Non-compact lattice Higgs model with Abelian discrete gauge groups: phase diagram and gauge symmetry enlargement

We study the phase diagram and phase transitions of the three dimensional multicomponent lattice Higgs model with non-compact Abelian discrete groups. The model with non-compact U(1) gauge group is known to undergo, for a sufficiently large number of scalar fields $N$, a continuous transition associated to the charged fixed point of the continuous Abelian Higgs field theory. We show that in the model with gauge group ${\mathbb{Z}}_q^{(nc)}\equiv 2\pi \mathbb{Z}\mathrm{/}q$ only critical transitions in the orthogonal universality classes are present for small values of $N$, while a symmetry enlargement to the continuous Abelian Higgs universality class happens when $q\ge 5$ and $N$ is large enough.