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

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:mi>O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>∂</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math> orders of the derivative expansion for the three-dimensional O ( N ) models with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:mi>N</mml:mi><mml:mtext> </mml:mtext><mml:mo>∈</mml:mo><mml:mtext> </mml:mtext><mml:mo fence="false" stretchy="false">{</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo fence="false" stretchy="false">}</mml:mo></mml:mrow></mml:math> . 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy="true">~</mml:mo></mml:mover></mml:mrow></mml:mrow></mml:math> grid is cut off at the value <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:msub><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy="true">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mtext>Max</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math> 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.