Reversal of the chiral anomaly bulk states with periodically staggered potential

A chiral anomaly Landau level emerges when a Weyl semimetal is subjected to an external magnetic field. Recently, it was demonstrated that similar chiral anomaly bulk states exist within two-dimensional Dirac semimetals if proper boundary conditions are applied. The resulting chiral bulk states disperse linearly with the slope determined by a combination of the boundary condition and chirality of the Dirac cone, i.e., whether it is at $K$ or $K$′ point. In this paper, we show that the slope (the sign of group velocity) near the $K$ and $K$′ points can be reversed under a periodically staggered potential. We first analyze the parameter dependence of this phenomenon with a nearest-neighbor tight-binding model. Then, we give a semianalytical solution for the dispersion of the chiral anomaly bulk states. In the end, we provide a photonic crystal system and prove with full-wave simulations that such a phenomenon can indeed be observed.

Figure 1

Dirac cone and possible CABS for a finite system. (a) First Brillouin zone of a graphene lattice with conical dispersion at the $K$ and $K$′ points. (b) A strip of graphene lattice which is finite along the $x$ direction and periodic along the $y$ direction. The number of unit cells along the $x$ direction is $N_x$. Color code denotes the field-intensity distribution of one typical CABS [marked by the yellow star in (d)]. (c), (d) Band dispersion of the graphene lattice in (b), where the onsite potential of the outmost atoms (marked by the bold black circles) is kept at zero in (c) and set as $U=-0.75$ in (d). Red lines in (c) highlight the edge states while the red lines in (d) denote the CABSs. Green and blue lines sketch the evolution of CABSs for later purpose. Except for the green and blue lines in (d), all the other lines are calculated within a nearest-neighbor tight-binding model in which the hopping constant is set as 1. The number of unit cells is fixed at $N_x=6$ in (c) and (d).

Figure 2

Evolution of CABSs under periodically staggered potential. (a) Sketch of the tight-binding model. Onsite potential is −$U$ for the left half and +$U$ for the right half. The lattice is infinite in the $y$ direction and periodic horizontally with a total unit number $2N_x$. (b)–(f) Band dispersions for different combinations of $U$ and $N_x$, where the CABSs are highlighted in red and blue. Only the nearest-neighbor hopping is considered here with the hopping constant set as 1. $q_y$ is the wave vector relative to the $K$ point in the $y$ direction.

Figure 3

Semianalytical solutions. (a) Band dispersion calculated with the semianalytical solution in Eq. (5), where gray lines represent the bulk bands, while the red and blue lines denote the CABSs. (b) Sign of $\mathrm{\Delta }$ defined in Eq. (6) as a function of $L$ and $U$. Red regions represent $\mathrm{\Delta }>0$, blue for $\mathrm{\Delta }<0$, and black dashed lines denote the transitions whereat $\mathrm{\Delta }=0$. $L=4$ and $U=0.5$ in (a), and $v_F=\sqrt{3}/2$ for (a), (b).

Figure 4

Realizations with PCs. (a) Unit cells of two 2D honeycomb lattices. Solid disks represent dielectric cylinders, where the radii of cylinders for the upper- and lower unit cells are slightly different. Here, black dashed circles share the same radius. (b) Band structure for the unit cells in (a), where $r_1=1.2$ mm, $r_2=1.3$ mm, and $a=8$ mm; relative permittivity of the dielectric cylinder is ${\varepsilon }_r=16$ and background is air with relative permittivity 1. (c) Dielectric cylinders with finite height $H=5\mathrm{mm}$ in honeycomb lattice are sandwiched by two PEC boundaries, and are then wrapped on a larger concentric cylindrical waveguide. Here, the outer PEC boundary is set as transparent for illustration. (d)–(f) Band structures of the system in (c) for different $N_x$, where the average lattice constant is kept at $a=8$ mm. All other parameters are the same as (b).

Figure 5

Dispersion of CABSs for different boundary decorations. Onsite potential of the outermost column is set as $U$, with $U=-0.75$ in (a) and $U=0.75$ in (b). Here, gray lines represent bulk bands while red lines denote the dispersion of the CABSs.

Figure 6

Band diagram for $r_1=r_2=1.2\mathrm{mm}$ with different $N_x$. Other parameters are the same as those in Fig. 4.

Figure 7

Geometry of the supercell for $N_x=4$ and $r_1=r_2$. Top and bottom surfaces (colored blue) are set as periodic boundary conditions (PBC). The outer- and inner-cylindrical surfaces (in gray color) are set as PEC boundaries. Yellow cylinders inside represent dielectric pillars.

Figure 8

Evolution of CABSs when increasing $U$. (a) Band structures of unit cells with different radii. Other parameters are the same as those in Fig. 4 and the radius is in millimeters. (b)–(d) Band diagram for supercells with fixed $N_x=4$ and different $r_2$.