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 wellknown 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.