Disordered Weak and Strong Topological Insulators

A global phase diagram of disordered weak and strong topological insulators is established numerically. As expected, the location of the phase boundaries is renormalized by disorder, a feature recognized in the study of the so-called topological Anderson insulator. Here, we report unexpected quantization, i.e., robustness against disorder of the conductance peaks on these phase boundaries. Another highlight of the work is on the emergence of two subregions in the weak topological insulator phase under disorder. According to the size dependence of the conductance, the surface states are either robust or “defeated” in the two subregions. The nature of the two distinct types of behavior is further revealed by studying the Lyapunov exponents.

Figure 1

The “global phase diagram” of the disordered ${\mathbb{Z}}_2$ topological insulator in the ($m_0/m_2$, $W/m_2$) plane determined by the behavior of two-terminal conductance. Solid lines on the phase boundaries are guides to the eyes. Dotted lines indicate the value of parameters relevant in Figs. 2, 3. The metallic ($M$) phase lies in the intermediate range of disorder strength, typically $10\lesssim W/m_2\lesssim 25$ in the parameter range of $m_0/m_2$ shown in this figure.

Figure 2

Two-terminal conductance $⟨G⟩$ (a) in the slab and (b) in the bulk geometries. $⟨G⟩$ is plotted against $m_0/m_2$ for a system of size $L=10$ under different strength of disorder: $W=1$ (green solid line), $W=3$ (red dashed line), and $W=5$ (blue dotted line). In the bulk geometry PBCs are imposed in both the $x$ and $y$ directions, while in the slab geometry, a FBC is applied to the $x$ direction.

Figure 3

Two-terminal conductance as a function of disorder with $m_0/m_2=-2.5$ for different system sizes: $L=10$ (dotted line), $L=14$ (dashed line), and $L=20$ (solid line). The upper panel (a) is for the slab geometry. The WTI surface states are stable for a certain value of disorder (about $W\lesssim 4$, which is actually not small), and defeated by disorder around $W=6$. For $W\gtrsim 7$, the system enters the metallic phase. The lower panel (b) shows the conductance in bulk geometry. There is no peak, which should appear at the topological phase transition point around $W\simeq 4$. The statistical error is less than 0.02.

Figure 4

Decay lengths ${\xi }_1$ (solid lines) and ${\xi }_2$ (dashed lines) in the STI $(m_0/m_2,W/m_2)=(-1,1)$ (blue open circles), the WTI $(m_0/m_2,W/m_2)=(-2.5,3.5)$ (red open squares), and DWTI $(m_0/m_2,W/m_2)=(-2.5,6)$ (green crosses) regions in the slab geometry with system width $L_y=6$. The higher decay lengths ${\xi }_{i\ge 3}$ appear in the shaded region, which indicates the typical range of decay lengths in the OI phase. The error bars are less than 5%.