Mean-field theories for disordered electrons: Diffusion pole and Anderson localization

TL;DR

This paper explores the conditions necessary for mean-field theories to accurately describe electron localization and transport in disordered systems, emphasizing electron-hole symmetry and self-consistency at the two-particle level.

Contribution

It introduces a mean-field framework derived from high-dimensional limits that captures the Anderson localization transition with key symmetry and self-consistency requirements.

Findings

The diffusion pole weight decreases with disorder and vanishes in the localized phase.

A local mean-field theory consistent with electron-hole symmetry is derived from high-dimensional asymptotics.

The theory provides insights into the vanishing of electron diffusion at localization.

Abstract

We discuss conditions to be put on mean-field-like theories to be able to describe fundamental physical phenomena in disordered electron systems. In particular, we investigate options for a consistent mean-field theory of electron localization and for a reliable description of transport properties. We argue that a mean-field theory for the Anderson localization transition must be electron-hole symmetric and self-consistent at the two-particle (vertex) level. We show that such a theory with local equations can be derived from the asymptotic limit to high spatial dimensions. The weight of the diffusion pole, i. e., the number of diffusive states at the Fermi energy, in this mean-field theory decreases with the increasing disorder strength and vanishes in the localized phase. Consequences of the disclosed behavior for our understanding of vanishing of electron diffusion are discussed.