Despite the rapid progress in the field of the quantum spin Hall (QSH) effect, most of the QSH systems studied up to now are based on crystalline materials. Here we propose that the QSH effect can be realized in quasicrystal lattices (QLs). We show that the electronic topology of aperiodic and amorphous insulators can be characterized by a spin Bott index $B_s$. The nontrivial QSH state in a QL is identified by a nonzero spin Bott index $B_s=1$, associated with robust edge states and quantized conductance. We also map out a topological phase diagram in which the QSH state lies in between a normal insulator and a weak metal phase due to the unique wave functions of QLs. Our findings not only provide a better understanding of electronic properties of quasicrystals but also extend the search of the QSH phase to aperiodic and amorphous materials that are experimentally feasible.

中文翻译：

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准晶格中的量子自旋霍尔效应和自旋博特指数

尽管在量子自旋霍尔（QSH）效应领域迅速发展，但迄今为止研究的大多数QSH系统都是基于晶体材料。在这里，我们建议可以在准晶格（QLs）中实现QSH效应。我们表明，非周期性和非晶态绝缘子的电子拓扑可以通过自旋博特指数来表征${乙}_s$。QL中的非平凡QSH状态由非零自旋Bott索引标识${乙}_s=\mathrm{1个}$与稳健的边缘状态和量化的电导相关。我们还绘制了一个拓扑相图，其中由于QL的独特波函数，其QSH状态位于正常绝缘体和弱金属相之间。我们的发现不仅提供了对准晶体电子性质的更好理解，而且将QSH相的搜索范围扩展到了实验上可行的非周期性和非晶态材料。