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 $⟨N⟩\gg 1$ is proved to behave as $⟨N{⟩}^{\gamma }$ 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.

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