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In the present article we analyze Non-Perturbative Renormalization Group flowequations in the order phase of $\mathbb{Z}_2$ and $O(N)$ invariant scalarmodels in the derivative expansion approximation scheme. We first address thebehavior of the leading order approximation (LPA), discussing for whichregulators the flow is smooth and gives a convex free energy and when itbecomes singular. We improve the exact known solutions in the "internal" regionof the potential and exploit this solution in order to implement a numericalalgorithm that is much more stable that previous ones for $N>1$. After that, westudy the flow equations at second order of the Derivative Expansion andanalyze how and when LPA results change. We also discuss the evolution of fieldrenormalization factors.

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