Scarring of Dirac fermions in chaotic 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.

Figure 1

Schematic picture of closed Dirac system with arbitrary geometry and zero outgoing flux boundary condition $\mathbf{j}·\mathbf{n}=0$. This boundary condition is equivalent to $\chi /\phi =i\exp \left(i{\theta }_{\mathbf{n}}\right)$ with ${\theta }_{\mathbf{n}}$ being the argument of the surface normal $\mathbf{n}$.

Figure 2

Proposed discretization scheme to eliminate any fermion-doubling effect. A two-dimensional domain, which exhibits chaos in the classical limit, is illustrated to show the discretized lattice. Red solid circles and blue open circles spaced by $\Delta$ represent the boundary and inner lattice points, respectively, where the Dirac spinor values are sampled. The actual Dirac equations are evaluated at black cross points, the centers of unit cells.

Figure 3

Comparison of numerical and analytical results of eigenenergies and eigenstates for the ring cavity. The lowest 80 positive eigenenergy levels and two examples of the eigenstates from numerics (blue square or above the energy spectrum) and theory (red cross or below the energy spectrum) are compared. The convenient unit convention $\hslash =v=1$ was used in the numerical computation.

Figure 4

Level-spacing statistics for eigenenergies of the ring domain. Inset (a) shows spectral staircase $N\left(E\right)$ as a function of $E^2$. The dashed lines represent the linear relationship between $N\left(E\right)$ and $E^2$. Horizontal axes are linear mappings of $E^2$ to the range $[0,N]$, where $N$ is the total number of eigenstates. Panel (b) is the cumulative distribution of the nearest-neighbor spacing  $F_S\left(S\right)$ as a function of spectral spacings $S_n=E_{n+1}^2-E_n^2$. Panel (c) represents the density distribution $f_S\left(S\right)$ of $F_S\left(S\right)$ in (b). In both the middle and bottom rows, green dash-dotted, red solid, and cyan dashed lines denote theoretical distribution curves for Poisson, GOE, and GUE statistics, respectively. The results of the ring are obtained through a polar coordinate version of our numerical method to preserve the perfect circular symmetry.

Figure 5

Level-spacing statistics for eigenenergies of the chaotic bow-tie domain. Figure legends are the same as for Fig. 4.

Figure 6

Level-spacing statistics for eigenenergies of the chaotic Africa domain. Figure legends are the same as for Fig. 4.

Figure 7

Spectral rigidity ${\Delta }_3\left(L\right)$ for the three domains in Figs. 4, 5, 6. Lines with different styles denote theoretical expectations of spectral rigidity for the respective statistics.

Figure 8

Examples of eigenstates of the Dirac equation for a bow-tie chaotic billiard. The maximum height (vertical distance from tip to base) is set to 1, and the distance between two tips is 2. Panels (a)–(d) are for $E=36.1335$, 43.0190, 27.9300, and 36.2729, respectively. Panels (a) and (b) show the $\phi$ component, while (c) and (d) show the $\chi$ component.

Figure 9

Examples of eigenstates of the Dirac equation for the chaotic Africa billiard. The domain is confined within a rectangular box $x\in [-0.99,1.35]$ and $y\in [-1.22,0.92]$. Panels (a)–(d) are for $E=15.0468$, 32.7638, 13.4838, and 15.2402, respectively. Panels (a) and (b) show the $\phi$ component, while (c) and (d) show the $\chi$ component.