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 correlation function on the boundary falls off as $\langle \vec{S}(x) \cdot \vec{S}(0)\rangle \sim \frac{1}{(\log x)^q}$. This universality class exists for a range $2 \leq N < N_c$ {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 $N \ge 2$. We additionally show, in agreement with our RG analysis, that the line of defect fixed points which is present at $N = \infty$ 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 $N > N_c$.