the field amplitude that minimizes Uk (ρ), Uk (ρm,k ) = 0. By introducing ˜ of a dimensionless correlator R(z) = Z k−2 k D−4 R(z), it can be shown that in the ˜ leading order of  = D − 4, R(z) satisfies the purturbative FRG equation (2.36), ˜ ˜ ˜ − (N − 1)z R˜  (1) R˜  (z) = (4 − D) R(z) + 2(N − 2) R˜  (1) R(z) ∂l R(z) 1 +(1 − z 2 ) R˜  (1) R˜  (z) + (N − 2 + z 2 ) R˜  (z)2 2 1 2 ˜  −z(1 − z ) R (z) R˜ (z) + (1 − z 2 )2 R˜  (z)2 . (3.132) 2

3.3 NP-FRG of Disordered Systems

69

Fig. 3.6 Phase diagram with respect to the spatial dimension D and the field component number N . In regions I and II, the dimensional reduction fails and holds, respectively. This figure is made based on Fig. 4 in Ref. [16]

The detailed discussion of the dimensionless quantities is postponed to Chap. 4. We consider when the dimensional reduction fails. The criterion for the breakdown of the dimensional reduction is that at the fixed point, ∂z V∗ (ρ1 , ρ2 , z) has a √ cusp of the form 1 − z near z = 1. By using this criterion, Fig. 3.6 shows the region of the parameters (spatial dimension D and field component number N ) in which the dimensional reduction holds or fails [16]. At D = 4 + , the dimensional reduction recovers when N ≥ 18, as shown in Sect. 2.3.3. For N = 1 (random field Ising model), it recovers when D ≥ 5.1. This is consistent with numerical simulations of the four and five dimensional random field Ising model (RFIM) [17–19]. In these simulations, it was shown that the critical exponents of the four-dimensional RFIM deviate from their dimensional reduction values, but in five dimensions, the dimensional reduction reasonably recovers. Unfortunately, even if ∂z V∗ (ρ1 , ρ2 , z) does not have any cusp, the critical exponents are not always equal to their dimensional reduction values. This is an artifact of the approximation Eqs. (3.128) and (3.129), and of the choice of th regulator (3.98). To overcome this problem, we need a more sophisticated formalism [20, 21].

3.4 Appendix 3.4.1 Exact Flow Equation for k In this appendix, we derive Eq. (3.14). The scale-dependent partition function Z k [J ] is defined by Eq. (3.6). Taking the k-derivative of Z k [J ] for a fixed J yields

70

3 Nonperturbative Renormalization Group

(∂k Z k )[J ] = − =−

1 2 1 2

 

   ∂k Rk (r − r  )φr φr  Dφ exp −H[φ] − Hk [φ] + J φ r

∂k Rk (r − r  )

r,r 

r,r 

δ Z k [J ] . δ Jr δ Jr  2

(3.133)

Thus, we have 1 (∂k Wk )[J ] = − 2

 r,r 

 δ 2 Wk [J ] δWk [J ] δWk [J ] , ∂k Rk (r − r ) + δ Jr δ Jr  δ Jr δ Jr  



(3.134)

where Wk [J ] = ln Z k [J ]. Taking the k-derivative of k [ψ] for a fixed ψ yields 

 δWk ∂k J + (∂k J )ψ − ∂k H[ψ] δJ r r  1 ∂k Rk (r − r  )ψr ψr  = −(∂k Wk )[J ] − 2

∂k k [ψ] = −(∂k Wk )[J ] −

=

1 2



r,r 

∂k Rk (r − r  )

r,r 

δ 2 Wk [J ] . δ Jr δ Jr 

(3.135)

We next attempt to express δ 2 Wk /δ J δ J in terms of k . To do this, we take the ψ-derivative of Eq. (3.9),  2 δ Wk [J ] δ Jr  δ 2 Wk [J ] = δψr  δ Jr δ Jr δ Jr  δψr  r   2   2 δ Wk [J ] δ k [ψ]   = + Rk (r − r ) , δ Jr δ Jr  δψr  δψr 

δ(r − r  ) =

(3.136)

r 

where we have used

Thus, we have

δk [ψ] = Jr − δψr



Rk (r − r  )ψr  .

(3.137)

r

−1 δ 2 Wk [J ]  (2) = k [ψ] + Rk  , r,r δ Jr δ Jr 

(3.138)

where k(2) = δ 2 k /δψδψ. Substituting the above equation into Eq. (3.135) leads to

3.4 Appendix

71

∂k k [ψ] = =

1 2 1 2



−1  ∂k Rk (r − r  ) k(2) [ψ] + Rk  r,r

r,r 



−1  ∂k Rk (q) k(2) [ψ] + Rk . q,−q

(3.139)

q

3.4.2 Exact Flow Equations for  p,k In this appendix, we derive the exact flow equations for  p,k , (3.107) and (3.108). The expansion of k [{ψa }] reads 1 1 2,k [ψa , ψb ] + 3,k [ψa , ψb , ψc ] − · · · . 2 a,b 3! a,b,c a (3.140) The exact flow equation of k [{ψa }] is given by k [{ψa }] =

1,k [ψa ] −

∂k k [{ψa }] =

 −1 1 Tr ∂k Rk (q) k(2) [{ψa }] + Rk I , 2

(3.141)

where Tr involves the summations with respect to the momenta, replica indices, and field component indices. A nontrivial point is the calculation of the inverse matrix of k(2) + Rk I with respect to the replica index. We rewrite it as   (2) k [{ψa }] + Rk I ab = P[ψa ]−1 δab − A[ψa ]δab − B[ψa , ψb ],

(3.142)

where P[ψ] is the one-replica propagator defined by Eq. (3.105), −1

(2) [ψ] + Rk . Pk [ψ] = 1,k

(3.143)

A[ψa ] and B[ψa , ψb ] can be expanded in increasing number of free replica sum, A[ψa ] =

c

A[1] [ψa |ψc ] +

1 [2] A [ψa |ψc , ψd ] + · · · , 2 c,d

B[ψa , ψb ] = B [0] [ψa , ψb ] +

(3.144)

B [1] [ψa , ψb |ψc ]

c

1 [2] + B [ψa , ψb |ψc , ψd ] + · · · , 2 c,d

(3.145)

where the vertical bar in each term A[ p] [a |c1 , . . . , c p ] is introduced to distinguish between the “explicit” index a and the dummy indices c1 , . . . , c p , which run from

72

3 Nonperturbative Renormalization Group

1 to n as the summation is taken. In the following, we use simplified notations such as, δ 2 3 [ψ1 , ψ2 , ψ3 ] , δψ1 δψ1 δ 2 3 [ψ1 , ψ2 , ψ3 ] . 3(110) [ψ1 , ψ2 , ψ3 ] = δψ1 δψ2

3(200) [ψ1 , ψ2 , ψ3 ] =

(3.146)

From Eq. (3.140), each term is written as A[1] [ψa |ψc ] = 2(20) [ψa , ψc ], A[2] [ψa |ψc , ψd ] = −3(200) [ψa , ψc , ψd ], ...,

(3.147)

and B [0] [ψa , ψb ] = 2(11) [ψa , ψb ], B [1] [ψa , ψb |ψc ] = −3(110) [ψa , ψb , ψc ], B [2] [ψa , ψb |ψc , ψd ] = 4(1100) [ψa , ψb , ψc , ψd ], ....

(3.148)

The inverse of Eq. (3.142) reads −1    (2) k [{ψa }] + Rk I ab = P[ψa ]δab + P[ψa ] A[ψa ]δab + B[ψa , ψb ] P[ψb ]   +P[ψa ] A[ψa ]δac + B[ψa , ψc ] P[ψc ]   × A[ψc ]δcb + B[ψc , ψb ] P[ψb ]   +P[ψa ] A[ψa ]δac + B[ψa , ψc ] P[ψc ]   × A[ψc ]δcd + B[ψc , ψd ] P[ψd ]   × A[ψd ]δdb + B[ψd , ψb ] P[ψb ] + · · · , (3.149) where the summation over the repeated indices is assumed. By substituting Eqs. (3.144) and (3.145) into the above equation, we have

 (2)

−1 1 k [{ψa }] + Rk I aa = Q 1 [ψa ] + Q 2 [ψa , ψb ] 2 a,b a a +

1 Q 3 [ψa , ψb , ψc ] + · · · , 3! a,b,c

where, for example, Q 1 and Q 2 are given by

(3.150)

3.4 Appendix

73

Fig. 3.7 Graphical representation of the flow equation for 1 . Reprinted figure with permission from Ref. [22]. Copyright (2018) by the American Physical Society. https://doi.org/10.1103/PhysRevE. 98.032122

Q 1 [ψa ] = P[ψa ] + P[ψa ]B [0] [ψa , ψa ]P[ψa ],

(3.151)

 Q 2 [ψa , ψb ] = P[ψa ] A[1] [ψa |ψb ] + B [1] [ψa , ψa |ψb ] +B [0] [ψa , ψb ]P[ψb ]B [0] [ψb , ψa ] +A[1] [ψa |ψb ]P[ψa ]B [0] [ψa , ψa ]  +B [0] [ψa , ψa ]P[ψa ]A[1] [ψa |ψb ] P[ψa ] +perm(ψa , ψb ).

(3.152)

Finally, from Eq. (3.141), we obtain the exact flow equation for 1 : ∂l 1 [ψ] =

1 1 γ1,a + γ1,b , 2 2

(3.153)

γ1,a = Tr∂l Rk P[ψ],

(3.154)

γ1,b = Tr∂l Rk P[ψ]2(11) [ψ, ψ]P[ψ],

(3.155)

where Tr involves the summations with respect to the momenta and field component indices. Figure 3.7 shows the graphical representation of the flow equation for 1 . The rule for the graphical representation is as follows: 1. An inner line denotes the propagator P[ψ]. 2. A filled circle represents a vertex obtained from a derivative of the one-replica ( p) action 1 [ψ]. 3. Two open dots linked by a dashed line represent a vertex obtained from a derivative (p p ) of the two-replica action 2 1 2 [ψ1 , ψ2 ]. 4. A cross symbol denotes ∂l Rk (q). The exact flow equation for 2 reads ∂l 2 [ψ1 , ψ2 ] = −

 1

γ2,a + γ2,b + 2γ2,c − γ2,d + perm , 2

(3.156)

74

3 Nonperturbative Renormalization Group

Fig. 3.8 Graphical representation of the flow equation for 2 . Reprinted figure with permission from Ref. [22]. Copyright (2018) by the American Physical Society. https://doi.org/10.1103/PhysRevE. 98.032122

γ2,a = Tr∂l Rk P[ψ1 ]2(20) [ψ1 , ψ2 ]P[ψ1 ],

(3.157)

γ2,b = Tr∂l Rk P[ψ1 ]2(11) [ψ1 , ψ2 ]P[ψ2 ]2(11) [ψ2 , ψ1 ]P[ψ1 ],

(3.158)

γ2,c = Tr∂l Rk P[ψ1 ]2(20) [ψ1 , ψ2 ]P[ψ1 ]2(11) [ψ1 , ψ1 ]P[ψ1 ],

(3.159)

γ2,d = Tr∂l Rk P[ψ1 ]3(110) [ψ1 , ψ1 , ψ2 ]P[ψ1 ].

(3.160)

Figure 3.8 shows the graphical representation of the flow equation for 2 . The exact flow equation for 3 reads ∂l 3 [ψ1 , ψ2 , ψ3 ] =

1

γ3,a + 2γ3,b−1 + γ3,b−2 + 2γ3,c−1 + γ3,c−2 + γ3,d 2  −γ3,e − 2γ3, f − 2γ3,g − γ3,h + γ3,i + perm , (3.161)

γ3,a = Tr∂l Rk P[ψ1 ]2(20) [ψ1 , ψ2 ]P[ψ1 ]2(20) [ψ1 , ψ3 ]P[ψ1 ],

(3.162)

γ3,b−1 = Tr∂l Rk P[ψ1 ]2(20) [ψ1 , ψ2 ]P[ψ1 ]2(11) [ψ1 , ψ3 ] ×P[ψ3 ]2(11) [ψ3 , ψ1 ]P[ψ1 ],

(3.163)

γ3,b−2 = Tr∂l Rk P[ψ1 ]2(11) [ψ1 , ψ2 ]P[ψ2 ]2(20) [ψ2 , ψ3 ] ×P[ψ2 ]2(11) [ψ2 , ψ1 ]P[ψ1 ],

(3.164)

3.4 Appendix

75

γ3,c−1 = Tr∂l Rk P[ψ1 ]2(20) [ψ1 , ψ2 ]P[ψ1 ]2(20) [ψ1 , ψ3 ] ×P[ψ1 ]2(11) [ψ1 , ψ1 ]P[ψ1 ],

(3.165)

γ3,c−2 = Tr∂l Rk P[ψ1 ]2(20) [ψ1 , ψ2 ]P[ψ1 ]2(11) [ψ1 , ψ1 ] ×P[ψ1 ]2(20) [ψ1 , ψ3 ]P[ψ1 ],

(3.166)

Fig. 3.9 Graphical representation of the flow equation for 3 . Reprinted figure with permission from Ref. [22]. Copyright (2018) by the American Physical Society. https://doi.org/10.1103/PhysRevE. 98.032122

76

3 Nonperturbative Renormalization Group

γ3,d = Tr∂l Rk P[ψ1 ]2(11) [ψ1 , ψ2 ]P[ψ2 ]2(11) [ψ2 , ψ3 ] ×P[ψ3 ]2(11) [ψ3 , ψ1 ]P[ψ1 ], γ3,e =

1 Tr∂l Rk P[ψ1 ]3(200) [ψ1 , ψ2 , ψ3 ]P[ψ1 ], 2

(3.167)

(3.168)

γ3, f = Tr∂l Rk P[ψ1 ]2(20) [ψ1 , ψ2 ]P[ψ1 ]3(110) [ψ1 , ψ1 , ψ3 ]P[ψ1 ],

(3.169)

γ3,g = Tr∂l Rk P[ψ1 ]2(11) [ψ1 , ψ2 ]P[ψ2 ]3(110) [ψ2 , ψ1 , ψ3 ]P[ψ1 ],

(3.170)

γ3,h = Tr∂l Rk P[ψ1 ]2(11) [ψ1 , ψ1 ]P[ψ1 ]3(200) [ψ1 , ψ2 , ψ3 ]P[ψ1 ],

(3.171)

γ3,i =

1 Tr∂l Rk P[ψ1 ]4(1100) [ψ1 , ψ1 , ψ2 , ψ3 ]P[ψ1 ]. 2

(3.172)

Figure 3.9 shows the graphical representation of the flow equation for 3 .

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Chapter 4

Dimensional Reduction and its Breakdown in the Driven Random Field O(N) Model

Let us summarize the main findings for disordered systems in equilibrium: • The standard perturbation theory predicts that the critical exponents of disordered systems in D dimensions are identical to those of pure systems in D − 2 dimensions. • This dimensional reduction breaks down when the renormalized disorder cumulant develops a linear cusp as a function of the field. • In such cases, the critical behavior is described by a nonanalytic fixed point of the renormalized disorder cumulant. We now proceed to our project mentioned in Sect. 1.4, namely, we attempt to establish a novel type of dimensional reduction property which relates the nonequilibrium steady states of driven disordered systems to the equilibrium states of lower dimensional pure systems. This property implies that the critical exponents of the dynamical reordering transitions in driven disordered systems are the same as those of the equilibrium phase transitions in the corresponding pure systems. However, this dimensional reduction does not always hold. Therefore, we also attempt to clarify the condition that the dimensional reduction holds or fails by the functional renormalization group (FRG) analysis. The outline of this chapter is as follows: 1. One of the well-studied driven disordered systems is interacting many-particle system driven in a random pinning potential. However, this system is too complicated for theoretical analysis. Therefore, we introduce a toy model, the driven random field O(N ) model (DRFO(N )M), which is the random field O(N ) model (RFO(N )M) driven at a uniform and steady velocity. This is the simplest model that exhibits a dynamical reordering transition. 2. From an intuitive argument, we derive a dimensional reduction property which predicts that the critical exponents of the D-dimensional DRFO(N )M at zero © Springer Nature Singapore Pte Ltd. 2019 T. Haga, Renormalization Group Analysis of Nonequilibrium Phase Transitions in Driven Disordered Systems, Springer Theses, https://doi.org/10.1007/978-981-13-6171-5_4

79

80

4 Dimensional Reduction and its Breakdown …

temperature are identical to those of the (D − 1)-dimensional pure O(N ) model in equilibrium. We also explain the reason why this dimensional reduction can break down. 3. By employing the nonperturbative functional renormalization group (NP-FRG) approach, we investigate the critical behavior of the model near three dimensions. Especially, the critical exponents are calculated as functions of N in the first order of  = D − 3. 4. As in equilibrium, the breakdown of the dimensional reduction is caused by a nonanalytic behavior of the renormalized disorder cumulant. For the DRFO(N )M, we show that the dimensional reduction breaks down for 2 < N < 10 near three dimensions. The results of this chapter are reported in Ref. [1].

4.1 Driven Random Field O(N) Model We consider the driven random field O(N ) model (DRFO(N )M) defined in Sect. 1.4. Let φ(r) = (φ1 (r), . . . , φ N (r)) be an N -component real vector field. The Hamiltonian of the O(N ) model with a quenched random field h(r) is given by  H [φ; h] =

D

d r

 1 α

2

 K |∇φ | + U (ρ) − h · φ , α 2

(4.1)

where ρ = |φ|2 /2 is the field amplitude and U (ρ) = (λ0 /2)(ρ − ρ0 )2 is a doublewell potential. The random field obeys a mean-zero Gaussian distribution with h α (r)h β (r  ) = h 20 δ αβ δ(r − r  ),

(4.2)

The dynamics are described by (∂t φα + v∂x φα ) = −

δ H [φ; h] + ξα, δφα

(4.3)

where v denotes a uniform time-independent driving velocity, and ξ α (r, t) represents the thermal noise that satisfies ξ α (r, t)ξ β (r  , t  ) = 2T δ αβ δ(r − r  )δ(t − t  ).

(4.4)

In this study, we consider the case that N ≥ 2. Since we are interested in the nonequilibrium steady states of this model, in the following, . . . denotes the average over the distribution function of the steady state,

4.1 Driven Random Field O(N ) Model

81

 A[φ] =

DφA[φ]Pst [φ; h],

(4.5)

where Pst is the probability distribution function of the steady state for a given realization of the random field. The disorder average is given by  A[φ] =

DhA[φ]PR [h],

(4.6)

where PR is the distribution function of the random field. We next consider the lower critical dimension of the DRFO(N )M, and denote the transverse fluctuation of the order parameter field from a completely ordered state as φT (r). Its equation of motion at zero temperature is given by (∂t φT + v∂x φT ) = K ∇ 2 φT + h T ,

(4.7)

where h T is the transverse component of the random field. Therefore, if the renormalization of the random field is ignored, the disconnected Green’s function is given by h 20 ) T T (q) = φ (q)φ (−q) = , (4.8) G (T d K 2 |q|4 +  2 v 2 qx2 whose q-integral exhibits an infrared-divergence below three dimensions and the lower critical dimension is three. Recall that the lower critical dimension of the random field O(N ) model (RFO(N )M) is four. Therefore, at D = 3 + , the DRFO(N )M is always in a disordered phase for v = 0, while it exhibits a dynamical reordering transition to a long-range ordered (LRO) phase at sufficiently large v. In particular, at D = 3 and N = 2, one may expect the existence of a quasi-long-range ordered (QLRO) phase and the Kosterlitz-Thouless (KT) transition [2, 3] from the analogy of the two-dimensional (2D) pure XY model. In fact, it is suggested from the one-loop perturbative renormalization group analysis that elastic lattices driven in a random potential exhibit an anisotropic QLRO at weak disorder [4–6]. However, since the analysis for this case requires a more careful treatment than that given in this chapter, the detailed investigation is postponed to the next chapter.

4.2 Dimensional Reduction We now derive a novel type of dimensional reduction property for driven disordered systems. We consider the following general Hamiltonian: 

 H[φ] =

dDr

 1 |∇φ(r)|2 + U (ρ(r)) + V (r; φ(r)) , 2

(4.9)

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4 Dimensional Reduction and its Breakdown …

where U (ρ) is a nonrandom potential and V (r; φ) is a random potential. The random force is then defined by (4.10) F α (r; φ) = −∂φα V (r; φ). We assume that the random force obeys a mean-zero Gaussian distribution satisfying F α (r; φ)F β (r  ; φ ) = αβ (φ − φ )δ(r − r  ).

(4.11)

From the O(N ) rotational symmetry, αβ (φ) is written as αβ (φ) = 0 (ρ)δ αβ + 1 (ρ)φα φβ .

(4.12)

For the random manifold, we set U (ρ) = 0, for the random field O(N ) model, we set U (ρ) = (λ0 /2)(ρ − ρ0 )2 and αβ (φ) = h 20 δ αβ . The dynamics are described by Eq. (4.3). At zero temperature, the equation of motion is given by (∂t φα (r, t) + v∂x φα (r, t)) = ∇ 2 φα (r, t) − U  (ρ(r, t))φα (r, t) + F α (r; φ(r, t)). (4.13) The solution of Eq. (4.13) reaches a stationary state φst (r) after a sufficiently long time. The stationary state satisfies the following equation: v∂x φαst (r) = ∇ 2 φαst (r) − U  (ρst (r))φαst (r) + F α (r; φst (r)),

(4.14)

where ρst = |φst |2 /2. In the large length scale, the longitudinal elastic term ∂x2 φ is negligible compared to the advection term v∂x φ. By ignoring the former term, we have the following equation: v∂x φα (r) = ∇⊥2 φα (r) − U  (ρ(r))φα (r) + F α (x, r ⊥ ; φ(r)),

(4.15)

where ∇⊥ is the derivative operator for the transverse directions and r ⊥ is the transverse coordinate. Equation (4.15) is identical to the dynamical equation of the (D − 1)-dimensional pure system by considering the spatial coordinate x as a fictitious time and the random force F α (x, r ⊥ ; φ) as a thermal noise. Although Eq. (4.15) has infinitely many solutions because one can obtain a solution by propagating an arbitrary (D − 1)-dimensional “initial” configuration φ(x = 0, r ⊥ ) along x-direction, there exist a solution φ∗ (x, r ⊥ ) such that its large-scale behavior is the same as that of φst (r). It is worth to note that the correlation of F α (x, r ⊥ ; φ) for different field values is irrelevant because, at a specific coordinate (x, r ⊥ ), the field φ feels the random force only once in the forward propagation along x-direction. In other words, F α (x, r ⊥ ; φ) can be replaced with a field-independent random force f α (x, r ⊥ ), whose cumulant is given by Eq. (4.11) evaluated at the same field φ = φ : f α (x, r ⊥ ) f β (x  , r ⊥ ) = 0 (0)δ αβ δ(x − x  )δ(r ⊥ − r ⊥ ).

