Physical Review Letters

Multiple-Correlation Functions and Scaling Laws

Cyril Domb

Published Jun 17, 1968• 11 citations

Abstract

The form of the multiple-correlation function in the Ornstein-Zernike approximation is investigated. As a result it is suggested that the index $\ensuremath{\nu}$, which describes the behavior of correlations in the critical region, is not determined by the indices which describe critical thermodynamic behavior. The "scaling law" relations for $\ensuremath{\nu}$ represent an approximation which is good for the Ising model in sufficiently few dimensions.

DOI: 10.1103/physrevlett.20.1425

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Physical Review Letters

Multiple-Correlation Functions and Scaling Laws

Cyril Domb

Published Jun 17, 1968• 11 citations

Publisher

Abstract

The form of the multiple-correlation function in the Ornstein-Zernike approximation is investigated. As a result it is suggested that the index

\ensuremath{\nu}

, which describes the behavior of correlations in the critical region, is not determined by the indices which describe critical thermodynamic behavior. The "scaling law" relations for

\ensuremath{\nu}

represent an approximation which is good for the Ising model in sufficiently few dimensions.

DOI: 10.1103/physrevlett.20.1425

Related papers

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Scaling laws and correlation functions near magnetic critical points

Robert Hecht

"Critical point phenomena in magnetic systems are studied with the aid of the scaling laws. These laws relate the critical indices which describe the behavior of thermodynamic and correlation functions near T. Kadanoff's derivation of the scaling 1aw is reviewed. An experimental survey of ferromagnetic and antiferromagnetic data near T is presented. Susceptibility, correlation length, magnectization, specific heat and a magnetic equation of state are discussed. It is found that the ""critical region"" can extend as far as [equation] although frequently it is restricted to E < 10-2 The critical indices S, y, v, ~ and ~, are close to the three dimensional ISing model values. Values for 0, y' and ~ are not known as conclusively. Specific heat behavior is discussed in terms -~ of a three-parameter fit, namely C = aE + b. Varying b allows one to estimate a possible range of a's. The experimental information is consistent with the scaling law relations 2 - alpha = 2 - alpha' = dV = d gamma / (2-pi) = gamma + 2B = B(d+1) The scaling law theory is extended to predict the temperature, field and spatial dependence of various spin correlation functions. These predictions are checked by analytic calculations for the two dimensional Ising model in zero field. Results are presented for the energy density-energy density and energy density-spin correlation functions in the limit E « 1, R » 1, but the product ER arbitrary. Results are also presented for a group of two, three and four spin correlations at cT. In each case the scaling law predictions are verified.""Critical point phenomena in magnetic systems are studied with the aid of the scaling laws. These laws relate the critical indices which describe the behavior of thermodynamic and correlation functions near T. Kadanoff's derivation of the scaling 1aw is reviewed. An experimental survey of ferromagnetic and antiferromagnetic data near T is presented. Susceptibility, correlation length, magnectization, specific heat and a magnetic equation of state are discussed. It is found that the ""critical region"" can extend as far as [equation] although frequently it is restricted to E < 10-2 The critical indices S, y, v, ~ and ~, are close to the three dimensional ISing model values. Values for 0, y' and ~ are not known as conclusively. Specific heat behavior is discussed in terms -~ of a three-parameter fit, namely C = aE + b. Varying b allows one to estimate a possible range of a's. The experimental information is consistent with the scaling law relations 2 - alpha = 2 - alpha' = dV = d gamma / (2-pi) = gamma + 2B = B(d+1) The scaling law theory is extended to predict the temperature, field and spatial dependence of various spin correlation functions. These predictions are checked by analytic calculations for the two dimensional Ising model in zero field. Results are presented for the energy density-energy density and energy density-spin correlation functions in the limit E « 1, R » 1, but the product ER arbitrary. Results are also presented for a group of two, three and four spin correlations at cT. In each case the scaling law predictions are verified."

