Flat-band ferromagnetism and spin waves in the Haldane-Hubbard model

We study the flat-band ferromagnetic phase of the Haldane-Hubbard model and, in particular, we calculate the spin-wave excitation spectrum. This is done via a previously developed bosonization scheme, that was successfully applied to describe the spin-wave excitations of the flat-band ferromagnetic phase of a square lattice topological Hubbard model. The noninteracting Hamiltonian has four bands, each with a nonzero Chern number. Here, we consider the lower (doubly degenerated) band at half filling, which can be made nearly flat with a particular choice of the hopping parameters. By applying the bosonization scheme, we map the Haldane-Hubbard Hamiltonian into an effective interacting boson model, whose quadratic term allow 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. To further characterize the flat-band ferromagnetic ground state, 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 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 a finite staggered on-site energy term. Finally, we also provide some additional results for an Z_2 interacting topological insulator on a honeycomb lattice derived within the bosonization scheme. In this case, our findings indicate a gapped spin wave spectrum.