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
An and
Bn factor as
An(Q)An(L) and
Bn(Q)Bn(L), respectively. If $\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}$ is an operator involving an odd number of closely spaced spins,
An(Q) tends to a finite limit and
Bn(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,
An(Q) tends to zero and
Bn(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.
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.
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
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.
The asymptotic decay of the two-point correlation function ${G}_{\mathrm{AB}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})$ at and near a phase-separation point is discussed for
d-dimensional, spin-\textonehalf{} Ising models at low temperature. The general behavior, even on the phase boundary (
H=0), is in agreement with extended Ornstein-Zernike predictions. It is shown why the nearest-neighbor two-dimensional model in zero field is an exception. The decay of ${G}_{\mathrm{AB}}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}}}_{1},{\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}}}_{2})$ near a free surface at high temperatures agrees with phenomenological predictions using a vanishing boundary condition. At low temperature, however, the decay of correlation near a surface is exponentially slower than in the bulk.
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=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=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> 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)
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)
r1(T) is a slowly-varying decreasing function near
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 of 0.03 for
d=2 and of 0.06 to 0.09 for
d=3. Explicit formulas are given for ${\ensuremath{\kappa}}_{1}$,
r1, and $\ensuremath{\varphi}$ as functions of
T. The correlations and the scattering are isotropic near
Tc. 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>Tc. Correspondingly, $\stackrel{^}{\ensuremath{\chi}}(\mathbf{k},T)$ exhibits a maximum for fixed k, at a temperature above
Tc; for
d=2 the maxima are very well marked, but for
d=3 they are smaller and occur closer to
Tc. The theory is compared favorably with recent neutron-scattering experiments on pure beta-brass.
The direct correlation function of the one-dimensional lattice-gas (Ising) model with nearest-neighbor and next-nearest-neighbor interactions is calculated exactly. It is shown that, depending on the strengths and signs of the coupling constants, the direct correlation function can either have a finite range equal to that of the interactions, or decay exponentially in a monotonic or oscillatory fashion. At the critical point, which occurs at zero temperature, the direct correlation function is found in all cases to have exactly the range of the interactions, while its values become unbounded; this is in contrast to earlier nonrigorous results. The second moment of the direct correlation function is found to diverge at the critical point, in the neighborhood of which it becomes proportional to the correlation length of the density (spin) fluctuations, confirming a prediction of scaling theory in one dimension.