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Eugene Helfand, James S. Langer

The Ising-model correlation function $C({R}_{12})=〈{\ensuremath{\mu}}_{1}{\ensuremath{\mu}}_{2}〉$ is studied in terms of a novel $N$ -fold integral representation. This formula stems from a procedure proposed by Montroll and Berlin. The integral is estimated by maximizing the integrand, an approximation related to the spherical-model assumptions. The correlation function is not of the Ornstein-Zernike type, just above the critical point, but rather $C(R)\ensuremath{\propto}{R}^{\ensuremath{-}1\ensuremath{-}\ensuremath{\eta}}$ for $R\ensuremath{\ll}\frac{1}{\ensuremath{\kappa}}$, and $C(R)\ensuremath{\propto}{\ensuremath{\kappa}}^{\ensuremath{\eta}}{R}^{\ensuremath{-}1}\mathrm{exp}(\ensuremath{-}\ensuremath{\kappa}R)$ for $R\ensuremath{\gg}\frac{1}{\ensuremath{\kappa}}$. The correlation length $\frac{1}{\ensuremath{\kappa}}$ becomes infinite at the critical point. The calculated value $\ensuremath{\eta}=0.646$ is too large, reflecting the omission of important terms in the evaluation of the integral. The unusual mechanism inducing the nonclassical behavior is carefully examined.The Ising-model correlation function $C({R}_{12})=〈{\ensuremath{\mu}}_{1}{\ensuremath{\mu}}_{2}〉$ is studied in terms of a novel $N$-fold integral representation. This formula stems from a procedure proposed by Montroll and Berlin. The integral is estimated by maximizing the integrand, an approximation related to the spherical-model assumptions. The correlation function is not of the Ornstein-Zernike type, just above the critical point, but rather $C(R)\ensuremath{\propto}{R}^{\ensuremath{-}1\ensuremath{-}\ensuremath{\eta}}$ for $R\ensuremath{\ll}\frac{1}{\ensuremath{\kappa}}$, and $C(R)\ensuremath{\propto}{\ensuremath{\kappa}}^{\ensuremath{\eta}}{R}^{\ensuremath{-}1}\mathrm{exp}(\ensuremath{-}\ensuremath{\kappa}R)$ for $R\ensuremath{\gg}\frac{1}{\ensuremath{\kappa}}$. The correlation length $\frac{1}{\ensuremath{\kappa}}$ becomes infinite at the critical point. The calculated value $\ensuremath{\eta}=0.646$ is too large, reflecting the omission of important terms in the evaluation of the integral. The unusual mechanism inducing the nonclassical behavior is carefully examined.

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Behavior of Two-Point Correlation Functions at High Temperatures

William J. Camp, Michael Ellis Fisher

The asymptotic decay of the general two-point correlation function ${G}_{\mathrm{AB}}(\stackrel{\ensuremath{\rightarrow}}{R})$ at high temperatures is analyzed on the basis of the $d$ -dimensional, spin-\textonehalf{} Ising model in general field $H$ . For $H\ensuremath{\ne}0$ the Ornstein-Zernike form $\ensuremath{\approx}\frac{{D}_{A}{D}_{B}{e}^{\ensuremath{-}\ensuremath{\kappa}R}}{{R}^{\frac{(d\ensuremath{-}1)}{2}}}$ is found for general $\stackrel{^}{A}$ and $\stackrel{^}{B}$. However for certain operators, including the energy, the amplitude of the Ornstein-Zernike term vanishes as ${H}^{2}$ and in zero field only the higher order decay $\ensuremath{\sim}\frac{{e}^{\ensuremath{-}2\ensuremath{\kappa}R}}{{R}^{d}}$ remains. The relation to approximate treatments and to critical-point phenomena is discussed briefly.The asymptotic decay of the general two-point correlation function ${G}_{\mathrm{AB}}(\stackrel{\ensuremath{\rightarrow}}{R})$ at high temperatures is analyzed on the basis of the $d$-dimensional, spin-\textonehalf{} Ising model in general field $H$. For $H\ensuremath{\ne}0$ the Ornstein-Zernike form $\ensuremath{\approx}\frac{{D}_{A}{D}_{B}{e}^{\ensuremath{-}\ensuremath{\kappa}R}}{{R}^{\frac{(d\ensuremath{-}1)}{2}}}$ is found for general $\stackrel{^}{A}$ and $\stackrel{^}{B}$. However for certain operators, including the energy, the amplitude of the Ornstein-Zernike term vanishes as ${H}^{2}$ and in zero field only the higher order decay $\ensuremath{\sim}\frac{{e}^{\ensuremath{-}2\ensuremath{\kappa}R}}{{R}^{d}}$ remains. The relation to approximate treatments and to critical-point phenomena is discussed briefly.

