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Quantum Phase Transitions: Second Edition Subir Sachdev Frontmatter More Information Cambridge University Press 978-0-521-51468-2 - Quantum Phase Transitions: Second Edition Subir Sachdev Frontmatter More information Quantum Phase Transitions Second Edition This is the first book to describe the physical properties of quantum materials near critical points with long-range many-body quantum entanglement. Readers are introduced to the basic theory of quantum phases, their phase transitions, and their observable properties. This second edition begins with nine chapters, six of them new, suitable for an introduc- tory course on quantum phase transitions, assuming no prior knowledge of quantum field theory. There are several new chapters covering important recent advances, such as the Fermi gas near unitarity, Dirac fermions, Fermi liquids and their phase transitions, quan- tum magnetism, and solvable models obtained from string theory. After introducing the basic theory, it moves on to a detailed description of the canonical quantum-critical phase diagram at nonzero temperatures. Finally, a variety of more complex models is explored. This book is ideal for graduate students and researchers in condensed matter physics and particle and string theory. Subir Sachdev is Professor of Physics at Harvard University and holds a Distinguished Research Chair at the Perimeter Institute for Theoretical Physics. His research has focused on a variety of quantum materials, and especially on their quantum phase transitions. © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-51468-2 - Quantum Phase Transitions: Second Edition Subir Sachdev Frontmatter More information © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-51468-2 - Quantum Phase Transitions: Second Edition Subir Sachdev Frontmatter More information Quantum Phase Transitions Second Edition t SUBIR SACHDEV Harvard University © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-51468-2 - Quantum Phase Transitions: Second Edition Subir Sachdev Frontmatter More information University Printing House, Cambridge CB2 8BS, United Kingdom Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9780521514682 c S. Sachdev 2011 This publication 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 2011 4th printing 2014 Printed in the United Kingdom by Print on Demand, World Wide A catalog record for this publication is available from the British Library Library of Congress Cataloging-in-Publication Data Sachdev, Subir, 1961- Quantum phase transitions / Subir Sachdev. – Second edition. p. cm Includes bibliographical references and index. ISBN 978-0-521-51468-2 (Hardback) 1. Phase transformations (Statistical physics) 2. Quantum theory. I. Title. QC175.16.P5S23 2011 530.4 74–dc22 2010050328 ISBN 978-0-521-51468-2 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-51468-2 - Quantum Phase Transitions: Second Edition Subir Sachdev Frontmatter More information To my parents and Menaka, Monisha, and Usha © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-51468-2 - Quantum Phase Transitions: Second Edition Subir Sachdev Frontmatter More information © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-51468-2 - Quantum Phase Transitions: Second Edition Subir Sachdev Frontmatter More information Contents From the Preface to the first edition page xiii Preface to the second edition xvii Part I Introduction 1 1 Basic concepts 3 1.1 What is a quantum phase transition? 3 1.2 Nonzero temperature transitions and crossovers 5 1.3 Experimental examples 8 1.4 Theoretical models 9 1.4.1 Quantum Ising model 10 1.4.2 Quantum rotor model 12 1.4.3 Physical realizations of quantum rotors 14 2 Overview 18 2.1 Quantum field theories 21 2.2 What’s different about quantum transitions? 