- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Testing the parquet equations and the U(1) Ward identity for real-frequency correlation functions from the multipoint numerical renormalization group

Nepomuk Ritz1, Anxiang Ge1, Markus Frankenbach1, Mathias Pelz1, Jan von Delft1, and Fabian B. Kugler2,3

PDF

Phys. Rev. Research 7, 033139 – Published 11 August, 2025

DOI: https://doi.org/10.1103/3jtq-5wf5

Abstract

Recently, it has become possible to compute real-frequency four-point correlation functions of quantum impurity models using a multipoint extension of the numerical renormalization group (mpNRG). In this work, we perform several numerical consistency checks of the output of mpNRG by investigating exact relations between two- and four-point functions. This includes the Bethe-Salpeter equations and the Schwinger-Dyson equation from the parquet formalism, which we evaluate in two formally identical but numerically nonequivalent ways. We also study the first-order U(1) Ward identity between the vertex and the self-energy for the first time in full generality in the real-frequency Keldysh formalism. We generally find good agreement of all relations, often up to a few percent, both at weak and at strong interaction.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (87)

  1. A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions, Rev. Mod. Phys. 68, 13 (1996).
  2. G. Rohringer, H. Hafermann, A. Toschi, A. A. Katanin, A. E. Antipov, M. I. Katsnelson, A. I. Lichtenstein, A. N. Rubtsov, and K. Held, Diagrammatic routes to nonlocal correlations beyond dynamical mean field theory, Rev. Mod. Phys. 90, 025003 (2018).
  3. F. B. Kugler, S.-S. B. Lee, and J. von Delft, Multipoint correlation functions: Spectral representation and numerical evaluation, Phys. Rev. X 11, 041006 (2021).
  4. S.-S. B. Lee, F. B. Kugler, and J. von Delft, Computing local multipoint correlators using the numerical renormalization group, Phys. Rev. X 11, 041007 (2021).
  5. E. Walter, Real-frequency dynamics of quantum impurity models studied with fRG, NRG, CFT, Ph.D. thesis, LMU München, 2021.
  6. A. Ge, N. Ritz, E. Walter, S. Aguirre, J. von Delft, and F. B. Kugler, Real-frequency quantum field theory applied to the single-impurity Anderson model, Phys. Rev. B 109, 115128 (2024).
  7. N. Ritz, A. Ge, E. Walter, S. Aguirre, J. von Delft, and F. B. Kugler, KeldyshQFT: A C++ codebase for real-frequency multiloop functional renormalization group and parquet computations of the single-impurity Anderson model, J. Chem. Phys. 161, 054118 (2024).
  8. R. Bulla, T. A. Costi, and T. Pruschke, Numerical renormalization group method for quantum impurity systems, Rev. Mod. Phys. 80, 395 (2008).
  9. A. Toschi, A. A. Katanin, and K. Held, Dynamical vertex approximation: A step beyond dynamical mean-field theory, Phys. Rev. B 75, 045118 (2007).
  10. K. Held, A. A. Katanin, and A. Toschi, Dynamical vertex approximation: An introduction, Prog. Theor. Phys. Suppl. 176, 117 (2008).
  11. N. E. Bickers, Self-consistent many-body theory for condensed matter systems, in Theoretical Methods for Strongly Correlated Electrons, CRM Series in Mathematical Physics, edited by D. Sénéchal, A.-M. Tremblay, and C. Bourbonnais (Springer, New York, 2004).
  12. F. B. Kugler and J. von Delft, Multiloop functional renormalization group that sums up all parquet diagrams, Phys. Rev. Lett. 120, 057403 (2018).
  13. F. B. Kugler and J. von Delft, Multiloop functional renormalization group for general models, Phys. Rev. B 97, 035162 (2018).
  14. F. B. Kugler and J. von Delft, Derivation of exact flow equations from the self-consistent parquet relations, New J. Phys. 20, 123029 (2018).
  15. W. Metzner, M. Salmhofer, C. Honerkamp, V. Meden, and K. Schönhammer, Functional renormalization group approach to correlated fermion systems, Rev. Mod. Phys. 84, 299 (2012).
  16. C. Taranto, S. Andergassen, J. Bauer, K. Held, A. Katanin, W. Metzner, G. Rohringer, and A. Toschi, From infinite to two dimensions through the functional renormalization group, Phys. Rev. Lett. 112, 196402 (2014).
