We study Hubbard models for ultracold bosonic or fermionic atoms loaded into an optical lattice. The atoms carry a high spin $F>1/2$, and interact on site via strong repulsive Van der Waals forces. Making convenient rearrangements of the interaction
terms, and exploiting their symmetry properties, we derive low energy effective models with nearest-neighbor interactions, and their properties. We apply our method to $F=3/2$, and 5/2 fermions on two-dimensional square lattice at quarter, and 1/6 fillings, respectively, and investigate mean-field equations for repulsive couplings. We find for $F=3/2$ fermions that the plaquette state appearing in the highly symmetric SU(4) case does not require 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 for repulsive interactions in absence of external magnetic field. The SU(2) flux state has, however, lower energy than the plaquette phase, and stabilizes in the presence of weak applied magnetic field. For $F=5/2$ fermions a similar SU(2) plaquette phase is found to be the ground state without external magnetic field.
Two-component fermionic superfluids on a lattice with an external non-Abelian gauge field give access to a variety of topological phases in presence of a sufficiently large spin imbalance. We address here the important issue of superfluidity breakdow
n induced by spin imbalance by a self-consistent calculation of the pairing gap, showing which of the predicted phases will be experimentally accessible. We present the full topological phase diagram, and we analyze the connection between Chern numbers and the existence of topologically protected and non-protected edge modes. The Chern numbers are calculated via a very efficient and simple method.
Ultra-cold atomic systems provide a new setting where to investigate the role of long-range interactions. In this paper we will review the basics features of those physical systems, in particular focusing on the case of Chromium atoms. On the exper
imental side, we report on the observation of dipolar effects in the expansion dynamics of a Chromium Bose-Einstein condensate. By using a Feshbach resonance, the scattering length characterising the contact interaction can be strongly reduced, thus increasing the relative effect of the dipole-dipole interaction. Such experiments make Chromium atoms the strongest candidates at present for the achievement of the strong dipolar regime. On the theoretical side, we investigate the behaviour of ultra-cold dipolar systems in the presence of a periodic potential. We discuss how to realise this situation experimentally and we characterise the system in terms of its quantum phases and metastable states, discussing in detail the differences with respect to the case of zero-range interactions.
