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Physical review. B, Condensed matter

Comment on "Variational approximations for renormalization-group transformations"

Yu Ming Shih, Dong-Chung Jou, C. K. Pan, W. S. Lee, Wen‐Den Chen, H. M. Huang, Chin‐Kun Hu, Kuo Gen Chen, Hsein Che Tseng

Published Jan 1, 1979• 7 citations
Publisher

Abstract

The variational method proposed by Kadanoff et al. is very useful for solving various statistical models. However, there are two major difficulties. Its numerical procedure is tedious and time consuming, and the authors considered that there exists an "internal contradiction." We have resolved both of them by developing a new variational procedure, and examining the physical meaning of the variational parameters.

DOI
10.1103/physrevb.19.529

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Unknow•Feb 1, 1982•0 citations
The variational method proposed by Kadanoff et al. is very useful for solving various statistical models. However, there are two major difficulties. Its numerical procedure is tedious and time consuming, and the authors considered that there exists an "internal contradiction." We have resolved both of them by developing a new variational procedure, and examining the physical meaning of the variational parameters.
The critical properties of Potts models with random bonds are considered in two dimensions. A position-space renormalization-group procedure, based on the Migdal-Kadanoff method, is developed. While all previous position-space calculations satisfied the Harris criterion and the resulting scaling relation only approximately, we found conditions under which these relations are exactly satisfied, and constructed our renormalization-group procedure accordingly. Numerical results for phase diagrams and thermodynamic functions for various random-bond Potts models are presented. In addition, some exact results obtained using a duality transformation, as well as an heuristic derivation of scaling properties that correspond to the percolation problem are given.
We present a systematic procedure for the direct calculation of the free energy and its first and second derivatives for the Ising model in one to three dimensions with a wide class of symmetry properties. This renormalization group method is based on the modified Kadanoff's variational method (MKUM). This method is not only general but also very accurate numerically both near and far from the critical point. Further it includes correction to scaling effects not present in the standard linearized renormalization group treatment. This work describes the techniques and presents some illustrative results for the square latttice and body-centered cubic lattice with nearest neighbor interactions. A full treatment of our results will be given elsewhere. (Author)