(4.16)

4.2 Dimensional Reduction

83

Therefore, we can conclude that the transverse section of the D-dimensional driven disordered system at zero temperature is identical to the corresponding (D − 1)dimensional pure system in equilibrium with a temperature Teff =

0 (0) . 2v

(4.17)

Unfortunately, this dimensional reduction is not always correct. For example, let us consider the case of the driven random field Ising model (DRFIM), which is defined by Eqs. (4.1) and (4.3) with N = 1. In equilibrium, the lower critical dimension of the random field Ising model is two (see Sect. 1.2.2). Since the advection term v∂x φ reduces the lower critical dimension, we expect that the 2D-DRFIM exhibits LRO at weak disorder. However, the dimensional reduction predicts that it does not because it is identical to the one-dimensional pure Ising model. The existence of LRO in the 2D-DRFIM can be also understood from the following argument. When we go to the moving frame, r  = r − vt and φ(r, t) = φ (r  , t), the advection term in Eq. (4.3) vanishes and the random field moves with a velocity −v, ∂t φ (r  , t) = ∇ 2 φ (r  , t) − U  (ρ (r  , t))φ (r  , t) + h(r  + vt).

(4.18)

√ We consider the fluctuations around the completely ordered state φ (r) = 2ρ0 . Due to the presence of a nonzero mass U  (ρ0 ) = λ0 , each spin feels the moving random −1/2 force in a finite range λ0 . This implies that the moving random force has the same effect as the thermal noise. Therefore, the 2D-DRFIM can exhibit LRO at weak disorder. Note that, when the model has massless modes, the moving random force is no longer considered as an effective thermal noise. As mentioned above, the lower critical dimension of the DRFO(N )M with N ≥ 2 is three, not two as the pure O(N ) model. The mechanism of the breakdown of the nonequilibrium dimensional reduction (χ) resembles that of the equilibrium case discussed in Sect. 1.2.3. Let {φst (r)}χ=1,...,N be the set of all stationary states that satisfy Eq. (4.14), where N is the number of the stationary states. In general, N increases exponentially with the system size. As in Eq. (1.40), we define a correlation function Cuni (r) by averaging over all stationary states with a equal weight,  Cuni (r) =

DV PR [V ]

N 1  (χ) (χ) sign(χ)φst (r) · φst (0), N χ=1

(4.19)

where “sign(χ)” represents the sign of the Jacobian det(v∂x + δ 2 H/δφδφ) evaluated (χ) at φ = φst and PR [V ] is the probability distribution function of the random potential V (r; φ). In the similar way as described in Sect. 1.2.3, Eq. (4.14) can be cast into a field theoretical action and one finds that Cuni (x, r ⊥ ) is equivalent to the dynamical correlation function for the equilibrium state of the (D − 1)-dimensional pure model

84

4 Dimensional Reduction and its Breakdown …

by considering the coordinate x as the time t. However, it is nontrivial whether Cuni (r) is identical to the actual correlation function at zero temperature. It is worth to note how the correlation function C(r) should be defined at zero temperature. In equilibrium, C(r) is defined by using the ground state. However, for the nonequilibrium cases, one cannot define the ground state because the advection term v∂x φ in Eq. (4.3) cannot be cast into the functional derivative of any potential. In the following, we consider two definitions of C(r). In the first definition, we choose an initial condition φi (r) according to the probability distribution function 2 Pi [φ; μ] ∼ e−(μ/2)|φ| and solve Eq. (4.13) to obtain a stationary state φst (r), which is one of the solutions of Eq. (4.14). Note that φst (r) depends on the initial condition φi (r). We then define C1 (r) by  C1 (r) = lim

μ→0

DV PR [V ]φst (r) · φst (0)i ,

(4.20)

where . . .i denotes the average over the probability distribution function of the initial condition Pi [φ; μ]. In the second definition, we go to the moving frame, r  = r − vt and φ(r, t) = φ (r  , t). The equation of motion is given by Eq. (4.18). We consider free boundary conditions, not periodic boundary conditions, to prevent the system from feeling the same disorder periodically. The new random force is generated in the front boundary of the system and moves with the velocity −v. Let φ(r, t) be a solution of Eq. (4.18) with the above boundary conditions. We then define C2 (r) by 1 C2 (r) = lim τ →∞ τ

τ φ(r, t) · φ(0, t)dt.

(4.21)

0

Although it is not trivial whether C1 (r) and C2 (r) are identical in the thermodynamic limit, we assume that it is true in this study. This ambiguity in the definition of the correlation function at zero temperature should be investigated carefully in future works.

4.3 NP-FRG Formalism We employ the NP-FRG approach, which is reviewed in Sect. 3.3, to investigate the large-scale behavior of the DRFO(N )M. We extend this formalism to driven disordered systems and derive the flow equation of the renormalized disorder cumulant in the first order of  = D − 3.

4.3 NP-FRG Formalism

85

4.3.1 Scale-Dependent Effective Action Our starting point is the field-theoretical representation of Eq. (4.3). We introduce n , with the same disorder. The dynamics are given by an n replicated system, {φa }a=1 (∂t φa + v∂x φa ) = K ∇ 2 φa − U  (ρa )φa + h + ξa ,

(4.22)

where the thermal noise satisfies β

ξaα (r, t)ξb (r  , t  ) = 2T δ αβ δab δ(r − r  )δ(t − t  ).

(4.23)

In the following, a superscript with a Greek alphabet letter represents the index of the field component α, β = 1, . . . , N and a subscript with a Roman alphabet letter represents the replica index a, b = 1, . . . , n. The average of a function of the field A[{φa }] over the thermal noise is written as  A[{φa }] =

 Dξ P[ξ]

Dφδ(φa − φa [ξ])A[{φa }],

(4.24)

where φa [ξ] is the solution of Eq. (4.22) for a realization of the noise ξa . According to the technique presented in Sect. 3.1.2, this average can be calculated as  A[{φa }] =

  ˆ ˆa DφDφJ [{φa }]A[{φa }] exp − iφ a rt

 ˆ a ) − K ∇ 2 φa + U  (ρa )φa − h} , (4.25) ·{(∂t φa + v∂x φa − T i φ where J [{φa }] is the Jacobian, which is set to unity [7]. We also take the average over the disorder,  A[{φa }] = DA[{φa }] exp(−S[{a }]), (4.26) ˆ a ) and the disorder averaged action S[{a }] is given by where a = t (φa , φ S[{a }] =



S1 [a ] −

a

S1 [] =

1 S2 [a , b ], 2 a,b

(4.27)

   ˆ · (∂t φ + v∂x φ − T φ) ˆ +φ ˆ · {−K ∇ 2 φ + U  (ρ)φ} , (4.28) φ rt

 S2 [a , b ] = r tt 

ˆ b,r t  . ˆ a,r t · φ h 20 φ

(4.29)

86

4 Dimensional Reduction and its Breakdown …

ˆ a. To simplify the notation, we have omitted i in i φ t ˆ By introducing source fields Ja = ( j a , j a ), the partition function and generating functional are defined as     t (4.30) Ja a , Z [{Ja }] = D exp −S[{a }] + a rt

W [{Ja }] = ln Z [{Ja }].

(4.31)

The effective action is given by a Legendre transformation of W [{Ja }], [{a }] = −W [{Ja }] +



t

Ja a ,

(4.32)

a rt

ˆ a ) and Ja are related by where a = t (ψ a , ψ a =

δW [{Ja }] . δ Ja

The definition of the scale-dependent effective action k [{a }] is straightforward according to the conventional formulation in Chap. 3. To suppress the contribution from the low-energy modes, a mass-like quadratic term is added to the bare action, 1 Sk [{a }] = 2 a

 t

a (q) Rk (q) a (−q),

(4.33)

q

where we have used the notation q = (q, ω) and



=



d D qdω/(2π) D+1 . A

q

frequency-independent 2N × 2N matrix Rk (q) is given by Rk (q) = Rk (q)

01 10

⊗ IN ,

(4.34)

where I N is the N × N unit matrix, which acts on the field component index. Rk (q) is a cutoff function, which has a constant value proportional to k 2 for |q|  k and rapidly decreases for |q| > k. The explicit form of Rk (q) will be given later. The partition function and generating functional with the running scale k read  Z k [{Ja }] =

  D exp −S[{a }] − Sk [{a }] +

t

 Ja a ,

(4.35)

a rt

Wk [{Ja }] = ln Z k [{Ja }].

(4.36)

We define the scale-dependent effective action through a Legendre transformation,

4.3 NP-FRG Formalism

87

k [{a }] = −Wk [{Ja }] +



t

Ja a − Sk [{a }],

(4.37)

a rt

where a and Ja are related by a =

δWk [{Ja }] . δ Ja

4.3.2 Exact Flow Equation for the Effective Action The exact evolution of k is described by ∂k  k =

 −1 1 ˆ k (q)  (2) + R ˆ k (q) Tr∂k R , k q,−q 2

(4.38)

where k(2) is the second functional derivative, μν

(k(2) )ab (q, q  ) =

δ 2 k , μ δa (q)δbν (q  )

(4.39)

and Tr represents an integration over momentum and frequency as well as a sum over ˆ the indices of the replica, the field component, and the two conjugate fields {ψ, ψ}. We have introduced a 2n N × 2n N matrix ˆ k (q) = Rk (q) ⊗ In , R

(4.40)

ˆ k (q) where In is the n × n unit matrix, which acts on the replica index. Note that R implicitly contains the delta function of momentum and frequency. According to Eq. (3.92), k is expanded by increasing the number of free replica sums, k [{a }] =

 a

+

1,k [a ] −

1 2,k [a , b ] 2 a,b

1  3,k [a , b , c ] − · · · . 3! a,b,c

(4.41)

We also define the one-replica propagator with the infrared cutoff,  (2) −1 [] + Rk (q) . Pk [] = 1,k The exact flow equations for 1,k [] and 2,k [1 , 2 ] read

(4.42)

88

4 Dimensional Reduction and its Breakdown …

∂k 1,k [] =

1 tr 2



  (11) ∂k Rk (q) Pk [] + Pk []2,k [, ]Pk []

q,−q

, (4.43)

q

1 ∂k 2,k [1 , 2 ] = − tr 2





(20) (110) ∂k Rk (q) Pk [1 ] 2,k [1 , 2 ] − 3,k [1 , 1 , 2 ]

q (11) (11) +2,k [1 , 2 ]Pk [2 ]2,k [2 , 1 ] (20) (11) [1 , 2 ]Pk [1 ]2,k [1 , 1 ] +2,k

 (11) (20) +2,k [1 , 1 ]Pk [1 ]2,k [1 , 2 ] Pk [1 ]  +perm(1 , 2 ) , q,−q

(4.44)

where tr in Eqs. (4.43) and (4.44) represents the sum over the indices of the field ˆ The derivation of these is completely component and the two conjugate fields {ψ, ψ}. the same as that of Eqs. (3.107) and (3.108).

4.3.3 Derivative Expansion To solve Eqs. (4.43) and (4.44), we introduce an approximation for the functional form of  p,k . For the one-replica part 1,k , we include the first order of the derivative expansion and employ the following functional form: 1,k [] =

 

ˆ · (∂t ψ + vk ∂x ψ − Tk ψ) ˆ Xkψ

rt

 ˆ · {−Z ,k ∂ 2 ψ − Z ⊥,k ∇ 2 ψ − Fk (ρ)ψ} , +ψ x ⊥

(4.45)

where two field renormalization factors Z ,k and Z ⊥,k are introduced because the system is anisotropic due to the driving. Although, in the equilibrium case (v = 0), a renormalized local force Fk (ρ) can be written as the derivative of a potential Fk (ρ) = −Uk (ρ), it cannot in the nonequilibrium case. ˆ · ψ∇ 2 ρ and From the rotational symmetry, the higher-order terms such as ψ ˆ · ∇ 4 ψ can be generated along the RG flow. At D = 3 + , the critical expoψ nents have values of O(), but the contributions from these terms are O(2 ). This ˆ · ψ∇ 2 ρ modifies can be understood from the following argument. The first term ψ the propagator of the longitudinal (massive) mode. However, as we will explain in Sect. 4.3.5, the contribution from the longitudinal mode is negligible in the first order ˆ · ∇ 4 ψ yields a term proportional to |q|4 in  (2) (q). of  = D − 3. The second one ψ 1,k (2) At a critical point, 1,k (q) is expected to behave as |q|2 (|q|2 + ck 2 )−η/2 near q = 0, where η is the anomalous dimension [8]. By expanding this expression around q = 0, one finds that the coefficients of the higher-order terms O(|q|4 ) are proportional to

4.3 NP-FRG Formalism

89

η. As we will show in Sect. 4.3.5, η has a value of O(). Therefore, the contribution ˆ · ∇ 2 ψ. ˆ · ∇ 4 ψ is sub-leading compared to that from the term ψ from the term ψ The two-replica part 2,k is given by  2,k [1 , 2 ] =

μ μν ν ψˆ 1,r t ψˆ 2,r t  k (ψ 1,r t , ψ 2,r t  ),

(4.46)

r tt  μν

where k (ψ 1 , ψ 2 ) is the second cumulant of the renormalized random field. From μν the rotational symmetry, k (ψ 1 , ψ 2 ) can be rewritten as μν

k (ψ 1 , ψ 2 ) = 00,k (ρ1 , ρ2 , z)δ μν

μ

μ

+(4ρ1 ρ2 )−1/2 [12,k (ρ1 , ρ2 , z)ψ1 ψ2ν + 21,k (ρ1 , ρ2 , z)ψ2 ψ1ν μ μ +11,k (ρ1 , ρ2 , z)ψ1 ψ1ν + 22,k (ρ1 , ρ2 , z)ψ2 ψ2ν ], (4.47)

√ where z = ψ 1 · ψ 2 / 4ρ1 ρ2 is the cosine of the angle between ψ 1 and ψ 2 . The μν bare cumulant is given by k= (ψ 1 , ψ 2 ) = h 20 δ μν . In the equilibrium case, it can be written as the derivative of the cumulant of the random potential, μν

k (ψ 1 , ψ 2 ) = ∂ψ1μ ∂ψ2ν Vk (ψ 1 , ψ 2 ).

(4.48)

This cannot be done for the nonequilibrium case. For simplicity, we ignore the higher order cumulants, 3,k = 4,k = · · · = 0. Let us consider the symmetry of the effective action in the absence of the disorder or driving. In the equilibrium case (v = 0), the bare action Eqs. (4.28) and (4.29) are invariant under a time-reversal transformation [7], ⎧ ⎪ ⎨t → −t, μ μ ψa → ψa , ⎪ ⎩ ˆμ μ μ ψa → ψˆa − (1/T )∂t ψa .

(4.49)

The effective action Eqs. (4.45) and (4.46) should be invariant under the same transformation, thus, in equilibrium we can conclude that the temperature is not renormalized, Tk = T . In contrast, in the presence of a finite velocity v = 0, Eq. (4.28) is not invariant under Eq. (4.49) because the advection term v∂x φ cannot be expressed as a functional derivative of a potential. Therefore, in the nonequilibrium case, the temperature can be renormalized, Tk = T . Next, we consider the zero random field case h 0 = 0 with a finite driving velocity, and Eq. (4.28) is invariant under a space-time-reversal transformation, ⎧ t → −t, ⎪ ⎪ ⎪ ⎨x → −x, ⎪ψaμ → ψaμ , ⎪ ⎪ ⎩ ˆμ μ μ μ ψa → ψˆa − (1/T )∂t ψa − (v/T )∂x ψa .

(4.50)

90

4 Dimensional Reduction and its Breakdown …

This implies that the driving velocity is not renormalized, vk = v, which can be easily understood by noting that, in the absence of the random field, the driven system can be mapped into the corresponding equilibrium system through the Galilei transformation. However, in the presence of the random field, Eq. (4.29) is not invariant under Eq. (4.50). Thus, the driving velocity can be renormalized, vk = v. From Eq. (4.45), the renormalized parameters are given by the following functional derivatives:  X k = ∂iω  (2)ˆ 2 2 (p, ω)p,ω=0 ,

(4.51)

 X k vk = ∂(−i px )  (2)ˆ 2 2 (p, ω)p,ω=0 ,

(4.52)

 2X k Tk = − (2)ˆ 2 ˆ 2 (p, ω)p,ω=0 ,

(4.53)

 Z ,k = ∂ px2  (2)ˆ 2 2 (p, px vk )p=0 , 1;ψ ψ  Z ⊥,k = ∂ p⊥2  (2)ˆ 2 2 (p, px vk )p=0 ,

(4.54)

1;ψ ψ

1;ψ ψ

1;ψ ψ

1;ψ ψ



 2ρFk (ρ) = − (1)ˆ 1 (p, ω)p,ω=0 , 1;ψ

(4.55)

where we have used the following notation: δ1,k [] , δ ψˆ μ (p, ω) δ 2 1,k []  (2)ˆ μ ν (p, ω) = . 1;ψ ψ δ ψˆ μ (p, ω)δψ ν (−p, −ω)  (1)ˆ μ (p, ω) = 1;ψ

The functional derivatives in Eqs. (4.51)–(4.55) are evaluated for a uniform field √ ˆ r t ≡ t (0, . . . , 0). Especially, for X k , Tk , vk , and Z k , the ψr t ≡ t ( 2ρ, 0, . . . , 0), ψ field amplitude ρ is set to ρm,k satisfying Fk (ρm,k ) = 0,

(4.56)

where we call ρm,k the renormalized spontaneous magnetization. In Eq. (4.54), we (2) have defined the field renormalization factors Z k as the momentum derivatives of 1,k evaluated at ω = px vk , and not at ω = 0 as in Eq. (4.51). Within the approximation Eq. (4.45), Z k should be almost independent of the momentum and frequency at which the derivatives are evaluated. However, in some cases, the contribution of other terms that do not appear in Eq. (4.45) might not be negligible. Thus, there is an ambiguity in the choice of the momentum and frequency in the definition of X k and Z k . In Eq. (4.54), we have chosen the frequency ω = px vk such that in the absence of the disorder the RG equation for Z k is identical to that of the

4.3 NP-FRG Formalism

91

corresponding equilibrium model. To derive the RG equations for X k , Tk , vk , Z k and (1) (2) and 1,k , which is obtained from Fk (ρ), we need the exact flow equations for 1,k the functional derivative of Eq. (4.43) (see Sects. 4.6.2 and 4.6.4). The renormalized cumulant in Eq. (4.46) is given by  μν  k (ψ 1 , ψ 2 ) =  (11) ˆ μ ˆ ν (p, ω) p,ω=0 , 2;ψ1 ψ2

(4.57)

where we have used the following notation:  (11) ˆ μ ˆ ν (p, ω) = 2;ψ1 ψ2

δ 2 2,k [1 , 2 ] . μ δ ψˆ 1 (p, ω)δ ψˆ2ν (−p, −ω)

The functional derivative in Eq. (4.57) is evaluated for a uniform field ψ 1,r t ≡ ˆ 1,r t ≡ 0 and ψ 2,r t ≡ ψ 2 , ψ ˆ 2,r t ≡ 0. The exact flow equation for  (11) , which ψ1 , ψ 2,k is obtained from the functional derivative of Eq. (4.44), yields the RG equation for μν k (ψ 1 , ψ 2 ) (see Sect. 4.6.3).

4.3.4 Dimensionless Quantities To obtain a fixed point, the RG equations should be expressed in terms of renormalized dimensionless quantities. For comparison between the equilibrium and nonequilibrium cases, we define the dimensionless quantities for both cases below. In the following, the cutoff  is set to unity. Equilibrium case For the equilibrium case (v = 0), we employ the definition introduced in Ref. [9]. Since the momentum q is measured in units of k, we introduce the dimensionless momentum, |q|2 y= 2 . (4.58) k The dimensionless quantities, which are denoted with a tilde, are defined as follows: ρ = Z k−1 k D−2 τk−1 ρ, ˜

(4.59)

Fk (ρ) = Z k k 2 F˜k (ρ), ˜

(4.60)

˜ k (ρ˜1 , ρ˜2 , z), k (ρ1 , ρ2 , z) = Z k k 2 τk−1 

(4.61)

˜ k attains a fixed point. To where Z k = Z ,k = Z ⊥,k , and τk is chosen such that  define τk , we introduce the renormalized disorder strength by m,k = 00,k (ρm,k , ρm,k , z = 1),

(4.62)

92

4 Dimensional Reduction and its Breakdown …

where ρm,k is given by Eq. (4.56). Then, τk is defined by τk =

(Z k /Z  )k 2 . m,k /m,

(4.63)

˜ 00,k (ρm,k , ρm,k , z = 1) is independent of the RG scale k. We also From Eq. (4.61),  define the running exponent associated with τk as θk = k∂k ln τk = 2 − ηk − k∂k ln m,k ,

(4.64)

where ηk is the running anomalous dimension ηk = −k∂k ln Z k .

