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