Spectral Correlations from the Metal to the Mobility Edge

TL;DR

This study numerically analyzes spectral correlations across metallic and transition phases, confirming theoretical predictions and revealing new behaviors of the two-point correlation function and form factor at the metal-insulator transition.

Contribution

First numerical calculation of diffusive corrections to spectral correlations at the metal-insulator transition, validating recent microscopic predictions.

Findings

Spectral correlations in the metallic phase match Random Matrix Theory.

Diffusive corrections to number variance are observed at the transition.

Correlation function decreases as a power law at large separations with specific parameters.

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 $R(s,s')$. 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 $<\delta n^2(s)>$ predicted by Al'tshuler and Shklovski\u{\i}. At the transition, at small energy scales, $R(s-s')$ starts linearly, with a slope larger than in a metal. At large separations $|s - s'| \gg 1$, it is found to decrease as a power law $R(s,s') \sim - c / |s -s'|^{2-\gamma}$ with $c \sim 0.041$ and $\gamma \sim 0.83$, in good agreement with recent microscopic predictions. At the transition, we have also calculated the form factor $\tilde K(t)$, Fourier transform of $R(s-s')$. At large $s$, the number variance contains two terms $<\delta n^2(s) >= B < n >^\gamma + 2 \pi \tilde K(0)< n > where $\tilde{K}(0)$ is the limit of the form factor for $t \to 0$.