Universal finite-size scaling in the extraordinary-log boundary phase of 3d \(O(N)\) 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 \(N\).