Surface criticality of the antiferromagnetic Potts model

Abstract

We study the three-state antiferromagnetic Potts model on the simple-cubic lattice, paying attention to the surface critical behaviors. When the nearest-neighboring interactions of the surface is tuned, we obtain a phase diagram similar to the XY model, owing to the emergent O(2) symmetry of the bulk critical point. For the ordinary transition, we get ${y}_{h1}=0.780(3), {\ensuremath{\eta}}_{\ensuremath{\parallel}}=1.44(1)$, and ${\ensuremath{\eta}}_{\ensuremath{\perp}}=0.736(6)$; for the special transition, we get ${y}_{s}=0.59(1), {y}_{h1}=1.693(2), {\ensuremath{\eta}}_{\ensuremath{\parallel}}=\ensuremath{-}0.391(4)$, and ${\ensuremath{\eta}}_{\ensuremath{\perp}}=\ensuremath{-}0.179(5)$; in the extraordinary-log phase, the surface correlation function ${C}_{\ensuremath{\parallel}}(r)$ decays logarithmically with decaying exponent $q=0.60(2)$, however, the correlation ${C}_{\ensuremath{\perp}}(r)$ still decays algebraically with critical exponent ${\ensuremath{\eta}}_{\ensuremath{\perp}}=\ensuremath{-}0.442(5)$. If the ferromagnetic next-nearest-neighboring surface interactions are added, we find two transition points, the first one is a special point between the ordinary phase and the extraordinary-log phase, the second one is a transition between the extraordinary-log phase and the ${Z}_{6}$ symmetry-breaking phase, with critical exponent ${y}_{\mathrm{s}}=0.41(2)$. The scaling behaviors of the second transition is very interesting, the surface spin-correlation function ${C}_{\ensuremath{\parallel}}(r)$, and the surface squared staggered magnetization at this point decays logarithmically with exponent $q=0.37(1)$; however, the surface structure factor with the smallest wave vector and the correlation function ${C}_{\ensuremath{\perp}}(r)$ satisfy power-law decaying, with critical exponents ${\ensuremath{\eta}}_{\ensuremath{\parallel}}=\ensuremath{-}0.69(1)$ and ${\ensuremath{\eta}}_{\ensuremath{\perp}}=\ensuremath{-}0.37(1)$, respectively.