  17. J.-M. Lihm, J. Halbinger, J. Shim, J. von Delft, F. B. Kugler, and S.-S. B. Lee, Symmetric improved estimators for multipoint vertex functions, Phys. Rev. B 109, 125138 (2024).
  18. P. W. Anderson, Localized magnetic states in metals, Phys. Rev. 124, 41 (1961).
  19. F. B. Anders and A. Schiller, Real-time dynamics in quantum-impurity systems: A time-dependent numerical renormalization-group approach, Phys. Rev. Lett. 95, 196801 (2005).
  20. K. G. Wilson, The renormalization group: Critical phenomena and the Kondo problem, Rev. Mod. Phys. 47, 773 (1975).
  21. H. R. Krishna-murthy, J. W. Wilkins, and K. G. Wilson, Renormalization-group approach to the Anderson model of dilute magnetic alloys. I. Static properties for the symmetric case, Phys. Rev. B 21, 1003 (1980).
  22. R. Bulla, A. C. Hewson, and T. Pruschke, Numerical renormalization group calculations for the self-energy of the impurity Anderson model, J. Phys.: Condens. Matter 10, 8365 (1998).
  23. T. Pruschke and R. Bulla, Hund's coupling and the metal-insulator transition in the two-band Hubbard model, Eur. Phys. J. B 44, 217 (2005).
  24. K. M. Stadler, Z. P. Yin, J. von Delft, G. Kotliar, and A. Weichselbaum, Dynamical mean-field theory plus numerical renormalization-group study of spin-orbital separation in a three-band Hund metal, Phys. Rev. Lett. 115, 136401 (2015).
  25. K. Stadler, G. Kotliar, A. Weichselbaum, and J. von Delft, Hundness versus Mottness in a three-band Hubbard–Hund model: On the origin of strong correlations in Hund metals, Ann. Phys. 405, 365 (2019).
  26. X. Deng, K. M. Stadler, K. Haule, A. Weichselbaum, J. von Delft, and G. Kotliar, Signatures of Mottness and Hundness in archetypal correlated metals, Nat. Commun. 10, 2721 (2019).
  27. F. B. Kugler, S.-S. B. Lee, A. Weichselbaum, G. Kotliar, and J. von Delft, Orbital differentiation in Hund metals, Phys. Rev. B 100, 115159 (2019).
  28. F. B. Kugler, M. Zingl, H. U. R. Strand, S.-S. B. Lee, J. von Delft, and A. Georges, Strongly correlated materials from a numerical renormalization group perspective: How the Fermi-liquid state of Sr2RuO4 emerges, Phys. Rev. Lett. 124, 016401 (2020).
  29. K. M. Stadler, G. Kotliar, S.-S. B. Lee, A. Weichselbaum, and J. von Delft, Differentiating Hund from Mott physics in a three-band Hubbard-Hund model: Temperature dependence of spectral, transport, and thermodynamic properties, Phys. Rev. B 104, 115107 (2021).
  30. F. B. Kugler and G. Kotliar, Is the orbital-selective Mott phase stable against interorbital hopping? Phys. Rev. Lett. 129, 096403 (2022).
  31. F. B. Kugler, C.-J. Kang, and G. Kotliar, Low-energy perspective on two-orbital Hund metals and the case of LaNiO2, Phys. Rev. B 110, 155101 (2024).
  32. M. Grundner, F. B. Kugler, O. Parcollet, U. Schollwöck, A. Georges, and A. Hampel, Liv2O4: Hund-assisted orbital-selective mottness, Phys. Rev. B 112, L041106 (2025).
  33. J. Lee-Hand, H. LaBollita, F. B. Kugler, L. V. Muñoz, J. Kaye, S. Beck, A. Hampel, A. Georges, and C. E. Dreyer, Fermi-liquid t2 resistivity: Dynamical mean-field theory meets experiment, arXiv:2412.16363.
  34. A. Gleis, S.-S. B. Lee, G. Kotliar, and J. von Delft, Emergent properties of the periodic anderson model: A high-resolution, real-frequency study of heavy-fermion quantum criticality, Phys. Rev. X 14, 041036 (2024).
  35. A. Gleis, S.-S. B. Lee, G. Kotliar, and J. von Delft, Dynamical scaling and planckian dissipation due to heavy-fermion quantum criticality, Phys. Rev. Lett. 134, 106501 (2025).
  36. L. D. Landau, A. A. Abrikosov, and I. M. Khalatnikov, An asymptotic expression for the photon Green function in quantum electrodynamics, Dokl. Akad. Nauk SSSR 95, 1177 (1954).
  37. C. De Dominicis and P. C. Martin, Stationary entropy principle and renormalization in normal and superfluid systems. I. Algebraic formulation, J. Math. Phys. 5, 14 (1964).
  38. C. De Dominicis and P. C. Martin, stationary entropy principle and renormalization in normal and superfluid systems. II. Diagrammatic formulation, J. Math. Phys. 5, 31 (1964).
