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arXiv:cond-mat/9802145 · Journal of Physics A Mathematical and General

The 3-state Potts antiferromagnet on the hexagonal lattice

Jesús Salas

Published Jul 17, 1998• 9 citations
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Abstract

We study the 3-state hexagonal-lattice Potts antiferromagnet by a Monte Carlo simulation using the Wang-Swendsen-Kotecký cluster algorithm. We study the staggered susceptibility and the correlation length, and we confirm that this model is disordered at all temperatures . We also measure the ground-state entropy density.
DOI
10.1088/0305-4470/31/28/011

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We study the 3-state hexagonal-lattice Potts antiferromagnet by a Monte Carlo simulation using the Wang-Swendsen-Koteck´y cluster algorithm. We study the staggered susceptibility and the correlation length, and we confirm that this model is disordered at all temperatures T ≥ 0. We also measure the Antiferromagnetic Potts models [1, 2, 3] are much less well understood than their ferromagnetic counterparts. One reason for this is that the behavior depends strongly on the microscopic lattice structure, in contrast to the universality typically enjoyed by ferromagnets. As a result, many basic questions have to be investigated case-bycase:
study the antiferromagnetic q-state Potts model on the square lattice for q = 3 and q = 4, using the Wang-Swendsen-Kotecky (WSK) Monte Carlo algorithm and a powerful finite-size-scaling extrapolation method. For q=3 we obtain good control up to correlation length xi similar to 5000; the data are consistent with xi(B)=Ae(2 beta)beta(P)(1 + a(1)e(-beta) + ...) as beta--> infinity, with p approximate to 1. The staggered susceptibility behaves as chi(stagg) similar to xi(5/3). For q = 4 the model is disordered (xi less than or similar to 2) even at zero temperature. In appendices we prove a correlation inequality for Potts antiferromagnets on a bipartite lattice, and we prove ergodicity of the WSK algorithm at zero temperature for Potts antiferromagnets on a bipartite lattice.
We study the antiferromagnetic q-state Potts model on the square lattice for q=3 and 4, using the Wang-Swendsen-Koteck\'y Monte Carlo algorithm and a finite-size-scaling extrapolation method. For q=3, we obtain good control up to correlation length \ensuremath{\xi}\ensuremath{\sim}5000; the data are consistent with \ensuremath{\xi}(\ensuremath{\beta})=${\mathit{Ae}}^{2\mathrm{\ensuremath{\beta}}}$\ensuremath{\beta}(1+a1{\mathit{a}}_{1}a1​${\mathit{e}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\beta}}}$+...) as \ensuremath{\beta}\ensuremath{\rightarrow}\ensuremath{\infty}. For q=4, the model is disordered even at zero temperature.
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.
We apply a newly proposed Monte Carlo method, the Wang-Landau algorithm, to the study of the three-dimensional antiferromagnetic q -state Potts models on a simple cubic lattice. We systematically study the phase transition of the models with q = 3, 4, 5 and 6. We obtain the finite-temperature phase transition for q = 3 and 4, whereas the transition temperature is down to zero for q = 5. For q = 6 there exists no order for any temperature. We also study the ground-state properties. The size dependence of the ground-state entropy is investigated. We find that the ground-state entropy is larger than the contribution from the typical configurations of the broken-sublattice-symmetry state for q = 3. The same situations are found for q = 4, 5 and 6.