In recent years there have been many measurements of the scaling-law equation of state for different materials, and the "scaling function" so obtained has generally been fit by an empirical equation involving the selection of several adjustable parameters. We propose a method for calculating, directly from high-temperature series expansions, the function
h(x) that determines the scaling-law equation of state $H={M}^{\ensuremath{\delta}}h(x)$. Previously,
h(x) has been calculated only for the
S=21 Ising model, but the method is not generalizable to the case of the Heisenberg model because it relies upon the use of low-temperature expansions as well, and these are not known for the Heisenberg model. First we calculate
h(x) for the Ising model (bcc, fcc, and simple cubic lattices) in order to assess the utility and credibility of our method. Our Ising model
h(x) agrees well with the previous calculation that used both high-and low-temperature expansions. Next we calculate
h(x) in its entire region of definition for the
S=21 Heisenberg model (fcc and bcc lattices) and the $S=\ensuremath{\infty}$ Heisenberg models (fcc lattice), where
S denotes the spin quantum number. The accuracy of our resulting expressions is limited by the finite number of known terms in the corresponding high-temperature series expansions, but it is generally of the order of a few percent. In Paper II the scaling functions calculated here are compared with experiment and with the predictions of the universality hypothesis.
The scaling equation of state of an Ising-like ferromagnet is derived by an expansion in $\ensuremath{\epsilon}=4\ensuremath{-}d$, where
d is the dimension of space. The result is compared with numerical calculations on the three-dimensional Ising model. It is also established that the "linear model" is exact up to order ${\ensuremath{\epsilon}}^{2}$.
A dispersion-theory approach provides an exact representation of the two-point correlation function $G(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}},t)=C{t}^{\ensuremath{-}\ensuremath{\gamma}}g({q}^{2}{\ensuremath{\xi}}^{2})$, where $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$ and $\ensuremath{\xi}$ is the correlation length, which describes the critical scattering near a second-order phase transition. The threshold property of the spectral weight function $F(\frac{x}{3})\ensuremath{\propto}\mathrm{Im}{g}^{\ensuremath{-}1}(\ensuremath{-}{x}^{2})$ is used to motivate an approximation to the scaling function
g(x2) based on the asymptotically exact Fisher-Langer approximant ${g}_{\mathrm{FL}}({x}^{2})=(\frac{{C}_{1}}{{x}^{2\ensuremath{-}\ensuremath{\eta}}})(1+\frac{{C}_{2}}{{x}^{\frac{(1\ensuremath{-}\ensuremath{\alpha})}{\ensuremath{\nu}}}}+\frac{{C}_{3}}{{x}^{\frac{1}{\ensuremath{\nu}}}})$, where $\ensuremath{\alpha}$, $\ensuremath{\eta}$, $\ensuremath{\nu}$ are the usual critical exponents. The approximation consists of truncating the spectral function associated with ${g}_{\mathrm{FL}}^{\ensuremath{-}1}({x}^{2})$ in a manner designed to simulate the known threshold property of the exact spectral function, namely
F(3x)=0 for $x\ensuremath{\le}3$. The new approximant is checked on the two-dimensional Ising model and shown to agree with the exact result to better than 0.03% for all
x. Near four dimensions, the agreement with exact results is also excellent. A phenomenological approach to the intermediate three-dimensional case is presented, and shown to be in good agreement with high-temperature series and $\ensuremath{\epsilon}$-expansion results. The dispersion-theory approach is also shown to provide a convenient framework for calculations within the $\ensuremath{\epsilon}$ expansion and is used to compute
g(x2) to $O({\ensuremath{\epsilon}}^{3})$, a new result.
Using a Monte Carlo procedure with a self-consistent-field boundary condition, the magnetization of a simple cubic classical Heisenberg ferromagnet with nearest-neighbor interactions only is calculated. For reduced temperatures and reduced fields in the range $0.02<~|\frac{T}{{T}_{c}}\ensuremath{-}1|$,
JH< 0.5 the data of this computer experiment can be described in terms of the "effective" critical exponents $0.31<~\ensuremath{\beta}<~0.35$, $1.33<~\ensuremath{\gamma}<~1.39$, and $4.9<~\ensuremath{\delta}<~5.3$. Although the present data are not very close to the critical point they do obey the homogeneity requirement and determine the scaling function rather precisely. This function agrees very well with the scaling function for the face-centered-cubic classical Heisenberg magnet derived recently by Milo\ifmmode \check{s}\else \v{s}\fi{}evi\ifmmode \acute{c}\else \'{c}\fi{} and Stanley using high-temperature---series-expansion techniques. This agreement supports their hypothesis that neither critical exponents nor the scaling function depend on the lattice structure in the Heisenberg model.
Thermodynamic functions in the vicinity of an ordinary critical point are expected to obey asymptotic scaling laws as $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$ and the ordering field,
h, approach zero. However, the optimal scaling variables are the nonlinear scaling fields, ${g}_{t}=t+{b}_{t}{h}^{2}+{c}_{t}{t}^{2}+\ensuremath{\cdots}$ and ${g}_{h}=h(1+{c}_{h}t+\ensuremath{\cdots})$. The nonlinearities yield correction factors to the leading power-law (and scaling) variation of thermodynamic quantities,
L, of the form ($1+{a}_{L}t+{b}_{L}{t}^{2}+\ensuremath{\cdots}$), etc., where the correction amplitudes
aL,bL,…, are uniquely determined by the nonlinear scaling-field coefficients. It follows that "analytic" corrections to, e.g., the susceptibility, are directly related to those for the free energy and magnetization (in zero field). The term
bth2 also generates nonanalytic contributions such as an additive, energylike term, varying as ${|t|}^{1\ensuremath{-}\ensuremath{\alpha}}$ in the zero-field susceptibility, and factors like ($1+{c}_{L}{|h|}^{2\ensuremath{-}\frac{1}{\ensuremath{\Delta}}}$) on the critical isotherm,
t=0. Irrelevant scaling fields yield further, in general distinct, nonanalytic corrections, and cause shifts in
Tc and various amplitudes although "universal" ratios remain constant.
The theory of critical phenomena in systems at equilibrium is reviewed at an introductory level with special emphasis on the values of the critical point exponents α, β, γ,..., and their interrelations. The experimental observations are surveyed and the analogies between different physical systems - fluids, magnets, superfluids, binary alloys, etc. - are developed phenomenologically. An exact theoretical basis for the analogies follows from the equivalence between classical and quantal `lattice gases' and the Ising and Heisenberg-Ising magnetic models. General rigorous inequalities for critical exponents at and below T c are derived. The nature and validity of the `classical' (phenomenological and mean field) theories are discussed, their predictions being contrasted with the exact results for plane Ising models, which are summarized concisely. Padé approximant and ratio techniques applied to appropriate series expansions lead to precise critical-point estimates for the three-dimensional Heisenberg and Ising models (tables of data are presented). With this background a critique is presented of recent theoretical ideas: namely, the `droplet' picture of the critical point and the `homogeneity' and `scaling' hypotheses. These lead to a `law of corresponding states' near a critical point and to relations between the various exponents which suggest that perhaps only two or three exponents might be algebraically independent for any system.