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Antiferromagnetic potts models on the square lattice: A high-precision Monte Carlo study

Ferreira, SJ, Alan D. Sokal

Published Aug 1, 1999• 33 citations
OpenAlex

Abstract

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.

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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.
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 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:
We investigate the finite-temperature corrections to scaling in the three-state square-lattice Potts antiferromagnet, close to the critical point at T=0. Numerical diagonalization of the transfer matrix on semi-infinite strips of width L sites, 4 < or = L < or = 14, yields finite-size estimates of the corresponding scaled gaps, which are extrapolated to L --> infinity. Owing to the characteristics of the quantities under study, we argue that the natural variable to consider is x identical with L e(-2 beta). For the extrapolated scaled gaps we show that square-root corrections, in the variable x, are present, and provide estimates for the numerical values of the amplitudes of the first- and second-order correction terms, for both the first and second scaled gaps. We also calculate the third scaled gap of the transfer matrix spectrum at T=0, and find an extrapolated value of the decay-of-correlations exponent, eta(3)=2.00(1). This is at odds with earlier predictions, to the effect that the third relevant operator in the problem would give eta(P(stagg))=3, corresponding to the staggered polarization.
It has been shown that the partition function of the q-state Potts model (QPM) is the generating function of a q-state bond-correlated percolation model (QBCPM) such that the spontaneous magnetization M, the magnetic susceptibility \ensuremath{\chi}, the internal energy U, and the specific heat Ch{C}_{h}Ch​ for the QPM are related to the percolation probability P, the mean cluster size S, the average number of occupied bond p\ifmmode\bar\else\textasciimacron\fi{}, and the fluctuations of occupied bonds F for the QBCPM. We use the Monte Carlo simulation method of Swendsen and Wang and the multiple labeling technique of Hoshen and Kopelman to calculate P, S, p\ifmmode\bar\else\textasciimacron\fi{}, and F for the QBCPM on the square and the simple cubic lattice. Our calculations give accurate critical points for the QPM.
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.