  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.
