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Physica A Statistical Mechanics and its Applications

On the equivalence of finite size scaling renormalization group and phenomenological renormalization

J. A. Plascak, J. Kamphorst Leal da Silva

Published Nov 1, 1996• 5 citations
Publisher
DOI
10.1016/s0378-4371(96)00227-0

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Physical review. B, Condensed matter•Nov 1, 1984•1 citation
Some renormalization group approaches have been proposed during the last few years which are close in spirit to the Nightingale phenomenological procedure. In essence, by exploiting the finite size scaling hypothesis, the approximate critical behavior of model on infinite lattice is obtained through the exact computation of some thermal quantities of the model on finite clusters. In this work some of these methods are reviewed, namely the mean field renormalization group, the effective field renormalization group and the finite size scaling renormalization group procedures. Although special emphasis is given to the mean field renormalization group (since it has been, up to now, much more used to study a wide variety of different systems) a discussion of their potentialities and interrelations to other methods is also presented.
We performed phenomenological renormalization-group calculations for ferroelectric 12- and 16-vertex models on a square lattice with periodic and helical boundary conditions. We considered strips of infinite length and finite widths ($n=1\ensuremath{-}7,8, or 9$). The extrapolated values for the transition temperature of the 12-vertex model, which has been used for assessing the transition in squaric acid, are lower than the predictions of the Bethe approximation. The estimates for the critical exponent $\ensuremath{\nu}$ do not allow a definite conclusion about its asymptotic behavior, although the Ising value $\ensuremath{\nu}=1$ seems more plausible. The estimates for the 16-vertex model considered in this paper, which is equivalent to an anisotropic nearest-neighbor Ising model, show an excellent convergence to the exact values. Also, we analyze the finite-size scaling behavior of the critical free energy of both models.