Numerical Accuracy of the Derivative-Expansion-Based Functional Renormalization Group

TL;DR

This paper assesses the numerical accuracy of the functional renormalization group method applied to three-dimensional $O(N)$ models, identifying error sources and providing guidelines for reliable implementation.

Contribution

It introduces a detailed error analysis and convergence tests for the derivative expansion-based functional renormalization group, with an open-source implementation for reproducibility.

Findings

Numerical schemes converge with errors much smaller than derivative expansion uncertainties.

Rounding errors can impair convergence, especially with coarse discretization grids.

Proper choice of the $ ilde ho$ grid cutoff is crucial for reliable results.

Abstract

We investigate the precision of the numerical implementation of the functional renormalization group based on extracting the eigenvalues from the linearized RG transformation. For this purpose, we implement the LPA and $O(\partial^2)$ orders of the derivative expansion for the three-dimensional $O(N)$ models with $N~\in~\{1,2,3\}$. We identify several categories of numerical error and devise simple tests to track their magnitude as functions of numerical parameters. Our numerical schemes converge properly and are characterized by errors of several orders of magnitude smaller than the error bars of the derivative expansion for these models. We highlight situations in which our methods cease to converge, most often due to rounding errors. In particular, we observe an impaired convergence of the discretization scheme when the $\tilde \rho$ grid is cut off at the value $\tilde \rho_{\text{Max}}$ smaller than $3.5$ times the local potential minimum. The program performing the numerical calculations for this study is shared as an open-source library accessible for review and reuse.