Topological corner states in graphene by bulk and edge engineering

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 the edge anticrossing deviates 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 multiorbital model. The gapless helical edge modes cross inside the bulk, where the magnetization-induced edge gap is also located. After demonstrating its second-order nontriviality in bulk topology by a series of evidence, we show that a difference in bulk-edge on-site 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.