The critical behavior of a semi-infinite
n-vector model with anisotropic ferromagnetic pair interactions on the surface and isotropic interactions everywhere else is studied with the use of renormalization-group methods for dimension $d=4\ensuremath{-}\ensuremath{\epsilon}$. The effects of surface anisotropies are analyzed near the isotropic ordinary transition (where they produce corrections to scaling), near the isotropic special transition (where they are relevant perturbations), and near a new class of "anisotropic special" transitions. The latter occur if the surface exchange constants associated with
me (easy-magnetization) components take a critical value above which
me-component surface order is possible while the interaction constants of the remaining ${m}_{h}=n\ensuremath{-}{m}_{e}$ (hard-magnetization) components are weaker. Of particular interest is the anisotropic special transition with an easy axis (
me=1) because it occurs also in three dimensions, in contrast to the isotropic special one. The associated critical, crossover, and correction-to-scaling exponents are given to second order in $\ensuremath{\epsilon}$.
The effects of irrelevant surface perturbations on the local critical behavior of ($4\ensuremath{-}\ensuremath{\epsilon}$)-dimensional
n-vector models with a free surface or an internal defect plane are investigated with particular attention paid to the corrections to scaling induced by the surface term $\frac{1}{2}c\ensuremath{\int}{\mathrm{surface}}^{}{\ensuremath{\varphi}}^{2}$ near the ordinary transition. A systematic expansion in ${c}^{\ensuremath{-}1}$ is developed. In the case of a semi-infinite system, ${c}^{\ensuremath{-}1}$ couples to the square of the normal derivative of $\ensuremath{\varphi}$. The associated crossover exponent is ${\ensuremath{\Phi}}^{o}=\ensuremath{-}\ensuremath{\nu}$ to all orders in $\ensuremath{\epsilon}$. In the case of an infinite-space model with a defect plane, ${c}^{\ensuremath{-}1}$ couples to the squares of the normal derivatives on either side of the plane and to the product of the normal derivative on one side with the normal derivative on the other side. The latter operator represents perpendicular interactions across the defect plane, while the other two describe parallel interactions between spins on the same side of the plane. The associated crossover exponents are ${\ensuremath{\Phi}}_{\ensuremath{\parallel}}^{o}=\ensuremath{-}\ensuremath{\nu}$ and ${\ensuremath{\Phi}}_{\ensuremath{\perp}}^{o}={\ensuremath{\gamma}}_{11}^{o}$. A reanalysis of Bray and Moore's derivation of the (incorrect) relation ${\ensuremath{\Phi}}^{o}={\ensuremath{\gamma}}_{11}^{o}=\ensuremath{\nu}\ensuremath{-}1$ is presented which reveals where their arguments go wrong.
Correlation-function exponents ${\ensuremath{\eta}}_{\ensuremath{\perp}}$ and ${\ensuremath{\eta}}_{\ensuremath{\parallel}}$ appropriate to the free-surface problem have been obtained by renormalization-group calculation to order ${\ensuremath{\epsilon}}^{2}$. By using the scaling relations ${\ensuremath{\gamma}}_{1}=\ensuremath{\nu}(2\ensuremath{-}{\ensuremath{\eta}}_{\ensuremath{\perp}})$ and ${\ensuremath{\gamma}}_{11}=\ensuremath{\nu}(1\ensuremath{-}{\ensuremath{\eta}}_{\ensuremath{\parallel}})$, expansions for ${\ensuremath{\gamma}}_{1}$ and ${\ensuremath{\gamma}}_{11}$ are obtained. These expansions are in agreement with the surface scaling relation $2{\ensuremath{\gamma}}_{1}\ensuremath{-}{\ensuremath{\gamma}}_{11}=\ensuremath{\gamma}+\ensuremath{\nu}$, but disagree with the relation ${\ensuremath{\gamma}}_{11}=\ensuremath{\nu}\ensuremath{-}1$ due to Bray and Moore.
The effects of weak symmetry-breaking surface fields on the surface critical behavior of semi-infinite systems with a scalar order parameter are investigated near a bulk critical point. Previous field-theoretic analyses in 4-\ensuremath{\epsilon} dimensions are generalized by incorporating an additional surface term
w0${\mathrm{\ensuremath{\varphi}}}_{\mathit{s}}^{3}$ in the Hamiltonian. The mapping of microscopic models of the lattice-gas type on a continuum field theory is reconsidered, and we conclude that the cubic surface field
w0 generically does not vanish both in the case of an Ising ferromagnet in an applied magnetic field H (where
w0\ensuremath{\sim}H as H\ensuremath{\rightarrow}0) as well as in the case of critical adsorption of fluids onto a wall or an interface. Explicit results of a perturbative renormalization-group analysis are given, which show that a fixed point with nonvanishing cubic surface field
w∗=O(\ensuremath{\surd}\ensuremath{\epsilon} ) exists but that at order \ensuremath{\epsilon} the usual
w∗=0 fixed point is stable in the w direction. The associated correction-to-scaling exponent is ${\mathrm{\ensuremath{\omega}}}_{\mathit{w}}$=\ensuremath{\epsilon}+O(${\mathrm{\ensuremath{\epsilon}}}^{2}$). It is shown that the ${\mathrm{\ensuremath{\varphi}}}_{\mathit{s}}^{3}$ term leads to a mixing of surface enhancement and surface field in the leading odd relevant surface scaling field. Furthermore, it is proven beyond perturbation theory that a (redundant) surface operator exists, that corresponds to a similar mixing in the leading even surface scaling field.The implications of this mixing for the critical behavior near the special surface transition are discussed both for the case of Ising ferromagnets as well as that of fluids. In the case of fluids, the mixing entails that quenched surface disorder of the random bond type is sufficient to prevent the occurrence of a (multicritical) special transition for bulk dimensions d\ensuremath{\le}3 A number of other redundant surface operators are identified, and a general redundancy criterion for surface operators (analogous to Wegner's for bulk operators) is given. Finally, a fixed point with
w∗=iO(\ensuremath{\surd}\ensuremath{\epsilon} ) is obtained, which describes the Yang-Lee edge singularities of an equivalent (d-1)-dimensional bulk system with long-range interactions. The associated critical exponents can be determined to all orders in \ensuremath{\epsilon}.
The 1/n expansion method is developed in the study of the critical behavior of the n-vector model with a free surface. The method is applied at the bulk critical temperature (T = Tc) in the absence of the surface ordering. The surface critical exponents η;// and η;⊥ are identified up to order 1/n for general spatial dimension d: 2 ≪ d ≪ 4 for the ordinary transition and 3 ≪ d ≪4 for the special transition. The results are compared with several scaling relations, other theories and some experiments.