凝聚态物理的格林函数理论 英文PDF电子书下载

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