# Vertex corrections to the dc conductivity in anisotropic multiband systems

## TL;DR

This paper investigates vertex corrections to dc conductivity in anisotropic multiband systems, establishing a consistent relation with semiclassical Boltzmann theory and extending understanding beyond isotropic single-band models.

## Contribution

It generalizes the relation between vertex corrections and transport relaxation times to anisotropic multiband systems, confirming consistency with semiclassical results.

## Key findings

- Derived the relation for transport relaxation times in anisotropic multiband systems.
- Verified the consistency between Kubo formula and Boltzmann transport theory.
- Extended the understanding of vertex corrections beyond isotropic single-band models.

## Abstract

For an isotropic single-band system, it is well known that the semiclassical Boltzmann transport theory within the relaxation time approximation and the Kubo formula with the vertex corrections provide the same result with the $(1-\cos\theta)$ factor in the inverse transport relaxation time. In anisotropic multiband systems, the semiclassical Boltzmann transport equation is generalized to coupled integral equations relating transport relaxation times at different angles in different bands. Using the Kubo formula, we study the vertex corrections to the dc conductivity in anisotropic multiband systems and derive the relation satisfied by the transport relaxation time for both elastic and inelastic scatterings, verifying that the result is consistent with the semiclassical approach.