A.C. Conductivity of a Disordered Metal

TL;DR

This paper calculates the frequency-dependent electrical conductivity of a disordered metal using a theoretical approach based on the free Fermi gas model and the Kubo formula, considering Gaussian disorder.

Contribution

It introduces a formal diagonalization method and a cumulant expansion technique to compute a.c. conductivity in disordered systems across different dimensions.

Findings

Derived a formula for a.c. conductivity in disordered metals

Validated the approach in the long-wavelength limit

Applicable to Gaussian disorder with linearization approximation

Abstract

The degenerate free Fermi gas coupled to a random potential is used to compute a.c. conductivity in various dimensions. We first formally diagonalise the hamiltonian using an appropriate basis that is a functional of the disorder potential. Then we compute the a.c. conductivity at zero temperature using the Kubo formula. This a.c. conductivity is a functional of the disordered potential. The wavefunction of extended states is written as exponential of the logarithm. We use the cumulant expansion to compute the disordered averaged a.c. conductivity for Gaussian disorder. The formula is valid if a certain linearization approximation is valid in the long-wavelength limit.