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Functional renormalization group meets computational fluid dynamics: RG flows in a multidimensional field space

Niklas Zorbach1,*, Adrian Koenigstein2, and Jens Braun1,3,4,†

  • *Contact author: niklas.zorbach@tu-darmstadt.de
  • †Contact author: jens.braun@tu-darmstadt.de
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Phys. Rev. D 113, 036011 – Published 10 February, 2026

DOI: https://doi.org/10.1103/z954-46md

Abstract

Within the functional renormalization group framework, we present a fluid-dynamical approach to solving flow equations for models living in a multidimensional 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. We demonstrate the effectiveness of our approach by performing explicit benchmark tests using (0+0)-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.

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References (101)

  1. C. Wetterich, Exact evolution equation for the effective potential, Phys. Lett. B 301, 90 (1993).
  2. U. Ellwanger, Flow equations for N point functions and bound states, Z. Phys. C 62, 503 (1994).
  3. T. R. Morris, The exact renormalization group and approximate solutions, Int. J. Mod. Phys. A 09, 2411 (1994).
  4. T. R. Morris, Derivative expansion of the exact renormalization group, Phys. Lett. B 329, 241 (1994).
  5. C. Wetterich, Effective average action in statistical physics and quantum field theory, Int. J. Mod. Phys. A 16, 1951 (2001).
  6. J. Berges, N. Tetradis, and C. Wetterich, Nonperturbative renormalization flow in quantum field theory and statistical physics, Phys. Rep. 363, 223 (2002).
  7. J. M. Pawlowski, Aspects of the functional renormalisation group, Ann. Phys. (Amsterdam) 322, 2831 (2007).
  8. H. Gies, Introduction to the functional RG and applications to gauge theories, Lect. Notes Phys. 852, 287 (2012).
  9. P. Kopietz, L. Bartosch, and F. Schütz, Introduction to the Functional Renormalization Group, Lecture Notes in Physics, Vol. 798 (Springer-Verlag, Berlin, Heidelberg, 2010).
  10. J. Braun, Fermion interactions and universal behavior in strongly interacting theories, J. Phys. G 39, 033001 (2012).
  11. N. Dupuis, L. Canet, A. Eichhorn, W. Metzner, J. M. Pawlowski, M. Tissier, and N. Wschebor, The nonperturbative functional renormalization group and its applications, Phys. Rep. 910, 1 (2021).
  12. N. Strodthoff, B.-J. Schaefer, and L. von Smekal, Quark-meson-diquark model for two-color QCD, Phys. Rev. D 85, 074007 (2012).
  13. N. Strodthoff and L. von Smekal, Polyakov-Quark-Meson-Diquark model for two-color QCD, Phys. Lett. B 731, 350 (2014).
  14. N. Khan, J. M. Pawlowski, F. Rennecke, and M. M. Scherer, The phase diagram of QC2D from functional methods, arXiv:1512.03673.
  15. P. Lakaschus, Inhomogeneous chiral condensates in low-energy color-superconductivity models of QCD, Ph.D. thesis, Universitätsbibliothek Johann Christian Senckenberg, 2021.
  16. B.-J. Schaefer and M. Wagner, The three-flavor chiral phase structure in hot and dense QCD matter, Phys. Rev. D 79, 014018 (2009).
  17. M. Mitter and B.-J. Schaefer, Fluctuations and the axial anomaly with three quark flavors, Phys. Rev. D 89, 054027 (2014).
  18. K. Otto, M. Oertel, and B.-J. Schaefer, Hybrid and quark star matter based on a nonperturbative equation of state, Phys. Rev. D 101, 103021 (2020).
  19. K. Otto, M. Oertel, and B.-J. Schaefer, Nonperturbative quark matter equations of state with vector interactions, Eur. Phys. J. Spec. Top. 229, 3629 (2020).
  20. K. Kamikado, N. Strodthoff, L. von Smekal, and J. Wambach, Fluctuations in the quark-meson model for QCD with isospin chemical potential, Phys. Lett. B 718, 1044 (2013).
  21. B. Hawashin, J. Rong, and M. M. Scherer, UV complete local field theory of persistent symmetry breaking in 2+1 dimensions, Phys. Rev. Lett. 134, 041602 (2025).
  22. M. Tissier, D. Mouhanna, and B. Delamotte, A nonperturbative approach of the principal chiral model between two-dimensions and four-dimensions, Phys. Rev. B 61, 15327 (2000).
