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

It is known that the classical O(N) O ( N ) model in dimension d > 3 d gt; 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 d = 3 , it is less clear what happens to the extra-ordinary and special fixed points when d = 3 d = 3 and N \ge 2 N ≥ 2 . Here we show that formally treating N N as a continuous parameter, there exists a critical value N_c > 2 N c gt; 2 separating two distinct regimes. For 2 \leq N < N_c 2 ≤ N < N c the extra-ordinary fixed point survives in d = 3 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 log r . For N > N_c N gt; N c there is no fixed point with order parameter correlations decaying slower than power law. We discuss several scenarios for the evolution of the phase diagram past N = N_c N = N c . Our findings appear to be consistent with recent Monte Carlo studies of classical models with N = 2 N = 2 and N = 3 N = 3 . We also compare our results to numerical studies of boundary criticality in 2+1D quantum spin models.