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arXiv:2510.15124 · Physical review. E

Cluster percolation and dynamical scaling in the Baxter-Wu model

Alexandros Vasilopoulos, M. Akritidis, Nikolaos G. Fytas, Martin Weigel

Published Jan 5, 2026• 1 citation
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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 multicluster and single-cluster algorithms resulting from the cluster construction, highlighting their efficiency and scaling behavior with system size.

DOI
10.1103/9425-hxtb

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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 multicluster and single-cluster algorithms resulting from the cluster construction, highlighting their efficiency and scaling behavior with system size.
We study the spatial overlap of successive spin configurations generated by Markov chain Monte Carlo simulations of the Baxter–Wu model. Using the Novotny–Evertz sublattice-freezing single-cluster update, we track the mean and variance of the algorithmic overlap across the critical region. We show that, even in this three-spin model, the overlap acts as an algorithmic observable that follows the thermodynamics of the transition: the single-cluster overlap mean behaves like an order parameter, dropping from a finite ordered-phase plateau toward zero across T c . The overlap does not diverge at criticality, instead it remains finite and its finite-size value decays as a clean power law, U 2 ( T c ) ∼ L − ψ , over eleven sizes with an exponent ψ = 0.378 ( 4 ) . The exponent is stable under removal of small and large sizes and under the correction-to-scaling terms known for this model, which enter with amplitudes consistent with zero. It differs by more than eight standard errors from the values found for the Ising, three-state Potts and four-state Potts models under standard Fortuin–Kasteleyn Wolff dynamics. The Novotny–Evertz clusters reproduce the susceptibility exponent γ / ν = 7 / 4 , which the Ising model shares, so the difference is not set by the static exponents or by the cluster-size exponent.
We apply generalizations of the Swendson-Wang and Wolff cluster algorithms, which are based on the construction of Fortuin-Kasteleyn clusters, to the three-dimensional ±1 random-bond Ising model. The behavior of the model is determined by the temperature T and the concentration p of negative (antiferromagnetic) bonds. The ground state is ferromagnetic for 0≤p<p_{c}, and a spin glass for p_{c}<p≤0.5 where p_{c}≃0.222. We investigate the percolation transition of the Fortuin-Kasteleyn clusters as a function of temperature for large system sizes up to N=200^{3} spins. Except for p=0 the Fortuin-Kasteleyn percolation transition occurs at a higher temperature than the magnetic ordering temperature. This was known before for p=1/2 but here we provide evidence for a difference in transition temperatures even for p arbitrarily small. Furthermore, for all values of p>0, our data suggest that the percolation transition is universal, irrespective of whether the ground state exhibits ferromagnetic or spin-glass order, and is in the universality class of standard percolation. This shows that correlations in the bond occupancy of the Fortuin-Kasteleyn clusters are irrelevant, except for p=0 where the clusters are strictly tied to Ising correlations so the percolation transition is in the Ising universality class.
The short-time behavior of the Baxter-Wu model is investigated through the relaxation of the order parameter at the critical temperature. We considered Monte Carlo simulations for this model on a triangular lattice, and we studied relaxation starting from the fourfold-degenerate ground state. Using the short-time scaling formalism we found the static critical exponents beta and nu of the model and the corresponding dynamical critical exponent z. The values of the static exponents we find agree with the exact ones. To the best of our knowledge, this is the first determination of the dynamical critical exponent of the Baxter-Wu model.
Abstract Using a Monte Carlo simulation and the single histogram reweighting technique, we study the critical behaviors and phase transitions of the Baxter–Wu (BW) model on a two-layer triangular lattice with Ising-type interlayer couplings. Via the finite-size analysis, we obtain the transition temperatures and critical exponents at repulsive and attractive interlayer couplings. The data for the repulsive interlayer coupling suggest continuous transitions, and the critical behaviors are the same as those of the 2D BW model, belonging to the four-state Potts universality class. The reduced energy cumulants and the histograms reveal that attractive coupling leads to weak first-order phase transitions. The pseudocritical exponents with the existence of the interlayer couplings indicate that the first-order transition is very close to the critical point of the 2D standard BW model.
Our research focuses on the creation of a 3D layered system model that adheres to a three-spin Baxter–Wu-like interaction. To address the phase transition problem, we use entropic sampling simulations to construct the density of states and analyze various quantities such as the total magnetization, average energy, magnetic susceptibility, energy probability distribution function, and inverse microcanonical temperature. Once we established that the system undergoes a first-order phase transition, we carried out a finite-size scaling procedure by carefully studying the critical behavior of the minimum temperature of cumulants for energy and magnetization. We also computed the pseudocritical temperature, Tc(L), which is the temperature where the energy density probability distribution has a double peak. Based on our analysis, we were able to obtain an estimation of the transition temperature as Tc≈3.94 and the latent heat value ΔE≈1.2 at this point.
We study the percolation properties of geometrical clusters defined in the overlap space of two statistically independent replicas of a square-lattice Ising model that are simulated at the same temperature. In particular, we consider two distinct types of clusters in the overlap, which we dub soft- and hard-constraint clusters, and which are subsets of the regions of constant spin overlap. By means of Monte Carlo simulations and a finite-size scaling analysis we estimate the transition temperature as well as the set of critical exponents characterizing the percolation transitions undergone by these two cluster types. The results suggest that both soft- and hard-constraint clusters percolate at the critical temperature of the Ising model and their critical behavior is governed by the correlation-length exponent ν=1 found by Onsager. At the same time, they exhibit nonstandard and distinct sets of exponents for the average cluster size and percolation strength.