One Fixed Point Can Hide Another One: Nonperturbative Behavior of the Tetracritical Fixed Point of O(N) Models at Large N

We show that at N = co and below its upper critical dimension, d < d(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 ? 8 limit and deriving them do not commute and that two relevant eigenperturbations show singularities. This invalidates both the e-and the 1/N-expansions. We also show how the Bardeen-Moshe-Bander line of tetracritical FPs at N = 8 and d = d(up )can be understood from a finite -N analysis.

Automated summary

This study shows that in the O(N) models, the critical and tetracritical behaviors are associated with the same FP potential when N = co and below its upper critical dimension, d < d(up). However, their derivatives introduce subtleties such as non-commutativity when taking the N ? 8 limit and deriving them, and two relevant eigenperturbations exhibit singularities. This invalidates both the e- and 1/N-expansions. Additionally, we demonstrate how the Bardeen-Moshe-Bander line of tetracritical FPs at N = 8 and d = d(up) can be understood through finite-N analysis.