(4.65)

It is worth to remark that τk may be considered as an effective temperature. In the static formulation, the replicated Hamiltonian with temperature τ is given by H [{φa }] =

    1 1 1  d D r K |∇φa |2 + U (ρa ) − 2 d D rh 20 φa · φb . τ a 2 2τ a,b

One finds that the ratio of the kinetic term |∇φa |2 to the disorder term h 20 φa · φb is proportional to the temperature. Since the kinetic term scales as Z k k 2 , Eq. (4.63) can be considered as an effective temperature. Note that τk should not be confused with Tk in Eq. (4.45), which is the strength of the renormalized thermal noise. In equilibrium, τk ∼ k θ and θ  2 near D = 4, while Tk is a constant along the RG flow as shown in Eq. (4.49). In the RG equations, Tk and τk always appear in the product Tk τk and its flow controls the critical behavior of the system. At a critical point, the connected and disconnected Green’s functions exhibit power-law behavior, G c (r − r  ) =

δφ(r) ∼ |r − r  |−(D−2+η) , δh(r  )

G d (r − r  ) = φ(r)φ(r  ) ∼ |r − r  |−(D−4+¯η) ,

(4.66) (4.67)

where η is the fixed point value of ηk and η¯ is a new exponent. We have defined the connected Green’s function as the response of the order parameter field with respect to the external field. From the fluctuation-dissipation theorem, it is proportional to the conventional definition φ(r)φ(r  ) − φ(r)φ(r  ). We now derive a relation between η, θ, and η. ¯ The renormalized spontaneous magnetization ρm,k , which is defined in Eq. (4.56), can be considered as the amplitude of the order parameter field averaged over a region V of linear dimension k −1 . Therefore, at the critical point, it behaves as

4.3 NP-FRG Formalism

93

 ρm,k ∼ k 2D

d D rd D r  G d (r − r  ) ∼ k D−4+¯η ,

V

where we have used Eq. (4.67). On the other hand, from Eq. (4.59), ρm,k scales as k D−2+η−θ at the fixed point. Thus, if we define η¯k = 2 + ηk − θk ,

(4.68)

η¯ is the fixed point value of η¯k . The cutoff function Rk (q) is written as Rk (q) = Z k k 2 r˜ (y),

(4.69)

where y = |q|2 /k 2 . In the following calculations, we employ the optimized cutoff function (3.54), r˜ (y) = (1 − y)(1 − y),

(4.70)

where (x) is the step function. Nonequilibrium case For the nonequilibrium case (v = 0), considering the anisotropy due to the driving, the transverse momentum q⊥ and longitudinal momentum qx are measured in units of k and Q k (defined below), respectively. Thus, we introduce the dimensionless momentum, |q⊥ |2 qx2 y⊥ = , y = . (4.71)  k2 Q 2k The dimensionless quantities, which are denoted with a tilde, are defined as follows: −1 D−3 k Q k τk−1 ρ, ˜ ρ = Z ⊥,k

(4.72)

˜ Fk (ρ) = Z ⊥,k k 2 F˜k (ρ),

(4.73)

˜ k (ρ˜1 , ρ˜2 , z). k (ρ1 , ρ2 , z) = Z ⊥,k k 2 τk−1 

(4.74)

The temperature scaling exponent θk is defined by Eq. (4.64) with η⊥,k = −k∂k ln Z ⊥,k .

(4.75)

We also introduce the dimensionless velocity v˜k and dimensionless longitudinal elastic constant z˜ ,k as X k vk Q k , (4.76) v˜k = Z ⊥,k k 2

94

4 Dimensional Reduction and its Breakdown …

z˜ ,k =

Z ,k Q 2k . Z ⊥,k k 2

(4.77)

Q k should be chosen such that v˜k attains a fixed point. Thus, we define Qk =

(Z ⊥,k /Z ⊥, )k 2 . X k vk /(X  v )

(4.78)

We also introduce the running exponent associated with Q k as ζk = k∂k ln Q k = 2 − η⊥,k − k∂k ln(X k vk ).

(4.79)

Since v˜k is constant along the RG flow, we omit the subscript k below. At a critical point, the connected and disconnected Green’s functions for the transverse direction exhibit power-law behavior, G c (r ⊥ ) ∼ |r ⊥ |−(D−2+η⊥ ) ,

(4.80)

G d (r ⊥ ) ∼ |r ⊥ |−(D−3+¯η⊥ ) ,

(4.81)

where η⊥ is the fixed point value of η⊥,k and η¯⊥ is a new exponent. For the longitudinal direction, they behave as −(D−2+η⊥ )/ζ G c (r ) ∼ r , (4.82) −(D−3+¯η⊥ )/ζ

G d (r ) ∼ r

,

(4.83)

where ζ is the fixed point value of ζk . Although the connected Green’s function (response function) is no longer equivalent to φ(r)φ(r  ) − φ(r)φ(r  ) due to the breakdown of the fluctuation-dissipation theorem, their asymptotic behaviors in the long distance are expected to be same. We now derive a relation between η⊥ , θ, ζ, and η¯⊥ . The renormalized spontaneous magnetization ρm,k can be considered as the amplitude of the order parameter field averaged over a region V of linear dimension k −1 for the perpendicular direction and Q −1 k for the parallel direction. Therefore, at the critical point, it behaves as  ρm,k ∼ k

2(D−1)

Q 2k

d D rd D r  G d (r − r  ) ∼ k D−3+¯η⊥ ,

V

where we have used Eqs. (4.81) and (4.83). On the other hand, from Eq. (4.72), ρm,k scales as k D−3+ζ+η⊥ −θ . Thus, if we define η¯⊥,k = ζk + η⊥,k − θk , η¯⊥ is the fixed point value of η¯⊥,k .

(4.84)

4.3 NP-FRG Formalism

95

We assume that Rk (q) = Rk (qx2 , |q⊥ |2 ) is independent of the longitudinal momentum qx , Rk (qx2 , |q⊥ |2 ) = Rk (|q⊥ |2 ) = Z ⊥,k k 2 r˜ (y⊥ ),

(4.85)

where r˜ (y) is given by Eq. (4.70).

4.3.5 RG Equations Near the Lower Critical Dimensions Let us derive the flow equations at D = 4 +  for the equilibrium case and at D = 3 +  for the nonequilibrium case, in the first order of . In the case of the pure O(N ) model at D = 2 + , the fixed point associated with the critical point has a value of ρ˜m that diverges as 1/. Therefore, one can organize a systematic expansion in powers of 1/ρ˜m . The first order of this expansion is considered below. The transverse and longitudinal components of the one-replica propagator are denoted as P (T ) (q) and P (L) (q), respectively, which are given in Sect. 4.6.1. If we assume that F˜k (ρ˜m ) has a value of O(1) at the fixed point, P (T ) (q) = O(1) and P (L) (q) ∼ ρ˜−1 . Therefore, retaining the first order of the expansion of 1/ρ˜m means that the contribution from the longitudinal mode is ignored. The RG flow of the one-replica force Fk (ρ) is given by Eqs. (4.162), (4.163), and (4.164) in Sect. 4.6.2. We introduce a scale parameter l = − ln(k/), which goes from 0 to ∞ as k decreases from  to 0. The flow equation for ρm can be derived from Fk (ρm )∂l ρm + ∂l Fk (ρ)|ρ=ρm = 0, where ∂l = −k∂k . In the leading order of ρ−1 m , the flow equation for ρm is given by   Tk (T ) (T ) L 2 (ρm ) + m I12 (ρm ) , ∂l ρm = −(N − 1) 2

(4.86)

where the integrals L and I are defined by Eqs. (4.156) and (4.157) in Sect. 4.6.1, (T ) (T ) (ρm ) = I12 (ρm , ρm ). The renorrespectively. We have used notations such as I12 malized disorder strength m is defined by Eq. (4.62). Next, we consider the RG flow of the cumulant of the random field μν (ψ 1 , ψ 2 ), which can be rewritten as Eq. (4.47). The details of the derivation of the flow equation are given in Sect. 4.6.3. It is worth noting that in the first order of ρ−1 m , the flow equations for 00 and 21 do not contain 12 , 11 , or 22 . Since the running anomalous dimensions ηk and η¯k depend on only 00 as shown later, it is sufficient to consider the flow of 00 and 21 . The flow equations for these are given by ∂l 00 (ρ1 , ρ2 , z) =

Tk ) (T ) A00 (ρ1 , ρ2 , z)L (T 2 (ρ1 ) + A00 (ρ1 , ρ2 , z)00 (ρ1 )I21 (ρ1 , ρ1 ) 2 (T ) (T ) (ρ1 , ρ2 ) + C00 (ρ1 , ρ2 , z)J21 (ρ1 , ρ2 ) +B00 (ρ1 , ρ2 , z)I21 +perm(ρ1 , ρ2 ), (4.87)

96

4 Dimensional Reduction and its Breakdown …

∂l 21 (ρ1 , ρ2 , z) =

Tk ) (T ) A21 (ρ1 , ρ2 , z)L (T 2 (ρ1 ) + A21 (ρ1 , ρ2 , z)00 (ρ1 )I21 (ρ1 , ρ1 ) 2 (T ) (T ) (ρ1 , ρ2 ) + C21 (ρ1 , ρ2 , z)J21 (ρ1 , ρ2 ) +B21 (ρ1 , ρ2 , z)I21 +perm(ρ1 , ρ2 ),

(4.88)

where the integral J is defined by Eq. (4.158) and 00 (ρ1 ) = 00 (ρ1 , ρ1 , z = 1). A, B, C in Eqs. (4.87) and (4.88) are given as follows: A00 = (2ρ1 )−1 [(N − 1)(2ρ1 ∂ρ1 − z∂z )00 + (1 − z 2 )∂z2 00 − 2z21 ], B00 = (4ρ1 ρ2 )−1/2 [(N − 2 + z 2 )00 ∂z 00 + z200 + (1 + z 2 )21 00 −z(1 − z 2 )(21 ∂z 00 + 00 ∂z2 00 ) + (1 − z 2 )2 21 ∂z2 00 ], C00 = −2(4ρ1 ρ2 )−1/2 (00 + z21 )(1 − z 2 )∂z 00 , A21 = (2ρ1 )−1 [(N − 1)(2ρ1 ∂ρ1 21 − 21 − z∂z 21 ) +(1 − z 2 )∂z2 21 − 4z∂z 21 − 2∂z 00 ], B21 = (4ρ1 ρ2 )−1/2 [(N − 2 + z 2 )00 ∂z 21 + 221 (z00 + z 2 21 ) −z(1 − z 2 )(21 ∂z 21 + 00 ∂z2 21 ) + (1 − z 2 )2 21 ∂z2 21 −2(∂z 00 + 21 + 2z∂z 21 ){(1 − z 2 )21 − z00 }], C21 = (4ρ1 ρ2 )−1/2 [(N − 2 + z 2 )(21 )2 + 2z 2 21 ∂z 00 + z 2 (∂z 00 )2 −2z(1 − z 2 )(221 ∂z 21 + ∂z 00 ∂z 21 ) − 2(1 − z 2 )00 ∂z 21 +(1 − z 2 )2 (∂z 21 )2 + (00 + z21 )(00 + 3z21 )]. Let us consider the RG flows of X k vk , Tk , and Z k . The flow equations for Z k and X k vk can be obtained from the momentum derivatives of ∂l 1(2) , which are given in Sect. 4.6.4. The flow equation of Z k is written as (1) (2) + ∂l Z ⊥ , ∂l Z ⊥ = ∂l Z ⊥

(4.89)

where the first and second terms represent the contributions from the one and tworeplica parts, respectively. They are written as

4.3 NP-FRG Formalism

97

  T ∂l Rk (q)M0 (q)−2 2ρm q   2 |q⊥ |2 Rk (q) 2 Z ⊥ + Rk (q) + D−1  4 |q⊥ |2 [Z ⊥ + Rk (q)]2 M0 (q)−1 , − D−1

(1) ∂l Z ⊥ = −

(2) ∂l Z ⊥

m = − 2ρm

 ∂l Rk (q)D0 (q)

−2



q



4M0 (q) Z ⊥ +

Rk (q)

(4.90)

2 |q⊥ |2 Rk (q) + D−1



 4 2  2 2 −1 − |q⊥ | [Z ⊥ + Rk (q)] [4M0 (q) D0 (q) − 1] , D−1

(4.91)

where M(q) and D(q) = D(q, ω = 0) are defined in Sect. 4.6.1, and Rk (q) = ∂|q⊥ |2 Rk (|q⊥ |2 ). The flow equation of X k vk is written as m ∂l (X v) = − Xv 2ρm



∂l Rk (q) 2M0 (q)D0 (q)−2 ,

(4.92)

q

where note that it does not contain the contribution from the temperature. The running critical exponents η⊥,k and ζk can be calculated from these equations. For the equilibrium case, the flow equations can be obtained by the replacements Z ⊥ → Z , |q⊥ |2 → |q|2 , (D − 1) → D, and Rk (q) → ∂|q|2 Rk (|q|2 ). The flow equation for Tk is given by ∂l T = −

T 2ρm



∂l Rk (q)[(N − 2)(∂z 00 − 21 )2M0 (q)D0 (q)−2

q

+(00 + ∂z 00 + N 21 )8qx2 v 2 X 2 M0 (q)D0 (q)−3 ],

(4.93)

where all (ρ1 , ρ2 , z) are evaluated at ρ1 = ρ2 = ρm and z = 1. The detailed derivation of Eq. (4.93) is presented in Sect. 4.6.4. Equilibrium case For the equilibrium case (v = 0), the flow equations of the dimensionless quantities introduced in Sect. 4.3.4 can be derived. The integrals in Eqs. (4.87) and (4.88) at ρ1 = ρ2 = ρm are given by (T ) (T ) (ρm , ρm ) = J21 (ρm , ρm ) = Z k−2 k D−4 I21

8 AD, D

(4.94)

98

4 Dimensional Reduction and its Breakdown …

where A D −1 = 2 D+1 π D/2 (D/2). We define δ00 (z) and δ21 (z) by ˜ 00 (ρ˜m , ρ˜m , z)  16 AD , D 2ρ˜m ˜ 21 (ρ˜m , ρ˜m , z) 16  δ21 (z) = AD . D 2ρ˜m

δ00 (z) =

(4.95)

We also introduce a renormalized temperature, 8 T˜k = A D (2ρ˜m )−1 Tk τk . D

(4.96)

From Eqs. (4.59) and (4.86), the flow equation for ρ˜m is given by ∂l ρ˜m = (D − 4 + η¯k )ρ˜m − (N − 1)[δ00 (1) + T˜k ]ρ˜m ,

(4.97)

where η¯k is determined from the condition that 2ρ˜m δ00 (1) is constant along the RG flow. From Eq. (4.89), the anomalous dimension ηk is calculated as ηk = δ00 (1) + T˜k .

(4.98)

From Eqs. (4.87) and (4.88), it can be shown that δ00 (z) and δ21 (z) are related by δ21 (z) = ∂z δ00 (z),

(4.99)

which is easily understood from the fact that 00 (z) and 21 (z) can be written as 00 = √

1 1 ∂z V, 21 = √ ∂ 2 V, 4ρ1 ρ2 4ρ1 ρ2 z

(4.100)

where V (ρ1 , ρ2 , z) is the cumulant of the random potential (4.48). By introducing a potential R(z) by δ00 (z) = ∂z R(z) and δ21 (z) = ∂z2 R(z), the flow equation for R(z) at D = 4 +  is given by ∂l R(z) = −R(z) + (R  (1) + T˜k )[2(N − 2)R(z) − (N − 1)z R  (z) 1 +(1 − z 2 )R  (z)] + (N − 2 + z 2 )R  (z)2 2 1 −z(1 − z 2 )R  (z)R  (z) + (1 − z 2 )2 R  (z)2 . (4.101) 2 This flow equation with T˜k = 0 was also derived from the one-loop perturbative FRG calculation in Sect. 2.3. From Eq. (4.93), Tk is not renormalized ∂l Tk = 0, thus the flow equation of T˜k is given by (4.102) ∂l T˜k = [2 − D + (N − 2)ηk ]T˜k ,

4.3 NP-FRG Formalism

99

where we have used Eqs. (4.96) and (4.97). Since η = O() at D = 4 + , the temperature is irrelevant. Note that, for N = 2, Eq. (4.102) is identical to the flow equation of the temperature in the random manifold with a periodic potential ζ = 0 (see Eq. (3.123)). Nonequilibrium case We next derive the flow equations for the nonequilibrium case. Note that the dimensionless longitudinal elastic constant z˜ ,k , which is defined by Eq. (4.77), scales as z˜ ,k ∼ k 2(ζ−1) . Thus, since ζ > 1, z˜ ,k can be set to zero in the large scale. The integrals in Eqs. (4.87) and (4.88) at ρ1 = ρ2 = ρm are calculated as (T ) −2 D−3 (ρm , ρm ) = Z ⊥,k k I21 (T ) J21 (ρm , ρm ) = 0.

2 A D−1 v˜ −1 , D−1 (4.103)

(T ) Note that the second integral J21 vanishes in contrast to the equilibrium case. This fact is independent of the choice for the cutoff function Rk (|q⊥ |2 ). We define δ00 (z) and δ21 (z) by

˜ 00 (ρ˜m , ρ˜m , z) 4  A D−1 v˜ −1 , D−1 2ρ˜m ˜ 21 (ρ˜m , ρ˜m , z) 4  δ21 (z) = . A D−1 v˜ −1 D−1 2ρ˜m

δ00 (z) =

(4.104)

We also define a renormalized temperature, T˜k =

2 −1/2 A D−1 (2ρ˜m )−1 Tk τk z˜ ,k . D−1

(4.105)

The flow equation for ρ˜m is given by ∂l ρ˜m = (D − 3 + η¯⊥,k )ρ˜m − (N − 1)[δ00 (1) + T˜k ]ρ˜m ,

(4.106)

and the anomalous dimension for the transverse direction η⊥,k is written as η⊥,k = δ00 (1) + T˜k .

(4.107)

The flow equations for δ00 (z) and δ21 (z) at D = 3 +  are given by ∂l δ00 (z) = −δ00 (z) + (δ00 (1) + T˜k )[(N − 3)δ00 (z) − (N − 1)z∂z δ00 (z) −2zδ21 (z) + (1 − z 2 )∂z2 δ00 (z)] + zδ00 (z)2 +(N − 2 + z 2 )δ00 (z)∂z δ00 (z) + (1 + z 2 )δ00 (z)δ21 (z) −z(1 − z 2 )[δ21 (z)∂z δ00 (z) + δ00 (z)∂z2 δ00 (z)] +(1 − z 2 )2 δ21 (z)∂z2 δ00 (z),

(4.108)

100

4 Dimensional Reduction and its Breakdown …

∂l δ21 (z) = −δ21 (z) + (δ00 (1) + T˜k )[−2∂z δ00 (z) − 2δ21 (z) −(N + 3)z∂z δ21 (z) + (1 − z 2 )∂z2 δ21 (z)] +2zδ00 (z)[∂z δ00 (z) + 2δ21 (z)] +(N − 2 + 5z 2 )δ00 (z)∂z δ21 (z) −2(1 − z 2 )δ21 (z)∂z δ00 (z) − 2(1 − 2z 2 )δ21 (z)2 −z(1 − z 2 )[δ00 (z)∂z2 δ21 (z) + 5δ21 (z)∂z δ21 (z)] +(1 − z 2 )2 δ21 (z)∂z2 δ21 (z).

(4.109)

Since the two integrals in Eq. (4.103) have different values, the relation in Eq. (4.99) does not hold. From Eqs. (4.79), (4.89), and (4.92), ζk = 2 − T˜k .

(4.110)

The flow equation of T˜k is given by ∂l T˜k = [2 − D + (N − 2)η⊥,k + δ00 (1) + (N − 1)∂z δ00 (1) + 2δ21 (1)]T˜k . (4.111) Note that the last three terms in the right-hand side of Eq. (4.111) result from the breakdown of the detailed balance condition and they do not exist for the equilibrium case (4.102).

4.4 Critical Exponents In this section, we calculate a fixed point by solving the flow equations derived in the previous section. We determine the anomalous dimensions η and η¯ as functions of N and compare them to those predicted from the dimensional reduction. In the following, we restrict our attention to the zero-temperature case Tk = 0.

4.4.1 Analytic Fixed Point First, we assume that the derivatives of δ00 (z) and δ21 (z) at z = 1 are finite. By taking the derivative of Eqs. (4.108) and (4.109), we have ∂l δ00 (1) = −δ00 (1) + (N − 2)δ00 (1)2 ,    (1) = −δ00 (1) + δ00 (1)2 + 2δ00 (1)δ00 (1) ∂l δ00  2  +(N − 1)δ00 (1) + 4δ00 (1)δ21 (1),

(4.112)

(4.113)

4.4 Critical Exponents

101

∂l δ21 (1) = −δ21 (1) + 2δ00 (1)δ21 (1) + 2δ21 (1)2 .

(4.114)

From these equations, we obtain the following fixed points:  , N −2 √ (N − 4) − (N − 2)(N − 10) ∗ , δ00 (1) = 2(N − 1)(N − 2) ∗ δ21 (1) = 0,

∗ (1) = δ00

(4.115)

∗ ∗ (1) and δ21 (1). In fact, Eq. (4.114) where we have chosen a stable solution for δ00 ∗ also has a fixed point δ21 (1) = (N − 4)/2(N − 2) , but it is unstable for N > 4. The fixed point Eq. (4.115) describes the second-order transition from a LRO phase to a ∗ (1) = /(N − 2) is the same as the anomalous disordered phase. Note that η⊥ = δ00 dimension of the (2 + )-dimensional pure O(N ) model. This analytic fixed point exists for N ≥ 10. Thus, NDR = 10 is the field component number above which the dimensional reduction property is recovered.

4.4.2 Nonanalytic Fixed Point We next consider the nonanalytic fixed points. For the equilibrium case, δ00 and δ21 have the following singularity near z = 1, (eq)

δ00 (z) ∼ (1 − z)1/2 , (eq)

(eq)

δ21 (z) = ∂z δ00 (z) ∼ −(1 − z)−1/2 . Thus, we expand δ00 and δ21 around z = 1, δ00 (z) = a0 + a1 (1 − z)1/2 + a2 (1 − z) + · · · δ21 (z) = b−1 (1 − z)−1/2 + b0 + b1 (1 − z)1/2 + · · · .

