Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Adrian Koenigstein, Martin J. Steil, Stefan Floerchinger

TL;DR

This work reframes FRG flows for two-point functions as viscous Hamilton-Jacobi-type PDEs, enabling stable, high-resolution numerical treatment via a KT-HJ finite-volume scheme. It starts from a zero-dimensional fermion-boson toy model to derive exact and numerical flow equations, then demonstrates the method on higher-dimensional truncations including a $d=3$ Z2 model and a $1+1$-dimensional Gross-Neveu–Yukawa system, highlighting how advection and diffusion terms govern field-space dynamics and how shocks, phase transitions, and sign problems manifest in the flows. The key contributions are the identification of the functional HHJ structure, the development of a hybrid conservative/nonconservative numerical scheme that handles field-dependent couplings, and the demonstration of convergence and stability across challenging regimes, with insights into convexity, Yang-Lee zeros, and stiffness. The results open avenues for more robust nonperturbative FRG calculations and suggest connections to stochastic control and viscosity solution theory, potentially guiding future numerical and analytical developments in FRG studies. The practical impact lies in enabling reliable, scalable simulations of complex FRG flows with field-dependent interactions, aiding exploration of phase structure and critical phenomena in quantum and statistical field theories.

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.

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

TL;DR

This work reframes FRG flows for two-point functions as viscous Hamilton-Jacobi-type PDEs, enabling stable, high-resolution numerical treatment via a KT-HJ finite-volume scheme. It starts from a zero-dimensional fermion-boson toy model to derive exact and numerical flow equations, then demonstrates the method on higher-dimensional truncations including a Z2 model and a -dimensional Gross-Neveu–Yukawa system, highlighting how advection and diffusion terms govern field-space dynamics and how shocks, phase transitions, and sign problems manifest in the flows. The key contributions are the identification of the functional HHJ structure, the development of a hybrid conservative/nonconservative numerical scheme that handles field-dependent couplings, and the demonstration of convergence and stability across challenging regimes, with insights into convexity, Yang-Lee zeros, and stiffness. The results open avenues for more robust nonperturbative FRG calculations and suggest connections to stochastic control and viscosity solution theory, potentially guiding future numerical and analytical developments in FRG studies. The practical impact lies in enabling reliable, scalable simulations of complex FRG flows with field-dependent interactions, aiding exploration of phase structure and critical phenomena in quantum and statistical field theories.

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 -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 -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.

Paper Structure

This paper contains 46 sections, 119 equations, 22 figures.

Table of Contents

  1. Introduction
  2. Research issue
  3. Scope, structure and conception of this work
  4. The Wetterich equation and its functional derivatives
  5. Functional flow equation for the one-point function
  6. Functional flow equation for the two-point function
  7. The FRG as an infinite dimensional stochastic optimal control problem
  8. Testbed: A zero-dimensional fermion-boson model
  9. Expectation values and generating functions
  10. Full field-dependence of the two-point vertex functions
  11. The FRG approach to the zero-dimensional model
  12. An exact truncation
  13. FRG flow equations
  14. Comments on the FRG flow equations and their nonconservative form
  15. FRG flow equations as coupled viscous Hamilton-Jacobi systems
  16. ...and 31 more sections

Figures (22)

  • Figure 1: Test 0: flow of $M ( t, \varphi )$ (colored) and the exact reference solution (dashed, black).
  • Figure 2: Test 0: Relative error \ref{['eq:relative_error']} of the numerical solution to the exact solution as a function of $\varphi$ for the potentials \ref{['eq:test_0_U_I']}-\ref{['eq:test_0_U_IV']} at a resolution of $N = 4001$ points for $\varphi_\mathrm{max} = 10$.
  • Figure 3: Test 0: $L^1$ and $L^\infty$ norm, see \ref{['eq:L1_norm', 'eq:Linf_norm']}, of the deviation between the numerical and the exact solution on the computational domain as a function of the spatial resolution for the potentials \ref{['eq:test_0_U_I']}-\ref{['eq:test_0_U_IV']}.
  • Figure 4: Test 1: flow for the initial conditions \ref{['eq:test_U_minimal']} and \ref{['eq:test_H_minimal']}. The exact reference solution for the last three times is shown black dashed.
  • Figure 5: Test 1: Relative error \ref{['eq:relative_error']} of the numerical solution to the exact solution as a function of $\varphi$ for the initial conditions \ref{['eq:test_U_minimal']} and \ref{['eq:test_H_minimal']} for different limiters at a resolution of $N = 4001$ points with $\varphi_\mathrm{max} = 10$.
  • ...and 17 more figures