Two coupled semi-infinite systems near criticality

We explicitly construct a continuum spin model realization of that suggested in an alternative form by Diehl et al. Our model consists of two inequivalent semi-infinite systems A and B with surface interaction parameters $c_A$ and $c_B$. The strength of the coupling between the two systems is dictated by a third surface interaction parameter ĉ. As ĉ ranges from zero to infinity, the model crosses over from that describing two noninteracting semi-infinite critical systems to that of Bray and Moore for an infinite system with a defect plane and with a single surface interaction parameter $c^{\mathrm{}\left(\mathrm{BM}\right)\mathrm{}=c_A+c_B}$. We evaluate surface susceptibilities to one-loop order when the two bulks A and B are identical, retaining the full crossover dependence on the two surface parameters ĉ and c=$c_A$=$c_B$. The theory is studied near the special, ordinary, and bulk fixed points.