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We consider the end states of a half-filled rectangular armchair grapheneribbon (RAGR) in a staggered potential. Taking electron-electron interactionsinto account we find that, as the strength of the staggered potential varies,three types of couplings between the end states can occur: antiferromagneticwithout or with spin splitting, and paramagnetic without spin-splitting. Wefind 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-splitantiferromagnetic state, and this spin-splitting disappears suddenly at$\Delta_c$. For small and large values of $\Delta$ the end charge of a RAGR canbe connected to the Zak phase of the periodic armchair graphene ribbon (PARG)with the same width, and it varies continuously as the strength of thepotential changes.

Topological end states and Zak phase of rectangular armchair ribbon