We estimate, at
d=3, from ${\ensuremath{\varphi}}^{4}$ field theory the universal ratios of confluent critical correction amplitudes relative to the correlation length, the susceptibility, and the specific heat above
Tc. Our central values are comparable to these available from analysis of high-temperature series and the $\ensuremath{\epsilon}$ expansion. Furthermore, experiments in progress on a binary mixture are in good agreement with one of our results, the ratio of the confluent corrections for the specific heat and susceptibility which have not been previously estimated. We also present, in detail, a derivation of the universality of these quantities in the framework of field theory.
The susceptibility of the spin-
s nearest-neighbor fcc Ising model is studied at zero magnetic field by twelve-term high-temperature series. We find that the data near criticality are consistent with $\ensuremath{\chi}=A(s){t}^{\frac{\ensuremath{-}5}{4}}[1+B(s){t}^{{\ensuremath{\Delta}}_{1}}]$, $t=1\ensuremath{-}\frac{{T}_{c}(s)}{T}$. The correction-to-scaling exponent has the universal value ${\ensuremath{\Delta}}_{1}=0.50\ifmmode\pm\else\textpm\fi{}0.05$ in good agreement with $\ensuremath{\epsilon}$-expansion estimates. The confluent amplitude
B(s) vanishes at
s=21 to within uncertainties. Values of the critical parameters are tabulated for a variety of spins.
The hypothesis of universality implies that for every scaling relation among critical exponents there exists a universal ratio among the corresponding critical amplitudes. If one writes $B{|t|}^{\ensuremath{\beta}}$, ${{A}_{F}}^{\ifmmode\pm\else\textpm\fi{}}{|t|}^{2\ensuremath{-}\ensuremath{\alpha}}$, ${C}^{\ifmmode\pm\else\textpm\fi{}}{|t|}^{\ensuremath{-}\ensuremath{\gamma}}$, and ${\ensuremath{\xi}}_{0}{|t|}^{\ensuremath{-}\ensuremath{\nu}}$ [where $t=\frac{({p}_{c}\ensuremath{-}p)}{{p}_{c}}$,
p being the concentration of nonzero bonds, and +(-) stands for
p<pc (
p>pc)] for the leading singular terms in the probability to belong to the infinite cluster, the mean number of clusters, the clusters' mean-square size, and the pair connectedness correlation length, then it is shown that the ratios $\frac{{{A}_{F}}^{+}}{{{A}_{F}}^{\ensuremath{-}}}$, $\frac{{C}^{+}}{{C}^{\ensuremath{-}}}$, ${{A}_{F}}^{+}{B}^{\ensuremath{-}2}{C}^{+}$, $\frac{{\ensuremath{\xi}}_{0}^{+}}{{\ensuremath{\xi}}_{0}^{\ensuremath{-}}}$, and ${{A}_{F}}^{+}{({\ensuremath{\xi}}_{0}^{+})}^{d}$ (
d is the dimensionality) are universal. Similar quantities are found for the behavior at
p=pc (as a function of a "ghost" field). All of these universal ratios are derived from a universal scaled equation of state, which is calculated to second order in $\ensuremath{\epsilon}=6\ensuremath{-}d$. The (extrapolated) results are compared with available information in dimensionalities
d=2,3,4,5, with reasonable agreements. The amplitude relations become exact at
d=6, when logarithmic corrections appear. Additional universal ratios are obtained for the confluent correction to scaling terms.
We have extended the series for the zero-field susceptibility of the spin-
s Ising model through tenth order in the reduced inverse temperature
K on the square, triangular, simple-cubic, body-centered-cubic, and face-centered-cubic lattices. The series coefficients
hn(s) are expressed as simple polynomials in
X=s(s+1). Using extended methods of analysis we have estimated the nature of the leading singularities on the face-centered lattice and conclude with good confidence that the susceptibility exponent ${\ensuremath{\gamma}}_{1}$ equals 1.25, independent of
s. The exponent of the leading correction term is estimated to be ${\ensuremath{\gamma}}_{2}\ensuremath{\simeq}0.75\ifmmode\pm\else\textpm\fi{}0.08$ in good agreement with renormalization-group theory. For
s=21 only, the amplitude of the confluent correction apparently vanishes. We have also studied the leading singularities on the triangular lattice and conclude that ${\ensuremath{\gamma}}_{1}=\frac{7}{4}$, independent of
s, with, however, much stronger corrections than in three dimensions. These results provide a very strong corroboration of the universality hypothesis.
Recently it has been found that for several two-dimensional classical models the amplitude of the correlation length of a finite system at a critical point is universal and related to the correlation-function exponent $\ensuremath{\eta}$. Here we propose that for quantum systems the ratios of two such amplitudes are universal and are related to the ratios of the corresponding exponents. This proposition is confirmed numerically for the transverse Ising model for spin
S=21,1,23,and2. It is used to calculate $\ensuremath{\eta}$ in a simplified way.
The hypothesis of two-scale-factor universality, originally proposed by Stauffer, Ferer, and Wortis, is shown to follow from the renormalization-group approach, for systems close to their critical point. Values of the universal ratios involving correlation length and specific-heat amplitudes are obtained from the $\ensuremath{\epsilon}$ expansion, for Ising, $X\ensuremath{-}Y$, and Heisenberg models. In the latter two cases the correlation function has a power-law behavior at large distances below
Tc, and the (transverse) correlation length is defined in terms of the stiffness constant ${\ensuremath{\rho}}_{s}$. Experimental values of the correlation lengths and amplitude ratios are determined for superfluid
4He, which is $X\ensuremath{-}Y$-like, and for the Heisenberg antiferromagnet
RbMnF3. Comparisons are made between the values of the amplitude ratios coming from $\ensuremath{\epsilon}$ expansions, series, and experiments.