Ferromagnetic Potts models with multisite interaction

We study the $q$-state Potts model with four-site interaction on a square lattice. Based on the asymptotic behavior of lattice animals, it is argued that when $q\le 4$ the system exhibits a second-order phase transition and when $q>4$ the transition is first order. The $q=4$ model is borderline. We find $1/lnq$ to be an upper bound on $T_c$, the exact critical temperature. Using a low-temperature expansion, we show that $1/(\theta lnq)$, where $\theta >1$ is a $q$-dependent geometrical term, is an improved upper bound on $T_c$. In fact, our findings support $T_c=1/(\theta lnq)$. This expression is used to estimate the finite correlation length in first-order transition systems. These results can be extended to other lattices. Our theoretical predictions are confirmed numerically by an extensive study of the four-site interaction model using the Wang-Landau entropic sampling method for $q=3,4,5$. In particular, the $q=4$ model shows an ambiguous finite-size pseudocritical behavior.

Figure 1

Portion of the square lattice showing a graph $G$ with $c\left(G\right)=4$ monochromatic clusters, $f\left(G\right)=18$ faces (colored squares), and $\nu \left(G\right)=43$ nodes residing in the corners of these squares. The three different colors (also denoted by c,g,o) represent a model with $q\ge 3$.

Figure 2

Variation of the specific heat of each model against temperature for $L=44$. While the $q=4$ model and especially the $q=5$ model display sharp and narrow peaks at the $q$-dependent position of the specific-heat maximum $T_L\left(q\right)$, the $q=3$ peak is an order of magnitude smaller and rather broad.

Figure 3

(a) Pseudocritical canonical energy distribution computed at $T_L\left(q\right)$ for $q=3,4,5$ and $L=44$. Note the peak width $1/L$ behavior when $q=5$, typical of normal distributions. Conversely, the distributions for the $q=4$ (and of course the $q=3)$ models are essentially not normal. (b) Scaling of the specific-heat maximum $c_L^{max}$ with $L$ on a log-log scale for $q=3\left(▴\right),q=4\left(•\right)$, and $q=5\left(\blacksquare \right)$.

Figure 4

Scaling of the position [temperature $T_L\left(q\right)]$ of the specific-heat maximum with $L$, for the three models: (a) $q=3$ and $T_L-T_c\propto L^{-1/\nu }{(lnL)}^{\alpha /\nu }$, (b) $q=4$ and $T_L-T_c\propto L^{-1/\nu +\mathcal{T}}$, and (c) $q=5$ and $T_L-T_c\propto L^{-d}$. Solid lines are presented to guide the eye.

Figure 5

Specific-heat universal scaling function $\mathcal{F}\left(x\right)$ for several lattice sizes $L$. The estimated values $T_c\left(4\right)\approx 0.689\left(9\right)$ and $\alpha /\nu \approx 1.832\left(7\right)$ are used in all the plots.

Figure 6

Reweighted PDF [24] (blue symbols) together with a double Gaussian fit for $L=44$. Note that the peaks are centered at points satisfying $P_L\left({\epsilon }_{-}\right)\approx q{P}_L\left({\epsilon }_{+}\right)$. The inset shows the difference between these points, as a function of $L^{-d}$ (closed squares). Absent error bars are smaller than the symbols. The estimated infinite volume $\mathrm{\Delta }{\epsilon }_{\infty }^{\mathrm{PDF}}\approx 0.813\left(9\right)$ is denoted by the closed circle. Lattices with $L<24$ have too noisy distributions around the peaks and are therefore omitted.