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Journal of Physics A Mathematical and General

The weight function and optimal transformations

Naeem Jan, L. L. Moseley

Published Jul 11, 1983• 0 citations
Publisher

Abstract

A class of optimal transformations is defined through the weight function which, when implemented into simple approximations of the renormalisation group, predicts values for the thermal exponent, y T , and the magnetic field exponent, y H , which are in better agreement with the exact results than those obtained with the majority sign rule. Specifically, the authors find that for the ferromagnetic triangular Ising lattice y T =0.914 and y H =1.908 (using the same approximation, the majority sign rule predicts y T =0.736 and y H =1.669). This weight function is used to calculate the properties of the ferromagnetic and antiferromagnetic square Ising lattice with a two-cell cluster and a one-hypercube approximation. The weight function is the first stage in the development of a real-space renormalisation group transformation free of the 'peculiarities' observed by Griffiths (1981).

DOI
10.1088/0305-4470/16/10/025

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A class of optimal transformations is defined through the weight function which, when implemented into simple approximations of the renormalisation group, predicts values for the thermal exponent, y T , and the magnetic field exponent, y H , which are in better agreement with the exact results than those obtained with the majority sign rule. Specifically, the authors find that for the ferromagnetic triangular Ising lattice y T =0.914 and y H =1.908 (using the same approximation, the majority sign rule predicts y T =0.736 and y H =1.669). This weight function is used to calculate the properties of the ferromagnetic and antiferromagnetic square Ising lattice with a two-cell cluster and a one-hypercube approximation. The weight function is the first stage in the development of a real-space renormalisation group transformation free of the 'peculiarities' observed by Griffiths (1981).
Using a continuous class of real-space renormalisation transformations the authors study the critical behaviour of the Ising model in a variety of two-dimensional lattices. They use the simplest possible cluster, with two cells, and a single renormalised coupling. Certain of these transformations-well separated from the majority spin rule-produce sharply more accurate critical properties than the rest. From this study they note an optimal set of conditions which characterise the successful transformations. The optimal transformations appear to have a certain 'locality' property: renormalised couplings beyond nearest neighbour are especially small.
From a consideration of the ground states of the antiferromagnetic Ising model a weighting function is proposed. This function preserves the ground states under the renormalisation group transformation. This form of the lower bound approximation leads to results which are in excellent agreement with the exact values in two dimensions and in fair agreement with the approximate results of the series expansion in three dimensions. The implication of this approach for other real-space approximations is noted.
The renormalisation group method of Kadanoff is applied to the spin- 1 / 2 Ising model in two dimensions using a transformation applicable to both the square and triangular lattices. Two types of cluster are studied for the triangular lattice.The first cluster yields a new fixed point but the results for the critical temperature and critical exponents are in poor agreement with the known exact values. The second cluster maps directly onto the fixed point found by Kadanoff for the square lattice yielding excellent values for both the critical exponents and the critical temperature.
A simple renormalisation-group approach based on self-dual clusters is proposed for the two-dimensional nearest-neighbour spin- 1 / 2 Ising model on the square lattice; it reproduces the exact critical point. The authors calculate the internal energy and the specific heat for vanishing external magnetic field, spontaneous magnetisation and the thermal (y T ) and magnetic (y H ) critical exponents. The results obtained from the first four smallest cluster sizes strongly suggest the convergence towards the exact values when the cluster size increases. Even for the smallest cluster, where the calculation is very simple, the results are quite accurate, particularly in the neighbourhood of the critical point.