The Level Spacing Distribution Near the Anderson Transition

TL;DR

This paper investigates the distribution of energy level spacings near the Anderson transition, revealing a universal asymptotic form that interpolates between metallic and insulating behaviors, depending on the system's dimensionality.

Contribution

It introduces a universal form for the level spacing distribution near the Anderson transition, characterized by a power-law repulsion with a critical exponent, derived via a classical gas analogy.

Findings

The level spacing distribution follows an exponential of a power-law form.

The asymptotics depend only on the system's dimensionality.

The interaction between levels is a power-law repulsion with exponent -γ.

Abstract

For a disordered system near the Anderson transition we show that the nearest-level-spacing distribution has the asymptotics $P(s)\propto \exp(-A s^{2-\gamma })$ for $s\gg \av{s}\equiv 1$ which is universal and intermediate between the Gaussian asymptotics in a metal and the Poisson in an insulator. (Here the critical exponent $0<\gamma<1$ and the numerical coefficient $A$ depend only on the dimensionality $d>2$). It is obtained by mapping the energy level distribution to the Gibbs distribution for a classical one-dimensional gas with a pairwise interaction. The interaction, consistent with the universal asymptotics of the two-level correlation function found previously, is proved to be the power-law repulsion with the exponent $-\gamma$.