Atiyah-Hirzebruch spectral sequence for topological insulators and superconductors: $E_2$ pages for 1651 magnetic space groups

Abstract

We compute the $E_2$ pages of the momentum-space and real-space Atiyah-Hirzebruch spectral sequence (AHSS) for topological crystalline insulators and superconductors up to three spatial dimensions, considering the cell decomposition in which if a group action fixes a cell setwise then its group action fixes the same cell pointwise. We provide a detailed description of the implementation for computing the $E_2$ pages of AHSS. Under a physically reasonable assumption, we enumerate all possible $K$-groups that are compatible with the $E_2$ pages for both momentum and real-space AHSS. As a result, we determine the $K$-groups for approximately 59\% of symmetry settings in three spatial dimensions. All the results can be found at this http \href{https://www2.yukawa.kyoto-u.ac.jp/~ken.shiozaki/ahss/e2.html}{URL}.