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
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=2 64
5.3.2 d=1 66
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 atransversefield 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-？expansion 79
6.2 Small-？expansion 80
6.3 The classical X Y 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
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 chainin atransversefield 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 Thed=1,0(N≥3)rotormodels 190
12.1 Scaling analysis at zero temperature 192
12.2 Low-temperature limit ofthe continuum theory,T《△+ 193
12.3 High-temperature limit of the continuum theory,△+《T《J 199
12.3.1 Field-theoretic renormalization group 201
12.3.2 Computation of xu 205
12.3.3 Dynamics 206
12.4 Summary 211
13 The d=2,0(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 τψ 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 Transportind=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
16.2 The dilute spinless Fermi gas 298
16.2.1 Dilute classical gas,kBT《|μ|,μ＜0 300
16.2.2 Fermi liquid,kBT《μ,μ＞0 301
16.2.3 High-T limit,kBT》|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？0 354
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
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 modelin 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 lsing models near the percolation transition 447
21.5.1 Percolation theory 447
21.5.2 Classieal 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-feld theory 470
22.3 Applications and extensions 477
References 479
Index 496