  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 $y_{h1}=0.780(3)$,
$\eta_\parallel=1.44(1)$, and $\eta_\perp=0.736(6)$; for the special
transition, we get $y_s=0.59(1)$, $y_{h1}=1.693(2)$,
$\eta_\parallel=-0.391(4)$, and $\eta_\perp=-0.179(5)$; in the
extraordinary-log phase, the surface correlation function $C_\parallel(r)$
decays logarithmically, with decaying exponent $q=0.60(2)$, however, the
correlation $C_\perp(r)$ still decays algebraically, with critical exponent
$\eta_\perp=-0.442(5)$. 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 $y_{\rm s}=0.41(2)$. The
scaling behaviors of the second transition is very interesting, the surface
spin correlation function $C_\parallel(r)$ 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 $C_\perp(r)$ satisfy power-law decaying, with critical
exponents $\eta_\parallel=-0.69(1)$ and $\eta_\perp=-0.37(1)$, respectively.
