Universal finite-size scaling in the extraordinary-log boundary phase of three-dimensional $O(N)$ model

Recent advances in boundary critical phenomena have led to the discovery of a new surface universality class in the three-dimensional $O(N)$ model. The newly found ``extraordinary-log" phase can be realized on a two-dimensional surface for $N< N_c$, with $N_c>3$, and on a plane defect embedded into a three-dimensional system, for any $N$. One of the key features of the extraordinary-log phase is the presence of logarithmic violations of standard finite-size scaling. In this work we study finite-size scaling in the extraordinary-log universality class by means of Monte Carlo simulations of an improved lattice model. We simulate the model with open boundary conditions, realizing the extraordinary-log phase on the surface for $N=2,3$, as well as with fully periodic boundary conditions and in the presence of a plane defect for $N=2,3,4$. In line with theory predictions, renormalization-group invariant observables studied here exhibit a logarithmic dependence on the size of the system. We numerically access not only the leading term in the $\beta$-function governing these logarithmic violations, but also the subleading term, which controls the evolution of the boundary phase diagram as a function of $N$.