Quantum field theory in condensed matter physics

Alexei M. Tsvelik9780511078590, 9780521529808, 9780521822848, 0521529808, 052182284X

This course in modern quantum field theory for condensed matter physics includes a derivation of the path integral representation, Feynman diagrams and elements of the theory of metals. Alexei Tsvelik also covers Landau Fermi liquid theory and gradually turns to more advanced methods used in the theory of strongly correlated systems. The book contains a thorough exposition of such non-perturbative techniques, as 1/N-expansion, bosonization (Abelian and non-Abelian), conformal field theory and theory of integrable systems. First edition Hb (1995): 0-521-45467-0 First edition Pb (1996): 0-521-58989-4

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Table of contents :
Half-title……Page 3
Title……Page 5
Copyright……Page 6
Dedication……Page 7
Contents……Page 9
Preface to the first edition……Page 13
Preface to the second edition……Page 17
Acknowledgements for the first edition……Page 19
Acknowledgements for the second edition……Page 20
I Introduction to methods……Page 21
1 QFT: language and goals……Page 23
2 Connection between quantum and classical: path integrals……Page 35
Green’s function for a harmonic oscillator……Page 38
3 Definitions of correlation functions: Wick’s theorem……Page 45
4 Free bosonic field in an external field……Page 50
Semiclassical approximation……Page 52
Elements of differential geometry……Page 54
5 Perturbation theory: Feynman diagrams……Page 61
6 Calculation methods for diagram series: divergences and their elimination……Page 68
7 Renormalization group procedures……Page 76
Appendix……Page 85
8 O(N)-symmetric vector model below the transition point……Page 86
9 Nonlinear sigma models in two dimensions: renormalization group and 1/N-expansion……Page 94
References……Page 101
10 O(3) nonlinear sigma model in the strong coupling limit……Page 102
Reference……Page 105
II Fermions……Page 107
11 Path integral and Wick’s theorem for fermions……Page 109
12 Interacting electrons: the Fermi liquid……Page 116
Appendix: calculation of the integral (12.17)……Page 121
Reference……Page 122
13 Electrodynamics in metals……Page 123
Electrodynamics in metals: the semiclassical approach……Page 124
Microscopic description……Page 126
Single-electron Green’s function……Page 133
References……Page 137
14 Relativistic fermions: aspects of quantum electrodynamics……Page 139
(1 + 1)-Dimensional quantum electrodynamics (Schwinger model)……Page 143
Reference……Page 148
15 Aharonov–Bohm effect and transmutation of statistics……Page 149
The index theorem……Page 155
Quantum Hall ferromagnet……Page 157
References……Page 159
III Strongly fluctuating spin systems……Page 161
Introduction……Page 163
References……Page 167
16 Schwinger–Wigner quantization procedure: nonlinear sigma models……Page 168
Continuous field theory for a ferromagnet……Page 169
Continuous field theory for an antiferromagnet……Page 170
References……Page 175
17 O(3) nonlinear sigma model in (2 + 1) dimensions: the phase diagram……Page 177
Ordered phase……Page 179
Quantum critical region……Page 180
Topological excitations: skyrmions……Page 182
References……Page 184
18 Order from disorder……Page 185
Model of spin nematic……Page 189
References……Page 191
19 Jordan–Wigner transformation for spin S = 1/2 models in D = 1, 2, 3……Page 192
References……Page 198
20 Majorana representation for spin S=1/2 magnets: relationship to Z2 lattice gauge theories……Page 199
References……Page 203
21 Path integral representations for a doped antiferromagnet……Page 204
References……Page 213
IV Physics in the world of one spatial dimension……Page 215
Introduction……Page 217
22 Model of the free bosonic massless scalar field……Page 219
Bosonization……Page 223
References……Page 225
23 Relevant and irrelevant fields……Page 226
References……Page 231
24 Kosterlitz–Thouless transition……Page 232
How the nonlinear sigma model on a torus becomes a linear one……Page 236
25 Conformal symmetry……Page 239
Gaussian model in the Hamiltonian formulation……Page 242
References……Page 245
26 Virasoro algebra……Page 246
Ward identities……Page 250
Subalgebra sl(2)……Page 251
Reference……Page 252
27 Differential equations for the correlation functions……Page 253
Coulomb gas construction for the minimal models……Page 259
References……Page 264
Ising model as a minimal model……Page 265
Quantum Ising model……Page 268
Order and disorder operators……Page 269
Correlation functions outside the critical point……Page 271
Two coupled Ising models……Page 272
Ising model in a magnetic field……Page 273
References……Page 274
29 One-dimensional spinless fermions: Tomonaga–Luttinger liquid……Page 275
Single-electron correlator in the presence of Coulomb interaction……Page 276
Spin S = 1/2 Heisenberg chain……Page 277
Explicit expression for the dynamical magnetic susceptibility……Page 281
References……Page 286
30 One-dimensional fermions with spin: spin-charge separation……Page 287
Bosonic form of the SU1(2) Kac–Moody algebra……Page 291
Spin S = 1/2 Tomonaga–Luttinger liquid……Page 293
Incommensurate charge density wave……Page 294
Half-filled band……Page 295
References……Page 296
31 Kac–Moody algebras: Wess–Zumino–Novikov–Witten model……Page 297
Knizhnik–Zamolodchikov (KZ) equations……Page 301
Conformal embedding……Page 302
SU1(2) WZNW model and spin S = 1/2 Heisenberg antiferromagnet……Page 306
SU2(2) WZNW model and the Ising model……Page 309
References……Page 311
32 Wess–Zumino–Novikov–Witten model in the Lagrangian form: non-Abelian bosonization……Page 312
Non-Abelian bosonization:nontrivial determinant……Page 317
References……Page 319
33 Semiclassical approach to Wess–Zumino–Novikov–Witten models……Page 320
Reference……Page 322
34 Integrable models: dynamical mass generation……Page 323
General properties of integrable models……Page 324
Generalities……Page 331
The sine-Gordon model……Page 332
Perturbations of spin S = 1/2 Heisenberg chain: confinement……Page 339
References……Page 342
35 A comparative study of dynamical mass generation in one and three dimensions……Page 343
Single-electron Green’s function in a one-dimensional charge density wave state……Page 347
References……Page 353
Spin ladder……Page 354
Correlation functions……Page 360
Staggered susceptibility of the conventional (Haldane) spin liquid……Page 362
Spontaneously dimerized spin liquid……Page 366
Chirally stabilized spin liquid……Page 367
Spin S = 1 antiferromagnets……Page 368
References……Page 369
37 Kondo chain……Page 370
References……Page 374
38 Gauge fixing in non-Abelian theories: (1 + 1)-dimensional quantum chromodynamics……Page 375
References……Page 377
Select bibliography……Page 378
Index……Page 379