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 $90^{\circ }$ 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 $90^{\circ }$ corners, it is uncertain whether a similar localized state can appear at $E=0$ near a complicated edge consisting of multiple $90^{\circ }$ and $270^{\circ }$ 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.

arXiv comment7 pages, 8 figures

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• Anisotropic Penetration Depths of Corner States in a Higher-Order Topological Insulator
• Gapless States Localized along a Staircase Edge in Second-Order Topological Insulators