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Fermions and the renormalization group at large Nf

Charlie Cresswell-Hogg1,* and Daniel F. Litim1,2,†

  • 1Department of Physics and Astronomy, University of Sussex, Brighton, England BN1 9QH, United Kingdom
  • 2Theoretical Physics Department, CERN, 1211 Geneva 23, Switzerland

  • *Contact author: c.cresswell-hogg@sussex.ac.uk
  • †Contact author: d.litim@sussex.ac.uk
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Phys. Rev. D 112, 025005 – Published 7 July, 2025

DOI: https://doi.org/10.1103/393b-lc5d

Abstract

We investigate fermionic quantum field theories using functional renormalization. In the limit of many fermion flavors N, we demonstrate that theories have exact solutions for their quantum effective actions given by quasilocal interaction functionals of fermion bilinears. The structure implies that local potential approximations are exact and exactly solvable and that field anomalous dimensions vanish. Theories with nontrivial anomalous dimensions may also arise under conditions that are identified. We further demonstrate that higher-derivative interactions are inevitably induced by pointlike ones, including at large N. The local potential flows for fermionic theories with the most general U(N) symmetric interactions are provided. For sample theories with scalar, pseudoscalar, vector, or axial-vector interactions, we identify conformal fixed points, scaling dimensions, conformal manifolds, and quantum-induced shifts in scaling dimensions of higher-derivative interactions. We also study fermion mass generation and subleading modifications due to finite N corrections. Implications for conformal field theories and applications in condensed matter and particle physics are indicated.

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References (114)

