Part Ⅰ Green's Functions in Mathematical Physics 3
Chapter 1 Time-Independent Green's Functions 3
1.1 Formalism 3
1.2 Examples 8
1.2.1 3-d case 9
1.2.2 2-d case 10
1.2.3 1-d case 11
Chapter 2 Time-dependent Green's Functions 13
2.1 First-Order Case of Time-Derivative 13
2.2 Second-Order Case of Time-Derivative 16
Part Ⅱ One-Body Green's Functions 25
Chapter 3 Physical Significance of One-Body Green's Functions 25
3.1 One-Body Green's Functions 25
3.2 The Free-Particle Case 27
3.2.1 3-d case 28
3.2.2 2-d case 28
3.2.3 1-d case 29
Chapter 4 Green's Functions and Perturbation Theory 31
4.1 Time-Independent Case 31
4.2 Time-Dependent Case 36
4.3 Application:Scattering Theory(E>0) 40
4.4 Application:Bound States in Shallow Potential Wells(E<0) 44
4.4.1 3-d space 44
4.4.2 2-d space 45
4.4.3 1-d space 46
Chapter 5 Green's Functions for Tight-Binding Hamiltonians 48
5.1 Tight-Binding Hamiltonians 48
5.2 Lattice Green's functions 52
5.2.1 1-d simple lattice 53
5.2.2 2-d square lattice 55
5.2.3 3-d simple cubic lattice 58
Chapter 6 Single Impurity Scattering 62
6.1 Formalism 62
6. 2 Applications 69
6.2.1 3-d case 69
6.2.2 1-d case 73
6.2.3 2-d case 75
Chapter 7 Extension Theory for Lattice Green's Functions 77
7.1 Introduction 77
7.2 Extension of Hamiltonians in Powers 79
7.3 Extension of Hamiltonians by Products 84
7.4 Extension by Lattice Constructions 90
Part Ⅲ Many-Body Green's Functions 99
Chapter 8 Field Operators and Three Pictures 99
8.1 Field Operators 99
8.2 Three Pictures 102
8.2.1 Schr？dinger picture 102
8.2.2 Heisenberg picture 102
8.2.3 Interaction picture 103
8.2.4 The relation between interaction and Heisenberg pictures 103
Chapter 9 Definition and Properties of Many-Body Green's Functions 109
9.1 Definition of the Many-Body Green's Functions 109
9.2 The Characteristics and Usage of the Green's Functions 116
9.2.1 The Lehmann representation and spectral function 116
9.2.2 Evaluation of physical quantities 126
9.3 The Physical Significance of the Green's Functions 132
9.3.1 Quasiparticles 132
9.3.2 Physical interpretation of the Green's function and its poles 136
9.4 The Green's functions of Noninteraction Systems 141
9.4.1 Fermions(Bosons) 141
9.4.2 Phonons 143
Chapter 10 The Diagram Technique for Zero-Temperature Green's Functions 147
10.1 Wick' Theorem 147
10.2 Diagram Rules in Real Space 152
10.2.1 Two-body interaction 152
10.2.2 External field 160
10.2.3 Electron-phonon interaction 161
10.3 Diagram Rules in Momentum Space 165
10.3.1 Two-body interaction 166
10.3.2 External field 168
10.3.3 Electron-phonon interaction 170
10.4 Proper Self-Energies and Dyson's Equations 172
Chapter 11 Definition and Properties of Matsubara Green's Functions 183
11.1 The Imaginary-Time Picture 183
11.2 The Definition and Properties of the Matsubara Green's Function 186
11.2.1 The definition of the Matsubara Green's function 186
11.2.2 A significant property of the Matsubara Green's functions 187
11.3 The Analytical Continuation and Evaluation of Physical Quantities 189
11.3.1 The analytical continuation 189
11.3.2 Evaluation of physical quantities 193
11.3.3 The Matsubara Green's functions for noninteracting systems 194
11.3.4 The formulas for frequency sums 195
Chapter 12 Diagram Technique for the Matsubara Green's Functions 200
12.1 Wick's Theorem at Finite Temperature 200
12.2 Diagram Rules in Real Space 205
