The celebrated tenfold way of Altland-Zirnbauer symmetry classes discern any quantum system by its pattern of nonspatial symmetries. It lays at the core of the periodic table of topological insulators and superconductors which provided a complete classification of weakly interacting electrons' noncrystalline topological phases for all symmetry classes. Over recent years, a plethora of topological phenomena with diverse surface states has been discovered in crystalline materials. In this paper, we obtain an exhaustive classification of topologically distinct ground states as well as topological phases with anomalous surface states of crystalline topological insulators and superconductors for key space-groups, layer-groups, and rod-groups. This is done in a unified manner for the full tenfold way of Altland-Zirnbauer nonspatial symmetry classes. We establish a comprehensive paradigm that harnesses the modern mathematical framework of equivariant spectra; it allows us to obtain results applicable to generic topological classification problems. In particular, this paradigm provides efficient computational tools that enable an inherently unified treatment of the full tenfold way.

中文翻译：

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晶体的十倍拓扑：晶体拓扑绝缘体和超导体的统一分类

Altland-Zirnbauer对称性类的著名十倍方式通过其非空间对称性模式来识别任何量子系统。它位于拓扑绝缘体和超导体的周期表的核心，它们为所有对称类别提供了弱相互作用电子的非晶态拓扑相的完整分类。近年来，在晶体材料中发现了具有多种表面状态的大量拓扑现象。在本文中，我们对关键的空间组，层组和棒组的拓扑上不同的基态以及具有结晶表面绝缘子和超导体异常表面状态的拓扑相进行了详尽的分类。对于Altland-Zirnbauer非空间对称类的全部十倍方式，这是以统一的方式完成的。我们建立了一个全面的范式，利用了等变谱的现代数学框架；它使我们可以获得适用于一般拓扑分类问题的结果。特别是，此范例提供了有效的计算工具，可对完整的十倍方式进行固有的统一处理。