Impact of localization on Dyson's circular ensemble

TL;DR

This paper investigates how localization affects the eigenvalue statistics of systems modeled by Dyson's circular ensemble, revealing a transition from universal Wigner to Poisson distribution due to localization.

Contribution

It introduces a solvable model that qualitatively captures the impact of localization on eigenvalue restrictions and spectrum gaps in Dyson's circular ensemble.

Findings

Localization induces a spectral gap in the eigenvalue distribution.

Transition from Wigner to Poisson distribution with increasing localization.

Model provides qualitative understanding of localization effects on eigenvalue statistics.

Abstract

A wide variety of complex physical systems described by unitary matrices have been shown numerically to satisfy level statistics predicted by Dyson's circular ensemble. We argue that the impact of localization in such systems is to provide certain restrictions on the eigenvalues. We consider a solvable model which takes into account such restrictions qualitatively and find that within the model a gap is created in the spectrum, and there is a transition from the universal Wigner distribution towards a Poisson distribution with increasing localization.