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arXiv:2404.18707 · Journal of Statistical Mechanics Theory and Experiment

Numerical accuracy of the derivative-expansion-based functional renormalization group

Andrzej Chlebicki

Published Sep 17, 2024• 1 citation
arXivHTMLPDF

Abstract

Abstract We investigate the precision of the numerical implementation of the functional renormalization group based on extracting the eigenvalues from the linearized renormalization group transformation. For this purpose, we implement the local potential approximation and O(∂2) orders of the derivative expansion for the three-dimensionalO(N) models with N ∈ {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 ρ~ grid is cut off at the value ρ~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.

DOI
10.1088/1742-5468/ad6c31

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Abstract We investigate the precision of the numerical implementation of the functional renormalization group based on extracting the eigenvalues from the linearized renormalization group transformation. For this purpose, we implement the local potential approximation and O(∂2) orders of the derivative expansion for the three-dimensionalO(N) models with N ∈ {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 ρ~ grid is cut off at the value ρ~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.
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In the last few years the derivative expansion of the nonperturbative renormalization group has proven to be a very efficient tool for the precise computation of critical quantities. In particular, recent progress in the understanding of its convergence properties allowed for an estimate of the error bars as well as the precise computation of many critical quantities. In this work we extend previous studies to the computation of several universal amplitude ratios for the critical regime of O(N) models using the derivative expansion of the nonperturbative renormalization group at order O(∂^{4}) for three-dimensional systems.
Precision calculation of universal amplitude ratios in O(N)O(N)O(N) universality classes: Derivative Expansion results at order O(∂4)\mathcal{O}(\partial^4)O(∂4)
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