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

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.

8 pages, 2 figures, accepted for publication in Phys. Rev. Lett

References in corpus (5)

• Random Matrix Theories in Quantum Physics: Common Concepts
• Quantum localization in rough billiards
• Weak Chaos in a Quantum Kepler Problem
• Emergence of Quantum Ergodicity in Rough Billiards
• Berry-Robnik level statistics in a smooth billiard system

Cited by in corpus (1)

• Criticality in the Quantum Kicked Rotor with a Smooth Potential