Chaos and integrability in triangular billiards

TL;DR

This work analyzes quantum dynamics in triangular billiards by classifying triangles via internal angles into integrable, pseudo-integrable, and non-integrable, and by applying five diagnostics that connect spectral behavior to Krylov-space dynamics. Through LSR, SC, Lanczos-variance, Krylov-eigenstate localisation, and spread complexity, the authors reveal systematic transitions: integrable cases show strong degeneracies, large Lanczos variance, and recurrences; generic non-integrable cases display GOE-like statistics with delocalized Krylov states and chaotic spread complexity; pseudo-integrable triangles sit between these regimes, with sector-specific GOE-like behavior in symmetry sectors. The study highlights the crucial role of symmetry (isosceles sectors) and translation-surface topology in shaping quantum chaos signatures, offering a unified Krylov-based framework for comparing integrability and chaos in polygonal billiards. The results provide a path toward applying these diagnostics to more complex quantum systems and quantum-field-theoretic contexts, including holographic models.

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.

Figures (12)

• Figure 1: 92nd (left) and 95th (right) energy eigenstate of $\kappa$=24 (irrational isosceles) triangle. The left (right) panel shows a symmetric (an anti-symmetric) eigenstates with respect to the reflection axis.
• Figure 2: Level spacing ratio (LSR) for all 27 triangles. The data points are marked by labels of the form $\chi:Tg$. Here, $\kappa$ is the index of the triangle from Table \ref{['tab:trig_list']}, $T$ is the triangle type ($E$ = equilateral, $I$ = isosceles, $R$ = right, $G$ = general), and $g$ is the genus of the translation surface of the triangle. For irrational triangles which have infinite genus, we replace $g$ by $2i$ or $3i$ for triangles with 2 and 3 irrational angles, respectively. Sky blue dashed line = LSR for the Gaussian Orthogonal Ensemble (GOE). Orange dot-dashed line = LSR value for the Poisson distribution. Integrable and isosceles triangles approach the Poisson result. Generic and right triangles approach the GOE result. We report lower and upper average LSRs for the three integrable triangles by replacing all ill-defined $0/0$ level spacing ratios with either $0$ or $1$.
• Figure 3: Level spacing ratio (LSR) for isosceles triangles. Data points marked using the same labels as Fig. \ref{['fig:LSRDistribution']}. PI = pseudo-integrable and NI = non-integrable. $\kappa$:S and $\kappa$:A denote the symmetric and anti-symmetric sectors of the triangle indexed by $\kappa$. The total spectra of isosceles triangles have LSRs closer to Poisson, but their symmetric and anti-symmetric sectors have LSRs closer to GOE.
• Figure 4: Spectral complexity of all the 27 billiards. The numbers on the right show the $\kappa$ index of the triangles as defined in Table \ref{['tab:trig_list']}. The labels P, G and D denote our analytical prediction of the spectral complexity growth for models with Poisson (proportional to $t$ at late times), GOE (logarithmic in $t$ at late times) and degenerate spectra (proportional to $t^2$ at late times) with the same degeneracy as that in the equilateral triangle $1$. Dotted lines are for isosceles triangles, dashed lines are for right triangles, and solid lines (except the three denoted P, G and D) are for general triangles. The integrable triangles with degenerate spectra show quadratic late time growth, while the non-integrable triangles match logarithmic late time growth except for the GOE. The isosceles triangles show linear late time growth similar to that expected for a Poisson spectrum.
• Figure 5: Spectral complexity for isosceles triangles. Labels match those in Fig. \ref{['fig:LSR_iso_sectors']}. The labels P and G denote the Poissonian and GOE spectral complexity, respectively. The isosceles triangles show linear growth at late times, but their anti-symmetric sectors more closely match the GOE behaviour, while their symmetric sectors also deviate dramatically from linear growth. Both symmetry sectors show large late time oscillations.