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Motivated by exploring the effect of interactions on Chern insulators, and byrecent experiments realizing topological bands for ultracold atoms in syntheticgauge fields, we study the honeycomb lattice Haldane-Hubbard model ofspin-$1/2$ fermions. Using an unrestricted mean field approach, we map out theinstability of the topological band insulator towards magnetically orderedinsulators which emerge with increasing Hubbard repulsion. In addition to thetopological N\'eel phase, we recover various chiral noncoplanar magnetic orderspreviously identified within a strong coupling approach. We compute the bandstructure of these ordered phases, showing that the triple-$Q$ tetrahedralphase harbors topological Chern bands with large Chern numbers.

Mean field study of the topological Haldane-Hubbard model of spin-$1/2$ fermions