  We study the flat-band ferromagnetic phase of the Haldane-Hubbard model on a
honeycomb lattice within a bosonization scheme for flat-band Chern insulators,
focusing on the calculation of the spin-wave excitation spectrum. We consider
the Haldane-Hubbard model with the noninteracting lower bands in a nearly-flat
band limit, previously determined for the spinless model, and at 1/4-filling of
its corresponding noninteracting limit. Within the bosonization scheme, the
Haldane-Hubbard model is mapped into an effective interacting boson model,
whose quadratic term allows us to determine the spin-wave spectrum at the
harmonic approximation. We show that the excitation spectrum has two branches
with a Goldstone mode and Dirac points at center and at the K and K' points of
the first Brillouin zone, respectively. We also consider the effects on the
spin-wave spectrum due to an energy offset in the on-site Hubbard repulsion
energies and due to the presence of an staggered on-site energy term, both
quantities associated with the two triangular sublattices. In both cases, we
find that an energy gap opens at the K and K' points. Moreover, we also find
some evidences for an instability of the flat-band ferromagnetic phase in the
presence of the staggered on-site energy term. We provide some additional
results for the square lattice topological Hubbard model previous studied
within the bosonization formalism and comment on the differences between the
bosonization scheme implementation for the correlated Chern insulators on both
square and honeycomb lattices.
