Random Walks through the Ensemble: Linking Spectral Statistics with Wavefunction Correlations in Disordered Metals

TL;DR

This paper introduces a Brownian motion model for energy levels in disordered metals, linking spectral statistics with wavefunction correlations through a stochastic approach, providing insights into spectral form factors at large energy scales.

Contribution

It develops a novel Langevin equation-based framework connecting spectral and wavefunction properties in disordered conductors, advancing understanding of spectral statistics in such systems.

Findings

Derived a Langevin equation relating spectral statistics to eigenfunction correlations.

Obtained the spectral form factor and its parametric dependence at large energy scales.

Established a stochastic model for impurity configuration ensembles in disordered metals.

Abstract

We use a random walk in the ensemble of impurity configurations to generate a Brownian motion model for energy levels in disordered conductors. Treating arc-length along the random walk as fictitous time, the resulting Langevin equation relates spectral statistics to eigenfunction correlations. Solving this equation at energy scales large compared with the mean level spacing, we obtain the spectral form factor, and its parametric dependence.