A plane defect in the 3d O(N) model

It was recently found that the classical 3d O (N) ( N ) model in the semi-infinite geometry can exhibit an “extraordinary-log” boundary universality class, where the spin-spin correlation function on the boundary falls off as \langle \vec{S}(x) \cdot \vec{S}(0)\rangle \sim \frac{1}{(\log x)^q} 〈 S → ( x ) ⋅ S → ( 0 ) ⟩ ∼ 1 ( log x ) q . This universality class exists for a range 2 ≤N 3 N c > 3 . In this work, we extend this result to the 3d O (N) ( N ) model in an infinite geometry with a plane defect. We use renormalization group (RG) to show that in this case the extraordinary-log universality class is present for any finite N ≥2 N ≥ 2 . We additionally show, in agreement with our RG analysis, that the line of defect fixed points which is present at N = ∞ N = ∞ is lifted to the ordinary, special (no defect) and extraordinary-log universality classes by 1/N 1 / N corrections. We study the “central charge” a a for the O(N) O ( N ) model in the boundary and interface geometries and provide a non-trivial detailed check of an a a -theorem by Jensen and O’Bannon. Finally, we revisit the problem of the O (N) ( N ) model in the semi-infinite geometry. We find evidence that at N = N_c N = N c the extraordinary and special fixed points annihilate and only the ordinary fixed point is left for N > N_c N > N c .