Spectral Statistics and Dynamical Localization: sharp transition in a generalized Sinai billiard

TL;DR

This paper investigates how changing the potential near a scatterer in a Sinai billiard causes a transition in spectral statistics from Poisson to Wigner-Dyson, linked to a dynamical localization-delocalization transition similar to metal-insulator transitions.

Contribution

It demonstrates a sharp transition in spectral statistics and dynamical localization in a generalized Sinai billiard based on the potential's decay rate.

Findings

Spectral statistics tend to Poisson for <2 and Wigner-Dyson for >2.

At =2, spectral statistics are energy-independent but depend on .

Transition is accompanied by a dynamical localization-delocalization transition.

Abstract

We consider a Sinai billiard where the usual hard disk scatterer is replaced by a repulsive potential with $V(r)\sim\lambda r^{-\alpha}$ close to the origin. Using periodic orbit theory and numerical evidence we show that its spectral statistics tends to Poisson statistics for large energies when $\alpha<2$ and to Wigner-Dyson statistics when $\alpha>2$, while for $\alpha=2$ it is independent of energy, but depends on $\lambda$. We apply the approach of Altshuler and Levitov [Phys. Rep. {\bf 288}, 487 (1997)] to show that the transition in the spectral statistics is accompanied by a dynamical localization-delocalization transition. This behaviour is reminiscent of a metal-insulator transition in disordered electronic systems.