Multitude of topological phase transitions in bipartite dice and Lieb lattices with interacting electrons and Rashba coupling

We report the results of a Hartree-Fock study applied to interacting electrons moving in two different bipartite lattices: the dice and the Lieb lattices, at half-filling. Both lattices develop ferrimagnetic order in the phase diagram $U$-$\lambda$, where $U$ is the Hubbard onsite repulsion and $\lambda$ the Rashba spin-orbit coupling strength. Our main result is the observation of an unexpected multitude of topological phases for both lattices. All these phases are ferrimagnetic, but they differ among themselves in their set of six Chern numbers (six numbers because the unit cells have three atoms). The Chern numbers $|C|$ observed in our study range from 0 to 3, showing that large Chern numbers can be obtained by the effect of electronic correlations, adding to the recently discussed methodologies to increase $|C|$ based on extending the hopping range in tight-binding models, using sudden quenches, or photonic crystals, all without including electronic interactions.