(4.116)

By substituting Eq. (4.116) into Eq. (4.109), the right-hand side of Eq. (4.109) leads to a term proportional to b−1 (1 − z)−1 , and we have b−1 = 0.

(4.117)

Therefore, for the nonequilibrium case, δ21 (z) is finite at z = 1, in contrast to the equilibrium case where it diverges as (1 − z)−1/2 . By substituting Eq. (4.116) into Eq. (4.108), we have the flow equation for a0 = δ00 (1), da0 1 = −a0 + (N − 2)a02 − (N − 2)a12 + 2a1 b−1 , (4.118) dl 2

102

4 Dimensional Reduction and its Breakdown …

(a)

(b)

Fig. 4.1 Anomalous dimensions η and η¯ for the DRFO(N )M and RFO(N )M as functions of N . The red solid lines represent η⊥ and η¯ ⊥ for the DRFO(N )M at D = 3 + . The green dashed lines represent η and η¯ for the RFO(N )M at D = 4 + .  is set to unity. The blue dotted line represents the dimensional reduction value ηDR = (N − 2)−1 . The insets show η/ηDR and η¯ /ηDR . Reprinted figure with permission from Ref. [1]. Copyright (2017) by the American Physical Society. https:// doi.org/10.1103/PhysRevB.96.184202

where the third and fourth terms of the right-hand side are absent if δ00 (z) and δ21 (z) are analytic. Since b−1 = 0, the fixed point behaves as δ ∗ (z) ∝ (N − 2)−1 near N = 2. This fixed point is unstable and it controlls to the second-order transition between the LRO phase and the disordered phase. Note that the Bragg glass phase does not exist for  < 0 and N > 2. ∗ (1) for the The upper panel of Fig. 4.1 shows the anomalous dimension η = δ00 nonequilibrium and equilibrium cases. The red solid line represents η⊥ for the

4.4 Critical Exponents

103

∗ (z) and (N − 2)δ ∗ (z) for N = 3, 4, and 5. Reprinted figure Fig. 4.2 Fixed functions (N − 2)δ00 21 with permission from Ref. [1]. Copyright (2017) by the American Physical Society https://doi.org/ 10.1103/PhysRevB.96.184202

nonequilibrium case at D = 3 +  calculated from Eqs. (4.108) and (4.109). The green dashed line represents η for the equilibrium case at D = 4 +  calculated from Eq. (4.101). We have plotted η corresponding to the unstable fixed point. Also shown is the anomalous dimension of the (2 + )-dimensional pure O(N ) model, ηDR = (N − 2)−1 , which is represented by the blue dotted line. The inset displays the ratio of η and ηDR . From Eqs. (4.97) and (4.106), the anomalous dimension for the disconnected Green’s function η¯ is given by η¯ = − + (N − 1)η,

(4.119)

which is shown in the lower panel of Fig. 4.1. For the equilibrium case, η diverges as N approaches to Nc = 2.83, while for the nonequilibrium case, η⊥ diverges as N approaches 2. The ratio η⊥ /ηDR exhibits nonmonotonic behavior with decreasing N . In particular, lim N →2 η⊥ /ηDR = 1, which will be shown below. Note that η⊥ /ηDR − 1 ∝ (N − 2) near N = 2, and lim (η⊥ − ηDR ) = χ,

N →2

(4.120)

∗ where χ  1.65 is a universal constant. Figure 4.2 shows the fixed functions δ00 (z) ∗ ∗ ∗ and δ21 (z) of the DRFO(N )M. Here, δ21 (z) significantly differs from ∂z δ00 (z), in contrast to the equilibrium √ case where they are identical. They exhibit nonanalytic behavior in the form of 1 − z near z = 1. The numerical method to obtain the fixed function is explained in Sect. 4.6.5.

4.4.3 Fixed Line in the Case that N = 2 and D = 3 Let us consider the case of the driven random field XY model (DRFXYM) in three √ dimensions. The phase parameter u is defined by φ = (φ1 , φ2 ) = 2ρ(cos u, sin u). We denote the renormalized random force as F α (φ), which satisfies

104

4 Dimensional Reduction and its Breakdown …

F α (φ1 )F β (φ2 ) = αβ (φ1 , φ2 ).

(4.121)

The tangential component of F α (φ) is written as F⊥ (u) = (2ρ)−1/2 (−F 1 (φ)φ2 + F 2 (φ)φ1 ),

(4.122)

and its second cumulant is given by (u 1 − u 2 ) = F⊥ (u 1 )F⊥ (u 2 ) = z00 − (1 − z 2 )21 .

(4.123)

Thus, we define a dimensionless cumulant δ(u) as δ(u) = zδ00 (z) − (1 − z 2 )δ21 (z),

(4.124)

with cos u = z. Then, Eqs. (4.108) and (4.109) can be reduced to the following equation: ∂l δ(u) = −δ(u) + δ  (u)[δ(0) − δ(u)],

(4.125)

which was also obtained in Ref. [5] by a one-loop perturbative calculation. In Ref. [5], Eq. (4.125) was derived for the driven random manifold model, (∂t u + v∂x u) = K ∇ 2 u + F(r; u) + ξ(r, t),

(4.126)

which describes the dynamics of the transverse displacement field u(r, t) of an elastic lattice moving in a random pinning potential. The random force F(r; u) satisfies F(r; u)F(r  ; u  ) = (u − u  )δ(r − r  ),

(4.127)

where (u) is a periodic function. If topological defects or vortices are ignored, the driven random manifold model Eq. (4.126) is equivalent to the DRFXYM. From Eq. (4.125), in three dimensions ( = 0) we find that the beta function of η⊥,l = δl (0) identically vanishes even if δ(u) has a cusp at u = 0. This result can be also understood from Eq. (4.118). The vanishing of the beta function suggests the existence of a QLRO characterized by a line of fixed points as in the two-dimensional pure XY model. The family of fixed points of Eq. (4.125) at  = 0 is given by δ ∗ (u) = C,

(4.128)

∗ δ00 (z) = C z, ∗ δ21 (z) = −C,

(4.129)

or 

4.4 Critical Exponents

105

where C is an arbitrary positive constant. These fixed points are stable in the sense that δ00 (z) and δ21 (z) flow to Eq. (4.129) with an initial value dependent C in the limit l → ∞. We will present a more detailed analysis of Eq. (4.125) in the next chapter. It is worth noting that the beta function of δ(0) does not vanish in the fourdimensional random field XY model. In this case, from Eq. (4.101), the flow equation of Eq. (4.124) is given by ∂l δ(u) = δ  (u)[δ(0) − δ(u)] − δ  (u)2 .

(4.130)

The last term on the right-hand side δ  (u)2 has a finite value at u = 0 if δ(u) has a linear cusp (see the end of Sect. 2.3.3). We show that lim N →2 η⊥ /ηDR = 1 at  = 1, as shown in Fig. 4.1. It is convenient to introduce δ˜00 (z) = (N − 2)δ00 (z) and δ˜21 (z) = (N − 2)δ21 (z). The fixed ∗ ∗ (z) and δ˜21 (z) are stationary solutions of Eqs. (4.108) and (4.109) with functions δ˜00 ∗ ∗  = (N − 2). Thus, in the limit N → 2, δ˜00 (z) and δ˜21 (z) are given by Eq. (4.129). The constant C can be determined by considering a stationary solution of Eq. (4.118) with  = (N − 2). Note that a1 vanishes as N → 2 because Eq. (4.129) does not have a cusp. Thus, we have a0 = C = 1, implying that lim N →2 η⊥ /ηDR = 1. The existence of the fixed line implies that the three-dimensional DRFXYM may exhibit a Kosterlitz-Thouless (KT) transition. However, there is a possibility that the beta function vanishes in the leading order but has a finite contribution in the higher order as in the case of the RFO(N )M at D = 4 and N = 2.83 [10]. To verify the existence of the fixed line, one must show that all higher order contributions of the beta function vanish. A detailed investigation concerning this problem will be presented in the next chapter.

4.4.4 Random Anisotropy Case We next consider the driven random anisotropy O(N ) model (DRAO(N )M). In this case, the disorder correlators satisfy δ00 (−z) = −δ00 (z),

δ21 (−z) = δ21 (z).

(4.131)

δ21 (z) = S21 (z 2 ).

(4.132)

It is convenient to rewrite them as δ00 (z) = zS00 (z 2 ),

If we assume that S00 (z 2 ) and S21 (z 2 ) are analytic at z = 1, from Eqs. (4.108) and (4.109), we have ∂l S00 (1) = −S00 (1) + (N − 2)S00 (1)2 ,

(4.133)

106

4 Dimensional Reduction and its Breakdown …

Fig. 4.3 Anomalous dimension η⊥ for the DRAO(N )M as a function of N . The ratio between η⊥ and ηDR = /(N − 2) is plotted. The inset shows the large-N region. The green dashed line represents lim N →∞ η⊥ /ηDR = 3/2. The rugged behavior shown in these data comes from numerical errors

   ∂l S00 (1) = −S00 (1) + 2S00 (1)2 + 2N S00 (1)S00 (1)   +2(N − 1)S00 (1)2 + 2[S00 (1) + 2S00 (1)]S21 (1),

∂l S21 (1) = −S21 (1) + 2S00 (1)S21 (1) + 2S21 (1)2 .

(4.134) (4.135)

∗ ∗ (1) = S00 (1) = /(N − Eq. (4.133) leads to the dimensional reduction fixed point δ00 2), and then Eq. (4.134) has solutions

∗ (1) S00

√ −(N + 2) ± (N − 2)(N − 10) . = 4(N − 1)(N − 2)

(4.136)

∗ Since these both solutions are negative, the flow of S00 (1) started from a nonnegative initial value always diverges for any N . Therefore, any analytic fixed point does not exist for all N , as in the random anisotropy O(N ) model (see Sect. 2.3.4). Figure 4.3 shows η⊥ /ηDR for the DRAO(N )M as a function of N . One can see that the dimensional reduction breaks down for all N . In the large N limit, η⊥ /ηDR converges to 3/2. This value is the same as that of the random anisotropy O(N ) model (2.65), because in the large N limit, the flow Eqs. (4.108) and (4.109) are identical to that of the equilibrium model (4.101).

4.5 Correlation Length in Three Dimensions The DRFO(N )M neither exhibits LRO nor QLRO for N > 2 at D = 3. However, the correlation length ξ rapidly diverges with decreasing the disorder strength to zero. In this section, let us consider the behavior of ξ in the weak disorder regime. We show that the anomalous dimension η at D = 3 +  is related to the diverging behavior of the correlation length at D = 3. For simplicity, the temperature is assumed to be zero.

4.5 Correlation Length in Three Dimensions

107

We define lc by the renormalization scale at which ρm,l vanishes. The correlation function satisfies C(r , r ⊥ ) = e−(D−3+¯η⊥ )lc C  (e−2lc r , e−lc r ⊥ ),

(4.137)

where we have used the fact that ζ = 2. C  (r , r ⊥ ) is the correlation function corresponding to the renormalized model described by   (∂t φ + v  ∂x φ) = K  ∇⊥2 φ − m 2 φ + h  (r; φ),

(4.138)

where   , v  , and K  are the renormalized relaxation coefficient, velocity, and elastic constant, respectively. We have omitted the longitudinal elastic term K  ∂x2 φ, which is negligible compared to   v  ∂x φ. m 2 = F˜l=lc (ρ˜ = 0) > 0 is the renormalized mass ˜ are ignored. Although the renormalized random and the higher-order terms of F˜l (ρ) field h  (r; φ) depends on φ, we can put φ = 0 in it due to ρm = 0. From Eq. (4.138) with ∂t φ = 0, C  (r , r ⊥ ) for steady states can be calculated as C



(r , r ⊥ )

=

h 2 0

 q





ei(qx r +q⊥ r⊥ ) , (K  |q⊥ |2 + m 2 )2 +  2 v 2 qx2

(4.139)

2   where h 2 0 is defined by h (r 1 ; φ = 0)h (r 2 ; φ = 0) = h 0 δ(r 1 − r 2 ). The asymptotic behaviors for C  (r , r ⊥ ) are given by

m 2  C = 0) ∼ exp −   r , v

m     −1/2  C (r = 0, r ⊥ ) ∼ |r ⊥ | exp − 1/2 |r ⊥ | . K 

(r , r ⊥

r−1

(4.140)

Since ρm = 0 at this scale, the correlation length for the transverse direction is the order of the cutoff length scale, K 1/2 /m  ∼ −1 . In addition, the renormalization of v and K can be ignored in the disordered phase. Thus, the above equations are written as

K 2  −1    r , C (r , r ⊥ = 0) ∼ r exp − v  C  (r = 0, r ⊥ ) ∼ |r ⊥ |−1/2 exp(−|r ⊥ |).

(4.141)

From Eq. (4.137), we have

r , = 0) ∼ exp − ξ

|r ⊥ | −1/2 C(r = 0, r ⊥ ) ∼ |r ⊥ | , exp − ξ⊥

C(r , r ⊥

r−1

(4.142)

108

4 Dimensional Reduction and its Breakdown …

where ξ⊥ and ξ are given by ξ⊥ = elc −1 ,

ξ =

v 2 ξ . K ⊥

(4.143)

Next, we calculate lc . Since 2ρ˜m δ00 (1) is constant along the RG flow, lc is the renormalization scale at which δ00,l diverges. Let lL be a renormalization scale at which the first derivative of δ(z) at z = 1 diverges. We call lL the Larkin scale. Note that the flow of δl (z) exhibits qualitatively different behaviors depending on whether l is smaller or larger than the Larkin scale lL (< lc ). For l < lL , δ00,l (1) obeys the RG equation Eq. (4.112), thus we have δ00,l (1) =

1 δB−1 − (N − 2)l

,

(4.144)

where δB = δ00,l=0 (1) = h 20 /(4π K v) is the bare disorder strength. For lL < l < lc , we assume the following forms for δ00 (z) and δ21 (z), ∗ (z) δ00 lc − l δ ∗ (z) δ21,l (z) = 21 . lc − l

δ00,l (z) =

(4.145)

∗ By substituting these equations into Eqs. (4.108) and (4.109), we find that δ00 (z) and ∗ δ21 (z) are the fixed function corresponding to  = 1. Thus, we have

δ00,l (1) =

η⊥ ( = 1) . lc − l

(4.146)

lc can be obtained by assuming Eqs. (4.144) and (4.146) are continuously connected at l = lL . Thus, we have lc = η⊥ ( = 1)δB−1 − [(N − 2)η⊥ ( = 1) − 1]lL .

(4.147)

By introducing new N -dependent constants ηL and ηc by lL = ηL δB−1 and lc = ηc δB−1 , we have ηc = η⊥ ( = 1) − [(N − 2)η⊥ ( = 1) − 1]ηL .

(4.148)

From Eq. (4.143), we finally obtain

4π K v , ξ⊥ ∼ exp ηc h 20

ξ ∼

8π K v v . exp ηc K h 20

(4.149)

4.5 Correlation Length in Three Dimensions

109

Therefore, the correlation length diverges exponentially in the weak disorder limit. It is worth noting that, in the 2D pure O(N ) model, the correlation length behaves as   2πK , (4.150) ξ ∼ exp (N − 2)T at low temperatures [11]. One can see a clear resemblance between Eqs. (4.149) and (4.150). In fact, if the dimensional reduction holds η⊥ ( = 1) = 1/(N − 2), Eq. (4.150) coincides with ξ⊥ in Eq. (4.149) by replacing T with Teff defined by Eq. (4.17).

4.6

Appendix

4.6.1

Propagators

In this section, we show the expression of the one-replica propagator Eq. (4.42). The (2) [] is evaluated for a uniform field ψr t ≡ t (ψ 1 , . . . , ψ N ) functional derivative 1,k ˆ r t ≡ 0. 1,k [] is given by Eq. (4.45). For simplicity, we omit the subscript k and ψ in the following. We introduce P(q, ω; ψ) as P[]q1 ,q2 = P(q1 , ω1 ; ψ)(2π) D+1 δ(q1 + q2 )δ(ω1 + ω2 ),

(4.151) μν

where q = (q, ω). P(q, ω; ψ) is a 2N × 2N matrix, thus we write its element as Pi j , ˆ and μ, ν = 1, . . . , N where i, j = 1, 2 represent the two conjugate fields ψ and ψ, are the field component indices. μν Pi j (q, ω; ψ) can be written as μν Pi j (q, ω; ψ)

=

) Pi(T j (q, ω; ρ)

δ

μν

ψμ ψν − 2ρ



+ Pi(L) j (q, ω; ρ)

ψμ ψν , (4.152) 2ρ

where the transverse and longitudinal parts are given by 2X T , D0 (q, ω; ρ) M0 (q; ρ) − i(ω − qx v)X (T ) P12 , (q, ω; ρ) = D0 (q, ω; ρ) M0 (q; ρ) + i(ω − qx v)X (T ) P21 , (q, ω; ρ) = D0 (q, ω; ρ)

(T ) P11 (q, ω; ρ) =

(T ) P22 (q, ω; ρ) = 0,

(4.153)

110

4 Dimensional Reduction and its Breakdown …

2X T , D1 (q, ω; ρ) M1 (q; ρ) − i(ω − qx v)X (L) P12 , (q, ω; ρ) = D1 (q, ω; ρ) M1 (q; ρ) + i(ω − qx v)X (L) P21 , (q, ω; ρ) = D1 (q, ω; ρ)

(L) P11 (q, ω; ρ) =

(L) P22 (q, ω; ρ) = 0.

(4.154)

We have defined the following notations: M0 (q; ρ) = Z  qx2 + Z ⊥ q⊥2 + Rk (q) − F(ρ), M1 (q; ρ) = Z  qx2 + Z ⊥ q⊥2 + Rk (q) − F(ρ) − 2ρF  (ρ), D0 (q, ω; ρ) = M0 (q; ρ)2 + (ω − qx v)2 X 2 , D1 (q, ω; ρ) = M1 (q; ρ)2 + (ω − qx v)2 X 2 .

(4.155)

In Sect. 4.3.5, we also use the simplified notation D(q; ρ) = D(q, ω = 0; ρ). To express the RG equations in a compact form, we introduce the following integrals: ) L (T n (ρ)

 =−

∂l Rk (q)M0 (q; ρ)−n ,

(4.156)

q (T ) Inn  (ρ1 , ρ2 ) = −



(T ) (T ) ∂l Rk (q)P21 (q; ρ1 )n P12 (q; ρ2 )n ,

(4.157)

(T ) (T ) ∂l Rk (q)P21 (q; ρ1 )n P21 (q; ρ2 )n ,

(4.158)



q (T ) Jnn  (ρ1 , ρ2 ) = −





q

where ∂l = −k∂k and the frequency ω in Eqs. (4.157) and (4.158) are set to zero. The integrals for the longitudinal mode are also defined by replacing M0 and D0 in Eqs. (4.156)–(4.158) with M1 and D1 , respectively.