12.2.1 Two-body interaction 206
12.2.2 External field 208
12.2.3 Electron-phonon interaction 209
12.3 Diagram Rules in Momentum Space 211
12.3.1 Two-body interaction 213
12.3.2 External field 215
12.3.3 Electron-phonon interaction 216
12.4 Proper Self-Energies and Dyson's Equations 218
12.5 Zero-Temperature Limit 220
Chapter 13 Three Approximation Schemes of the Diagram Technique 224
13.1 The Formal and Partial Summations of Diagrams 224
13.1.1 Formal summations and framework diagrams 224
13.1.2 Polarized Green's functions 229
13.1.3 Partial summation of diagrams 232
13.2 Self-Consistent Hartree-Fock Approximation 233
13.2.1 Self-consistent Hartree-Fock approximation method 233
13.2.2 Zero temperature 236
13.2.3 Finite temperature 241
13.3 Ring-Diagram Approximation 244
13.3.1 High-density electron gases 244
13.3.2 Zero temperature 245
13.3.3 Equivalence to random phase approximation 262
13.4 Ladder-Diagram Approximation 265
13.4.1 Rigid-ball model 265
13.4.2 Ladder-diagram approximation 268
13.4.3 Physical quantities 281
Chapter 14 Linear Response Theory 287
14.1 Linear Response Functions 287
14.2 Matsubara Linear Response Functions 294
14.3 Magnetic Susceptibility 297
14.3.1 Magnetic susceptibility expressed by the retarded Green's function 297
14.3.2 Magnetic susceptibility of electrons 299
14.3.3 Enhancement of magnetic susceptibility 300
14.3.4 Dynamic and static susceptibilities of paramagnetic states 300
14.3.5 Stoner criterion 301
14.4 Thermal Conductivity 302
14.5 Linear Response of Generalized Current 306
14.5.1 Definitions of several generalized currents 306
14.5.2 Linear response 307
14.5.3 Response coefficients expressed by correlation functions 311
14.5.4 Electric current 313
Chapter 15 The Equation of Motion Technique for the Green's Functions 317
15.1 The Equation of Motion Technique 318
15.1.1 Hartree approximation 321
15.1.2 Hartree-Fock approximation 322
15.2 Spectral Theorem 324
15.2.1 Spectral theorem 324
15.2.2 The procedure of solving Green's functions by equation of motion 328
15.3 Application:Hubbard Model 329
15.3.1 Hubbard Hamiltonian 330
15.3.2 Exact solution of Hubbard model in the case of zero bandwidth 332
15.3.3 Strong-correlation effect in a narrow energy band 335
15.4 Application:Interaction Between Electrons Causes the Enhancement of Magnetic Susceptibility 341
15.5 Equation of Motion Method for the Matsubara Green's Functions 343
Chapter 16 Magnetic Systems Described by Heisenberg Model 348
16.1 Spontaneous Magnetization and Heisenberg Model 348
16.1.1 Magnetism of materials 348
16.1.2 Heisenberg model 350
16.2 One Component of Magnetization For S=1/2 Ferromagnetism 354
16.3 One Component of Magnetization for a Ferromagnet With Arbitrary Spin Quantum Number 358
16.4 Explanation to the Experimental Laws of Ferromagnets 363
16.4.1 Spontaneous magnetization at very low temperature 363
16.4.2 Spontaneous magnetization when temperature closes to Curie point 364
16.4.3 Magnetic susceptibility of paramagnetic phase 365
16.5 One Component of Magnetization for an Antiferromagnet With Arbitrary Spin Quantum Number 366
16.5.1 Spin quantum number S=1/2 367
16.5.2 Magnetic field is absent 372
16.5.3 Arbitrary spin quantum number S 373
16.6 One Component of Magnetization for Ferromagnetic and Antiferromagnetic Films 374
16.6.1 Ferromagnetic films 374
