Spectral correlations in finite-size Anderson insulators

TL;DR

This paper derives an analytical expression for spectral correlations in finite-size Anderson insulators, bridging classical diffusive and quantum localized regimes, and discusses the transition from Poisson to Wigner-Dyson statistics.

Contribution

It introduces a non-perturbative method to analytically describe spectral statistics across different regimes in finite Anderson insulators.

Findings

Derived a closed-form expression for spectral correlations.

Identified the transition from Poisson to Wigner-Dyson statistics.

Bridged known asymptotic behaviors in different regimes.

Abstract

We investigate spectral correlations in quasi one-dimensional Anderson insulators with broken time-reversal symmetry. While energy levels are uncorrelated in the thermodynamic limit of infinite wire-length, some correlations remain in finite-size Anderson insulators. Asymptotic behaviors of level-level correlations in these systems are known in the large- and small-frequency limits, corresponding to the regime of classical diffusive dynamics and the deep quantum regime of strong Anderson localization. Employing non-perturbative methods and a mapping to the Coulomb-scattering problem, recently introduced by {\it M.~A.~Skvortsov} and {\it P.~M.~Ostrovsky}, we derive a closed analytical expression for the spectral statistics in the classical-to-quantum region bridging the known asymptotic behaviors. We further discuss how Poisson statistics at large energies develop into Wigner-Dyson statistics as the wire-length decreases.