A semiclassical theory of the Anderson transition

TL;DR

This paper develops a semiclassical analytical framework for the Anderson transition, deriving critical exponents and disorder thresholds as functions of spatial dimensions, and analyzing level statistics in disordered conductors.

Contribution

It combines self-consistent localization theory with scaling theory to explicitly determine critical parameters and exponents for the Anderson transition across dimensions.

Findings

Critical exponent ν = 1/2 + 1/(d-2) for localization length divergence.

Upper critical dimension for Anderson transition is infinity.

Level correlation functions decay exponentially with increasing dimensionality.

Abstract

We study analytically the metal-insulator transition in a disordered conductor by combining the self-consistent theory of localization with the one parameter scaling theory. We provide explicit expressions of the critical exponents and the critical disorder as a function of the spatial dimensionality, $d$. The critical exponent $\nu$ controlling the divergence of the localization length at the transition is found to be $\nu = {1 \over 2}+ {1 \over {d-2}}$. This result confirms that the upper critical dimension is infinity. Level statistics are investigated in detail. We show that the two level correlation function decays exponentially and the number variance is linear with a slope which is an increasing function of the spatial dimensionality.