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arXiv:1801.02786 · Journal of the Physical Society of Japan

Detecting Signals of Weakly First-order Phase Transitions in Two-dimensional Potts Models

Shumpei Iino, Satoshi Morita, Naoki Kawashima, Anders W. Sandvik

Published Feb 28, 2019• 62 citations
arXivHTMLPDF

Abstract

We investigate the first-order phase transitions of the $q$-state Potts models with $q = 5, 6, 7$, and $8$ on the two-dimensional square lattice, using Monte Carlo simulations. At the very weakly first-order transition of the $q=5$ system, the standard data-collapse procedure for the order parameter, carried out with results for a broad range of system sizes, works deceptively well and produces non-trivial critical exponents different from the trivial values expected for first-order transitions. However, a more systematic study reveals significant drifts in the `pseudo-critical' exponents as a function of the system size. For this purpose, we employ two methods of analysis: the data-collapse procedure with narrow range of the system size, and the Binder-cumulant crossing technique for pairs of system sizes. In both methods, the estimates start to drift toward the trivial values as the system size used in the analysis exceeds a certain `cross-over' length scale. This length scale is remarkably smaller than the correlation length at the transition point for weakly first-order transitions, e.g., less than one tenth for $q=5$, in contrast to the naive expectation that the system size has to be comparable to or larger than the correlation length to observe the correct behavior. The results overall show that proper care is indispensable to diagnose the nature of a phase transition with limited system sizes.

DOI
10.7566/jpsj.88.034006

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A new kind of finite-size phenomenon in the five-state Potts model in two dimensions, a ``false'' ground state, was recently reported by Katznelson and Lauwers based on Monte Carlo studies of large lattices. We reexamine the behavior of this weakly first-order system by performing extensive Monte Carlo simulations, and demonstrate that the ``novel'' states previously observed in internal-energy-distribution histograms tend to disappear when runs of sufficient length are used. We present evidence that very strong critical slowing down and a pseudodivergent correlation length are responsible for the observed effects. Our additional analysis of finite-size effects in the ten-state Potts model on small lattices (L\ensuremath{\le}10) shows a crossover of scaling from ${L}^{\mathrm{\ensuremath{-}}d}$ behavior (observed earlier on larger lattices) to a nonanalytic form. This result is used to discuss the observed nonanalytic scaling of thermodynamic quantities in the five-state Potts model.
The Monte Carlo method is used to study phase transitions in the two-dimensional Potts model with the number of states spin q=3 on square and hexagonal lattices. Considered systems with linear dimensions Lx L=N, L=21/102. The obtained numerical data indicate that in the considered In the Potts model, a second-order phase transition is observed in accordance with the analytical theory. The Binder cumulants method of the fourth order determines the values of the critical points of the Potts model on different lattices. Keywords: Phase transitions, Potts model, Monte Carlo, thermodynamic parameters, critical temperature.
Monte Carlo simulations are performed for first-order phase-transition models. The three-dimensional three-state Potts model has a weak first-order transition. For this model we calculate the density of states on L3{\mathit{L}}^{3}L3 block lattices (L up to size 36) and obtain high-precision estimates for the leading partition-function zeros. The finite-size-scaling analysis of the first zero exhibits the expected convergence of the critical exponent \ensuremath{\nu} toward 1/D for large L; in particular, we find \ensuremath{\nu}=2.955(26) from our two largest lattices. Analysis of our specific-heat Cv{\mathit{C}}_{\mathit{v}}Cv​ data yields l=0.080 31(26) for the latent heat. Along another line of approach, we calculate the mass gap m=1/\ensuremath{\xi} (\ensuremath{\xi} is the correlation length) for cylindrical L2{\mathit{L}}^{2}L2Lz{\mathit{L}}_{\mathit{z}}Lz​ lattices (L up to 24 and Lz{\mathit{L}}_{\mathit{z}}Lz​=256). The finite size-scaling analysis of these results is also consistent with the convergence of \ensuremath{\nu} toward 1/D, but that the limiting value is 1/D is not yet conclusively established. Some theoretical arguments favor \ensuremath{\nu}\ensuremath{\rightarrow}0 in case of a first-order transition in a cylindrical ${\mathit{L}}^{\mathit{D}\mathrm{\ensuremath{-}}1}$\ensuremath{\infty} geometry. Therefore, we also applied our approach to the 2D ten-state Potts model, which is known to have a strong first-order transition. In this case we find unambiguous evidence in favor of 1/D as the limiting value.
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.
It is known that finite size scaling theory is well established for the systems undergoing a strong first-order transition [1-3]. However, if the correlation length is larger than the system size, methods using finite size scaling theory may fail to give indication of the first-order transition. The aim in this work is to observe how the system approaches to the first-order regime (where the correlation length is smaller but comparable to the size of the system) in two-dimensional 7-state Potts model which is known to exhibit first-order transition [4,5]. Cluster algorithm [6,7] is used to calculate commonly used equilibrium averages and the energy histogram, as well as cluster size related measurables like the average cluster size, fluctuations in the cluster size and the distribution of the cluster sizes [8,9]. The behavior of these measurables on different sizes varying from 8 \times 8 to 64 \times 64 is observed and the effect of this behavior on the order of the transition is studied.
Motivated by recent experiments on phase behavior of systems confined in porous media, we have studied the effect of randomness on the nature of the phase transition in the two-dimensional Potts model. To model the effects of the porous matrix we introduce a random distribution of couplings P(J(ij))=pdelta(J(ij)-J1)+(1-p)delta(J(ij)-J2) in the q state Potts Hamiltonian. An extensive Monte Carlo study is made on this system for q=5. We studied two different cases of disorder (a) J(1)/J(2)-->infinity and p=0.8 and (b) J(1)/J(2)=10 and p=0.5. We observed, in both cases, that the weak first order transition that appears in the pure case, changes to a second-order transition. A finite size scaling analysis shows that the correlation length exponent nu is close to 1 and the best fit to the dependence of the specific heat on system size is logarithmic. This suggests that both cases belong to the universality class of the Ising model. In contrast, the magnetic exponents point to a different universality class.