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Journal of Physics C Solid State Physics

A real space renormalisation transformation for lattice systems

Leigh Sneddon

Published Oct 14, 1977• 2 citations
Publisher

Abstract

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.

DOI
10.1088/0022-3719/10/19/006

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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.
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.
A simple hybrid of real space renormalisation group theory and finite-lattice calculations is described. The method is based on a lattice-restructuring transformation, with the criterion that the free energies of the original and restructured lattices be equal. The authors approximate the free energy per spin of the infinite lattice by an exact evaluation of the partition function for a small cluster. As a result of this transformation, the coupling constant of the restructured lattice is found to be temperature dependent. The method is illustrated by studying the Ising model in two and three dimensions, and two-dimensional Potts models. The calculations involve little more than is required for a Migdal-Kadanoff transformation, yet, the critical properties are found to be significantly improved in comparison. Detailed comparisons are also made with the global thermodynamic properties calculated within this approximation (free energy, internal energy, specific heat, spontaneous magnetisation) and exact or best known results for the models. The simplicity of the method allows for the study of more complex lattice models, and several possible applications are discussed.