Topological end states and Zak phase of rectangular armchair ribbon

TL;DR

This paper studies the end states and topological properties of a rectangular armchair graphene ribbon under a staggered potential, revealing phase transitions and the relationship between end charge and Zak phase.

Contribution

It introduces a detailed analysis of how electron interactions and staggered potential influence end states and topological phases in RAGRs, including the connection to Zak phase.

Findings

Identification of three coupling types between end states

Discovery of a critical potential $\

and the connection between end charge and Zak phase

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.