4.6.2

Flow Equation for Fk (ρ)

In this section, the flow equation for Fk (ρ) is derived, which is given by Eq. (4.55). To do this, the exact flow equation for 1(1) is required. It is convenient to introduce a graphical representation. The flow equation for 1 is rewritten as ∂l 1 =

1 [γ1,a + γ1,b ], 2

(4.159)

4.6 Appendix

111

(1)

Fig. 4.4 Graphical representations for the flow equations of 1 . Reprinted figure with permission from Ref. [1]. Copyright (2017) by the American Physical Society. https://doi.org/10.1103/ PhysRevB.96.184202

where γ1,a and γ1,b are given by Eqs. (3.154) and (3.155), respectively. A scale parameter l = − ln(k/) has been introduced. The flow equation for  (1)ˆ 1 = δ1 /δ ψˆ 1 is 1;ψ then written as 1 (1) (1) (1) ∂l  (1)ˆ 1 = [−γ1,a−1 − 2γ1,b−1 + 2γ1,b−2 ]. (4.160) 1;ψ 2 (1) (1) (1) (1) where γ1,a−1 , γ1,b−1 , and γ1,b−2 are given in Fig. 4.4. For example, γ1,b−1 is written as (1) = Tr∂l Rk (q)P[]2(11) [, ]P[] (3)ˆ 1 []P[], γ1,b−1 1;ψ

(4.161)

where  (3)ˆ 1 = δ1(2) /δ ψˆ 1 and Tr represents an integration over momentum and fre1;ψ quency as well as a sum over the field component and the two conjugate fields. √ All functional derivatives are evaluated for a uniform field ψr t ≡ t ( 2ρ, 0, . . . , 0), ˆ r t ≡ t (0, . . . , 0). ψ The following notation is introduced:  (3) 2

δ 2 1 []

(q1 , q2 , q3 ) =

, δψ 2 (q1 )δψ 2 (q2 )δ ψˆ 1 (q3 ) δ 2 2 [1 , 2 ] .  (21)2 ˆ 2 ˆ 1 (q1 , q2 , q3 ) = 2;ψ1 ψ1 ψ2 δψ 2 (q1 )δ ψˆ 2 (q2 )δ ψˆ 1 (q3 ) 1;ψ

ψ 2 ψˆ 1

1

1

2

From Eqs. (4.45) and (4.46), they are calculated as  (3) 1 1;ψ

ψ 1 ψˆ 1

 (q1 , q2 , q3 ) = − 2ρ[3F  (ρ) + 2ρF  (ρ)](2π) D+1 δ(q1 + q2 + q2 ) ×δ(ω1 + ω2 + ω2 ),

 (3) ν

1;ψ ψ ν ψˆ 1

 (q1 , q2 , q3 ) = − 2ρF  (ρ)(2π) D+1 δ(q1 + q2 + q2 ) ×δ(ω1 + ω2 + ω2 ), (ν = 2, . . . , N ),

 D+2  δ(q1 + q2 )δ(ω1 )δ(ω2 ),  (11) ˆ 1 ˆ 1 (q1 , q2 ) 1 =2 = =  L (ρ)(2π) 2;ψ1 ψ2

112

4 Dimensional Reduction and its Breakdown …

 D+2   (11) δ(q1 + q2 )δ(ω1 )δ(ω2 ), ˆ ν ˆ ν (q1 , q2 ) 1 =2 = = T (ρ)(2π) 2;ψ1 ψ2

(ν = 2, . . . , N ),  1  (21)1 ˆ 1 ˆ 1 (q1 , q2 , q3 )1 =2 = = 2ρL (ρ)(2π) D+2 δ(q1 + q2 + q3 ) 2;ψ1 ψ1 ψ2 2 ×δ(ω1 + ω2 )δ(ω3 ),  1  (21)ν ˆ 1 ˆ ν (q1 , q2 , q3 )1 =2 = = √ [21 (ρ) + 11 (ρ)](2π) D+2 δ(q1 + q2 + q3 ) 2;ψ1 ψ1 ψ2 2ρ ×δ(ω1 + ω2 )δ(ω3 ), (ν = 2, . . . , N ), where we have used the notations ... (ρ) = ... (ρ, ρ, z = 1), T (ρ) = 22 (ψ, ψ) = 00 (ρ),  L (ρ) = 11 (ψ, ψ) = 00 (ρ) + 12 (ρ) + 21 (ρ) + 11 (ρ) + 22 (ρ). From these expressions, we obtain (1) = γ1,a−1 (1) = γ1,b−1

(1) γ1,b−2



(T )  2ρT {[3F  (ρ) + 2ρF  (ρ)]L (L) 2 (ρ) + (N − 1)Fk (ρ)L 2 (ρ)},

 (L) (T ) 2ρ{[3F  (ρ) + 2ρF  (ρ)] L (ρ)I12 (ρ) + (N − 1)F  (ρ)T (ρ)I12 (ρ)},    1  1 (L) (T ) = − 2ρ  L (ρ)J11 (ρ) + (N − 1) [21 (ρ) + 11 (ρ)]J11 (ρ) , 2 2ρ

where we have already calculated the ω-integral. The integrals L, I , and J are defined by Eqs. (4.156), (4.157), and (4.158), respectively, and simplified notations (T ) (T ) such as Inn  (ρ) = Inn  (ρ, ρ) are used. From Eqs. (4.55) and (4.160), we have the flow equation for F(ρ), ∂l F(ρ) = ∂l F (1) (ρ) + ∂l F (2) (ρ),

(4.162)

where ∂l F (1) (ρ) and ∂l F (2) (ρ) are the contributions from the one and two-replica parts, respectively, ∂l F (1) (ρ) =

1 (T )  T {[3F  (ρ) + 2ρF  (ρ)]L (L) 2 (ρ) + (N − 1)F (ρ)L 2 (ρ)}, (4.163) 2

(L) (T ) ∂l F (2) (ρ) = [3F  (ρ) + 2ρF  (ρ)] L (ρ)I12 (ρ) + (N − 1)F  (ρ)T (ρ)I12 (ρ) 1  1 (L) (T ) −  L (ρ)J11 (ρ) − (N − 1) [21 (ρ) + 11 (ρ)]J11 (ρ). (4.164) 2 2ρ

It can be easily checked that, in the equilibrium case (v = 0), the equation can be reduced to that of the RFO(N )M, which is given in Ref. [9].

4.6 Appendix

113

Flow Equation for k (ψ 1 , ψ 2 )

4.6.3

In this section, we derive the RG equation for k (ψ 1 , ψ 2 ), which is given by Eq. (4.57). To do this, the exact flow equation for 2(11) is required. The flow equation for 2 is rewritten as 1 ∂l 2 [1 , 2 ] = − [γ2,a + γ2,b + 2γ2,c + perm], 2

(4.165)

where γ2,a , γ2,b , and γ2,c are given by Eqs. (3.157), (3.158), and (3.159), respectively. 2 ˆμ ˆν The flow equation for  (11) ˆ μ ˆ ν = δ 2 /δ ψ1 δ ψ2 is then written as 2;ψ1 ψ2

1 (11) (11) (11) (11) (11) ∂l  (11) [1 , 2 ] = − [−2γ2,a−1 + γ2,a−2 − 2γ2,b−1 − 2γ2,b−2 + 2γ2,b−3 μ 2;ψˆ 1 ψˆ 2ν 2 (11) (11) (11) (11) (11) +2γ2,b−4 − 2γ2,b−5 + 2γ2,b−6 − 2γ2,c−1 − 2γ2,c−2 (11) (11) (11) (11) −2γ2,c−3 + 2γ2,c−4 + 2γ2,c−5 + 2γ2,c−6 + perm], (4.166) (11) (11) , . . . , γ2,c−6 are given in Fig. 4.5 and “perm” denotes the permutation where γ2,a−1 (11) between the indices 1 and 2, μ and ν. For example, γ2,b−1 is written as (11) γ2,b−1 = Tr∂l Rk (q)Pk [1 ] (3)ˆ μ [1 ]Pk [1 ] (12) ˆ ν [1 , 2 ] 1;ψ1

2;ψ2

×Pk [2 ]2(11) [2 , 1 ]Pk [1 ].

(4.167)

ˆ 1,r t ≡ 0, All functional derivatives are evaluated for a uniform field ψ 1,r t ≡ ψ 1 , ψ ˆ 2,r t ≡ 0. ψ 2,r t ≡ ψ 2 , and ψ The functional derivatives of 1 [] and 2 [1 , 2 ] are calculated as  (3) α

1;ψ ψ β ψˆ μ

(q1 , q2 , q3 ) = [−F  (ρ)ψ α ψ β ψ μ −F  (ρ)(δ αβ ψ μ + δ αμ ψ β + δ βμ ψ α )] ×(2π) D+1 δ(q1 + q2 + q2 )δ(ω1 + ω2 + ω2 ),

μν D+2 δ(q1 + q2 )δ(ω1 )δ(ω2 ),  (11) ˆ μ ˆ ν (q1 , q2 ) =  (ψ 1 , ψ 2 )(2π) 2;ψ1 ψ2

 (21)α ˆ β ˆ μ (q1 , q2 , q3 ) = ∂ψ1α βμ (ψ 1 , ψ 2 )(2π) D+2 δ(q1 + q2 + q3 )δ(ω1 + ω2 )δ(ω3 ), 2;ψ1 ψ1 ψ2

 (31)α

2;ψ1 ψ1 ψˆ 1 ψˆ 2ν β

μ

(q1 , q2 , q3 , q4 ) = ∂ψ1α ∂ψβ μν (ψ 1 , ψ 2 )(2π) D+2 δ(q1 + q2 + q3 + q4 ) 1

×δ(ω1 + ω2 + ω3 )δ(ω4 ),

114

4 Dimensional Reduction and its Breakdown …

(2)

Fig. 4.5 Graphical representation for the flow equation of 2 . Reprinted figure with permission from Ref. [1]. Copyright (2017) by the American Physical Society. https://doi.org/10.1103/ PhysRevB.96.184202

 (22)α ˆ μ

β 2;ψ1 ψ1 ψ2 ψˆ 2ν

(q1 , q2 , q3 , q4 ) = ∂ψ1α ∂ψβ μν (ψ 1 , ψ 2 )(2π) D+2 δ(q1 + q2 + q3 + q4 ) 2

×δ(ω1 + ω2 )δ(ω3 + ω4 ), where μν (ψ 1 , ψ 2 ) is expressed as Eq. (4.47). As mentioned in Sect. 4.3.5, near the lower critical dimension D = Dlc + , γ2(11) is expanded in terms of ρ−1 ∼ . In addition, γ2(11) is rewritten as

4.6 Appendix

115 μ

μ

(11) μν (11) (11) γ2(11) = γ2,00 δ + (4ρ1 ρ2 )−1/2 [γ2,12 ψ1 ψ2ν + γ2,21 ψ2 ψ1ν μ

μ

(11) (11) +γ2,11 ψ1 ψ1ν + γ2,22 ψ2 ψ2ν ].

(4.168)

By noting αβ

1 ψβ , 2ρF  (ρ)

(4.169)

αβ

2X T ψβ , 4ρ2 F  (ρ)2

(4.170)

(L) β ψ − ψ α P12 = P12

and (L) β ψ α P11 = P11 ψ 

each coefficients of Eq. (4.168) can be calculated in the leading order of ρ−1 as follows:  (11) ) = T (2ρ1 )−1 ( ρ1 /ρ2 11 + z21 )L (T γ2,a−1,00 2 (ρ1 ), (11) = T (2ρ1 )−1 [(N − 1)(2ρ1 ∂ρ1 − z∂z )00 + (1 − z 2 )∂z2 00 γ2,a−2,00  ) +2 ρ1 /ρ2 11 ]L (T 2 (ρ1 ), (11) = (4ρ1 ρ2 )−1/2 (12 + γ2,b−1,00



(T ) ρ2 /ρ1 z22 )00 I21 (ρ1 , ρ2 ),

(11) = (4ρ1 ρ2 )−1/2 (1 − z 2 )(00 + z21 + γ2,b−2,00



(T ) (ρ1 , ρ2 ), ρ1 /ρ2 11 )∂z 00 J21

(11) = (4ρ1 ρ2 )−1/2 [z00 + z 2 21 + 12 + γ2,b−3,00  (T ) + ρ2 /ρ1 z22 ]00 I21 (ρ1 , ρ2 ),



ρ1 /ρ2 z11

(11) = (4ρ1 ρ2 )−1/2 [(N − 2 + z 2 )00 ∂z 00 γ2,b−4,00

−z(1 − z 2 )(21 ∂z 00 + 00 ∂z2 00 ) (T ) +(1 − z 2 )2 21 ∂z2 00 + (21 + 12 )00 ]I21 (ρ1 , ρ2 ), (11) = (4ρ1 ρ2 )−1/2 (12 + γ2,b−5,00



(T ) ρ1 /ρ2 z11 )00 I21 (ρ1 , ρ2 )  (T ) −1/2 2 +(4ρ1 ρ2 ) (1 − z )(00 + z21 + ρ2 /ρ1 22 )∂z 00 J21 (ρ1 , ρ2 ),

  (11) (T ) = (4ρ1 ρ2 )−1/2 (1 − z 2 )( ρ1 /ρ2 11 + ρ2 /ρ1 22 )∂z 00 J21 (ρ1 , ρ2 ), γ2,b−6,00 (11) = (2ρ1 )−1 (z21 + γ2,c−1,00

 (T ) ρ1 /ρ2 11 )00 (ρ1 , ρ1 , z = 1)I21 (ρ1 , ρ1 ), (11) = 0, γ2,c−2,00

116

4 Dimensional Reduction and its Breakdown … (11) γ2,c−3,00 = (2ρ1 )−1 (z21 +

 (T ) ρ1 /ρ2 11 )00 (ρ1 , ρ1 , z = 1)I21 (ρ1 , ρ1 ),

 (11) = (2ρ1 )−1 [(N − 1)(2ρ1 ∂ρ1 − z∂z )00 + (1 − z 2 )∂z2 00 + 2 ρ1 /ρ2 11 ] γ2,c−4,00 (T ) ×00 (ρ1 , ρ1 , z = 1)I21 (ρ1 , ρ1 ), (11) (11) γ2,c−5,00 = γ2,c−6,00 = 0, (11) = T (2ρ1 )−1 (∂z 00 + z∂z 21 + γ2,a−1,21

(11) γ2,b−1,21



) ρ1 /ρ2 ∂z 11 )L (T 2 (ρ1 ),

(11) = T (2ρ1 )−1 [(1 − z 2 )∂z2 21 − 2z∂z 21 γ2,a−2,21 +(N − 1)(2ρ1 ∂ρ1 21 − 21 − z∂z 21 )  ) +2 ρ1 /ρ2 ∂z 11 ]L (T 2 (ρ1 ),  = −(4ρ1 ρ2 )−1/2 {21 (12 + ρ2 /ρ1 z22 ) + [−z00 + (1 − z 2 )21 ]  (T ) (ρ1 , ρ2 ), ×(∂z 00 + 21 + z∂z 21 + ρ1 /ρ2 ∂z 11 )}I21

 (11) = (4ρ1 ρ2 )−1/2 (00 + z21 + ρ1 /ρ2 11 )[(1 − z 2 )∂z 21 − z21 γ2,b−2,21  (T ) + ρ2 /ρ1 22 ]J21 (ρ1 , ρ2 ),  (11) = (4ρ1 ρ2 )−1/2 (z00 + 12 + z 2 21 + ρ1 /ρ2 z11 γ2,b−3,21  (T ) + ρ2 /ρ1 z22 )21 I21 (ρ1 , ρ2 )  −1/2 +(4ρ1 ρ2 ) (00 + z21 + ρ1 /ρ2 11 )  (T ) ×(00 + z21 + ρ2 /ρ1 22 )J21 (ρ1 , ρ2 ), (11) = (4ρ1 ρ2 )−1/2 {(N − 2 + z 2 )00 ∂z 21 γ2,b−4,21

−z(1 − z 2 )(00 ∂z2 21 + 21 ∂z 21 ) +2∂z 21 [z 2 00 − z(1 − z 2 )21 ] + 21 (z00 + 12 + z 2 21 )  + ρ1 /ρ2 ∂z 11 [−00 + (1 − z 2 )21 ]  + ρ2 /ρ1 ∂z 22 [−00 + (1 − z 2 )21 ] (T ) +(1 − z 2 )2 21 ∂z2 21 }I21 (ρ1 , ρ2 ), (11) = (4ρ1 ρ2 )−1/2 {[−z00 + (1 − z 2 )21 ](∂z 00 + 21 + z∂z 21 + ∂z 22 ) γ2,b−5,21  (T ) (ρ1 , ρ2 ) +21 (12 + ρ1 /ρ2 z11 )}I21  −1/2 2 +(4ρ1 ρ2 ) [(1 − z )∂z 21 − z21 + ρ1 /ρ2 11 ]  (T ) ×(00 + z21 + ρ2 /ρ1 22 )J21 (ρ1 , ρ2 ),

4.6 Appendix

117

(11) γ2,b−6,21 = (4ρ1 ρ2 )−1/2 {(N − 2 + z 2 )(21 )2 + 11 22 − 2z(1 − z 2 )21 ∂z 21

+(1 − z 2 )2 (∂z 21 )2 + z 2 (∂z 00 )2 + [z21 − (1 − z 2 )∂z 21 ]   (T ) (ρ1 , ρ2 ), ×(− ρ1 /ρ2 11 − ρ2 /ρ1 22 + 2z∂z 00 )}J21 (11) = (2ρ1 )−1 (∂z 00 + z∂z 21 + γ2,c−1,21

×00 (ρ1 , ρ1 , z =



ρ1 /ρ2 ∂z 11 )

(T ) 1)I21 (ρ1 , ρ1 ),

(11) γ2,c−2,21 = 0, (11) = (2ρ1 )−1 (∂z 00 + z∂z 21 + γ2,c−3,21

×00 (ρ1 , ρ1 , z =



ρ1 /ρ2 ∂z 11 )

(T ) 1)I21 (ρ1 , ρ1 ),

(11) = (2ρ1 )−1 [(1 − z 2 )∂z2 21 − 2z∂z 21 γ2,c−4,21 +(N − 1)(2ρ1 ∂ρ1 21 − 21 − z∂z 21 )  (T ) (ρ1 , ρ1 ), + ρ1 /ρ2 ∂z 11 ]00 (ρ1 , ρ1 , z = 1)I21 (11) (11) γ2,c−5,21 = γ2,c−6,21 = 0, (11) (11) and γ2,21 yield the flow equations for 00 and 21 , respectively. In the γ2,00 (11) (11) leading order of ρ−1 , 12 , 11 , and 22 do not appear in γ2,00 and γ2,21 . Thus, the flow equations for 00 and 21 compose a closed set of equations.

4.6.4

Flow Equations for X k , vk , Zk , and Tk

In this section, we derive the flow equations for X k , vk , Z k , and Tk , which are given by Eqs. (4.51), (4.52), (4.54) and (4.53), respectively. From Eq. (4.159), we have the (2) 2   μ exact flow equation for 1;( p),  ( p ) = δ 1 /δ( p)δ ( p ), where  represents ψ or ψˆ μ and p = (p, ω p ), as follows: (2) ∂l 1;( p),  ( p ) =

1 (2) (2) (2) [2γ (2) − γ1,a−2 + 2γ1,b−1(+) + 2γ1,b−1(−) 2 1,a−1 (2) (2) (2) (2) +2γ1,b−2 − 2γ1,b−3(+) − 2γ1,b−3(−) − 2γ1,b−4(+) (2) (2) (2) (2) −2γ1,b−4(−) − 2γ1,b−5 + 2γ1,b−6 + 2γ1,b−7 ],

(4.171)

where the terms on the right-hand side are given in Fig. 4.6. All functional derivatives √ ˆ r t ≡ t (0, . . . , 0). are evaluated for a uniform field ψr t ≡ t ( 2ρm , 0, . . . , 0) and ψ Calculation of ∂l (X k vk ) and ∂l Z k

118

4 Dimensional Reduction and its Breakdown …

(2)

Fig. 4.6 Graphical representation for the flow equation of 1 . Reprinted figure with permission from Ref. [1]. Copyright (2017) by the American Physical Society https://doi.org/10.1103/ PhysRevB.96.184202

4.6 Appendix

119

We set  = ψˆ 2 ,   = ψ 2 , and p  = − p for the calculation of ∂l (X k vk ), ∂l Z k , and ∂l X k . We denote each term in Eq. (4.171) as γ (2) ˆ , and so on. Note that γ (2) ˆ , 1,a−1,ψψ

1,a−2,ψψ

γ (2) ˆ , and γ (2) ˆ do not depend on the external momentum p and frequency 1,b−5,ψψ 1,b−7,ψψ ω p , thus they do not contribute to the flow equations for X k vk , Z k , and X k . The other terms are given by γ (2)

ˆ 1,a−1,ψψ



= 4ρm F  (ρm )2 X T

 (L) ∂l Rk (q) P21 (q, ωq )2 D0 (p + q, ω p + ωq )−1

q,ωq (T ) +2M1 (q)D1 (q, ωq )−2 P12 (p + q, ω p + ωq ) (T ) +P21 (q, ωq )2 D1 (p + q, ω p + ωq )−1

γ (2)

 (L) +2M0 (q)D0 (q, ωq )−2 P12 (p + q, ω p + ωq ) ,    2 ∂l Rk (q) ˆ = 2ρm F (ρm )

(4.172)

1,b−1(±),ψψ

q (T ) (T ) (L) (q, 0)2 P12 (q, 0)P12 (p T (ρm )P21

± q, ω p )

 (L) (L) (T ) (q, 0)2 P12 (q, 0)P12 (p ± q, ω p ) , (4.173) + L (ρm )P21

γ (2)

ˆ 1,b−2,ψψ



= 2ρm F  (ρm )2

∂l Rk (q)



q (T ) (T ) (L) T (ρm )P12 (−p + q, 0)P21 (−p + q, 0)P12 (q, ω p )2  (L) (L) (T ) + L (ρm )P12 (−p + q, 0)P21 (−p + q, 0)P12 (q, ω p )2 , (4.174)

γ (2)

ˆ 1,b−3(+),ψψ

= F  (ρm )

 ∂l Rk (q)



q (L) (T ) (q, 0)2 P12 (p + q, ω p ) (12 (ρm ) + 11 (ρm ))P12  (T ) (L) +ρm T (ρm )P12 (q, 0)2 P12 (p + q, ω p ) , (4.175)

γ (2)

ˆ 1,b−3(−),ψψ

= F  (ρm )



∂l Rk (q)



q (L) (T ) (q, 0)2 P12 (p − q, ω p ) (12 (ρm ) + 11 (ρm ))P12

 (T ) (L) +(21 (ρm ) + 11 (ρm ))P12 (q, 0)2 P12 (p − q, ω p ) , (4.176)

γ (2)

ˆ 1,b−4(+),ψψ

= F  (ρm )

 ∂l Rk (q)



q (T ) (L) (q, ω p )2 P12 (−p + q, 0) (12 (ρm ) + 11 (ρm ))P12  (L)  2 (T ) (4.177) +ρm T (ρm )P12 (q, ω p ) P12 (−p + q, 0) ,

120

4 Dimensional Reduction and its Breakdown …

γ (2)

ˆ 1,b−4(−),ψψ

= F  (ρm )

 ∂l Rk (q)



q (T )

(L)

(12 (ρm ) + 11 (ρm ))P12 (q, ω p )2 P21 (−p + q, 0)

 (L) (T ) +(21 (ρm ) + 11 (ρm ))P12 (q, ω p )2 P21 (−p + q, 0) , (4.178)

γ (2)

ˆ 1,b−6,ψψ

= (2ρm )−1 [∂z 00 (ρm ) + 12 (ρm ) + (N − 1)21 (ρm )]  (T ) (q, ω p )2 , (4.179) × ∂l Rk (q)P12 q

where we have used the notations ... (ρ) = ... (ρ, ρ, z = 1), T (ρ) = 22 (ψ, ψ) = 00 (ρ), and  L (ρ) = 11 (ψ, ψ) = 00 (ρ) + 12 (ρ) + 21 (ρ) + 11 (ρ) + 22 (ρ). The flow equations of X k vk and Z k can be obtained from the momentum derivative of γ (2)ˆ . We next expand ∂(−i px ) γ (2)ˆ and ∂ p⊥2 γ (2)ˆ in terms of ρ−1 and retain the 1,ψψ

1,ψψ

1,ψψ

(T ) (L) leading order. By noting that P12 (q) = O(1) and P12 (q) = O(ρ−1 ), one finds that the leading order contributions to ∂l (X k vk ) and ∂l Z k come from γ (2) ˆ , γ (2) ˆ , 1,a−1,ψψ

