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 ${S}^{4}$ model. A sharp separation between noncorrelated and correlated regimes is also found for the Gaussian model, well above the critical temperature. For the ${S}^{4}$ model, the results suggest that $\ensuremath{\xi}$ is characterized by a high-temperature Gaussian branch and by a genuine ${S}^{4}$ branch at low temperatures, connected by a ``plateau'' in the intermediate region.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 ${S}^{4}$ model. A sharp separation between noncorrelated and correlated regimes is also found for the Gaussian model, well above the critical temperature. For the ${S}^{4}$ model, the results suggest that $\ensuremath{\xi}$ is characterized by a high-temperature Gaussian branch and by a genuine ${S}^{4}$ branch at low temperatures, connected by a ``plateau'' in the intermediate region.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsMar 1, 19982 citations

Block-spin renormalization group in the large- n limit

J. E. Hirsch, Stephen H. Shenker

We present a real-space renormalization-group (RG) study of the $n$ -component classical Heisenberg model in the large- $n$ limit. We obtain exact expressions for correlation functions of block spins, given those of the original spins. We show how to extract the critical behavior of the model from such information. The model provides a good testing ground for evaluating the effectiveness of Monte Carlo RG techniques. We study numerically the implementation of the real-space RG transformation on finite lattices. In two dimensions, we obtain the $\ensuremath{\beta}$ function as a function of temperature. In three dimensions, we calculate the critical temperature and the thermal exponent. We discuss the effects of the use of finite lattices and of the use of different forms for the initial action on the accuracy of the results.We present a real-space renormalization-group (RG) study of the $n$-component classical Heisenberg model in the large-$n$ limit. We obtain exact expressions for correlation functions of block spins, given those of the original spins. We show how to extract the critical behavior of the model from such information. The model provides a good testing ground for evaluating the effectiveness of Monte Carlo RG techniques. We study numerically the implementation of the real-space RG transformation on finite lattices. In two dimensions, we obtain the $\ensuremath{\beta}$ function as a function of temperature. In three dimensions, we calculate the critical temperature and the thermal exponent. We discuss the effects of the use of finite lattices and of the use of different forms for the initial action on the accuracy of the results.

Physical review. B, Condensed matterFeb 1, 198336 citations

Renormalization Group

G. Benfatto, Giovanni Gallavotti

Princeton University Press eBooksNov 10, 20207 citations

s

shows that there is a stationarity value

{s}^{*}

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

{S}^{4}

model. A sharp separation between noncorrelated and correlated regimes is also found for the Gaussian model, well above the critical temperature. For the

{S}^{4}

model, the results suggest that $\ensuremath{\xi}$ is characterized by a high-temperature Gaussian branch and by a genuine

{S}^{4}

branch at low temperatures, connected by a ``plateau'' in the intermediate region.

We present a real-space renormalization-group (RG) study of the

n

-component classical Heisenberg model in the large-

n

limit. We obtain exact expressions for correlation functions of block spins, given those of the original spins. We show how to extract the critical behavior of the model from such information. The model provides a good testing ground for evaluating the effectiveness of Monte Carlo RG techniques. We study numerically the implementation of the real-space RG transformation on finite lattices. In two dimensions, we obtain the $\ensuremath{\beta}$ function as a function of temperature. In three dimensions, we calculate the critical temperature and the thermal exponent. We discuss the effects of the use of finite lattices and of the use of different forms for the initial action on the accuracy of the results.