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Physical review. B, Condensed matter

Analysis of the lower-bound renormalization-group approximation

Naeem Jan

Published Sep 1, 1979• 0 citations
Publisher

Abstract

The lower-bound variational renormalization-group approximation of Kadanoff is reformulated and applied to the triangular Ising lattice. The basic inconsistencies inherent in the method are removed and the results thus obtained are in fair agreement with the exact values.
DOI
10.1103/physrevb.20.2201

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The use of variational approximations in the study and application of renormalization groups is discussed. In particular, a simple approximation is derived which yields an upper bound to the free energy of Ising models on d-dimensional lattices. The optimal transformation, which yields the least upper bound, is determined analytically. The criterion proposed by Kadanoff (1975) to determine the 'best' approximation to the fixed point is found to fail in this case. The reasons for this failure and several of the basic problems posed by variational approximations are discussed.
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).
A previously developed real-space renormalization-group approach to critical dynamics is examined. A correction to the formalism has to be introduced. The need for this correction is demonstrated through an exactly soluble one-dimensional Ising model. The new version imposes additional serious limitations on the usefulness of the method within standard approximation schemes. The second-order cumulant approximation of the two-dimensional kinetic Ising model on a triangular lattice is used to demonstrate these limitations.