Time-reversal-invariant hexagonal billiards with a point symmetry.

Abstract

A biparametric family of hexagonal billiards enjoying the C_{3} point symmetry is introduced and numerically investigated. First, the relative measure r(ℓ,θ;t) in a reduced phase space was mapped onto the parameter plane ℓ×θ for discrete time t up to 10^{8} and averaged in tens of randomly chosen initial conditions in each billiard. The resulting phase diagram allowed us to identify fully ergodic systems in the set. It is then shown that the absolute value of the position autocorrelation function decays like |C_{q}(t)|∼t^{-σ}, with 0<σ⩽1 in the hexagons. Following previous examples of irrational triangles, we were able to find billiards for which σ∼1. This is further evidence that, although not chaotic (all Lyapunov exponents are zero), billiards in polygons might exhibit a near strongly mixing dynamics in the ergodic hierarchy. Quantized counterparts with distinct classical properties were also characterized. Spectral properties of singlets and doublets of the quantum billiards were investigated separately well beyond the ground state. As a rule of thumb, for both singlet and doublet sequences, we calculate the first 120000 energy eigenvalues in a given billiard and compute the nearest neighbor spacing distribution p(s), as well as the cumulative spacing function I(s)=∫_{0}^{s}p(s^{'})ds^{'}, by considering the last 20000 eigenvalues only. For billiards with σ∼1, we observe the results predicted for chaotic geometries by Leyvraz, Schmit, and Seligman, namely, a Gaussian unitary ensemble behavior in the degenerate subspectrum, in spite of the presence of time-reversal invariance, and a Gaussian orthogonal ensemble behavior in the singlets subset. For 0<σ<1, formulas for intermediate quantum statistics have been derived for the doublets following previous works by Brody, Berry and Robnik, and Bastistić and Robnik. Different regimes in a given energy spectrum have been identified through the so-called ergodic parameter α=t_{H}/t_{C}, the ratio between the Heisenberg time and the classical diffusive-like transport time, which signals the possibility of quantum dynamical localization when α<1. A good quantitative agreement is found between the appropriate formulas with parameters extracted from the classical phase space and the data from the calculated quantum spectra. A rich variety of standing wave patterns and corresponding Poincaré-Husimi representations in a reduced phase space are reported, including those associated with lattice modes, scarring, and high-frequency localization phenomena.