Surface critical properties of the three-dimensional clock model

Surface effects are significant at the bulk critical point due to divergence of correlation length. Recently, a novel extraordinary-log surface behavior was proposed in 3D O($N$) model. Here, authors study properties $q$-state clock model, where ${Z}_{q}$ anisotropy is prototype dangerously irrelevant perturbation. They identify existence an phase between ordinary and extraordinary phases as consequence emergent O(2) symmetry.