Scaling density profiles of critical O(N)-symmetric systems confined to a film geometry are analyzed by field-theoretic renormalization-group techniques in d=4-\ensuremath{\varepsilon} spatial dimensions. At bulk criticality we explicitly calculate the energy density profile for five different boundary conditions and determine their corresponding universal scaling functions. In order to discuss the influence of the boundary conditions at the two surfaces of the film on the global shape of the profiles, a suitable exponentiation scheme is constructed. Using these exponentiated scaling functions as well as results from conformal field theory, the shapes of the energy density profiles in d=2,3, and 4 are compared. Finally, we discuss the short-distance behavior of the energy density profiles close to the surfaces of the film as it follows from the knowledge of the full profiles in first order \ensuremath{\varepsilon}.
Scaling density profiles of critical O(N)-symmetric systems confined to a film geometry are analyzed by field-theoretic renormalization-group techniques in d=4-\ensuremath{\varepsilon} spatial dimensions. At bulk criticality we explicitly calculate the energy density profile for five different boundary conditions and determine their corresponding universal scaling functions. In order to discuss the influence of the boundary conditions at the two surfaces of the film on the global shape of the profiles, a suitable exponentiation scheme is constructed. Using these exponentiated scaling functions as well as results from conformal field theory, the shapes of the energy density profiles in d=2,3, and 4 are compared. Finally, we discuss the short-distance behavior of the energy density profiles close to the surfaces of the film as it follows from the knowledge of the full profiles in first order \ensuremath{\varepsilon}.
Scaling density profiles O(N) symmetric systems confined between two parallel plates with a large separation L are analyzed by field-theoretical methods for a continuous phase transition at bulk criticality Tc,b in spatial dimension d=4-\ensuremath{\varepsilon}. At distances z\ensuremath{\ll}L from one wall a the profile contains a perturbation Ba,b(z/L)d induced by the distant wall b. In contrast to previous expectations the ratio of the universal amplitude Ba,b and the corresponding Casimir amplitude is not hyperuniversal, i.e., not independent of both surface universality classes characterizing the walls a and b.
The critical behavior near a surface that breaks the order parameter symmetry is analyzed with field-theoretic renormalization-group methods. We study the short-distance expansions of the order parameter and the energy density about the surface and investigate various density profiles and correlation functions in the half-space and film geometries. Our results confirm Cardy's conjecture that the stress-energy tensor appears as the leading surface density. Its universal prefactor C in the expansions determines several physical properties of interest in the critical adsorption of fluids: In a film at the bulk critical temperature Tc, C determines the amplitude ratio of the Fisher-de Gennes distant-wall correction to the order-parameter profile and the Casimir correction to the free energy. In the half-space close to Tc, C determines the amplitude ratio of the ${(\frac{z}{\ensuremath{\xi}})}^{d}$ term in the order parameter profile to the Ferer-Stauffer-Wortis term in the bulk free energy. Here z is the distance from the surface and $\ensuremath{\xi}$ the bulk correlation length. In the half-space at Tc, C also determines the universal short-distance amplitudes of layer susceptibilities, which should be observable in computer experiments. We calculate C to second order in $\ensuremath{\epsilon}=4\ensuremath{-}d$ and estimate its value for d=3. C is closely related to the stress-stress correlation in a half-space, for which we obtain the bulk to surface crossover. We present a simple argument showing why this crossover depends, in general, on the surface universality class except in two dimensions.
An introduction into the theory of boundary critical phenomena and the application of the field-theoretical renormalization group method to these is given. The emphasis is on a discussion of surface critical behavior at bulk critical points of magnets, binary alloys, and fluids. Yet a multitude of related phenomena are mentioned. The most important distinct surface universality classes that may occur for a given universality class of bulk critical behavior are described, and the respective boundary conditions of the associated field theories are discussed. The short-distance singularities of the order-parameter profile in the diverse asymptotic regimes are surveyed.
Finite-size effects are investigated in the Gaussian model with isotropic and anisotropic short-range interactions in film geometry with nonperiodic boundary conditions (bc) above, at, and below the bulk critical temperature Tc. We have obtained exact results for the free energy and the Casimir force for antiperiodic, Neumann, Dirichlet, and Neumann-Dirichlet mixed bc in 1<d<4 dimensions. For the Casimir force, finite-size scaling is found to be valid for all bc. For the free energy, finite-size scaling is valid in 1<d<3 and 3<d<4 dimensions for antiperiodic, Neumann, and Dirichlet bc, but logarithmic deviations from finite-size scaling exist in d=3 dimensions for Neumann and Dirichlet bc. This is explained in terms of the borderline dimension d*=3 , where the critical exponent 1-α-ν=(d-3)∕2 of the Gaussian surface energy density vanishes. For Neumann-Dirichlet bc, finite-size scaling is strongly violated above Tc for 1<d<4 because of a cancelation of the leading scaling terms. For antiperiodic, Dirichlet, and Neumann-Dirichlet bc, a finite film critical temperature Tc,film(L)<Tc exists at finite film thickness L . Our results include an exact description of the dimensional crossover between the d -dimensional finite-size critical behavior near bulk Tc and the (d-1) -dimensional critical behavior near Tc,film(L). This dimensional crossover is illustrated for the critical behavior of the specific heat. Particular attention is paid to an appropriate representation of the free energy in the region Tc,film(L)≤T≤Tc. For 2<d<4 , the Gaussian results are renormalized and reformulated as one-loop contributions of the φ4 field theory at fixed dimension d and are then compared with the ε=4-d expansion results at ε=1 as well as with d=3 Monte Carlo data. For d=2 , the Gaussian results for the Casimir force scaling function are compared with those for the Ising model with periodic, antiperiodic, and free bc; unexpected exact relations are found between the Gaussian and Ising scaling functions. For both the d -dimensional Gaussian model and the two-dimensional Ising model it is shown that anisotropic couplings imply nonuniversal scaling functions of the Casimir force that depend explicitly on microscopic couplings. Our Gaussian results provide the basis for the investigation of finite-size effects of the mean spherical model in film geometry with nonperiodic bc above, at, and below the bulk critical temperature.
The critical behavior of the energy density near surfaces belonging to the surface universality classes called ordinary (O) and special (SB) transitions is investigated field theoretically. Both energy density profiles in a film and correlation functions in a half space of the energy density with other operators are considered right at O or SB criticality. We also discuss half-space profiles off criticality. Their properties are in accordance with the operator short-distance expansion of the energy density about the surface. Particular attention is paid to the occurrence of the component perpendicular to the surface of the stress tensor as a surface operator in these expansions.
Off-diagonal profiles of local densities (e.g. order parameter or energy density) are calculated at the bulk critical point, by conformal methods, on a strip with transverse coordinate v, for different types of boundary conditions (free, fixed and mixed). Such profiles, which are defined by the non-vanishing matrix element of the appropriate operator between the ground state and the corresponding lowest excited state of the strip Hamiltonian, enter into the expression of two-point correlation functions on a strip. They are of interest in the finite-size scaling study of bulk and surface critical behaviour since they allow the elimination of regular contributions. The conformal profiles, which are obtained through a conformal transformation of the correlation functions from the half-plane to the strip, are in agreement with the results of a direct calculation, for the energy density of the two-dimensional Ising model.