Functional Renormalization Group meets Computational Fluid Dynamics: RG flows in a multi-dimensional field space

TL;DR

This paper introduces a novel fluid-dynamical method for solving Functional Renormalization Group flow equations in multi-dimensional field spaces, leveraging CFD techniques for improved computational efficiency and applicability.

Contribution

It reformulates FRG flow equations as nonlinear advection-diffusion equations and applies CFD numerical schemes, enabling effective solutions for complex multi-invariant models.

Findings

Successfully benchmarked with zero-dimensional models

Demonstrated applicability to fermion-boson systems in three dimensions

Validated the fluid-dynamical approach for multi-invariant flow equations

Abstract

Within the Functional Renormalisation Group (FRG) approach, we present a fluid-dynamical approach to solving flow equations for models living in a multi-dimensional field space. To this end, the underlying exact flow equation of the effective potential is reformulated as a set of nonlinear advection-diffusion-type equations which can be solved using the Kurganov-Tadmor central scheme, a modern finite-volume discretization from computational fluid dynamics (CFD). We demonstrate the effectiveness of our approach by performing explicit benchmark tests using zero-dimensional models with two discretized field space directions or two symmetry invariants. Our techniques can be directly applied to flow equations of effective potentials of general (fermion-)boson systems with multiple invariants or condensates, as we also demonstrate for two concrete examples in three spacetime dimensions.