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

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

1999-07-12

arxiv.org/abs/cond-mat/9907174v1

Physics

Condensed Matter

Disordered Systems and Neural Networks

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

Scientific paper

10.1103/PhysRevLett.83.1139

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.

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