Functional Renormalization Group flows as diffusive Hamilton-Jacobi-type equations

TL;DR

The paper reformulates functional renormalization group equations as Hamilton-Jacobi PDEs and introduces a numerical scheme for solving them, demonstrated on fermionic models in various dimensions and conditions.

Contribution

It presents a novel approach by interpreting FRG flow equations as Hamilton-Jacobi equations and developing a numerical method for their solution.

Findings

Successfully applied to zero-dimensional fermion-boson system

Extended to three-dimensional bosonic Z2 model with field-dependent functions

Analyzed (1+1)-dimensional Gross-Neveu model at finite temperature and chemical potential

Abstract

In order to find reliable and efficient numerical approximation schemes, we suggest to identify the Functional Renormalization Group flow equations of one-particle irreducible two-point functions as Hamilton-Jacobi(-Bellman)-type partial differential equations. Based on this reformulation and reinterpretation we adopt a numerical scheme for the solution of field-dependent flow equations as nonlinear partial differential equations. We demonstrate this novel approach by first applying it to a simple fermion-boson system in zero spacetime dimensions - which itself presents as an interesting playground for method development. Afterwards, we show, how the gained insights can be transferred to more interesting problems: One is the bosonic $\mathbb{Z}_2$-symmetric model in three Euclidean dimensions within a truncation that involves the field-dependent effective potential and field-dependent wave-function renormalization. The other example is the $(1 + 1)$-dimensional Gross-Neveu model within a truncation that involves a field-dependent potential and a field-dependent fermion mass/Yukawa coupling at nonzero temperature, chemical potential, and finite fermion number.