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

Abstract

We present a gauge-invariant effective action for the Abelian-Higgs model (scalar electrodynamics) with a chemical potential $\ensuremath{\mu}$ on a ($1+1$)-dimensional lattice. This formulation provides an expansion in the hopping parameter $\ensuremath{\kappa}$ which we test with Monte Carlo simulations for a broad range of the inverse gauge coupling ${\ensuremath{\beta}}_{pl}=1/{g}^{2}$ and small values of the scalar self-coupling $\ensuremath{\lambda}$. In the opposite limit of infinitely large $\ensuremath{\lambda}$, 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 $\ensuremath{\mu}$ 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 ${\ensuremath{\beta}}_{pl}$ 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.