Gapless States Localized along a Staircase Edge in Second-Order Topological Insulators

TL;DR

This paper investigates the emergence of gapless one-dimensional edge states along staircase edges in second-order topological insulators, revealing symmetry protections and topological invariants that ensure their robustness.

Contribution

It introduces the concept of staircase edges in second-order topological insulators and demonstrates how symmetry and winding numbers protect gapless edge states.

Findings

Staircase edges host hybridized one-dimensional edge states.

Symmetry ensures the gapless nature of these edge states.

Winding numbers guarantee the bulk-boundary correspondence.

Abstract

A second-order topological insulator on a two-dimensional square lattice hosts zero-dimensional states inside a band gap. They are localized near $90^{\circ}$ and $270^{\circ}$ corners constituting an edge of the system. When the edge is in a staircase form consisting of these two corners, two families of edge states (i.e., one-dimensional states localized near the edge) appear as a result of the hybridization of zero-dimensional states. We identify symmetry that makes them gapless. We also show that a pair of nontrivial winding numbers associated with this symmetry guarantee a gapless spectrum of edge states, indicating that bulk--boundary correspondence holds in this topological insulator with a staircase edge.