Topological classification under nonmagnetic and magnetic point group symmetry: Application of real-space Atiyah-Hirzebruch spectral sequence to higher-order topology

TL;DR

This paper classifies topological crystalline insulators with various point group symmetries using real-space K-homology and the Atiyah-Hirzebruch spectral sequence, revealing relationships among higher-order topological phases.

Contribution

It introduces a real-space K-homology approach with AHSS to classify magnetic and nonmagnetic topological insulators, including higher-order phases, for all point groups.

Findings

Complete classification of topological phases for each point group.

Identification of possible higher-order topological insulators.

Discovery of smooth deformation between different topological phases.

Abstract

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.