Sublattice extraordinary-log phase and special points of the antiferromagnetic Potts model

We study the surface criticality of a three-state antiferromagnetic Potts model on a simple cubic lattice, whose bulk critical behavior belongs to the universality class of the XY model because of emergent O(2) symmetry. We find that the surface antiferromagnetic next-nearest-neighboring interactions can drive the extraordinary-log phase to an ordinary phase, and the transition between the two phases belongs to the universality class of the well-known special transition of the XY model. Further enhancing the surface next-nearest-neighboring interactions, the extraordinary-log phase reappears, but the main critical behaviors are shown in the sublattices; the special point between the ordinary phase and the sublattice extraordinary-log phase belongs to a different universality class.