We compute exactly the spin-spin correlation functions $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{M,N}〉$ for the two-dimensional Ising model on a square lattice in zero magnetic field for
T>Tc and
T<Tc. We then analyze the correlation functions in the scaling limit $T\ensuremath{\rightarrow}{T}_{c},{M}^{2}+{N}^{2}\ensuremath{\rightarrow}\ensuremath{\infty}$ such that $(T\ensuremath{-}{T}_{c})$ is fixed. In this scaling limit $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{M,N}〉={R}^{\ensuremath{-}\frac{1}{4}}{F}_{\ifmmode\pm\else\textpm\fi{}}(t)+{R}^{\ensuremath{-}\frac{5}{4}}{F}_{1\ifmmode\pm\else\textpm\fi{}}(t)+o({R}^{\ensuremath{-}\frac{5}{4}})$, where
t is the scaling variable $\frac{R}{\ensuremath{\xi}}$ and ${F}_{\ifmmode\pm\else\textpm\fi{}}(t)$ and ${F}_{1\ifmmode\pm\else\textpm\fi{}}(t)$ are the scaling functions ($\ensuremath{\xi}$ is the correlation length). We derive exact expressions for these scaling functions, in terms of a Painlev\'e function of the third kind and analyze both the small- and large-
t behavior. A table of values for ${F}_{\ifmmode\pm\else\textpm\fi{}}(t)$ (good to ten significant digits) is also given. As an application we computer the coefficients ${C}_{0\ifmmode\pm\else\textpm\fi{}}$ and ${C}_{1\ifmmode\pm\else\textpm\fi{}}$ in the expansion ${k}_{B}T\ensuremath{\chi}(T)={C}_{0\ifmmode\pm\else\textpm\fi{}}{|1\ensuremath{-}\frac{{T}_{c}}{T}|}^{\ensuremath{-}\frac{7}{4}}+{C}_{1\ifmmode\pm\else\textpm\fi{}}{|1\ensuremath{-}\frac{{T}_{c}}{T}|}^{\ensuremath{-}\frac{3}{4}}+O(1)$ of the zero-field susceptibility $\ensuremath{\chi}(T)$ as $T\ensuremath{\rightarrow}{T}_{c}^{\ifmmode\pm\else\textpm\fi{}}$.
The energy-density---energy-density and energy-density---spin correlation functions are evaluated for the two-dimensional square Ising model. Results are obtained at zero magnetic field and near the critical point, i.e., for $\ensuremath{\epsilon}=(\frac{4J}{\mathrm{kT}})|(T\ensuremath{-}\frac{{T}_{c}}{{T}_{c}})|\ensuremath{\ll}1.$ The correlation functions are ${〈{E}_{R}{E}_{0}〉\ensuremath{-}{〈E〉}^{2},={(\frac{2J}{\ensuremath{\pi}})}^{2}{\ensuremath{\epsilon}}^{2}[{{K}_{1}}^{2}(\ensuremath{\epsilon}R)\ensuremath{-}{{K}_{0}}^{2}(\ensuremath{\epsilon}R)],}{〈{E}_{R}{\ensuremath{\sigma}}_{0}〉\ensuremath{-}〈E〉〈\ensuremath{\sigma}〉,=(\frac{2J}{\ensuremath{\pi}})\ensuremath{\epsilon}〈\ensuremath{\sigma}〉\ensuremath{\int}{2\ensuremath{\epsilon}R}^{\ensuremath{\infty}}\mathrm{ds} {s}^{\ensuremath{-}2}{e}^{\ensuremath{-}8},}$ where
R is the distance between the points in question and
K0 and
K1 are modified Bessel functions of the second kind. Very similar results hold if the coupling constants are different in the two directions. The above results agree in form with the structure predicted by the "scaling-law" approach.
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>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=Tc and for $T\ensuremath{\gtrless}{T}_{c}$ with
H=0.
