Semi-Poisson statistics in quantum chaos

Abstract

We investigate the quantum properties of a non-random Hamiltonian with a step-like singularity. It is shown that the eigenfunctions are multifractals and, in a certain range of parameters, the level statistics is described exactly by Semi-Poisson statistics (SP) typical of pseudo-integrable systems. It is also shown that our results are universal; namely, they depend exclusively on the presence of the step-like singularity and are not modified by smooth perturbations of the potential or the addition of a magnetic flux. Although the quantum properties of our system are similar to those of a disordered conductor at the Anderson transition, we report important quantitative differences in both the level statistics and the multifractal dimensions controlling the transition. Finally the study of quantum transport properties suggests that the classical singularity induces quantum anomalous diffusion. We discuss how these findings may be experimentally corroborated by using ultra cold atoms techniques.