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

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 Z4 topological index µ1 is µ1 = 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 Z4 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 Z4 = 2 and the weak topological indices are ν1 = ν2 = ν3 = 0.

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

“…We note that, in this paper, we consider a process of cutting the three-dimensional HOTI along a plane. In another paper, we consider a process of cutting the threedimensional HOTI along two planes 43 . In the paper, we discuss allowed positions of the hinge states, which is determined only if we consider a cutting along two planes.…”

General corner charge formula in two-dimensional C_n-symmetric higher-order topological insulators

“…Recent studies have revealed that some topological crystalline insulators exhibit higher-order bulk boundary correspondence, which are known as higher-order topological insulators . For example, three-dimensional second-order topological insulators with inversion symmetry (and time-reversal symmetry) have protected anomalous gapless mode along the hinges 9,[12][13][14]16,[21][22][23] , and two-dimensional second-order topological insulators with rotation symmetry have protected quantized corner charges [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42] . Such fractionally quantized corner charges are generalizations of the quantized surface charge caused by the quantized bulk electric polarization 43 .…”

Out of equilibrium chiral higher order topological insulator on a $π$-flux square lattice

“…Recently, the notion of conventional TI's is generalized to higher order topological insulators (HOTI) with n > 1 [32][33][34][35][36][37][38][39][40][41][42][43]. The HOTI's are protected by the crystalline (spatial) symmetries such as the inversion, the mirror reflection, and the four fold rotation or space-time symmetries of both the bulk and the boundaries [32][33][34][44][45][46][47][48].…”