Universal and nonuniversal critical behavior of then-vector model with a defect plane in the limitn→∞

The local critical behavior of the $d$-dimensional $n$-vector model near an internal $d-1\mathrm{}$ dimensional plane of weakened interactions in the limit $n\to \infty$ changes qualitatively at $d=3\mathrm{}$. For $d>\mathrm{}3\mathrm{}$ the critical exponent ${\eta }_{\parallel }$, which characterizes the correlation of spins in the defect plane, is universal, i.e., is independent of the strength of the interactions in the defect plane. We argue that ${\eta }_{\parallel }$ is nonuniversal for $d=3\mathrm{}$ and calculate its dependence on the difference between the defect and bulk couplings to second order in perturbation theory. We show that the excess energy density at a distance $z$ from the defect plane decays at the critical point to the bulk value as $z^{-2\mathrm{}}$ for $d\ge 3$ and faster than $z^{-2\mathrm{}}$ for $d<\mathrm{}3\mathrm{}$. Near the free surface of a semi-infinite system one finds a $z^{-2\mathrm{}}$ dependence for $d>\mathrm{}3\mathrm{}$ as well as for $d\ge 3$.