Boundary operator expansion and extraordinary-log phase in the tricritical O(N) model

Boundary criticality exhibits rich behavior; for example, in the boundary 3D O(N) model, in addition to the ordinary and special transitions, an extraordinary-log phase transition is observed for certain values of N. In this paper, we study a 3D tricritical O(N) model with a boundary. We first compute the mean-field Green's function with a general coupling of φ²ⁿ (with n=3 corresponding to the tricritical point) at the extraordinary phase transition. Then, by employing the technique of layer susceptibility, we solve the boundary operator expansion using the ε=3-d expansion. Based on these results, we demonstrate for the first time that when 1c, the tricritical point exhibits an extraordinary-log transition characterized by logarithmic decay of correlation functions.