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Journal of Physics C Solid State Physics

Application of the Legendre transformation to the Potts model

L.G. Marland

Published Aug 14, 1978• 1 citation
Publisher

Abstract

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.

DOI
10.1088/0022-3719/11/15/001

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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.
The transformation is tested on the two-dimensional Ising model. The transformation is formed by compounding the decimation and Legendre transformations, as suggested by van Leeuwen (1975). The Legendre transformation involves an adjustable parameter p. The cluster approximation is used, as well as a decoupling approximation which is exact in the limit of p to 1. The procedure is seen to be capable of estimating both the thermal and magnetic critical exponents, v and eta respectively, for two-dimensional systems. The accuracy obtained is compared to that of other methods and the applicability of the procedure to three-dimensional systems is discussed.
A real-space rescaling technique (decimation) is used to study the critical properties of the bond-dilute, S state Ashkin-Teller-Potts model on the square lattice. Fixed points and critical exponents are obtained and critical curves plotted for values of S from 1 to 4. For S=1, 2 the dilute model exhibits the same critical behaviour as the pure system. However, for S=3, 4, there is crossover to a new dilute fixed point. The transition remains second order but exhibits new values of the critical exponents. This behaviour is in accord with the Harris criterion (1974).
The two-dimensional four-state Potts model at finite temperature can be transformed, via the transfer matrix, into a one-dimensional quantum mechanical model at zero temperature. The duality invariant renormalisation group introduced by Fernandez-Pacheco (1979) is then employed to study the ground-state critical properties of this model. The fixed point is located at exactly the self-dual critical point K*=1. The thermal exponent is calculated to be y T =1.3219. It is in excellent agreement with the recent series value of Ditzian and Kadanoff (1979) (y T =1.33). Although it is not inconsistent with den Nijs's (1979) conjectured exact value of 3 / 2 , the difference is nevertheless substantial.