Correlation length and the scaling parameter in the Renormalization Group

The basic procedure of renormalization group theory is used to split the free energy into a Kadanoff block formation part, and a renormalized block-block interaction part. The study of this redistribution as a function of the scaling parameter s shows that there is a stationarity value s* of s, which turns out to have the same critical behavior as the correlation length. It is suggested that s* can be used as an appropriate measure and definition of the correlation length, even for noncritical regions. The calculation of s* is thereby performed explicitly for the Gaussian, and numerically for the S4 model. A sharp separation between noncorrelated and correlated regimes is also found for the Gaussian model, well above the critical temperature. For the S4 model, the results suggest that ξ is characterized by a high-temperature Gaussian branch and by a genuine S4 branch at low temperatures, connected by a “plateau” in the intermediate region.