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It was recently found that the classical 3d O(N) model in the semi-infinite geometry can exhibit an “extraordinary-log” boundary universality class, where the spin-spin correlation function on the boundary falls off as < S(x) S(0)> ~ 1/ (log x)^q. This universality class exists for a range 2 ≤ N < Nc and Monte-Carlo simulations and conformal bootstrap indicate Nc > 3. In this work, we extend this result to the 3d O(N) model in an infinite geometry with a plane defect. We use renormalization group (RG) to show that in this case the extraordinary-log universality class is present for any finite N ≥ 2. We additionally show, in agreement with our RG analysis, that the line of defect fixed points which is present at infinite N is lifted to the ordinary, special (no defect) and extraordinary-log universality classes by 1/N corrections. We study the “central charge” a for the O(N) model in the boundary and interface geometries and provide a non-trivial detailed check of an a-theorem by Jensen and O’Bannon. Finally, we revisit the problem of the O(N) model in the semi-infinite geometry. We find evidence that at N = Nc the extraordinary and special fixed points annihilate and only the ordinary fixed point is left for N > Nc . 

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 ordinarytransition the bulk and the boundary order simultaneously; theextra-ordinary fixed point corresponds to the bulk transition occurringin the presence of an ordered boundary, while the special fixed pointcorresponds to a boundary phase transition between the ordinary and theextra-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 fixedpoints 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 decayingslower than power law. We discuss several scenarios for the evolution ofthe phase diagram past N = N_c N = N c .Our findings appear to be consistent with recent Monte Carlo studies ofclassical models with N = 2 N = 2 and N = 3 N = 3 .We also compare our results to numerical studies of boundary criticalityin 2+1D quantum spin models. 

This paper studies the critical behavior of the 3d classicalO (N) ( N ) model with a boundary. Recently, one of us established that upontreating N N as a continuous variable, there exists a critical value N_c > 2 N c > 2 such that for 2 \leq N < N_c 2 ≤ N < N c the model exhibits a new extraordinary-log boundary universality class,if the symmetry preserving interactions on the boundary are enhanced. N_c N c is determined by a ratio of universal amplitudes in the normaluniversality class, where instead a symmetry breaking field is appliedon the boundary. We study the normal universality class using thenumerical conformal bootstrap. We find truncated solutions to thecrossing equation that indicate N_c \approx 5 N c ≈ 5 .Additionally, we use semi-definite programming to place rigorous boundson the boundary CFT data of interest to conclude that N_c > 3 N c > 3 ,under a certain positivity assumption which we check in variousperturbative limits.