Recently, we have found that the zeros of the diagonal components of the Green functions are useful quantities for detecting a wide range of topological insulators [1]. In particular, we have shown that the zeros of the Green functions traverse the band gap due to band inversions in the topological phases. Utilizing this feature, we can distinguish topological phases by seeing whether the zeros traverse the band gap. For microscopic models of the conventional six classes of topological insulators, we show that the traverses of the zeros universally occur in the topological phases. We also show that higher-order topological insulators, which have recently attracted much attention, can also be detected by the zeros of the Green functions.

Interestingly, the recently rediscovered eigenvector-eigenvalue identity [2], which is a simple but long-time-overlooked mathematical formula in linear algebra, plays an important role in the analysis of the zeros of the Green functions. Furthermore, by using the zeros of the Green function, we find that a conventional antiferromagnetic Mott insulator in κ-(BEDT-TTF)2Cu[N(CN)2]Cl can be regarded as a correlated topological insulator [3].

References
[1] T. Misawa and Y. Yamaji, Phys. Rev. Research 4, 023177 (2022).
[2] P. Denton, S. Parke, T. Tao, and X. Zhang, Bull. Am. Math. Soc. 59, 31 (2022). The story on the finding of the eigenvector-eigenvalue identity is available at https://www.quantamagazine.org/neutrinos-lead-to-unexpected-discovery-in-basic-math-20191113/
[3] T. Misawa and M. Naka, arXiv:2301.04490.

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