Recent Developments of the Classification of Topological Insulators and Superconductors

Yukawa Institute, Kyoto University

A band insulator is a gapped phase of electronic systems with a crystal structure. In particular, when electron-electron interactions can be neglected, the classification problem becomes equivalent to the classification of continuous deformations of Hermitian matrices with a finite energy gap, parameterized by the Brillouin zone torus. This classification can be approximately determined by topological K-theory. In superconductors, the Bogoliubov-de Gennes Hamiltonian, introduced under the mean-field approximation, can be formulated within the same classification framework.

The classification problem of topological insulators and superconductors, considering the 1651 types of magnetic space groups and the symmetry of the superconducting gap function, involves tens of thousands of independent symmetry classes. Thus, a comprehensive and automated computational method for classification is required. We are developing the computational implementation of the Atiyah-Hirzebruch spectral sequence (AHSS), a tool for calculating general (co)homology theories, from both the perspectives of topological physics and mathematical structure behind. In particular, understanding the physical interpretation of the connected homomorphism is crucial in both the K-cohomology theory on k-space and the K-homology theory on real space.

In this talk, after discussing the interpretation of the connected homomorphism, I introduce some general structure of AHSS and the computational method of the E2 page. By comparing the E2 pages obtained from real-space and k-space K-theories, we got about 60% of the K-groups of interest in condensed matter physics in three-dimensional space. 

References:
KS, Masatoshi Sato, Kiyonori Gomi, arXiv:1802.06694.
KS, Charles Zhaoxi Xiong, Kiyonori Gomi, arXiv:1810.00801.
KS, Seishiro Ono, arXiv:2304.01827.