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arXiv:2504.15055 · Physical Review Letters

Failure of the Conformal-Map Method for Relativistic Quantum Billiards

Barbara Dietz

Published Jun 26, 2025• 1 citation
arXivHTMLPDF

Abstract

In H. Xu et al. [Phys. Rev. Lett. 110, 064102 (2013)PRLTAO0031-900710.1103/PhysRevLett.110.064102], a numerical method is introduced-an extension of the conformal-map method of Robnik [J. Phys. A 17, 1049 (1984)JPHAC50305-447010.1088/0305-4470/17/5/027] for nonrelativistic quantum billiards-for the quantization of relativistic neutrino billiards consisting of a massless noninteracting spin-1/2 particle confined to a two-dimensional domain. We demonstrate in this Letter that this method does not provide solutions of the associated Weyl (Dirac) equation, nor does it fulfill the boundary conditions imposed on the spinor eigenfunctions to ensure confinement of the particle to the domain of the billiard. We review in detail the wave equation, boundary conditions, and quantization of neutrino billiards and derivation of relevant equations to make the proof comprehensible for the general reader. Our results are corroborated with numerical ones for nonrelativistic and relativistic quantum billiards whose shapes depend on a parameter, which allows the study of the properties of their eigenstates as the classical dynamics experiences a transition from regular to chaotic dynamics.

