Ferromagnetism and spin excitations in topological Hubbard models with a flat band

We study the spin-1 excitation spectra of the flat-band ferromagnetic phases in interacting topological insulators. As a paradigm, we consider a quarter-filled square lattice Hubbard model whose free part is the $\pi$-flux state with topologically nontrivial and nearly flat electron bands, which can realize either the Chern or $Z_2$ Hubbard model. By using the numerical exact diagonalization method with a projection onto the nearly flat band, we obtain the ferromagnetic spin-1 excitation spectra for both the Chern and $Z_2$ Hubbard models, consisting of spin waves and Stoner continuum. The spectra exhibit quite distinct dispersions for both cases; in particular the spin wave is gapless for the Chern Hubbard model, while gapped for the $Z_2$ Hubbard model. Remarkably, in both cases, the nonflatness of the free electron bands introduces dips in the lower boundary of the Stoner continuum. It significantly renormalizes the energies of the spin waves around these dips downward and leads to roton-like spin excitations. We elaborate that it is the softening of the roton-like modes that destabilizes the ferromagnetic phase, and determine the parameter region where the ferromagnetic phase is stable.