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We study Hubbard models for ultracold bosonic or fermionic atoms loaded intoan optical lattice. The atoms carry a high spin $F>1/2$, and interact on sitevia strong repulsive Van der Waals forces. Making convenient rearrangements ofthe interaction terms, and exploiting their symmetry properties, we derive lowenergy effective models with nearest-neighbor interactions, and theirproperties. We apply our method to $F=3/2$, and 5/2 fermions on two-dimensionalsquare lattice at quarter, and 1/6 fillings, respectively, and investigatemean-field equations for repulsive couplings. We find for $F=3/2$ fermions thatthe plaquette state appearing in the highly symmetric SU(4) case does notrequire fine tuning, and is stable in an extended region of the phase diagram.This phase competes with an SU(2) flux state, that is always suppressed forrepulsive interactions in absence of external magnetic field. The SU(2) fluxstate has, however, lower energy than the plaquette phase, and stabilizes inthe presence of weak applied magnetic field. For $F=5/2$ fermions a similarSU(2) plaquette phase is found to be the ground state without external magneticfield.

Exotic magnetic orders for high spin ultracold fermions