New formulation of the Kubo Optical Conductivity: a Shortcut to Transport Properties

TL;DR

This paper introduces a new formulation of the Kubo optical conductivity that simplifies the calculation of transport properties by expressing it as a sum over eigenstates with their own relaxation times, bridging the gap between Kubo and Boltzmann approaches.

Contribution

The paper presents a novel eigenstate-based formulation of the Kubo conductivity that makes static properties more accessible and clarifies the connection to Boltzmann transport theory.

Findings

The new formulation simplifies access to dc conductivity.

It correctly recovers the Boltzmann result with appropriate scattering times.

Applicable to systems with elastic and inelastic scattering at lowest order.

Abstract

The Kubo formula for the electrical conductivity is rewritten in terms of a sum of Drude-like contributions associated to the exact eigenstates of the interacting system, each characterized by its own frequency-dependent relaxation time. The structure of the novel and equivalent formulation, weighting the contribution from each eigenstate by its Boltzmann occupation factor, simplifies considerably the access to the static properties (dc conductivity) and resolves the long standing difficulties to recover the Boltzmann result for dc conductivity from the Kubo formula. It is shown that the Boltzmann result, containing the correct transport scattering time instead of the electron lifetime determined by the Green function, can be recovered in problems with elastic and inelastic scattering at the lowest order of interaction.