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

It was recently found that the classical 3d O(N) model in the semi-infinite geometry can exhibit an "extraordinary-log" boundary universality class, where the spin-spin correla-tion function on the boundary falls off as < S(x) <middle dot> S(0)) similar to 1/(log x)(q) . This universality class exists for a range 2 <= N < N-c and Monte-Carlo simulations and conformal bootstrap indicate N-c > 3. In this work, we extend this result to the 3d O(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. We addition-ally show, in agreement with our RG analysis, that the line of defect fixed points which is present at N = infinity is lifted to the ordinary, special (no defect) and extraordinary-log universality classes by 1/N corrections. We study the "central charge" a for the O(N) model in the boundary and interface geometries and provide a non-trivial detailed check of an a-theorem by Jensen and O'Bannon. Finally, we revisit the problem of the O(N) model in the semi-infinite geometry. We find evidence that at N = N-c the extraordinary and special fixed points annihilate and only the ordinary fixed point is left for N > N-c.