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

Understand

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.