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The surface states of insulators and fully gapped superconductors can be detected by topological invariants in momentum space (k-space). Well-known topological invariants include the Chern number, winding number, and the number of irreducible representations at high-symmetry points. However, many topological invariants remain unknown in the presence of crystalline symmetry, representing a significant challenge in the search for topological materials.

In this talk, I will discuss an attempt to systematically and comprehensively construct topological invariants using K-theory and spectral sequences[1]. During the spectral sequence calculation, k-space is divided into high-symmetry points, lines, polygons, and fundamental domain. Each region is provided with an appropriate set of irreducible representations for the construction of topological invariants. We propose a method to construct topological invariants defined on one-dimensional subspaces of momentum space from the spectral sequence data[2].

In particular, for time-reversal symmetric superconductors with trivial point group representations of the gap function, detecting topological superconductors by counting the number of irreducible representations (symmetry indicators) is silent, although many topological phases exist[3]. Using our one-dimensional subspace invariants, many classes of topological superconductors can be numerically detected. In the seminar, I will introduce the construction method and discuss its practical applications.

 

[1] K. Shiozaki, S. Ono, arXiv:2304.01827.

[2] S. Ono, K. Shiozaki, arXiv:2311.15814.

[3] S Ono, K Shiozaki, H Watanabe, PRB 109 (21), 214502.

 

Registration　：　https://forms.gle/tZAhMsJJATF4ndy58

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