Critical phenomena in semi-infinite systems. I. $\epsilon$ expansion for positive extrapolation length

The Wilson-Fisher $\epsilon$ expansion is used to calculate critical exponents to first order in $\epsilon =4-d$ for $n$-dimensional classical spins on a semi-infinite lattice with surface exchange such that the extrapolation length is positive. It is found that to first order in $\epsilon$, all surface exponents can be calculated from bulk exponents and a single surface exponent, $\tilde{\eta }=\frac{\left(\frac{1}{2}\right)\epsilon \mathrm{}(n+2\mathrm{})\mathrm{}}{\mathrm{}(n+8\mathrm{})\mathrm{}}$, describing the rate at which bulk correlation functions are approached when all coordinates are far from the surface. The exponents ${\eta }_{\perp }$ and ${\eta }_{\parallel }$ introduced by Binder and Hohenberg are, respectively, $1-\tilde{\eta }$ and $2(1-\tilde{\eta })$. A form for the fixed-point spin correlation valid for all dimensions containing only the exponents $\eta$ and $\tilde{\eta }$ is proposed. With this form, all critical exponents for a semi-infinite system can be obtained from $\eta$, $\nu$, and $\tilde{\eta }$ if scaling is assumed.