  1. W. Metzner, M. Salmhofer, C. Honerkamp, V. Meden, and K. Schonhammer, Functional renormalization group approach to correlated fermion systems, Rev. Mod. Phys. 84, 299 (2012).
  2. J. Braun, Fermion interactions and universal behavior in strongly interacting theories, J. Phys. G 39, 033001 (2012).
  3. B. Rosenstein, H.-L. Yu, and A. Kovner, Critical exponents of new universality classes, Phys. Lett. B 314, 381 (1993).
  4. I. F. Herbut, Interactions and phase transitions on Graphene’s honeycomb lattice, Phys. Rev. Lett. 97, 146401 (2006).
  5. S. Hands and C. Strouthos, Quantum critical behaviour in a Graphene-like model, Phys. Rev. B 78, 165423 (2008).
  6. I. F. Herbut, V. Juricic, and B. Roy, Theory of interacting electrons on the Honeycomb lattice, Phys. Rev. B 79, 085116 (2009).
  7. O. Vafek and A. Vishwanath, Dirac fermions in solids: From high-Tc cuprates and graphene to topological insulators and Weyl semimetals, Annu. Rev. Condens. Matter Phys. 5, 83 (2014).
  8. T. O. Wehling, A. M. Black-Schaffer, and A. V. Balatsky, Dirac materials, Adv. Phys. 63, 1 (2014).
  9. L. Classen, I. F. Herbut, L. Janssen, and M. M. Scherer, Mott multicriticality of Dirac electrons in graphene, Phys. Rev. B 92, 035429 (2015).
  10. R. Boyack, H. Yerzhakov, and J. Maciejko, Quantum phase transitions in Dirac fermion systems, Eur. Phys. J. Special Topics 230, 979 (2021).
  11. N. Parthenios and L. Classen, Twisted bilayer graphene at charge neutrality: Competing orders of SU(4) Dirac fermions, Phys. Rev. B 108, 235120 (2023).
  12. J. Biedermann and L. Janssen, Twist-tuned quantum criticality in moiré bilayer graphene, arXiv:2412.16042.
  13. K. G. Wilson, Quantum field theory models in less than four-dimensions, Phys. Rev. D 7, 2911 (1973).
  14. G. Parisi, The theory of nonrenormalizable interactions. 1. The large N expansion, Nucl. Phys. B100, 368 (1975).
  15. T. Eguchi, A new approach to collective phenomena in superconductivity Models, Phys. Rev. D 14, 2755 (1976).
  16. K.-i. Shizuya, 1/N expansion and the theory of composite particles, Phys. Rev. D 21, 2327 (1980).
  17. K. Gawedzki and A. Kupiainen, Renormalizing the nonrenormalizable, Phys. Rev. Lett. 55, 363 (1985).
  18. B. Rosenstein, B. J. Warr, and S. H. Park, The four Fermi theory is renormalizable in (2+1)-dimensions, Phys. Rev. Lett. 62, 1433 (1989).
  19. Y. Kikukawa and K. Yamawaki, Ultraviolet fixed point structure of renormalizable four Fermion theory in less than four-dimensions, Phys. Lett. B 234, 497 (1990).
  20. C. de Calan, P. A. Faria da Veiga, J. Magnen, and R. Seneor, Constructing the three-dimensional Gross-Neveu model with a large number of flavor components, Phys. Rev. Lett. 66, 3233 (1991).
  21. S. Hands, A. Kocic, and J. B. Kogut, Compositeness, anomalous dimensions and renormalizability in four Fermi theories, Phys. Lett. B 273, 111 (1991).
  22. S. Hands, A. Kocic, and J. B. Kogut, Four Fermi theories in fewer than four-dimensions, Ann. Phys. (N.Y.) 224, 29 (1993).
  23. J. A. Gracey, Computation of critical exponent Eta at O(1/N3) in the four Fermi model in arbitrary dimensions, Int. J. Mod. Phys. A 09, 727 (1994).
  24. H. Gies and L. Janssen, UV fixed-point structure of the three-dimensional Thirring model, Phys. Rev. D 82, 085018 (2010).
  25. A. Jakovác, A. Patkós, and P. Pósfay, Non-Gaussian fixed points in Fermionic field theories without auxiliary Bose-fields, Eur. Phys. J. C 75, 2 (2015).
  26. F. Gehring, H. Gies, and L. Janssen, Fixed-point structure of low-dimensional relativistic Fermion field theories: Universality classes and emergent symmetry, Phys. Rev. D 92, 085046 (2015).
  27. L. Dabelow, H. Gies, and B. Knorr, Momentum dependence of quantum critical Dirac systems, Phys. Rev. D 99, 125019 (2019).
