Quantum chaos in triangular billiards

Abstract

Triangular billiards whose angles are rational multiples of π are one of the simplest examples of pseudo-integrable models with intriguing classical and quantum properties. We perform an extensive numerical study of spectral statistics of eight quantized rational triangles, six belonging to the family of right-angled Veech triangles and two obtuse rational triangles. Large spectral samples of up to one million energy levels were calculated for each triangle which permits to determine their spectral statistics with great accuracy. It is demonstrated that they are of the intermediate type, sharing some features with chaotic systems, like level repulsion and some with integrable systems, like exponential tails of the level spacing distributions. Another distinctive feature of intermediate spectral statistics is a finite value of the level compressibility. The short range statistics such as the level spacing distributions, and long-range statistics such as the number variance and spectral form factors were analyzed in detail. An excellent agreement between the numerical data and the model of gamma distributions is revealed.

Dynamical quantum ergodicity from energy level statistics

“…For Wigner-Dyson random matrix ensembles as well as individual systems with Wigner-Dyson (local) level statistics, it is convenient to choose Σ d to be an energy shell spanned by d consecutive energy levels. There is numerical evidence that the mode fluctuation distribution is Gaussian [33,55,56] (as well as analytical evidence for a related measure, number fluctuations [74]) especially near ∆ ≈ 0, suggestive of an idealized Gaussian persistence. In Refs.…”

“…In this case, the ∆ n are essentially what have been called mode fluctuations in the spectrum 1 [33,55,56]; the Gaussianity of their distribution has been conjectured to be a signature of chaos [55,56]. A minor, but important, technical distinction between ∆ n and conventional mode fluctuations, is that there is no unfolding [7,8] of the energy levels to make Ω(Σ d ) appear uniform prior to calculating the ∆ n .…”

“…Sec. 3 defines their analogues in quantum mechanics, and proves that cyclic ergodicity and aperiodicity are directly determined by a specific measure of level statistics and spectral rigidity (namely, mode fluctuations [33,55,56]).…”

“…[29][30][31][32]), assuming the dominance of isolated periodic orbits in semiclassical contributions to level statistics, a certain uniform distribution of these periodic orbits, and a K-mixing classical system [7]. However, extremely recent numerical studies of systems without K-mixing show that Wigner-Dyson level statistics can emerge even on quantization of merely mixing [33] or merely ergodic [34] classical systems. At minimum, this merits a theoretical explanation of spectral rigidity that connects to the classical limit but does not rely on K-mixing.…”

Solitary wave billiards

“…Even though it was proved that if classical billiard is ergodic, then the corresponding quantum billiard is quantum ergodic, so-called quantum "scars" are observed in most cases, which demonstrate that quantum billiards are more complicated. The Bunimovich stadium [15,16] has recently become a very popular quantum billiard model, while the study of quantum chaos in different billiards is generally a topic that is gaining considerable traction [19][20][21].…”