Robustness of Helical Hinge States of Weak Second-Order Topological Insulators

Abstract: Robustness of helical hinge states of three-dimensional weak second-order topological i sulators (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 inter-valley 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.