  We study the flat-band ferromagnetic phase of a spinfull and time-reversal
symmetric Haldane-Hubbard model on a honeycomb lattice within a bosonization
formalism for flat-band Z$_2$ topological insulators. Such a study extend our
previous one [L. S. G. Leite and R. L. Doretto, Phys. Rev. B {\bf 104}, 155129
(2021)] concerning the flat-band ferromagnetic phase of a correlated Chern
insulator described by a Haldane-Hubbard model. We consider the topological
Hubbard model at $1/4$ filling of its corresponding noninteracting limit and in
the nearly flat band limit of its lower free-electronic bands. We show that it
is possible to define boson operators associated with two distinct spin-flip
excitations, one that changes (mixed-lattice excitations) and a second one that
preserves (same-lattice excitations) the index related with the two triangular
sublattices. Within the bosonization scheme, the fermionic model is mapped into
an effective interacting boson model, whose quadratic term is considered at the
harmonic approximation in order to determine the spin-wave excitation spectrum.
For both mixed and same-lattice excitations, we find that the spin-wave
spectrum is gapped and has two branches, with an energy gap between the lower
and the upper bands at the $K$ and $K'$ points of the first Brillouin zone.
Such a behavior is distinct from the one of the corresponding correlated Chern
insulator, whose spin-wave spectrum has a Goldstone mode at the center of the
first Brillouin zone and Dirac points at $K$ and $K'$ points. We also find some
evidences that the spin-wave bands for the same-lattice excitations might be
topologically nontrivial even in the completely flat band limit.
