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 extraordinary and the special. For the ordinarytransition the bulk and the boundary order simultaneously; theextraordinary 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 theextraordinary classes. While the ordinary fixed point survives in d = 3 d = 3 ,it is less clear what happens to the extraordinary 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 extraordinary fixed point survives in d = 3 d = 3 ,albeit in a modified form: the longrange 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.
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The extraordinary boundary transition in the 3d O(N) model via conformal bootstrap
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 extraordinarylog 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 semidefinite 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.
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 Award ID(s):
 1847861
 NSFPAR ID:
 10394236
 Date Published:
 Journal Name:
 SciPost Physics
 Volume:
 12
 Issue:
 6
 ISSN:
 25424653
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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