  We propose to go beyond the usual Hubbard model description of atoms in
optical lattices and show how few-body physics can be used to simulate
many-body phenomena, e.g., an electron-phonon system. We take one atomic
species to be trapped in a deep optical lattice at full filling and another to
be untrapped spin-polarized fermions (which do not see the optical lattice) but
has an s-wave contact interaction with the first species. For large positive
scattering length on the order of lattice spacing, the usual two-body bound
(dimer) states overlap forming giant orbitals extending over the entire
lattice, which can be viewed as an "electronic" band for the untrapped species
while the trapped atoms become the "ions" with their own on-site dynamics,
thereby simulating an electron-phonon system with renormalization of the phonon
frequencies and Peierls transitions. This setup requires large scattering
lengths but minimises losses, does not need higher bands and adds new degrees
of freedom which cannot easily be described in terms of lattice variables, thus
opening up intriguing possibilities to explore interesting physics at the
interface between few-body and many-body systems.
