  We theoretically examine the possible spin ordered states in zigzag graphene
nanoribbon in a large supercell by the self-consistent mean field method as
well as the first principle calculation. In addition to the well-known
anti-ferromagnetic (AF) and ferromagnetic (FM) edge states, we find that there
are also some excited spin density wave (ESDW) states, the energy of which are
close to the AF edge state (ground state). We thus argue that these ESDW states
may coexist in experiment when the temperature is not too low. Our numerical
results indicate that the allowed ESDW are commensurate; and their dispersion
curve is plotted. This interesting feature results from the fact the there is a
flat edge band near the Fermi surface in the graphene zigzag ribbon. Finally we
use the non-equilibrium Green function theory combined with the Hubbard model
to calculate some possible spin polarization states localized in the middle and
terminal parts of graphene nanoribbons.
