A diagrammer's note on DC conductivity of anisotropic Fermi liquids for beginners: Maebashi-Fukuyama formula and Taylor formula

TL;DR

This paper reviews exact DC conductivity formulas for anisotropic Fermi liquids, comparing the Maebashi-Fukuyama and Taylor formulas, and discusses the limitations of current-vertex-correction approximations.

Contribution

It clarifies the formal exact formulas for anisotropic Fermi liquids and highlights issues with common approximation methods.

Findings

Maebashi-Fukuyama formula based on Fermi-surface harmonics

Taylor formula based on scattering eigenfunction

Current-vertex-correction approximation is inconsistent

Abstract

Formal but exact DC conductivity formulae for anisotropic Fermi liquids are reviewed. One is the Maebashi-Fukuyama formula based on the Fermi-surface harmonics. The other is the Taylor formula based on the scattering eigenfunction. In comparison to these two formulae the current-vertex-correction in the fluctuation-exchange approximation is shown to be a bad vision caused by an inconsistent approximation.