A mean-field theory of Anderson localization

TL;DR

This paper develops a mean-field theory for Anderson localization in high dimensions, identifying a disorder-driven transition where diffusion vanishes, and introduces an order parameter for localization.

Contribution

It simplifies the complex parquet equations to a single algebraic equation, revealing the bifurcation point indicating localization onset in high-dimensional systems.

Findings

Disorder-driven bifurcation signals localization transition.

No bifurcation occurs in 1D and 2D, where all states are localized.

A natural order parameter for Anderson localization is identified.

Abstract

Anderson model of noninteracting disordered electrons is studied in high spatial dimensions. We find that off-diagonal one- and two-particle propagators behave as gaussian random variables w.r.t. momentum summations. With this simplification and with the electron-hole symmetry we reduce the parquet equations for two-particle irreducible vertices to a single algebraic equation for a local vertex. We find a disorder-driven bifurcation point in this equation signalling vanishing of diffusion and onset of Anderson localization. There is no bifurcation in $d=1,2$ where all states are localized. A natural order parameter for Anderson localization pops up in the construction.