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The 3-State Potts Antiferromagnet on the Hexagonal Lattice

Jesús Salas, Materia Condensada, Facultad De Ciencias

Published Jan 1, 2008• 7 citations
OpenAlex

Abstract

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:

Related papers

The 3-state Potts antiferromagnet on the hexagonal lattice

Jesús Salas

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.

arXiv:cond-mat/9802145•Journal of Physics A Mathematical and General•Jul 17, 1998•9 citations

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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.

•Aug 1, 1999•33 citations

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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+${\mathit{a}}_{1}$${\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.

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Yutaka Okabe, Hiromi Otsuka

Using the Wang–Landau Monte Carlo method, we study the antiferromagnetic (AF) three-state Potts model with a staggered polarization field on the square lattice. We obtain two phase transitions: one belongs to the ferromagnetic three-state Potts universality class and the other to the Ising universality class. The phase diagram obtained is quantitatively consistent with the transfer matrix calculation. The Ising transition in the large nearest-neighbour interaction limit has been made clear by the detailed analysis of the energy density of states.

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Three-state Potts model on the maple leaf lattice

Z. F. Wang, B. W. Southern

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.

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Simulation of the Three-Component Potts Model on a Hexagonal Lattice by the Monte Carlo Method

Albert Babaevich Babaev, Akai Kurbanovich Murtazaev

Abstract Computer simulation of the three-component Potts model on a hexagonal lattice was carried out using the Monte Carlo method. Systems with linear dimensions L × L = N, L = 20–320 in units of interatomic distances are considered. Based on the theory of finite-size scaling, the static critical exponents of heat capacity α, susceptibility γ, magnetization β, and correlation radius ν are calculated. The data we obtained confirm that in the considered Potts model on a hexagonal lattice, a second-order phase transition is observed with critical exponents corresponding to the universality class of the three-component Potts model.

The Physics of Metals and Metallography•Jul 1, 2023•5 citations
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.
Using the Wang–Landau Monte Carlo method, we study the antiferromagnetic (AF) three-state Potts model with a staggered polarization field on the square lattice. We obtain two phase transitions: one belongs to the ferromagnetic three-state Potts universality class and the other to the Ising universality class. The phase diagram obtained is quantitatively consistent with the transfer matrix calculation. The Ising transition in the large nearest-neighbour interaction limit has been made clear by the detailed analysis of the energy density of states.
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.
Abstract Computer simulation of the three-component Potts model on a hexagonal lattice was carried out using the Monte Carlo method. Systems with linear dimensions L × L = N, L = 20–320 in units of interatomic distances are considered. Based on the theory of finite-size scaling, the static critical exponents of heat capacity α, susceptibility γ, magnetization β, and correlation radius ν are calculated. The data we obtained confirm that in the considered Potts model on a hexagonal lattice, a second-order phase transition is observed with critical exponents corresponding to the universality class of the three-component Potts model.