Some difficulties associated with parameter-dependent renormalization-group transformations

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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.