The Zak phase and the existence of edge states in graphene

TL;DR

This paper presents a method to predict edge states in graphene ribbons using the Zak phase, establishing a bulk-edge correspondence and exploring topological transitions caused by anisotropy.

Contribution

It introduces a geometrical approach based on the Zak phase to determine edge states in graphene, generalizing previous results and including effects of anisotropy.

Findings

Edge states exist for specific momentum ranges depending on edge orientation.

Anisotropy induces a topological transition affecting edge state localization.

The method is rigorously demonstrated for zigzag edges and a toy model.

Abstract

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, it confirms and generalizes the results of several previous works.