Recent work on local functional theories of critical inhomogeneous fluids and Ising-like magnets has shown them to be a potentially exact, or near exact, description of universal finite-size effects associated with the excess free energy and scaling of one-point functions in critical thin films. This approach is extended to predict the two-point correlation function $G$ in critical thin films with symmetric surface fields in arbitrary dimension $d$. In $d=2\mathrm{}$ we show there is exact agreement with the predictions of conformal invariance for the complete spectrum of correlation lengths ${\xi }^{\left(n\right)}$ as well as the detailed position dependence of the asymptotic decay of $G$. In $d=3\mathrm{}$ and $d\ge 4$ we present new numerical predictions for the universal finite-size correlation length and scaling functions determining the structure of $G$ across the thin film. Highly accurate analytical closed form expressions for these universal properties are derived in arbitrary dimension.

• Received 20 April 2000

DOI:https://doi.org/10.1103/PhysRevLett.85.4108