On the Stability of the O(N)-Invariant and the Cubic-Invariant 3-Dimensional $N$-Component Renormalization Group Fixed Points in the Hierarchical Approximation

TL;DR

This paper analyzes the stability of O(N) and cubic-invariant fixed points in a 3D N-component field model using renormalization group methods within an ultralocal approximation, identifying a critical N value near 2.2194.

Contribution

It provides a precise computation of the critical N value where the O(N) fixed point becomes unstable, and compares it with the cubic-invariant fixed point within the hierarchical approximation.

Findings

Critical N_c is approximately 2.219435 to 2.219436.

The O(N) fixed point is stable below N_c and unstable above.

The critical N values for both fixed points coincide within computational accuracy.

Abstract

We compute renormalization group fixed points and their spectrum in an ultralocal approximation. We study a case of two competing non-trivial fixed points for a three-dimensional real $N$-component field: the O(N)-invariant fixed point vs.~the cubic-invariant fixed point. We compute the critical value $N_{c}$ of the cubic $\phi^{4}$-perturbation at the O(N)-fixed point. The O(N) fixed point is stable under a cubic $\phi^{4}$-perturbation below $N_{c}$, above $N_{c}$ it is unstable. The critical value comes out as $2.219435<N_{c}< 2.219436$ in the ultralocal approximation. We also compute the critical value of $N$ at the cubic invariant fixed point. Within the accuracy of our computations, the two values coincide.