Critical phenomena in semi-infinite systems. I.εexpansion for positive extrapolation length

The Wilson-Fisher ε expansion is used to calculate critical exponents to first order in ε=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 ε, all surface exponents can be calculated from bulk exponents and a single surface exponent, ̃ \~η=(1/2)ε(n+2)(n+8), describing the rate at which bulk correlation functions are approached when all coordinates are far from the surface. The exponents η_⊥ and η_∥ introduced by Binder and Hohenberg are, respectively, 1- ̃ \~η and 2(1- ̃ \~η). A form for the fixed-point spin correlation valid for all dimensions containing only the exponents η and ̃ \~η is proposed. With this form, all critical exponents for a semi-infinite system can be obtained from η, ν, and ̃ \~η if scaling is assumed.