Conformal Invariance of the large-$N$ limit of the $O(N)$ universality class

TL;DR

This paper proves conformal invariance in the large-$N$ limit of the $O(N)$ universality class using non-perturbative renormalization group methods, providing insights into the structure needed for conformal symmetry.

Contribution

It offers two non-perturbative proofs of conformal invariance in the large-$N$ limit of the $O(N)$ model, clarifying the theory's structure for conformal symmetry realization.

Findings

Two proofs of conformal invariance at large $N$

Unveiled the theory's structure enabling conformal symmetry

Provided expectations for conformal invariance in general theories

Abstract

Conformal symmetry is expected to be realized in many equilibrium statistical mechanical systems at criticality. Although this is certainly true in two-dimensional systems, the three-dimensional case is subtler, and only a few proofs exist, only so in very specific cases. In this work, we give two proofs for the large $N$ limit of the $O(N)$ universality class within the non-perturbative renormalization group framework: one functional, and one vertex-by-vertex in Fourier space. While doing so, we unveil how the theory is structured in order for conformal symmetry to be realized. As a consequence, we shed light on what to expect, on rather general grounds, for a theory to be conformally invariant.