A fixed point can hide another one: the nonperturbative behavior of the tetracritical fixed point of the O($N$) models at large $N$

Abstract

We show that at $N=\infty$ and below its upper critical dimension, $d<d_{\mathrm{up}}$, the critical and tetracritical behaviors of the O($N$) models are associated with the same renormalization group fixed point (FP) potential. Only their derivatives make them different with the subtleties that taking their $N\to \infty$ limit and deriving them do not commute and that two relevant eigenperturbations show singularities. This invalidates both the $\epsilon -$ and the $1/N-$ expansions. We also show how the Bardeen-Moshe-Bander line of tetracritical FPs at $N=\infty$ and $d=d_{\mathrm{up}}$ can be understood from a finite-$N$ analysis.

arXiv comment8 pages, 11 figures

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