Surface criticality of the antiferromagnetic Potts model

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

Figure 1

Surface phase diagrams of the antiferromagnetic Potts model (1): (a) the phase diagram with NNN interactions $J_{\mathrm{s}}^{\prime }=0$; (b) the phase diagram with NN interactions $J_{\mathrm{s}}=1$.

Figure 2

Squared magnetization $m_{\mathrm{s}}^2$ and correlation ratio $\xi /L$ of the simple-cubic antiferromagnetic Potts model (1).

Figure 3

Surface squared magnetization $m_{\mathrm{s}1}^2$ and correlation ratio ${\xi }_1/L$ of the antiferromagnetic Potts model (1) with $T=T_c^{\mathrm{bulk}}=1.22603$ and the NNN surface interaction $J_{\mathrm{s}}^{\prime }=0$.

Figure 4

(a) Log-log plot of $m_{\mathrm{s}1}^2, C_{\parallel }(L/2)$, and $F_1$ versus $ln(L/L_0)$ for the antiferromagnetic Potts model (1) with $T=T_c^{\mathrm{bulk}}=1.22603, J_{\mathrm{s}}=5$, and $J_{\mathrm{s}}^{\prime }=0$; $L_0=1.54$ is taken for all three variables. (b) Log-log plot of the surface correlation $C_{\perp }(L/2)$.

Figure 5

Surface critical behaviors of the antiferromagnetic Potts model (1) with $T=T_c=1.22603$ and $J_{\mathrm{s}}=1$: (a) surface susceptibility ${\chi }_{\mathrm{s}1}$; (b) surface squared magnetization $m_{\mathrm{s}1}^2$; (c) surface structure factor $F_1$; and (d) the surface Binder ratio $Q_{\mathrm{s}1}$.

Figure 6

(a) Log-log plot of $F_1$ versus $ln(L/L_0)$ using the values of $L_0$ fit from $m_{s1}^2$, listed in Table 1; (b) log-log plot of $F_1$ versus $L$.

Figure 7

(a) $F_1{L}^{-1-{\eta }_{\parallel }}=F_1{L}^{0.31}$ versus $J_{\mathrm{s}}^{\prime }$; the dashed line is the critical point; (b) data collapse: $F_1{L}^{0.31}$ versus $(J_{\mathrm{s}}^{\prime }-0.815)L^{y_s}$ with $y_s=0.41$.

Figure 8

Histograms of the surface staggered magnetization of the antiferromagnetic Potts model (1); in all the cases, the strength of the surface NN interactions is $J_{\mathrm{s}}=1$.