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〉\ensuremath{\gg}1 is proved to behave as 〈N${\mathrm{〉}}^{\ensuremath{\gamma}}$ where \ensuremath{\gamma}=1-(\ensuremath{\nu}d${)}^{\mathrm{\ensuremath{-}}1}$ and \ensuremath{\nu} is the correlation length exponent. The inequality \ensuremath{\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.