  Mean-field theories have since long predicted edge magnetism in graphene
nanoribbons, where the order parameter is given by the local magnetization.
However, signatures of edge magnetism appears elusive in the experiments,
suggesting another class of solutions. We employ a self-consistent mean field
approximation within a multi-orbital tight-binding model and obtain
particle-hole symmetry broken solutions, where the local filling plays the role
of the order parameter. Unlike the magnetic edge solutions, these are
topologically non-trivial and show zero local magnetization. A small and a
large doping regime are studied, and a free energy minimum for finite hole
doping is encountered, which may serve as an explanation for the absence of
experimental evidence for magnetic edge states in zigzag graphene nanoribbons.
The electronic interaction may increase the finite \(d\)-orbital occupation,
which leads to a change of the effective Coulomb interaction of the dominant
\(p_z\)-orbitals. Our findings indicate that the non-magnetic solution for
finite hole doping becomes energetically preferred, compared to the magnetic
phases at half-filling, once thermal fluctuations or unintentional doping from
the substrate are considered. This result persists even in the presence of the
\(d\)-orbitals and the Coulomb interaction therein.