1,b−1,ψψ

and γ (2) ˆ . The corresponding flow equations are given by Eqs. (4.90), (4.91), 1,b−2,ψψ and (4.92). Calculation of ∂l Tk We set  = ψˆ 2 ,   = ψˆ 2 and p = p  = 0 for the calculation of ∂l (X k Tk ). We denote each term in Eq. (4.171) as γ (2) ˆ ˆ , and so on. They are given as follows: 1,a−1,ψ ψ

γ (2)

1,a−2,ψˆ ψˆ

γ (2) ˆ ˆ 1,a−1,ψ ψ

= γ (2)

1,b−5,ψˆ ψˆ





= 2ρm F (ρm )

= γ (2)

1,b−7,ψˆ ψˆ

= 0,

(4.180)

  (L) (T ) (L) ∂l Rk (q) P11 (q, ωq ) P12 (q, ωq )P11 (q, ωq )

2 q,ωq

 (L) (L) (q, ωq )P21 (q, ωq ) +P11

(L) + P11 (q, ωq )

 (T )  (T ) (T ) (T ) × P12 (q, ωq )P11 (q, ωq ) + P11 (q, ωq )P21 (q, ωq ) , (4.181) γ (2) 1,b−1(±),ψˆ ψˆ



= 2ρm F (ρm )

 2

 (T ) (T ) (L) ∂l Rk (q) T (ρm )P21 (q)2 P12 (q)P11 (q)

q

 (L) (L) (T ) + L (ρm )P21 (q)2 P12 (q)P11 (q) ,

(4.182)

4.6 Appendix

121

γ (2)

1,b−2,ψˆ ψˆ

= 2ρm F  (ρm )2



 (T ) (T ) ∂l Rk (q) T (ρm )P12 (q)P21 (q)

q



 (L) (L) (L) (L) × P12 (q)P11 (q) + P11 (q)P21 (q) (L) (L) + L (ρm )P12 (q)P21 (q)

 (T )  (T ) (T ) (T ) × P12 (q)P11 (q) + P11 (q)P21 (q) , γ (2)

1,b−3(±),ψˆ ψˆ

= F  (ρm )



 (L) (T ) ∂l Rk (q) (12 (ρm ) + 11 (ρm ))P12 (q)2 P11 (q)

q (T ) (L) +ρm T (ρm )P12 (q)2 P11 (q)

γ (2)

1,b−4(±),ψˆ ψˆ

= F  (ρm ) 



 ,

(4.184)

 (L) ∂l Rk (q) (12 (ρm ) + 11 (ρm ))P12 (q)

q

(T ) (T ) × P12 (q)P11 (q) + (T )  +ρm T (ρm )P12 (q)

(T ) (T ) P11 (q)P21 (q)



 (L)  (L) (L) (L) × P12 (q)P11 (q) + P11 (q)P21 (q) , γ (2)

1,b−6,ψˆ ψˆ

(4.183)

(4.185)

= (2ρm )−1 [(N − 1)∂z 00 (ρm ) + 12 (ρm ) + 21 (ρm )]   (T ) (T ) (T ) (T ) (q)P11 (q) + P11 (q)P21 (q) , (4.186) × ∂l Rk (q) P12 q

where the frequency ωq in Eqs. (4.182)–(4.186) is set to zero. (T ) (L) (L) (q) = O(1), P12 (q) = O(ρ−1 ), and P11 (q) = O(ρ−2 ), one By noting that P12 finds that the leading order contributions to ∂l (X k Tk ) come from γ (2) ˆ ˆ , γ (2) ˆ ˆ , γ (2)

, 1,b−4,ψˆ ψˆ

1,b−1,ψ ψ

and γ (2)

. 1,b−6,ψˆ ψˆ

1,b−2,ψ ψ

To obtain the flow equation for Tk , we also need the flow

equation for X k . The leading order contributions to ∂l X k come from γ (2) γ (2) ˆ , 1,b−2,ψψ

from

γ (2) ˆ , 1,b−4,ψψ

and

γ (2) ˆ . 1,b−6,ψψ

ˆ 1,b−1,ψψ

,

Finally, the flow equation for Tk is calculated

  ∂l Tk = X k−1 ∂l (X k Tk ) − Tk ∂l X k   

1   , (4.187) = X k−1 − ∂l  (2)ˆ ˆ  −Tk ∂iω p ∂l  (2)ˆ  1;ψ( p),ψ( p) p=0 1;ψ( p),ψ(− p) p=0 2 which yields Eq. (4.93).

122

4.6.5

4 Dimensional Reduction and its Breakdown …

Numerical Scheme to Calculate the Fixed Point

∗ In this section, the numerical method used to obtain the fixed functions δ00 (z) and ∗ δ21 (z) is presented. Since the solution exhibits a nonanalytic behavior near z = 1, standard numerical techniques are not applicable. We define δ˜00 (z) = (N − 2)δ00 (z) (t) (t) (z) and δ˜21 (z) as follows, and δ˜21 (z) = (N − 2)δ21 (z), and the trial functions δ˜00 n max 

(t) (z) = a0 + δ˜00

an (1 − z)n/2

n=1 (t) (z) = b0 + δ˜21

n max 

bn (1 − z)n/2 .

(4.188)

∂l δ00 (z) = β00 [δ00 , δ21 ; ](z), ∂l δ21 (z) = β21 [δ00 , δ21 ; ](z).

(4.189)

n=1

We rewrite Eqs. (4.108) and (4.109) as

∗ ∗ (z) and δ˜21 (z) satisfy Then, the fixed functions δ˜00 ∗ ˜∗ , δ21 ;  = N − 2](z) = 0, β00 [δ˜00 ∗ ∗ ;  = N − 2](z) = 0. β21 [δ˜00 , δ˜21

(4.190)

The integral S({an }, {bn }) is introduced, 1 S({an }, {bn }) =

(t) ˜(t) {β00 [δ˜00 , δ21 ; N − 2](z)2

−1 (t) ˜(t) +β21 [δ˜00 , δ21 ; N − 2](z)2 }dz,

(4.191)

which vanishes if the true fixed functions are attained. The set of optimal parameters {an } and {bn } can be obtained by minimizing S({an }, {bn }). From Eq. (4.118), a0 and a1 satisfy −a0 + a02 − (1/2)a12 = 0. Since a0 = 1 when a1 = 0, we obtain the following constraint,   1 1 + 1 + 2a12 , (4.192) a0 = 2 which enables us to avoid the trivial solution {an } = {bn } = 0. Note that the integral S({an }, {bn }) has several local minima. One is chosen such that it recovers the fixed function Eq. (4.129) at N = 2. The truncation number is fixed at n max = 4. The ∗ (1) by less than one percent. inclusion of the higher order terms only changes η⊥ = δ00 For the case of the random anisotropy, the disorder correlators satisfy

4.6 Appendix

123 ∗ ∗ δ00 (−z) = −δ00 (z),

∗ ∗ δ21 (−z) = δ21 (z).

(4.193)

Therefore, we introduce the following trial functions: (t) (z) = a0 + δ˜00

(t) (z) = b0 + δ˜21

n max 

an z(1 − z 2 )n/2

n=1 n max 

bn (1 − z 2 )n/2 .

(4.194)

n=1

With these trial functions, Eq. (4.192) becomes a0 =

1 1+ 2



 1 + 4a12 .

(4.195)

By employing a similar method, the anomalous dimensions η for the random field and random anisotropy O(N ) models can be also calculated from Eq. (4.101). We have checked that they agree with the known values given in Refs. [12] and [13].

References 1. Haga T (2017) Dimensional reduction and its breakdown in the driven random-field O(N ) model. Phy. Rev B 96:184202 2. Kosterlitz JM, Thouless DJ (1973) Ordering, metastability and phase transitions in twodimensional systems. J Phys C 6:1181 3. Jose JV, Kadanoff LP, Kirkpatrick S, Nelson DR (1977) Renormalization, vortices, and symmetry-breaking perturbation in the two-dimensional planar model. Phys Rev B 16:1217 4. Giamarchi T, Le Doussal P (1996) Moving glass phase of driven lattices. Phys Rev Lett 76:3408 5. Le Doussal P, Giamarchi T (1998) Moving glass theory of driven lattices with disorder. Phys Rev B 57:11356 6. Balents L, Marchetti MC, Radzihovsky L (1998) Nonequilibrium steady states of driven periodic media. Phys Rev B 57:7705 7. Canet L, Chaté H, Delamotte B (2011) General framework of the non-perturbative renormalization group for non-equilibrium steady states. J Phys A Math Gen 44:495001 8. Berges J, Tetradis N, Wetterich C (2002) Non-perturbative renormalization flow in quantum field theory and statistical physics. Phys Rep 363:223 9. Tarjus G, Tissier M (2008) Nonperturbative functional renormalization group for random field models and related disordered systems. I. Effective average action formalism. Phys Rev B 78:024203 10. Tissier M, Tarjus G (2006) Unified picture of ferromagnetism, quasi-long-range order, and criticality in random-field models. Phys Rev Lett 96:087202 11. Zinn-Justin J (1989) Quantum field theory and critical phenomena. Clarendon Press, Oxford 12. Feldman DE (2000) Quasi-long-range order in the random anisotropy Heisenberg model: Functional renormalization group in 4 −  dimensions. Phys. Rev. B 61:382 13. Feldman DE (2002) Critical exponents of the random-field O(N ) model. Phys Rev Lett 88:177202

Chapter 5

Nonequilibrium Kosterlitz-Thouless Transition in the Three-Dimensional Driven Random Field XY Model

In Sect. 4.2, we have derived the dimensional reduction property, which states that the cross-section of the D-dimensional driven random field O(N ) model (DRFO(N )M) is essentially identical to the (D − 1)-dimensional pure O(N ) model. This property predicts that the three-dimensional (3D) driven random field XY model (DRFXYM) exhibits quasi-long-range order (QLRO) in the weak disorder regime and the Kosterlitz-Thouless (KT) transition at a critical disorder strength. However, it is not trivial. Recall the case of the four-dimensional (4D) random field XY model (RFXYM). The conventional dimensional reduction in equilibrium predicts that the 4D-RFXYM exhibits the KT transition. As shown in the end of Sect. 2.3.3, this prediction is not correct. In fact, the 4D-RFXYM exhibits logarithmic QLRO in the weak disorder regime and second-order transition at a critical disorder strength. To verify the existence of the QLRO and KT transition in the 3D-DRFXYM, a detailed investigation of the functional renormalization group (FRG) equation is required. First, let us briefly explain the theory of the conventional KT transition in the pure 2D XY model. At low temperatures, this model exhibits QLRO characterized by a continuously varying exponent. This QLRO phase is controlled by a line of fixed points, which is a consequence of the fact that the spin-wave model of the 2D XY model can be reduced to the massless free field theory. In the QLRO phase, tightly bounded vortex-antivortex pairs are formed and the only effect of them is to renormalize the elastic constant (helicity modulus) of the spin-wave model. At the transition temperature, the dissociation of such pairs leads to the vanishing of the helicity modulus and the QLRO is destroyed. Therefore, the strategy of our study is as follows. First, we show that the large-scale physics of the spin-wave model corresponding to the DRFXYM is identical to that of the massless free field theory. Second, we take into account the effect of the vortices by the renormalization of the helicity modulus.

© Springer Nature Singapore Pte Ltd. 2019 T. Haga, Renormalization Group Analysis of Nonequilibrium Phase Transitions in Driven Disordered Systems, Springer Theses, https://doi.org/10.1007/978-981-13-6171-5_5

125

126

5 Nonequilibrium Kosterlitz-Thouless Transition in the Three-Dimensional …

The outline of this chapter is as follows: 1. The original DRFXYM is too complicated for the FRG analysis. Therefore, by eliminating the amplitude fluctuations of the order-parameter field, we introduce a spin-wave model as an effective model valid in the weak disorder regime. We show that a naive calculation of the spin-wave model leads to a power-law decay of the correlation function in three dimensions. This is consistent with the prediction of the dimensional reduction. 2. We perform numerical simulations of the 3D-DRFXYM [1]. The correlation function in the steady states is calculated by numerically integrating the original equation of motion. The results suggest the existence of QLRO and the KT transition. By using the nonequilibrium relaxation method, we also determine the transition temperature as a function of the disorder and driving velocity. 3. We show that the spin-wave model flows to the massless free field theory in the RG procedure, and that it exhibits QLRO, wherein the correlation function shows power-law decay with an exponent that depends on the disorder strength and the driving velocity [2]. In Sect. 4.4.3, we have already shown that, within the first order of the disorder, the FRG equation of the 3D-DRFXYM has a line of fixed points, which corresponds to the massless free field theory. We go beyond this first order FRG calculation. By applying the nonperturbative functional renormalization group (NP-FRG) approach to the spin-wave model, we find that the fixed line is stable even if the higher-order contributions are considered. 4. The spin-wave model cannot describe vortices. Therefore, we next attempt to develop a phenomenological theory of the KT transition by taking into account the effect of the vortices [2]. The main idea of this theory is to introduce an effective elastic constant, which is assumed to evolve according to the RG equation similar to that of the 2D XY model. We also discuss the structural change of the vortices at the KT transition with the aid of the dimensional reduction. The vortices in the 3D XY model are lines, not points in contrast to the 2D case. The dissociation of the vortex-antivortex pairs in the 2D XY model corresponds to the breakdown of vortex rings in the 3D-DRFXYM.

5.1 Spin-Wave Calculation We recall the definition of the DRFXYM. Let φ(r) = (φ 1 (r), φ 2 (r)) be a two component real vector field. The Hamiltonian of the XY model with a quenched random field h(r) = (h 1 (r), h 2 (r)) is given by  H [φ; h] =

dD r

 α

 1 K |∇φ α |2 + U (ρ) − h · φ , 2

(5.1)

where ρ = |φ|2 /2 is the field amplitude and U (ρ) = (λ0 /2)(ρ − 1/2)2 is a doublewell potential. The random field obeys a mean-zero Gaussian distribution with

5.1 Spin-Wave Calculation

127

h α (r)h β (r  ) = h 20 δ αβ δ(r − r  ).

(5.2)

The dynamics are described by ∂t φ + v∂x φ = −

δ H [φ; h] + ξ, δφ

(5.3)

where the thermal noise satisfies ξ α (r, t)ξ β (r  , t  ) = 2T δ αβ δ(r − r  )δ(t − t  ).

(5.4)

The original model defined by Eqs. (5.1) and (5.3) is too complicated for renormalization group analysis. Since we are interested in the phase structure in the weak disorder regime, it is convenient to introduce the spin-wave model of the DRFXYM. We ignore the amplitude fluctuations of φ and define the single-valued phase parameter u ∈ (−∞, ∞) by (φ 1 , φ 2 ) = (cos u, sin u). The dynamics of u(r, t) are described by ∂t u + v∂x u = K ∇ 2 u + F(r; u) + ξ(r, t),

(5.5)

where F(r; u) = −h 1 (r) sin u + h 2 (r) cos u

(5.6)

is a random force, whose second cumulant is given by F(r; u)F(r  ; u  ) = h 20 cos(u − u  )δ(r − r  ).

(5.7)

The thermal noise ξ(r, t) satisfies ξ(r, t)ξ(r  , t  ) = 2T δ(r − r  )δ(t − t  ).

(5.8)

This model was also introduced in the context of the moving Bragg glass in Refs. [3– 5] to discuss the dynamics of the transverse displacement field of an elastic lattice driven in a random pinning potential. The spin-wave model is valid only when the order parameter varies slowly in space. Especially, it cannot describe topological defects, namely, vortices. Thus, the phase transition cannot be captured within the spin-wave model. In the following, we show that the correlation function, which is anisotropic due to the driving, decays with power-law form r −η from a naive calculation analogous to the case of the 2D pure XY model [6]. At zero temperature, after a sufficiently long time the solution of Eq. (5.5) reaches a stationary state, v∂x u = K ∇ 2 u − h 1 (r) sin u + h 2 (r) cos u.

(5.9)

128

5 Nonequilibrium Kosterlitz-Thouless Transition in the Three-Dimensional …

As discussed in Sect. 4.2, by introducing anti-commuting fields one can show that the random force F(r; u) can be replaced with the field-independent random force. Equation (5.9) then becomes a linear equation and one can easily calculate the correlation function. In this section, we present a different derivation of the QLRO without the anti-commuting fields. We follow the scheme presented in Ref. [7], wherein the correlation function for the random field XY model is calculated. Equation (5.9) can be formally solved as  u(r) =

  d 3 r  G(r − r  ) −h 1 (r  ) sin u(r  ) + h 2 (r  ) cos u(r  ) .

(5.10)

We have defined the Green’s function G(r) by its Fourier transform G(q) =

1 . K |q|2 + ivqx

(5.11)

The mean square relative displacement B(r1 − r2 ) = (u(r1 ) − u(r2 ))2  is calculated as  B(r1 − r2 ) =

 d 3 r  d 3 r  G(r1 − r  ) − G(r2 − r  )  × G(r1 − r  ) − G(r2 − r  )

× h 1 (r  )h 1 (r  ) sin u(r  ) sin u(r  )

+ h 2 (r  )h 2 (r  ) cos u(r  ) cos u(r  )

 −2 h 1 (r  )h 2 (r  ) sin u(r  ) cos u(r  ) .

(5.12)

We use factorization approximations such as h α (r)h β (r  ) sin u(r) sin u(r  )  h α (r)h β (r  ) × sin u(r) sin u(r  ).

(5.13)

Although this approximation is nontrivial, one can naively expect that it can be justified when the correlation length of u(r) is much larger than that of the random field. This condition is satisfied in the weak disorder regime h 20 K v. Then, we have   2 B(r1 − r2 ) = h 20 d 3 r  G(r1 − r  ) − G(r2 − r  ) . (5.14) By substituting the explicit form of the Green’s function, we have  B(r1 − r2 ) =

2h 20

d 3 q 1 − cos {q · (r1 − r2 )} . (2π )3 K 2 |q|4 + v 2 qx2

(5.15)

5.1 Spin-Wave Calculation

129

We investigate the asymptotic behavior of the mean square relative displacement Eq. (5.15) over a large distance r K /v. First, we consider the case in which r1 − r2 is parallel to v. Equation (5.15) is then rewritten in terms of the polar coordinate ∞ B(r1 − r2 ) =

2h 20 0

=

h 20 2π 2

dq (2π )3

∞

2π

dθq 2 sin θ

dφ 0

0

1 dq

0

π

ds −1

1 − cos {q|r1 − r2 | cos θ } K 2 q 4 + v 2 q 2 cos2 θ

1 − cos {q|r1 − r2 |s} , K 2q 2 + v2 s 2

(5.16)

where we have changed the variable s = cos θ . Since we are interested in the contribution from the small wave number regime q v/K , the range of the s-integral can be extended to (−∞, ∞). Thus, we have h 20 B(r1 − r2 )  2π K v

v/K 0



 K q2 1 1 − exp − |r1 − r2 | . dq q v

The integrand takes the maximal value at q = qm ∼ as q −1 for q qm . Therefore, we obtain h 20 B(r1 − r2 )  2π K v

v/K

√ v/(K |r1 − r2 |), and it decays

dq + const q

qm

v/K ln + const 2π K v qm h 20 = ln|r1 − r2 | + const. 4π K v

=

(5.17)

h 20

(5.18)

This logarithmic dependence on the distance is similar to that of the QLRO in the 2D XY model [6]. We next consider the case in which r1 − r2 is perpendicular to v. Equation (5.15) is then rewritten as  d 3 q 1 − cos {qz |r1 − r2 |} , (5.19) B(r1 − r2 ) = 2h 20 (2π )3 K 2 |q|4 + v 2 qx2 where r1 − r2 is assumed to be parallel to the z-axis. Since we are interested in the contribution from the small wave number regime q v/K , the term containing qx in q 4 of the denominator can be ignored. Thus, we have

130

5 Nonequilibrium Kosterlitz-Thouless Transition in the Three-Dimensional …

 B(r1 − r2 ) 

2h 20 q Tc . The relaxation of the magnetization M(t) and its scaling plot are shown in the panels (a) and (b) of Fig. 5.3. The system size is 603 . M(t) is obtained by averaging over 100 independent runs. From the analogy to the KT transition in the 2D XY model √ [9], the correlation length is expected to diverge exponentially as ξ ∼ exp(a/ T − Tc ). Thus, we assume that the relaxation time diverges in the same way, 

A . (5.29) τ (T ) = B exp √ T − Tc By fitting τ (T ) into Eq. (5.29) with parameters A, B, and Tc , we can estimate the transition temperature. The best fitting is shown in the panel (c) of Fig. 5.3. Figure 5.4 shows the schematic phase diagram with respect to the strength of the disorder h 0 , driving velocity v, and temperature T . The QLRO phase appears in the large-v and low-T regime. In the region wherein h 0 = 0, the LRO phase exists because the model is identical to the 3D pure XY model in the moving frame. The high-T regime corresponds to the disordered phase. The phase boundary between the QLRO and disordered phases at T = 0 is approximately given by h 20 ∼ K v. The inset displays the transition temperature Tc determined from the nonequilibrium relaxation method. Note that the infinitesimally small random field breaks the LRO and leads to the QLRO. For an arbitrarily large value of h 0 , the QLRO is observed for sufficiently large values of v.

5.3 RG Analysis of the Spin-Wave Model

135

5.3 RG Analysis of the Spin-Wave Model In the previous section, we have shown that numerical results support the existence of the 3D KT transition. However, it is difficult to obtain a definite conclusion concerning the large-scale behavior of the model only from the numerical simulations. Therefore, in this section, we perform a FRG analysis of the spin-wave model of the DRFXYM and show the existence of the QLRO. In Sect. 4.4.3, we have shown that, at least in the first order of the disorder strength, the flow equation of the renormalized disorder cumulant (4.125) has a fixed line characterizing the QLRO in three dimensions. However, one cannot exclude a possibility that the beta function of the disorder cumulant vanishes in the first order but has a finite contribution in the higher order. To verify the existence of the QLRO, we have to show that the fixed line is stable with respect to these higher-order contributions. Therefore, we are required to go beyond the first order calculation. This is because we consider the spin-wave model, not the original DRFXYM. As explained in the end of Sect. 3.2.3, the NPRG approach fails to predict the exact fixed line of the 2D pure XY model. The higher order contributions to the beta function do not vanish due to an artificial effect associated with the massive (amplitude) mode. When we perform the NP-FRG analysis of the original DRFXYM, it is difficult to distinguish whether nonvanishing higher-order contributions are artifact or not. To avoid such a problem, we consider the spin-wave model in which the massive mode is eliminated.

