Universal relations among critical amplitudes of surface quantities are discussed for d-dimensional semi-infinite systems near their bulk critical temperature
Tcb. Based on the field-theoretic renormalization-group approach, a general derivation of the expected multi-scale-factor universality is given and various universal ratios involving critical amplitudes of the ordinary and special transitions both above and below
Tcb are calculated to first order in \ensuremath{\epsilon}=4-d. In addition, an error in the published one-loop result for the universal order-parameter profile at the ordinary transition of the semi-infinite Ising model is corrected.
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.
The critical behaviour of a semi-infinite system with O(n) spin symmetry is investigated using four techniques: (i) mean-field theory, (ii) exact solution for n= infinity , (iii) the epsilon expansion, and (iv) scaling arguments. The surface equation of state is computed using mean-field theory, and the four phase transitions defined by Lubensky and Rubin, the 'ordinary', 'surface', 'extraordinary', and ' lambda = infinity ' ('special') transitions, are identified. The first three may be observed in the bulk system by adding a suitable 'surface' perturbation which destroys translational invariance. Scaling arguments give their critical exponents exactly in terms of bulk exponents. Critical exponents for the ' lambda = infinity ' transition are determined for n= infinity and also to O( epsilon ). Spin-spin correlation functions at T=T c for the 'ordinary', 'extraordinary' and ' lambda = infinity ' transitions are calculated exactly for n= infinity .
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.
The critical behavior of a continuous spin system in a semi-infinite sample is studied for all values of the extrapolation length $\ensuremath{\lambda}$ using mean-field theory. A new transition, which we have called the extraordinary transition, is found for $\ensuremath{\lambda}<0$ in which the bulk orders at a temperature below the surface ordering temperature. In this paper we have calculated the magnetic susceptibilities $\ensuremath{\chi}(z)$ and $\ensuremath{\chi}(z, z)$ and the correlation function $\ensuremath{\Gamma}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}, {\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}}^{\ensuremath{'}})$ at all the phase transitions. We have used these results to compute the various $\ensuremath{\gamma}$ and $\ensuremath{\eta}$ exponents and to study the scaling relations introduced by Binder and Hohenberg.