  23. M. Tissier, B. Delamotte, and D. Mouhanna, Heisenberg frustrated magnets: A nonperturbative approach, Phys. Rev. Lett. 84, 5208 (2000).
  24. M. Tissier, B. Delamotte, and D. Mouhanna, An exact renormalization group approach to frustrated magnets, Int. J. Mod. Phys. A 16, 2131 (2001).
  25. M. Tissier, B. Delamotte, and D. Mouhanna, XY frustrated systems: Continuous exponents in discontinuous phase transitions, Phys. Rev. B 67, 134422 (2003).
  26. B. Delamotte, D. Mouhanna, and M. Tissier, Nonperturbative renormalization group approach to frustrated magnets, Phys. Rev. B 69, 134413 (2004).
  27. B. Delamotte, M. Dudka, D. Mouhanna, and S. Yabunaka, Functional renormalization group approach to noncollinear magnets, Phys. Rev. B 93, 064405 (2016).
  28. T. Debelhoir and N. Dupuis, First-order phase transitions in spinor Bose gases and frustrated magnets, Phys. Rev. A 94, 053623 (2016).
  29. S. Yabunaka and B. Delamotte, Global fixed point potential approach to frustrated antiferromagnets, J. Stat. Mech. (2025) 023204.
  30. C. A. Sánchez-Villalobos, B. Delamotte, and N. Wschebor, O(N)×O(2) scalar models: Including O(∂2) corrections in the functional renormalization group analysis, Phys. Rev. E 111, 034104 (2025).
  31. L. Classen, I. F. Herbut, L. Janssen, and M. M. Scherer, Competition of density waves and quantum multicritical behavior in Dirac materials from functional renormalization, Phys. Rev. B 93, 125119 (2016).
  32. E. Torres, L. Classen, I. F. Herbut, and M. M. Scherer, Fermion-induced quantum criticality with two length scales in Dirac systems, Phys. Rev. B 97, 125137 (2018).
  33. J. Berges and C. Wetterich, Equation of state and coarse grained free energy for matrix models, Nucl. Phys. B487, 675 (1997).
  34. A. Eichhorn, D. Mesterházy, and M. M. Scherer, Multicritical behavior in models with two competing order parameters, Phys. Rev. E 88, 042141 (2013).
  35. A. Eichhorn, L. Janssen, and M. M. Scherer, Critical O(N) models above four dimensions: Small-N solutions and stability, Phys. Rev. D 93, 125021 (2016).
  36. A. Eichhorn, T. Koslowski, and A. D. Pereira, Status of background-independent coarse-graining in tensor models for quantum gravity, Universe 5, 53 (2019).
  37. A. Koenigstein, M. J. Steil, N. Wink, E. Grossi, J. Braun, M. Buballa, and D. H. Rischke, Numerical fluid dynamics for FRG flow equations: Zero-dimensional QFTs as numerical test cases. I. The O(N) model, Phys. Rev. D 106, 065012 (2022).
  38. A. Koenigstein, M. J. Steil, N. Wink, E. Grossi, and J. Braun, Numerical fluid dynamics for FRG flow equations: Zero-dimensional QFTs as numerical test cases. II. Entropy production and irreversibility of RG flows, Phys. Rev. D 106, 065013 (2022).
  39. M. J. Steil and A. Koenigstein, Numerical fluid dynamics for FRG flow equations: Zero-dimensional QFTs as numerical test cases. III. Shock and rarefaction waves in RG flows reveal limitations of the N→∞ limit in O(N)-type models, Phys. Rev. D 106, 065014 (2022).
  40. E. Grossi and N. Wink, Resolving phase transitions with discontinuous Galerkin methods, SciPost Phys. Core 6, 071 (2023).
  41. E. Grossi, F. J. Ihssen, J. M. Pawlowski, and N. Wink, Shocks and quark-meson scatterings at large density, Phys. Rev. D 104, 016028 (2021).
  42. J. Stoll, N. Zorbach, A. Koenigstein, M. J. Steil, and S. Rechenberger, Bosonic fluctuations in the (1+1)-dimensional Gross-Neveu(-Yukawa) model at varying μ and T and finite N, arXiv:2108.10616.
  43. F. Ihssen and J. M. Pawlowski, Functional flows for complex effective actions, SciPost Phys. 15, 074 (2023).
  44. F. Ihssen, J. M. Pawlowski, F. R. Sattler, and N. Wink, Local disontinuous Galerkin for the functional renormalisation group, Comput. Phys. Commun. 300, 109182 (2024).
  45. L. Batini, E. Grossi, and N. Wink, Dissipation dynamics of a scalar field, Phys. Rev. D 108, 125021 (2023).
