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

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.