Dynamical "breaking" of time reversal symmetry and converse quantum ergodicity

TL;DR

This paper demonstrates that certain quantum billiards with specific geometric properties defy the typical GOE energy level statistics, exhibiting GUE-type distributions due to broken ergodicity in momentum space.

Contribution

It introduces a class of convex billiards with constant width that show GUE statistics despite time reversal symmetry, highlighting a new mechanism for quantum ergodicity failure.

Findings

Convex billiards of constant width exhibit GUE-type energy level statistics.

Lack of ergodicity in momentum space causes deviation from classical predictions.

Certain multiply connected billiards are quantum ergodic but not classically ergodic.

Abstract

It is a common assumption that quantum systems with time reversal invariance and classically chaotic dynamics have energy spectra distributed according to GOE-type of statistics. Here we present a class of systems which fail to follow this rule. We show that for convex billiards of constant width with time reversal symmetry and "almost" chaotic dynamics the energy level distribution is of GUE-type. The effect is due to the lack of ergodicity in the "momentum" part of the phase space and, as we argue, is generic in two dimensions. Besides, we show that certain billiards of constant width in multiply connected domains are of interest in relation to the quantum ergodicity problem. These billiards are quantum ergodic, but not classically ergodic.