Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical review. B, Solid state

Decay of Order in Classical Many-Body Systems. II. Ising Model at High Temperatures

William J. Camp

Published Aug 1, 1972• 23 citations
Publisher

Abstract

We employ the transfer-matrix formalism expounded in Paper I to study the decay of pair correlation functions at high temperatures in the $d$-dimensional Ising model in an arbitrary magnetic field $H$. A general correlation function decays according to $〈\ensuremath{\delta}L(\stackrel{\ensuremath{\rightarrow}}{0})\ensuremath{\delta}Q(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})〉\ensuremath{\approx}{R}^{\frac{(d\ensuremath{-}1)}{2}} {e}^{\ensuremath{-}\ensuremath{\kappa}R}({A}_{0}+{A}_{1}{R}^{\ensuremath{-}1}+\ensuremath{\cdots})+{R}^{\ensuremath{-}x}{e}^{\ensuremath{-}{\ensuremath{\kappa}}^{\ensuremath{'}}R}({B}_{0}+{B}_{1}{R}^{\ensuremath{-}1}+\ensuremath{\cdots})+\ensuremath{\cdots}$ as $R\ensuremath{\rightarrow}\ensuremath{\infty}$. For sufficiently small $H$ and high temperature $T$, the exponent $x$ is equal to the dimension of the lattice and ${\ensuremath{\kappa}}^{\ensuremath{'}}=2\ensuremath{\kappa}$. The coefficients ${A}_{n}$ and ${B}_{n}$ factor as ${A}_{n}(Q){A}_{n}(L)$ and ${B}_{n}(Q){B}_{n}(L)$, respectively. If $\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}$ is an operator involving an odd number of closely spaced spins, ${A}_{n}(Q)$ tends to a finite limit and ${B}_{n}(Q)$ tends to zero as $H$ tends to zero. In contrast, if $\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}$ involves an even number of closely spaced spins, ${A}_{n}(Q)$ tends to zero and ${B}_{n}(Q)$ to a finite limit as $H$ tends to zero. Thus, for finite $H$ an arbitrary pair correlation function verifies the Ornstein-Zernike (OZ) prediction; whereas in the zero field, (i) if both $\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}$ and $\stackrel{\ensuremath{\rightarrow}}{\mathrm{L}}$ are products of odd numbers of spin operators, ${G}_{\mathrm{LQ}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})$ will verify the OZ prediction in zero field; (ii) if both $\stackrel{\ensuremath{\rightarrow}}{\mathrm{L}}$ and $\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}$ involve even numbers of spins, ${G}_{\mathrm{LQ}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})$ fails to verify the OZ prediction in zero field; and (iii) if $\stackrel{\ensuremath{\rightarrow}}{\mathrm{L}}$ involves an odd number of spins while $\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}$ involves an even number of spins, ${G}_{\mathrm{LQ}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})$ is identically zero in zero field for all $\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}}$. The failure of the zero-field Ising model to completely verify the OZ prediction at high temperatures is due to the symmetry of this model about the $H=0$ line.

DOI
10.1103/physrevb.6.960

Related papers

Decay of Order in Classical Many-Body Systems. III. Ising Model at Low Temperature

William J. Camp

In this work the decay of correlation in the $d$-dimensional Ising model is studied at low temperatures as a function of dimensionality of the lattice and magnetic field $h$. Except for the special case of the two-dimensional zero-field nearest-neighbor lattices, the decay of correlation verifies the Ornstein-Zernike prediction ${G}_{\mathrm{AB}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})\ensuremath{\approx}{D}_{\mathrm{AB}}(d,h){R}^{\ensuremath{-}\frac{(d\ensuremath{-}1)}{2}}{e}^{\ensuremath{-}\ensuremath{\kappa}R}$. For the two-dimensional zero-field case, the Ornstein-Zernike form is replaced by the "anomalous" form ${G}_{\mathrm{AB}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})\ensuremath{\approx}{D}_{\mathrm{AB}}{R}^{\ensuremath{-}2}{e}^{\ensuremath{-}\ensuremath{\kappa}R}$. This "anomalous" result is shown to arise from the peculiarities of the spectrum of the transfer matrix in this case and is replaced by the Ornstein-Zernike result when further-neighbor forces are present. The results presented herein agree with the previously obtained exact results for the zero-field two-dimensional Ising model.

