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Journal of Physics A Mathematical and General

Renormalisation-group results for the three-state Potts model

Theodore W. Burkhardt, H. J. F. Knops, Marcel den Nijs

Published Dec 1, 1976• 53 citations
Publisher

Abstract

A lower-bound renormalisation transformation of the type introduced by Kadanoff (see J. Statist. Phys., vol.14, p.171, 1976) is applied to the three-state Potts model. For both d=2 and d=3 the Kadanoff transformation predicts a continuous transition. Values for the critical exponents and the critical temperature are reported. The consistency of the d=2 results with series estimates gives the authors some confidence in the predictions for d=3.

DOI
10.1088/0305-4470/9/12/002

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A lower-bound renormalisation transformation of the type introduced by Kadanoff (see J. Statist. Phys., vol.14, p.171, 1976) is applied to the three-state Potts model. For both d=2 and d=3 the Kadanoff transformation predicts a continuous transition. Values for the critical exponents and the critical temperature are reported. The consistency of the d=2 results with series estimates gives the authors some confidence in the predictions for d=3.
The critical behavior of the sss-state Ashkin-Teller-Potts model on a square lattice is studied for s=3,4,and5s=3, 4, \mathrm{and} 5s=3,4,and5, using a variational renormalization-group transformation of the type introduced by Kadanoff. For the s=3s=3s=3 and s=4s=4s=4 models, the transformation correctly predicts a continuous phase transition. The calculated values of the critical exponents and the transition temperature are consistent with the results obtained from other approaches. For s=5s=5s=5, however, the transformation fails to detect the first-order transition that is known to occur.
Critical properties of the qqq-state Potts and Potts-lattice-gas models are calculated by means of the Kadanoff variational renormalization group for dimensions 1.58, 2, and 2.32, and the Migdal approximation for arbitrary dimensions. Accurate results for the critical and tricritical exponents are obtained as well as for qc{q}_{c}qc​, the largest value of qqq for which the Potts transition is continuous. It is concluded unambiguously that the three-state Potts model in three dimensions undergoes a first-order phase transition.
A Kadanoff lower-bound renormalization transformation with three variational parameters has been applied to the Blume-Capel model by Burkhardt. We report some additional results obtained in a particular one-parameter subspace of the three variational parameters. In this subspace the transformation treats the three possible Blume-Capel ground-state configurations with all the spins in the same state on an equal footing and preserves the three-state permutational symmetry of the Potts model. There are fixed points associated with first-order, critical, and tricritical transitions. The variational parameters maximizing the free energy at the critical and tricritical fixed points differ by only a few percent. The critical exponents associated with the different transitions and various points on the critical surfaces are calculated for d=2,3d=2,3d=2,3 dimensions.
The Legendre transformation, a modification of the decimation method used in the real space renormalisation group, is employed to study the critical behaviour of the two-dimensional S-state Potts model which exhibits a second-order phase transition for S<or=4. Thermal and magnetic critical exponents are calculated for the percolation (S to 1), Ising (S=2) and three- and four-state Potts models. Decimation estimates of the critical exponents and critical points for S=2, 3 and 4 are also presented.