Incompleteness of the large-$N$ analysis of the $O(N)$ models: Nonperturbative cuspy fixed points and their nontrivial homotopy at finite $N$

TL;DR

This paper reveals that traditional large-$N$ analyses of $O(N)$ models miss important fixed points due to singularities, and introduces new fixed points with complex homotopy structures using nonperturbative renormalization group methods.

Contribution

It identifies nonperturbative cuspy fixed points in $O(N)$ models and explores their homotopy, extending understanding beyond standard large-$N$ approaches.

Findings

Four new fixed points discovered across all dimensions and $N>0$

Cuspy singularities form boundary layers at large but finite $N$

Homotopy structures of fixed points are nontrivial and bi-valued

Abstract

We summarize the usual implementations of the large $N$ limit of $O(N)$ models and show in detail why and how they can miss some physically important fixed points when they become singular in the limit $N\to\infty$. Using Wilson's renormalization group in its functional nonperturbative versions, we show how the singularities build up as $N$ increases. In the Wilson-Polchinski version of the nonperturbative renormalization group, we show that the singularities are cusps, which become boundary layers for finite but large values of $N$. The corresponding fixed points being never close to the Gaussian, are out of reach of the usual perturbative approaches. We find four new fixed points and study them in all dimensions and for all $N>0$ and show that they play an important role for the tricritical physics of $O(N)$ models. Finally, we show that some of these fixed points are bi-valued when they are considered as functions of $d$ and $N$ thus revealing important and nontrivial homotopy structures. The Bardeen-Moshe-Bander phenomenon that occurs at $N=\infty$ and $d=3$ is shown to play a crucial role for the internal consistency of all our results.