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Based on consideration of the system symmetry and its Hilbert space, we showthat strongly interacting fermions in an optical lattice or superlattice can begenerically described by a lattice resonance Hamiltonian. The latter can bemapped to a general Hubbard model with particle assisted tunneling rates. Weinvestigate the model under population imbalance and show the attractive andthe repulsive models have the same complexity in phase diagram under theparticle-hole mapping. Using this mapping, we propose an experimental method todetect possible exotic superfluid/magnetic phases for this system.

General Hubbard model for strongly interacting fermions in an optical lattice and its phase detection