Chaos and integrability in triangular billiards

Abstract

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) average LSRs increase; (2) SC 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 characteristics 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.