Electron Self-Energy and Generalized Drude Formula for Infrared Conductivity of Metals

TL;DR

This paper explores the relationship between the memory function used in infrared conductivity models of metals and the electron self-energy, providing a derivation using Matsubara Green's functions to clarify their connection.

Contribution

It offers a detailed derivation linking the memory function to the electron self-energy, clarifying their relation beyond approximate treatments in the context of electron-phonon interactions.

Findings

The imaginary part of the memory function is twice that of the self-energy in approximate models.

A precise relation between the memory function and self-energy is established for electron-phonon interactions.

The derivation uses Matsubara Green's functions to rigorously connect these quantities.

Abstract

Goetze and Woelfle (GW) wrote the conductivity in terms of a memory function M as (ine2/m)/(omega+M(omega)), where M=i/tau in the Drude limit. The analytic properties of -M are the same as those of the self-energy of a retarded Green's function. In the approximate treatment of GW, -M closely resembles a self-energy, with differences, e.g., the imaginary part is twice too large. The correct relation between -M and the self-energy is known for the electron-phonon case and is conjectured to be similar for other perturbations. When vertex corrections are ignored there is a known relation. A derivation using Matsubara temperature Green's functions is given.