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Physical Review

Critical Correlations in the Ising Model

Eugene Helfand, James S. Langer

Published Aug 10, 1967• 24 citations
Publisher

Abstract

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.

DOI
10.1103/physrev.160.437

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The Ising-model correlation function C(R12)=〈\ensuremathμ1\ensuremathμ2〉C({R}_{12})=〈{\ensuremath{\mu}}_{1}{\ensuremath{\mu}}_{2}〉C(R12​)=〈\ensuremathμ1​\ensuremathμ2​〉 is studied in terms of a novel NNN-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∝R\ensuremath−1\ensuremath−\ensuremathηC(R)\ensuremath{\propto}{R}^{\ensuremath{-}1\ensuremath{-}\ensuremath{\eta}}C(R)\ensuremath∝R\ensuremath−1\ensuremath−\ensuremathη for R\ensuremath≪1\ensuremathκR\ensuremath{\ll}\frac{1}{\ensuremath{\kappa}}R\ensuremath≪\ensuremathκ1​, and C(R)\ensuremath∝\ensuremathκ\ensuremathηR\ensuremath−1exp(\ensuremath−\ensuremathκR)C(R)\ensuremath{\propto}{\ensuremath{\kappa}}^{\ensuremath{\eta}}{R}^{\ensuremath{-}1}\mathrm{exp}(\ensuremath{-}\ensuremath{\kappa}R)C(R)\ensuremath∝\ensuremathκ\ensuremathηR\ensuremath−1exp(\ensuremath−\ensuremathκR) for R\ensuremath≫1\ensuremathκR\ensuremath{\gg}\frac{1}{\ensuremath{\kappa}}R\ensuremath≫\ensuremathκ1​. The correlation length 1\ensuremathκ\frac{1}{\ensuremath{\kappa}}\ensuremathκ1​ becomes infinite at the critical point. The calculated value \ensuremathη=0.646\ensuremath{\eta}=0.646\ensuremathη=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.
Power-series expansions of the spin-pair correlation functions of the square, sc, and bcc Ising lattices have been obtained using the semi-invariant approach for general field HHH and temperature TTT. In three dimensions P\'ade-approximant analyses indicate $2{v}^{\ensuremath{'}}=1.285\ifmmode\pm\else\textpm\fi{}0.020$, in agreement with scaling, but $2{v}^{c}\ensuremath{\simeq}0.835\ifmmode\pm\else\textpm\fi{}0.02$ which is some (1-3)% above the scaling prediction ${v}^{c}=\frac{v}{\ensuremath{\beta}\ensuremath{\delta}}\ensuremath{\simeq}0.823$. However, ratio techniques reveal that this discrepancy can be attributed to significant confluent critical singularities. Cubic and quintic parametric representations of the critical equation of state, and corresponding expressions for the correlation length, ${\ensuremath{\xi}}_{1}(H, T)$, are developed, which are considerable improvements over the linear model. The universality and spherical symmetry of the critical scattering intensity $\stackrel{^}{\ensuremath{\chi}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}, H, T)$ is confirmed to within (1-2)% by estimating suitable invariant combinations of amplitudes. The deviations from Ornstein-Zernike theory for general HHH and TTT are found to be considerably greater than for H=0H=0H=0, T>TcT>{T}_{c}T>Tc​. Complete parametric scaling representations of $\stackrel{^}{\ensuremath{\chi}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}, H, T)$ are developed for three dimensions; corresponding scaling approximants are constructed in two dimensions but only for T=TcT={T}_{c}T=Tc​ and for $T\ensuremath{\gtrless}{T}_{c}$ with H=0H=0H=0.
We show the way perturbation procedures originated by Brout and Hemmer and further developed by Lebowitz and co-workers can be used to study the critical behavior of lattice systems with a potential v(r)v(\mathrm{r})v(r) having a weak, long-ranged tail that approaches ${\ensuremath{\gamma}}^{d}\ensuremath{\phi}(\ensuremath{\gamma}\mathrm{r})$ as $\ensuremath{\gamma}$ approaches zero, where ddd is dimensionality. We consider both the Ising and the spherical models, and note that the results for the latter model are also those that follow from the Ornstein-Zernike assumption that the direct correlation function behaves like $\ensuremath{-}\frac{v(\mathrm{r})}{\mathrm{kT}}$, when $v(\mathrm{r})\ensuremath{\ll}\mathrm{kT}$. We recover by a graph technique the previously known result that in one dimension, quantities of interest such as the inverse correlation length $\ensuremath{\kappa}$ cannot be expanded in $\ensuremath{\gamma}$ at the mean-field critical temperature T0{T}_{0}T0​ and density ${\ensuremath{\rho}}_{0}$ that characterize the critical point in the $\ensuremath{\gamma}\ensuremath{\rightarrow}0$ limit. Instead, at (${T}_{0}, {\ensuremath{\rho}}_{0}$), $\ensuremath{\kappa}\ensuremath{\sim}{\ensuremath{\gamma}}^{\frac{4}{3}}$ as $\ensuremath{\gamma}\ensuremath{\rightarrow}0$. This is true for both the Ising and the spherical models. In the two-dimensional spherical model, we find $\ensuremath{\kappa}$ to vary as ${\ensuremath{\gamma}}^{2}{[\mathrm{ln}(\frac{1}{\ensuremath{\gamma}})]}^{\frac{1}{2}}$ when $\ensuremath{\gamma}\ensuremath{\rightarrow}0$, while in three dimensions $\ensuremath{\kappa}\ensuremath{\sim}{\ensuremath{\gamma}}^{\frac{5}{2}}$. In the Ising case for d=1d=1d=1, we characterize topologically the infinite sum of graphs that contribute to the $\ensuremath{\kappa}\ensuremath{\sim}{\ensuremath{\gamma}}^{\frac{4}{3}}$ term in the expansion of the pair correlation function. (In the spherical model, a chain graph gives the entire contribution to the pair correlation function for all ddd). For d=3d=3d=3 we do not attempt to deal with the correlation length in the Ising case, but instead consider the shift in the critical temperature as $\ensuremath{\gamma}\ensuremath{\rightarrow}0$. We find that the spherical-model result that gives a shift of order ${\ensuremath{\gamma}}^{3}$ is exact to that order in $\ensuremath{\gamma}$ for the Ising model. We also find that the lowest-order correction to the spherical-model result is of the form ($\mathrm{const}{\ensuremath{\gamma}}^{6}\mathrm{ln}(\mathrm{const}\ensuremath{\gamma})$, in agreement with recent work by Thouless; but in general we expect to find terms of order ${\ensuremath{\gamma}}^{4}$ and ${\ensuremath{\gamma}}^{5}$ from the spherical-model result dominating this correction.
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We study the asymptotic behavior, for large separations, of the spin-spin correlation function $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{M,N}〉$ in the two-dimensional Ising model, where the two spins are not necessarily on the same row. Besides the limiting value for infinite separation, which is the square of the spontaneous magnetization, we evaluate the two leading terms in the asymptotic expression in each of the two cases T<TcT<{T}_{c}T<Tc​ and T>TcT>{T}_{c}T>Tc​. It is found that the nearest singularity of the generating function for the correlation is quite simple in the case T>TcT>{T}_{c}T>Tc​, but much more complicated for T<TcT<{T}_{c}T<Tc​. In an Appendix, we also give exactly in a very simple form the correlation $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{N,N}〉$ for symmetrical Ising lattice at the critical temperature Tc{T}_{c}Tc​.
The partition function of the Ising model with NNN spins, written as a functional integral, is approximated by replacing the field ${\ensuremath{\psi}}_{l}$ at the lattice site lll by the root-mean-square value of these fields, and evaluating the integral by a saddle-point method. A phase transition, whose mechanism is mathematically similar to that of the spherical model, is predicted at a temperature lower than the Weiss critical temperature. After parametrizing the mean-square field, one may apply a general variational principle to choose the parameters which minimize the difference between the approximate free energy and the exact Ising-model free energy. This optimum choice leads to the Horwitz-Callen theory.