Density and current response functions in strongly disordered electron systems: Diffusion, electrical conductivity and Einstein relation

TL;DR

This paper investigates the fundamental relations between density and current response functions in strongly disordered electron systems, establishing exact relations and extending classical formulas to quantum nonequilibrium regimes, including diffusion and Einstein relations.

Contribution

It derives exact relations between response functions, extends the Einstein relation to quantum systems, and introduces a quantum diffusion framework in disordered electron gases.

Findings

Exact relations between density and current response functions.

Extension of the Einstein relation to quantum nonequilibrium systems.

Proof of the diffusion pole in zero-temperature electron-hole correlation functions.

Abstract

We study consequences of gauge invariance and charge conservation of an electron gas in a strong random potential perturbed by a weak electromagnetic field. We use quantum equations of motion and Ward identities for one- and two-particle averaged Green functions to establish exact relations between density and current response functions. In particular we find precise conditions under which we can extract the current-current correlation function from the density-density correlation function and vice versa. We use these results in two different ways to extend validity of a formula associating the density response function with the electrical conductivity from semiclassical equilibrium to quantum nonequilibrium systems. Finally we introduce quantum diffusion via a response relating the current with the negative gradient of the charge density. With the aid of this response function we derive a quantum version of the Einstein relation and prove the existence of the diffusion pole in the zero-temperature electron-hole correlation function with the the long-range spatial fluctuations controlled by the static diffusion constant.