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We study the $O(N)$ model in dimension three (3$d$) at large and infinite $N$and show that the line of fixed points found at $N=\infty$ --theBardeen-Moshe-Bander (BMB) line-- has an intriguing origin at finite $N$. Thelarge $N$ limit that allows us to find the BMB line must be taken on particulartrajectories in the $(d,N)$-plane: $d=3-\alpha/N$ and not at fixed dimension$d=3$. Our study also reveals that the known BMB line is only half of the trueline of fixed points, the second half being made of singular fixed points. Thepotentials of these singular fixed points show a cusp for a finite value of thefield and their finite $N$ counterparts a boundary layer.

The finite $N$ origin of the Bardeen-Moshe-Bander phenomenon and its extension at $N=\infty$ by singular fixed points