Universal Spectral Correlations at the Mobility Edge

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.