O. Introduction

Abstract

The variational principles introduced by Kadanoff et al in the renormahzatmn theory are analyzed. It is shown that the values for the specific heat critical exponent a which can be found by a variational method are restricted to a < 0 or t~ = 1 (first order transition). The reason is the confluence of the singularities in the free energy and in the variational parameters. A full implementation of the variational principle changes for the square Ising lattice the earlier obtained a = 0.001756 to t~ =-0.123413. The renormalization theory for critical phenomena has provided the means to calculate the critical exponents. The first attempts 1'2) to obtain accurate exponents by a position space method involved a considerable amount of computational effort. In the search for a simple and accurate method the variational approach of Kadanoff, Houghton and Yalabik (KHY) 3) seems to be a breakthrough and it has been applied to a variety of models4). The idea of KHY is to use the freedom in defining a renormalization transformation to optimize the free energy. They showed that approximate renormalization transformations can be defined generating a free energy which is a rigorous upper or lower bound to the true free energy. Then the free parameters in the renormalization transformation can be chosen such as to optimize the bounds. Thus, an impressively accurate bound to the free energy results and also the reported critical exponents are very close to what is known exactly or may be expected from other sources. It has been noted by KnopsS), however, that the variational principle has not been applied fully around the fixed point and that a more consequent application changes the value of a from a = 0.001756 to a value around 323