Time-reversal-invariant hexagonal billiards with a point symmetry

Area of Science:

• Mathematical Physics
• Dynamical Systems
• Quantum Chaos

Background:

• Billiards in polygonal domains are models for studying classical and quantum dynamics.
• Understanding the relationship between classical chaos and quantum spectral statistics is a key challenge.
• Hexagonal billiards with C3 symmetry offer a unique system for exploring these dynamics.

Purpose of the Study:

• To introduce and numerically investigate a biparametric family of hexagonal billiards.
• To map the phase diagram and identify fully ergodic systems.
• To characterize the spectral properties of quantum counterparts and their relation to classical dynamics.

Main Methods:

• Numerical investigation of relative measure in reduced phase space for discrete time.
• Analysis of position autocorrelation function decay rates (σ).
• Calculation of energy eigenvalues, nearest neighbor spacing distribution, and cumulative spacing function for quantum billiards.

Main Results:

• Identification of fully ergodic systems within the hexagonal billiard family.
• Observation of position autocorrelation function decay |Cq(t)|∼t−σ, with 0 < σ ≤ 1.
• Quantum spectra exhibit Gaussian Unitary Ensemble (GUE) and Gaussian Orthogonal Ensemble (GOE) behaviors for σ∼1, akin to chaotic systems.
• Formulas for intermediate quantum statistics derived for 0 < σ < 1.
• Ergodic parameter α identifies regimes of quantum dynamical localization.

Conclusions:

• Hexagonal billiards, despite being non-chaotic (zero Lyapunov exponents), can exhibit near strongly mixing dynamics.
• Quantum spectral statistics in these billiards mimic those of chaotic systems, providing evidence for universality.
• The study establishes a quantitative link between classical phase space properties and quantum spectral behavior.