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 Z(2) 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. We show that the Z(4) indicators, which characterize three-dimensional inversion-symmetric HOTIs in classes A and AII, are directly related to the Z(2) indicators for the corresponding two-dimensional slabs with inversion symmetry, i.e., the Chern number parity and the Z(2) topological invariant, for classes A and AII, respectively. 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- and two-dimensional inversion-symmetric insulators.