Strong side of weak topological insulators

Three-dimensional topological insulators are classified into “strong” (STI) and “weak” (WTI) according to the nature of their surface states. While the surface states of the STI are topologically protected from localization, this does not hold for the WTI. In this work, we show that the surface states of the WTI are actually protected from any random perturbation that does not break time-reversal symmetry, and does not close the bulk energy gap. Consequently, the conductivity of metallic surfaces in the clean system remains finite even in the presence of strong disorder of this type. In the weak disorder limit, the surfaces are found to be perfect metals, and strong surface disorder only acts to push the metallic surfaces inwards. We find that the WTI differs from the STI primarily in its anisotropy, and that the anisotropy is not a sign of its weakness but rather of its richness.

Figure 1

Topologically trivial and nontrivial pair switching. (a) The WTI is adiabatically connected to stacked layers of 2D TI. We consider a WTI of dimension $L^3$ with $\nu =\left(001\right)$ and $\widehat{z}$ as the stacking direction. The boundary conditions (green dotted lines) are periodic in the $\widehat{z}$ direction and are twisted by an Aharonov-Bohm flux in the $\widehat{x}$ direction (green thick arrow). The remaining $xz$ surfaces are metallic. (b), (c) Typical patterns of energies of surface state as a function of the Aharonov-Bohm flux $\phi$ for (b) trivial and (c) nontrivial surfaces. Kramer's theorem assures that at the time-reversal-invariant fluxes $\phi =0,\pi$, states come in degenerate pairs. On a trivial surface, the pairs remain the same between these two values, while on a nontrivial surface the pair switch partners. The mean level spacing of the surfaces states is $\Delta /L^2$, where $\Delta$ is the bulk gap. We show that nontrivial pair switching implies the existence of at least $O\left(L\right)$ extended states.

Figure 2

Renormalization group flow of the conductivity. The $\beta$ function of the dimensionless conductivity ${\tilde{\sigma }}_{xx}$ for a nontrivial surface of the WTI, compared with that of a 2D metal with strong spin-orbit coupling that belongs to the symplectic class (blue). According to Eq. (5), in the limit of high conductivity the flow is toward a perfect metal. We showed that the conductivity of a WTI surface can not drop below $e^2/h$. Consequently, two types of flows are possible: (I) always flowing towards a perfect metal (yellow), and (II) flowing with a stable fixed point of finite conductivity (red).

Figure 3

Insulating and metallic surfaces. (a) The WTI has both trivial and nontrivial surfaces. A surface with Miller indices $\mathbf{h}$ is trivial if $(\mathbf{h}-\nu )\mathrm{mod}2=0$, denoted by $\mathbf{h}\sim \nu$, since any such $\mathbf{h}$ can denote a stacking direction. The figure depicts two trivial surfaces ${\mathbf{h}}_1=\left(001\right)$ and ${\mathbf{h}}_2=\left(201\right)$ for a cubic crystal with assumed $\nu =\left(001\right)$. Both ${\mathbf{h}}_1$ and ${\mathbf{h}}_2$ are legitimate stacking directions. Alternatively, for a stacking along ${\mathbf{h}}_1$, the ${\mathbf{h}}_2$ surface is composed of steps of two layers. The coupling between the layers gaps theirs edge states. For $\mathbf{h}\nsim \nu$, the steps will be of odd number of layers and will therefore conduct. (b) An example of the surface anisotropy in the Fu, Kane, and Mele model of the weak $\nu =\left(001\right)$ phase. Depicted is the local density of surface states integrated over an energy window $\left|E\right|<0.1\Delta$, with disorder strength comparable to the bulk gap. The surfaces of the parallelepiped are spanned by the primitive vectors. The two faces with Miller indices equal to $\nu$ are gapped, while the other four, with orthogonal Miller indices, are metallic. By controlling the cleavage process, the conductance of each face of the WTI can be engineered.

Figure 4

The conductance $g_{xx}$ for a WTI with $\nu =\left(001\right)$ of the Fu, Kane, and Mele model as estimated from the flux sensitivity of surface-state energies multiplied by the density of states. Each line corresponds to a given number of layers ($L_z$) and shows $g_{xx}$ as a function of $L_x$, where $L_y$ is fixed to 10. Periodic boundary conditions were imposed on $\widehat{z}$ and $\widehat{x}$. Samples with an odd number of layers have a topologically protected minimal conductance of $e^2/h$. Samples with an even number of layers are (strictly speaking) topologically trivial and show a localization behavior. However, their conductance converges to that of the odd layers as the number of channels is increased.

Figure 5

The leading self-energy diagrams in the Born approximation: (a) first order, and (b), (c) second order.

Figure 6

Diagrammatic representation of the self-consistent equation for (a) the dressed vertex and (b) the Cooperon.

Figure 7

The leading-order quantum corrections to the conductivity. These diagrams can be viewed as a combination of a dressed Hikami box with the Cooperon.