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Journal of Physics A Mathematical and General

Resolving the order of phase transitions in Monte Carlo simulations

M. Fukugita, H. Mino, Masanori Okawa, Akira Ukawa

Published Jun 7, 1990• 29 citations
Publisher

Abstract

The authors made a Monte Carlo simulation of the two-dimensional Potts model with q=3, 4 and 5 to examine the question whether numerical methods can distinguish the order of a phase transition for the subtle cases that this model exemplifies. They found that the finite-size scaling test for susceptibilities has sufficient power for this purpose, whereas the simple method of detecting metastability signals often fails.

DOI
10.1088/0305-4470/23/11/009

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The authors made a Monte Carlo simulation of the two-dimensional Potts model with q=3, 4 and 5 to examine the question whether numerical methods can distinguish the order of a phase transition for the subtle cases that this model exemplifies. They found that the finite-size scaling test for susceptibilities has sufficient power for this purpose, whereas the simple method of detecting metastability signals often fails.
Using the ``multihit'' Swendsen-Wang cluster flipping method, we performed extensive Monte Carlo simulations to investigate the critical behavior of the two-dimensional (2D) eight-state random-bond Potts model. We applied finite-size-scaling techniques to extract the critical exponents for two different sets of bond strengths, from which we concluded that the transition is second order with critical exponents for both sets falling into same universality class, that of a 2D Ising model. A variation of the Lee-Kosterlitz method for determining the order of a phase transition was also applied. The double-peaked structure of the specific heat, which was found in some of the bond configurations, was also studied by simulation on periodic arrangements of strong and weak bonds.
The influence of uncorrelated, quenched disorder on the phase transition of two dimensional Potts models will be reviewed. After an introduction where the conditions of relevance of quenched randomness on phase transitions are exemplified by experimental measurements, the results of perturbative and numerical investigations in the case of the Potts model will be presented. The Potts model is of particular interest, since it can have in the pure case a second-order or a first-order transition, depending on the number of states per spin. In 2D, transfer matrix calculations and Monte Carlo simulations were used in order to check the validity of conformal invariance methods in disordered systems. These techniques were then used to investigate the universality class of the disordered Potts model, in both regimes of the pure model phase transitions. A test of replica symmetry became possible through a study of multiscaling properties and a detailed analysis of the probability distribution of the correlation functions was also made possible.
We present a new method for using the data from Monte Carlo simulations that can increase the efficiency by 2 or more orders of magnitude. A single Monte Carlo simulation is sufficient to obtain complete thermodynamic information over the entire scaling region near a phase transition. The accuracy of the method is demonstrated by comparison with exact results for the d=2 Ising model. New results for d=2 eight-state Potts model are also presented. The method is generally applicable to statistical models and lattice-gauge theories.
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.