  46. F. J. Ihssen, Resolving the QCD phase structure, Ph.D. thesis, University of Heidelberg, 2023.
  47. F. Ihssen, F. R. Sattler, and N. Wink, Numerical RG-time integration of the effective potential: Analysis and benchmark, Phys. Rev. D 107, 114009 (2023).
  48. F. Ihssen and J. M. Pawlowski, Flowing fields and optimal RG-flows, arXiv:2305.00816.
  49. F. Ihssen, J. M. Pawlowski, F. R. Sattler, and N. Wink, Towards quantitative precision for QCD at large densities, Phys. Rev. D 111, 036030 (2025).
  50. F. Ihssen, J. M. Pawlowski, F. R. Sattler, and N. Wink, Towards quantitative precision in functional QCD I, arXiv:2408.08413.
  51. F. Ihssen and J. M. Pawlowski, Physics-informed renormalisation group flows, Ann. Phys. (Amsterdam) 481, 170177 (2025).
  52. N. Tetradis and D. F. Litim, Analytical solutions of exact renormalization group equations, Nucl. Phys. B464, 492 (1996).
  53. D. Litim and N. Tetradis, Approximate solutions of exact renormalization group equations, arXiv:hep-th/9501042.
  54. K.-I. Aoki, S.-I. Kumamoto, and D. Sato, Weak solution of the non-perturbative renormalization group equation to describe dynamical chiral symmetry breaking, Prog. Theor. Exp. Phys. 2014, 043B05 (2014).
  55. K.-I. Aoki, S.-I. Kumamoto, and M. Yamada, Phase structure of NJL model with weak renormalization group, Nucl. Phys. B931, 105 (2018).
  56. V. Rivasseau, Constructive field theory in zero dimension, Adv. Theor. Math. Phys. 2010, 180159 (2009).
  57. J. Keitel and L. Bartosch, The zero-dimensional O(N) vector model as a benchmark for perturbation theory, the large-N expansion and the functional renormalization group, J. Phys. A 45, 105401 (2012).
  58. D. Skinner, Lecture notes: Quantum Field Theory II (2018), [Online; accessed 2021.01.12].
  59. K. Fraboulet, Path-integral approaches to strongly-coupled quantum many-body systems, Ph.D. thesis, LPMC—Palaiseau—IJCLab, Orsay, 2021.
  60. C. Wetterich, The average action for scalar fields near phase transitions, Z. Phys. C 57, 451 (1993).
  61. U. Wolff, The phase diagram of the infinite-N Gross-Neveu model at finite temperature and chemical potential, Phys. Lett. 157B, 303 (1985).
  62. O. Schnetz, M. Thies, and K. Urlichs, Full phase diagram of the massive Gross-Neveu model, Ann. Phys. (Amsterdam) 321, 2604 (2006).
  63. M. D’Attanasio and T. R. Morris, Large N and the renormalization group, Phys. Lett. B 409, 363 (1997).
  64. G. ’t Hooft, A two-dimensional model for mesons, Nucl. Phys. B75, 461 (1974).
  65. B. Delamotte, An introduction to the nonperturbative renormalization group, Lect. Notes Phys. 852, 49 (2012).
  66. J.-P. Blaizot, J. M. Pawlowski, and U. Reinosa, Functional renormalization group and 2PI effective action formalism, Ann. Phys. (Amsterdam) 431, 168549 (2021).
  67. J. M. Pawlowski, Solving integrals with flow equations, Slides for the lecture Non-perturbative aspects of gauge theories winter term 2012/2013 (2013), [Online; accessed 2020.10.29].
  68. S. Flörchinger, Functional renormalization and ultracold quantum gases, Springer Theses, 2010, 10.1007/978-3-642-14113-3.
  69. S. Kemler and J. Braun, Towards a renormalization group approach to density functional theory—general formalism and case studies, J. Phys. G 40, 085105 (2013).
  70. A. G. Catalano, Application of renormalization group techniques to the solution of integrals and Schrödinger eigenvalue equations, Master’s thesis, Politecnico di Torino, 2019.
  71. J. Braun, M. Leonhardt, and J. M. Pawlowski, Renormalization group consistency and low-energy effective theories, SciPost Phys. 6, 056 (2019).
  72. S. Moroz, Few-body physics with functional renormalization, Ph.D. thesis, University of Heidelberg, 2011.
  73. J. S. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, 1st ed. (Springer, New York, 2007).
  74. L. Rezzolla and O. Zanotti, Relativistic hydrodynamics (Oxford University Press, Oxford, England, UK, 2018).
  75. R. J. LeVeque, Numerical Methods for Conservation Laws, 2nd ed. (Birkhäuser, Basel, 1992).
