Flat-band ferromagnetism in a correlated topological insulator on a honeycomb lattice

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 ${\mathrm{Z}}_{2}$ topological insulators. Such a study extends our previous one [Phys. Rev. B 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 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 to the two triangular sublattices. Within the bosonization scheme, the fermion 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 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 ${K}^{\ensuremath{'}}$ points of the first Brillouin zone. We find that the same-lattice excitations are indeed the lowest-energy (elementary) excitations that characterize the flat-band ferromagnetic phase, a feature that contrasts with the behavior of a previously studied correlated topological insulator on a square lattice, whose flat-band ferromagnetic phase is characterized by mixed-lattice excitations. 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.