Contents More Information

Cambridge University Press 978-0-521-51468-2 - Quantum Phase Transitions: Second Edition Subir Sachdev Table of Contents 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 Table of Contents More information

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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

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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

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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

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16.2 The dilute spinless Fermi gas 298 16.2.1 Dilute classical gas, kB T |μ|, μ<0 300 16.2.2 Fermi liquid, kB T μ, μ>0 301 16.2.3 High-T limit, kB T |μ| 304 16.3 The dilute Bose gas 305 16.3.1 d < 2 307 16.3.2 d = 3 310 16.3.3 Correlators of ZB in d = 1 314 16.4 The dilute spinful Fermi gas: the Feshbach resonance 320 16.4.1 The Fermi–Bose model 323 16.4.2 Large-N expansion 327 16.5 Applications and extensions 331

17 Phase transitions of Dirac fermions 332 17.1 d-wave superconductivity and Dirac fermions 332 17.2 Time-reversal symmetry breaking 335 17.3 Field theory and RG analysis 338 17.4 Ising-nematic ordering 342

18 Fermi liquids, and their phase transitions 346 18.1 Fermi liquid theory 347 18.1.1 Independence of choice of k0 354 18.2 Ising-nematic ordering 355 18.2.1 Hertz theory 356 18.2.2 Fate of the fermions 358 18.2.3 Non-Fermi liquid criticality in d = 2 360 18.3 Spin density wave order 363 18.3.1 Mean-ﬁeld theory 364 18.3.2 Continuum theory 365 18.3.3 Hertz theory 367 18.3.4 Fate of the fermions 368 18.3.5 Critical theory in d = 2 369 18.4 Nonzero temperature crossovers 370 18.5 Applications and extensions 374

19 Heisenberg spins: ferromagnets and antiferromagnets 375 19.1 Coherent state path integral 375 19.2 Quantized ferromagnets 380 19.3 Antiferromagnets 385 19.3.1 Collinear antiferromagnetism and the quantum nonlinear sigma model 385 19.3.2 Collinear antiferromagnetism in d = 1 388 19.3.3 Collinear antiferromagnetism in d = 2 390

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19.3.4 Noncollinear antiferromagnetism in d = 2: deconﬁned spinons and visons 395 19.3.5 Deconﬁned criticality 401 19.4 Partial polarization and canted states 403 19.4.1 Quantum paramagnet 405 19.4.2 Quantized ferromagnets 406 19.4.3 Canted and Néel states 406 19.4.4 Zero temperature critical properties 408 19.5 Applications and extensions 410

20 Spin chains: bosonization 412 20.1 The XX chain revisited: bosonization 413 20.2 Phases of H12 423 20.2.1 Sine–Gordon model 425 20.2.2 Tomonaga–Luttinger liquid 428 20.2.3 Valence bond solid order 428 20.2.4 Néel order 431 20.2.5 Models with SU(2) (Heisenberg) symmetry 431 20.2.6 Critical properties near phase boundaries 433 20.3 O(2) rotor model in d = 1 435 20.4 Applications and extensions 436

21 Magnetic ordering transitions of disordered systems 437 21.1 Stability of quantum critical points in disordered systems 438 21.2 Grifﬁths–McCoy singularities 440 21.3 Perturbative ﬁeld-theoretic analysis 442 21.4 Metallic systems 445 21.5 Quantum Ising models near the percolation transition 447 21.5.1 Percolation theory 447 21.5.2 Classical dilute Ising models 448 21.5.3 Quantum dilute Ising models 449 21.6 The disordered quantum Ising chain 453 21.7 Discussion 460 21.8 Applications and extensions 461

22 Quantum spin glasses 463 22.1 The effective action 464 22.1.1 Metallic systems 469 22.2 Mean-ﬁeld theory 470 22.3 Applications and extensions 477 References 479 Index 496