Discrete honeycombs, rational edges and edge states

Abstract

Consider the tight binding model of graphene, sharply terminated along an edge l parallel to a direction of translational symmetry of the underlying period lattice. We classify such edges l into those of "zigzag type" and those of "armchair type", generalizing the classical zigzag and armchair edges. We prove that zero energy/flat band edge states arise for edges of zigzag type, but never for those of armchair type. We exhibit explicit formulas for flat band edge states when they exist. We produce strong evidence for the existence of dispersive (non flat) edge state curves of nonzero energy for most l.

Wave propagation in quasiperiodic media

“…In fact, since one can no longer use the Floquet-Bloch transform along σ, the behaviour of a guided mode along the interface is not clear. This issue has been investigated in [FFW22] for a discrete Schrödinger model by means of rational approximations. Another idea in the continuous setting would be to use the lifting approach.…”