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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 $μ$ on a 1+1 dimensional lattice. This formulation provides an expansion in the hopping parameter $κ$ which we test with Monte Carlo simulations for a broad range of the inverse gauge coupling $β_{pl}$ and small values of the scalar self-coupling $λ$. In the opposite limit of infinitely large $λ$, the partition function can be written as a traced product of local tensors which allows us to write exact blocking formulas. Their numerical implementation requires truncations but there is no sign problem for arbitrary values of $μ$. We show that the time continuum limit of the blocked transfer matrix can be obtained numerically and, in the limit of infinite $β_{pl}$ and with 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 onsite repulsion. We extend this procedure for finite $β_{pl}$ 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.