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Physical review. B, Condensed matter

Critical behavior of the three-state Potts model: Monte Carlo renormalization group

Robert H. Swendsen, A. Nihat Berker

Published Oct 1, 1983• 7 citations
Publisher

Abstract

The critical behavior of the three-state Potts model in two dimensions is investigated with the use of the Monte Carlo renormalization group (MCRG), and the results are compared with conjectured values as well as with other methods. Recent advances in the MCRG enable greatly improved convergence, as well as greater insight into the structure of the renormalization transformation. The introduction of a four-point interaction is used to control the chemical potential of effective vacancies in a purely Potts Hamiltonian.

DOI
10.1103/physrevb.28.3897

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The critical behavior of the three-state Potts model in two dimensions is investigated with the use of the Monte Carlo renormalization group (MCRG), and the results are compared with conjectured values as well as with other methods. Recent advances in the MCRG enable greatly improved convergence, as well as greater insight into the structure of the renormalization transformation. The introduction of a four-point interaction is used to control the chemical potential of effective vacancies in a purely Potts Hamiltonian.
The thermal critical exponent of the four-state Potts model in two dimensions is evaluated as ${\ensuremath{\nu}}^{\ensuremath{-}1}=1.49\ifmmode\pm\else\textpm\fi{}0.01$, using the Monte Carlo renormalization-group method. The presence of a marginal scaling direction is also indicated. These results confirm previous conjectures, universality, and logarithmic corrections. A purely Potts Hamiltonian is considered. A four-point interaction is used to control the chemical potential of effective vacancies, without the introduction of explicit vacancy states.
We have undertaken a large-scale computer simulation of the 3d Ising model using the Monte Carlo Renormalization Group (MCRG) method. Our calculation improves on past calculations in three ways. Firstly, larger lattices are used -- we have run on 128{sup 3} as well as 64{sup 3}. Secondly, many more spin operators -- 53 even and 46 odd -- are measured. Lastly, we have incorporated more efficient ways of spin update -- Wolff's single-cluster variant of the Swendsen and Wang algorithm. In addition to calculating the critical temperature the thermal and magnetic exponents, and improving the estimate of the correction-to-scaling exponent, we are aiming to verify the presence of redundant operators. We do not yet have sufficient statistics to give final results so here we describe our calculation and present preliminary results which demonstrate that we have control over both finite size effects and systematic errors arising from truncation. 16 refs., 7 figs.
The d=3d=3d=3 planar model is analyzed for the first time with the Monte Carlo renormalization-group (MCRG) method. To increase the efficiency of the calculation, an operator is introduced in analogy with the "vacancy" operator used in the MCRG analysis of the q=4q=4q=4 Potts model. This operator is shown to be important in understanding the effect of the renormalization transformation on the Hamiltonian, and in improving the convergence of the method.
The qqq-state Potts models in two dimensions are studied with the Monte Carlo renormalization-group method. The results are in agreement with known properties. Combining our data with the qualitative picture of Nienhuis, Berker, Riedel, and Schick, we obtain a consistent description of the behavior of these models, including nearly singular behavior for q>4q>4q>4 (as proposed by Kim and Joseph) and suggesting logarithmic corrections to scaling for q=4q=4q=4.