Quantum chaos in triangular billiards

Abstract

We present an extensive numerical study of spectral statistics and eigenfunctions of quantized triangular billiards. We compute two million consecutive eigenvalues for six representative cases of triangular billiards, three with generic angles with irrational ratios with π, whose classical dynamics is presumably mixing, and three with exactly one angle rational with π, which are presumably only weakly mixing or even only non-ergodic in case of right-triangles. We find excellent agreement of short and long range spectral statistics with the Gaussian orthogonal ensemble of random matrix theory for the most irrational generic triangle, while the other cases show small but significant deviations which are attributed either to scarring or super-scarring mechanism. This result, which extends the quantum chaos conjecture to systems with dynamical mixing in the absence of hard (Lyapunov) chaos, has been corroborated by analysing distributions of phase-space localisation measures of eigenstates and inspecting the structure of characteristic typical and atypical eigenfunctions.

Statistical and dynamical properties of the quantum triangle map

“…The classical dynamics is not chaotic (even though it can become so in a generalized version of the map, discussed later in our paper, where the cusps in the potential are rounded-off) and, depending on the model parameters, is mixing, ergodic, quasi-ergodic [7], or quasi-integrable. Our results show that ergodicity is a sufficient condition to obtain spectral statistics as well as eigenfunction properties in agreement with RMT (see [9] for recent similar results for triangle billiards), while the quasi-ergodic case, where a single trajectory fills in the classical phase space extremely slowly in time [6], exhibits a different behavior depending on the quantity under scrutiny. That is, level spacing statistics is in good agreement with the Wigner-Dyson distribution in the semi-classical limit, while there exist eigenfunctions localized in phase space, incompatible with the predictions of RMT.…”

Statistical and dynamical properties of the quantum triangle map

“…The classical dynamics is not chaotic (even though it can become so in a generalized version of the map, discussed later in our paper, where the cusps in the potential are rounded-off ) and, depending on the model parameters, is mixing, ergodic, quasi-ergodic 8 , or quasi-integrable. Our results show that ergodicity is a sufficient condition to obtain spectral statistics as well as eigenfunction properties in agreement with RMT (see [12] for recent similar results for triangle billiards), while the quasi-ergodic case, where a single trajectory fills in the classical phase space extremely slowly in time [10], exhibits a different behavior depending on the quantity under scrutiny. That is, level spacing statistics is in good agreement with the Wigner-Dyson distribution in the semi-classical limit, while there exist eigenfunctions localized in phase space, incompatible with the predictions of RMT.…”

Chaos enhancement in large-spin chains

“…For systems with clear classical or semi-classical limits, quantum chaos refers to signatures found in the quantum domain, such as level statistics as in full random matrices [20], that indicate whether the classical system is chaotic in the sense of positive Lyapunov exponent and mixing. While this correspondence holds well for some systems with a small number of degrees of freedom, such as Sinai's billiard [1,2], it has recently been shown to be violated in triangular billiards [21] and quantum triangle maps [22]. As one moves to systems with many interacting particles, this issue gets even more complicated, since the classical limit is not always straightforward [23].…”