Surface critical properties of the three-dimensional clock model

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