Confined Electronic States on Grain Boundaries in Topologically Gapped Graphene

Propagating edge modes are a signature of the topological order of the bulk spectrum. In this work, we consider a grain boundary embedded in a topologically ordered bulk as an interface between topologically identical mismatched domains. At such interfaces, the edge modes of the individual domains can hybridize to form an electronic state confined to the grain boundary. These states are not topologically protected, but for ordered grain boundaries backscattering can be suppressed by specific momentum selection rules. We study grain boundary modes in two models on the honeycomb lattice: the Haldane model and the Kane-Mele model, where the bulk is characterized by nontrivial Chern and Z₂ invariants, respectively. We discuss the possibility of using grain boundary modes as transport channels and highlight the effects of grain boundary symmetry on band structure and spin degeneracies. Finally, we sketch out an extension of the spin orbit coupled model to the transition metal dichalcogenide quantum spin Hall insulators (TMD QSHIs), which suggests similarities between grain boundary modes in TMD QSHIs and those in topologically ordered graphene models.