  We present a gauge-invariant effective action for the Abelian-Higgs model in
1+1 dimensions. It is constructed by integrating out the gauge field and then
using the hopping parameter expansion. The latter is tested with Monte Carlo
simulations for small values of the scalar self-coupling. In the opposite
limit, at infinitely large self-coupling, the Higgs mode is frozen and the
partition function can be written in terms of local tensors and the tensor
renormalization group blocking can be applied. The numerical implementation
requires truncations and the time continuum limit of the blocked transfer
matrix can be obtained numerically. At zero gauge coupling 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. The procedure is extended to finite gauge coupling and we derive a
spin-1 approximation of the Hamiltonian which involves terms corresponding to
transitions among the two species in the Bose-Hubbard model. An optical lattice
implementation involving a ladder structure is proposed.