16.6.2 Antiferromagnetic films 379
16.7 More Than One Spin in Every Site 384
16.7.1 The model Hamiltonian and formalism 384
16.7.2 Properties of the system 388
16.8 Three Components of Magnetization for a Ferromagnet with Arbitrary Spin Quantum Number 401
16.8.1 Single-ion anisotropy along z direction 402
16.8.2 Single-ion anisotropy along any direction 412
16.8.3 The solution of the ordinary differential equation 419
16.9 Three Components of Magnetizations for Antiferromagnets and Magentic Films 422
16.9.1 Three components of magnetization for an antiferromagnet 422
16.9.2 Three components of magnetization for ferromagnetic films 425
16.9.3 Three components of magnetization for antiferromagnetic films 439
Chapter 17 The Green's Functions for Boson Systems with Condensation 453
17.1 The Properties of Boson Systems with Condensation 454
17.1.1 Noninteracting ground state 454
17.1.2 Interacting ground state 454
17.1.3 The energy spectrum of weakly excited states 456
17.2 The Normal and Anomalous Green's functions 457
17.2.1 The Green's functions 457
17.2.2 The anomalous Green's functions 459
17.2.3 The Green's functions for noninteracting systems 460
17.3 Diagram Technique 462
17.4 Proper Self-Energies and Dyson's Equations 469
17.4.1 Dyson's equations 469
17.4.2 Solutions of Dyson's equations 471
17.4.3 The energy spectrum of weakly excited states 473
17.5 Low-Density Bosonic Rigid-Ball Systems 476
17.6 Boson Systems at Very Low Temperature 481
Chapter 18 Superconductors With Weak Interaction Between Electrons 489
18.1 The Hamiltonian 490
18.2 The Green's and Matsubara Green's Functions in the Nambu Representation 491
18.2.1 Nambu Green's functions 491
18.2.2 Nambu Matsubara Green's functions 493
18.3 Equations of Motion of Nambu Matsubara Green's functions and Their Solutions 494
18.4 Evaluation of Physical Quantities 499
18.4.1 The self-consistent equation and the gap function 499
18.4.2 Energy gap at zero temperature 500
18.4.3 Critical temperature Tc 501
18.4.4 Energy gap as a function of temperature△(T) 502
18.4.5 Density of states of excitation spectrum 502
18.5 Mean-Field Approximation 502
18.5.1 Mean-field approximation of the Hamiltonian 502
18.5.2 Expressions of the Heisenberg operators 504
18.5.3 Construction of the Green's functions 506
18.6 Some Remarks 508
18.6.1 Strongly coupling Hamiltonian 508
18.6.2 The coexistence of superconducting and magnetic states 509
18.6.3 Off-diagonal long-range order 510
18.6.4 Two-fluid model 511
18.6.5 The electromagnetic properties 512
18.6.6 High Tc superconductivity 513
Chapter 19 Nonequilibrium Green's Functions 516
19.1 Definitions and Properties 516
19.2 Diagram Technique 519
19.3 Proper Self-Energies and Dyson's Equations 528
19.4 Langreth Theorem 533
Chapter 20 Electronic Transport through a Mesoscopic Structure 541
20.1 Model Hamiltonian 541
20.1.1 Model Hamiltonian 541
20.1.2 Unitary transformation 543
20.2 Formula of Electric Current 546
20.3 Tunnelling Conductance 550
20.4 Magnetoresistance Effect of a FM/I/FM Junction 558
Appendix A Wick's Theorem in the Macroscopic Limit 568
Appendix B The Hamiltonian of the Jellium Model of an Electron Gas in a Metal 571
Appendix C An Alternative Derivation of the Regularity Condition 574
Appendix D Identities Valid for Both Trigonometric and Hyperbolic Chebyshev Functions 576
Index 577