Physical Review LettersJan 11, 197153 citations

The form of the multiple-correlation function in the Ornstein-Zernike approximation is investigated. As a result it is suggested that the index

\ensuremath{\nu}

, which describes the behavior of correlations in the critical region, is not determined by the indices which describe critical thermodynamic behavior. The "scaling law" relations for

\ensuremath{\nu}

represent an approximation which is good for the Ising model in sufficiently few dimensions.

d=2

and

d=3

Ising models. The generalized equation changes the usual Ornstein-Zernike values of various quantities in qualitatively the correct way; however, quantitative discrepancies remain.

{T}_{c}

than above. Numerical comparison with series-expansion results agree surprisingly well. A powerful technique of evaluating diagrams using the Fourier transform of the propagator is also presented.

We consider in detail the critical behavior of a

d

-dimensional system that is finite in one of its dimensions. Such a system can exhibit four different types of critical behavior:

d

-dimensional mean field,

d

-dimensional critical, ($d\ensuremath{-}1$)-dimensional mean field, ($d\ensuremath{-}1$)-dimensional critical. By matching its behavior in the various regions we obtain equalities connecting the critical indices in

d

and $d\ensuremath{-}1$ dimensions. Assuming the usual two-dimensional critical indices of the Ising model, these relations lead to the following indices for the three-dimensional Ising model: $\ensuremath{\nu}=\frac{2}{3}$, $\ensuremath{\alpha}=0$, $\ensuremath{\beta}=\frac{3}{8}$, and $\ensuremath{\gamma}=\frac{5}{4}$.

The Ising-model correlation function $C({R}_{12})=〈{\ensuremath{\mu}}_{1}{\ensuremath{\mu}}_{2}〉$ is studied in terms of a novel

N

-fold integral representation. This formula stems from a procedure proposed by Montroll and Berlin. The integral is estimated by maximizing the integrand, an approximation related to the spherical-model assumptions. The correlation function is not of the Ornstein-Zernike type, just above the critical point, but rather $C(R)\ensuremath{\propto}{R}^{\ensuremath{-}1\ensuremath{-}\ensuremath{\eta}}$ for $R\ensuremath{\ll}\frac{1}{\ensuremath{\kappa}}$, and $C(R)\ensuremath{\propto}{\ensuremath{\kappa}}^{\ensuremath{\eta}}{R}^{\ensuremath{-}1}\mathrm{exp}(\ensuremath{-}\ensuremath{\kappa}R)$ for $R\ensuremath{\gg}\frac{1}{\ensuremath{\kappa}}$. The correlation length $\frac{1}{\ensuremath{\kappa}}$ becomes infinite at the critical point. The calculated value $\ensuremath{\eta}=0.646$ is too large, reflecting the omission of important terms in the evaluation of the integral. The unusual mechanism inducing the nonclassical behavior is carefully examined.

The asymptotic decay of the general two-point correlation function ${G}_{\mathrm{AB}}(\stackrel{\ensuremath{\rightarrow}}{R})$ at high temperatures is analyzed on the basis of the

d

-dimensional, spin-\textonehalf{} Ising model in general field

H

. For $H\ensuremath{\ne}0$ the Ornstein-Zernike form $\ensuremath{\approx}\frac{{D}_{A}{D}_{B}{e}^{\ensuremath{-}\ensuremath{\kappa}R}}{{R}^{\frac{(d\ensuremath{-}1)}{2}}}$ is found for general $\stackrel{^}{A}$ and $\stackrel{^}{B}$. However for certain operators, including the energy, the amplitude of the Ornstein-Zernike term vanishes as

{H}^{2}

and in zero field only the higher order decay $\ensuremath{\sim}\frac{{e}^{\ensuremath{-}2\ensuremath{\kappa}R}}{{R}^{d}}$ remains. The relation to approximate treatments and to critical-point phenomena is discussed briefly.