Physical review. B, Solid state•Apr 1, 1973•22 citations

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.

Physical Review Letters•Jan 11, 1971•53 citations

Decay of Order in Classical Many-Body Systems. I. Introduction and Formal Theory

William J. Camp, Michael Ellis Fisher

In this work we briefly review the Ornstein-Zernike prediction for the decay of correlation functions, extend it to treat the decay of correlation near surfaces, and then contrast this prediction with the exactly known results for the two-dimensional Ising model. We develop the transfer-matrix approach to classical statistical mechanics in sufficient generality for its use in later papers in this series, where it is employed to derive general forms for the decay of correlation functions in Ising models away from the critical point, which provide a clear explanation of the failure of the Ornstein-Zernike theory for the two-dimensional Ising model. In particular, we show that the thermodynamic behavior of a classical system with short-range interactions reduces, when the system becomes infinite in at least one dimension, to the calculation of the largest eigenvalue of the transfer matrix. Using the Perron-Fr\"obenius theorem, we show that for a system infinite in no more than one dimension, an arbitrary correlation function defined on the system decays at least exponentially fast. One is able to predict whether the decay of correlation is monotone or oscillatory on the basis of the largest few eigenvalues of the transfer matrix.

Physical review. B, Solid state•Aug 1, 1972•69 citations

Theory of critical point scattering and correlations. III. The Ising model below Tc and in a field

Howard B. Tarko, Michael Ellis Fisher

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 $H$ and temperature $T$. 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 $H$ and $T$ are found to be considerably greater than for $H=0$, $T>{T}_{c}$. 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 $T={T}_{c}$ and for $T\ensuremath{\gtrless}{T}_{c}$ with $H=0$.

Physical review. B, Solid state•Feb 1, 1975•237 citations

Scaling function for two-point correlations. II. Expansion to order 1n

Amnon Aharony A

The zero-field two-point correlation function of an $n$-vector system in $d$ dimensions is calculated to order $\frac{1}{n}$ for $T\ensuremath{\ge}{T}_{c}$ and $2<d<4$. The critical-point scaling function is obtained as a closed cutoff-independent integral. As $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}\ensuremath{\rightarrow}0$ at fixed wave vector $q$, the variation of the Fourier transform of the correlation function is ${\stackrel{^}{E}}_{1}(q){t}^{1\ensuremath{-}\ensuremath{\alpha}}+{\stackrel{^}{E}}_{2}(q)t+{\stackrel{^}{E}}_{3}(q){t}^{2}+\ensuremath{\cdots}$, where $\ensuremath{\alpha}$ is the specific-heat exponent and ${\stackrel{^}{E}}_{2}$, ${\stackrel{^}{E}}_{3}$ are of order $\frac{1}{n}$. At $d=3$ the term ${t}^{2}$ is modified by logarithmic corrections. As $q\ensuremath{\rightarrow}0$ at fixed nonzero $t$, deviations of the scaling function from the Ornstein-Zernike form ${(1+{\ensuremath{\xi}}^{2}{q}^{2})}^{\ensuremath{-}1}$ are of order $\frac{{10}^{\ensuremath{-}3}{\ensuremath{\xi}}^{4}{q}^{4}}{n}$. Results are compared with high-temperature series-expansion estimates and with the $\ensuremath{\epsilon}$-expansion results of previous work.

