Critical Behavior of then-Vector Model with a Free Surface

Abstract

Correlation-function exponents ${\eta }_{\perp }$ and ${\eta }_{\parallel }$ appropriate to the free-surface problem have been obtained by renormalization-group calculation to order ${\epsilon }^2$. By using the scaling relations ${\gamma }_1=\nu (2-{\eta }_{\perp })$ and ${\gamma }_{11}=\nu (1-{\eta }_{\parallel })$, expansions for ${\gamma }_1$ and ${\gamma }_{11}$ are obtained. These expansions are in agreement with the surface scaling relation $2{\gamma }_1-{\gamma }_{11}=\gamma +\nu$, but disagree with the relation ${\gamma }_{11}=\nu -1\mathrm{}$ due to Bray and Moore.