Bulk-edge and bulk-hinge correspondence in inversion-symmetric insulators

We show that a slab of a three-dimensional inversion-symmetric higher-order topological insulator (HOTI) in class A is a 2D Chern insulator, and that in class AII is a 2D Z₂ topological insulator. We prove it by considering a process of cutting the three-dimensional inversion-symmetric HOTI along a plane, and study the spectral flow in the cutting process [1]. We show that the Z₄ indicators, which characterize three-dimensional inversion-symmetric HOTIs in classes A and AII, are directly related to the Z₂ indicators for the corresponding two-dimensional slabs with inversion symmetry, i.e. the Chern number parity and the Z₂ topological invariant, for classes A and AII respectively [2]. The existence of the gapless hinge states is understood from the conventional bulk-edge correspondence between the slab system and its edge states. Moreover, we also show that the spectral-flow analysis leads to another proof of the bulk-edge correspondence in one-dimensional inversion-symmetric insulators.
[1] J. C. Y. Teo, L. Fu, and C. L. Kane, Phys. Rev. B 78, 045426 (2008).
[2] R. Takahashi, Y. Tanaka, and S. Murakami, arXiv:1910.08290 (2019).