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 .
The Wilson-Fisher $\ensuremath{\epsilon}$ expansion is used to calculate critical exponents to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$ for
n-dimensional classical spins on a semi-infinite lattice with surface exchange such that the extrapolation length is positive. It is found that to first order in $\ensuremath{\epsilon}$, all surface exponents can be calculated from bulk exponents and a single surface exponent, $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}}=\frac{(\frac{1}{2})\ensuremath{\epsilon}(n+2)}{(n+8)}$, describing the rate at which bulk correlation functions are approached when all coordinates are far from the surface. The exponents ${\ensuremath{\eta}}_{\ensuremath{\perp}}$ and ${\ensuremath{\eta}}_{\ensuremath{\parallel}}$ introduced by Binder and Hohenberg are, respectively, $1\ensuremath{-}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}}$ and $2(1\ensuremath{-}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}})$. A form for the fixed-point spin correlation valid for all dimensions containing only the exponents $\ensuremath{\eta}$ and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}}$ is proposed. With this form, all critical exponents for a semi-infinite system can be obtained from $\ensuremath{\eta}$, $\ensuremath{\nu}$, and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}}$ if scaling is assumed.
We calculate the surface critical exponents of the ordinary transition occurring in semi-infinite, quenched dilute Ising-like systems. This is done by applying the field theoretic approach directly in d=3 dimensions up to the two-loop approximation as well as in 4-epsilon dimensions. At d=4-epsilon we extend, up to the next-to-leading order, the previous first-order results of the square root of epsilon expansion by Ohno and Okabe [Phys. Rev. B 46, 5917 (1992)]. In both cases numerical estimates for surface exponents are computed using Padé approximants extrapolating the perturbation theory expansions. The obtained results indicate that the critical behavior of semi-infinite systems with quenched bulk disorder is characterized by the new set of surface critical exponents.
By the use of the Migdal–Kadanoff renormalization group technique and the mean field approximation, we have explored the critical behavior of the semi-infinite mixed spin-7/2 and spin-1/2 Blume–Capel model. As a function of the computation ratios (bulk-surface) R and Y, different phase diagrams in the bulk and on the surface are classified and determined in the (surface anisotropy, temperature) plane. We have found four types of phase diagrams characterized by ordinary, extraordinary, surface, and special phase transitions. The derivative of the free energy and the behavior of the bulk and surface magnetizations are plotted at very low temperatures proving the existence of first-order transitions for both the surface and bulk. We have also presented the related fixed points and the critical exponents manifesting several classes of universality at the surface. Otherwise, a comparison was made between the two methods as well as with previous studies.
With simple, exact arguments we show that the surface magnetization
m1 at the extraordinary and normal transitions and the surface energy density ${\ensuremath{\epsilon}}_{1}$ at the ordinary, extraordinary, and normal transitions of semi-infinite
d-dimensional Ising systems have leading thermal singularities $B\ifmmode\pm\else\textpm\fi{}{|t|}^{2\ensuremath{-}\ensuremath{\alpha}}$, with the same critical exponent and amplitude ratio as the bulk free energy
fb(t,0). The derivation is carried out in three steps: (i) By tracing out the surface spins, the semi-infinite Ising model with supercritical surface enhancement
g and vanishing surface magnetic field
h1 is mapped exactly onto a semi-infinite Ising model with subcritical surface enhancement, a nonzero surface field, and irrelevant additional surface interactions. This establishes the equivalence of the extraordinary (
h1=0,g>0) and normal (${h}_{1}\ensuremath{\ne}0, g<0$) transitions. (ii) The magnetization
m1 at the interface of an infinite system with uniform temperature
t and a nonzero magnetic field
h in the half-space
z>0 only is shown to be proportional to ${f}_{b}(t,0)\ensuremath{-}{f}_{b}(t,h)$. (iii) The energy density ${\ensuremath{\epsilon}}_{1}$ at the interface of an infinite system with temperatures
t+ and
t in the half-spaces
z>0 and
z<0 and no magnetic field is shown to be proportional to ${f}_{b}(t,0)\ensuremath{-}{f}_{b}({t}_{+},0)$.