  By using a modulated magnetic field in a Feshbach resonance for ultracold
fermionic atoms in optical lattices, we show that it is possible to engineer a
class of models usually referred to as correlated-hopping models. These models
differ from the Hubbard model in exhibiting additional density-dependent
interaction terms that affect the hopping processes. In addition to the
spin-SU(2) symmetry, they also possess a charge-SU(2) symmetry, which opens the
possibility of investigating the $\eta$-pairing mechanism for superconductivity
introduced by Yang for the Hubbard model. We discuss the known solution of the
model in 1D (where $\eta$ states have been found in the degenerate manifold of
the ground state) and show that, away from the integrable point, quantum Monte
Carlo simulations at half filling predict the emergence of a phase with
coexisting incommensurate spin and charge order.
