Critical behavior near a symmetry-breaking surface and the stress tensor.

Abstract

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 ${T}_{c}$, $C$ 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 ${T}_{c}$, $C$ 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 ${T}_{c}$, $C$ 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.