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Journal of Physics A Mathematical and General

Long-range one-dimensional Potts models: a cluster mean-field and extrapolation approach

James L. Monroe

Published Sep 28, 1999• 12 citations
Publisher

Abstract

We obtain approximations for the critical temperature ( T c ) of q -state ferromagnetic Potts models on one-dimensional lattices with algebraically decaying ferromagnetic pair interactions, i.e. decaying as 1/ r , with 1< 2. Initially we use a cluster mean-field method to get approximations to the critical temperature and we look at increasingly large clusters. This gives us a sequence of ever more accurate approximate values which can be used as input to various extrapolation algorithms. We see that as the interaction decreases more slowly and the value of q increases we obtain more and more accurate T c estimates. Our best estimates we believe to be correct to four-figure accuracy.

DOI
10.1088/0305-4470/32/41/304

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We obtain approximations for the critical temperature ( T c ) of q -state ferromagnetic Potts models on one-dimensional lattices with algebraically decaying ferromagnetic pair interactions, i.e. decaying as 1/ r , with 1< 2. Initially we use a cluster mean-field method to get approximations to the critical temperature and we look at increasingly large clusters. This gives us a sequence of ever more accurate approximate values which can be used as input to various extrapolation algorithms. We see that as the interaction decreases more slowly and the value of q increases we obtain more and more accurate T c estimates. Our best estimates we believe to be correct to four-figure accuracy.
The one-dimensional q-state Potts model with ferromagnetic pair interactions which decay with the distance r as is considered. We calculate, through a real-space renormalization group technique using Kadanoff blocks of length b, the critical temperature and the correlation length critical exponent as a function of for different values of q. Some of the very few known rigorous results for general q are reproduced by our approach. Several asymptotic behaviours are derived analytically for q = 2, 3 in the limit. We also obtain extrapolated critical temperatures for arbitrary values of and for q = 2, 3, 4, which we believe approximate the exact ones well, except in the region near . Furthermore, the use of another extrapolation procedure suitable only in the vicinity of led us to conjecture that the exact critical temperature is the same for any value of q . We also verify that , which is consistent with a recent conjecture of Tsallis.
We consider Ising models with long-range ferromagnetic pair interactions decaying as for . We first find approximate values for the critical temperature. We use a cluster mean-field approach combined with finite-size scaling and Vanden Broeck and Schwartz transformations. For we find which can be compared with recent results of Luijten and Blöte who found , and which is two orders of magnitude more accurate than any previous results. Since we use a mean-field cluster approximation as part of our approach, the accuracy for larger values of decreases significantly. In addition to we obtain approximate values for the critical exponents , and using the coherent anomaly method. For we obtain , , and - all extremely close to the predictions of renormalization group calculations which say that these exponents should take on their classical values for this value of .
We study the nature of the phase transition of the qqq-state Potts model with long-range ferromagnetic interactions decaying as $1/{r}^{d+\ensuremath{\sigma}}$, in dimension d=1d\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1d=1, using a histogram Monte Carlo (MC) technique. The model can exhibit a first-order transition or a second-order phase transition with nonstandard critical exponents. The critical value of qqq above which a first-order transition occurs decreases with decreasing $\ensuremath{\sigma}$, from qc=8{q}_{c}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}8qc​=8 for $\ensuremath{\sigma}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$ to qc=2{q}_{c}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2qc​=2 for $\ensuremath{\sigma}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.3$. Detailed results for various $\ensuremath{\sigma}$ will be shown and discussed. Mean-field calculation confirms the tendency of our MC results.
The mean-field approximation and the Bethe approximation are two of the most often used approximations when one wants to obtain approximations of the phase diagrams and the critical temperature of lattice spin systems. Both can and have often been generalized to produce what are known as cluster mean-field and Bethe approximations. Generally, three characteristics are associated with these approximations. First, they give upper bounds to the critical temperature; second, considering larger clusters will result in better approximations; and third, the Bethe approximation is better than the corresponding mean-field approximation. We show what we believe to be a rather surprising result that, for one-dimensional Ising models with algebraically decaying interactions falling off slowly enough, the Bethe cluster approximations violate all three of these characteristics.
Ferromagnetism in one dimension is a observation which has been reported in a recent work [Gambardella et al., Nature (London) 416, 301 (2002)], and it is thought that anisotropy barriers are responsible in that relevant effect. In the present work, transitions between two different magnetic ordering phases are obtained as a result of an alternative approach. The critical temperature has been estimated by the Binder method. Ferromagnetic long-range interactions have been included in a special Hamiltonian through a power law that decays at large interparticle distance rrr as ${r}^{\ensuremath{-}\ensuremath{\alpha}}$ for $\ensuremath{\alpha}\ensuremath{\geqslant}0$. If the range of interactions decreases $(\ensuremath{\alpha}\ensuremath{\rightarrow}\ensuremath{\infty})$, the trend of the critical temperature disappears, but if the range of interactions increases $(\ensuremath{\alpha}\ensuremath{\rightarrow}0)$, the trend of the critical temperature approaches to the mean field approximation. The crossover between these two limiting situations is discussed.
The phase transition temperature T c of the q-state Potts model with ferromagnetic interactions J x and antiferromagnetic ones J y in the two respective directions of a 2D lattice is calculated by the high-temperature series expansion (HTSE) of the appropriate staggered susceptibility as well a by the cluster variate approximation (CVA). For q=3 the HTSE result in the range of mod J y mod /J x <or approximately=1.5 shows a good coincidence with the Kosterlitz-Thouless-type phase transition temperature calculated on the prescription by Ostlund which is given by 1/16 pi =exp(-2 beta c J x ) coth(- beta c J y /2), beta =1/kT c . The CVA result shows similar behaviour of T c to that of the Migdal-Kadanoff renormalisation group method obtained by Kinzel et al, but it is considered to be less reliable than the HTSE result.