Topological Corner States in Graphene by Bulk and Edge Engineering

TL;DR

This paper demonstrates how to identify topological corner states in graphene-based topological insulators by tuning bulk and edge parameters, revealing a higher-order bulk-boundary correspondence in a realistic model.

Contribution

It introduces a method to locate topological corner states in graphene with spin-orbit coupling and magnetization by bulk-edge energy tuning, advancing higher-order topological insulator understanding.

Findings

Bulk-edge onsite energy difference tunes edge mode crossings.

Corner states are identifiable in a realistic graphene model.

Obtuse-angle corner states are more stable than acute-angle ones.

Abstract

Two-dimensional higher-order topology is usually studied in (nearly) particle-hole symmetric models, so that an edge gap can be opened within the bulk one. But more often deviates the edge anticrossing even into the bulk, where corner states are difficult to pinpoint. We address this problem in a graphene-based $\mathbb{Z}_2$ topological insulator with spin-orbit coupling and in-plane magnetization both originating from substrates through a Slater-Koster multi-orbital model. The gapless helical edge modes cross inside the bulk, where is also located the magnetization-induced edge gap. After demonstrating its second-order nontriviality in bulk topology by a series of evidence, we show that a difference in bulk-edge onsite energy can adiabatically tune the position of the crossing/anticrossing of the edge modes to be inside the bulk gap. This can help unambiguously identify two pairs of topological corner states with nonvanishing energy degeneracy for a rhombic flake. We further find that the obtuse-angle pair is more stable than the acute-angle one. These results not only suggest an accessible way to "find" topological corner states, but also provide a higher-order topological version of "bulk-boundary correspondence".