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

Understand

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.