  39. G. Rohringer, New routes towards a theoretical treatment of nonlocal electronic correlations, Ph. D. Thesis, Technische Universität Wien, 2013.
  40. N. Wentzell, G. Li, A. Tagliavini, C. Taranto, G. Rohringer, K. Held, A. Toschi, and S. Andergassen, High-frequency asymptotics of the vertex function: Diagrammatic parametrization and algorithmic implementation, Phys. Rev. B 102, 085106 (2020).
  41. F. Krien, Efficient evaluation of the polarization function in dynamical mean-field theory, Phys. Rev. B 99, 235106 (2019).
  42. F. Krien, A. Valli, and M. Capone, Single-boson exchange decomposition of the vertex function, Phys. Rev. B 100, 155149 (2019).
  43. F. Krien and A. Valli, Parquetlike equations for the Hedin three-leg vertex, Phys. Rev. B 100, 245147 (2019).
  44. F. Krien, A. I. Lichtenstein, and G. Rohringer, Fluctuation diagnostic of the nodal/antinodal dichotomy in the Hubbard model at weak coupling: A parquet dual fermion approach, Phys. Rev. B 102, 235133 (2020).
  45. F. Krien, A. Valli, P. Chalupa, M. Capone, A. I. Lichtenstein, and A. Toschi, Boson-exchange parquet solver for dual fermions, Phys. Rev. B 102, 195131 (2020).
  46. F. Krien, A. Kauch, and K. Held, Tiling with triangles: parquet and GWγ methods unified, Phys. Rev. Res. 3, 013149 (2021).
  47. M. Gievers, E. Walter, A. Ge, J. v. Delft, and F. B. Kugler, Multiloop flow equations for single-boson exchange fRG, Eur. Phys. J. B 95, 108 (2022).
  48. L. V. Keldysh, Diagram technique for nonequilibrium processes, Sov. Phys. JETP 20, 1018 (1965).
  49. J. Schwinger, Brownian motion of a quantum oscillator, J. Math. Phys. 2, 407 (1961).
  50. L. P. Kadanoff and G. A. Baym, Quantum Statistical Mechanics (Benjamin, New York, 1962).
  51. A. Tsvelick and P. Wiegmann, Exact results in the theory of magnetic alloys, Adv. Phys. 32, 453 (1983).
  52. N. Ritz and A. Ge, KeldyshQFT: A C++ Codebase for real-frequency multiloop functional renormalization group and parquet computations for the single impurity Anderson model, https://github.com/NepomukRitz/KeldyshQFT.
  53. E. Wang and U. Heinz, Generalized fluctuation-dissipation theorem for nonlinear response functions, Phys. Rev. D 66, 025008 (2002).
  54. A. Ge, Analytic continuation of correlators from the Matsubara to the Keldysh formalism, Master's thesis, LMU München, 2020.
  55. A. Ge, J. Halbinger, S.-S. B. Lee, J. von Delft, and F. B. Kugler, Analytic continuation of multipoint correlation functions, Ann. Phys. (Leipzig) 536, 2300504 (2024).
  56. F. B. Kugler, Improved estimator for numerical renormalization group calculations of the self-energy, Phys. Rev. B 105, 245132 (2022).
  57. J.-M. Lihm and S.-S. Lee (private communication).
  58. L. Hedin, New method for calculating the one-particle Green's function with application to the electron-gas problem, Phys. Rev. 139, A796 (1965).
  59. M. Patricolo, M. Gievers, K. Fraboulet, A. Al-Eryani, S. Heinzelmann, P. M. Bonetti, A. Toschi, D. Vilardi, and S. Andergassen, Single-boson exchange formulation of the Schwinger-Dyson equation and its application to the functional renormalization group, SciPost Phys. 18, 078 (2025).
  60. F. Krien, Conserving dynamical mean-field approaches to strongly correlated systems, Ph.D. thesis, Hamburg University, 2018.
  61. A. A. Katanin, Fulfillment of Ward identities in the functional renormalization group approach, Phys. Rev. B 70, 115109 (2004).
  62. P. Kopietz, L. Bartosch, L. Costa, A. Isidori, and A. Ferraz, Ward identities for the Anderson impurity model: Derivation via functional methods and the exact renormalization group, J. Phys. A 43, 385004 (2010).
  63. F. Krien, E. G. C. P. van Loon, H. Hafermann, J. Otsuki, M. I. Katsnelson, and A. I. Lichtenstein, Conservation in two-particle self-consistent extensions of dynamical mean-field theory, Phys. Rev. B 96, 075155 (2017).
  64. P. Chalupa-Gantner, F. B. Kugler, C. Hille, J. von Delft, S. Andergassen, and A. Toschi, Fulfillment of sum rules and Ward identities in the multiloop functional renormalization group solution of the Anderson impurity model, Phys. Rev. Res. 4, 023050 (2022).
