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Critical Phenomena in a Three-Layer Coupled q=3 Potts Model on the Hexagonal Lattice: Discovery of a New Universality Class with η > 1

坤晉 江

Published Mar 17, 2026• 0 citations
Publisher

Abstract

We report Monte Carlo simulations of a three-layer coupled q=3 Potts model on the hexagonal lattice (z=6) using Metropolis single-spin-flip dynamics. The model features intra-layer coupling β, inter-layer coupling α, and a quenched memory field w that explicitly breaks Z₃ symmetry. Through finite-size scaling analysis of system sizes L = 9, 18, 27, 36, 54, 81, 96, and 128, we extract a set of critical exponents centered on the three-body order parameter C₃. The anomalous dimension η ≈ 1.41 and effective correlation dimension D_eff = 2 − η ≈ 0.59 fall outside all known two-dimensional universality classes, suggesting a new universality class driven by the irreducible three-body coupling. Five independent scaling laws (Fisher, Josephson, Rushbrooke, Widom, hyperscaling) are satisfied to residuals < 10⁻³. L = 96 and L = 128 data are preliminary; high-statistics runs are ongoing.

DOI
10.5281/zenodo.19058772

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We report Monte Carlo simulations of a three-layer coupled q=3 Potts model on the hexagonal lattice (z=6) using Metropolis single-spin-flip dynamics. The model features intra-layer coupling β, inter-layer coupling α, and a quenched memory field w that explicitly breaks Z₃ symmetry. Through finite-size scaling analysis of system sizes L = 9, 18, 27, 36, 54, 81, 96, and 128, we extract a set of critical exponents centered on the three-body order parameter C₃. The anomalous dimension η ≈ 1.41 and effective correlation dimension D_eff = 2 − η ≈ 0.59 fall outside all known two-dimensional universality classes, suggesting a new universality class driven by the irreducible three-body coupling. Five independent scaling laws (Fisher, Josephson, Rushbrooke, Widom, hyperscaling) are satisfied to residuals < 10⁻³. L = 96 and L = 128 data are preliminary; high-statistics runs are ongoing.
Computer simulation of the three-component Potts model on a hexagonal lattice was carried out using the Monte Carlo method. Systems with linear dimensions L × L = N, L = 10–320 in units of interatomic distances are considered. Based on the theory of finite-size scaling, the static critical exponents of heat capacity α, susceptibility γ, magnetization β, and correlation radius ν are calculated. The data we obtained confirm that in the considered Potts model on a hexagonal lattice, a second-order phase transition is observed with critical exponents corresponding to the universality class of the three-component Potts model.
Abstract Computer simulation of the three-component Potts model on a hexagonal lattice was carried out using the Monte Carlo method. Systems with linear dimensions L × L = N, L = 20–320 in units of interatomic distances are considered. Based on the theory of finite-size scaling, the static critical exponents of heat capacity α, susceptibility γ, magnetization β, and correlation radius ν are calculated. The data we obtained confirm that in the considered Potts model on a hexagonal lattice, a second-order phase transition is observed with critical exponents corresponding to the universality class of the three-component Potts model.
Scaling, universality, and renormalization are three pillars of modern critical phenomena. According to the hypothesis of universality, continuous phase transitions fall into classes mainly determined by spatial dimensionality and symmetry of the order parameter. The latter is usually reflected by the degeneracy of the ground state of the Hamiltonian. However, for certain systems at criticality, a higher symmetry emerges in the order parameter, and the associated critical behavior may become very rich. Such emergent symmetry has been found in spin ice systems, deconfined quantum critical points, high-Tc{T}_{c}Tc​ superconductors, and so forth, and are often accompanied by very interesting critical phenomena. New results presented here are based on Monte Carlo simulations and finite-size scaling of a series of qqq-state Potts models on the simple cubic lattice with ferromagnetic interactions in one lattice direction and antiferromagnetic interactions in the other two directions. The staggered magnetization appears to display an emergent continuous O(nnn) symmetry with nnn=qqq-1, as illustrated by two-dimensional intersections of the distribution functions. Also the estimated critical exponents are consistent with the O(nnn=qqq-1) universality classes.
We study the two-dimensional anisotropic three-state Potts ferromagnet on the square lattice using Wolff single-cluster Monte Carlo simulations and finite-size scaling. For coupling ratios λ=Jy/Jx=0.5λ=J_y/J_x=0.5λ=Jy​/Jx​=0.5, 0.750.750.75, and 111, finite-size scaling of the correlation ratio yields critical behavior consistent with the two-dimensional three-state Potts universality class. The scaling of the leading Fisher zeros gives a correlation-length exponent consistent with ν=5/6ν=5/6ν=5/6, while their cumulative density is consistent with the expected specific-heat exponent α=1/3α=1/3α=1/3. We further characterize the anisotropy at criticality using directional correlation ratios RxR_xRx​ and RyR_yRy​ together with directional FK wrapping probabilities. For λ=0.5λ=0.5λ=0.5 on a physically square lattice, RxR_xRx​ and RyR_yRy​ approach distinct critical values while yielding a common correlation-length exponent. From the wrapping probabilities, we independently determine an effective aspect ratio ρe□=0.6413(5)ρ_e^\square=0.6413(5)ρe□​=0.6413(5), in close agreement with the theoretical value ρe□,th≃0.64150030ρ_e^{\square,\mathrm{th}}\simeq0.64150030ρe□,th​≃0.64150030 obtained from the isoradial representation. Using the theoretical value to set the physical aspect ratio restores directional equivalence, Rx≃RyR_x\simeq R_yRx​≃Ry​, and brings the overall correlation ratio toward the isotropic-square reference. The results show that spatial anisotropy changes the effective critical geometry without altering the bulk three-state Potts universality class.
Phase transitions and critical properties in the antiferromagnetic Heisenberg model on a layered cubic lattice with allowance for intralayer next nearest neighbor interactions have been studied using the replica Monte Carlo algorithm. The character of phase transitions has been analyzed using the histogram method and the Binder cumulant method. It has been found that a transition from the collinear to paramagnetic phase in the model under study occurs as a second order phase transition. The statistical critical exponents of the specific heat α, susceptibility γ, order parameter β, and correlation radius ν, as well as the Fisher index η, have been calculated using the finite-size scaling theory. It has been shown that the three-dimensional Heisenberg model on the layered cubic lattice with allowance for the next nearest neighbor interaction belongs to the same universality class of the critical behavior as the antiferromagnetic Heisenberg model on a layered triangular lattice.
The influence of quenched nonmagnetic impurities on phase transitions in the three-dimensional Potts model with the number of spin states q = 3 is investigated using the Wolff single-cluster algorithm of the Monte Carlo method. The systems with linear sizes L = 20–44 at the spin concentrations p = 1.0, 0.9, 0.8, and 0.7 are analyzed. It is demonstrated with the use of the method of fourth-order Binder cumulants that the second-order phase transition occurs in the model under consideration at the spin concentrations p = 0.9, 0.8, and 0.7 and that the first-order phase transition is observed in the pure model ( p = 1.0). The static critical exponents α (heat capacity), γ (susceptibility), β (magnetization), and ν (correlation length) are calculated in the framework of the finite-size scaling theory. The problem regarding the universality classes of the critical behavior of weakly diluted systems is discussed.