Fermions in optical lattices near a Feshbach resonance: from band insulator to Mott insulator

We study a model of an equal mixture of two species of fermions in a deep optical lattice at a filling of two fermions per site. At weak inter-species interaction, the system is a band insulator. When the inter-species interaction is tuned via a Feshbach resonance to be larger than an energy related to the energy separation of the first and second Bloch band, atoms populate equally the two Bloch bands. With weak tunneling between sites of the optical lattice, the system becomes a Mott insulator with the low energy effective Hamiltonian of a spin-1 Heisenberg antiferromagnet, because of a Hund's rule like coupling between the two bands. We discuss experimental signatures of these two types of insulators.