Normal-Fermi-Liquid Theory for Electrons in Solids

Abstract

A Landau-type Fermi-liquid theory for interacting electrons moving in a fixed periodic potential is derived. Two methods are used: the Green's-function method, which gives a more physical description of the system but does not give all the desired results, and the many-body perturbation-theory approach, which is complete but is more complicated. The Green's-function formalism of Kadanoff and Baym is adapted to inhomogeneous systems by rewriting it in operator form. A kinetic equation for an energy-dependent, single-particle density matrix is then derived. The quasiparticle concept is introduced and a formal solution of the kinetic equation is given and discussed. These developments are specialized to the periodic potential and a transport equation of the Landau-Boltzmann type is derived for quasiparticles moving within a single energy band. Expressions for the density and current density are then identified from a continuity equation. Although the present work does not treat real transitions between energy bands, the Green's-function method suggests a plausible form for a Landau-like theory which includes such transitions. The perturbation-theory treatment follows the work of Luttinger and Nozi\`eres, who derived Landau's theory for systems without a lattice potential. When the perturbation theory is expressed in terms of Bloch functions, the quantities involved become matrices between different energy bands. However, it is shown that many of the key results of Luttinger and Nozi\`eres can be suitably generalized. The perturbation-theory calculation shows that when the periodic potential is present the change in quasiparticle energy can still be expressed in terms of the variation of the quasiparticle distribution function, and that the results obtained from the Green's-function approach are exact in the case of slowly varying external fields.