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

TL;DR

This paper demonstrates that three-dimensional inversion-symmetric HOTIs exhibit 2D topological phases when cut into slabs, linking bulk invariants to edge states and providing new insights into bulk-edge correspondence.

Contribution

It establishes a direct relation between 3D inversion-symmetric HOTI indicators and 2D topological invariants, and offers a spectral-flow-based proof of bulk-edge correspondence.

Findings

3D HOTIs in class A become 2D Chern insulators when cut into slabs.

3D HOTIs in class AII become 2D Z2 topological insulators in slabs.

Spectral flow analysis confirms bulk-edge correspondence in inversion-symmetric insulators.

Abstract

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.