Semi-infinite O ( n ) systems are considered at a bulk tricritical point. A continuum model with a |ϕ| 6 bulk nonlinearity and a |ϕ| 4 surface nonlinearity is used to analyse the critical behaviour at the special transition in d = 3 and d = 3−ε dimensions. For n = 0 the model describes the statistics of polymers at the θ-point, in the presence of a wall. It is shown that the surface critical exponents have a systematic expansion in powers of ε 1/2 . For d = 3, logarithmic corrections of an unusual form are found for surface quantities. In the special case of vanishing bulk nonlinearity the surface exponents can be expressed in terms of known bulk exponents for systems with long-range interactions.
Semi-infinite O ( n ) systems are considered at a bulk tricritical point. A continuum model with a |ϕ| 6 bulk nonlinearity and a |ϕ| 4 surface nonlinearity is used to analyse the critical behaviour at the special transition in d = 3 and d = 3−ε dimensions. For n = 0 the model describes the statistics of polymers at the θ-point, in the presence of a wall. It is shown that the surface critical exponents have a systematic expansion in powers of ε 1/2 . For d = 3, logarithmic corrections of an unusual form are found for surface quantities. In the special case of vanishing bulk nonlinearity the surface exponents can be expressed in terms of known bulk exponents for systems with long-range interactions.
We study the surface critical behavior of semi-infinite systems with an n-component order parameter and short-range interactions at a bulk tricritical point. The special surface transition is investigated in detail for space dimensionalities d=3-\ensuremath{\epsilon}\ensuremath{\le}3 using field-theoretic renormalization-group methods. For \ensuremath{\epsilon}>0 its infrared-stable fixed point is described by a Hamiltonian which contains a \ensuremath{\Vert}\ensuremath{\varphi}\ensuremath{\rightarrow}${\ensuremath{\Vert}}^{4}$ surface interaction of strength O(${\ensuremath{\epsilon}}^{1/2}$) in addition to the familiar \ensuremath{\Vert}\ensuremath{\varphi}\ensuremath{\rightarrow}${\ensuremath{\Vert}}^{6}$ bulk interaction of strength O(\ensuremath{\epsilon}). For d\ensuremath{\equiv}3, logarithmic corrections of a non-power-law form appear. The renormalization of a proper continuum model is expounded and the implications of the results are discussed, especially with regard to the adsorption of polymers from a \ensuremath{\theta} solution.
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}.
We use a three-dimensional massive field theory up to the two-loop approximation to study the critical behavior of semi-infinite quenched random Ising-like systems at the special surface transition. Besides, we extend up to the next-to leading order, the previous first-order results of the sqrt[epsilon] expansion obtained by Ohno and Okabe [Phys. Rev. B 46, 5917 (1992)]. The numerical estimates for surface critical exponents in both cases are computed by means of the Padé analysis. Moreover, in the case of the massive field theory we perform Padé-Borel resummation of the resulting two-loop series expansions for surface critical exponents. The most reliable estimates for critical exponents of semi-infinite systems with quenched bulk randomness at the special surface transition, which we can obtain in the frames of the present approximation scheme, are eta(//)=-0.238, Delta(1)=1.098, eta( perpendicular )=-0.104, beta(1)=0.258, gamma(11)=0.839, gamma(1)=1.426, delta(1)=6.521, and delta(11)=4.249. These values are different from critical exponents for pure semi-infinite Ising-like systems and show that in a system with quenched bulk randomness the plane boundary is characterized by a new set of critical exponents at the special surface transition.
We study the time evolution of classical spin systems with purely relaxational dynamics, quenched from $T\ensuremath{\gg}{T}_{c}$ to the critical point, in the semi-infinite geometry. Shortly after the quench, as in the bulk, a nonequilibrium regime governed by universal power laws is also found near the surface. We show for "ordinary" and "special" transitions that the corresponding critical exponents differ from their bulk values, but can be expressed via scaling relations in terms of known bulk and surface exponents. To corroborate our scaling analysis, we present perturbative ($\ensuremath{\epsilon}$-expansion) and Monte Carlo results.
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.
A review is given of recent work on the ordinary surface critical behaviour of systems in two dimensions. Several models of interest in statistical mechanics are considered: Potts model, percolation, Ising clusters, ZN-model, O(n) model and polymers. Numerical results for surface exponents, obtained by suitable finite size scaling extrapolations, are discussed in the light of recent advances based on the conformal invariance approach. Surface exponents are often seen as important tests of conformal invariance predictions. In other cases these exponents provide important information for a location of the problem within the classification schemes offered by the conformal approach, and a determination of its universality class. A relevant example of the first aspect is the study of the q-state Potts model with q near 4, for which an analytical study of logarithmic scaling corrections is needed to achieve a successful test. The latter point of view applies, e.g., to the more controversial cases of polymers at the theta point and critical Ising clusters. Emphasis is put on the importance of an integrated study of both bulk and surface properties. Relevant issues, like the possible existence of analytical expressions for the indices in particular model families, or of general relationships between bulk and surface exponents, are critically discussed. The new problem of critical behaviour at fractal boundaries is also considered for random (RW) and self-avoiding walks (SAW). From the numerical analysis of this problem remarkable universalities of the surface exponents seem to emerge, which, in the case of SAW’s, are still far from being understood.