Cluster percolation and dynamical scaling in the Baxter--Wu model

TL;DR

This study analyzes Fortuin–Kasteleyn–type clusters in the spin-$\tfrac{1}{2}$ Baxter–Wu model by freezing one sublattice to form an effective two-body problem on a honeycomb lattice, enabling FKCK cluster updates. It demonstrates that the resulting stochastic clusters percolate exactly at the thermal critical point $T_c$ and that cluster observables reproduce the Baxter–Wu universality class exponents, with $1/\nu \approx 1.50$, $\beta/\nu = 1/8$, and $\gamma/\nu = 7/4$. The authors also quantify the dynamical scaling of multi- and single-cluster algorithms via integrated autocorrelation times, finding $z\approx 1.16$ and $z\approx 1.25$ respectively, both substantially reducing critical slowing down relative to Metropolis updates. These findings validate the cluster approach for the Baxter–Wu model and suggest pathways for applying similar percolation analyses to related spin systems, including spin-1 variants and gonihedric-like models.

Abstract

We investigate the percolation behavior of Fortuin-Kasteleyn--type clusters in the spin-$1/2$ Baxter--Wu model with three-spin interactions on a triangular lattice. The considered clusters are constructed by randomly freezing one of the three sublattices, resulting in effective pairwise interactions among the remaining spins. Using Monte Carlo simulations combined with a finite-size scaling analysis, we determine the percolation temperature of these stochastic clusters and show that it coincides with the exact thermal critical point of the model. The critical exponents derived from cluster observables are consistent with those of the underlying thermal phase transition. Finally, we analyze the dynamical scaling of the multi-cluster and single-cluster algorithms resulting from the cluster construction, highlighting their efficiency and scaling behavior with system size.

Figures (7)

• Figure 1: Example of a $9 \times 9$ triangular lattice with the three sublattices distinguished by color. Each spin interacts with its nearest neighbors on the other two sublattices, illustrating the triplet-interaction structure characteristic of the BW model.
• Figure 2: Local mapping of the triangular lattice onto a honeycomb lattice by freezing one of the three sublattices. In the example shown, the spins on the frozen C sublattice lead to effective pair interactions between the spins on the A and B sublattices, denoted by $J^{\prime}$. For the bond $(0,2)$ the opposite spins on sublattice C are $\sigma_{\perp(0,2),+1} = \sigma_3$ and $\sigma_{\perp(0,2),-1} = \sigma_1$.
• Figure 3: Comparison of simulation results obtained using the Metropolis, single-cluster, and multi-cluster updates for the specific heat and magnetic susceptibility of the spin-$1/2$ BW model at various temperatures for a system of linear size $L=48$. The main panel shows the specific heat $C(T)$, while the inset displays the magnetic susceptibility $\chi_2(T)$. The excellent agreement between the methods confirms the correctness of the cluster-algorithm implementation.
• Figure 4: (a) Wrapping probability as a function of temperature for different system sizes. (b) Extrapolation of the percolation temperature $T_{\rm p}$ from the crossings of the wrapping probability curves according to Eq. \ref{['eq:scaling_crossing']}. The horizontal dashed line indicates the exact critical temperature $T_{\rm c}$ of the BW model. The inset shows the difference $T_{\rm p} - T_{\rm c}$ as a function of the minimum system size $L_{\rm min}$ included in the fits. The excellent agreement between the two temperatures supports identifying the percolation transition with the thermal phase transition.
• Figure 5: FSS of the derivatives of the wrapping probability evaluated at their respective maxima (main panel). The inset in the top left shows the temperature dependence of the derivative of the wrapping probability for several system sizes. The bottom-right inset displays the estimated exponent $1/\nu$ as a function of $1/L_{\rm min}$ for fits including corrections and a free $\omega$ parameter; the horizontal dashed line marks the exact value $1/\nu = 3/2$.