Ergodicity and quantum correlations in irrational triangular billiards

Abstract

Pseudochaotic properties are systematically investigated in a one-parameter family of irrational triangular billiards (all angles irrational with π). The absolute value of the position correlation function C(x)(t) decays like ~t(-α). Fast (α≈1) and slow (0<α<1) decays are observed, thus indicating that the irrational triangles do not share a unique ergodic dynamics, which, instead, may vary smoothly between the opposite limits of strong mixing (α=1) and regular behaviors (α=0). Upgrading previous data, spectral statistical properties of the quantized counterparts are computed from 150000 energy eigenvalues numerically calculated for each billiard. Gaussian orthogonal ensemble spectral fluctuations are observed when α≈1 and intermediate statistics are found otherwise. Our irrational billiards have zero Kolmogorov-Sinai entropy and essentially infinity genus. Thus, differently from previous works on rational (pseudointegrable) enclosures, our results provide a missing classical-quantum correspondence regarding the ergodic hierarchy for a set of nonchaotic systems that might enjoy the strong mixing property.

Universal level statistics of the out-of-time-ordered operator

“…Moreover, quantum systems with merely mixing (non-chaotic) classical counterparts can obey Wigner-Dyson distribution even without classical exponential instabilities (see, e.g., Ref. [19]). Such cases are considered outside of the BGS characterization.…”

Entanglement Entropy in a Triangular Billiard

“…In the usual sense, building triangular billiards can be done easily by assigning rational or irrational values to the ratio between the inner angles of the triangle and π. Here, we follow the approach used by F. M. de Aguiar et al [8,9]. In order to construct the irrational triangular billiard, he has considered acute triangles with sides N, N + 1, and N + 2, where N is an integer.…”

“…Positive K-S entropy or the chaotic property is assured only for the Kolmogorov and Bernoulli systems while pseudochaos is a partiular case of weak chaos with zero K-S entropy. In [8,9], it has been shown that the irrational triangular billiard can indeed exhibit the pseudochaotic property. Taking this into account, we would like to explore how the irrationality of a triangle or the pseudochaotic can affect the bipartite entanglement of the eigenmodes.…”

“…Then it was shown in [8] that the irrational triangles may vary smoothly between the limits of strong mixing and regular behaviors. Furthermore, in [9] it was also shown that the level statistics of the irrational triangle indeed obey the Gaussian orthogonal ensemble (GOE) distribution and its intermediate statistics.…”

“…A quantum particle inside a triangular potential is another interesting problem studied by several researchers [5][6][7][8][9]. It was Krishnamurthy et al [10] who found a transformation to map the three-particle collision problem in one dimension into the problem of a quantum particle inside a two-dimensional triangular domain.…”