Apatite as a higher-order topological insulator with 2/3-filled hinge states

In our previous work we have shown that electrides are suitable for achieving various topological semimetal phases [1]. In our presentation, 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 metallic hinge states. This is considered to be in the class of C_n-protected higher-order topological insulators proposed by Benalcazar et al. [2], but the hinge charge is different from that naively expected from their theory. This difference comes from the difference in the fundamental units constituting the crystal; the apatite crystal consists of triangular units, unlike hexagonal units assumed in Ref. [2]. We show how this difference affects topological properties of the system via crystal terminations. [1] M.Hirayama, S. Matsuishi, H. Hosono, and S. Murakami, Phys. Rev. X 8, 031067 (2018). [2] W. A. Benalcazar, T. Li, T. L. Hughes, Phys. Rev.B 99, 245151 (2019).