Gauge-invariant implementation of the Abelian-Higgs model on optical lattices

We present a gauge-invariant effective action for the Abelian-Higgs model (scalar electrodynamics) with a chemical potential on a (1+1) -dimensional lattice. This formulation provides an expansion in the hopping parameter which we test with Monte Carlo simulations for a broad range of the inverse gauge coupling ₏₋=1/g^2 and small values of the scalar self-coupling. In the opposite limit of infinitely large, the partition function can be written as a traced product of local tensors which allows us to write exact blocking formulas. Gauss's law is automatically satisfied and the introduction of has consequences only if we have an external electric field, g^2=0 or an explicit gauge symmetry breaking. The time-continuum limit of the blocked transfer matrix can be obtained numerically and, for g^2=0 and a spin-1 truncation, the small volume energy spectrum is identical to the low energy spectrum of a two-species Bose-Hubbard model in the limit of large on-site repulsion. We extend this procedure for finite ₏₋ and derive a spin-1 approximation of the Hamiltonian. It involves new terms corresponding to transitions among the two species in the Bose-Hubbard model. We propose an optical lattice implementation involving a ladder structure.