Physical review. B, Solid state•Oct 1, 1974•20 citations

Theory of Critical-Point Scattering and Correlations. I. The Ising Model

Michael Ellis Fisher, Robert J. Burford

The theory of the correlations and critical scattering of two- and three-dimensional nearest-neighbor Ising models is discussed critically. A distinction is drawn between $\ensuremath{\kappa}(T)$, the true inverse range of exponential decay of the correlations, and ${\ensuremath{\kappa}}_{1}(T)$, the effective range determined from the low-angle scattering intensity. Ten to eleven terms of appropriate high-temperature series exapnsions for $\ensuremath{\kappa}$ and ${\ensuremath{\kappa}}_{1}$ are determined for the square and simple cubic lattices, and shorter series are given for the triangular, fcc, and bcc lattices. For the former lattices, the complete correlation expansions are obtained to the same order. It is shown that $\ensuremath{\kappa}$ and ${\ensuremath{\kappa}}_{1}$ vary as ${(T\ensuremath{-}{T}_{0})}^{\ensuremath{\nu}}$ when $T\ensuremath{\rightarrow}{T}_{c}$, with $\ensuremath{\nu}=1$ for dimensionality $d=2$, but $\ensuremath{\nu}=0.6430\ifmmode\pm\else\textpm\fi{}0.0025\ensuremath{\simeq}\frac{9}{14}$ for $d=3$. The asymptotic decay of correlation at $T={T}_{c}$ is found to be $\frac{1}{{r}^{d\ensuremath{-}2+\ensuremath{\eta}}}$, where $\ensuremath{\eta}$ is related to the exponent $\ensuremath{\gamma}$ of the divergence of the susceptibility by $(2\ensuremath{-}\ensuremath{\eta})\ensuremath{\nu}=\ensuremath{\gamma}$, Numerical values are $\ensuremath{\eta}=\frac{1}{4}$ for $d=2$ and $\ensuremath{\eta}=0.056\ifmmode\pm\else\textpm\fi{}0.008\ensuremath{\simeq}\frac{1}{18} \mathrm{for} d=3$. The relative scattering intensity $\stackrel{^}{\ensuremath{\chi}}$ as a function of wave number k is given to high accuracy for all $T>~{T}_{c}$ by $\stackrel{^}{\ensuremath{\chi}}(\mathbf{k},T)\ensuremath{\simeq}\left(\frac{a}{{r}_{1}}\right)\frac{^{2\ensuremath{-}\ensuremath{\eta}}[{({\ensuremath{\kappa}}_{1}a)}^{2}+{\ensuremath{\varphi}}^{2}{a}^{2}{K}^{2}(\mathbf{k})]^{\frac{\ensuremath{\eta}}{2}}}{[{({\ensuremath{\kappa}}_{1}a)}^{2}+\ensuremath{\psi}{a}^{2}{K}^{2}(\mathbf{k})]},$ where (i) $a$ is the lattice spacing, (ii) ${a}^{2}{K}^{2}=2d[1\ensuremath{-}{q}^{\ensuremath{-}1}\ensuremath{\Sigma}\mathrm{exp}(i\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{r})]\ensuremath{\simeq}{(\mathrm{ka})}^{2}$, the sum being over the $q$ nearest-neighbor lattice sites, (iii) ${r}_{1}(T)$ is a slowly-varying decreasing function near ${T}_{c}$, (iv) $\ensuremath{\psi}=1+\frac{1}{2}\ensuremath{\eta}{\ensuremath{\varphi}}^{2}$, and $(\mathrm{v}) \ensuremath{\varphi}(T)$ is slowly varying with a magnitude at ${T}_{c}$ of 0.03 for $d=2$ and of 0.06 to 0.09 for $d=3$. Explicit formulas are given for ${\ensuremath{\kappa}}_{1}$, ${r}_{1}$, and $\ensuremath{\varphi}$ as functions of $T$. The correlations and the scattering are isotropic near ${T}_{c}$. The critical scattering isotherm is curved for low $k$ according to ${\stackrel{^}{\ensuremath{\chi}}}^{\ensuremath{-}1}\ensuremath{\sim}{k}^{2\ensuremath{-}\ensuremath{\eta}}$ and it intersects the isotherms for $T>{T}_{c}$. Correspondingly, $\stackrel{^}{\ensuremath{\chi}}(\mathbf{k},T)$ exhibits a maximum for fixed k, at a temperature above ${T}_{c}$; for $d=2$ the maxima are very well marked, but for $d=3$ they are smaller and occur closer to ${T}_{c}$. The theory is compared favorably with recent neutron-scattering experiments on pure beta-brass.

