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

TL;DR

This paper reveals that at large N, the critical and tetracritical behaviors in O(N) models are governed by the same fixed point, with subtle differences arising from derivatives and non-commuting limits, challenging traditional expansion methods.

Contribution

It demonstrates that the critical and tetracritical fixed points coincide at infinite N and below the upper critical dimension, and clarifies the nature of their differences and the Bardeen-Moshe-Bander line.

Findings

Critical and tetracritical behaviors share the same fixed point at N=∞.

Derivatives of the fixed point potential distinguish critical from tetracritical behavior.

The standard epsilon and 1/N expansions are invalidated by non-commuting limits.

Abstract

We show that at $N=\infty$ and below its upper critical dimension, $d<d_{\rm up}$, 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 $d=d_{\rm up}$ can be understood from a finite-$N$ analysis.