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arXiv:cond-mat/9907388 · Physical review. B, Condensed matter

Ordered phase and scaling in Z n models and the three-state antiferromagnetic Potts model in three dimensions

Masaki Oshikawa

Published Feb 1, 2000• 73 citations
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Abstract

Based on a renormalization-group picture of ${Z}_{n}$ symmetric models in three dimensions, we derive a scaling law for the ${Z}_{n}$ order parameter in the ordered phase. An existing Monte Carlo calculation on the three-state antiferromagnetic Potts model, which has the effective ${Z}_{6}$ symmetry, is shown to be consistent with the proposed scaling law. It strongly supports the renormalization-group picture that there is a single massive ordered phase, although an apparently rotationally symmetric region in the intermediate temperature was observed numerically.

DOI
10.1103/physrevb.61.3430

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Based on a renormalization-group picture of Zn{Z}_{n}Zn​ symmetric models in three dimensions, we derive a scaling law for the Zn{Z}_{n}Zn​ order parameter in the ordered phase. An existing Monte Carlo calculation on the three-state antiferromagnetic Potts model, which has the effective Z6{Z}_{6}Z6​ symmetry, is shown to be consistent with the proposed scaling law. It strongly supports the renormalization-group picture that there is a single massive ordered phase, although an apparently rotationally symmetric region in the intermediate temperature was observed numerically.
The antiferromagnetic three-state Potts model on the simple-cubic lattice is studied using Monte Carlo simulations. The ordering in a medium temperature range below the critical point is investigated in detail. Two different regimes have been observed: the so-called broken sublattice-symmetry phase dominates at sufficiently low temperatures, while the phase just below the critical point is characterized by an effectively continuous order parameter and by a fully restored rotational symmetry. However, the latter phase is not the permutationally sublattice symmetric phase recently predicted by the cluster variation method.
We present a study of the three-state antiferromagnetic Potts model in two and three dimensions, using a cluster-flip Monte Carlo simulation algorithm. The new approach enables us to perform simulations with greatly improved efficiency. We have obtained results for the ground-state entropy and critical exponents in two and three dimensions. The low-temperature phase in three dimensions is shown to have long-range order with a finite-size dependence of the magnetization and susceptibility similar to that of the XY model.
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.
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.