Surface critical properties of the three-dimensional clock model

Using Monte Carlo simulations and finite-size scaling analysis, we show that the $q$-state clock model with $q=6$ on the simple cubic lattice with open surfaces has a rich phase diagram; in particular, it has an extraordinary-log phase, besides the ordinary and extraordinary transitions at the bulk critical point. We prove numerically that the presence of the intermediate extraordinary-log phase is due to the emergence of an O(2) symmetry in the surface state before the surface enters the $Z_{q}$ symmetry-breaking region as the surface coupling is increased at the bulk critical point, while O(2) symmetry emerges for the bulk. The critical behaviors of the extraordinary-log transition, as well as the ordinary and the special transition separating the ordinary and the extraordinary-log transition are obtained.