Better Integrators for Functional Renormalization Group Calculations

TL;DR

This paper evaluates and optimizes numerical integration schemes for functional renormalization group calculations, demonstrating significant reductions in computational effort while maintaining accuracy, using the Hubbard model as a test case.

Contribution

It introduces an error estimate for FRG ODE solutions and recommends integrators that reduce computational cost by threefold.

Findings

Threefold reduction in integration steps needed

Error estimate for FRG solutions developed

Recommended integrators improve efficiency

Abstract

We analyze a variety of integration schemes for the momentum space functional renormalization group calculation with the goal of finding an optimized scheme. Using the square lattice $t-t'$ Hubbard model as a testbed we define and benchmark the quality. Most notably we define an error estimate of the solution for the ordinary differential equation circumventing the issues introduced by the divergences at the end of the FRG flow. Using this measure to control for accuracy we find a threefold reduction in number of required integration steps achievable by choice of integrator. We herewith publish a set of recommended choices for the functional renormalization group, shown to decrease the computational cost for FRG calculations and representing a valuable basis for further investigations.