  28. C. Cresswell-Hogg and D. F. Litim, Line of fixed points in Gross-Neveu theories, Phys. Rev. Lett. 130, 201602 (2023).
  29. C. Cresswell-Hogg and D. F. Litim, Critical Fermions with spontaneously broken scale symmetry, Phys. Rev. D 107, L101701 (2023).
  30. S. Weinberg, Ultraviolet divergences in quantum theories of gravitation, in General Relativity: An Einstein centenary survey, edited by S. W. Hawking and W. Israel (Cambridge University Press, 1980), pp. 790–831.
  31. M. Reuter, Nonperturbative evolution equation for quantum gravity, Phys. Rev. D 57, 971 (1998).
  32. M. Reuter and F. Saueressig, Renormalization group flow of quantum gravity in the Einstein-Hilbert truncation, Phys. Rev. D 65, 065016 (2002).
  33. D. F. Litim, Fixed points of quantum gravity, Phys. Rev. Lett. 92, 201301 (2004).
  34. H. Gies, S. Rechenberger, and M. M. Scherer, Towards an asymptotic-safety scenario for chiral Yukawa systems, Eur. Phys. J. C 66, 403 (2010).
  35. J. Braun, H. Gies, and D. D. Scherer, Asymptotic safety: A simple example, Phys. Rev. D 83, 085012 (2011).
  36. D. F. Litim and F. Sannino, Asymptotic safety guaranteed, J. High Energy Phys. 12 (2014) 178.
  37. A. D. Bond and D. F. Litim, Theorems for asymptotic safety of gauge theories, Eur. Phys. J. C 77, 429 (2017).
  38. A. D. Bond and D. F. Litim, Price of asymptotic safety, Phys. Rev. Lett. 122, 211601 (2019).
  39. H. Gies and K. K. K. Tam, Fermionic fixed-point structure of asymptotically safe QED with a Pauli term, Eur. Phys. J. C 84, 477 (2024).
  40. D. Poland, S. Rychkov, and A. Vichi, The conformal bootstrap: Theory, numerical techniques, and applications, Rev. Mod. Phys. 91, 015002 (2019).
  41. S. Giombi and X. Yin, The higher spin/Vector model duality, J. Phys. A 46, 214003 (2013).
  42. J. Polchinski, Renormalization and effective Lagrangians, Nucl. Phys. B231, 269 (1984).
  43. C. Wetterich, Exact evolution equation for the effective potential, Phys. Lett. B 301, 90 (1993).
  44. U. Ellwanger, Flow equations for N point functions and bound states, Z. Phys. C 62, 503 (1994).
  45. T. R. Morris, The exact renormalization group and approximate solutions, Int. J. Mod. Phys. A 09, 2411 (1994).
  46. K. G. Wilson, Renormalization group and critical phenomena. 1. Renormalization group and the Kadanoff scaling picture, Phys. Rev. B 4, 3174 (1971).
  47. K. G. Wilson, Renormalization group and critical phenomena. 2. Phase space cell analysis of critical behavior, Phys. Rev. B 4, 3184 (1971).
  48. N. Dupuis, L. Canet, A. Eichhorn, W. Metzner, J. M. Pawlowski, M. Tissier et al., The nonperturbative functional renormalization group and its applications, Phys. Rep. 910, 1 (2021).
  49. J. Comellas, Y. Kubyshin, and E. Moreno, Exact renormalization group study of Fermionic theories, Nucl. Phys. B490, 653 (1997).
  50. D. U. Jungnickel and C. Wetterich, Effective action for the chiral quark-meson model, Phys. Rev. D 53, 5142 (1996).
  51. H. Gies and C. Wetterich, Renormalization flow of bound states, Phys. Rev. D 65, 065001 (2002).
  52. H. Gies and C. Wetterich, Universality of spontaneous chiral symmetry breaking in gauge theories, Phys. Rev. D 69, 025001 (2004).
  53. H. Gies, J. Jaeckel, and C. Wetterich, Towards a renormalizable standard model without fundamental Higgs scalar, Phys. Rev. D 69, 105008 (2004).
  54. J. M. Pawlowski, Aspects of the functional renormalisation group, Ann. Phys. (N.Y.) 322, 2831 (2007).
  55. K.-I. Aoki and D. Sato, Weak solution method of the non-perturbative renormalization group equation to describe dynamical chiral symmetry breaking and its application to beyond the Ladder analysis in QCD, in Kmi-Gcoe Workshop on Strong Coupling Gauge Theories in the Lhc Perspective (2014), pp. 448–451.
