Basic Aspects of the Quantum Theory of Solids

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Foreword......Page 11
General introduction......Page 15
1.1 Gibbs distribution function and partition function......Page 17
1.2 Thermodynamic functions......Page 18
1.3 Systems with variable number of particles; grand partition function......Page 20
2.1 Second-order phase transitions (Landau theory)......Page 22
2.2 (Weak) First-order phase transitions......Page 27
2.2.1 Another possibility of getting a first-order phase transition......Page 29
2.3 Interaction with other degrees of freedom......Page 30
2.4 Inhomogeneous situations (Ginzburg–Landau theory)......Page 32
2.5 Fluctuations at the second-order phase transitions......Page 35
2.5.1 Critical indices and scaling relations......Page 37
2.6 Quantum phase transitions......Page 39
2.7.2 General principle......Page 41
2.7.3 Broken symmetry and driving force of phase transitions......Page 42
2.7.4 The Goldstone theorem......Page 43
2.7.5 Critical points......Page 44
3 Bose and Fermi statistics......Page 47
4.1 Harmonic oscillator......Page 50
4.2 Second quantization......Page 51
4.3 Physical properties of crystals in the harmonic approximation......Page 54
4.4 Anharmonic effects......Page 57
4.4.1 Thermal expansion......Page 59
4.4.2 Melting......Page 61
4.4.3 Another approach to melting. Quantum melting......Page 64
4.4.4 Low-dimensional solids; why is our world three-dimensional?......Page 67
5.1 Bose condensation......Page 70
5.2 Weakly interacting Bose gas......Page 74
5.3 Bose condensation and superfluidity......Page 78
5.3.1 Landau criterion of superfluidity......Page 81
5.3.2 Vortices in a superfluid......Page 83
6.1 Basic notions; different types of magnetic response......Page 86
6.1.1 Susceptibility of noninteracting spins......Page 90
6.2 Interacting localized moments; magnetic ordering......Page 92
6.2.1 Mean field approximation......Page 93
6.2.2 Landau theory for ferromagnets......Page 96
6.2.3 Antiferromagnetic interactions......Page 100
6.2.4 General case......Page 103
6.3 Quantum effects: magnons, or spin waves......Page 107
6.3.1 Magnons in ferromagnets......Page 108
Another way to treat spin waves......Page 111
6.3.2 Antiferromagnetic magnons. Zero-point oscillations and their role......Page 114
Bose condensation......Page 118
6.4 Some magnetic models......Page 120
(1a) 1d Ising model with nearest-neighbour interaction......Page 121
(1b) 1d xy model, spins 1/2......Page 123
(1c) 1d Heisenberg model for S = 1/2......Page 124
6.4.2 Resonating valence bonds, spinons and holons......Page 125
(3a) 2d Ising model......Page 133
(3b) 2d xy model. Topological excitations (vortices)......Page 135
6.5 Defects and localized states in magnetic and other systems......Page 139
7.1 General properties of Fermi systems......Page 143
7.1.1 Specific heat and susceptibility of free electrons in metals......Page 145
8.1 Introduction to field-theoretical methods in condensed matter physics......Page 149
8.2 Representations in quantum mechanics......Page 152
8.3 Green functions......Page 155
8.4 Green functions of free (noninteracting) electrons......Page 157
8.5 Spectral representation of Green functions......Page 159
8.5.1 Physical meaning of the poles of G(p, w)......Page 160
8.5.2 Physical meaning of the spectral function A(p, w)......Page 162
8.6 Phonon Green functions......Page 163
8.7 Diagram techniques......Page 165
8.7.1 Dyson equations, self-energy and polarization operators......Page 169
8.7.2 Effective mass of the electron excitation......Page 172
9.1 Dielectric function, screening: random phase approximation......Page 175
9.2 Nesting and giant Kohn anomalies......Page 182
9.3 Frequency-dependent dielectric function; dynamic effects......Page 185
10.1 The foundations of the Fermi-liquid theory......Page 191
10.2.1 Marginal Fermi liquid......Page 199
10.2.2 Non-Fermi-liquid close to a quantum critical point......Page 200
10.2.3 Microscopic mechanisms of non-Fermi-liquid behaviour; Luttinger liquid......Page 202
11.1.1 Qualitative considerations......Page 204
11.1.2 Peierls instability in the general case......Page 206
11.1.3 Different theoretical ways to treat Peierls distortion......Page 208
11.1.4 Peierls distortion and some of its physical consequences in real systems......Page 214
11.2 Spin-Peierls transition......Page 218
11.3 Charge-density waves and structural transitions, higher-dimensional systems......Page 222
11.4 Excitonic insulators......Page 223
11.5 Intermezzo: BCS theory of superconductivity......Page 228
11.6 Spin-density waves......Page 232
11.7 Different types of CDW and SDW......Page 236
11.8 Weakly and strongly interacting fermions. Wigner crystallization......Page 238
12 Strongly correlated electrons......Page 245
12.2 Mott insulators......Page 246
12.3 Magnetic ordering in Mott insulators......Page 250
12.4 One-particle spectrum of strongly correlated systems......Page 251
12.4.1 Aproximate treatment (Hubbard I decoupling)......Page 252
12.4.2 Dealing with Hubbard bands. Spectral weight transfer......Page 254
12.4.3 Motion of electrons and holes in an antiferromagnetic background......Page 255
12.6 Phase diagram of the Hubbard model......Page 260
12.7 Phase separation......Page 263
12.8 t-J model......Page 267
12.9 Orbital ordering in the degenerate Hubbard model......Page 268
12.10 Charge-transfer insulators......Page 274
12.11 Insulator-metal transition......Page 281
13.1 Localized magnetic moments in metals......Page 288
13.2 Kondo effect......Page 292
13.3 Heavy fermion and mixed-valence systems......Page 298
13.4 Kondo insulators......Page 304
13.5 Ferromagnetic Kondo lattice and double exchange mechanism of ferromagnetism......Page 306
Bibliography......Page 312
Index......Page 314