Exact Methods in Low-dimensional Statistical Physics and Quantum Computing

Contents......Page 14
List of participants......Page 20
PART I: LONG LECTURES......Page 28
1 Quantum impurity problems in condensed matter physics......Page 30
1.1 Quantum impurity problems and the renormalization group......Page 31
1.2 Multichannel Kondo model......Page 39
1.3 Quantum dots: Experimental realizations of one- and two-channel Kondo models......Page 51
1.4 Quantum impurity problems in Luttinger liquids......Page 60
1.5 Quantum impurity entanglement entropy......Page 68
1.6 Y-junctions of quantum wires......Page 75
1.7 Boundary-condition-changing operators and the X-ray edge singularity......Page 81
1.8 Conclusions......Page 89
References......Page 90
2 Conformal field theory and statistical mechanics......Page 92
2.2 Scale invariance and conformal invariance in critical behavior......Page 93
2.3 The role of the stress tensor......Page 97
2.4 Radial quantization and the Virasoro algebra......Page 103
2.5 CFT on the cylinder and torus......Page 107
2.6 Height models, loop models, and Coulomb gas methods......Page 113
2.7 Boundary conformal field theory......Page 117
2.8 Further reading......Page 125
3 The quantum Hall effect......Page 126
4 Topological phases and quantum computation......Page 128
4.1 Introduction: The quest for protected qubits......Page 129
4.2 Topological phenomena in 1D: Boundary modes in the Majorana chain......Page 130
4.3 The two-dimensional toric code......Page 135
4.4 Abelian anyons and quasiparticle statistics......Page 138
4.5 The honeycomb lattice model......Page 144
References......Page 152
5 Four lectures on computational statistical physics......Page 154
5.1 Sampling......Page 155
5.2 Classical hard-sphere systems......Page 162
5.3 Quantum Monte Carlo simulations......Page 168
5.4 Spin systems: Samples and exact solutions......Page 175
References......Page 183
6 Loop models......Page 186
6.1 Historical perspective......Page 187
6.2 Brief summary of renormalization theory......Page 188
6.3 Loop models......Page 193
6.4 The Coulomb gas......Page 203
6.5 Summary and perspective......Page 220
References......Page 221
7 Lectures on the integrability of the six-vertex model......Page 224
7.2 Classical integrable spin chains......Page 225
7.3 Quantization of local integrable spin chains......Page 230
7.4 The spectrum of transfer matrices......Page 240
7.5 The thermodynamic limit......Page 243
7.6 The six-vertex model......Page 245
7.7 The six-vertex model on a torus in the thermodynamic limit......Page 253
7.8 The six-vertex model at the free-fermionic point......Page 255
7.9 The free energy of the six-vertex model......Page 261
7.10 Some asymptotics of the free energy......Page 269
7.11 The Legendre transform of the free energy......Page 273
7.12 The limit shape phenomenon......Page 275
7.13 Semiclassical limits......Page 281
7.14 The free-fermionic point and dimer models......Page 283
7.A Appendix......Page 285
References......Page 291
8 Mathematical aspects of 2D phase transitions......Page 294
PART II: SHORT LECTURES......Page 296
9 Numerical simulations of quantum statistical mechanical models......Page 298
9.1 Introduction......Page 299
9.2 A rapid survey of methods......Page 300
9.3 Path integral and related methods......Page 306
9.4 Classical worm algorithm......Page 308
9.5 Projection methods......Page 315
9.6 Valence bond projection method......Page 320
References......Page 332
10 Rapidly rotating atomic Bose gases......Page 336
10.1 Introduction......Page 337
10.2 Rapidly rotating atomic Bose gases......Page 341
10.3 Strongly correlated phases......Page 348
10.4 Conclusions......Page 361
References......Page 362
11 The quantum Hall effect......Page 366
12 The dimer model......Page 368
12.1 Overview......Page 369
12.2 Dimer definitions......Page 370
12.3 Gibbs measures......Page 375
12.4 Kasteleyn theory......Page 376
12.5 Partition function......Page 379
12.6 General graphs......Page 382
References......Page 388
13 Boundary loop models and 2D quantum gravity......Page 390
13.2 Continuous world-sheet description: Liouville gravity......Page 391
13.3 Discrete models of 2D gravity......Page 400
13.4 Boundary correlation functions......Page 415
13.A Appendices......Page 428
References......Page 432
14 Real-space condensation in stochastic mass transport models......Page 434
14.1 Introduction......Page 435
14.2 Three simple mass transport models......Page 436
14.3 A generalized mass transport model......Page 441
14.4 Condensation in mass transport models with a factorizable steady state......Page 445
14.5 Interpretation as sums and extremes of random variables......Page 451
14.6 Conclusion......Page 452
References......Page 454
15 Quantum spin liquids......Page 458
15.1 Introduction: Band and Mott insulators......Page 459
15.2 Some materials without magnetic order at T = 0......Page 460
15.3 Spin wave theory, zero modes, and breakdown of the 1/S expansion......Page 462
15.4 Lieb–Schultz–Mattis theorem, and Hastings’s extension to D > 1: Ground state degeneracy in gapped spin liquids......Page 466
15.5 Anderson’s short-range resonating-valence-bond picture......Page 469
15.6 Schwinger bosons, large-N limit, and Z[sub(2)] topological phase......Page 471
References......Page 480
16 Superspin chains and supersigma models: A short introduction......Page 482
16.1 Introduction......Page 483
16.2 Some mathematical aspects: The gl(1|1) case......Page 484
16.3 The two simplest sigma models......Page 491
16.4 From gl(N—N) spin chains to sigma models......Page 496
16.5 A conformal sigma model at c = –2......Page 503
References......Page 507
17 Integrability and combinatorics: Selected topics......Page 510
17.1 Free-fermionic methods......Page 511
17.2 The six-vertex model......Page 527
17.3 Razumov–Stroganov conjecture......Page 539
References......Page 550
PART III: SEMINARS......Page 554
18 A rigorous perspective on Liouville quantum gravity and the KPZ relation......Page 556
18.1 Introduction......Page 557
18.2 GFF regularization......Page 565
18.3 Random measure and Liouville quantum gravity......Page 570
18.4 Proof of the KPZ relation......Page 572
18.5 Boundary KPZ relation......Page 575
18.6 Liouville quantum duality......Page 580
References......Page 584
19 Topologically protected qubits based on Josephson junction arrays......Page 590
19.1 Introduction......Page 591
19.2 Topological superconductor......Page 593
19.3 Ground state, excitations, and topological order......Page 594
19.4 Effect of physical perturbations......Page 598
19.5 Topological insulator......Page 601
19.6 Quantum manipulations......Page 606
19.7 Physical properties of small arrays......Page 608
19.8 XZ array......Page 609
19.9 Rhombus chain: Quantitative analysis......Page 624
19.10 Recent developments......Page 627
References......Page 628
20 On some quantum Hall states with negative flux......Page 630
20.1 Introduction......Page 631
20.2 Classical hierarchies......Page 632
References......Page 640
21 Supersolidity and what soluble models can tell us about it......Page 642
21.2 Some old theory......Page 643
21.3 Some recent experimental results......Page 644
21.4 Classical and nonclassical inertia......Page 645
21.5 One-dimensional models......Page 646
21.6 Two-dimensional flow......Page 649
References......Page 650