Dirac quantum walks on triangular and honeycomb lattices

In this paper, we present a detailed study on discrete-time Dirac quantum walks (DQWs) on triangular and honeycomb lattices. At the continuous limit, these DQWs coincide with the Dirac equation. Their differences in the discrete regime are analyzed through the dispersion relations, with special emphasis on Zitterbewegung. An extension which couples these walks to arbitrary discrete electromagnetic field is also proposed and the resulting Bloch-like oscillations are discussed.

Figure 1

First Brillouin zone of the equilateral triangle lattice (gray) and the periodicity domain of the three-step walk (light blue)

Figure 2

Dispersion relations $\omega \left(\mathbf{k}\right)$ for six-step DQW on the equilateral triangular lattice. (a) $m=0$, (b) $m=\pi /3$, (c) $m=\pi /2$, (d) $m=2\pi /3$, (e) $m=\pi$, (f) $m=4\pi /3$.

Figure 3

Dispersion relations $\omega \left(\mathbf{k}\right)$ for three-step DQW on the equilateral triangular lattice. (a) $m=0$, (b) $m=\pi /3$, (c) $m=\pi /2$, (d) $m=2\pi /3$, (e) $m=\pi$, (f) $m=4\pi /3$.

Figure 4

Dispersion relations $\omega \left(\mathbf{k}\right)$ for the DQW on the isosceles triangular lattice. (a) $m=0$, (b) $m=\pi /3$, (c) $m=\pi /2$, (d) $m=2\pi /3$, (e) $m=\pi$, (f) $m=4\pi /3$.

Figure 5

Dispersion relations $\omega \left(\mathbf{k}\right)$ for the DQW on the hexagonal honeycomb lattice. (a) $m=0$, (b) $m=2\pi /3\sqrt{3}$, (c) $m=\pi /\sqrt{3}$, (d) $m=4\pi /3\sqrt{3}$, (e) $m=2\pi /\sqrt{3}$, (f) $m=8\pi /3\sqrt{3}$.

Figure 6

Density plots in $(t,x)$ of (a) no electric field (free space), and (b, c) electric field in the $x$ direction with $E_x=0.1$ and $E_x=0.05$, respectively.

Figure 7

Density plots in $(t,y)$ of (a) no electric field (free space), and (b), (c) electric field in the $x$ direction with $E_x=0.1$ and $E_x=0.05$, respectively.

Figure 8

Density plot of electric field in the $y$ direction: $E_y=0.1$

Figure 9

Density plots of (a) electric field in both the $x$-direction and $y$-direction: $E_x=0.1$ and $E_y=0.1$, and (b) electric field only in the $x$ direction of equivalent magnitude ($E_x=\frac{\sqrt{2}}{10}$)