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

Abstract

We study the O(N ) model in dimension three (3d) at large and infinite N and show that the line of fixed points found at N = ∞ -the Bardeen-Moshe-Bander (BMB) line-has an intriguing origin at finite N . The large N limit that allows us to find the BMB line must be taken on particular trajectories in the (d, N )-plane: d = 3 − α/N and not at fixed dimension d = 3. Our study also reveals that the known BMB line is only half of the true line of fixed points, the second half being made of singular fixed points. The potentials of these singular fixed points show a cusp for a finite value of the field and their finite N counterparts a boundary layer.

The fate of $O(N)$ multi-critical universal behaviour

“…Applying this proce-dure to the n = 2 fixed point and its UV stable partner projects each of them on single point in the Bardeen-Moshe-Bander (BMB) line found at N = ∞ in d = 3 [59]. Moreover, any fixed point involved in the annihilation mechanism of the n = 2 case appears to have an N = ∞ counterpart in d = 3 [60]. Repeating this analysis for any n ∈ 2N shall yield the same result, as infinite generalisations of the BMB phenomenon are expected to appear at the upper critical dimensions of even order multi-critical universality classes [61].…”

The background field method and critical vector models

“…The instability effect is, however, nonperturbative in 1/N . Moreover, it is not clear whether it will persist at any finite N , see [28] and references therein for a recent discussion. In this section we introduce our notation, review the derivation of the UV fixed point, and calculate the anomalous dimensions of φ and φ 2 .…”