Boundary criticality of the O(N) model in d = 3 critically revisited

It is known that the classical $O(N)$ model in dimension $d > 3$ at its bulk critical point admits three boundary universality classes: the ordinary, the extra-ordinary and the special. For the ordinary transition the bulk and the boundary order simultaneously; the extra-ordinary fixed point corresponds to the bulk transition occurring in the presence of an ordered boundary, while the special fixed point corresponds to a boundary phase transition between the ordinary and the extra-ordinary classes. While the ordinary fixed point survives in $d = 3$, it is less clear what happens to the extra-ordinary and special fixed points when $d = 3$ and $N \ge 2$. Here we show that formally treating $N$ as a continuous parameter, there exists a critical value $N_c > 2$ separating two distinct regimes. For $N < N_c$ the extra-ordinary fixed point survives in $d = 3$, albeit in a modified form: the long-range boundary order is lost, instead, the order parameter correlation function decays as a power of $\log r$. In particular, for $N=2$, starting in the surface phase with quasi-long-range order and approaching the bulk phase transition, the stiffness of the surface order parameter diverges logarithmically. For $N > N_c$ there is no fixed point with order parameter correlations decaying slower than power law; we discuss two scenarios for the evolution of the phase diagram past $N = N_c$. Our findings appear to be consistent with recent Monte-Carlo studies of classical models with $N = 2$ and $N = 3$. We also compare our results to numerical studies of boundary criticality in 2+1D quantum spin models.