Physical Review•Apr 10, 1967•753 citations
We employ the transfer-matrix formalism expounded in Paper I to study the decay of pair correlation functions at high temperatures in the ddd-dimensional Ising model in an arbitrary magnetic field HHH. A general correlation function decays according to 〈\ensuremathδL(0\ensuremath→)\ensuremathδQ(R\ensuremath→)〉\ensuremath≈R(d\ensuremath−1)2e\ensuremath−\ensuremathκR(A0+A1R\ensuremath−1+\ensuremath⋯)+R\ensuremath−xe\ensuremath−\ensuremathκ\ensuremath′R(B0+B1R\ensuremath−1+\ensuremath⋯)+\ensuremath⋯〈\ensuremath{\delta}L(\stackrel{\ensuremath{\rightarrow}}{0})\ensuremath{\delta}Q(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})〉\ensuremath{\approx}{R}^{\frac{(d\ensuremath{-}1)}{2}} {e}^{\ensuremath{-}\ensuremath{\kappa}R}({A}_{0}+{A}_{1}{R}^{\ensuremath{-}1}+\ensuremath{\cdots})+{R}^{\ensuremath{-}x}{e}^{\ensuremath{-}{\ensuremath{\kappa}}^{\ensuremath{'}}R}({B}_{0}+{B}_{1}{R}^{\ensuremath{-}1}+\ensuremath{\cdots})+\ensuremath{\cdots}〈\ensuremathδL(0\ensuremath→​)\ensuremathδQ(R\ensuremath→​)〉\ensuremath≈R2(d\ensuremath−1)​e\ensuremath−\ensuremathκR(A0​+A1​R\ensuremath−1+\ensuremath⋯)+R\ensuremath−xe\ensuremath−\ensuremathκ\ensuremath′R(B0​+B1​R\ensuremath−1+\ensuremath⋯)+\ensuremath⋯ as R\ensuremath→\ensuremath∞R\ensuremath{\rightarrow}\ensuremath{\infty}R\ensuremath→\ensuremath∞. For sufficiently small HHH and high temperature TTT, the exponent xxx is equal to the dimension of the lattice and \ensuremathκ\ensuremath′=2\ensuremathκ{\ensuremath{\kappa}}^{\ensuremath{'}}=2\ensuremath{\kappa}\ensuremathκ\ensuremath′=2\ensuremathκ. The coefficients An{A}_{n}An​ and Bn{B}_{n}Bn​ factor as An(Q)An(L){A}_{n}(Q){A}_{n}(L)An​(Q)An​(L) and Bn(Q)Bn(L){B}_{n}(Q){B}_{n}(L)Bn​(Q)Bn​(L), respectively. If Q\ensuremath→\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}Q\ensuremath→​ is an operator involving an odd number of closely spaced spins, An(Q){A}_{n}(Q)An​(Q) tends to a finite limit and Bn(Q){B}_{n}(Q)Bn​(Q) tends to zero as HHH tends to zero. In contrast, if Q\ensuremath→\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}Q\ensuremath→​ involves an even number of closely spaced spins, An(Q){A}_{n}(Q)An​(Q) tends to zero and Bn(Q){B}_{n}(Q)Bn​(Q) to a finite limit as HHH tends to zero. Thus, for finite HHH an arbitrary pair correlation function verifies the Ornstein-Zernike (OZ) prediction; whereas in the zero field, (i) if both Q\ensuremath→\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}Q\ensuremath→​ and L\ensuremath→\stackrel{\ensuremath{\rightarrow}}{\mathrm{L}}L\ensuremath→​ are products of odd numbers of spin operators, GLQ(R\ensuremath→){G}_{\mathrm{LQ}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})GLQ​(R\ensuremath→​) will verify the OZ prediction in zero field; (ii) if both L\ensuremath→\stackrel{\ensuremath{\rightarrow}}{\mathrm{L}}L\ensuremath→​ and Q\ensuremath→\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}Q\ensuremath→​ involve even numbers of spins, GLQ(R\ensuremath→){G}_{\mathrm{LQ}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})GLQ​(R\ensuremath→​) fails to verify the OZ prediction in zero field; and (iii) if L\ensuremath→\stackrel{\ensuremath{\rightarrow}}{\mathrm{L}}L\ensuremath→​ involves an odd number of spins while Q\ensuremath→\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}Q\ensuremath→​ involves an even number of spins, GLQ(R\ensuremath→){G}_{\mathrm{LQ}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})GLQ​(R\ensuremath→​) is identically zero in zero field for all R\ensuremath→\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}}R\ensuremath→​. The failure of the zero-field Ising model to completely verify the OZ prediction at high temperatures is due to the symmetry of this model about the H=0H=0H=0 line.
