Why Might the Standard Large $N$ Analysis Fail in the O($N$) Model: The Role of Cusps in the Fixed Point Potentials

TL;DR

This paper reveals that the standard large N analysis of the O(N) model misses certain fixed points with cusp singularities at N=infinity, which significantly influence finite N multicritical behavior and are likely a general phenomenon.

Contribution

It demonstrates the failure of traditional large N methods to capture fixed points with cusp singularities in the O(N) model and related models, highlighting a new mechanism affecting multicritical physics.

Findings

Identification of fixed points with cusp singularities at N=infinity

Demonstration of the physical impact of these fixed points at finite N

Evidence that the mechanism is likely generic across similar models

Abstract

The large $N$ expansion plays a fundamental role in quantum and statistical field theory. We show on the example of the O$(N)$ model that at $N=\infty$, its traditional implementation misses in all dimensions below four some fixed points of the renormalization group. These new fixed points show singularities at $N=\infty$ in their effective potential that become a boundary layer at finite $N$. We show that they have a physical impact on the multicritcal physics of the $O(N$) model at finite $N$. We also show that the mechanism at play holds also for the O($N$)$\otimes$O(2) model and is thus probably generic.