Spectral Correlations from the Metal to the Mobility Edge

Abstract

We have studied numerically the spectral correlations in a metallic phase and at the metal-insulator transition. We have calculated directly the two-point correlation function of the density of states $R(s,s&#39;)$. In the metallic phase, it is well described by the Random Matrix Theory (RMT). For the first time, we also find numerically the diffusive corrections for the number variance $<δn^2(s)>$ predicted by Al&#39;tshuler and Shklovski\uı. At the transition, at small energy scales, $R(s-s&#39;)$ starts linearly, with a slope larger than in a metal. At large separations $|s - s&#39;| \gg 1$, it is found to decrease as a power law $R(s,s&#39;) \sim - c / |s -s&#39;|^{2-γ}$ with $c \sim 0.041$ and $γ\sim 0.83$, in good agreement with recent microscopic predictions. At the transition, we have also calculated the form factor $\tilde K(t)$, Fourier transform of $R(s-s&#39;)$. At large $s$, the number variance contains two terms $<δn^2(s) >= B < n >^γ+ 2 π\tilde K(0)< n > where $\tilde{K}(0)$ is the limit of the form factor for $t \to 0$.