Classical singularities and Semi-Poisson statistics in quantum chaos and disordered systems

TL;DR

This paper demonstrates that semi-Poisson level statistics in quantum systems arise from non-analytic dispersion relations and are robust against microscopic details, differing fundamentally from critical statistics at the Anderson transition.

Contribution

It establishes the origin of semi-Poisson statistics in non-analyticities and distinguishes it from critical statistics, supported by analysis of disordered Hamiltonians and kicked rotators.

Findings

Semi-Poisson statistics are robust to microscopic details.

Non-analyticity in dispersion relations causes semi-Poisson statistics.

Semi-Poisson differs fundamentally from critical statistics at the Anderson transition.

Abstract

We investigate a 1D disordered Hamiltonian with a non analytical step-like dispersion relation whose level statistics is exactly described by Semi-Poisson statistics(SP). It is shown that this result is robust, namely, does not depend neither on the microscopic details of the potential nor on a magnetic flux but only on the type of non-analyticity. We also argue that a deterministic kicked rotator with a non-analytical step-like potential has the same spectral properties. Semi-Poisson statistics (SP), typical of pseudo-integrable billiards, has been frequently claimed to describe critical statistics, namely, the level statistics of a disordered system at the Anderson transition (AT). However we provide convincing evidence they are indeed different: each of them has its origin in a different type of classical singularities.