Zero modes and edge states of the honeycomb lattice

The honeycomb lattice in the cylinder geometry with zigzag edges, bearded edges, zigzag and bearded edges (zigzag-bearded), and armchair edges are studied. The tight-binding model with nearest-neighbor hoppings is used. Edge states are obtained analytically for these edges except the armchair edges. It is shown, however, that edge states for the armchair edges exist when the system is anisotropic. These states have not been known previously. We also find strictly localized states, uniformly extended states, and states with macroscopic degeneracy.

Figure 1

(Color online) The honeycomb lattice. The open and closed circles show sublattices A and B, respectively. The wave functions are ${\varphi }_{n,m}$, ${\varphi }_{n+1∕2,m+1∕2}$, ${\psi }_{n,m}$, and ${\psi }_{n+1∕2,m+1∕2}$, where $n$ and $m$ are both integers or both half-integers. Hoppings are $t_a$, $t_b$, and $t_c$.

Figure 2

(Color online) The honeycomb lattices with (a) zigzag $(L_x=4)$, (b) bearded $(L_x=4)$, (c) zigzag-bearded $(L_x=4)$, and (d) armchair $(L_y=6)$ edges, which we call zigzag, bearded, zigzag-bearded, and armchair, respectively.

Figure 3

(Color online) Energy spectrum of zigzag for $L_x=14$. The edge states are on the blue lines.

Figure 4

(Color online) An example of the edge states.

Figure 5

(Color online) Energy spectrum for bearded. The energy mode for the edge states are on the blue line.

Figure 6

(Color online) Energy spectrum for zigzag-bearded for $L_x=40$. The zero energy mode for the edge states are on the blue line.

Figure 7

(Color online) Energy spectrum for armchair in the isotropic case. Neither Eq. (31) nor Eq. (33) is satisfied in this case. There is no edge state.

Figure 8

(Color online) Energy spectrum for armchair in an anisotropic case. The condition Eq. (33) is satisfied for certain $k_x$’s and edge states exist on the blue line.