  76. R. J. LeVeque, Finite-Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics (Cambridge University Press, Cambridge, England, 2002).
  77. M. Reuter and C. Wetterich, Effective average action for gauge theories and exact evolution equations, Nucl. Phys. B417, 181 (1994).
  78. D. F. Litim and J. M. Pawlowski, Wilsonian flows and background fields, Phys. Lett. B 546, 279 (2002).
  79. H. Gies, Running coupling in Yang-Mills theory: A flow equation study, Phys. Rev. D 66, 025006 (2002).
  80. A. Kurganov and E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. Comput. Phys. 160, 241 (2000).
  81. W. F. Ames, Numerical Methods for Partial Differential Equations, 3rd ed., Computer Science and Scientific Computing (Academic Press, Boston [u.a.], 1992).
  82. A. Kurganov, S. Noelle, and G. Petrova, Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton–Jacobi equations, SIAM J. Sci. Comput. 23, 707 (2001).
  83. A. Kurganov and D. Levy, A third-order semi-discrete central scheme for conservation laws and convection-diffusion equations, SIAM J. Sci. Comput. 22, 1461 (2000).
  84. A. Kurganov, G. Petrova, and B. Popov, Adaptive semidiscrete central-upwind schemes for nonconvex hyperbolic conservation laws, SIAM J. Sci. Comput. 29, 2381 (2007).
  85. T. Jourdan, G. Stoltz, F. Legoll, and L. Monasse, An accurate scheme to solve cluster dynamics equations using a Fokker-Planck approach, Comput. Phys. Commun. 207, 170 (2016).
  86. H. Tang and T. Tang, Adaptive mesh methods for one- and two-dimensional hyperbolic conservation laws, SIAM J. Numer. Anal. 41, 487 (2003).
  87. Wikipedia contributors, Flux limiter, 2020, https://en.wikipedia.org/wiki/Flux_limiter [Online; accessed 2020.10.15].
  88. J. Rais, A. Koenigstein, N. Zorbach, and C. Greiner, Solving the Lindblad equation with methods from computational fluid dynamics, Phys. Rev. E 111, 055308 (2025).
  89. Wolfram Research, Inc., Mathematica, Version 12.2 (2020).
  90. P. Virtanen et al. (scipy1.0 Contributors), scipy1.0: Fundamental algorithms for scientific computing in python, Nat. Methods 17, 261 (2020).
  91. D. F. Litim, Optimized renormalization group flows, Phys. Rev. D 64, 105007 (2001).
  92. D. F. Litim, Mind the gap, Int. J. Mod. Phys. A 16, 2081 (2001).
  93. D. F. Litim, J. M. Pawlowski, and L. Vergara, Convexity of the effective action from functional flows, arXiv:hep-th/0602140.
  94. N. Zorbach, J. Stoll, and J. Braun, Optimization and stabilization of functional renormalization group flows, Phys. Rev. D 111, 096022 (2025).
  95. J. Braun, M. Leonhardt, and M. Pospiech, Fierz-complete NJL model study III: Emergence from quark-gluon dynamics, Phys. Rev. D 101, 036004 (2020).
  96. W.-j. Fu, J. M. Pawlowski, and F. Rennecke, QCD phase structure at finite temperature and density, Phys. Rev. D 101, 054032 (2020).
  97. N. Zorbach, A. Koenigstein, and J. Braun, Functional renormalization group meets computational fluid dynamics: RG flows in a multidimensional field space, Technische Universität data library, 2025, https://tudatalib.ulb.tu-darmstadt.de/items/60aa2899-bf38-4694-86ce-a7f78a7b5fc3.
  98. C. R. Harris et al., Array programming with numpy, Nature (London) 585, 357 (2020).
  99. A. Kurganov and G. Petrova, A third-order semi-discrete genuinely multidimensional central scheme for hyperbolic conservation laws and related problems, Numer. Math. 88, 683 (2001).
  100. A. Kurganov and E. Tadmor, Solution of two-dimensional Riemann problems for gas dynamics without Riemann problem solvers, Numer. Methods Partial Differ. Eqn. 18, 584 (2002).
  101. A. Kurganov, S. Noelle, and G. Petrova, Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton–Jacobi equations, J. Comput. Phys. SIAM J. Sci. Comput. 23, 707 (2000).

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Phys. Rev. D 113, 036011– Published 10 February, 2026
  • Received 11 September 2025
  • Accepted 9 November 2025
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