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Physical Review Letters

Is the Phase Transition of the Three-State Potts Model Continuous in Three Dimensions?

Svend J. Knak Jensen, Ole G. Mouritsen

Published Dec 3, 1979• 101 citations
Publisher

Abstract

The internal energy and the order parameter of the three-state Potts model in a simple cubic lattice are calculated using the Monte Carlo technique. Both properties exhibit metastabilities in a small temperature region demonstrating that the phase transition is of first order. This result is in agreement with the prediction of the $\ensuremath{\epsilon}$ expansion but in contrast to the position-space renormalization-group calculations which lead to a continuous transition.

DOI
10.1103/physrevlett.43.1736

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The internal energy and the order parameter of the three-state Potts model in a simple cubic lattice are calculated using the Monte Carlo technique. Both properties exhibit metastabilities in a small temperature region demonstrating that the phase transition is of first order. This result is in agreement with the prediction of the \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ expansion but in contrast to the position-space renormalization-group calculations which lead to a continuous transition.
We have used the Monte Carlo renormalization group method to study the order-disorder phase transition of the three and four dimensional, three-state Potts model. In both cases, the transition appears to be first order. However, no evidence for a discontinuity fixed point associated with these transitions was found. Instead, the existence of a nearby second order fixed point just beyond the first order transition could be inferred from the RG flow in the ordered as well as in the disordered state. These results imply that the thermodynamic quantities are analytically continuable into the metastable regions, and that apart from discontinuities caused by the intersection of the two free-energy branches, no additional singularities occur at the transition.
Critical properties of the qqq-state Potts and Potts-lattice-gas models are calculated by means of the Kadanoff variational renormalization group for dimensions 1.58, 2, and 2.32, and the Migdal approximation for arbitrary dimensions. Accurate results for the critical and tricritical exponents are obtained as well as for qc{q}_{c}qc​, the largest value of qqq for which the Potts transition is continuous. It is concluded unambiguously that the three-state Potts model in three dimensions undergoes a first-order phase transition.
The three-state antiferromagnetic Potts model on the simple cubic lattice is studied by the Monte Carlo method using a new order parameter and employing the histogram technique. It is shown that apart from the low-temperature broken-sublattice-symmetry phase with nonequivalent sublattices, another type of ordering with equivalent sublattices does exist in the model between the broken-sublattice-symmetry and disordered phases. The temperature of the transition between the ordered phases is estimated from the histogram of the order parameter. The value obtained is in good agreement with that obtained from cluster variation circulations.
Monte Carlo simulations and renormalization-group ideas are used to analyze the behavior of the antiferromagnetic Potts model and the Ashkin-Teller model in three dimensions. A variety of continuous transitions, observed in the Monte Carlo data on the antiferromagnetic Potts model for simple cubic and body-centered-cubic lattices, are found to be consistent with $\ensuremath{\epsilon}$-expansion analyses of these models. A glassy "plastic crystal" phase is observed, along with the analog of the glass transition in the four-state antiferromagnetic Potts model. An $\ensuremath{\epsilon}$-expansion analysis of the Ashkin-Teller model is found to yield results consistent with those obtained by Ditzian et al. using series analysis and Monte Carlo simulations.