5.3.1 Exact Flow Equation for the Effective Action First, the equation of motion (5.5) is cast into a field theoretical formalism in the similar way described in Sect. 4.3.1. We introduce a replicated field Ua = t (u a , uˆ a ), a = 1, . . . , n to take the average over the disorder, and then we have the bare action, S[{Ua }] =

 a rt

−

uˆ a [∂t u a − T uˆ a + v∂x u a − K ∇ 2 u a ]

1 2 a,b

 uˆ a,t uˆ b,t  B (u a,t − u b,t  ),

(5.30)

r tt 

where B (u) = h 20 cos u is the second cumulant of the bare random force. By introducing source fields Ja = t ( ja , jˆa ), a = 1, . . . , n, the partition function Z [{Ja }] and generating functional W [{Ja }] read  Z [{Ja }] =

  DU exp −S[{Ua }] + a

rt

t

 Ja Ua ,

(5.31)

136

5 Nonequilibrium Kosterlitz-Thouless Transition in the Three-Dimensional …

W [{Ja }] = ln Z [{Ja }].

(5.32)

The effective action is defined as a Legendre transform: [{Ua }] = −W [{Ja }] +



t

Ja Ua ,

(5.33)

a rt

where Ua and Ja are related by δW [{Ja }] . δ Ja

Ua =

(5.34)

We define the scale-dependent effective action k by adding, to the original action, a momentum-dependent mass term 1 Sk [{Ua }] = 2 a

 t

Ua (q, ω) Rk (q) Ua (−q, −ω),

(5.35)

 0 Rk (q) , Rk (q) 0

(5.36)

q,ω

where Rk (q) is a 2 × 2 matrix,

Rk (q) =

where Rk (q) is an infrared cutoff function, which has a constant value proportional to k 2 for |q| k and rapidly decreases for |q| > k. The scale-dependent partition function Z k [{Ja }] and generating functional Wk [{Ja }] read  Z k [{Ja }] =

  DU exp −S[{Ua }] − Sk [{Ua }] + a

t

 Ja Ua ,

(5.37)

rt

Wk [{Ja }] = ln Z k [{Ja }].

(5.38)

Then, the scale-dependent effective action is defined as k [{Ua }] = −Wk [{Ja }] +



t

Ja Ua − Sk [{Ua }],

(5.39)

a rt

where Ua and Ja are related by Ua =

δWk [{Ja }] . δ Ja

(5.40)

5.3 RG Analysis of the Spin-Wave Model

137

It can be shown that k=0 =  and limk→∞ k = S. The exact flow equation for k is given by

−1 1 ˆ k (q)  (2) + R ˆ k (q) , Tr∂k R k 2

∂k  k =

(5.41)

where k(2) is the second functional derivative of k and Tr denotes an integration over momentum and frequency as well as a sum over replica indices and the two conjugate fields {u, u}. ˆ We have introduced a 2n × 2n matrix ˆ k (q) = Rk (q) ⊗ In , R

(5.42)

where In is the n × n unit matrix, which acts on the space of the replica index. k is expanded by increasing number of free replica sums as k [{Ua }] =

∞   (−1) p−1  p,k [Ua1 , . . . , Ua p ]. p! p=1 a ,...,a 1

(5.43)

p

The exact flow equations for  p,k are given in Sect. 3.4.2.

5.3.2 Flow Equations of the Disorder Cumulants Let us introduce an approximation for the functional form of  p,k . We employ the following functional form for the one-replica part,  1,k [U ] =

u[X ˆ k (∂t u − Tk u) ˆ + v∂x u − K ∇ 2 u],

(5.44)

rt

where X k and Tk are the scale-dependent relaxation coefficient and temperature, respectively. Note that the definition of v is slightly different from that in Eq. (4.45), here, X k vk in Eq. (4.45) is redefined as v. For the multi-replica part,   p,k [U1 , . . . , U p ] =

uˆ a1 ,t1 . . . uˆ a p ,t p  p,k (u a1 ,t1 , . . . , u a p ,t p ),

(5.45)

r t1 ...t p

where  p,k (u 1 , . . . , u p ) is the p-th cumulant of the renormalized random force, which is a fully symmetric and satisfying  p,k (u 1 + 2π, u 2 , . . . , u p ) =  p,k (u 1 , u 2 , . . . , u p ),

(5.46)

 p,k (u 1 + λ, . . . , u p + λ) =  p,k (u 1 , . . . , u p ),

(5.47)

138

5 Nonequilibrium Kosterlitz-Thouless Transition in the Three-Dimensional …

for an arbitrary λ. Since the bare random force is chosen as Gaussian, the higher order cumulants vanish at k = . However, they can be finite in the coarse-grained model due to its nonlinearity. Within Eqs. (5.44) and (5.45), the elastic constant K and the (2) is independent driving velocity v do not change along the RG flow because ∂l 1,k of the external momenta, while the relaxation coefficient X k and temperature Tk can be renormalized. From the functional form Eq. (5.45), the flow equation for  p can be obtained from ∂k  p,k (u 1 , . . . , u p ) =

δp ∂k  p,k [U1 , . . . , U p ], δ uˆ 1 . . . δ uˆ p

(5.48)

where the functional derivative in the right-hand side is evaluated for a uniform field configuration: u 1,r t ≡ u 1 , . . . , u p,r t ≡ u p and uˆ 1,r t ≡ 0, . . . , uˆ p,r t ≡ 0. Insertion of Eqs. (5.44) and (5.45) into Eqs. (3.156) and (3.161) leads to the flow equation of  p . In the following, we write the flow equation in terms of a renormalization scale l = − ln(k/), which goes from 0 to ∞ as k approaches to 0. The exact flow equations of  p have the one-replica propagator, −1  (2) Pk [U ] = 1,k [U ] + Rk .

(5.49)

We denote the matrix elements of Pk [U ] as Pi j (q, ω), where i and j represent the two conjugate fields u and u. ˆ Pi j (q, ω) is given by 2X k Tk , + (X k ω − qx v)2 1 , P12 (q, ω) = M(q) + i(X k ω − qx v) 1 P21 (q, ω) = , M(q) − i(X k ω − qx v) P22 (q, ω) = 0, P11 (q, ω) =

M(q)2

(5.50)

where M(q) = K |q|2 + Rk (q). Below, we employ simplified notations such as P12 (q) = P12 (q, ω = 0). It is convenient to define the following integrals, L− n

1 =− 2

 ∂l Rk (q)[n P21 (q)n+1 + n P12 (q)n+1 ],

(5.51)

q

L+ n =−

1 2

 ∂l Rk (q) q

n 

2P21 (q)n+1− j P12 (q) j ,

(5.52)

j=1

where ∂l = −k∂k . In the following, we omit the subscript k in  p,k . For the zero temperature case T = Tk = 0, the flow equation for 2 is given as follows:

5.3 RG Analysis of the Spin-Wave Model

∂l 2 (u 1 , u 2 ) =

1 [(1) + (2) + (3) + (4) + perm(u 1 , u 2 )], 2

139

(5.53)

+ (1) = (11) 2 (u 1 , u 2 )2 (u 1 , u 2 )L 2 ,

(5.54)

(01) − (2) = (10) 2 (u 1 , u 2 )2 (u 1 , u 2 )L 2 ,

(5.55)

+ (3) = (20) 2 (u 1 , u 2 )2 (u 1 , u 1 )L 2 ,

(5.56)

(4) = −2(100) (u 1 , u 1 , u 2 )L − 3 1,

(5.57)

where we have used simplified notations such as (11) 2 (u a , u b ) = ∂u 1 ∂u 2 2 (u a , u b ), (20) 2 (u a , u b ) = ∂u 1 ∂u 1 2 (u a , u b ).

(5.58)

The flow equation for 3 is given as follows: ∂l 3 (u 1 , u 2 , u 3 ) = (A − 1) + · · · + (A − 5) + (B − 1) + · · · + (B − 6) + (C − 1) + perm(u 1 , u 2 , u 3 ), (5.59) (01) + (A − 1) = −(20) 2 (u 1 , u 2 )2 (u 1 , u 3 )2 (u 1 , u 3 )L 3 ,

(5.60)

(11) + (A − 2) = −(10) 2 (u 1 , u 2 )2 (u 1 , u 3 )2 (u 1 , u 3 )L 3 ,

(5.61)

(10) (01) − (A − 3) = −(10) 2 (u 1 , u 2 )2 (u 1 , u 3 )2 (u 1 , u 3 )L 3 ,

(5.62)

(10) + (A − 4) = −(20) 2 (u 1 , u 2 )2 (u 1 , u 3 )2 (u 1 , u 1 )L 3 ,

(5.63)

(01) + (A − 5) = −(11) 2 (u 1 , u 2 )2 (u 2 , u 3 )2 (u 1 , u 3 )L 3 ,

(5.64)

(B − 1) =

1 (20)  (u 1 , u 2 )3 (u 1 , u 1 , u 3 )L + 2, 2 2

(100) (B − 2) = (10) (u 1 , u 1 , u 3 )L − 2 (u 1 , u 2 )3 2,

(B − 3) =

1 (11)  (u 1 , u 2 )3 (u 1 , u 2 , u 3 )L + 2, 2 2

(010) (B − 4) = (10) (u 1 , u 2 , u 3 )L − 2 (u 1 , u 2 )3 2,

(5.65) (5.66) (5.67) (5.68)

140

5 Nonequilibrium Kosterlitz-Thouless Transition in the Three-Dimensional …

(B − 5) =

1 2 (u 1 , u 2 )(110) (u 1 , u 2 , u 3 )L + 3 2, 2

(5.69)

(B − 6) =

1 2 (u 1 , u 1 )(200) (u 1 , u 2 , u 3 )L + 3 2, 4

(5.70)

1 (C − 1) = − (1000) (u 1 , u 1 , u 2 , u 3 )L − 1. 2 4 5.3.2.1

(5.71)

Equilibrium Case

For the equilibrium case (v = 0), the cutoff function Rk (|q|2 ) is written as

Rk (|q|2 ) = K k 2 r˜

 |q|2 . k2

(5.72)

In the following calculation, the optimized cutoff function is employed, r˜ (y) = (1 − y)(1 − y),

(5.73)

where (x) is the step function (see Sect. 3.2.2). If one uses this cutoff function, the integrals in the flow equations are calculated as + −n D−2n k L− n = L n = 2n K

4 AD, D

(5.74)

where A D −1 = 2 D+1 π D/2 (D/2). We next express the flow equations in terms of renormalized dimensionless quantities. In the following, the cutoff scale  is set to unity. The renormalized dimensionless cumulants are defined as follows: δ2 (u 1 , u 2 ) =

δ3 (u 1 , u 2 , u 3 ) =

16 A D K −2 k D−4 2 (u 1 , u 2 ), D

16 D

2

A D 2 K −3 k 2D−6 3 (u 1 , u 2 , u 3 ),

(5.75)

(5.76)

where note that the scaling dimension of u is zero (ζ = 0) due to its periodicity. We also introduce the notation

δ3 (u a − u b ) =

δ(u a − u b ) = δ2 (u a , u b ),

(5.77)

1 {∂u δ3 (u a , u a , u b ) + ∂u 1 δ3 (u b , u b , u a )}. 2 1

(5.78)

5.3 RG Analysis of the Spin-Wave Model

141

The flow equations for δ(u) and δ3 (u) are given by ∂l δ(u) = −(D − 4)δ(u) + δ  (u)[δ(0) − δ(u)] − δ  (u)2 − δ3 (u).

(5.79)

3 ∂l δ3 (u) = −(2D − 6)δ3 (u) − {[δ(u) − δ(0)]δ  (u)2 − δ  (0)2 δ(u)} 2 + O(δ2 δ3 ) + O(δ4 ). (5.80) Equation (5.79) without δ3 has been already derived in Sect. 3.3.2. Note that an “anomalous” term δ  (0)2 in Eq. (5.80) vanishes if the second derivative of δ(u) is finite at u = 0. If one considers a fixed point δ∗ (u) at D = 4 + , δ3 (u) in Eq. (5.79) can be eliminated, and finally we have 0 = −δ∗ (u) + δ∗ (u)[δ∗ (0) − δ∗ (u)] − δ∗ (u)2 + C{[δ∗ (u) − δ∗ (0)]δ∗ (u)2 − δ∗ (0)2 δ∗ (u)} ,

(5.81)

The constant C depends on the functional form of Rk (q). For the optimized cutoff function Eq. (5.73), C = 3/4. Equation (5.81) was also obtained from the two-loop perturbative calculation [10, 11], however, C = 1/2 in this case.

5.3.2.2

Nonequilibrium Case

For the nonequilibrium case, the transverse and longitudinal momenta q⊥ , qx are measured in units of k and k 2 , respectively, due to the anisotropy of the system. For simplicity, we assume that an infrared cutoff function is independent of qx ,  |q⊥ |2 , Rk (q) = K k r˜ k2

2

(5.82)

with Eq. (5.73). The integrals in the flow equations are calculated as follows: L± n =

4 A D−1 K −n+1 k D−2n+1 v −1ln± (z k ), D−1

(5.83)

where z k = v −2 K 2 k 2 = v −2 K 2 e−2l , which is the ratio of the longitudinal elastic term K ∂x2 u to the advection term v∂x u, and ln− (z)

n = π

∞ −∞

d x (1 + zx 2 + i x)−(n+1) ,

(5.84)

142

5 Nonequilibrium Kosterlitz-Thouless Transition in the Three-Dimensional …

ln+ (z)

1 = π

∞ dx

n 

(1 + zx 2 + i x)−(n+1− j) (1 + zx 2 − i x)− j .

(5.85)

j=1

−∞

One can easily confirm that ln+ (0) = 1 while ln− (z) ∼ z n for a small z. The renormalized dimensionless cumulants are defined as follows: 4 A D−1 K −1 v −1 k D−3 2 (u 1 , u 2 ), D−1 2

4 δ3 (u 1 , u 2 , u 3 ) = A D−1 2 K −1 v −2 k 2D−4 3 (u 1 , u 2 , u 3 ). D−1 δ2 (u 1 , u 2 ) =

(5.86)

(5.87)

The flow equations for δ(u) and δ3 (u) are given by ∂l δ(u) = −(D − 3)δ(u) + l2+ (zl )δ  (u)[δ(0) − δ(u)] − l2− (zl )δ  (u)2 − 2l1− (zl )δ3 (u),

(5.88)

∂l δ3 (u) = −(2D − 4)δ3 (u) − 2l3+ (zl ){δ  (u)δ  (u)[δ(u) − δ(0)]} − l3− (zl ){δ  (u)3 − δ  (0)2 δ  (u)} + O(δ4 ) + O(δ2 δ3 ).

(5.89)

5.3.3 RG Evolution of the Disorder Cumulants The flow equation of the p-th cumulant δ p contains the ( p + 1)-th cumulant δ p+1 . Therefore, we obtain an infinite series of the coupled flow equations. However, in the nonequilibrium case (5.88), the contribution of δ3 vanishes in the large scale limit because l1− (zl ) ∼ e−2l . Moreover, one can easily check that the flow equation of δ p always contains δ p+1 in the form of l1− (zl )δ p+1 . In fact,  p+1 appears in the flow equation for  p as ∂l  p [U1 , . . . , U p ] =

n Tr∂l Rk P[U1 ] (110...0) [U1 , U1 , U2 , . . . , U p ]P[U1 ],(5.90) p+1 2

and this term leads to l1− (zl )δ p+1 . This fact implies that, rather surprisingly, the infinite series of the flow equations can be decoupled in the large scale limit. This conclusion does not depend on the explicit functional form of the cutoff function r˜ (y) in Eq. (5.82). It should be recalled that such a decoupling does not occur in the equilibrium case (see Eq. (5.80)). Let us consider the RG evolution of the disorder correlator in the weak disorder regime. In the large length scale, one can set zl to zero. Thus, at D = 3, we have a simple FRG equation,

5.3 RG Analysis of the Spin-Wave Model

143

Fig. 5.5 RG evolution of δ(u) corresponding to the 3D-DRFXYM (a) and 4D-RFXYM (b). The bare cumulant (red solid curve) is given by the cosine function. The renormalization scale l increases from the bottom to the top at u = π . For the 3D-DRFXYM, δ(u) converges to a finite constant, while for the 4D-RFXYM, it goes to zero in the large-scale limit. Reprinted figure with permission from Ref. [2]. Copyright (2018) by the American Physical Society. https://doi.org/10.1103/PhysRevE. 98.032122

∂l δ(u) = δ  (u)[δ(0) − δ(u)],

(5.91)

which has been already derived in the previous chapter, Eq. (4.125). Remarkably, from the decoupling property mentioned above, Eq. (5.91) is found to be exact, at least within the ansatz Eqs. (5.44) and (5.45). Let us consider the RG evolution of δ(u) described by Eq. (5.91). First note that, if δ(u) is analytic around u = 0, we obtain the flow equation for the second derivative δ  (0) from Eq. (5.91), ∂l δ  (0) = −δ  (0)2 .

(5.92)

Therefore, δ  (0) diverges to minus infinity at a finite renormalization scale l L , which corresponds to the (dynamical) Larkin scale, and a linear cusp is then generated at the origin, δ(u)  δ(0) + δ  (0+ )|u|. We numerically integrate Eq. (5.91) to obtain the RG evolution of δ(u), which is shown in the panel (a) in Fig. 5.5. The bare cumulant is given by the cosine function δB (u) = h 20 /(4π K v) cos u, where (4π K v)−1 comes from the factor in Eq. (5.86). At the Larkin scale, linear cusps are generated at u = 2π m and δ(u) evolves into a parabolic profile. In the large length limit l → ∞, δ(u) eventually becomes a constant function δl=∞ (u) = δB (0). Since the flow equations of the higher-order cumulants δ p ( p ≥ 3) contain only the derivative of the same and lower-order cumulants δ2 , . . . , δ p in this limit, we can easily see that all higher-order cumulants vanish. Therefore, there exists a family of stable fixed points δl=∞ (u) = const. and δ3 = δ4 = · · · = 0. At the fixed points, the system is considered as the massless free field theory in the sense that the renormalized random force is independent of the field u and Gaussian. It is worth noting that the fixed points δl=∞ (u) = const. = 0 is peculiar to the nonequilibrium case because it does not satisfy the potentiality condition,  2π

0

δl (u)du = 0.

(5.93)

144

5 Nonequilibrium Kosterlitz-Thouless Transition in the Three-Dimensional …

Let us show that this fixed point corresponds to QLRO, wherein the correlation function for the transverse direction (r ⊥ ex ) exhibits power-law decay C(r) = ei(u(r )−u(0))  ∼ |r|−η⊥ ,

(5.94)

with η⊥ = δl=∞ (0). From Eq. (5.5) with T = 0, a steady state satisfies v∂x u = K ∇ 2 u + F(r; u).

(5.95)

Since we are interested in the large-scale behavior of the steady state, the bare random ˜ force F(r; u) can be replaced with the renormalized random force F(r), which is Gaussian and satisfies ˜ F(r ˜  ) = l=∞ (0)δ(r − r  ) = 4π K vδl=∞ (0)δ(r − r  ). F(r)

(5.96)

˜ Note that F(r) is independent of the field u because the stable fixed point δl=∞ (u) is just a constant function. Therefore, we have ˜ r ⊥ ), v∂x u = K ∇⊥2 u + F(x,

(5.97)

where we have omitted the longitudinal elastic term K ∂x2 u because it is small compared to the advection term v∂x u in the large length scale. As in Sect. 4.2, by con˜ sidering the coordinate x as a fictitious time and F(x, r ⊥ ) as a thermal noise, one finds that Eq. (5.97) is identical to the 2D Edwards-Wilkinson equation, which is the spin-wave equation of the 2D pure XY model, with a temperature Teff =

l=∞ (0) = 2π K δl=∞ (0). 2v

(5.98)

Therefore, we conclude that this model exhibits QLRO with an exponent η⊥ = Teff /(2π K ) = δl=∞ (0). Since η⊥ continuously depends on the bare disorder strength h 0 and velocity v, this QLRO is qualitatively different from the Bragg glass phase in the random field XY model at D = 4 − , wherein the exponent is universal (see Sect. 2.3.3). To make comparison between the equilibrium and nonequilibrium cases, we also show the RG evolution of δ(u) for the 4D-RFXYM in the panel (b) of Fig. 5.5. From Eq. (5.79), the flow equation is given by ∂l δ(u) = δ  (u)[δ(0) − δ(u)] − δ  (u)2 ,

(5.99)

at weak disorder. The difference between Eqs. (5.99) and (5.91) is the presence of the last term −δ  (u)2 . Note that δl (u) satisfies the potentiality condition Eq. (5.93) for all renormalization scale l. The bare cumulant is given by the cosine function δB (u) = h 20 /(8π 2 K 2 ) cos u. At the Larkin scale, linear cusps are generated at u = 2π m and δ(u) evolves into a parabolic profile. Finally, δ(u) converges to zero according to

5.3 RG Analysis of the Spin-Wave Model

145

δ(u) ∼ l −1 (see the end of Sect. 2.3.3 for the detailed analysis). Thus, the disorder is marginally irrelevant. This means that the correlation function of the 4D-RFXYM behaves as C(r ) ∼ (ln r )−α , which decays more slowly than usual power-law. Let us emphasize the different points between our results and the previous ones such as Refs. [4] and [5]. These authors derived the one-loop FRG equation (5.91) and they investigated the transverse pinning of vortex lattices driven in impure type-II superconductors. At zero temperature, the cuspy behavior of the disorder cumulant δ(u) leads to trapping of the phase u, in the context of the vortex lattices, which is considered as the displacement field for the perpendicular direction to the driving velocity. The critical force to depin the phase is related to the amplitude of the discontinuity in δ  (u) at u = 0. In Refs. [4] and [5], by using Eq. (5.91) the authors calculated this transverse depinning threshold force of the driven vortex lattices. However, these authors did not mention the crucial fact that this FRG equation has a fixed line at the critical dimension D = 3, in contrast to the RFXYM whose FRG equation has only a trivial fixed point at D = 4. Especially, our study shows that there is a significant difference between the topologically ordered phase in the 3D-DRFXYM and the Bragg glass phase in the (4 − )-dimensional RFXYM. Furthermore, by employing the nonperturbative FRG formalism, we calculated the higher-order corrections to Eq. (5.91) and concluded that they have no effect on the stability of the fixed line. Such an analysis of the higher-order terms is difficult to perform within the framework of the perturbative field theory presented in the previous studies. We have found that, in the weak disorder regime, the spin-wave model of the DRFXYM shows the QLRO in three dimensions. When the strength of the disorder reaches a some critical value, we expect that the 3D-DRFXYM exhibits a phase transition to a disordered phase, wherein the correlation function C(r) shows exponential decay. However, it should be recalled that the spin-wave model is not valid at strong disorder because it cannot describe the topological defects (vortices). The QLRO phase with a large η⊥ can be destroyed by proliferation of the vortices. To describe the KT transition in the 3D-DRFXYM, we are required to consider the effect of the vortices.

