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Physical review. E

Two-dimensional Potts model with invisible states on a square lattice

Jae Hwan Lee, Wanki Park, J. M. Kim

Published May 22, 2025• 0 citations
Publisher

Abstract

We study the two-dimensional (q+r)-state ferromagnetic Potts model with q interacting visible states and r noninteracting invisible states on a square lattice. We concentrate on the case q=2 for various r values. The density of states is obtained by the Wang-Landau Monte Carlo simulation method, by which the specific heat, the partition function zeros, and the probability distribution of the internal energy are calculated. There exists a second-order phase transition for r<r_{c}, while a first-order phase transition appears for r>r_{c}, where r_{c} is the critical value. From all analyses, we conclude that r_{c}≈29 for q=2. Calculating the density of visible spins using the Metropolis algorithm, we find that the first-order phase transition is due to the abrupt density change of the visible spins. The critical temperatures are measured to compare with a conjecture.

DOI
10.1103/physreve.111.054131

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We study the two-dimensional (q+r)-state ferromagnetic Potts model with q interacting visible states and r noninteracting invisible states on a square lattice. We concentrate on the case q=2 for various r values. The density of states is obtained by the Wang-Landau Monte Carlo simulation method, by which the specific heat, the partition function zeros, and the probability distribution of the internal energy are calculated. There exists a second-order phase transition for r<r_{c}, while a first-order phase transition appears for r>r_{c}, where r_{c} is the critical value. From all analyses, we conclude that r_{c}≈29 for q=2. Calculating the density of visible spins using the Metropolis algorithm, we find that the first-order phase transition is due to the abrupt density change of the visible spins. The critical temperatures are measured to compare with a conjecture.
The q-state Potts model on a square lattice and in the presence of an external magnetic field that couples to only one state is studied by using extensive Monte Carlo simulations. A hybrid algorithm that combines the single spin-flip Metropolis method with Wolff cluster updates is employed. Single-histogram reweighting techniques and the field-mixing approach are also used to obtain the corresponding universal probability distributions. By analyzing the finite-size scaling behavior of the order parameters and universal probability distributions, the phase diagram in the temperature-versus-external field plane is obtained in the thermodynamic limit for some values of state q≥5. For positive fields and higher values of q, there is a clear first-order transition line that starts at the zero-field transition point and ends at an isolated critical point, which is in the two-dimensional Ising universality class. As the value of state q decreases, the corresponding critical field also decreases. The present results strongly suggest that the critical field vanishes when q→5^{+}. However, for negative fields, another first-order transition line occurs without any critical or multicritical behavior for q≥6. The special case with q=5 undergoes a weak first-order transition at zero field and a line of second-order transitions for negative fields. This second-order transition line is in the same universality class as the q=4 Potts model.
Phase transitions are ubiquitous phenomena, exemplified by the melting of ice and spontaneous magnetization of magnetic material. In general, a phase transition is associated with a symmetry breaking of a system; occurs due to the competition between coupling interaction and external fields such as thermal energy. If the phase transition occurs with no latent heat, the system experiences continuous transition, also known as second order phase transition. The ferromagnetic q-state Potts model with r extra invisible states, introduced by Tamura, Tanaka, and Kawashima [Prog. Theor. Phys. 124, 381 (2010)], is studied by using the Wang-Landau method. The density of states difference (DOSD), ln g(E + ΔE) − ln g(E), is used to investigate the order of the phase transition and examine the critical value of r changing the second to the first order transition.
The phase transitions in the two-dimensional ferro- and antiferromagnetic Potts models with q = 3 states of spin on a triangular lattice are studied using cluster algorithms and the classical Monte Carlo method. Systems with linear sizes L = 20–120 are considered. The method of fourth-order Binder cumulants and histogram analysis are used to discover that a second-order phase transition occurs in the ferromagnetic Potts model and a first-order phase transition takes place in the antiferromagnetic Potts model. The static critical indices of heat capacity (α), magnetic susceptibility (γ), magnetization (β), and correlation radius index (ν) are calculated for the ferromagnetic Potts model using the finite-size scaling theory.
Abstract It has been proven that the phase transition of the q-state ferromagnetic Potts model on the square lattice is of the second order for q=4q=4q=4. However, some observables of the q=4q=4q=4 model show features of a first-order phase transition. In particular, a recent large-scale Monte Carlo investigation of the 4-state Potts model observed that the two-peak structure of the energy density distribution becomes more noticeable when L increases. This finding suggests that the signal of coexistence of phases grows stronger with L. Motivated by these unusual critical behaviors, we study the energy density E and the specific heat CvC_vCv​ of the 4-state Potts model on the square lattice using the technique of bond-weighted tensor renormalization group (BTRG). As a comparison, we also investigate the q=2q=2q=2 and q=5q=5q=5 ferromagnetic Potts models on the square lattice using the same method. Remarkably, our results imply that there may be a small energy gap for the q=4q=4q=4 model. Although the appearance of the small energy gap can be explained and would disappear with a more sophisticated investigation, our finding suggests that further investigation is required to determine whether a message of a first-order phase transition is genuine or an artificial effect.
Abstract The Wang–Landau algorithm of the Monte Carlo method is used to study the magnetic structures of the ground state, phase transitions, and thermodynamic properties of the two-dimensional antiferromagnetic Potts model with number q = 4 of spin states in a hexagonal lattice with the interactions of the nearest (J1) and next-to-nearest (J2) neighbors. The exchange interaction ratios in the range of 0.0 ≤ r = |J2/J1| ≤ 1.0 are studied. A phase diagram of the dependence of the critical temperature on the exchange interaction value of the next-to-nearest neighbors is constructed. The nature of phase transitions is analyzed. It is found that a first-order phase transition is observed in the range of 0.1 ≤ r ≤ 1.0, while there is no phase transition in the system and a strong degeneracy of the ground state is observed at r = 0.0.