Statistical and dynamical properties of the quantum triangle map

TL;DR

This paper investigates the quantum triangle map, revealing that ergodicity influences spectral statistics aligning with Random Matrix Theory, while chaos diagnostics like OTOC differ due to the non-chaotic classical dynamics.

Contribution

It demonstrates that spectral properties reflect ergodicity even without classical chaos, and distinguishes chaos diagnostics from spectral analysis.

Findings

Spectral statistics follow Random Matrix Theory predictions for ergodic systems.

OTOC and harmonics growth rates vanish in the semiclassical limit.

Classical dynamics has zero Lyapunov exponent, indicating no chaos.

Abstract

We study the statistical and dynamical properties of the quantum triangle map, whose classical counterpart can exhibit ergodic and mixing dynamics, but is never chaotic. Numerical results show that ergodicity is a sufficient condition for spectrum and eigenfunctions to follow the prediction of Random Matrix Theory, even though the underlying classical dynamics is not chaotic. On the other hand, dynamical quantities such as the out-of-time-ordered correlator (OTOC) and the number of harmonics, exhibit a growth rate vanishing in the semiclassical limit, in agreement with the fact that classical dynamics has zero Lyapunov exponent. Our finding show that, while spectral statistics can be used to detect ergodicity, OTOC and number of harmonics are diagnostics of chaos.