εExpansion in Semi-infinite Ising Systems

The Wilson-Fisher $\epsilon$ expansion is used to calculate critical exponents in a semi-infinite Ising model to first order in $\epsilon$. It is found that potentials renormalize just as in the bulk, and all surface information is contained in the wave-function renormalization. $\nu$ is $\frac{1}{2}+\frac{1}{12}\epsilon$ and $\eta$ is 0, just as in infinite systems. ${\eta }_{\perp }$ and ${\eta }_{\parallel }$ are, respectively, $1-\frac{1}{6}\epsilon$ and $2-\frac{1}{3}\epsilon$ and the surface gap exponent ${\Delta }_1$ is $\frac{1}{2}-\frac{1}{12}\epsilon$ if scaling is assumed.