Nonrelativistic versus relativistic quantum scars in billiard systems

We study the features of scarred eigenstates of relativistic neutrino billiards (NBs), graphene billiards (GBs) and Haldane graphene billiards (HGBs) and recapitulate those for nonrelativistic quantum billiards (NRQBs) with the shapes of a full- and quarter-stadium billiard. Here, we restrict for the GBs and HGBs to the region of linear dispersion around the Fermi energy, where they are effectively described by the same Dirac equation for massless spin-1/2 particles as NBs. Scarred wave functions of the nonrelativistic billiards and spinor functions of the relativistic ones are localized along the same types of periodic orbits, the most prominent ones being bouncing-ball orbits. The objective is to demonstrate that the properties of the scarred eigenstates observed in the full- and quarter-stadium GB \emph{do not comply} with those of relativistic quantum systems. For this we apply the semiclassical approach associated with such non-generic contributions, which was developed for the spectral density of NRQBs and NBs. It provides semiclassical trace formulas in terms of the periodic orbits associated with a scarred wave function and a procedure to extract such contributions from the eigenvalue spectra. Furthermore, we analyze momentum distributions and Husimi functions of such scarred states and employ them to classify scarred wave functions according to the periodic orbits along which they are localized. We show that for the GB the semiclassical approach, the spectral properties, the symmetry properties and generally properties of the wave functions all comply with those of the NRQB, whereas for the HGB they agree well with those of the NB. Thus, even though around the Fermi energy GBs are described by the relativistic Dirac equation the quantum scars, or generally, the quantum scarred eigenstates observed in GBs do not exhibit those of relativistic ones.

Barbara Dietz, Dung Xuan Nguyen, Tilen Čadež · 2026