Appearance of hinge states in second-order topological insulators via the cutting procedure

TL;DR

This paper proves that second-order topological insulators with inversion symmetry and a specific topological index always host hinge states, revealing their emergence through boundary condition manipulations and topological invariants.

Contribution

It provides a general proof linking the $ u_1=2$ topological index to the existence of hinge states in 3D insulators with gapped surfaces.

Findings

Hinge states appear when the $ u_1=2$ topological index is present.

Boundary condition changes reveal the topological origin of hinge states.

Gapless hinge states are guaranteed under specified topological conditions.

Abstract

In recent years, second-order topological insulators have been proposed as a new class of topological insulators. Second-order topological insulators are materials with gapped bulk and surfaces, but with topologically protected gapless states at the intersection of two surfaces. These gapless states are called hinge states. In this paper, we give a general proof that any insulators with inversion symmetry and gapped surface in class A always have hinge states when the $\mathbb{Z}_{4}$ topological index $\mu_{1}$ is $\mu_{1}=2$. We consider a three-dimensional insulator whose boundary conditions along two directions change by changing the hopping amplitudes across the boundaries. We study behaviors of gapless states through continuously changing boundary conditions along the two directions, and reveal that the behaviors of gapless states result from the $\mathbb{Z}_{4}$ strong topological index. From this discussion, we show that gapless states inevitably appear at the hinge of a three-dimensional insulator with gapped surfaces when the strong topological index is $\mathbb{Z}_{4}=2$ and the weak topological indices are $\nu_{1}=\nu_{2}=\nu_{3}=0$.