Crossover scaling functions for a two-component spin system with quadratic anisotropy
g and cubic anisotropy $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{v}$ are calculated to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$ and in $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{v}$, near the transitions within the ordered phases below the multicritical point (
T=Tm,g=0). Emphasis is placed on the mechanism by which the system is driven to bicritical or tetracritical behavior by the dangerous irrelevant variable $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{v}$. Explicit expressions and graphs for the various thermodynamic scaling functions are presented.
The thermodynamic properties of an anisotropic multicomponent spin system are investigated near the bicritical point in the disordered phase using a renormalization group approach. Explicit forms for the cross-over scaling functions for the specific heat and the susceptibilities are found to lowest orders in epsilon =4-d in terms of z=gt - phi where g is the non-ordering field and t=(T-T b )/T b with T b the bicritical temperature. The scaling functions have the expected properties in the limits z to O and z to zt/dt, the value on the critical lines, and interpolate smoothly between these limits.
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 zero-field two-point correlation function of an
n-vector system in
d dimensions is calculated to order
n1 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
n1. At
d=3 the term
t2 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.
Renormalization-group recursion relations are used to calculate to order $\ensuremath{\epsilon}=4\ensuremath{-}d$ the free-energy, ordering-susceptibility, and magnetization-crossover scaling functions for tricritical points in isotropic
n-vector models. The equations of state describe critical and tricritical behavior in both ordered and disordered phases.
We extend our previous determinations of nonasymptotic critical behavior of Phys. Rev. B 32, 7209 (1985) and 35, 3585 (1987) to accurate expressions of the complete classical-to-critical crossover (in three-dimensional field theory) in terms of the temperaturelike scaling field (i.e., along the critical isochore) for (1) the correlation length, the susceptibility, and the specific heat in the homogeneous phase for the n-vector model (n=1 to 3) and (2) the spontaneous magnetization (coexistence curve), the susceptibility, and the specific heat in the inhomogeneous phase for the Ising model (n=1). The present calculations include the seventh-loop order of Murray and Nickel and closely account for the up-to-date estimates of universal asymptotic critical quantities (exponents and amplitude combinations) provided by Guida and Zinn-Justin [J. Phys. A 31, 8103 (1998)].