A Supersymmetry Approach to Billiards with Randomly Distributed Scatterers II: Correlations

TL;DR

This paper extends previous work on billiards with random scatterers by calculating the k-point correlation function, revealing that correlations in the bulk match GUE predictions and change near the band edges, indicating a mobility edge.

Contribution

It introduces a supersymmetric method to compute correlation functions in disordered billiards, showing a transition from GUE-like correlations to uncorrelated behavior near band edges.

Findings

Correlations in the bulk match GUE predictions.

Density of states depletes near band edges.

Transition from GUE to uncorrelated eigenvalues observed.

Abstract

In a previous contribution (H.J. Stoeckmann, J. Phys. A35, 5165 (2002)), the density of states was calculated for a billiard with randomly distributed delta-like scatterers, doubly averaged over the positions of the impurities and the billiard shape. This result is now extended to the k-point correlation function. Using supersymmetric methods, we show that the correlations in the bulk are always identical to those of the Gaussian Unitary Ensemble (GUE) of random matrices. In passing from the band centre to the tail states, The density of states is depleted considerably and the two-point correlation function shows a gradual change from the GUE behaviour to that found for completely uncorrelated eigenvalues. This can be viewed as similar to a mobility edge.