  56. A. Jakovac and A. Patkos, Local potential approximation for the renormalization group flow of Fermionic field theories, Phys. Rev. D 88, 065008 (2013).
  57. 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).
  58. G. P. Vacca and L. Zambelli, Multimeson Yukawa interactions at criticality, Phys. Rev. D 91, 125003 (2015).
  59. H. Gies, P. Heinzel, J. Laufkötter, and M. Picciau, Quantum field theories of relativistic Luttinger fermions, Phys. Rev. D 110, 065001 (2024).
  60. J. Hubbard, Calculation of partition functions, Phys. Rev. Lett. 3, 77 (1959).
  61. R. L. Stratonovich, On a method of calculating quantum distribution functions, Sov. Phys. Dokl. 2, 416 (1957).
  62. M. Moshe and J. Zinn-Justin, Quantum field theory in the large N limit: A review, Phys. Rep. 385, 69 (2003).
  63. F. J. Wegner and A. Houghton, Renormalization group equation for critical phenomena, Phys. Rev. A 8, 401 (1973).
  64. S.-k. Ma, Introduction to the renormalization group, Rev. Mod. Phys. 45, 589 (1973).
  65. J. F. Nicoll, T. S. Chang, and H. E. Stanley, Exact and approximate differential renormalization group generators, Phys. Rev. A 13, 1251 (1976).
  66. T. Chang, D. Vvedensky, and J. Nicoll, Differential renormalization-group generators for static and dynamic critical phenomena, Phys. Rep. 217, 279 (1992).
  67. M. D’Attanasio and T. R. Morris, Large N and the renormalization group, Phys. Lett. B 409, 363 (1997).
  68. D. F. Litim and J. M. Pawlowski, On gauge invariant Wilsonian flows, in Workshop on the Exact Renormalization Group (1998), pp. 168–185.
  69. D. F. Litim, Universality and the renormalisation group, J. High Energy Phys. 07 (2005) 005.
  70. T. R. Morris, Equivalence of local potential approximations, J. High Energy Phys. 07 (2005) 027.
  71. D. F. Litim and M. J. Trott, Asymptotic safety of scalar field theories, Phys. Rev. D 98, 125006 (2018).
  72. D. F. Litim and J. M. Pawlowski, Perturbation theory and renormalization group equations, Phys. Rev. D 65, 081701 (2002).
  73. D. F. Litim and J. M. Pawlowski, Completeness and consistency of renormalisation group flows, Phys. Rev. D 66, 025030 (2002).
  74. F. Freire, D. F. Litim, and J. M. Pawlowski, Gauge invariance and background field formalism in the exact renormalization group, Phys. Lett. B 495, 256 (2000).
  75. D. F. Litim, Mind the gap, Int. J. Mod. Phys. A 16, 2081 (2001).
  76. D. F. Litim, Derivative expansion and renormalization group flows, J. High Energy Phys. 11 (2001) 059.
  77. D. F. Litim and D. Zappala, Ising exponents from the functional renormalisation group, Phys. Rev. D 83, 085009 (2011).
  78. I. Balog, H. Chaté, B. Delamotte, M. Marohnic, and N. Wschebor, Convergence of nonperturbative approximations to the renormalization group, Phys. Rev. Lett. 123, 240604 (2019).
  79. A. Baldazzi, K. Falls, Y. Kluth, and B. Knorr, Robustness of the derivative expansion in asymptotic safety, arXiv:2312.03831.
  80. S. Bornholdt and C. Wetterich, Average action for models with fermions, Z. Phys. C 58, 585 (1993).
  81. J. Jaeckel and C. Wetterich, Flow equations without mean field ambiguity, Phys. Rev. D 68, 025020 (2003).
  82. S. Weinberg, Effective field theories in the large N limit, Phys. Rev. D 56, 2303 (1997).
  83. J. Jaeckel, Effective Actions for strongly interacting Fermionic systems, Other thesis (2003).
  84. J. F. Nicoll, T. S. Chang, and H. E. Stanley, Approximate renormalization group based on the Wegner-Houghton differential generator, Phys. Rev. Lett. 33, 540 (1974).
  85. G. R. Golner, Nonperturbative renormalization group calculations for continuum spin systems, Phys. Rev. B 33, 7863 (1986).
  86. R. D. Ball, P. E. Haagensen, J. I. Latorre, and E. Moreno, Scheme independence and the exact renormalization group, Phys. Lett. B 347, 80 (1995).
  87. D. Litim and N. Tetradis, Approximate solutions of exact renormalization group equations, arXiv:hep-th/9501042.