We calculate the scaled constant-magnetization and constant magnetic field specific heats $\frac{[{C}_{M}(\ensuremath{\epsilon},M)\ensuremath{-}{C}_{M}(\ensuremath{\epsilon},0)]}{{M}^{\ensuremath{-}\frac{\ensuremath{\alpha}}{\ensuremath{\beta}}}}$ and $\frac{[{C}_{H}(\ensuremath{\epsilon},H)\ensuremath{-}{C}_{H}(\ensuremath{\epsilon},0)]}{{H}^{\ensuremath{-}\frac{\ensuremath{\alpha}}{\ensuremath{\beta}\ensuremath{\delta}}}}$ as functions of the scaled variable $x\ensuremath{\equiv}\frac{\ensuremath{\epsilon}}{{M}^{\frac{1}{\ensuremath{\beta}}}}$ and $y\ensuremath{\equiv}\frac{\ensuremath{\epsilon}}{{H}^{\frac{1}{\ensuremath{\beta}\ensuremath{\delta}}}}$, respectively [where $\ensuremath{\epsilon}\ensuremath{\equiv}\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$ and
Tc is the critical temperature],for the spin-\textonehalf{} Ising model (bcc lattice) and the spin-\textonehalf{}, $\ensuremath{\infty}$ Heisenberg models (fcc lattice). Our calculations are based on previously calculated scaling functions for the
M(H,T) equation of state. We also calculate the amplitudes of the zero-field specific heat for
T>Tc and
T<Tc. Thus, we obtain the functions ${C}_{M}(\ensuremath{\epsilon},M)$ and ${C}_{H}(\ensuremath{\epsilon},H)$, though for the Heisenberg models we cannot obtain the finite nonzero constant
CM(0,0)=CH(0,0). We compare our calculated functions with the data of Connelly, Loomis, and Mapother on nickel.
We study the critical properties of the random field Ising model in general dimension d using high-temperature expansions for the susceptibility, \ensuremath{\chi}=${\ensuremath{\sum}}_{\mathit{j}}$ [〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}$${\mathrm{\ensuremath{\sigma}}}_{\mathit{j}}$
〉T-〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}$
〉T〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{j}}$
〉T]h and the structure factor, G=${\ensuremath{\sum}}_{\mathit{j}}$[〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}$${\mathrm{\ensuremath{\sigma}}}_{\mathit{j}}$
〉T]h, where 〈
〉T indicates a canonical average at temperature T for an arbitrary configuration of random fields and [
]h indicates an average over random fields. We treated two distributions of random fields, the bimodal in which each
hi=\ifmmode\pm\else\textpm\fi{}
h0 and a Gaussian distribution in which each
hi has variance
h02. We obtained series for \ensuremath{\chi} and G in the form ${\ensuremath{\sum}}_{\mathit{n}=1,15}$
an(g,d)(J/T
)n, where J is the exchange constant and the coefficients
an(g,d) are polynomials in g\ensuremath{\equiv}
h02/
J2 and in d. We assume that as T approaches its critical value,
Tc, one has \ensuremath{\chi}\ensuremath{\sim}(T-
Tc${)}^{\mathrm{\ensuremath{-}}\ensuremath{\gamma}}$ and G\ensuremath{\sim}(T-
Tc${)}^{\mathrm{\ensuremath{-}}\ensuremath{\gamma}\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}$. For dimensions above d=2 we find a range of values of g for which the critical exponents obtained from our series seem not to depend on g.For large values of g our results show a g dependence which is attributable to either a tricritical point or a first-order transition. All our results for critical exponents suggest that \ensuremath{\gamma}\ifmmode\bar\else\textasciimacron\fi{}=2\ensuremath{\gamma}, in agreement with the two-exponent scaling picture. In addition we have also constructed series for the amplitude ratio, A=(G/${\mathrm{\ensuremath{\chi}}}^{2}$)(
T2)/(
gJ2). We find that A approaches a constant value as T\ensuremath{\rightarrow}
Tc (consistent with \ensuremath{\gamma}\ifmmode\bar\else\textasciimacron\fi{}=2\ensuremath{\gamma}) with A\ensuremath{\approxeq}1. It appears that A is somewhat larger for the bimodal than for the Gaussian model, in agreement with a recent analysis at high d. \textcopyright{} 1996 The American Physical Society.
We present a calculation of the energy-density correlation ${d}_{\mathrm{\ensuremath{-}}}$(r,h) of the two-dimensional Ising model in a magnetic field under the limit T\ensuremath{\rightarrow}
Tc-, H\ensuremath{\rightarrow}0, and R\ensuremath{\rightarrow}\ensuremath{\infty} where h=const\ifmmode\times\else\texttimes\fi{}H\ensuremath{\Vert}T-
Tc${\ensuremath{\Vert}}^{\mathrm{\ensuremath{-}}15/8}$ is fixed and small, while r=R\ensuremath{\Vert}T-
Tc\ensuremath{\Vert} is fixed and large. We find ${\mathit{d}}_{\mathrm{\ensuremath{-}}}$(r,h)\ensuremath{\sim}2${\mathrm{\ensuremath{\pi}}}^{\mathrm{\ensuremath{-}}1/2}$${\mathrm{hr}}^{\mathrm{\ensuremath{-}}1/2}$${\mathrm{e}}^{\mathrm{\ensuremath{-}}2}$, where ${\ensuremath{\lambda}}_{l}$ are the positive solutions of
J1/3(${\ensuremath{\lambda}}_{\mathrm{l}}$/3)+${\mathrm{J}}_{\mathrm{\ensuremath{-}}1/3}$ (${\ensuremath{\lambda}}_{1/3}$)=0.