  We connect explicitly the classical $O(2)$ model in 1+1 dimensions, a model
sharing important features with $U(1)$ lattice gauge theory, to physical models
potentially implementable on optical lattices and evolving at physical time.
Using the tensor renormalization group formulation, we take the time continuum
limit and check that finite dimensional projections used in recent proposals
for quantum simulators provide controllable approximations of the original
model. We propose two-species Bose-Hubbard models corresponding to these finite
dimensional projections at strong coupling and discuss their possible
implementations on optical lattices using a $^{87}$Rb and $^{41}$K Bose-Bose
mixture.
