Robustness of helical hinge states of weak second-order topological insulators

Robustness of helical hinge states of three-dimensional weak second-order topological insulators (WSOTIs) against disorders is studied. The pure WSOTI is obtained from a weak ${\mathbb{Z}}_2$ first-order topological insulator through a surface band inversion. Both bulk states and surface states in the WSOTI are gapped, and in-gap valley-momentum locked helical hinge states are topologically protected by the surface valley-Chern number. In the presence of weak disorders, helical hinge states are robust against disorders while the quantized conductance of the states is fragile due to the intervalley scattering. As disorder increases, the system undergoes a series of quantum phase transitions: from the WSOTI to the weak first-order topological insulator, then to a diffusive metal and finally to an Anderson insulator. Our results thus fully establish the WSOTI phase as a genuine state of matters and open a door for the second-order valleytronics that allows one to control the valley degree of freedom through helical hinge states.

Figure 1

(a) Schematic plot of a WSOTI. Hinge states (black arrows) of different valleys are antiparallel. The dark yellow plane is $x=0$. (b) Energy spectrum $E\left(k_3\right)$ of Eq. (1) for $M=t,B=0.2t$. Colors encode ${log}_{10}{p}_2$. Red dotted line locates $E=0.02t$. (b) Spatial distribution of the in-gap helical states $k_3=k_{a,b,c,d}$ shown in (b). Colors encode ${log}_{10}{\left|{\psi }_{\mathit{i}}\right|}^2$.

Figure 2

(a) ${\eta }_{W,L}$ as a function of $W$ for various $L$. Each data is averaged over more than 100 samples. (b) $\rho \left(E\right)$ for $L=200$ and various $W$: solid (dash) lines are for $W<W_{c1}$ ($W>W_{c1}$). Here, $M=t, B=0.2t$, and $L_{\parallel }=L_{\perp }=L$.

Figure 3

(a) $\langle R\rangle$ as a function of $W$ for $L_{\perp }=L_{\parallel }=L=16,20,24,32$ (from down to up). (b) $\langle R\rangle$ versus $L_{\perp }$ for $W/t=0.3,0.5,1,0.8,1.2$ (from down to up) and $L_{\parallel }=10$. Dotted lines are fitted by Eq. (6). Inset: The obtained $L_m$ as a function $W$. Black solid line is a fit of $L_m=c{t}^2/W^2$ with $c=92$. Cyan dashed lines locate the intrinsic resistance $h/\left(2C_{\text{valley}}{e}^2\right)$. Here, $M=t$ and $B=0.2t$. Each data point is averaged over 1000 samples.

Figure 4

(a) ${\rho }_{\text{bulk}}\left(E\right)$ for various $W>W_{c1}$ and $L=L_{\parallel }=L_{\perp }=200$. Solid (dashed) lines are for $W<W_{c2}$ ($W>W_{c2}$). (b) ${\zeta }_{W,L}$ versus $W$ for various $L=L_{\parallel }=L_{\perp }$. Here, $M=t$ and $B=0.2t$.

Figure 5

(a) $ln{Y}_L$ as a function of $W$ for $M=t$ and $B=0.2t$. (b) Scaling function $ln{Y}_L=lnf\left(L\right|W-W_{c3}{|}^{\nu })$.