Anisotropic Penetration Depths of Corner States in a Higher-Order Topological Insulator

Abstract

Higher-order topological insulators in two dimensions have states that localize at their corners, called corner states. In this paper, we reveral characteristics of the penetration depth of their corner states by using the Benalcazar-Bernevig-Hughes model. First, we show that when we change the energy of the corner states toward the end of the edge gap by adding an on-site potential to the corner site, the penetration depth along the edge diverges toward infinity while the penetration depth into the bulk remaining finite. We analytically derive the corner-state wavefunction in a form of elliptic integrals, which reproduces this anisotropic behavior of corner states. This means that corner states have two kinds of penetration depths, and they behave differently. At last, we show that hybridizations between corner states are governed by the penetration depth through interference between the corner states. It is because the corner states almost do not interfere with edge states or bulk states.