  We study the flat-band ferromagnetic phase of a topological Hubbard model
within a bosonization formalism and, in particular, determine the spin-wave
excitation spectrum. We consider a square lattice Hubbard model at 1/4-filling
whose free-electron term is the \pi-flux model with topologically nontrivial
and nearly flat energy bands. The electron spin is introduced such that the
model either explicitly breaks time-reversal symmetry (correlated flat-band
Chern insulator) or is invariant under time-reversal symmetry (correlated
flat-band $Z_2$ topological insulator). We generalize for flat-band Chern and
topological insulators the bosonization formalism [Phys. Rev. B 71, 045339
(2005)] previously developed for the two-dimensional electron gas in a uniform
and perpendicular magnetic field at filling factor \nu=1. We show that, within
the bosonization scheme, the topological Hubbard model is mapped into an
effective interacting boson model. We consider the boson model at the harmonic
approximation and show that, for the correlated Chern insulator, the spin-wave
excitation spectrum is gapless while, for the correlated topological insulator,
gapped. We briefly comment on the possible effects of the boson-boson
(spin-wave--spin-wave) coupling.
