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It was recently realized that the three-dimensional O($N$) model possesses anextraordinary boundary universality class for a finite range of $N \ge 2$. Fora given $N$, the existence and universal properties of this class are predictedto be controlled by certain amplitudes of the normal universality class, whereone applies an explicit symmetry breaking field to the boundary. In thisLetter, we study the normal universality class for $N = 2, 3$ using Monte Carlosimulations on an improved lattice model and extract these universalamplitudes. Our results are in good agreement with direct Monte Carlo studiesof the extraordinary universality class serving as a nontrivial quantitativecheck of the connection between the normal and extraordinary classes.

Boundary Criticality of the 3D O($N$) Model: From Normal to Extraordinary