Universal order-parameter profiles for critical adsorption and the extraordinary transition: a comparison of epsilon-expansion and Monte Carlo results

Abstract

The universal, scaled order parameter profiles $P_{\pm}(z/ξ)$ for critical adsorption of a fluid or fluid mixture onto a wall or interface, and for the extraordinary transition of the semi-infinite Ising model, are discussed theoretically, where $z$ is the distance from the interface, $ξ(T)$ is the bulk correlation length, and the subscript $+$ ($-$) refers to the approach from above (below) $T_c$. Recent results to first order in the $ε=4-d$ expansion are extrapolated to $d=3$ space dimensions and compared with new Monte Carlo results. In order to obtain meaningful extrapolations it is crucial that both the exponential decay at large $ζ$ as well as the known algebraic behavior $P_{\pm}(ζ)\simζ^{-β/ν}$ at small $ζ$ be correctly reproduced. To this end a recently developed novel RG scheme involving a $z$ dependent amplitude renormalization is used. Reasonable agreement of our extrapolations with the Monte Carlo results and some experimental results is obtained.