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Physical review. B, Condensed matter

Some difficulties associated with parameter-dependent renormalization-group transformations

Michel Droz, Marcel Ausloos, J. D. Gunton

Published Jul 1, 1978• 5 citations
Publisher

Abstract

Certain difficulties associated with using a parameter-dependent real-space renormalization group are illustrated by a study of the one-dimensional Blume-Emery-Griffiths (BEG) model. In particular, it is noted that renormalization trajectories which are determined by optimizing the parameter-dependent free energy do not necessarily preserve the symmetries of the initial Hamiltonian and therefore can lead to incorrect results. This difficulty arises in a variational treatment of the BEG model but can be resolved by using a transformation which is the product of two successive variational renormalization transformations.

DOI
10.1103/physrevb.18.388

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Certain difficulties associated with using a parameter-dependent real-space renormalization group are illustrated by a study of the one-dimensional Blume-Emery-Griffiths (BEG) model. In particular, it is noted that renormalization trajectories which are determined by optimizing the parameter-dependent free energy do not necessarily preserve the symmetries of the initial Hamiltonian and therefore can lead to incorrect results. This difficulty arises in a variational treatment of the BEG model but can be resolved by using a transformation which is the product of two successive variational renormalization transformations.
We investigate the three-dimensional semi-infinite Blume-Emery-Griffiths (BEG) model using the Migdal-Kadanoff renormalization-group method. The parameter space is six dimensional. We find 69 fixed points describing a large variety of critical behaviors and discuss various semi-infinite models which are special cases of the BEG model such as the Blume-Capel model. For this last model we determine the various generic types of phase diagrams according to the values of the interactions on the surface and in the bulk of the system.
Under certain conditions, the renormalization-group flow of models in statistical mechanics can change dramatically under just very small changes of given external parameters. This can typically occur close to bifurcations of fixed points, close to the complete disappearance of fixed points, or in regions where the renormalization-group flow becomes chaotic. We present some explicit examples of these phenomena for the case of a Lie group valued spin-model analyzed by means of a variational real-space renormalization group. By directly computing the free energy of these models around the parameter regions in which such nontrivial modifications of the renormalization-group flow occur, we can extract the physical consequences of these phenomena.
For a Landau-Ginzburg-Wilson Hamiltonian of any given symmetry we show how one can find a group GT{G}_{T}GT​ of orthogonal transformations in parameter space, which commute with renormalization-group transformations. Then a renormalization-group transformation may be expanded into covariants of GT{G}_{T}GT​. We also present a systematic procedure for finding fixed points; they are most likely to decouple the Hamiltonian or to increase its symmetry. The merit of the conclusions obtained is illustrated using an example of a system with C4{C}_{4}C4​ symmetry. Agreement with the results of $\ensuremath{\epsilon}$-expansion calculations has been found.