Weak topological insulator with protected gapless helical states

A workable model for describing dislocation lines introduced into a three-dimensional topological insulator is proposed. We show how fragile surface Dirac cones of a weak topological insulator evolve into protected gapless helical modes confined to the vicinity of a dislocation line. It is demonstrated that surface Dirac cones of a topological insulator (either strong or weak) acquire a finite-size energy gap when the surface is deformed into a cylinder penetrating the otherwise surfaceless system. We show that, when a dislocation with a nontrivial Burgers vector is introduced, the finite-size energy gap plays the role of stabilizing the one-dimensional gapless states.

Figure 1

A pair of screw dislocations (upper) with Burgers vector $\mathit{b}=(0,0,\pm b)$ inserted between the two cuts (lower). The dislocation line is along the $z$ axis and parallel to the Burgers vector. The system is translationally invariant in the $z$ direction.

Figure 2

Edge dislocations, a concrete implementation between the two cuts. Here, the Burgers vector $\overrightarrow{b}$ is parallel to the $x$ axis; $\overrightarrow{b}=(b,0,0)$.

Figure 3

Calculated spectrum of surface Dirac cones in the WTI phase ($\Delta /B=6$, $A=B=1$). The periodic boundary condition in the $y$ direction is relaxed; the system forms a slab or a torus of finite thickness. The two Dirac cones are located at TRIM: ($\pi ,0$) and ($0,\pi$).

Figure 4

Energy spectrum $E(k)$ of WTI ($\Delta /B=6$) in the presence of screw dislocations. Here, the 1D momentum $k$ is chosen to be along the cuts; $k=k_z$. The Burgers vectors are given as $\overrightarrow{b}=(0,0,\pm b)$ with $b=0$ (dislocation is absent), $b=1$, $b=2$, and $b=3$, respectively, in the top, second, third, and bottom panels. The calculation is performed for a system of size $N_x\times N_y=16\times 16$ and cut width $N_c=8$. The other parameters are set as $A=B=1$.

Figure 5

Here, similar plots as the last two panels of Fig. 4 in the case of edge dislocations. Here, the Burgers vector is chosen as $\overrightarrow{b}=(\pm b,0,0)$ with $b=2$ and $b=3$, respectively, in the upper and lower panels.

Figure 6

Protected gapless helical modes along a pair of screw dislocations ($b=1$). The spectrum is calculated at $\Delta /B=6$ for two different values of the cut width $N_c$; $N_c=0$ (upper) and $N_c=16$ (lower). The size of the system is chosen as $N_x\times N_y=32\times 8$. See the text for discussions.

Figure 7

Protected gapless helical modes along a pair of edge dislocations ($b=1$). The spectrum is calculated at $\Delta /B=6$ for two different values of the cut width $N_c$; $N_c=1$ (upper) and $N_c=16$ (lower). $N_x\times N_y=32\times 8$.

Figure 8

Parity eigenvalues determining both the indices (${\nu }_0,{\nu }_1{\nu }_2{\nu }_3$) and the position of surface Dirac cones on the projected 2D plane; here, chosen as the ($k_z,k_x$) plane. Case 1. Screw dislocation [$\overrightarrow{b}=(0,0,b)$]—${\delta }_1^{″}{\delta }_2$ and ${\delta }_2^{\prime }{\delta }_3$ occurring at $(k_z,k_x)=(\pi ,0)$ and $(k_z,k_x)=(\pi ,\pi )$ are relevant. Case 2. Edge dislocation [$\overrightarrow{b}=(b,0,0)$]—${\delta }_1{\delta }_2^{″}$ and ${\delta }_2^{\prime }{\delta }_3$ occurring at $(k_z,k_x)=(0,\pi )$ and $(k_z,k_x)=(\pi ,\pi )$ are relevant.

Figure 9

(a) Necessary arrangement of the 2D indices introduced in Fig. 8 for the appearance of protected 1D helical modes along a screw dislocation in the $z$ direction. Only relative signs are relevant. Column (b), WTI example satisfying the condition in (a); $4<\Delta /B<8$. Column (c), STI example satisfying the condition in (a); $8<\Delta /B<12$.

Figure 10

Plots similar to Figs. 4 and 5 here in the STI phase ($\Delta /B=10$). The upper (lower) panel corresponds to the case of screw (edge) dislocations, $b$ is chosen as $b=3$ and $b=2$, respectively, in the two cases.