Spectral statistics and localization properties of a $C_3$-symmetric billiard

Abstract

We revisit the spectral statistics of the C$_3$--symmetric billiard introduced by Dembowski [Phys. Rev. E, R4516 (2000)], which exhibits both GOE and GUE statistics depending on the symmetry block. Using high--precision Beyn's contour--integral method for the nonlinear Fredholm eigenvalue problem with built-in separation of irreducible subspaces, we compute 2.8x10$^5$ eigenvalues in each symmetry subspace, enabling statistically meaningful comparisons with random matrix theory. The improved spectra reveal clear GOE--GUE correspondence and resolve previously observed deviations in long--range spectral correlations. Furthermore, we analyze phase--space eigenstate localization through the distribution of entropy localization measures, which, for chaotic states follow a Beta distribution whose standard deviation decays as a power--law with energy, consistent with the onset of quantum ergodicity as described by Schnirelman's theorem.

Figures (15)

• Figure 1: Primitive two- and three-bounce periodic orbits of the $C_3$ billiard at $a=0.2$. The boundary is shown in black, the three-bounce orbit in blue, the two-bounce orbit in green, and a six--bounce orbit in orange. Red dashed lines mark the symmetry axes defining the fundamental domain.
• Figure 2: Finite-time maximal Lyapunov exponent heatmaps $\lambda_{\max}$lyapunov_exponent for the $C_3$ billiard on the Poincaré section in the fundamental domain $s/L\in[0,1/3)$. For visual clarity the color scale is capped at $\lambda_{\max}=1$ (values above the cap are shown with the same maximal color). Dark regions ($\lambda_{\max}\approx0$) indicate regular motion, whereas brighter regions correspond to chaotic dynamics with positive $\lambda_{\max}$. Panels show increasing deformation amplitude:(a) $a=0.005$, (b) $a=0.01$, (c) $a=0.05$, (d) $a=0.2$, (e) $a=0.35$, (f) $a=0.5$. For $0.05 \lesssim a \lesssim 0.31$, the phase space is predominantly chaotic; for larger $a$, it becomes mixed. Our case $a=0.2$$(d)$ has very small stability islands (barely visible, encircled in $(d)$) corresponding to the orange PO in Fig.\ref{['fig:geometry_sketch']}. The heatmaps were constructed on a $1000 \times 1000$ grid with $5000$ collisions for each initial condition.
• Figure 3: Nearest-neighbor level spacing distributions for the desymmetrized spectrum of the $C_3$ billiard. (a) $m=0$ symmetry sector: the numerical histogram (approximately $2.8\times10^5$ unfolded eigenvalues) is compared with the Wigner surmise of the Gaussian Orthogonal Ensemble (GOE), reflecting the presence of time-reversal symmetry in this real irreducible representation. (b) $m=1$ symmetry sector: comparison with the Gaussian Unitary Ensemble (GUE) Wigner surmise, indicating effective time-reversal symmetry breaking in the complex irreducible representations.
• Figure 4: $\Delta_3(L)$ for the $m=0$ irreducible representation of the billiard with $a=0.2$ at various wavenumbers $k$. The dashed red line is the theoretical curve for the $\Delta_3^{GOE}(L)$, the blue vertical represents the length of the 3--bounce periodic orbit and the green line represents the 2--bounce periodic orbit, as shown in Fig.\ref{['fig:geometry_sketch']}.It is observed that the spectrum starts to deviate from RMT around $L_{max} \approx\frac{A_{fund} \times k_{max}}{l_0}$, which matches the value of $l_0 = 1.2415$ corresponding to the $2$--bounce orbit (green vertical line, shown in green in Fig.\ref{['fig:geometry_sketch']}), where $A_{fund}$ is the area of the desymmetrized domain ($A_{fund} = \frac{A}{3}$) and $k_{max}$ the largest wavenumber present in the spectrum (title in each subfigure) Berry1985prosen_shortest_PO. The blue vertical line corresponding to the $3$--bounce orbit with $l_0 = 1.6834$ (shown in blue in Fig.\ref{['fig:geometry_sketch']}) which was used in dietz_c3_new does not correctly capture the observed saturation trend.
• Figure 5: $\Delta_3(L)$ for the $m=1$ irreducible representation of the billiard with $a=0.2$ at various wavenumbers $k$. The plotted quantities are the same as in Fig.\ref{['fig:delta_prog_0']}, where importantly the value of $L_{max}$ is irrep agnostic, allowing for a direct comparison between saturation trend. In contrast to the $m=0$ case the deviation from $\Delta_3^{GUE}(L)$ RMT is observed already before the prediction given by the length of the shortest periodic orbit (green vertical line), while the saturation is still observed to start at the calculated $L_{max}$.