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We study two models realized by two-component Fermi gases loaded in opticallattices. We clarify that multi-band effects inevitably caused by the opticallattices generate a rich structure, when the systems crossover from the regionof weakly bound molecular bosons to the region of strongly bound atomic bosons.Here the crossover can be controlled by attractive fermion interaction. One ofthe present models is a case with attractive fermion interaction, where aninsulator-superfluid transition takes place. The transition is characterized asthe transition between a band insulator and a Bose-Einstein condensate (BEC)superfluid state. Differing from the conventional BCS superfluid transition,this transition shows unconventional properties. In contrast to the oneparticle excitation gap scaled by the superfluid order parameter in theconventional BCS transition, because of the multi-band effects, a large gap ofone-particle density of states is retained all through the transition althoughthe superfluid order grows continuously from zero. A reentrant transition withlowering temperature is another unconventionality. The other model is the casewith coexisting attractive and repulsive interactions. Within a mean fieldtreatment, we find a new insulating state, an orbital ordered insulator. Thisinsulator is one candidate for the Mott insulator of molecular bosons and isthe first example that the orbital internal degrees of freedom of molecularbosons appears explicitly. Besides the emergence of a new phase, a coexistingphase also appears where superfluidity and an orbital order coexist just bydoping holes or particles. The insulating and superfluid particles showdifferentiation in momentum space as in the high-Tc cuprate superconductors.

Transition from Band insulator to Bose-Einstein Condensate superfluid and Mott State of Cold Fermi Gases with Multiband Effects in Optical Lattices