Universal Spectral Correlations at the Mobility Edge

TL;DR

This paper shows that spectral level statistics near the Anderson transition in higher dimensions are universal and differ from known regimes, with a specific variance behavior linked to the correlation length exponent.

Contribution

It establishes the universality of spectral correlations at the mobility edge and derives the variance scaling law involving the correlation length exponent.

Findings

Level statistics are universal near the Anderson transition.

Variance of the number of levels scales as $raket{N}^A3$ with $A3=1-( u d)^{-1}$.

Spectral correlations differ from both metallic and insulating regimes.

Abstract

We demonstrate the level statistics in the vicinity of the Anderson transition in $d>2$ dimensions to be universal and drastically different from both Wigner-Dyson in the metallic regime and Poisson in the insulator regime. The variance of the number of levels $N$ in a given energy interval with $\langle N\rangle\gg1$ is proved to behave as $\langle N\rangle^\gamma$ where $\gamma=1-(\nu d)^{-1}$ and $\nu$ is the correlation length exponent. The inequality $\gamma<1$, shown to be required by an exact sum rule, results from nontrivial cancellations (due to the causality and scaling requirements) in calculating the two-level correlation function.