Topological end states and Zak phase of rectangular armchair ribbon

Abstract

We consider the end states of a half-filled rectangular armchair graphene ribbon (RAGR) in a staggered potential. Taking electron-electron interactions into account we find that, as the strength of the staggered potential varies, three types of couplings between the end states can occur: antiferromagnetic without or with spin splitting, and paramagnetic without spin-splitting. We find that a spin-splitting is present only in the staggered potential region $0<\Delta <{\Delta }_c$. The transition from the antiferromagnetic state at $\Delta =0$ to the paramagnetic state goes through an intermediate spin-split antiferromagnetic state, and this spin-splitting disappears suddenly at ${\Delta }_c$. For small and large values of $\Delta$ the end charge of a RAGR can be connected to the Zak phase of the periodic armchair graphene ribbon (PARG) with the same width, and it varies continuously as the strength of the potential changes.

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• Soliton fractional charge of disordered graphene nanoribbon
• Topological entanglement entropy of interacting disordered zigzag graphene ribbons