Spectral correlations in systems undergoing a transition from periodicity to disorder

TL;DR

This paper investigates how spectral statistics evolve in quasi-1D systems transitioning from periodic to disordered states, revealing a smooth interpolation between Poissonian and universal periodic behaviors.

Contribution

It provides a theoretical framework for spectral correlations during the transition, validated by numerical simulations on chaotic billiards and graphs.

Findings

Spectral two-point form factor depends on disorder degree.

Smooth interpolation between Poissonian and periodic spectral statistics.

Excellent agreement between theory and numerical results.

Abstract

We study the spectral statistics for extended yet finite quasi 1-d systems which undergo a transition from periodicity to disorder. In particular we compute the spectral two-point form factor, and the resulting expression depends on the degree of disorder. It interpolates smoothly between the two extreme limits -- the approach to Poissonian statistics in the (weakly) disordered case, and the universal expressions derived for the periodic case. The theoretical results agree very well with the spectral statistics obtained numerically for chains of chaotic billiards and graphs.