  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.
