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Physical review. B, Condensed matter

Antiferromagnetic Potts and Ashkin-Teller models in three dimensions

Jayanth R. Banavar, Gary S. Grest, David Jasnow

Published Apr 1, 1982• 66 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.25.4639

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Physical Review Letters•Dec 3, 1979•101 citations
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 \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ-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 \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ-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.
The phase transition of four-state antiferromagnetic Potts model on simple cubic and body-centered-cubic lattice is studied by cluster Monte Carlo simulations of finite systems up to 963{96}^{3}963 lattice sizes. The result indicates that the size 963{96}^{3}963 is still sufficient to extract asymptotic behavior. However, if one assumes that both models are well described by the usual Ginzburg-Landau Hamiltonian, the result is consistent with the prediction of the field-theoretical analysis that the phase transition is first-order one.
High- and low-temperature series, Monte Carlo simulations, mean-field calculations, and renormalization-group ideas are used to analyze the behavior of the Ashkin-Teller model in three dimensions. The data generated by series and Monte Carlo methods in the neighborhood of the decoupling point are consistent with the renormalization-group conclusion that this system cannot show a line of continuously varying exponents. Thus the phase diagram for the d=3d=3d=3 system is quite different from that in d=2d=2d=2. Indeed a new phase (not seen in d=2d=2d=2) is observed, one in which the system spontaneously breaks its natural symmetry between its two kinds of Ising spins. A variety of continuous transitions are observed and explained in terms of an analysis which is fully consonant with renormalization-group concepts.
The results of the large-scale Monte Carlo simulations of the continuous phase transitions in the three-dimensional Ashkin-Teller model on a cubic lattice are presented in the two-parameter space. The invariance of the ratio of the square of the second moment of the order parameter to its fourth moment in the critical region and the Metropolis algorithm are exploited. Using the universality hypothesis and the finite-size-scaling analysis, the accurate positions of the continuous phase-transition lines are calculated and their Ising character is confirmed. Approximate locations of the tricritical points are also estimated.
The critical behavior of the sss-state Ashkin-Teller-Potts model on a square lattice is studied for s=3,4,and5s=3, 4, \mathrm{and} 5s=3,4,and5, using a variational renormalization-group transformation of the type introduced by Kadanoff. For the s=3s=3s=3 and s=4s=4s=4 models, the transformation correctly predicts a continuous phase transition. The calculated values of the critical exponents and the transition temperature are consistent with the results obtained from other approaches. For s=5s=5s=5, however, the transformation fails to detect the first-order transition that is known to occur.