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Journal of Statistical Physics

Finite-size scaling and the renormalization group

Joseph Rudnick, Hong Min Guo, David Jasnow

Published Nov 1, 1985• 157 citations
Publisher
DOI
10.1007/bf01009013

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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 present a status report on the ongoing analysis of the 3D Ising model with nearest-neighbor interactions using the Monte Carlo Renormalization Group (MCRG) and finite size scaling (FSS) methods on 643, 1283, and 2563 simple cubic lattices. Our MCRG estimates are [Formula: see text] and ν = 0.625(1). The FSS results for Kc are consistent with those from MCRG but the value of ν is not. Our best estimate η = 0.025(6) covers the spread in the MCRG and FSS values. A surprise of our calculation is the estimate ω ≈ 0.7 for the correction-to-scaling exponent. We also present results for the renormalized coupling gR along the MCRG flow and argue that the data support the validity of hyperscaling for the 3D Ising model.