In this work the decay of correlation in the ddd-dimensional Ising model is studied at low temperatures as a function of dimensionality of the lattice and magnetic field hhh. Except for the special case of the two-dimensional zero-field nearest-neighbor lattices, the decay of correlation verifies the Ornstein-Zernike prediction ${G}_{\mathrm{AB}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})\ensuremath{\approx}{D}_{\mathrm{AB}}(d,h){R}^{\ensuremath{-}\frac{(d\ensuremath{-}1)}{2}}{e}^{\ensuremath{-}\ensuremath{\kappa}R}$. For the two-dimensional zero-field case, the Ornstein-Zernike form is replaced by the "anomalous" form ${G}_{\mathrm{AB}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})\ensuremath{\approx}{D}_{\mathrm{AB}}{R}^{\ensuremath{-}2}{e}^{\ensuremath{-}\ensuremath{\kappa}R}$. This "anomalous" result is shown to arise from the peculiarities of the spectrum of the transfer matrix in this case and is replaced by the Ornstein-Zernike result when further-neighbor forces are present. The results presented herein agree with the previously obtained exact results for the zero-field two-dimensional Ising model.
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 ddd-dimensional, spin-\textonehalf{} Ising model in general field HHH. 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 H2{H}^{2}H2 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.
In this work we briefly review the Ornstein-Zernike prediction for the decay of correlation functions, extend it to treat the decay of correlation near surfaces, and then contrast this prediction with the exactly known results for the two-dimensional Ising model. We develop the transfer-matrix approach to classical statistical mechanics in sufficient generality for its use in later papers in this series, where it is employed to derive general forms for the decay of correlation functions in Ising models away from the critical point, which provide a clear explanation of the failure of the Ornstein-Zernike theory for the two-dimensional Ising model. In particular, we show that the thermodynamic behavior of a classical system with short-range interactions reduces, when the system becomes infinite in at least one dimension, to the calculation of the largest eigenvalue of the transfer matrix. Using the Perron-Fr\"obenius theorem, we show that for a system infinite in no more than one dimension, an arbitrary correlation function defined on the system decays at least exponentially fast. One is able to predict whether the decay of correlation is monotone or oscillatory on the basis of the largest few eigenvalues of the transfer matrix.
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.
The zero-field two-point correlation function of an nnn-vector system in ddd dimensions is calculated to order 1n\frac{1}{n}n1​ for $T\ensuremath{\ge}{T}_{c}$ and 2<d<42<d<42<d<4. The critical-point scaling function is obtained as a closed cutoff-independent integral. As $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}\ensuremath{\rightarrow}0$ at fixed wave vector qqq, the variation of the Fourier transform of the correlation function is ${\stackrel{^}{E}}_{1}(q){t}^{1\ensuremath{-}\ensuremath{\alpha}}+{\stackrel{^}{E}}_{2}(q)t+{\stackrel{^}{E}}_{3}(q){t}^{2}+\ensuremath{\cdots}$, where $\ensuremath{\alpha}$ is the specific-heat exponent and ${\stackrel{^}{E}}_{2}$, ${\stackrel{^}{E}}_{3}$ are of order 1n\frac{1}{n}n1​. At d=3d=3d=3 the term t2{t}^{2}t2 is modified by logarithmic corrections. As $q\ensuremath{\rightarrow}0$ at fixed nonzero ttt, deviations of the scaling function from the Ornstein-Zernike form ${(1+{\ensuremath{\xi}}^{2}{q}^{2})}^{\ensuremath{-}1}$ are of order $\frac{{10}^{\ensuremath{-}3}{\ensuremath{\xi}}^{4}{q}^{4}}{n}$. Results are compared with high-temperature series-expansion estimates and with the $\ensuremath{\epsilon}$-expansion results of previous work.