24 Part II A first course 27 3 Classical phase transitions 29 3.1 Mean-field theory 30 3.2 Landau theory 33 3.3 Fluctuations and perturbation theory 34 3.3.1 Gaussian integrals 36 3.3.2 Expansion for susceptibility 39 Exercises 42 4 The renormalization group 45 4.1 Gaussian theory 46 4.2 Momentum shell RG 48 4.3 Field renormalization 53 4.4 Correlation functions 54 Exercises 56 vii © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-51468-2 - Quantum Phase Transitions: Second Edition Subir Sachdev Frontmatter More information viii Contents 5 The quantum Ising model 58 5.1 Effective Hamiltonian method 58 5.2 Large-g expansion 59 5.2.1 One-particle states 60 5.2.2 Two-particle states 61 5.3 Small-g expansion 64 5.3.1 d = 264 5.3.2 d = 166 5.4 Review 67 5.5 The classical Ising chain 67 5.5.1 The scaling limit 70 5.5.2 Universality 71 5.5.3 Mapping to a quantum model: Ising spin in a transverse field 72 5.6 Mapping of the quantum Ising chain to a classical Ising model 74 Exercises 77 6 The quantum rotor model 79 6.1 Large-g˜ expansion 79 6.2 Small-g˜ expansion 80 6.3 The classical XY chain and an O(2) quantum rotor 82 6.4 The classical Heisenberg chain and an O(3) quantum rotor 88 6.5 Mapping to classical field theories 89 6.6 Spectrum of quantum field theory 90 6.6.1 Paramagnet 91 6.6.2 Quantum critical point 92 6.6.3 Magnetic order 92 Exercises 95 7 Correlations, susceptibilities, and the quantum critical point 96 7.1 Spectral representation 97 7.1.1 Structure factor 98 7.1.2 Linear response 99 7.2 Correlations across the quantum critical point 101 7.2.1 Paramagnet 101 7.2.2 Quantum critical point 103 7.2.3 Magnetic order 104 Exercises 107 8 Broken symmetries 108 8.1 Discrete symmetry and surface tension 108 8.2 Continuous symmetry and the helicity modulus 110 8.2.1 Order parameter correlations 112 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-51468-2 - Quantum Phase Transitions: Second Edition Subir Sachdev Frontmatter More information ix Contents 8.3 The London equation and the superfluid density 112 8.3.1 The rotor model 115 Exercises 115 9 Boson Hubbard model 117 9.1 Mean-field theory 119 9.2 Coherent state path integral 123 9.2.1 Boson coherent states 125 9.3 Continuum quantum field theories 126 Exercises 130 Part III Nonzero temperatures 133 10 The Ising chain in a transverse field 135 10.1 Exact spectrum 137 10.2 Continuum theory and scaling transformations 140 10.3 Equal-time correlations of the order parameter 146 10.4 Finite temperature crossovers 149 10.4.1 Low T on the magnetically ordered side, >0, T 151 10.4.2 Low T on the quantum paramagnetic side, <0, T || 157 10.4.3 Continuum high T , T || 162 10.4.4 Summary 168 11 Quantum rotor models: large-N limit 171 11.1 Continuum theory and large-N limit 172 11.2 Zero temperature 174 11.2.1 Quantum paramagnet, g > gc 175 11.2.2 Critical point, g = gc 177 11.2.3 Magnetically ordered ground state, g < gc 178 11.3 Nonzero temperatures 181 11.3.1 Low T on the quantum paramagnetic side, g > gc, T + 186 11.3.2 High T , T +,− 186 11.3.3 Low T on the magnetically ordered side, g < gc, T − 187 11.4 Numerical studies 188 12 The d = 1, O(N ≥ 3) rotor models 190 12.1 Scaling analysis at zero temperature 192 12.2 Low-temperature limit of the continuum theory, T + 193 12.3 High-temperature limit of the continuum theory, + T J 199 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-51468-2 - Quantum Phase Transitions: Second Edition Subir Sachdev Frontmatter More information x Contents 12.3.1 Field-theoretic renormalization group 201 12.3.2 Computation of χu 205 12.3.3 Dynamics 206 12.4 Summary 211 13 The d = 2, O(N ≥ 3) rotor models 213 13.1 Low T on the magnetically ordered side, T ρs 215 13.1.1 Computation of ξc 216 13.1.2 Computation of τϕ 220 13.1.3 Structure of correlations 222 13.2 Dynamics of the quantum paramagnetic and high-T regions 225 13.2.1 Zero temperature 227 13.2.2 Nonzero temperatures 231 13.3 Summary 234 14 Physics close to and above the upper-critical dimension 237 14.1 Zero temperature 239 14.1.1 Tricritical crossovers 239 14.1.2 Field-theoretic renormalization group 240 14.2 Statics at nonzero temperatures 242 14.2.1 d < 3 244 14.2.2 d > 3 248 14.3 Order parameter dynamics in d = 2 250 14.4 Applications and extensions 257 15 Transport in d = 2 260 15.1 Perturbation theory 264 15.1.1 σI 268 15.1.2 σII 269 15.2 Collisionless transport equations 269 15.3 Collision-dominated transport 273 15.3.1 expansion 273 15.3.2 Large-N limit 279 15.4 Physical interpretation 281 15.5 The AdS/CFT correspondence 283 15.5.1 Exact results for quantum critical transport 285 15.5.2 Implications 288 15.6 Applications and extensions 289 Part IV Other models 291 16 Dilute Fermi and Bose gases 293 16.1 The quantum XX model 296 © in this
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