E2 page

User Guide

• This web page lists the E₂ pages of the Atiyah-Hirzebruch spectral sequence for topological insulators and superconductors in momentum (k) space and real space.
• The table titled "Candidate K-groups" shows the set of K-groups that are compatible with both the E₂ page of the k-space AHSS and the E² page of the real-space AHSS, under the assumption that second or higher order derivatives do not change the rank of K-groups.
• "3D", "2D", and "1D" denote the dimensions of real space: 3, 2, and 1, respectively.
• "Magnetic space group", "Magnetic layer group", and "Magnetic rod group" indicate the magnetic crystal symmetry groups with translational symmetries in 3D, 2D, and 1D, respectively. "Magnetic point group" refers to magnetic symmetry groups without translations.
• "TI" specifies insulators, and "SC" specifies superconductors.
• "Spinless" and "Spinful" denote the factor systems of electrons, corresponding to integer and half-integer spin systems, respectively. "Spinless" also applies to cases with SU(2) internal symmetry in spin-1/2 systems.
• In "SC", "D" and "C" indicate the properties of particle-hole symmetry, with the square of the particle-hole symmetry operator corresponding to 1 and -1, respectively. They correspond to the Altland-Zirnbauer symmetry classes D and C, respectively.

The calculations are based on the following reference:

Ken Shiozaki and Seishiro Ono, "Atiyah-Hirzebruch spectral sequence for topological insulators and superconductors: E₂ pages for 1651 magnetic space groups", arXiv:2304.01827 .

If you have any questions or comments, please contact ken.shiozaki[at]yukawa.kyoto-u.ac.jp