Green's Function in. Condensed Matter Physics. Wang Huaiyu. Alpha Science International Ltd. SCIENCE PRESS 2 Beijing \S7 Oxford, U.K.
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1 Green's Function in Condensed Matter Physics Wang Huaiyu SCIENCE PRESS 2 Beijing \S7 Oxford, U.K. Alpha Science International Ltd.
2 CONTENTS Part I Green's Functions in Mathematical Physics Chapter 1 Time-Independent Green's Functions Formalism Examples d case d case d case 11 Chapter 2 Time-dependent Green's Functions First-Order Case of Time-Derivative Second-Order Case of Time-Derivative 16 Part II One-Body Green's Functions Chapter 3 Physical Significance of One-Body Green's Functions One-Body Green's Functions The Free-Particle Case d case d case d case 29 Chapter 4 Green's Functions and Perturbation Theory Time-Independent Case Time-Dependent Case Application; Scattering Theory (E>0) Application; Bound States in Shallow Potential Wells (E<C0) d space d space d space 46 Chapter 5 Green's Functions for Tight-Binding Hamiltonians Tight-Binding Hamiltonians Lattice Green's functions 52
3 Vi * Green's Function in Condensed Matter Physics d simple lattice d square lattice d simple cubic lattice 58 Chapter 6 Single Impurity Scattering Formalism Applications d case d case d case 75 Chapter 7 Extension Theory for Lattice Green's Functions Introduction Extension of Hamiltonians in Powers Extension of Hamiltonians by Products Extension by Lattice Constructions 90 Part III Many-Body Green's Functions Chapter 8 Field Operators and Three Pictures Field Operators Three Pictures Schrodinger picture Heisenberg picture Interaction picture The relation between interaction and Heisenberg pictures 103 Chapter 9 Definition and Properties of Many-Body Green's Functions Definition of the Many-Body Green's Functions The Characteristics and Usage of the Green's Functions The Lehmann representation and spectral function Evaluation of physical quantities The Physical Significance of the Green's Functions Quasiparticles Physical interpretation of the Green's function and its poles The Green's functions of Noninteraction Systems Fermions (Bosons) Phonons 143
4 CONTENTS Vii Chapter 10 The Diagram Technique for Zero-Temperature Green's Functions Wick' Theorem Diagram Rules in Real Space Two-body interaction External field Electron-phonon interaction Diagram Rules in Momentum Space Two-body interaction External field Electron-phonon interaction Proper Self-Energies and Dyson's Equations 172 Chapter 11 Definition and Properties of Matsubara Green's Functions The Imaginary-Time Picture The Definition and Properties of the Matsubara Green's Function The definition of the Matsubara Green's function A significant property of the Matsubara Green's functions The Analytical Continuation and Evaluation of Physical Quantities The analytical continuation Evaluation of physical quantities The Matsubara Green's functions for noninteracting systems The formulas for frequency sums 195 Chapter 12 Diagram Technique for the Matsubara Green's Functions Wick's Theorem at Finite Temperature Diagram Rules in Real Space Two-body interaction External field Electron-phonon interaction Diagram Rules in Momentum Space Two-body interaction External field Electron-phonon interaction Proper Self-Energies and Dyson's Equations Zero-Temperature Limit 220
5 Vili * Green's Function in Condensed Matter Physics Chapter 13 Three Approximation Schemes of the Diagram Technique The Formal and Partial Summations of Diagrams Formal summations and framework diagrams Polarized Green's functions Partial summation of diagrams Self-Consistent Hartree-Fock Approximation Self-consistent Hartree-Fock approximation method Zero temperature Finite temperature Ring-Diagram Approximation High-density electron gases Zero temperature Equivalence to random phase approximation Ladder-Diagram Approximation Rigid-ball model Ladder-diagram approximation Physical quantities 281 Chapter 14 Linear Response Theory Linear Response Functions Matsubara Linear Response Functions Magnetic Susceptibility Magnetic susceptibility expressed by the retarded Green's function Magnetic susceptibility of electrons Enhancement of magnetic susceptibility Dynamic and static susceptibilities of paramagnetic states Stoner criterion Thermal Conductivity Linear Response of Generalized Current Definitions of several generalized currents Linear response Response coefficients expressed by correlation functions Electric current 313 Chapter 15 The Equation of Motion Technique for the Green's Functions The Equation of Motion Technique 318
6 CONTENTS ix Hartree approximation Hartree-Fock approximation Spectral Theorem Spectral theorem The procedure of solving Green's functions by equation of motion Application; Hubbard Model Hubbard Hamiltonian Exact solution of Hubbard model in the case of zero bandwidth Strong-correlation effect in a narrow energy band Application; Interaction Between Electrons Causes the Enhancement of Magnetic Susceptibility Equation of Motion Method for the Matsubara Green's Functions 343 Chapter 16 Magnetic Systems Described by Heisenberg Model Spontaneous Magnetization and Heisenberg Model Magnetism of materials Heisenberg model One Component of Magnetization For S=l/2 Ferromagnetism One Component of Magnetization for a Ferromagnet With Arbitrary Spin Quantum Number Explanation to the Experimental Laws of Ferromagnets Spontaneous magnetization at very low temperature Spontaneous magnetization when temperature closes to Curie point Magnetic susceptibility of paramagnetic phase One Component of Magnetization for an Antiferromagnet With Arbitrary Spin Quantum Number Spin quantum number S= 1/ Magnetic field is absent Arbitrary spin quantum number S One Component of Magnetization for Ferromagnetic and Antiferromagnetic Films Ferromagnetic films Antiferromagnetic films 379
7 X * Green's Function in Condensed Matter Physics More Than One Spin in Every Site The model Hamiltonian and formalism Properties of the system Three Components of Magnetization for a Ferromagnet with Arbitrary Spin Quantum Number Single-ion anisotropy along z direction Single-ion anisotropy along any direction The solution of the ordinary differential equation Three Components of Magnetizations for Antiferromagnets and Magentic Films Three components of magnetization for an antiferromagnet Three components of magnetization for ferromagnetic films Three components of magnetization for antiferromagnetic films 439 Chapter 17 The Green's Functions for Boson Systems with Condensation The Properties of Boson Systems with Condensation Noninteracting ground state Interacting ground state The energy spectrum of weakly excited states The Normal and Anomalous Green's functions The Green's functions The anomalous Green's functions The Green's functions for noninteracting systems Diagram Technique Proper Self-Energies and Dyson's Equations Dyson's equations Solutions of Dyson's equations The energy spectrum of weakly excited states Low-Density Bosonic Rigid-Ball Systems Boson Systems at Very Low Temperature 481 Chapter 18 Superconductors With Weak Interaction Between Electrons The Hamiltonian The Green's and Matsubara Green's Functions in the Nambu Representation Nambu Green's functions Nambu Matsubara Green's functions 493
8 CONTENTS xi Equations of Motion of Nambu Matsubara Green's functions and Their Solutions Evaluation of Physical Quantities The self-consistent equation and the gap function Energy gap at zero temperature Critical temperature Tc Energy gap as function of a temperature A(T) Density of states of excitation spectrum Mean-Field Approximation Mean-field approximation of the Hamiltonian Expressions of the Heisenberg operators Construction of the Green's functions Some Remarks Strongly coupling Hamiltonian The coexistence of superconducting and magnetic states Off-diagonal long-range order Two-fluid model The electromagnetic properties High Tc superconductivity 513 Chapter 19 Nonequilibrium Green's Functions Definitions and Properties Diagram Technique Proper Self-Energies and Dyson's Equations Langreth Theorem 533 Chapter 20 Electronic Transport through a Mesoscopic Structure Model Hamiltonian Model Hamiltonian Unitary transformation Formula of Electric Current Tunnelling Conductance 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
9 xii Green's Function in Condensed Matter Physics Appendix C An Alternative Derivation of the Regularity Condition 574 Appendix D Identities Valid for Both Trigonometric and Hyperbolic Chebyshev Functions 576 Index 577
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