  We discuss the U(1)-Higgs model in two dimensions in the strongly coupled
regime. If we neglect the plaquette interactions, we generate an effective
theory where link variables are integrated out, producing 4-field operators.
Plaquette interactions can be restored order by order as in recent calculations
with staggered fermions. In the case of a SU(2) gauge theory with fermions,
this strong coupling expansion can be related to the strong coupling expansion
of Fermi-Hubbard models possibly implementable on optical lattice. We would
like to provide a similar construction relating the U(1)-Higgs model to some
Bose-Hubbard model. As a first step in this direction, we discuss a recent
proposal to implement the O(2) model on optical lattices using a 87Rb and 41K
Bose-Bose mixture of cold atoms.
