1 Introduction ................................................. 1
1.1 More is different ....................................... 1
1.2 'Elementary' particles and physics laws ................. 3
1.3 Corner-stones of condensed matter physics ............... 5
1.4 Topological order and quantum order ..................... 7
1.5 Origin of light and fermions ............................ 9
1.6 Novelty is more important than correctness ............. 11
1.7 Remarks: evolution of the concept of elementary
particles .............................................. 12
2 Path integral formulation of quantum mechanics .............. 14
2.1 Semiclassical picture and path integral ................ 14
2.2 Linear responses and correlation functions ............. 26
2.3 Quantum spin, the Berry phase, and the path integral ... 43
2.4 Applications of the path integral formulation .......... 50
3 Interacting boson systems ................................... 68
3.1 Free boson systems and second quantization ............. 68
3.2 Mean-field theory of a superfluid ...................... 71
3.3 Path integral approach to interacting boson systems .... 77
3.4 Superfluid phase at finite temperatures ................ 99
3.5 Renormalization group ................................. 105
3.6 Boson superfluid to Mott insulator transition ......... 120
3.7 Superfluidity and superconductivity ................... 124
3.8 Perturbative calculation of the thermal potential ..... 139
4 Free fermion systems ....................................... 144
4.1 Many-fermion systems .................................. 144
4.2 Free fermion Green's function ......................... 152
4.3 Two-body correlation functions and linear responses ... 164
4.4 Quantized Hall conductance in insulators .............. 179
5 Interacting fermion systems ................................ 187
5.1 Orthogonality catastrophe and X-ray spectrum .......... 187
5.2 Hartree-Fock approximation ............................ 198
5.3 Landau Fermi liquid theory ............................ 203
5.4 Perturbation theory and the validity of Fermi liquid
theory ................................................ 216
5.5 Symmetry-breaking phase and the spin-density-wave
state ................................................. 229
5.6 Nonlinear σ-model ..................................... 240
6 Quantum gauge theories ..................................... 250
6.1 Simple gauge theories ................................. 250
6.2 Z2 lattice gauge theory ............................... 252
6.3 U(l) gauge theory and the XY-model in 1 + 2
dimensions ............................................ 258
6.4 The quantum U(l) gauge theory on a lattice ............ 267
7 Theory of quantum Hall states .............................. 275
7.1 The Aharonov-Bohm effect and fractional statistics .... 275
7.2 The quantum Hall effect ............................... 282
7.3 Effective theory of fractional quantum Hall liquids ... 294
7.4 Edge excitations in fractional quantum Hall liquids ... 308
8 Topological and quantum order .............................. 335
8.1 States of matter and the concept of order ............. 336
8.2 Topological order in fractional quantum Hall states ... 338
8.3 Quantum orders ........................................ 348
8.4 A new classification of orders ........................ 352
9 Mean-field theory of spin liquids and quantum order ........ 354
9.1 Projective construction of quantum spin-liquid
states ................................................ 355
9.2 The SU(2) projective construction ..................... 373
9.3 Topological orders in gapped spin-liquid states ....... 393
9.4 Quantum orders in symmetric spin liquids .............. 397
9.5 Continuous phase transitions without symmetry
breaking .............................................. 411
9.6 The zoo of symmetric spin liquids ..................... 414
9.7 Physical measurements of quantum orders ............... 421
9.8 The phase diagram of the J1-J2 model in the large-N
limit ................................................. 425
9.9 Quantum order and the stability of mean-field spin
liquids ............................................... 430
9.10 Quantum order and gapless gauge bosons and fermions ... 434
10 String condensation—an unification of light and fermions ... 440
10.1 Local bosonic models .................................. 443
10.2 An exactly soluble model from a projective
construction .......................................... 444
10.3 Z2 spin liquids and string-net condensation ........... 451
10.4 Classification of string-net condensations ............ 458
10.5 Emergent fermions and string-net condensation ......... 466
10.6 The quantum rotor model and U(1) lattice gauge
theory ................................................ 472
10.7 Emergent light and electrons from an SU(Nƒ) spin
model ................................................. 487
INDEX ......................................................... 501
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