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[资源] Basic Aspects of the Quantum Theory of Solids

Foreword and general introduction page ix
1 Some basic notions of classical and quantum statistical physics 1
1.1 Gibbs distribution function and partition function 1
1.2 Thermodynamic functions 2
1.3 Systems with variable number of particles; grand partition
function 4
2 General theory of phase transitions 6
2.1 Second-order phase transitions (Landau theory) 6
2.2 (Weak) First-order phase transitions 11
2.2.1 Another possibility of getting a first-order phase
transition 13
2.3 Interaction with other degrees of freedom 14
2.4 Inhomogeneous situations (Ginzburg–Landau theory) 16
2.5 Fluctuations at the second-order phase transitions 19
2.5.1 Critical indices and scaling relations 21
2.6 Quantum phase transitions 23
2.7 General considerations 25
2.7.1 Different types of order parameters 25
2.7.2 General principle 25
2.7.3 Broken symmetry and driving force of phase transitions 26
2.7.4 The Goldstone theorem 27
2.7.5 Critical points 28
3 Bose and Fermi statistics 31
4 Phonons in crystals 34
4.1 Harmonic oscillator 34
4.2 Second quantization 35
4.3 Physical properties of crystals in the harmonic approximation 38
v
vi Contents
4.4 Anharmonic effects 41
4.4.1 Thermal expansion 43
4.4.2 Melting 45
4.4.3 Another approach to melting. Quantum melting 48
4.4.4 Low-dimensional solids; why is our world
three-dimensional? 51
5 General Bose systems; Bose condensation 54
5.1 Bose condensation 54
5.2 Weakly interacting Bose gas 58
5.3 Bose condensation and superfluidity 62
5.3.1 Landau criterion of superfluidity 65
5.3.2 Vortices in a superfluid 67
6 Magnetism 70
6.1 Basic notions; different types of magnetic response 70
6.1.1 Susceptibility of noninteracting spins 74
6.2 Interacting localized moments; magnetic ordering 76
6.2.1 Mean field approximation 77
6.2.2 Landau theory for ferromagnets 80
6.2.3 Antiferromagnetic interactions 84
6.2.4 General case 87
6.3 Quantum effects: magnons, or spin waves 91
6.3.1 Magnons in ferromagnets 92
6.3.2 Antiferromagnetic magnons. Zero-point oscillations
and their role 98
6.4 Some magnetic models 104
6.4.1 One-dimensional models 105
6.4.2 Resonating valence bonds, spinons and holons 109
6.4.3 Two-dimensional models 117
6.5 Defects and localized states in magnetic and other systems 123
7 Electrons in metals 127
7.1 General properties of Fermi systems 127
7.1.1 Specific heat and susceptibility of free electrons
in metals 129
8 Interacting electrons. Green functions and Feynman diagrams
(methods of field theory in many-particle physics) 133
8.1 Introduction to field-theoretical methods in condensed
matter physics) 133
8.2 Representations in quantum mechanics 136
8.3 Green functions 139
8.4 Green functions of free (noninteracting) electrons 141
Contents vii
8.5 Spectral representation of Green functions 143
8.5.1 Physical meaning of the poles of G( p, ω) 144
8.5.2 Physical meaning of the spectral function A( p, ω) 146
8.6 Phonon Green functions 147
8.7 Diagram techniques 149
8.7.1 Dyson equations, self-energy and polarization
operators 153
8.7.2 Effective mass of the electron excitation 156
9 Electrons with Coulomb interaction 159
9.1 Dielectric function, screening: random phase approximation 159
9.2 Nesting and giant Kohn anomalies 166
9.3 Frequency-dependent dielectric function; dynamic effects 169
10 Fermi-liquid theory and its possible generalizations 175
10.1 The foundations of the Fermi-liquid theory 175
10.2 Non-Fermi-liquid states 183
10.2.1 Marginal Fermi liquid 183
10.2.2 Non-Fermi-liquid close to a quantum critical point 184
10.2.3 Microscopic mechanisms of non-Fermi-liquid
behaviour; Luttinger liquid 186
11 Instabilities and phase transitions in electronic systems 188
11.1 Peierls structural transition 188
11.1.1 Qualitative considerations 188
11.1.2 Peierls instability in the general case 190
11.1.3 Different theoretical ways to treat Peierls distortion 192
11.1.4 Peierls distortion and some of its physical
consequences in real systems 198
11.2 Spin-Peierls transition 202
11.3 Charge-density waves and structural transitions,
higher-dimensional systems 206
11.4 Excitonic insulators 207
11.5 Intermezzo: BCS theory of superconductivity 212
11.6 Spin-density waves 216
11.7 Different types of CDW and SDW 220
11.8 Weakly and strongly interacting fermions. Wigner
crystallization 222
12 Strongly correlated electrons 229
12.1 Hubbard model 230
12.2 Mott insulators 230
12.3 Magnetic ordering in Mott insulators 234
viii Contents
12.4 One-particle spectrum of strongly correlated systems 235
12.4.1 Aproximate treatment (Hubbard I decoupling) 236
12.4.2 Dealing with Hubbard bands. Spectral weight
transfer 238
12.4.3 Motion of electrons and holes in an
antiferromagnetic background 239
12.5 Ferromagnetism in the Hubbard model? 244
12.6 Phase diagram of the Hubbard model 244
12.7 Phase separation 247
12.8 t–J model 251
12.9 Orbital ordering in the degenerate Hubbard model 252
12.10 Charge-transfer insulators 258
12.11 Insulator–metal transition 265
13 Magnetic impurities in metals, Kondo effect, heavy fermions
and mixed valence 272
13.1 Localized magnetic moments in metals 272
13.2 Kondo effect 276
13.3 Heavy fermion and mixed-valence systems 282
13.4 Kondo insulators 288
13.5 Ferromagnetic Kondo lattice and double exchange
mechanism of ferromagnetism 290
Bibliography 296
Index 298

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