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arXiv:0802.2270 · Physical Review Letters

Phase Transition in the Three-State Potts Antiferromagnet on the Diced Lattice

Roman Kotecký, Jesús Salas, Alan D. Sokal

Published Jul 17, 2008• 23 citations
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Abstract

We prove that the 3-state Potts antiferromagnet on the diced lattice (dual of the kagome lattice) has entropically driven long-range order at low temperatures (including zero). We then present Monte Carlo simulations, using a cluster algorithm, of the 3-state and 4-state models. The 3-state model has a phase transition to the high-temperature disordered phase at v=e;{J}-1=-0.860 599+/-0.000 004 that appears to be in the universality class of the 3-state Potts ferromagnet. The 4-state model is disordered throughout the physical region, including at zero temperature.

DOI
10.1103/physrevlett.101.030601

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We prove that the 3-state Potts antiferromagnet on the diced lattice (dual of the kagome lattice) has entropically driven long-range order at low temperatures (including zero). We then present Monte Carlo simulations, using a cluster algorithm, of the 3-state and 4-state models. The 3-state model has a phase transition to the high-temperature disordered phase at v=e;{J}-1=-0.860 599+/-0.000 004 that appears to be in the universality class of the 3-state Potts ferromagnet. The 4-state model is disordered throughout the physical region, including at zero temperature.
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.
We use Monte Carlo methods to study the three-state Potts model on a two-dimensional lattice with fivefold coordination. The lattice consists of triangular and hexagonal units and has properties intermediate between the triangular and kagom\'e lattices. The critical behavior of both ferromagnetic and antiferromagnetic cases is considered. As expected, the critical behavior of the ferromagnetic model is independent of the lattice details and lies in the universality class of the three-state ferromagnetic Potts model. A critical end point is found at the coexistence between the ferromagnetic and antiferromagnetic phases, and the transition to the antiferromagnetic state is always first order.
We study the q-state Potts model on the simple cubic lattice with ferromagnetic interactions along one lattice direction, and antiferromagnetic interactions along the two other directions. As the temperature T decreases, the system undergoes a second-order phase transition that fits in the universality class of the 3D O(n) model with n=q-1. This conclusion is based on the estimated critical exponents, and histograms of the order parameter. At even smaller T we find, for q=4 and 5, a first-order transition to a phase with a different type of long-range order. This long-range order dissolves at T=0, and the system effectively reduces to a disordered two-dimensional Potts antiferromagnet. These results are obtained by means of Monte Carlo simulations and finite-size scaling.
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 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:
Abstract We study the antiferromagnetic 3-state Potts model on general (periodic) plane quadrangulations Γ. Any quadrangulation can be built from a dual pair ( G , G * ). Based on the duality properties of G , we propose a new criterion to predict the phase diagram of this model. If Γ is of self-dual type (i.e. if G is isomorphic to its dual G * ), the model has a zero-temperature critical point with central charge c = 1, and it is disordered at all positive temperatures. If Γ is of non-self-dual type (i.e. if G is not isomorphic to G * ), three ordered phases coexist at low temperature, and the model is disordered at high temperature. In addition, there is a finite-temperature critical point (separating these two phases) which belongs to the universality class of the ferromagnetic 3-state Potts model with central charge c = 4/5. We have checked these conjectures by studying four (resp. seven) quadrangulations of self-dual (resp. non-self-dual) type, and using three complementary high-precision techniques: Monte-Carlo simulations, transfer matrices, and critical polynomials. In all cases, we find agreement with the conjecture. We have also found that the Wang–Swendsen–Kotecký Monte Carlo algorithm does not have (resp. does have) critical slowing down at the corresponding critical point on quadrangulations of self-dual (resp. non-self-dual) type.