  65. J. Heyder, F. Bauer, D. Schimmel, and J. von Delft, Derivation of Oguri's linear conductance formula for interacting fermions within the Keldysh formalism, Phys. Rev. B 96, 125141 (2017).
  66. A. Oguri, Y. Teratani, K. Tsutsumi, and R. Sakano, Current noise and Keldysh vertex function of an Anderson impurity in the fermi-liquid regime, Phys. Rev. B 105, 115409 (2022).
  67. H. Shinaoka, M. Wallerberger, Y. Murakami, K. Nogaki, R. Sakurai, P. Werner, and A. Kauch, Multiscale space-time ansatz for correlation functions of quantum systems based on quantics tensor trains, Phys. Rev. X 13, 021015 (2023).
  68. Y. Núñez Fernández, M. Jeannin, P. T. Dumitrescu, T. Kloss, J. Kaye, O. Parcollet, and X. Waintal, Learning Feynman diagrams with tensor trains, Phys. Rev. X 12, 041018 (2022).
  69. M. K. Ritter, Y. Núñez Fernández, M. Wallerberger, J. von Delft, H. Shinaoka, and X. Waintal, Quantics tensor cross interpolation for high-resolution parsimonious representations of multivariate functions, Phys. Rev. Lett. 132, 056501 (2024).
  70. Y. N. Fernández, M. K. Ritter, M. Jeannin, J.-W. Li, T. Kloss, T. Louvet, S. Terasaki, O. Parcollet, J. von Delft, H. Shinaoka, and X. Waintal, Learning tensor networks with tensor cross interpolation: new algorithms and libraries, SciPost Phys. 18, 104 (2025).
  71. S. Rohshap, M. K. Ritter, H. Shinaoka, J. von Delft, M. Wallerberger, and A. Kauch, Two-particle calculations with quantics tensor trains – solving the parquet equations, Phys. Rev. Res. 7, 023087 (2025).
  72. M. Frankenbach, A. Ge, M. Ritter, N. Ritz, and J. von Delft, Compressing local vertex functions from MuNRG using tensor cross interpolation, arXiv:2506.13359.
  73. O. Kovalska, A. Gleis, and J. von Delft, Projected Krylov Approach for efficient matrix product state based computation of real-frequency spectral functions (unpublished).
  74. S. R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett. 69, 2863 (1992).
  75. F. D. Picoli, M. Huang, A. Gleis, and J. von Delft, Tangent-space Krylov solver for quantum impurity systems (unpublished).
  76. S.-S. B. Lee and A. Weichselbaum, Adaptive broadening to improve spectral resolution in the numerical renormalization group, Phys. Rev. B 94, 235127 (2016).
  77. S.-S. B. Lee, J. von Delft, and A. Weichselbaum, Doublon-holon origin of the subpeaks at the Hubbard band edges, Phys. Rev. Lett. 119, 236402 (2017).
  78. A. Weichselbaum, Non-Abelian symmetries in tensor networks: A quantum symmetry space approach, Ann. Phys. 327, 2972 (2012).
  79. A. Weichselbaum, Tensor networks and the numerical renormalization group, Phys. Rev. B 86, 245124 (2012).
  80. A. Weichselbaum, X-symbols for non-Abelian symmetries in tensor networks, Phys. Rev. Res. 2, 023385 (2020).
  81. A. Weichselbaum, QSpace - An open-source tensor library for Abelian and non-Abelian symmetries, SciPost Phys. Codebases 40 (2024).
  82. A. Weichselbaum, Codebase release 4.0 for QSpace, SciPost Phys. Codebases 40 (2024).
  83. N. Ritz, A. Ge, M. K. Frankenbach, M. Pelz, J. von Delft, and F. B. Kugler, Data for: Testing the parquet equations and the U(1) Ward identity for real-frequency correlation functions from the multipoint numerical renormalization group [Data set], LMU Munich, Faculty of Physics (2025), doi: https://doi.org/10.57970/at1en-t9f23.
  84. R. Peters, T. Pruschke, and F. B. Anders, Numerical renormalization group approach to Green's functions for quantum impurity models, Phys. Rev. B 74, 245114 (2006).
  85. A. Weichselbaum and J. von Delft, Sum-rule conserving spectral functions from the numerical renormalization group, Phys. Rev. Lett. 99, 076402 (2007).
  86. R. Žitko and T. Pruschke, Energy resolution and discretization artifacts in the numerical renormalization group, Phys. Rev. B 79, 085106 (2009).
  87. R. Žitko, Adaptive logarithmic discretization for numerical renormalization group methods, Comput. Phys. Commun. 180, 1271 (2009).

Outline

Information

Phys. Rev. Research 7, 033139– Published 11 August, 2025
  • Received 9 April 2025
  • Accepted 16 June 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation