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IMPERIAL COLLEGE PRESS eBooks

Green's Functions for Solid State Physicists

Sebastian Doniach, E. H. Sondheimer

Published Jun 1, 1998• 618 citations
Publisher

Abstract

Lattice dynamics in the harmonic approximation lattice dynamics at finite tempertature the Fenyman-Dyson expansion the scattering of fermions by a localized pertubation electrons in the presence of many impurities - the theory of electrical resistance in metals the interacting electron gas the magnetic instability of the interacting electron gas interacting electrons in the atomic limit transient response of the fermi gas - the X-ray and Kondo problems conventional superconductivity electron correlations and heavy fermions high Tc superconductivity.

DOI
10.1142/p067

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CRC Press eBooks•May 4, 2018•0 citations
Lattice dynamics in the harmonic approximation lattice dynamics at finite tempertature the Fenyman-Dyson expansion the scattering of fermions by a localized pertubation electrons in the presence of many impurities - the theory of electrical resistance in metals the interacting electron gas the magnetic instability of the interacting electron gas interacting electrons in the atomic limit transient response of the fermi gas - the X-ray and Kondo problems conventional superconductivity electron correlations and heavy fermions high Tc superconductivity.
Part 1 Fundamentals and examples: systems of identical particles second quantization for fermions second quantization for bosons unitary transformations and field operators example - the Hamiltonian of translationally invariant systems in second quantization density operators the Hartree-Fock approximation restricted Hartree-Fock approximation and the symmetry dilemma Hartree-Fock for translationally invariant systems the homogeneous electron gas in the Hartree-Fock approximation long-range correlations in the electron-gas - plasmons phonons superconductivity. Part 2 Green's functions: pictures the single-particle Green's function and the hierarchy of equations of motion. Part 3 Perturbation theory: time-dependent perturbation theory with adiabatic turning-on of the interaction particle and hole operators and Wick's theorem Feynman diagrams diagrammatic calculation of the vacuum amplitude an example - the Gell-Mann-Brueckner correlation energy of a dense electron gas diagrammatic calculation of the single-particle Green's function - Dyson's equation diagrammatic analysis of the Green's function G(kappa, omega) self-consistent perturbation theory, higher-order Hartree-Fock theory the quasi-particle concept diagrammatic calculation of the two-particle Green's function and the polarization propagator effective interaction and dressed vertices plasmons in higher order perturbation theory at finite temperatures. Part 4 Fermi liquid theory: introduction equilibrium properties transport equation and collective modes microscopic derivation of the Landau fermi liquid theory applications to the Kondo problem.
1. First and second quantization 2. The electron gas 3. Phonons: coupling to electrons 4. Mean field theory 5. Time evolution pictures 6. Linear response theory 7. Transport in mesoscopic systems 8. Green's functions 9. Equation of motion theory 10. Transport in interacting mesoscopic systems 11. Imaginary time Green's functions 12. Feynman diagrams and external potentials 13. Feynman diagrams and pair interactions 14. The interacting electron gas 15. Fermi liquid theory 16. Impurity scattering and conductivity 17. Green's functions and phonons 18. Superconductivity 19. 1D electron gases and Luttinger liquids A. Fourier transformations B. Exercises C. Index
The ways of introducing and handling renormalizations in the many-body perturbation theory are reviewed. We stress the indispensable role the technique of Green functions plays in extrapolating the weak-coupling perturbative approaches to intermediate and strong couplings. We separately discuss mass and charge renor- malizations. The former is incorporated in a self-consistent equation for the self-energy derived explicitly from generating Feynman diagrams within the Baym and Kadanoff approach. The latter amounts to self-consistent equations for two-particle irreducible vertices. We analyze the charge renormalization initiated by De Domi- nicis and Martin and demonstrate that its realization via the parquet approach may become a powerful and viable way of using the many-body diagrammatic approach reliably in non-perturbative regimes with coopera- tive phenomena induced by either strong interaction or strong static randomness. Electric and magnetic properties of solids are determined by the behavior of valence electrons weakly bound to the atoms forming the crystalline lattice. The behavior of the gas of valence electrons in solids is most profound in metals with an open Fermi surface. Conduction electrons in metals can be described in a number of situations and for various purposes quite accurately by Bloch waves, eigenstates of the Fermi-gas Hamiltonian. We know, however, that electrons even in metals are not a noninteracting gas. They are generally exposed to forces driving them out of the equilibrium state of the Fermi gas. Firstly, no metal is a pure crystal and hence the electrons do not move freely in space. They are scattered on randomly distributed impurities and lattice defects. Secondly, the electrons are neither free and their mutual repulsion cannot be neglected particularly for transition and heavy metals. It is the most important and long standing task of condensed-matter theorists to comprehend and qualitatively capture the deviations in the behavior of electrons in real metals from their idealized representation via Bloch waves. The problem with quantification of the behavior of electrons beyond the almost free-electron picture is in paucity of available tools for accomplishing this task. Even if we can reduce the description of the electron gas in metals to rather simple and generic models, we are unable to solve them exactly, except for a few limiting cases that mostly do not correspond to situations of physical interest. With the increasing power of modern computers the role of numerical solutions and numerical brute force approaches has increased. Numerical solutions are usually unbiased and can reach a rather high level of quantitative precision. In some aspects they substitute the missing exact analytic solutions. Numerical solutions are suitable and even indispensable in showing and displaying the trends and some global features of the models studied. They, however, fail in critical regions of phase transitions in the vicinity of singularities in correlation functions. In these
The discovery of high-Tc superconductivity in cuprates in 1986 moved strongly correlated systems from exotic worlds interesting only for pure theorists to the focus of solid-state research. In recent decades, the majority of hot topics in condensed matter physics (high-Tc superconductivity, colossal magnetoresistance, multiferroicity, ferromagnetism in diluted magnetic semiconductors, etc.) have been related to strongly correlated transition metal compounds. The highly successful electronic structure calculations based on density functional theory lose their predictive power when applied to such compounds. It is necessary to go beyond the mean field approximation and use the many-body theory. The methods and models that were developed for the description of strongly correlated systems are reviewed together with the examples of response function calculations that are needed for the interpretation of experimental information (inelastic neutron scattering, optical conductivity, resonant inelastic X-ray scattering, electron energy loss spectroscopy, angle-resolved photoemission, electron spin resonance, and magnetic and magnetoelectric properties). The peculiarities of (quasi-) 0-, 1-, 2-, and 3- dimensional systems are discussed.
Identical Particles Second Quantization Independent-Particle Model for Fermions in Finite Systems Two-Particle States and Interactions Noninteracting Bosons and Fermions Propagators in One-Particle Quantum Mechanics Single-Particle Propagator in Many-Particle Systems Perturbation Expansion of the Single-Particle Propagator Dyson Equation and Self-consistent Green's Functions Mean-Field or Hartree-Fock Approximation Beyond the Mean-Field Approximation Interacting Boson Systems Excited States in Finite Systems Excited States in Infinite Systems Excited States in N 2 Systems and In-Medium Scattering Dynamical Treatment of the Self-energy in Infinite Systems Dynamical Treatment of the Self-energy in Finite Systems Bogoliubov Perturbation Expansion for the Bose Gas Boson Perturbation Theory Applied to Physical Systems In-Medium Interaction and Scattering of Dressed Particles Conserving Approximations and Excited States Pairing Phenomena Propagator Framework for Phenomenological Description of Many-Body Systems Finite Temperature Formalism and Applications.
This chapter considers the interactions between the constituents of say a metal, i.e., electrons and ions. By adopting a mean field approach, the dynamics of the electrons can be obtained by perturbation theory from the properties of the non-interacting electrons, and the effective electron-electron interaction in good conductors is considered. Electron diffraction or X-ray experiments reveal the grainy character of a metal. Charge is separated spatially into two pieces: the nuclei and the tightly bound electrons, and the conduction electrons, which have their density spread throughout the solid. The mean field Hamiltonian is specified in terms of the mean field, which in turn is specified in terms of the average charge density. The diagrammatic interpretation of the degenerate free Fermi gas density response function allows us also to get a diagrammatic interpretation of the interacting Fermi gas density response function.