Topological classification under nonmagnetic and magnetic point group symmetry: Application of real-space Atiyah-Hirzebruch spectral sequence to higher-order topology
Topological classification under nonmagnetic and magnetic point group symmetry: Application of real-space Atiyah-Hirzebruch spectral sequence to higher-order topology
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DOI：
10.1103/physrevb.99.085127
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发表时间：
2018-10
影响因子：
3.7
通讯作者：
N. Okuma;M. Sato;Ken Shiozaki
中科院分区：
文献类型：
--
作者：
N. Okuma;M. Sato;Ken Shiozaki
我们分类时间反转破缺（A类）spinful拓扑晶体绝缘体与晶体非磁性（32种）和磁性（58种）的点群。该分类包括所有可能的磁性拓扑晶体绝缘体保护的点群对称。虽然拓扑绝缘体的分类是已知的，在动量空间中的$K$-理论，计算的$K$-理论一直是一项艰巨的任务，在复杂的晶体对称性的存在。这里我们考虑这个问题在真实的空间中的$K$-同调，而不是在动量空间中的$K$-理论，两者都给出了相同的拓扑分类。我们应用Atiyah-Hirzebruch谱序列（AHSS）来计算$K$-同调，这是一个研究广义（上）同调的数学工具.在真实的空间图像中，AHSS同时自然地给出了高阶拓扑绝缘体的分类。通过解决组扩展问题的基础上的物理参数的AHSS，我们完全确定可能的拓扑相位，包括高阶的每个点群。不同的高阶拓扑相位之间的关系争论的AHSS在$K$-同调。我们发现，在某些磁点群和磁点群中，两个$\mathbb{Z}_2$二阶拓扑绝缘体的堆叠可以光滑地变形为非平凡的四阶拓扑绝缘体，这意味着AHSS中的非平凡群扩张.
We classify time-reversal breaking (class A) spinful topological crystalline insulators with crystallographic non-magnetic (32 types) and magnetic (58 types) point groups. The classification includes all possible magnetic topological crystalline insulators protected by point group symmetry. Whereas the classification of topological insulators is known to be given by the $K$-theory in the momentum space, computation of the $K$-theory has been a difficult task in the presence of complicated crystallographic symmetry. Here we consider the $K$-homology in the real space for this problem, instead of the $K$-theory in the momentum space, both of which give the same topological classification. We apply the Atiyah-Hirzebruch spectral sequence (AHSS) for computation of the $K$-homology, which is a mathematical tool for generalized (co)homology. In the real space picture, the AHSS naturally gives the classification of higher-order topological insulators at the same time. By solving the group extension problem in the AHSS on the basis of physical arguments, we completely determine possible topological phases including higher-order ones for each point group. Relationships among different higher-order topological phases are argued in terms of the AHSS in the $K$-homology. We find that in some nonmagnetic and magnetic point groups, a stack of two $\mathbb{Z}_2$ second-order topological insulators can be smoothly deformed into non-trivial fourth-order topological insulators, which implies non-trivial group extensions in the AHSS.