The theory of the correlations and critical scattering of two- and three-dimensional nearest-neighbor Ising models is discussed critically. A distinction is drawn between $\ensuremath{\kappa}(T)$, the true inverse range of exponential decay of the correlations, and ${\ensuremath{\kappa}}_{1}(T)$, the effective range determined from the low-angle scattering intensity. Ten to eleven terms of appropriate high-temperature series exapnsions for $\ensuremath{\kappa}$ and ${\ensuremath{\kappa}}_{1}$ are determined for the square and simple cubic lattices, and shorter series are given for the triangular, fcc, and bcc lattices. For the former lattices, the complete correlation expansions are obtained to the same order. It is shown that $\ensuremath{\kappa}$ and ${\ensuremath{\kappa}}_{1}$ vary as ${(T\ensuremath{-}{T}_{0})}^{\ensuremath{\nu}}$ when $T\ensuremath{\rightarrow}{T}_{c}$, with $\ensuremath{\nu}=1$ for dimensionality d=2d=2d=2, but $\ensuremath{\nu}=0.6430\ifmmode\pm\else\textpm\fi{}0.0025\ensuremath{\simeq}\frac{9}{14}$ for d=3d=3d=3. The asymptotic decay of correlation at T=TcT={T}_{c}T=Tc​ is found to be $\frac{1}{{r}^{d\ensuremath{-}2+\ensuremath{\eta}}}$, where $\ensuremath{\eta}$ is related to the exponent $\ensuremath{\gamma}$ of the divergence of the susceptibility by $(2\ensuremath{-}\ensuremath{\eta})\ensuremath{\nu}=\ensuremath{\gamma}$, Numerical values are $\ensuremath{\eta}=\frac{1}{4}$ for d=2d=2d=2 and $\ensuremath{\eta}=0.056\ifmmode\pm\else\textpm\fi{}0.008\ensuremath{\simeq}\frac{1}{18} \mathrm{for} d=3$. The relative scattering intensity $\stackrel{^}{\ensuremath{\chi}}$ as a function of wave number k is given to high accuracy for all T> TcT>~{T}_{c}T> Tc​ by $\stackrel{^}{\ensuremath{\chi}}(\mathbf{k},T)\ensuremath{\simeq}\left(\frac{a}{{r}_{1}}\right)\frac{^{2\ensuremath{-}\ensuremath{\eta}}[{({\ensuremath{\kappa}}_{1}a)}^{2}+{\ensuremath{\varphi}}^{2}{a}^{2}{K}^{2}(\mathbf{k})]^{\frac{\ensuremath{\eta}}{2}}}{[{({\ensuremath{\kappa}}_{1}a)}^{2}+\ensuremath{\psi}{a}^{2}{K}^{2}(\mathbf{k})]},$ where (i) aaa is the lattice spacing, (ii) ${a}^{2}{K}^{2}=2d[1\ensuremath{-}{q}^{\ensuremath{-}1}\ensuremath{\Sigma}\mathrm{exp}(i\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{r})]\ensuremath{\simeq}{(\mathrm{ka})}^{2}$, the sum being over the qqq nearest-neighbor lattice sites, (iii) r1(T){r}_{1}(T)r1​(T) is a slowly-varying decreasing function near Tc{T}_{c}Tc​, (iv) $\ensuremath{\psi}=1+\frac{1}{2}\ensuremath{\eta}{\ensuremath{\varphi}}^{2}$, and $(\mathrm{v}) \ensuremath{\varphi}(T)$ is slowly varying with a magnitude at Tc{T}_{c}Tc​ of 0.03 for d=2d=2d=2 and of 0.06 to 0.09 for d=3d=3d=3. Explicit formulas are given for ${\ensuremath{\kappa}}_{1}$, r1{r}_{1}r1​, and $\ensuremath{\varphi}$ as functions of TTT. The correlations and the scattering are isotropic near Tc{T}_{c}Tc​. The critical scattering isotherm is curved for low kkk according to ${\stackrel{^}{\ensuremath{\chi}}}^{\ensuremath{-}1}\ensuremath{\sim}{k}^{2\ensuremath{-}\ensuremath{\eta}}$ and it intersects the isotherms for T>TcT>{T}_{c}T>Tc​. Correspondingly, $\stackrel{^}{\ensuremath{\chi}}(\mathbf{k},T)$ exhibits a maximum for fixed k, at a temperature above Tc{T}_{c}Tc​; for d=2d=2d=2 the maxima are very well marked, but for d=3d=3d=3 they are smaller and occur closer to Tc{T}_{c}Tc​. The theory is compared favorably with recent neutron-scattering experiments on pure beta-brass.