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arXiv:cond-mat/9902295

Height representation, critical exponents, and ergodicity in the four-state triangular Potts antiferromagnet

Cristopher Moore, Mark E. J. Newman

Published Feb 22, 1999• 4 citations

Abstract

We study the four-state antiferromagnetic Potts model on the triangular lattice. We show that the model has six types of defects which diffuse and annihilate according to certain conservation laws consistent with their having a vector-valued topological charge. Using the properties of these defects, we deduce a 2+2-dimensional height representation for the model and hence show that the model is equivalent to the three-state Potts antiferromagnet on the Kagome lattice and to bond-coloring models on the triangular and hexagonal lattices. We also calculate critical exponents for the ground state ensemble of the model. We find that the exponents governing the spin-spin correlation function and spin fluctuations violate the Fisher scaling law because of constraints on path length which increase the effective wavelength of the spin operator on the height lattice. We confirm our predictions by extensive Monte Carlo simulations of the model using the Wang-Swendsen-Kotecky cluster algorithm. Although this algorithm is not ergodic on lattices with toroidal boundary conditions, we prove that it is ergodic on lattices with free boundary conditions, or more generally on lattices possessing no non-contractible loops of infinite order. To guard against biases introduced by lack of ergodicity, we therefore perform our simulations on both the torus and the projective plane.

DOI: 10.48550/arxiv.cond-mat/9902295

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We study the four-state antiferromagnetic Potts model on the triangular lattice. We show that the model has six types of defects which diffuse and annihilate according to certain conservation laws consistent with their having a vector-valued topological charge. Using the properties of these defects, we deduce a 2+2-dimensional height representation for the model and hence show that the model is equivalent to the three-state Potts antiferromagnet on the Kagome lattice and to bond-coloring models on the triangular and hexagonal lattices. We also calculate critical exponents for the ground state ensemble of the model. We find that the exponents governing the spin-spin correlation function and spin fluctuations violate the Fisher scaling law because of constraints on path length which increase the effective wavelength of the spin operator on the height lattice. We confirm our predictions by extensive Monte Carlo simulations of the model using the Wang-Swendsen-Kotecky cluster algorithm. Although this algorithm is not ergodic on lattices with toroidal boundary conditions, we prove that it is ergodic on lattices with free boundary conditions, or more generally on lattices possessing no non-contractible loops of infinite order. To guard against biases introduced by lack of ergodicity, we therefore perform our simulations on both the torus and the projective plane.

We define a 4-state Potts model ensemble on the square lattice, with the constraints that neighbouring spins must have different values, and that no plaquette may contain all four states. The spin configurations may be mapped into those of a two-dimensional interface in a 2 + 5 dimensional space. If this interface is in a Gaussian rough phase (as is the case for most other models with such a mapping) then the spin correlations are critical and their exponents can be related to the stiffness governing the interface fluctuations. Results of our Monte Carlo simulations show height fluctuations with an anomalous dependence on the wavevector, intermediate between the behaviours expected in a rough and in a smooth phase; we argue that the smooth phase (which would imply long-range spin order) is the best interpretation.

By introducing a small amount of non-magnetic impurities into an antiferromagnetic (AF) two-dimensional (2D) Potts model on a triangular lattice it is that the impurities in spin systems described by this model result in the change of a first order to a second-order phase transition. The systems with linear sizes L × L = N, L = 9-144 are considered. Investigations are performed using the standard Metropolis algorithm along with Monte-Carlo single-cluster Wolff algorithm. On the basis of the theory of finite-size scaling, critical exponents (CE) are calculated: the heat capacity α, the susceptibility γ, the order parameter β, and the CE of the correlation radius ν.

The critical behavior of a two-dimensional weakly diluted antiferromagnetic Potts model with spin states q = 3 on a triangular lattice is computationally simulated. The systems with periodic boundary conditions were calculated, when spin concentrations p were 1.0 and 0.9. Systems with linear dimensions L × L = N and L = 9–144 were considered. Static critical exponents of heat capacity α, the susceptibility γ, the order parameter β, and critical exponent ν for a correlation radius are calculated based on a theory of finite-dimensional scaling.

The critical behavior of the disordered two-dimensional antiferromagnetic Potts model with the number of spin states q= 3 on a triangular lattice with disorder in the form of nonmagnetic impurities is studied by the Monte Carlo method. The critical exponents for the susceptibility γ, magnetization β, specific heat α, and correlation radius ν are calculated in the framework of the finite-size scaling theory at spin concentrations p = 0.90, 0.80, 0.70, and 0.65. It is found that the critical exponents increase with the degree of disorder, whereas the ratios and do not change, thus holding the scaling equality

\frac{{2\beta }}{\nu } + \frac{\gamma }{\nu } = d

. Such behavior of the critical exponents is related to the weak universality of the critical behavior characteristic of disordered systems. All results are obtained using independent Monte Carlo algorithms, such as the Metropolis and Wolff algorithms.

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.