Cutting a honeycomb lattice (HCL) can end up with three types of edges
(zigzag, bearded and armchair), as is well known in the study of graphene edge
states. Here we theoretically investigate and experimentally demonstrate a
class of graphene edges, namely, the twig-shaped edges, using a photonic
platform, thereby observing edge states distinctive from those observed before.
Our main findings are: (i) the twig edge is a generic type of HCL edges
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 along the boundary, manifesting both flat band and topological
features. Such new edge states are realized in a laser-written photonic
graphene and well corroborated by numerical simulations. Our results may
broaden the understanding of graphene edge states, bringing about new
possibilities for wave localization in artificial Dirac-like materials.Comment: 13 pages, 4 figure