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We present a gauge-invariant effective action for the Abelian-Higgs model in1+1 dimensions. It is constructed by integrating out the gauge field and thenusing the hopping parameter expansion. The latter is tested with Monte Carlosimulations for small values of the scalar self-coupling. In the oppositelimit, at infinitely large self-coupling, the Higgs mode is frozen and thepartition function can be written in terms of local tensors and the tensorrenormalization group blocking can be applied. The numerical implementationrequires truncations and the time continuum limit of the blocked transfermatrix can be obtained numerically. At zero gauge coupling and with a spin-1truncation, the small volume energy spectrum is identical to the low energyspectrum of a two-species Bose-Hubbard model in the limit of large onsiterepulsion. The procedure is extended to finite gauge coupling and we derive aspin-1 approximation of the Hamiltonian which involves terms corresponding totransitions among the two species in the Bose-Hubbard model. An optical latticeimplementation involving a ladder structure is proposed.

Effective action for the Abelian-Higgs model for a gauge-invariant implementation on optical lattices