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arXiv:1709.08368 · Physical review. E

Ferromagnetic Potts models with multisite interaction

Nir Schreiber, Reuven Cohen C, Simi Haber

Published Mar 8, 2018• 11 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreve.97.032106

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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.
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.
We use Monte Carlo method to study three-state Potts model on maple leaf lattice with pure three-site interaction. The critical behavior of both ferromagnetic and antiferromagnetic cases is studied. Our results confirm that the critical behavior of the ferromagnetic model is independent of the lattice details and lies in the universality class of the three-state ferromagnetic Potts model. For the antiferromagnetic case the transition is of the first order. We have calculated the energy jump and critical temperature in this area. We find there is a tricritical point separating the first order and second order phases for this system.
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.
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.
We study three-state Potts spins on a square lattice, in which all bonds are ferromagnetic along one of the lattice directions, and antiferromagnetic along the other. Numerical transfer-matrix methods are used on infinite strips of width L sites, 4 ⩽ L ⩽ 14. Based on the analysis of the ratio of scaled mass gaps (inverse correlation lengths) and scaled domain-wall free energies, we provide strong evidence that a critical (Kosterlitz–Thouless) phase is present, whose upper limit is, in our best estimate, T c = 0.29 ± 0.01. From an analysis of the (extremely anisotropic) nature of excitations below T c , we argue that the critical phase extends all the way down to T = 0. While domain walls parallel to the ferromagnetic direction are soft for the whole extent of the critical phase, those along the antiferromagnetic direction seem to undergo a softening transition at a finite temperature. Assuming a bulk correlation length varying, for T > T c , as ξ( T ) = a ξ exp[ b ξ ( T − T c ) −σ ], σ ≃ 1/2, we attempt finite-size scaling plots of our finite-width correlation lengths. Our best results are for T c = 0.50 ± 0.01. We propose a scenario in which such inconsistency is attributed to the extreme narrowness of the critical region.