Figure 1.

Level spacing statistics of the massless Dirac billiards with the boundary being a circle, the Africa shape, the heart shape for left, middle, and right, respectively. The first row shows the unfolded level-spacing distribution P(S), and the second row shows the spectral rigidity Δ3(L). The green dashed–dotted line, cyan dashed line, and blue solid line are for Poisson, GOE, and GUE. Red staircase curves and symbols are numerical results from 13000 energy levels for each shape by diagonalizing the operator $\hat{H}$ given by equation (1). Adapted from [535] with permission."

Figure 2.

(a) Chaotic graphene billiard with Africa shape cut from a graphene sheet. The system has 42 505 carbon atoms. The outline is determined by the equation $x+{\rm{i}}{y}=70a$ $(z+0.2{z}^{2}+0.2{z}^{3}{{\rm{e}}}^{{\rm{i}}\pi /3})$, where z is the unit circle in the complex plane, a = 2.46 Å is the lattice constant for graphene. The area is A = 1117 nm2. (b)–(d) are the level spacing distribution, integrated level spacing distribution, and the spectral rigidity, respectively, for 664 energy levels in the range $0.02\lt {E}_{n}/t\lt 0.4$, where t is the hopping energy between nearest neighboring atoms. Dashed line is Poisson, solid line is GOE, and dotted line is GUE. The results show clear evidence of GOE. Adopted from [539] with permission."

Figure 3.

Level spacing statistics of graphene billiard with the shape of a 60° sector with armchair edges. (a) With perfect edges and 227 254 atoms. (b) With one row of atoms removed along one edge so the structure is no longer symmetric (as indicated by the red dots in the inset) and 226 315 atoms. The energy range is 0.02 < En/t < 0.2. Dashed line is Poisson, solid line is GOE. Insets show the magnified view of the lattice structure close to the tip of the sector to illustrate the differences. Adopted from [564] with permission."

Figure 4.

(a) A stadium shaped graphene billiard with 11814 atoms. (b) and (c) show $| {\psi }_{n}{| }^{2}$ with ${E}_{n}/t=0.363\,58$ and 0.576 65, respectively. Adopted from [565] with permission."

Figure 5.

For two representative scars with orbit length 263a (left) and 275a (right), the energy values where it appears. Vertical axis shows the relative index of these scars. From [464] with permission."

Figure 6.

For scars on two representative orbits, period-4 for the left panels, and period-3 for the right panels, the upper panels show the corresponding eigenenergies of the scars, and the lower panels show η (see text) for these sates. Adopted from [568] with permission."

Figure 7.

Definition of the angles. ${\delta }_{i}^{+}=({\theta }_{i}-{\theta }_{i-1})/2$ is the extra phase due to the rotation of the spin, where ‘+’ indicates counterclockwise orientation. Typically, the phase associated with the time reversed reflection, e.g. ${\delta }_{i}^{-}$, from $-{{\boldsymbol{k}}}_{i}$ to $-{{\boldsymbol{k}}}_{i-1}$, would be different from ${\delta }_{i}^{+}$."

Figure 8.

Validation of the quantization rule equation (4). Shown are the relations between wavenumber k and magnetic flux α, for (a) the period-4 scar in figure 6(a), and (b) the period-3 scar in figure 6(c). The orange up-triangles indicate scars with a counterclockwise flow, and the blue down-triangles are those with a clockwise flow. The gray squares mark the scars whose flow orientations cannot be identified, which typically occur close to the cross points of the two orientation cases. The solid lines are theoretical predictions of equation (4). Vertical lines indicate the position of α = π/2. The step in the variation of α is 0.01. Adapted from [569] with permission."

Figure 9.

Δγ between counterclockwise and clockwise scaring states on a period-3 orbit (a) and a period-4 orbit (b). The massless Dirac regime is $m\to 0$, while $k\to 0$ is effectively $m\to \infty $ and is the Schrödinger limit. Adapted from [570] with permission."

Figure 10.

The imaginary part γ of the eigenenergies due to coupling between the dot and the leads, which is an effective indicator of the resonance width. The left panels are for 2DEG quantum dots, and the right panels are for graphene quantum dots. Upper panels are for the cases with mixed dynamics, and lower panels are for classically chaotic dynamics. Adapted from [573] with permission."

Figure 11.

Typical local density of states for quantum dots with classically mixed (a), (d) and chaotic (b) ,(c), (e), (f) dynamics. The upper panels are for 2DEG quantum dots, and lower panels are for graphene quantum dots."