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Condensates, Crystals, and Renormalons in the Gross-Neveu Model at Finite Density

Francesco Benini*, Ohad Mamroud†, Tomás Reis‡, and Marco Serone§

  • *Contact author: fbenini@sissa.it
  • †Contact author: omamroud@sissa.it
  • ‡Contact author: treis@sissa.it
  • §Contact author: serone@sissa.it
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Phys. Rev. Lett. 135, 141601 – Published 29 September, 2025

DOI: https://doi.org/10.1103/trj9-r9j8

Abstract

We study the O(2N) symmetric Gross-Neveu model at finite density in the presence of a U(1) chemical potential h for a generic number a≤N−2 of fermion fields. By combining perturbative quantum field theory, semiclassical large N, and Bethe ansatz techniques, we show that at finite N two new dynamically generated scales Λn and Λc appear in the theory, governing the mass gap of neutral and charged fermions, respectively. Above a certain threshold value for h, a-fermion bound states condense and form an inhomogeneous configuration, which at infinite N is a crystal spontaneously breaking translations. At large h, this crystal has mean Λn and spatial oscillations of amplitude Λc. The two scales also control the nonperturbative corrections to the free energy, resolving a puzzle concerning fractional-power renormalons and predicting new ones.

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References (45)

  1. G. ’t Hooft, in The Whys of Subnuclear Physics, The Subnuclear Series Vol. 15, edited by A. Zichichi (Springer, Boston, MA, 1979), pp. 943–982, 10.1007/978-1-4757-3069-2.
  2. M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Nucl. Phys. B147, 385 (1979).
  3. G. Parisi, Nucl. Phys. B150, 163 (1979).
  4. D. J. Gross and A. Neveu, Phys. Rev. D 10, 3235 (1974).
  5. D. K. Campbell and A. R. Bishop, Nucl. Phys. B200, 297 (1982).
  6. M. Thies, J. Phys. A 39, 12707 (2006).
  7. A. B. Zamolodchikov and A. B. Zamolodchikov, Ann. Phys. (N.Y.) 120, 253 (1979).
  8. M. Karowski and H. J. Thun, Nucl. Phys. B190, 61 (1981).
  9. A. M. Polyakov and P. B. Wiegmann, Phys. Lett. 131B, 121 (1983).
  10. P. Forgacs, F. Niedermayer, and P. Weisz, Nucl. Phys. B367, 123 (1991).
  11. D. Volin, Phys. Rev. D 81, 105008 (2010).
  12. M. Mariño and T. Reis, J. High Energy Phys. 04 (2020) 160.
  13. L. Di Pietro, M. Mariño, G. Sberveglieri, and M. Serone, J. High Energy Phys. 10 (2021) 166.
  14. M. Mariño, R. Miravitllas, and T. Reis, J. High Energy Phys. 08 (2022) 279.
  15. Z. Bajnok, J. Balog, and I. Vona, Phys. Lett. B 844, 138075 (2023).
  16. D. Sauzin, arXiv:1405.0356.
  17. I. Aniceto, G. Başar, and R. Schiappa, Phys. Rep. 809, 1 (2019).
  18. M. Serone, arXiv:2405.02224.
  19. V. Schön and M. Thies, Phys. Rev. D 62, 096002 (2000).
  20. M. