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Microscopic model for Feshbach interacting fermions in an optical lattice with arbitrary scattering length and resonance width

We numerically study the problem of two fermions in a three dimensional optical lattice interacting via a zero-range Feshbach resonance, and display the dispersions of the bound states as a two-particle band structure with unique features compared to typical single-particle band structures. We show that the exact two-particle solutions of a projected Hamiltonian may be used to define an effective two-channel, few-band model for the low energy, low density physics of many fermions at arbitrary s-wave scattering length. Our method applies to resonances of any width, and can be adapted to multichannel situations or higher-$\ell$ pairing. In strong contrast to usual Hubbard physics, we find that pair hopping is significantly altered by strong interactions and the presence of the lattice, and the lattice induces multiple molecular bound states.