Chaos and integrability in triangular billiards

TL;DR

This paper investigates quantum dynamics in triangular billiards, revealing how spectral and eigenstate properties evolve from integrable to non-integrable cases, highlighting differences in complexity, localization, and spectral statistics.

Contribution

It provides a comprehensive characterization of quantum chaos indicators in triangular billiards based on classical geometry and spectral analysis, connecting classical integrability with quantum spectral properties.

Findings

Level spacing ratios increase from integrable to non-integrable triangles.

Spectral complexity growth slows down in non-integrable cases.

Eigenstates delocalize in the Krylov basis as chaos increases.

Abstract

We characterize quantum dynamics in triangular billiards in terms of five properties: (1) the level spacing ratio (LSR), (2) spectral complexity (SC), (3) Lanczos coefficient variance, (4) energy eigenstate localisation in the Krylov basis, and (5) dynamical growth of spread complexity. The billiards we study are classified as integrable, pseudointegrable or non-integrable, depending on their internal angles which determine properties of classical trajectories and associated quantum spectral statistics. A consistent picture emerges when transitioning from integrable to non-integrable triangles: (1) LSRs increase; (2) spectral complexity growth slows down; (3) Lanczos coefficient variances decrease; (4) energy eigenstates delocalize in the Krylov basis; and (5) spread complexity increases, displaying a peak prior to a plateau instead of recurrences. Pseudo-integrable triangles deviate by a small amount in these charactertistics from non-integrable ones, which in turn approximate models from the Gaussian Orthogonal Ensemble (GOE). Isosceles pseudointegrable and non-integrable triangles have independent sectors that are symmetric and antisymmetric under a reflection symmetry. These sectors separately reproduce characteristics of the GOE, even though the combined system approximates characteristics expected from integrable theories with Poisson distributed spectra.