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Journal of Physics C Solid State Physics

Kadanoff's variational approximation applied to the two- and three-dimensional antiferromagnetic Ising lattices

Naeem Jan

Published Sep 28, 1979• 1 citation
Publisher

Abstract

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.

DOI
10.1088/0022-3719/12/18/022

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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.
The lower-bound renormalization group approximation developed by Kadanoff and co-workers is modified by the introduction of a novel form of the weight function which preserves the ground states of the Baxter–Wu model. The approximation is applied to this model and the problems encountered are discussed together with the implications of this approach for other real-space approximations. The relevant exponents (exact) obtained are 1.363 (1.50) and 1.879 (1.875) and the critical coupling KC31 is 0.4558 (0.4407).
The Kadanoff lower-bound renormalization transformation is used to obtain approximate values of the critical exponents for the spin-1/2 Ising model on a simple hypercubic ddd-dimensional lattice for all integral values of z=2dz={2}^{d}z=2d between 3 and 16. The results for $\ensuremath{\gamma}, \ensuremath{\delta}$, and the thermal and magnetic correction to scaling exponents are compared with corresponding results obtained from the $\ensuremath{\epsilon}$ expansion and series calculations.
The authors apply the variational lower-bound renormalization group transformation of Kadanoff (1975) to an SU (2) lattice spin model in two and three dimensions. Even in the one-hypercube framework of this renormalization group transformation the present model is characterized by having an infinite basis of fundamental operators. They investigate whether the lower-bound variational renormalization group transformation yields results stable under different truncations of this operator basis. Their results show that for this particular spin model this is not the case.
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).