5.4 Effect of Vortices In this section, we attempt to construct a phenomenological theory for the 3D-KT transition by taking into account the effect of the vortices. First, note that the dimensional reduction holds in the large length scale. The criterion for its breakdown is that the renormalized disorder correlator δl=∞ (u) has a linear cusp at the fixed point. For the spin-wave model, it was shown that δl=∞ (u) is analytic because it is just a constant. This implies that the dimensional reduction recovers in the spin-wave model. Since the fluctuations of the field amplitude, which are ignored in the spinwave approximation, do not affect the nonanalytic nature of the disorder correlator, one can expect that the dimensional reduction also holds in the presence of the vortices. Therefore, the transverse section of the 3D-DRFXYM is equivalent to the 2D

146

5 Nonequilibrium Kosterlitz-Thouless Transition in the Three-Dimensional …

pure XY model by considering the longitudinal coordinate x as a fictitious time (see Sect. 4.2). This leads to the conclusion that the value of the exponent η⊥ is equal to 1/4 at the KT transition point. This prediction is consistent with the numerical simulation in Sect. 5.2. Figure 5.2 shows η⊥ as a function of the temperature for fixed disorder strength and driving velocity. We have already remarked that the value of η⊥ is close to 1/4 at the transition temperature. Although the argument based on the dimensional reduction makes sense only at the zero-temperature, we expect that at the transition point the exponent η⊥ is universal even when temperature is finite. To develop a quantitative theory of the vortices for the whole range of the disorder strength is too ambitious at this time. Therefore, we attempt to establish the simplest theory that correctly predicts η⊥ in the weak disorder regime and near the transition. More precisely, this theory can reproduce the results of the spin-wave model in the weak disorder limit and recovers η⊥ = 1/4 at the transition point. It gives a simple interpolation between the weak and strong disorder regimes. To derive the flow equations of this theory, we employ the following assumptions. • The main effect of the vortices is to modify the elastic constant K . In other words, the phenomenological theory is defined by replacing K in the spin-wave model with K eff,⊥ and K eff, for the transverse and longitudinal directions, respectively. These effective elastic constants are smaller than K . • The effective elastic constant for the transverse direction K eff,⊥ obeys the RG equation similar to that of the 2D pure XY model because the dimensional reduction recovers in the large length scale. • We ignore the anisotropy of the effective elastic constant, K eff,  K eff,⊥ = K eff . The RG equation of the 2D pure XY model is given by  T = 2π 3 y 2 , K

 K dy = 2−π y, dl T d dl

(5.100) (5.101)

where y is the fugacity of the vortices, which is proportional to the density of the vortices [12]. The dimensional reduction says that, in the 3D-DRFXYM, the ratio of the disorder strength to the driving velocity corresponds to the temperature in the 2D pure XY model. Thus, the RG equation for K eff can be obtained by replacing T in the first equation of Eq. (5.100) with Teff = (0)/(2v), (0) d 2v dl

1 K eff

 = 2π 3 y 2 .

(5.102)

In Eq. (5.102), the derivative operator d/dl acts only on K eff , not on the dimensionfull disorder (0), which is not renormalized by the vortices. Note that, in the absence of the vortices (y = 0), d K eff /dl = 0. In terms of the dimensionless disorder δ(0) =

5.4 Effect of Vortices

147

(0)/(4π K eff v), Eq. (5.102) can be rewritten as −δ(0)

d ln K eff = π 2 y 2 . dl

(5.103)

From Eq. (5.101), the RG equation of the vortex fugacity can be obtained from the same procedure,

 1 dy = 2− y. dl 2δ(0)

(5.104)

The RG equations for δ(u) and δ3 (u) are obtained from Eqs. (5.88) and (5.89) by replacing K with K eff , ∂l δ(u) = l2+ (zl )δ  (u)[δ(0) − δ(u)] − l2− (zl )δ  (u)2 d − 2l1− (zl )δ3 (u) − δ(u) ln K eff , dl ∂l δ3 (u) = −2δ3 (u) − 2l3+ (zl ){δ  (u)δ  (u)[δ(u) − δ(0)]} d − l3− (zl ){δ  (u)3 − δ  (0)2 δ  (u)} − δ3 (u) ln K eff , dl

(5.105)

(5.106)

2 −2l e . The last terms proportional to d ln K eff /dl in the rightwhere zl = v −2 K eff hand sides of Eqs. (5.105) and (5.106) result from the flow of K eff in δ(u) = (u)/(4π K eff v) and δ3 (u) = k 2 3 (u)/(16π 2 K eff v 2 ). The bare value of y is also obtained from the replacement T → 2π K δB (0) in the 2D pure XY model [12]. Therefore, we have 

π (5.107) y0 = exp − δB (0)−1 . 4

Equations (5.103), (5.104), (5.105), and (5.106) constitute a closed set of flow equations. Unfortunately, Eq. (5.106) is not numerically stable because it contains the third derivative. Since δ(u) and δ3 (u) starting from any analytic initial cumulants eventually develop linear cusps beyond the Larkin scale, let us consider a modified model bare cumulant of which already has a linear cusp, δB (0− ) = δB (0+ ). We write the nonanalytic cumulants as follows, δl (u)  δ3,l (u)

= a1 (l)(u − π )2 + b1 (l), = a2 (l)(u − π )2 + b2 (l),

(5.108)

148

5 Nonequilibrium Kosterlitz-Thouless Transition in the Three-Dimensional …

for u ∈ [0, 2π ]. By using the periodicity δ(u ± 2π ) = δ(u) and δ3 (u ± 2π ) = δ3 (u), Eq. (5.108) is extended to the whole region u ∈ (−∞, ∞). By substituting Eq. (5.108) into Eqs. (5.105) and (5.106), we have d da1 = −{2l2+ (zl ) + 4l2− (zl )}a12 − 2l1− (zl )a2 − a1 ln K eff , dl dl

(5.109)

da2 d = −2a2 − 24{l3+ (zl ) + l3− (zl )}a13 − a2 ln K eff , dl dl

(5.110)

d d δ(0) = −4π 2 l2− (zl )a12 − 2l1− (zl )δ3 (0) − δ(0) ln K eff , dl dl

(5.111)

d  d δ3 (0) = −2δ3 (0) − 16π 2 {l3+ (zl ) + l3− (zl )}a13 − δ3 (0) ln K eff . (5.112) dl dl The anomalous term δ  (0)2 in Eq. (5.106), which vanishes for an arbitrary analytic δ(u), can be evaluated as 4π 2 a12 . It is worth noting that, as the parabolic functions Eq. (5.108) are substituted, the right-hand sides of Eqs. (5.105) and (5.106) neither lead to cubic nor quartic terms of u. Thus, the set of equations (5.109)–(5.112) is exact, provided Eqs. (5.105), (5.106), and (5.108). We numerically solve this set of ordinary differential equations with the following initial condition: δ(0) = δB (0) =

h 20 , 4π K v

δ3 (0) = 0, 3 a1 = δB (0), 2π 2 a2 = 0,

(5.113)

where the initial value of a1 is chosen such that the potentiality condition Eq. (5.93) is satisfied. Figure 5.6 shows the RG trajectories in the parameter space of the disorder δ(0) and vortex fugacity y. When the bare disorder strength is smaller than a critical value, δB (0) < δKT , the RG trajectories flow to the fixed line. This regime corresponds to the QLRO phase. In the weak disorder limit, the critical exponent η⊥ = δl=∞ (0) is identical to that of the naive spin-wave calculation Eq. (5.23). When the bare disorder strength is larger than the critical value, δB (0) > δKT , the trajectories diverge δ(0), y → ∞. This regime corresponds to a disordered phase. At δB (0) = δKT , η⊥ = 1/4 as in the conventional KT transition. When η⊥ = δl=∞ (0) is plotted as a function of the bare disorder δB (0), one can see a small deviation from Eq. (5.23) due to two distinct effects. One of them is the nontrivial renormalization of the dimensionful disorder strength (0), which is a consequence of the cusp in the renormalized cumulant. This effect lowers the value of η⊥ slightly, compared to Eq. (5.23). The second effect that leads to the deviation in η⊥ is the reduction of the elastic constant

5.4 Effect of Vortices

149

Fig. 5.6 RG trajectories for the spin-wave model with the correction of the vortices. The horizontal and vertical axes represent δ(0) and y, respectively. The thin dashed curve depicts the bare value of the vortex fugacity (5.107). The thick line on the horizontal axis represents the fixed line corresponding to the QLRO. The inset displays the trajectories near the endpoint of the fixed line. Reprinted figure with permission from Ref. [2]. Copyright (2018) by the American Physical Society. https://doi.org/10.1103/PhysRevE.98.032122

due to the vortices. Since the vortex density rapidly increases near the transition point δKT , η⊥ also shows strong dependence on the disorder strength. Note that this theory cannot make the precise prediction of the exponent η⊥ , while it may be useful to understand the qualitative features of η⊥ as a function of the bare disorder. It may be recalled that in the conventional KT transition in the 2D XY model the effective elastic constant (helicity modulus) K eff exhibits a discontinuous jump at the transition point and its amplitude satisfies the following universal relation [12]: 2 K eff = . T π

(5.114)

Let us derive the similar relation for the 3D KT transition. If the renormalization of the dimensionfull disorder (0) is small, one can obtain the universal relation for the helicity modulus by replacing the temperature T in Eq. (5.114) with h 20 /(2v), K eff v 1 = . 2 π h0

(5.115)

This relation is consistent with the fact that the exponent η⊥ = h 20 /(4π K eff v) is identical to 1/4 at the transition point. Equation (5.115) will be readily confirmed by numerical experiments. The helicity modulus is numerically estimated by calculating the force required to slightly twist the phase at the boundary of the system.

150

5 Nonequilibrium Kosterlitz-Thouless Transition in the Three-Dimensional …

Fig. 5.7 Schematic picture of the vortex dissociation process in the 3D-DRFXYM. The lines represent vortex lines. The cross section with respect to a plane perpendicular to the driving direction corresponds to a snapshot of the 2D vortices. a QLRO phase. The small vortex rings are dilutely dispersed over the space, whose volume density is extremely low at weak disorder. b Disordered phase. The whole system is filled with tangled long vortex lines. Reprinted figure with permission from Ref. [2]. Copyright (2018) by the American Physical Society. https://doi.org/10.1103/ PhysRevE.98.032122

We next consider the structural changes of the vortex at the transition point. It may be recalled that the KT transition in the 2D XY model is understood as the breakdown of vortex-antivortex pairs. Since the vortex is a line in three dimensions, this pair dissociation picture should be modified. The dimensional reduction property states that a field configuration of the 3D-DRFXYM at a specific time can be identified to a space-time trajectory of the 2D pure XY model by considering the longitudinal coordinate x as a fictitious time. Note that, even in the QLRO phase of the 2D XY model, vortex-antivortex pairs can be created by the thermal activation processes. These vortices and antivortices immediately collide and annihilate each other because of an attractive force between them. By using the mapping of the 3D-DRFXYM to the 2D pure XY model, the 3D-QLRO phase is considered as a dilute gas of tiny vortex loops, which correspond to the space-time trajectories of the creation and annihilation of the vortex-antivortex pairs. The volume density of the vortex loops is extremely low in the weak disorder regime and the renormalization of the elastic constant is also small. The vortex density increases with the disorder strength, and eventually, at the critical disorder, isolated vortex loops merge into tangled long vortex lines, which percolate to the whole space. The schematic picture of this vortex dissociation process is depicted in Fig. 5.7.

References 1. Haga T (2015) Nonequilibrium quasi-long-range order of a driven random-field O(N ) model. Phys Rev E 92:062113 2. Haga T (2018) Nonequilibrium Kosterlitz-Thouless transition in a three-dimensional driven disordered system. Phys Rev E 98:032122

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3. Giamarchi T, Le Doussal P (1996) Moving glass phase of driven lattices. Phys Rev Lett 76:3408 4. Le Doussal P, Giamarchi T (1998) Moving glass theory of driven lattices with disorder. Phys Rev B 57:11356 5. Balents L, Marchetti MC, Radzihovsky L (1998) Nonequilibrium steady states of driven periodic media. Phys Rev B 57:7705 6. Goldenfeld N (1992) Lectures on phase transitions and the renormalization group. Westview Press 7. Garanin DA, Chudnovsky EM, Proctor T (2013) Random field XY model in three dimensions. Phys Rev B 88:224418 8. Ozeki Y, Ito N (2007) Nonequilibrium relaxation method. J Phys A Math Gen 40:R149 9. Ozeki Y, Ogawa K, Ito N (2003) Nonequilibrium relaxation analysis of Kosterlitz-Thouless phase transition. Phys Rev E 67:026702 10. Le Doussal P, Wiese KJ, Chauve P (2004) Functional renormalization group and the field theory of disordered elastic systems. Phys Rev E 69:026112 11. Le Doussal P, Wiese KJ (2006) Random-field spin models beyond 1 loop. Phys Rev Lett 96:197202 12. Jose JV, Kadanoff LP, Kirkpatrick S, Nelson DR (1977) Renormalization, vortices, and symmetry-breaking perturbation in the two-dimensional planar model. Phys Rev B 16:1217

Chapter 6

Summary and Future Perspectives

The dimensional reduction is the most important concept in the statistical mechanics of disordered systems. Since the standard perturbation theory leads to the dimensional reduction, its failure implies that in disordered systems there is a nontrivial physics beyond the perturbative field theory. Therefore, to understand the physics underlying the breakdown of the dimensional reduction has been one of the central issues in this area and the remarkable progresses have been achieved in the last decade. In this thesis, we investigated what types of phase transitions and critical phenomena arise when disordered systems are driven out of equilibrium. From an intuitive argument, we first derived a novel type of dimensional reduction for driven disordered systems in the first part of Chap. 4. It states that a cross-section of D-dimensional driven disordered systems at zero temperature is identical to (D − 1)-dimensional pure systems in equilibrium. As in equilibrium, this new dimensional reduction can also fail in low enough dimensions due to a nonperturbative effect, e.g., the driven random field Ising model in two dimensions. To elucidate the mechanism of this breakdown, we employed the functional renormalization group (FRG) approach, which is known to be useful for describing the long-distance physics of disordered systems in equilibrium. By combining this approach with the nonperturbative formalism, we investigated the critical phenomena of the driven random field O(N ) model (DRFO(N )M). For disordered systems in thermal equilibrium, it is known that the dimensional reduction fails when the renormalized disorder cumulant develops a cusp as a function of the field. Through the FRG analysis of the DRFO(N )M, we have shown that the similar relation between the breakdown of the dimensional reduction and the nonanalyticity of the disorder cumulant also holds for the nonequilibrium cases. The critical exponents near three dimensions were calculated by constructing the cuspy fixed point of the FRG equation. We found that the dimensional reduction recovers for N = 2 and sufficiently large N . © Springer Nature Singapore Pte Ltd. 2019 T. Haga, Renormalization Group Analysis of Nonequilibrium Phase Transitions in Driven Disordered Systems, Springer Theses, https://doi.org/10.1007/978-981-13-6171-5_6

153

154

6 Summary and Future Perspectives

The case that N = 2 (XY model) deserves to be investigated in detail because the dimensional reduction predicts that the driven random field XY model (DRFXYM) exhibits the Kosterlitz-Thouless (KT) transition in three dimensions. Chapter 5 was devoted to the numerical and renormalization group (RG) studies of this threedimensional (3D) KT transition. In numerical simulation, the equation of motion for the DRFXYM was numerically integrated and the correlation function was calculated in the steady states. We observed the phase transition between a quasi-long-range order (QLRO) phase with a continuous varying exponent and a disordered phase. The value of the critical exponent reasonably agrees with the theoretical prediction of the dimensional reduction. These results support the existence of the 3D KT transition. In the rest of Chap. 5, we performed the RG analysis of the DRFXYM. Since the amplitude fluctuations of the order parameter field is expected to be irrelevant in the QLRO phase, we defined the spin-wave version of the DRFXYM, which is more analytically tractable than the original model. By applying the FRG treatment to the spin-wave model, we found that this model becomes the massless free field theory in the large-scale limit, thus it has a fixed line characterizing the QLRO phase with a nonuniversal exponent. This result is consistent with that of the numerical simulations. We next consider the effect of the vortices, which can destroy the QLRO in the strong disorder regime. As in the conventional KT theory in two dimensions, we introduced an effective elastic constant (helicity modulus). The flow equation of the effective elastic constant was derived with the aid of the dimensional reduction and the RG trajectories are calculated. We finally discuss the vortex structure at the 3D KT transition. The significant difference from the two-dimensional (2D) cases is that the vortex is a line, not a point. In the QLRO phase at weak disorder, the vortices form into small rings the density of which is extremely low. At the transition point, such vortex rings merge each other and long vortex lines fill the whole space. In thermal equilibrium, there is no example of the higher-dimensional KT transition induced by a quenched disorder. In fact, the Bragg glass phase in the random field XY model is quite different from the QLRO phase in the 2D XY model, namely, the former is controlled by a single stable fixed point, while the latter is a consequence of a line of fixed points. The 3D KT transition in the DRFXYM is the first example of topological phase transitions resulting from the interplay between the disorder and the nonequilibrium driving. Therefore, our findings may shed light on a novel aspect of topological defects in nonequilibrium phase transitions and stimulate further studies in this direction. In the rest of this section, we remark on the future perspectives and related topics. There are many unsettled issues concerning phase transitions in driven disordered systems. • Driven random field Ising model: It is intriguing to investigate whether the 2D driven random field Ising model (DRFIM) exhibits long-range order (LRO). The dimensional reduction predicts that it does not because it is identical to the one-dimensional pure Ising model. However, the lower critical dimension of the random field Ising model is two. Since the driving reduces the lower critical dimension, it is expected that the 2D-DRFIM

6 Summary and Future Perspectives

•

•

•

•

155

should exhibit LRO in the weak disorder and strong driving regime. To resolve this question is suggestive because the breakdown of the dimensional reduction is most prominent in this case. Hidden symmetry associated with the dimensional reduction: The conventional dimensional reduction in equilibrium is a consequence of the supersymmetry (super-rotational invariance) of the stochastic field equation (see Sect. 1.2.3) and its breakdown can be understood as the spontaneous breaking of the supersymmetry [1, 2]. However, in Sect. 4.2, the dimensional reduction for driven disordered systems is derived from an intuitive argument, without introducing the supersymmetric formalism. Thus, for a clear understanding of the dimensional reduction and its breakdown, a hidden symmetry responsible for the dimensional reduction in driven disordered systems should be identified. Avalanche dynamics around nonequilibrium steady states: In Sect. 1.4, we discussed the distinction between the longitudinal and transverse driving of elastic interfaces in a random potential. In the longitudinal driving, when a symmetry breaking force slightly exceeds the critical threshold of the depinning transition, the interface exhibits avalanche dynamics. However, in the transverse driving, the system does not exhibit any depinning transition and avalanche dynamics because the advection term does not break the symmetry of the order parameter. Suppose that the interface is transversely driven at a finite velocity and a nonequilibrium steady state is realized. Let us consider what happens when an additional symmetry breaking force is applied to the interface. One expects that it exhibits avalanche dynamics, while its statistical properties would be totally different from the usual cases without the transverse driving. Thus, it is an interesting problem to understand this avalanche dynamics around the nonequilibrium steady states. Effect of temperature: As mentioned in Sect. 3.3.2, the thermal fluctuations lead to the local averaging of the disorder and thermally activated transitions between different meta-stable states. At finite temperature, the diffusion-like terms appear in the flow equation of the disorder correlator and they smooth out the cusp. In thermal equilibrium, the temperature is irrelevant near the fixed point controlling the critical behavior [see Eq. (4.102)]. However, in the nonequilibrium cases, there is a possibility that a fixed point with nonzero temperature may appear due to the additional terms, which result from the violation of the fluctuation-dissipation theorem, in the flow equation for Tk [see Eq. (4.111)]. This implies that an infinitesimally weak thermal noise can drastically change the critical behavior of driven disordered systems. For example, such a new fixed point at finite temperature is found in the elastic manifold driven by a nonpotential static random force [3]. It is interesting to investigate the property of the finite-temperature fixed point in the driven random field O(N ) model. Critical slowing down in driven disordered systems In disordered systems such as the random field Ising model, the characteristic time scale is expected to diverge as exp(ξ θ ) as one approaches the critical point, where ξ is the correlation length [4]. The temperature exponent θ is given by

156

6 Summary and Future Perspectives

θ = 2 + η − η, ¯ where the anomalous dimensions η and η¯ are defined by Eqs. (4.66) and (4.67), respectively. It is a natural question to ask whether the critical slowing down in driven disordered systems is also described by the same way as in the equilibrium cases.

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