Generalized corner-charge formula in higher-order topological insulators and its application to one-dimensional electrides

In C_n-symmetric higher-order topological insulators, the corner charge is quantized to fractional values [1,2], calculated from irreducible representations at high-symmetry k points. Unlike previous works, which rely on assumptions that wavefunctions are periodic even close to boundaries [2], we obtain formulae for the corner charge for general cases with minimal assumptions. We also show that the corner charge quantization holds in more general cases, such as systems with edges gapped by surface reconstructions. As an application of these formulae, we show that the apatite A₆B₄(SiO₄)₆, one of the one-dimensional electrides, realizes a higher-order topological insulator with 2/3-filled one-dimensional hinge states [3,4], which is explained with our formula. [1] W. A. Benalcazar, T. Li, T. L. Hughes, Phys. Rev.B 99, 245151 (2019). [2] H. Watanabe, S. Ono, Phys. Rev. B 102, 165120 (2020). [3] M. Hirayama, R. Takahashi, S. Matsuishi, H. Hosono, and S. Murakami, Phys. Rev. Research 2, 043131 (2020). [4] M.Hirayama, S. Matsuishi, H. Hosono, and S. Murakami, Phys. Rev. X 8, 031067 (2018).