  88. N. Tetradis and D. F. Litim, Analytical solutions of exact renormalization group equations, Nucl. Phys. B464, 492 (1996).
  89. J. Comellas and A. Travesset, O (N) models within the local potential approximation, Nucl. Phys. B498, 539 (1997).
  90. T. R. Morris and M. D. Turner, Derivative expansion of the renormalization group in O(N) scalar field theory, Nucl. Phys. B509, 637 (1998).
  91. D. F. Litim, Optimization of the exact renormalization group, Phys. Lett. B 486, 92 (2000).
  92. D. F. Litim, Optimized renormalization group flows, Phys. Rev. D 64, 105007 (2001).
  93. D. F. Litim, Critical exponents from optimized renormalization group flows, Nucl. Phys. B631, 128 (2002).
  94. D. J. Gross and A. Neveu, Dynamical symmetry breaking in asymptotically free field theories, Phys. Rev. D 10, 3235 (1974).
  95. Y. Nambu and G. Jona-Lasinio, Dynamical model of elementary particles based on an analogy with superconductivity. 1, Phys. Rev. 122, 345 (1961).
  96. Y. Nambu and G. Jona-Lasinio, Dynamical model of elementary particles based on an analogy with superconductivity. II, Phys. Rev. 124, 246 (1961).
  97. W. A. Bardeen, C. T. Hill, and M. Lindner, Minimal dynamical symmetry breaking of the standard model, Phys. Rev. D 41, 1647 (1990).
  98. W. E. Thirring, A soluble relativistic field theory?, Ann. Phys. (N.Y.) 3, 91 (1958).
  99. R. F. Dashen and Y. Frishman, Four Fermion interactions and scale invariance, Phys. Rev. D 11, 2781 (1975).
  100. E. Oevermann, A. Koenigstein, and S. Floerchinger, Functional renormalization of QCD in 1+1 dimensions: Four-fermion interactions from quark-gluon dynamics, Phys. Rev. D 111, 07400 (2025).
  101. C. Cresswell-Hogg and D. F. Litim, Scale symmetry breaking and generation of mass at quantum critical points, J. High Energy Phys. 07 (2024) 066.
  102. G. ’t Hooft, Naturalness, chiral symmetry, and spontaneous chiral symmetry breaking, NATO Sci. Ser. B 59, 135 (1980).
  103. C. Cresswell-Hogg and D. F. Litim, Fermion mass generation without symmetry breaking, Phys. Rev. D 110, 065014 (2024).
  104. C. Cresswell-Hogg, D. F. Litim, and R. Zwicky, Dilaton physics from asymptotic freedom, arXiv:2502.00107.
  105. J. L. Cardy, Scaling and Renormalization in Atatistical Physics (Cambridge University Press, Cambridge, England, 1996).
  106. W. Chen, Y. Makeenko, and G. W. Semenoff, Four fermion theory and the conformal bootstrap, Ann. Phys. (N.Y.) 228, 341 (1993).
  107. D. Karateev, P. Kravchuk, M. Serone, and A. Vichi, Fermion conformal bootstrap in 4d, J. High Energy Phys. 06 (2019) 088.
  108. R. S. Erramilli, L. V. Iliesiu, P. Kravchuk, A. Liu, D. Poland, and D. Simmons-Duffin, The Gross-Neveu-Yukawa archipelago, J. High Energy Phys. 02 (2023) 036.
  109. P. Cvitanovic, Group Theory: Birdtracks, Lie’s, and Exceptional Groups (Princeton University Press, Princeton, NJ, 2020).
  110. T. Schäfer and E. V. Shuryak, Instantons in QCD, Rev. Mod. Phys. 70, 323 (1998).
  111. C. C. Nishi, Simple derivation of general Fierz-like identities, Am. J. Phys. 73, 1160 (2005).
  112. L. Huang, M. Kawaguchi, Y. Maitiniyazi, S. Matsuzaki, A. Tomiya, and M. Yamada, Functional renormalization group study of a four-fermion model with CP violation: Implications to spontaneous CP violation models, Phys. Rev. D 111, 074015 (2025).
  113. C. Wetterich, Spinors in Euclidean field theory, complex structures and discrete symmetries, Nucl. Phys. B852, 174 (2011).
  114. B. S. DeWitt, Supermanifolds, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, England, 2012).

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Phys. Rev. D 112, 025005– Published 7 July, 2025
  • Received 10 April 2025
  • Accepted 15 June 2025
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