Critical Behavior of the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>n</mml:mi></mml:math>-Vector Model with a Free Surface

Abstract

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