Quantum Transport and Two-Parameter Scaling at the Surface of a Weak Topological Insulator

Weak topological insulators have an even number of Dirac cones in their surface spectrum and are thought to be unstable to disorder, which leads to an insulating surface. Here we argue that the presence of disorder alone will not localize the surface states; rather, the presence of a time-reversal symmetric mass term is required for localization. Through numerical simulations, we show that in the absence of the mass term the surface always flow to a stable metallic phase and the conductivity obeys a one-parameter scaling relation, just as in the case of a strong topological insulator surface. With the inclusion of the mass, the transport properties of the surface of a weak topological insulator follow a two-parameter scaling form.

Figure 1

The phase diagram of the Hamiltonian (1) as a function of mass $m$ and disorder strength $g_{\alpha \beta }=g$. The solid line marks the metal-insulator transition at $\mu =0$, whereas the dashed line marks the transition at finite $\mu$. At the clean Dirac point ($g=\mu =0$), there is a topological phase transition between the two types of insulators. With increasing disorder or chemical potential $\mu$, a metallic phase appears separating the two topological sectors.

Figure 2

Demonstration of one-parameter scaling at $m=0$. Conductivity as a function of system size for various parameters all collapsed (by shifting the raw data horizontally) onto one scaling curve. At large $\sigma$, the slope $d\sigma /d\ln L$ approaches $1/\pi$ (gray line) consistent with weak antilocalization. Here $g_{00}=g$ for dotted lines and $g_{00}=0$ for dashed lines. For all other $\alpha \beta$, $g_{\alpha \beta }=g$. (Inset) Raw data $\sigma$ vs $L/\xi$.

Figure 3

(a) Metal-insulator transition as $m$ is varied. Conductivity is plotted vs system size for fixed $\mu \xi =1$ and $g_{x0}=g_{yz}=2$. For large $m$ the system flows to an insulating state, while for small $m$ the system is conducting. Among the conducting curves, the slope $d\sigma /d\ln L$ approaches $1/\pi$ at large $\sigma$. The data show a metal-insulator transition at ${\sigma }_c$ consistent with the known value of $1.42e^2/h$ [11]. (b) Metal-insulator transition as disorder strength $g_{\alpha \beta }=g$ is varied. The plot is $\sigma$ vs $L/\xi$ for fixed $m\xi =0.05$ and $\mu =0$. Increasing disorder increases the conductivity, inducing a transition from an insulating phase to a metallic one at some critical $g$. The dashed line indicates a slope of $1/\pi$. These figures are consistent with the phase diagram in Fig. 1.

Figure 4

(a) Two-parameter flow which captures the QSH-metal-insulator transition in the AII class. The scaling variables are $\sigma$ and $j$, the latter of which separates the normal or QSH insulator phases. (b) Numerical data $\sigma (L/{\xi }^{*})$, demonstrating that the conductivity curves may be collapsed onto each other. The gray line is the data at $m=0$ from Fig. 2. The raw data and parameters are given in the Supplemental Material [16].