Zak phase and the existence of edge states in graphene

We develop a method to predict the existence of edge states in graphene ribbons for a large class of boundaries. This approach is based on the bulk-edge correspondence between the quantized value of the Zak phase Z (k_), which is a Berry phase across an appropriately chosen one-dimensional Brillouin zone, and the existence of a localized state of momentum k_ at the boundary of the ribbon. This bulk-edge correspondence is rigorously demonstrated for a one-dimensional toy model as well as for graphene ribbons with zigzag edges. The range of k_ for which edge states exist in a graphene ribbon is then calculated for arbitrary orientations of the edges. Finally, we show that the introduction of an anisotropy leads to a topological transition in terms of the Zak phase, which modifies the localization properties at the edges. Our approach gives a new geometrical understanding of edge states, and it confirms and generalizes the results of several previous works.