Zero-Energy State Localized near an Arbitrary Edge in Quadrupole Topological Insulators
Abstract
A two-dimensional quadrupole topological insulator on a square lattice is a typical example of a higher-order topological insulator. It hosts an edge state localized near each of its 90 corners at an energy E inside the band gap, where E is set equal to zero for simplicity. Although the appearance of an edge state has been shown in simple systems with only 90 corners, it is uncertain whether a similar localized state can appear at E = 0 near a complicated edge consisting of multiple 90 and 270 corners. Here, we present a numerical method to determine the wavefunction of a zero-energy state localized near an arbitrary edge. This method enables us to show that one localized state appears at E = 0 if the edge consists of an odd number of corners. In contrast, the energy of localized states inevitably deviates from E = 0 if the edge includes an even number of corners.