Perfectly conducting channel on the dark surface of weak topological insulators

A weak topological insulator (WTI) bears, generally, an even number of Dirac cones on its surface; they are susceptible to doubling, while on the surface of a certain orientation it shows no Dirac cone. On this “dark” surface of a WTI, we predict that a single pair of isolated one-dimensional perfectly conducting channels emerges and forms either a closed loop or a segment of a line. The former is associated typically with a single-atomic-layer-thick island formed on the dark surface, while the latter is shown to be the consequence of a pair of crystal (screw) dislocations terminating on the dark surface.

Figure 1

Upper and lower terraces separated by a step of height $h$ are examples of the dark surface of the WTI. On the lower terrace is an island formed with the same height.

Figure 2

A pseudogapless 1D mode emergent along the periphery of an island. The island has a size of ${\xi }_x\times {\xi }_y=9\times 11$, and an altitude $h=1$. The system sizes are $N_x=N_y=16,N_z=7$.

Figure 3

Profile of the lowest-energy wave function in atomic-scale terraces of a WTI. (a) When $h=1$, 1D modes appear at the edge of the terrace. (b) When $h=2$, the wave function has a vanishing amplitude along the edge of the terraces. $N_x=N_y=16,N_z=7$.

Figure 4

A numerical experiment: verification of the perfectly conducting character of the 1D channels circulating the periphery of the island (case of $h=1$). The spectrum of the 1D modes is plotted against the Aharonov-Bohm phase $\varphi$.

Figure 5

A finite segment of 1D gapless modes penetrating into the bulk WTI at the ends of the segment along a screw dislocation line with $\mathit{b}=b\widehat{\mathit{z}}$ $(b=\pm 1)$. The two dislocation lines are separated by a distance of $\zeta =8$. $N_x=16,N_y=N_z=10$.

Figure 6

The “flux experiment” applied to a configuration with a pair of screw dislocation lines.