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 honeycomb lattice within bosonization formalism for ${\mathrm{Z}}_{2}$ topological insulators. Such extends our previous one [Phys. Rev. B 104, 155129 (2021)] concerning correlated Chern insulator described by model. consider Hubbard at $1/4$ filling its corresponding noninteracting limit in nearly flat band lower free-electronic bands. define boson operators associated with two distinct spin-flip excitations, that changes (mixed-lattice excitations) second preserves (same-lattice index related to triangular sublattices. Within scheme, fermion is mapped into an effective interacting model, whose quadratic term considered harmonic approximation order determine spin-wave spectrum. For both mixed- same-lattice we find spectrum gapped has branches, energy gap between upper bands $K$ ${K}^{\ensuremath{'}}$ points first Brillouin zone. excitations are indeed lowest-energy (elementary) characterize phase, feature contrasts behavior previously studied square lattice, characterized mixed-lattice excitations. also some evidences might be topologically nontrivial even completely limit.