Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Progress of Theoretical and Experimental Physics

Phase transitions of the four-state Potts model on the square and honeycomb lattices

Jhao-Hong Peng, Fu-Jiun Jiang

Published Dec 22, 2023• 4 citations
Publisher

Abstract

Abstract It is widely believed that the phase transition for the four-state ferromagnetic Potts model on the square lattice is of pseudo-first order. Specifically, it is expected that first-order phase transition behavior is found on small lattices and that the true nature of the second-order phase transition only emerges with large system sizes. It is also intuitively expected that, for other geometries, the types of the associated phase transitions should be identical to that of the square lattice. However, after simulating more than 16 million spins for the four-state Potts model, we observe that a feature of the first-order phase transition persists on the square lattice. Additionally, a characteristic of the second-order phase transition already appears on a small honeycomb lattice. Indications of a pseudo-first-order phase transition were not found in our investigation. This suggests that a thorough analytic calculation may be required to develop a better understanding of the presented results.

DOI
10.1093/ptep/ptad158

Related papers

Unexpected results of the phase transitions of four-state Potts model on the square and the honeycomb lattices

Jhao-Hong Peng, Fu-Jiun Jiang

It is widely believed that the phase transition for the four-state ferromagnetic Potts model on the square lattice is of the pseudo-first order. Specifically, it is expected that first-order phase transition behavior is found on small lattices and that the true nature of second-order phase transition only emerges with large system sizes. It is also intuitively expected that for other geometries, the types of the associated phase transitions should be identical to that of the square lattice. However, after simulating more than 16 million spins for the four-state Pott model, we observe that a feature of first-order phase transition persists on the square lattice. Additionally, a characteristic of second-order phase transition already appears on a small honeycomb lattice. Indications of a pseudo-first-order phase transition were not found in our investigation. This suggests that a thorough analytic calculation may be required to develop a better understanding of the presented results.

arXiv:2212.12272•Dec 23, 2022•0 citations

A Bond-Weighted Tensor Renormalization Group Study of the q-State Ferromagnetic Potts Models on the Square Lattice

Yuan-Heng Tseng, Shang-Wei Li, Fu-Jiun Jiang

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 $q=4$. However, some observables of the $q=4$ 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 $C_v$ 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 $q=2$ and $q=5$ 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 $q=4$ 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.

Progress of Theoretical and Experimental Physics•Jun 14, 2025•3 citations

Phase Transitions and Critical Phenomena in the 2D Potts Impurity Model on a Square Lattice

Akai Kurbanovich Murtazaev, Albert Babaevich Babaev, G. Ya. Ataeva +1 more

Abstract Phase transitions and critical behavior in the 2D dilute Potts model with number q = 4 of spin states on a square lattice are investigated by computer simulation. Systems with linear sizes L × L = N, l = 10–160 with spin concentrations p = 1.00, 0.95, 0.80 are considered. It is shown that the introduction of magnetic impurities plays a stabilizing role in the realization of a second-order phase transition in the Potts model under investigation. Based on the finite-size scaling theory, statistical critical indices of heat capacity α, susceptibility γ, magnetization β, and correlation radius index ν are calculated for the range of concentrations p considered here.

Journal of Experimental and Theoretical Physics•Sep 1, 2022•4 citations

Phase transitions in the two-dimensional ferro- and antiferromagnetic potts models on a triangular lattice

Akai Kurbanovich Murtazaev, Albert Babaevich Babaev

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.

Journal of Experimental and Theoretical Physics•Dec 1, 2012•17 citations

Monte Carlo study of the phase transitions in 2D ferro- and antiferromagnetic Potts models on a triangular lattice

Akai Kurbanovich Murtazaev, Albert Babaevich Babaev

The phase transitions in 2D ferro- and antiferromagnetic Potts models with number of spin states q = 3 on a triangular lattice are investigated by the cluster and classical Monte Carlo methods. Systems with linear sizes L = 20–120 are considered. Fourth-order Binder cumulants and histogram data analysis are used to show that second- and first-order phase transitions are observed in the ferromagnetic and antiferromagnetic Potts models, respectively. The static critical indices are calculated for specific heat α, susceptibility γ, magnetization β, and correlation length ν on the basis of finite-size scaling theory for a ferromagnetic Potts model.

Bulletin of the Russian Academy of Sciences Physics•Jan 1, 2014•2 citations

Ferromagnetic Potts models with multisite interaction

Nir Schreiber, Reuven Cohen C, Simi Haber

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≤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/(θlnq), where θ>1 is a q-dependent geometrical term, is an improved upper bound on T_{c}. In fact, our findings support T_{c}=1/(θlnq). 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.

arXiv:1709.08368•Physical review. E•Mar 8, 2018•11 citations
Abstract It is widely believed that the phase transition for the four-state ferromagnetic Potts model on the square lattice is of pseudo-first order. Specifically, it is expected that first-order phase transition behavior is found on small lattices and that the true nature of the second-order phase transition only emerges with large system sizes. It is also intuitively expected that, for other geometries, the types of the associated phase transitions should be identical to that of the square lattice. However, after simulating more than 16 million spins for the four-state Potts model, we observe that a feature of the first-order phase transition persists on the square lattice. Additionally, a characteristic of the second-order phase transition already appears on a small honeycomb lattice. Indications of a pseudo-first-order phase transition were not found in our investigation. This suggests that a thorough analytic calculation may be required to develop a better understanding of the presented results.
It is widely believed that the phase transition for the four-state ferromagnetic Potts model on the square lattice is of the pseudo-first order. Specifically, it is expected that first-order phase transition behavior is found on small lattices and that the true nature of second-order phase transition only emerges with large system sizes. It is also intuitively expected that for other geometries, the types of the associated phase transitions should be identical to that of the square lattice. However, after simulating more than 16 million spins for the four-state Pott model, we observe that a feature of first-order phase transition persists on the square lattice. Additionally, a characteristic of second-order phase transition already appears on a small honeycomb lattice. Indications of a pseudo-first-order phase transition were not found in our investigation. This suggests that a thorough analytic calculation may be required to develop a better understanding of the presented results.
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 Phase transitions and critical behavior in the 2D dilute Potts model with number q = 4 of spin states on a square lattice are investigated by computer simulation. Systems with linear sizes L × L = N, l = 10–160 with spin concentrations p = 1.00, 0.95, 0.80 are considered. It is shown that the introduction of magnetic impurities plays a stabilizing role in the realization of a second-order phase transition in the Potts model under investigation. Based on the finite-size scaling theory, statistical critical indices of heat capacity α, susceptibility γ, magnetization β, and correlation radius index ν are calculated for the range of concentrations p considered here.
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.
The phase transitions in 2D ferro- and antiferromagnetic Potts models with number of spin states q = 3 on a triangular lattice are investigated by the cluster and classical Monte Carlo methods. Systems with linear sizes L = 20–120 are considered. Fourth-order Binder cumulants and histogram data analysis are used to show that second- and first-order phase transitions are observed in the ferromagnetic and antiferromagnetic Potts models, respectively. The static critical indices are calculated for specific heat α, susceptibility γ, magnetization β, and correlation length ν on the basis of finite-size scaling theory for a ferromagnetic Potts model.
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≤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/(θlnq), where θ>1 is a q-dependent geometrical term, is an improved upper bound on T_{c}. In fact, our findings support T_{c}=1/(θlnq). 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.