Nonrelativistic versus relativistic quantum scars in billiard systems

TL;DR

The paper addresses whether quantum scars in graphene-based billiards are relativistic or nonrelativistic by comparing graphene billiards (GBs) and Haldane graphene billiards (HGBs) to their nonrelativistic (QB) and relativistic (NB) counterparts. It combines boundary-integral methods for eigenstate computation with semiclassical trace formulas and momentum/Husimi analyses to identify scarred states and their underlying periodic orbits, including non-generic contributions from bouncing-ball orbits. The main finding is that scars in GBs conform to nonrelativistic QB properties, while scars in HGBs align with relativistic NB behavior, with BBOs providing the dominant non-generic spectral contributions in both cases; around band edges and Dirac points the spectral fingerprints are captured by the appropriate trace formulas and orbit families. These results clarify the interpretation of quantum scars in graphene-like systems and point to experimental platforms, such as photonic crystals implementing Haldane-like models, for realizing relativistic quantum scars.

Abstract

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 (QBs) 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 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 QBs 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 nonrelativistic QB, whereas for the HGB they agree well with those of the NB, implying that the quantum scars observed in GBs are not relativistic.

Figures (18)

• Figure 1: Left: The honeycomb structure of graphene. Right: The conduction and valence bands of graphene resulting from tight-binding model calculations. They touch each other conically at the corners of the first Brillouin zone.
• Figure 2: Same as Fig. \ref{['BandStr_GB']} for graphene subject to the Haldane-model onsite potential $M=0.3$ and next-nearest neighbor tunneling at the critical point $t_2=M/(3\sqrt{3})$.
• Figure 3: On-shell momentum distribution of eigenstates $\Psi_m(\boldsymbol{r})$ of the QB versus momentum direction $\theta_k$ for a) $m=1048$, b) $m=1005$, c) $m=1102$, d) $m=1112$, e) $m=1020$, f) $m=1040$. To the left and right are exhibited intensity distributions of the corresponding wave functions $\psi_m(x,y)$.
• Figure 4: On-shell momentum distribution of the first eigenspinor component $\Psi_{1,m}(\boldsymbol{r})$ of the NB versus momentum direction $\theta_k$ for a) $m=1136$, b) $m=1141$, c) $m=482$, d) $m=1119$, e) $m=1001$, f) $m=1130$. To the left and right are exhibited the distributions of the corresponding local current $\vert\boldsymbol{u}_m(\boldsymbol{r})\vert$.
• Figure 5: Enhanced localization is observed along orbits from the family of the (a) BBO [cf. Figs. \ref{['Mom_QB']} and \ref{['Mom_NB']} a)], (b) the diameter orbit [cf. Figs. \ref{['Mom_QB']} and \ref{['Mom_NB']} c)], (c) the diamond orbit reflected at centers of the semicircular parts and at the the centers of the straight parts of the boundary (d) the bow-tie orbit constructed from reflections at opposite and diagonal corners, respectively [cf. Figs. \ref{['Mom_QB']} and \ref{['Mom_NB']} d)] (e) orbits that, like the orbit (c) look similar to those of a rectangular billiard with side lengths $(2L)\times (2L+2r_0)$, and are reflected at the centers of the semicircular parts and at the straight parts of the boundary [cf. Figs. \ref{['Mom_QB']} and \ref{['Mom_NB']} b)] and (f) edge orbits composed of whispering gallery orbits of the semicircular parts and diagonal orbits that connect two corners [cf. Figs. \ref{['Mom_QB']} and \ref{['Mom_NB']} b)].