Emergence of U(1) symmetry in the 3D XY model with Zq anisotropy

We study the three-dimensional XY model with a Z(q) anisotropic term. At temperatures T < T-c this dangerously irrelevant perturbation is relevant only above a length scale Lambda, which diverges as a power of the correlation length, Lambda similar to xi(a)(q). Below Lambda the order parameter is U(1) symmetric. We derive the full scaling function controlling the emergence of U(1) symmetry and use Monte Carlo results to extract the exponent a(q) for q=4,...,8. We find that a(q)approximate to a(4)(q/4)(2), with a(4) only marginally larger than 1. We discuss these results in the context of U(1) symmetry at deconfined quantum-critical points separating antiferromagnetic and valence-bond-solid states in quantum spin systems.