Asymptotic symmetry: Enhancement and stability

The scaling-limit symmetry of Zq-invariant spin systems is studied by renormalization-group methods based on qualitative bifurcation theory rather than a series expansion in. Under the plausible assumption of the absence of secondary bifurcations, it is shown that an ordinary critical phase with q 2 or 4 must have full SO2 invariance in the scaling limit; moreover, any such asymptotically isotropic critical phase is stable under Zq-invariant perturbations for q>4. Previously obtained results about two-dimensional Kosterlitz-Thouless phases for clock models have a natural interpretation within this group-theoretic bifurcation analysis.