The ordered phase of $O(N)$ model within the Non-Perturbative Renormalization Group

TL;DR

This paper investigates the phase structure of $O(N)$ models using Non-Perturbative Renormalization Group equations, improving numerical stability and analyzing effects of higher-order approximations on the flow and field renormalization.

Contribution

It introduces a more stable numerical algorithm for $O(N)$ models and analyzes the impact of second-order derivative expansion on the flow and renormalization factors.

Findings

Stable numerical algorithm for $O(N)$ models in the NPRG framework.

Analysis of the influence of second-order derivatives on flow equations.

Insights into the behavior of the potential and renormalization factors.

Abstract

In the present article we analyze Non-Perturbative Renormalization Group flow equations in the order phase of $\mathbb{Z}_2$ and $O(N)$ invariant scalar models in the derivative expansion approximation scheme. We first address the behavior of the leading order approximation (LPA), discussing for which regulators the flow is smooth and gives a convex free energy and when it becomes singular. We improve the exact known solutions in the "internal" region of the potential and exploit this solution in order to implement a numerical algorithm that is much more stable that previous ones for $N>1$. After that, we study the flow equations at second order of the Derivative Expansion and analyze how and when LPA results change. We also discuss the evolution of field renormalization factors.