Photonic Realization of a Generic Type of Graphene Edge States Exhibiting Topological Flat Band

Cutting a honeycomb lattice (HCL) ends up with three types of edges (zigzag, bearded, and armchair), as is well known in the study of graphene edge states. Here, we propose and demonstrate a distinctive twig-shaped edge, thereby observing new edge states using a photonic platform. Our main findings are (i) the twig edge is a generic type of HCL edge complementary to the armchair edge, formed by choosing the right primitive cell rather than simple lattice cutting or Klein edge modification; (ii) the twig edge states form a complete flat band across the Brillouin zone with zero-energy degeneracy, characterized by nontrivial topological winding of the lattice Hamiltonian; (iii) the twig edge states can be elongated or compactly localized at the boundary, manifesting both flat band and topological features. Although realized here in a photonic graphene, such twig edge states should exist in other synthetic HCL structures. Moreover, our results may broaden the understanding of graphene edge states, as well as new avenues for realization of robust edge localization and nontrivial topological phases based on Dirac-like materials.