Zero-Energy State Localized near an Arbitrary Edge in Quadrupole Topological Insulators

• Published in: arXiv.org
• Main Author: Takane, Yositake
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 20.01.2020
• Subjects: Corners; Numerical methods; Physics - Mesoscale and Nanoscale Physics; Quadrupoles; Topological insulators; Topology; Wave functions
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2001.06928

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.