Discrete symmetries and spectral statistics

Semi-Poisson Statistics in Relativistic Quantum Billiards with Shapes of Rectangles

Rectangular billiards have two mirror symmetries with respect to perpendicular axes and a twofold (fourfold) rotational symmetry for differing (equal) side lengths. The eigenstates of rectangular neutrino billiards (NBs), which consist of a spin-1/2 particle confined through boundary conditions to a planar domain, can be classified according to their transformation properties under rotation by π (π/2) but not under reflection at mirror-symmetry axes. We analyze the properties of these symmetry-projected eigenstates and of the corresponding symmetry-reduced NBs which are obtained by cutting them along their diagonal, yielding right-triangle NBs. Independently of the ratio of their side lengths, the spectral properties of the symmetry-projected eigenstates of the rectangular NBs follow semi-Poisson statistics, whereas those of the complete eigenvalue sequence exhibit Poissonian statistics. Thus, in distinction to their nonrelativistic counterpart, they behave like typical quantum systems with an integrable classical limit whose eigenstates are non-degenerate and have alternating symmetry properties with increasing state number. In addition, we found out that for right triangles which exhibit semi-Poisson statistics in the nonrelativistic limit, the spectral properties of the corresponding ultrarelativistic NB follow quarter-Poisson statistics. Furthermore, we analyzed wave-function properties and discovered for the right-triangle NBs the same scarred wave functions as for the nonrelativistic ones.

Time-reversal-invariant hexagonal billiards with a point symmetry

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 120 000 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 20 000 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.

Quantization and interference of a quantum billiard with fourfold rotational symmetry

Systems with discrete symmetries are highly important in quantum mechanics. We consider a two-dimensional quantum billiard with fourfold rotational symmetry, where the eigenenergies and eigenstates can be grouped into four symmetry subspaces. Unlike the threefold rotational symmetry case, here the interference of the scarring states on the fundamental domain orbits (FDO) is clean, that they either interfere constructively or annihilate completely. We shall show the complex behavior of the interference revealed in the length spectra for eigenenergies that belong to a particular symmetry subspace and combinations of different symmetry subspaces. We then provide detailed analysis of phase accumulation along the FDOs, which are the keys to determine the interference and could explain the enhancement or annihilation of the peaks well. The quantization condition for the scarring states belonging to different symmetry subspaces is discussed and used to reveal the time-reversal symmetry broken for a particular subspace. An experimental scheme to observe such complex behaviors is also proposed.

Gaussian orthogonal ensemble statistics in graphene billiards with the shape of classically integrable billiards

A crucial result in quantum chaos, which has been established for a long time, is that the spectral properties of classically integrable systems generically are described by Poisson statistics, whereas those of time-reversal symmetric, classically chaotic systems coincide with those of random matrices from the Gaussian orthogonal ensemble (GOE). Does this result hold for two-dimensional Dirac material systems? To address this fundamental question, we investigate the spectral properties in a representative class of graphene billiards with shapes of classically integrable circular-sector billiards. Naively one may expect to observe Poisson statistics, which is indeed true for energies close to the band edges where the quasiparticle obeys the Schrödinger equation. However, for energies near the Dirac point, where the quasiparticles behave like massless Dirac fermions, Poisson statistics is extremely rare in the sense that it emerges only under quite strict symmetry constraints on the straight boundary parts of the sector. An arbitrarily small amount of imperfection of the boundary results in GOE statistics. This implies that, for circular-sector confinements with arbitrary angle, the spectral properties will generically be GOE. These results are corroborated by extensive numerical computation. Furthermore, we provide a physical understanding for our results.

Spectral properties and dynamical tunneling in constant-width billiards

We determine with unprecedented accuracy the lowest 900 eigenvalues of two quantum constant-width billiards from resonance spectra measured with flat, superconducting microwave resonators. While the classical dynamics of the constant-width billiards is unidirectional, a change of the direction of motion is possible in the corresponding quantum system via dynamical tunneling. This becomes manifest in a splitting of the vast majority of resonances into doublets of nearly degenerate ones. The fluctuation properties of the two respective spectra are demonstrated to coincide with those of a random-matrix model for systems with violated time-reversal invariance and a mixed dynamics. Furthermore, we investigate tunneling in terms of the splittings of the doublet partners. On the basis of the random-matrix model we derive an analytical expression for the splitting distribution which is generally applicable to systems exhibiting dynamical tunneling between two regions with (predominantly) chaotic dynamics.

A semiclassical theory for nonseparable rovibrational motions in curved space and its application to energy quantization of nonrigid molecules

The nonseparability of vibrational and rotational motions of a nonrigid molecule placed in the rotationally isotropic space induces several important effects on the dynamics of intramolecular energy flow and chemical reaction. However, most of these studies have been performed within the framework of classical mechanics. We present a semiclassical theory for the motions of such nonrigid molecules and apply to the energy quantization of three body atomic cluster. It is shown numerically that the semiclassical spectum given without the correct account of the rotational symmetry suffers from unnecessary broadening of the resultant spectral lines and moreover from spurious peaks.

Symmetry-adapted correlation function for semiclassical quantization

We study a very simple method to incorporate quantum-mechanical symmetries, including the permutational symmetry on an equal footing with spatial symmetries, into the semiclassical calculation of correlation functions. This method is applied to the calculation of energy spectra to verify its validity by reproducing quantum energy levels for systems of bosons (symmetrized) and fermions (antisymmetrized). The mechanism of how the phase-space structure of classical dynamics is linked with the relevant quantum symmetry is discussed.

Transition from Gaussian-orthogonal to Gaussian-unitary ensemble in a microwave billiard with threefold symmetry

Recently it has been shown that time-reversal invariant systems with discrete symmetries may display, in certain irreducible subspaces, spectral statistics corresponding to the Gaussian-unitary ensemble (GUE) rather than to the expected orthogonal one (GOE). A Kramers-type degeneracy is predicted in such situations. We present results for a microwave billiard with a threefold rotational symmetry and with the option to display or break a reflection symmetry. This allows us to observe the change from GOE to GUE statistics for one subset of levels. Since it was not possible to separate the three subspectra reliably, the number variances for the superimposed spectra were studied. The experimental results are compared with a theoretical and numerical study considering the effects of level splitting and level loss.

Gaussian unitary ensemble statistics in a time-reversal invariant microwave triangular billiard

The spectrum of a chaotic two-dimensional quantum billiard with threefold symmetry has been studied in an experiment with a superconducting microwave cavity. In total 622 eigenvalues were identified experimentally and compared with numerical calculations. The statistical analysis of the data shows that Gaussian unitary ensemble statistics can be observed for a spectrum of a time-reversal invariant system.