Macroscopic Conductivity of Free Fermions in Disordered Media

TL;DR

This paper analyzes the macroscopic electrical conductivity of free fermions in disordered media, establishing its mathematical properties, and showing how it converges to different measures depending on disorder strength and temperature.

Contribution

It provides a rigorous mathematical analysis of the macroscopic conductivity measure for free fermions, including its convergence behavior in various disorder regimes.

Findings

Conductivity measure exists and is finite at macroscopic scale.

Convergence to trivial measure in strong disorder (localization).

Convergence to atomic measure at zero frequency in perfect conductors.

Abstract

We conclude our analysis of the linear response of charge transport in lattice systems of free fermions subjected to a random potential by deriving general mathematical properties of its conductivity at the macroscopic scale. The present paper belongs to a succession of studies on Ohm and Joule's laws from a thermodynamic viewpoint. We show, in particular, the existence and finiteness of the conductivity measure $\mu _{\mathbf{\Sigma }}$ for macroscopic scales. Then we prove that, similar to the conductivity measure associated to Drude's model, $\mu _{\mathbf{\Sigma }}$ converges in the weak$^{\ast } $-topology to the trivial measure in the case of perfect insulators (strong disorder, complete localization), whereas in the limit of perfect conductors (absence of disorder) it converges to an atomic measure concentrated at frequency $\nu =0$. However, the AC--conductivity $\mu _{\mathbf{\Sigma }}|_{\mathbb{R}\backslash \{0\}}$ does not vanish in general: We show that $\mu _{\mathbf{\Sigma }}(\mathbb{R}\backslash \{0\})>0$, at least for large temperatures and a certain regime of small disorder.