| We study the possibility of realizing strong coupled many-body quantum phases in ultracold atomic systems. Motivated by recent experiments, we first analyze the phase diagram of a Bose-Fermi mixture across a Feshbach resonance, and offer an explanation for the collapse of the system observed close to the Feshbach resonance: we find that phase separation leads to a high density phase which causes the collapse. We then focus on the recent attempts to realize the three-dimensional fermionic Hubbard model in an optical lattice. One milestone on the experimentalists' agenda is to access the antiferromagnetic ordered Neel phase, which has so far been hindered by the low ordering temperature. We ask which experimental parameters maximize the antiferromagnetic interactions, which set the scale for the ordering temperature. We find that the maximum is obtained in a regime where the effective Hamiltonian describing the system no longer corresponds to a simple one-band Hubbard model, and we characterize the physics of the system in this regime. The final system we consider is mass imbalanced polarized two-component Fermi gases interacting via a Feshbach resonance. By going to the strongly polarized limit, we use a recently developed method to obtain results which have been shown to be accurate in the mass balanced case, and we find an intriguing set of competing phases in this limit. We discuss what these results imply for the full phase diagram. |
