Topological Gap States of Rectangular Armchair Ribbon

Abstract

We consider a rectangular graphene armchair ribbon with an excitation gap. The boundary of this system consists of two short zigzag edges and two long armchair edges. Within such a ribbon, topological gap states exist that are localized along the zigzag edges. The end charge of these states is an integer, which can be related to the Zak phase of the periodic armchair ribbon constructed from the rectangular armchair ribbon by connecting its zigzag edges together. In this paper, we provide an explicit analytical computation of the Zak phase of a periodic armchair ribbon, and show that its value is consistent with the integer end charges that are computed numerically. In the presence of a staggered potential, non-integer end charges are possible. We discuss its relation to the Zak phase.