  We explore the edge properties of rectangular graphene nanoribbons featuring
two zigzag edges and two armchair edges. Although the self-consistent
Hartree-Fock fields break chiral symmetry, our work demonstrates that graphene
nanoribbons maintain their status as short-range entangled symmetry-protected
topological insulators. The relevant symmetry involves combined mirror and
time-reversal operations. In undoped ribbons displaying edge ferromagnetism,
the band gap edge states with a topological charge form on the zigzag edges. An
analysis of the anomalous continuity equation elucidates that this topological
charge is induced by the gap term. In low-doped zigzag ribbons, where the
ground state exhibits edge spin density waves, this topological charge appears
as a nearly zero-energy edge mode. Our system is outside the conventional
calssification for topological insulators.
