Theory of edge states in graphene-like systems

Systems that can be described with the same mathematical models that account for the properties of electrons in graphene are known as graphene-like systems. These include magnons, photons, polaritons, acoustic waves, and electrons in honeycomb lattices, either natural or artificial. All of them feature an outstanding property, the existence of states localized at the edges. We distinguish two classes of edge states depending on whether or not a topological band gap is present in the two-dimensional energy bands. We introduce the theory of edge states in terms of tight-binding models and discuss their properties, with an emphasis on the interplay between their one-way character and the topological nature of the bulk phase.