We theoretically study the quantum spin Hall states in an Ammann-Beenker-type octagonal quasicrystal and a periodic snub-square crystal, both sharing the same basic building blocks. Although the bulk states show significant differences in localization and transport properties, the topological phases manifest similarly in the two systems. This indicates the robustness of the topological properties regardless of symmetry and periodicity. We characterize the topological nature of the two systems with a nonzero topological invariant (spin Bott index $B_s$ and ${\mathbb{Z}}_2$ invariant), robust metallic edge states, and quantized conductance. In spite of some quantitative differences, the topological phase diagram of the two systems also exhibits similar behaviors, indicating that the topological phase transition is mainly determined by similar interactions in the two systems regardless of their structural difference. This is also reflected by the observation that the transition point between the normal insulator and the quantum spin Hall state in both systems follows a universal linear scaling relation for topological phase transitions.

• Received 15 April 2019

DOI:https://doi.org/10.1103/PhysRevB.100.085119