The thermodynamic functions of an
n-component
d-dimensional ${\ensuremath{\sigma}}^{4}$ theory are written in the form ${f}_{i}={A}_{i}{|t|}^{\ensuremath{-}{\ensuremath{\lambda}}_{i}}(1+{a}_{i}{|t|}^{\ensuremath{\Delta}})$, where $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$. The correction-to-scaling amplitudes
ai are calculated to leading order in
n1 for
2<d<4, and the universality of their ratios is explicitly shown. The effective exponents ${\ensuremath{\lambda}}_{i,\mathrm{eff}}={\ensuremath{\lambda}}_{i}\ensuremath{-}{a}_{i}\ensuremath{\Delta}{|t|}^{\ensuremath{\Delta}}$ violate the thermodynamic scaling laws already at this leading order. The amplitudes
ai have singularities (as functions of
d) at the dimensionalities ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{d}}_{l}=2+\frac{4}{l},l=3,4,5,\dots{}$. Above
Tc, the specific heat and the susceptibility correction amplitudes
ac+ and ${a}_{\ensuremath{\chi}}^{+}$ are discontinuous at ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{d}}_{l}$ and at ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{d}}_{2l}(l\ensuremath{\ge}2)$, respectively. Below
Tc, the magnetization and the stiffness correction amplitudes are zero, ${a}_{M}={a}_{{\ensuremath{\rho}}_{s}}=0$, while ${a}_{c}^{\ensuremath{-}}$ diverges at ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{d}}_{2l+1}(l\ensuremath{\ge}1)$ but is continuous at
d2l (in particular at
d=3). Results are also given for universal ratios among corrections-to-scaling amplitudes for the dependence of thermo-dynamic functions on the magnetization at
T=Tc and at a nonzero field.
A general derivation of the two-scale-factor universality is given using the renormalized ${\ensuremath{\varphi}}^{4}$ theory. As a consequence ten universal relations among critical amplitudes are obtained, any other universal relations being a combination of them. The situation is shown to be the same as for the scaling laws for critical exponents. The calculations up to order ${\ensuremath{\epsilon}}^{2}$ for systems with continuous symmetry are performed for some of these relations such as the specific-heat amplitudes ratio and ${R}_{\ensuremath{\xi}}^{+}$, which relates the amplitudes of the correlation length and of the specific heat, etc. Comparison with experiments and series results and, in particular, the discussion of the superfluid helium and RbMn
F3 cases, are repeated with our improved numerical results.
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.
Writing thermodynamic quantities near a critical point as ${f}_{i}\ensuremath{\sim}{A}_{i}{|t|}^{\ensuremath{-}{\ensuremath{\lambda}}_{i}}(1+{a}_{i}{|t|}^{\ensuremath{\Delta}})$, where $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$, we show that to leading order in $\ensuremath{\epsilon}=4\ensuremath{-}d$ one has $\frac{{a}_{i}}{{a}_{j}}=\frac{({\ensuremath{\lambda}}_{i}\ensuremath{-}{{\ensuremath{\lambda}}_{i}}^{0})}{({\ensuremath{\lambda}}_{j}\ensuremath{-}{{\ensuremath{\lambda}}_{j}}^{0})}$, where ${{\ensuremath{\lambda}}_{i}}^{0}$ is the mean-field value of ${\ensuremath{\lambda}}_{i}$. This ratio is also equal to $\frac{({\ensuremath{\lambda}}_{i, \mathrm{eff}}\ensuremath{-}{\ensuremath{\lambda}}_{i})}{({\ensuremath{\lambda}}_{j, \mathrm{eff}}\ensuremath{-}{\ensuremath{\lambda}}_{j})}$, where ${\ensuremath{\lambda}}_{i, \mathrm{eff}}$ is derived from a fit of data with ${f}_{i}\ensuremath{\sim}{|t|}^{\ensuremath{-}{\ensuremath{\lambda}}_{i, \mathrm{eff}}}$. Various experiments are analyzed and compared to these predictions.
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.
Monte Carlo simulations and series expansions data for the energy, specific heat, magnetization, and susceptibility of the three-state Potts model on the square lattice are analyzed in the vicinity of the critical point in order to estimate universal combinations of critical amplitudes. We estimate these amplitudes using the correction-to-scaling exponents predicted by conformal field theory. We also form effective ratios of the observables close to the critical point and analyze how they approach the universal critical-amplitude ratios. In particular, using the duality relation, we show analytically that for the Potts model with a number of states $q\ensuremath{\leqslant}4$, the effective ratio of the energy critical amplitudes always approaches unity linearly with respect to the reduced temperature. This fact leads to the prediction of relations among the amplitudes of correction-to-scaling terms of the specific heat in the low- and high-temperature phases. We present numerical and analytical support for the form of the first two correction-to-scaling terms. Our results for the amplitude ratios closely agree with the theoretical predictions and the earlier numerical estimates of the specific heat and the susceptibility amplitude ratios.