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Quantum Phase Transitions P1: GKW/UKS P2: GKW/UKS QC: GKW CB196-FM July 31, 1999 15:48 Quantum Phase Transitions SUBIR SACHDEV Professor of Physics Yale University iii P1: GKW/UKS P2: GKW/UKS QC: GKW CB196-FM July 31, 1999 15:48 PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge CB2 2RU, UK http://www.cup.cam.ac.uk 40 West 20th Street, New York, NY 10011-4211, USA http: //www.cup.org 10 Stamford Road, Oakleigh, Melbourne 3166, Australia Ruiz de Alarc´on 13, 28014 Madrid, Spain C Subir Sachdev 1999 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1999 Printed in the United States of America Typeface in Times Roman 10/12.5 pt. System LATEX2ε [TB] A catalog record for this book is available from the British Library. Library of Congress Cataloging-in-Publication Data Sachdev, Subir, 1961– Quantum phase transitions / Subir Sachdev. p. cm. Includes bibliographical references and index. ISBN 0-521-58254-7 1. Phase transformations (Statistical physics) 2. Quantum theory. I. Title. QC175. 16.P5S23 2000 530.4014 – dc21 99-12280 CIP ISBN 0 521 58254 7 hardback iv P1: GKW/UKS P2: GKW/UKS QC: GKW CB196-FM July 31, 1999 15:48 Contents Preface page xi Acknowledgments xv Part I: Introduction 1 1 Basic Concepts 3 1.1 What Is a Quantum Phase Transition? 3 1.2 Quantum Versus Classical Phase Transitions 5 1.3 Experimental Examples 6 1.4 Theoretical Models 8 1.4.1 Quantum Ising Model 8 1.4.2 Quantum Rotor Model 10 2 The Mapping to Classical Statistical Mechanics: Single-Site Models 13 2.1 The Classical Ising Chain 13 2.1.1 The Scaling Limit 16 2.1.2 Universality 17 2.1.3 Mapping to a Quantum Model: Ising Spin in a Transverse Field 18 2.2 The Classical XY Chain and an O(2) Quantum Rotor 20 2.3 The Classical Heisenberg Chain and an O(3) Quantum Rotor 26 3 Overview 28 3.1 Quantum Field Theories 30 3.2 What’s Different about Quantum Transitions? 33 Part II: Quantum Ising and Rotor Models 37 4 The Ising Chain in a Transverse Field 39 4.1 Limiting Cases at T = 041 4.1.1 Strong Coupling, g 142 4.1.2 Weak Coupling, g 145 4.2 Exact Spectrum 46 4.3 Continuum Theory and Scaling Transformations 49 vii P1: GKW/UKS P2: GKW/UKS QC: GKW CB196-FM July 31, 1999 15:48 viii Contents 4.4 Equal-Time Correlations of the Order Parameter 54 4.5 Finite-Temperature Crossovers 57 4.5.1 Low T on the Magnetically Ordered Side, 1>0, T 1 59 4.5.2 Low T on the Quantum Paramagnetic Side, 1<0, T |1| 65 4.5.3 Continuum High T , T |1| 69 4.5.4 Summary 75 4.6 Applications and Extensions 77 5 Quantum Rotor Models: Large-N Limit 78 5.1 Limiting Cases 79 5.1.1 Strong Coupling, g˜ 180 5.1.2 Weak Coupling, g˜ 182 5.2 Continuum Theory and Large-N Limit 83 5.3 Zero Temperature 85 5.3.1 Quantum Paramagnet, g > gc 86 5.3.2 Critical Point, g = gc 87 5.3.3 Magnetically Ordered Ground State, g < gc 89 5.4 Nonzero Temperatures 91 5.4.1 Low T on the Quantum Paramagnetic Side, g > gc, T 1+ 96 5.4.2 High T , T 1+,1 97 5.4.3 Low T on the Magnetically Ordered Side, g < gc, T 1 97 5.5 Applications and Extensions 99 6 The d =1,O(N3) Rotor Models 101 6.1 Scaling Analysis at Zero Temperature 103 6.2 Low-Temperature Limit of Continuum Theory, T 1+ 104 6.3 High-Temperature Limit of Continuum Theory, 1+ T J 110 6.3.1 Field-Theoretic Renormalization Group 112 6.3.2 Computation of u 115 6.3.3 Dynamics 116 6.4 Summary 121 6.5 Applications and Extensions 121 7 The d =2,O(N3) Rotor Models 123 7.1 Low T on the Magnetically Ordered Side, T s 125 7.1.1 Computation of c 126 7.1.2 Computation of ϕ 129 7.1.3 Structure of Correlations 131 7.2 Dynamics of the Quantum Paramagnetic and High-T Regions 134 7.2.1 Zero Temperature 136 7.2.2 Nonzero Temperatures 140 7.3 Summary 143 7.4 Applications and Extensions 144 8 Physics Close to and above the Upper-Critical Dimension 145 8.1 Zero Temperature 147 8.1.1 Perturbation Theory 147 P1: GKW/UKS P2: GKW/UKS QC: GKW CB196-FM July 31, 1999 15:48 Contents ix 8.1.2 Tricritical Crossovers 149 8.1.3 Field-Theoretic Renormalization Group 150 8.2 Statics at Nonzero Temperatures 151 8.2.1 d < 3 153 8.2.2 d > 3 157 8.3 Order Parameter Dynamics in d = 2 159 8.4 Applications and Extensions 165 9 Transport in d =2 168 9.1 Perturbation Theory 172 9.1.1 I 175 9.1.2 II 176 9.2 Collisionless Transport Equations 176 9.3 Collision-Dominated Transport 180 9.3.1 Expansion 180 9.3.2 Large-N Limit 185 9.4 Physical Interpretation 188 9.5 Applications and Extensions 189 Part III: Other Models 191 10 Boson Hubbard Model 193 10.1 Mean-Field Theory 195 10.2 Continuum Quantum Field Theories 198 10.3 Applications and Extensions 201 11 Dilute Fermi and Bose Gases 203 11.1 The Quantum XX Model 205 11.2 The Dilute Spinless Fermi Gas 207 11.2.1 Dilute Classical Gas, T ||,<0 209 11.2.2 Fermi Liquid, kB T , >0 210 11.2.3 High-T Limit, T || 213 11.3 The Dilute Bose Gas 214 11.3.1 d < 2 216 11.3.2 d = 3 218 11.4 Correlators of Z B in d = 1 222 11.4.1 Dilute Classical Gas, T ||,<0 223 11.4.2 Tomonaga–Luttinger Liquid, T , >0 225 11.4.3 High-T Limit, T || 226 11.4.4 Summary 227 11.5 Applications and Extensions 228 12 Phase Transitions of Fermi Liquids 229 12.1 Effective Field Theory 230 12.2 Finite-Temperature Crossovers 234 12.3 Applications and Extensions 238 P1: GKW/UKS P2: GKW/UKS QC: GKW CB196-FM July 31, 1999 15:48 x Contents 13 Heisenberg Spins: Ferromagnets and Antiferromagnets 240 13.1 Coherent State Path Integral 240 13.2 Quantized Ferromagnets 245 13.3 Antiferromagnets 250 13.3.1 Collinear Order 251 13.3.2 Noncollinear Ordering and Deconfined Spinons 260 13.4 Partial Polarization and Canted States 265 13.4.1 Quantum Paramagnet 267 13.4.2 Quantized Ferromagnets 268 13.4.3 Canted and N´eel States 268 13.4.4 Zero Temperature Critical Properties 270 13.5 Applications and Extensions 272 14 Spin Chains: Bosonization 274 14.1 The XX Chain Revisited: Bosonization 275 14.2 Phases of H12 283 14.2.1 Sine–Gordon Model 286 14.2.2 Tomonaga–Luttinger Liquid 287 14.2.3 Spin-Peierls Order 288 14.2.4 N´eel Order 291 14.2.5 Models with SU(2) (Heisenberg) Symmetry 292 14.2.6 Critical Properties near Phase Boundaries 294 14.3 O(2) Rotor Model in d = 1 295 14.4 Applications and Extensions 296 15 Magnetic Ordering Transitions of Disordered Systems 298 Note: Chapter Co-authored with T. Senthil 15.1 Stability of Quantum Critical Points in Disordered Systems 299 15.2 Griffiths–McCoy Singularities 300 15.3 Perturbative Field-Theoretic Analysis 303 15.3.1 Metallic Systems 305 15.4 Quantum Ising Models Near the Percolation Transition 306 15.4.1 Percolation Theory 306 15.4.2 Classical Dilute Ising Models 307 15.4.3 Quantum Dilute Ising Models 308 15.5 The Disordered Quantum Ising Chain 311 15.6 Discussion 318 15.7 Applications and Extensions 319 16 Quantum Spin Glasses 320 16.1 The Effective Action 321 16.1.1 Metallic Systems 325 16.2 Mean-Field Theory 326 16.3 Applications and Extensions 333 References 335 Index 349 P1: FJP CB196-01 July 8, 1999 11:23 1 Basic Concepts 1.1 What Is a Quantum Phase Transition? Consider a Hamiltonian, H(g), whose degrees of freedom reside on the sites of a lattice, and which varies as a function of a dimensionless coupling g. Let us follow the evolution of the ground state energy of H(g) as a function of g. For the case of a finite lattice, this ground state energy will generically be a smooth, analytic function of g. The main possibility of an exception comes from the case when g couples only to a conserved quantity (i.e., H(g) = H0 + gH1, where H0 and H1 commute). This means that H0 and H1 can be simultaneously diagonalized and so the eigenfunctions are independent of g even though the eigenvalues vary with g; then there can be a level-crossing where an excited level becomes the ground state at g = gc (say), creating a point of nonanalyticity of the ground state energy as a function of g (see Fig. 1.1). The possibilities for an infinite lattice are richer. An avoided level-crossing between the ground and an excited state in a finite lattice could become progressively sharper as the lattice size increases, leading to a nonanalyticity at g = gc in the infinite lattice limit.
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