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

Robustness of helical hinge states three-dimensional weak second-order topological insulators (WSOTIs) against disorders is studied. The pure WSOTI obtained from a ${\mathbb{Z}}_{2}$ first-order insulator through surface band inversion. Both bulk and in the are gapped, in-gap valley-momentum locked topologically protected by valley-Chern number. In presence disorders, robust while quantized conductance fragile due to intervalley scattering. As disorder increases, system undergoes series quantum phase transitions: insulator, then diffusive metal finally an Anderson insulator. Our results thus fully establish as genuine state matters open door for valleytronics that allows one control valley degree freedom states.