  The topological phases of graphene with spin-orbit coupling, an exchange
field, and a staggered-sublattice potential determine the properties of the
edge states of the zigzag nanoribbon. In the presence of the Hubbard
interaction, the spontaneous magnetization at the zigzag terminations induces
sizable magnetic moments at the lattice sites in the bulk region. Thus, the
exchange field and staggered-sublattice potential in the bulk region are
effectively changed, which in turn change the topological phase. Within a
certain parameter regime, quasi-stable excited states of the zigzag nanoribbon
exist, which have a different magnetism configuration at the zigzag
terminations from the ground state. The quasi-stable excited states could
effectively suppress the finite size effect of the topological edge states. The
investigation of the topological edge states in the presence of interaction
helps the engineering of spintronic nanodevices based on realistic materials.
