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Exact Methods in Low-Dimensional Statistical Physics and Quantum
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Contents List of participants xix PART I LONG LECTURES 1 Quantum impurity problems in condensed matter physics 3 by Ian AFFLECK 3 1.1 Quantum impurity problems and the renormalization group 4 1.2 Multichannel Kondo model 12 1.3 Quantum dots: Experimental realizations of one- and two-channelKondo models 24 1.4 Quantum impurity problems in Luttinger liquids 33 1.5 Quantum impurity entanglement entropy 41 1.6 Y-junctions of quantum wires 48 1.7 Boundary-condition-changing operators and the X-ray edge singularity 54 1.8 Conclusions 62 References 63 2 Conformal field theory and statistical mechanics 65 by John CARDY 65 2.1 Introduction 66 2.2 Scale invariance and conformal invariance in critical behavior 66 2.3 The role of the stress tensor 70 2.4 Radial quantization and the Virasoro algebra 76 2.5 CFT on the cylinder and torus 80 2.6 Height models, loop models, and Coulomb gas methods 86 2.7 Boundary conformal field theory 90 2.8 Further reading 98 3 The quantum Hall effect 99 by F. Duncan M. HALDANE 99 4 Topological phases and quantum computation 101 by Alexei KITAEV 101 4.1 Introduction: The quest for protected qubits 102 4.2 Topological phenomena in 1D: Boundary modes in the Majorana chain 103 4.3 The two-dimensional toric code 108 4.4 Abelian anyons and quasiparticle statistics 111 4.5 The honeycomb lattice model 117 References 125 Contents 5 Four lectures on computational statistical physics 127 by Werner KRAUTH 127 5.1 Sampling 128 5.2 Classical hard-sphere systems 135 5.3 Quantum Monte Carlo simulations 141 5.4 Spin systems: Samples and exact solutions 148 References 156 6 Loop models 159 by Bernard NIENHUIS 159 6.1 Historical perspective 160 6.2 Brief summary of renormalization theory 161 6.3 Loop models 166 6.4 The Coulomb gas 176 6.5 Summary and perspective 193 References 194 7 Lectures on the integrability of the six-vertex model 197 by Nicolai RESHETIKHIN 197 7.1 Introduction 198 7.2 Classical integrable spin chains 198 7.3 Quantization of local integrable spin chains 203 7.4 The spectrum of transfer matrices 213 7.5 The thermodynamic limit 216 7.6 The six-vertex model 218 7.7 The six-vertex model on a torus in the thermodynamic limit 226 7.8 The six-vertex model at the free-fermionic point 228 7.9 The free energy of the six-vertex model 234 7.10 Some asymptotics of the free energy 242 7.11 The Legendre transform of the free energy 246 7.12 The limit shape phenomenon 248 7.13 Semiclassical limits 254 7.14 The free-fermionic point and dimer models 256 7.A Appendix 258 References 264 8 Mathematical aspects of 2D phase transitions 267 byWendelin WERNER 267 PART II SHORT LECTURES 9 Numerical simulations of quantum statistical mechanical models 271 by Fabien ALET 271 9.1 Introduction 272 9.2 A rapid survey of methods 273 9.3 Path integral and related methods 279 9.4 Classical worm algorithm 281 Contents 9.5 Projection methods 288 9.6 Valence bond projection method 293 References 305 10 Rapidly rotating atomic Bose gases 309 by Nigel R. COOPER 309 10.1 Introduction 310 10.2 Rapidly rotating atomic Bose gases 314 10.3 Strongly correlated phases 321 10.4 Conclusions 334 References 335 11 The quantum Hall effect 339 by Ju¨rg FRO¨ HLICH 339 12 The dimer model 341 by Richard KENYON 341 12.1 Overview 342 12.2 Dimer definitions 343 12.3 Gibbs measures 348 12.4 Kasteleyn theory 349 12.5 Partition function 352 12.6 General graphs 355 References 361 13 Boundary loop models and 2D quantum gravity 363 by Ivan KOSTOV 363 13.1 Introduction 364 13.2 Continuous world-sheet description: Liouville gravity 364 13.3 Discrete models of 2D gravity 373 13.4 Boundary correlation functions 388 13.A Appendices 401 References 405 14 Real-space condensation in stochastic mass transport models 407 by Satya N. MAJUMDAR 407 14.1 Introduction 408 14.2 Three simple mass transport models 409 14.3 A generalized mass transport model 414 14.4 Condensation in mass transport models with a factorizable steady state 418 14.5 Interpretation as sums and extremes of random variables 424 14.6 Conclusion 425 References 427 15 Quantum spin liquids 431 by Gr´egoire MISGUICH 431 15.1 Introduction: Band and Mott insulators 432 15.2 Some materials without magnetic order at T = 0 433 Contents 15.3 Spin wave theory, zero modes, and breakdown of the 1/S expansion 435 15.4 Lieb–Schultz–Mattis theorem, and Hastings’s extension to D > 1: Ground state degeneracy in gapped spin liquids 439 15.5 Anderson’s short-range resonating-valence-bond picture 442 15.6 Schwinger bosons, large-N limit, and Z2 topological phase 444 References 453 16 Superspin chains and supersigma models: A short introduction 455 by Hubert SALEUR 455 16.1 Introduction 456 16.2 Some mathematical aspects: The gl(1|1) case 457 16.3 The two simplest sigma models 464 16.4 From gl(N—N) spin chains to sigma models 469 16.5 A conformal sigma model at c = −2 476 16.6 Conclusions 480 References 480 17 Integrability and combinatorics: Selected topics 483 by Paul ZINN-JUSTIN 483 17.1 Free-fermionic methods 484 17.2 The six-vertex model 500 17.3 Razumov–Stroganov conjecture 512 References 523 PART III SEMINARS 18 A rigorous perspective on Liouville quantum gravity and the KPZ relation 529 by Bertrand DUPLANTIER 529 18.1 Introduction 530 18.2 GFF regularization 538 18.3 Random measure and Liouville quantum gravity 543 18.4 Proof of the KPZ relation 545 18.5 Boundary KPZ relation 548 18.6 Liouville quantum duality 553 References 557 19 Topologically protected qubits based on Josephson junction arrays 563 by Mikhail V. FEIGEL’MAN 563 19.1 Introduction 564 19.2 Topological superconductor 566 19.3 Ground state, excitations, and topological order 567 19.4 Effect of physical perturbations 571 19.5 Topological insulator 574 19.6 Quantum manipulations 579 19.7 Physical properties of small arrays 581 19.8 XZ array 582 Contents 19.9 Rhombus chain: Quantitative analysis 597 19.10Recent developments 600 19.11Conclusion 601 References 601 20 On some quantum Hall states with negative flux 603 by Thierry JOLICOEUR 603 20.1 Introduction 604 20.2 Classical hierarchies 605 References 613 21 Supersolidity and what soluble models can tell us about it 615 by David THOULESS 615 21.1 Introduction 616 21.2 Some old theory 616 21.3 Some recent experimental results 617 21.4 Classical and nonclassical inertia 618 21.5 One-dimensional models 619 21.6 Two-dimensional flow 622 21.7 Conclusions 623 References 623 |
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