DOI
10.1103/vj94-kc98

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In H. Xu et al. [Phys. Rev. Lett. 110, 064102 (2013)PRLTAO0031-900710.1103/PhysRevLett.110.064102], a numerical method is introduced-an extension of the conformal-map method of Robnik [J. Phys. A 17, 1049 (1984)JPHAC50305-447010.1088/0305-4470/17/5/027] for nonrelativistic quantum billiards-for the quantization of relativistic neutrino billiards consisting of a massless noninteracting spin-1/2 particle confined to a two-dimensional domain. We demonstrate in this Letter that this method does not provide solutions of the associated Weyl (Dirac) equation, nor does it fulfill the boundary conditions imposed on the spinor eigenfunctions to ensure confinement of the particle to the domain of the billiard. We review in detail the wave equation, boundary conditions, and quantization of neutrino billiards and derivation of relevant equations to make the proof comprehensible for the general reader. Our results are corroborated with numerical ones for nonrelativistic and relativistic quantum billiards whose shapes depend on a parameter, which allows the study of the properties of their eigenstates as the classical dynamics experiences a transition from regular to chaotic dynamics.
Neutrino billiards serve as a model system for the study of aspects of relativistic quantum chaos. These are relativistic quantum billiards consisting of a spin-1/2 particle which is confined to a planar domain by imposing boundary conditions on the spinor components which were proposed in [Berry and Mondragon 1987, {\it Proc. R. Soc.} A {\bf 412} 53) . We review their general features and the properties of neutrino billiards with shapes of billiards with integrable dynamics. Furthermore, we review the features of two neutrino billiards with the shapes of billiards generating a chaotic dynamics, whose nonrelativistic counterpart exhibits particular properties. Finally we briefly discuss possible experimental realizations of relativistic quantium billiards based on graphene billiards, that is, finite size sheets of graphene.
Quantum signatures of classical chaos in the spectral properties of non-relativistic quantum billiards are well understood by now based on the semiclassical and the random matrix theory aproach.The experimental detection of relativistic phenomena featured by graphene implicated an increasing interest in the properties of graphene billiards and relativistic quantum billiards consisting of massless spin-1/2 particles governed by the Dirac equation and confined to a planar domain.Confinement is attained by imposing appropriate boundary conditions on the wave function components.We chose those proposed for neutrino billiards.By now numerical studies have been performed on the spectral properties of neutrino billiards of various shapes.Our main focus is the question to what extent the semiclassical approach in terms of trace formulae applies to such systems.We derive trace formulae for circular and elliptical neutrino billiards.The associated eigenvalue equations have been solved analytically for the circular billiard.We present analytical solutions for elliptical neutrino billiards.
Scarring in quantum systems with classical chaotic dynamics is one of the most remarkable phenomena in modern physics. Previous works were concerned mostly with nonrelativistic quantum systems described by the Schrödinger equation. The question remains outstanding of whether truly relativistic quantum particles that obey the Dirac equation can scar. A significant challenge is the lack of a general method for solving the Dirac equation in closed domains of arbitrary shape. In this paper, we develop a numerical framework for obtaining complete eigensolutions of massless fermions in general two-dimensional confining geometries. The key ingredients of our method are the proper handling of the boundary conditions and an efficient discretization scheme that casts the original equation in a matrix representation. The method is validated by (1) comparing the numerical solutions to analytic results for a geometrically simple confinement and (2) verifying that the calculated energy level-spacing statistics of integrable and chaotic geometries agree with the known results. Solutions of the Dirac equation in a number of representative chaotic geometries establish firmly the existence of scarring of Dirac fermions.
Abstract According to the Berry–Tabor conjecture, the spectral properties of typical nonrelativistic quantum systems with an integrable classical counterpart agree with those of Poissonian random numbers. We investigate to what extend it applies to relativistic neutrino billiards (NBs) consisting of a spin-1/2 particle confined to a bounded planar domain by imposing suitable boundary conditions (BCs). In distinction to nonrelativistic quantum billiards (QBs), NBs do not have a well-defined classical counterpart. However, the peaks in the length spectra, that is, the modulus of the Fourier transform of the spectral density from wave number to length, of NBs are just like for QBs at the lengths of periodic orbits of the classical billiard (CB). This implies that there must be a connection between NBs and the dynamic of the CB. We demonstrate that NBs with shapes of circle- and ellipse-sectors with an integrable classical dynamic, obtained by cutting the circle and ellipse NB along symmetry lines, have no common eigenstates with the latter and that, indeed, their spectral properties can be similar to those of classically chaotic QBs. These features orginate from the intermingling of symmetries of the spinor components and the discontinuity in the BCs leading to contradictory conditional equations at corners connecting curved and straight boundary parts. To corroborate the necessity of the curved boundary part in order to generate GOE-like behavior, we furthermore consider the right-angled triangle NB constructed by halving the equilateral-triangle NB along a symmetry axis. For an understanding of these findings in terms of purely classical quantities we use the semiclassical approach recently developed for massive NBs, and Poincaré–Husimi distributions of the eigenstates in classical phase space. The results indicate, that in the ultrarelativistic limit these NBs do not show the behavior expected for classically chaotic QBs.
Abstract We investigate the spectral properties and Husimi functions of relativistic quantum billiards (QBs) consisting of a spin-1/2 particle governed by the Dirac equation and confined to a planar domain of constant-width (CW) by imposing boundary conditions (BCs) on the spinor components. We consider those of neutrino billiards (NBs) proposed in (Berry and Mondragon 1987 Proc. R. Soc. A 412 53). The classical dynamics of billiards of corresponding shape is predominantly chaotic. CW billiards attracted particular attention because they exhibit unusual properties. Their classical dynamics features unidirectionality, whereas in the corresponding nonrelativistic QB a change of the rotational direction of motion is possible via dynamical tunneling, and the spectral properties coincide with those of typical quantum systems with violated time-reversal invariance. Unidirectionality of the quantum dynamics would arise in the structure of the Husimi functions. We analyze them for two realizations of CW NBs in the ultra-relativistic, i.e. the massless case and for massive cases and come to the result, that the modes can be separated into clockwise and counterclockwise modes and dynamical tunneling is absent. This is attributed to the BCs and the unidrectionality of the local current arising from them.
Abstract Thirty-five years ago, Sir Michael Berry and his collaborator Mondragon studied the behaviors of neutrino, a massless relativistic quantum particle, in a classically chaotic billiard—the neutrino billiard problem. To celebrate Sir Michael Berry’s eightieth birthday, here we report results on the role of geometric symmetries of the billiard system in relativistic quantum scarring. In particular, we investigate a Dirac billiard system with a four-fold rotational symmetry whose classical dynamics are fully chaotic. The system is described by the massless Dirac equation in the fundamental domain that consists of one fourth of the full billiard, with proper boundary conditions on the symmetry lines to preserve the physical properties under the symmetry operations. We show that the relativistic quantum characteristics of spin induced phase play a fundamental role in the quantum behaviors of the Dirac particle in the billiard. We find that the peaks in the length spectra are due to the interference of states circling the fundamental domain orbits (FDOs) in opposite propagating directions, which can be constructive or destructive depending on the accumulated phases. In addition, we derive the quantization conditions of the scarring states about the unstable periodic orbits within the fundamental domain from the phase along the FDOs. Our work is a vivid demonstration that relativistic quantum scarring, or more generally quantum manifestations of classical chaos, can be fully understood by analyzing the behaviors of the geometric phase—a powerful approach in modern physics pioneered by Sir Michael Berry.