Theory of spin Bott index for quantum spin Hall states in non-periodic systems

This is a joint publication with the Letter by H. Huang and F. Liu [Phys. Rev. Lett. 121, 126401 (2018)]. In this work, we propose the spin Bott index to identify the quantum spin Hall (QSH) state in both crystalline and non-periodic systems. The applicability of the spin Bott index is confirmed by analyzing the periodic and disorder Kane-Mele models. As an example of non-periodic systems, we systematically investigate the QSH effect in a Penrose-type quasicrystal lattice (QL). We characterized the nontrivial electronic topology of the QL by directly calculating the spin Bott index. In addition, the topological edge states, the localization of wavefunctions and quantized transport signatures are also studied in detail.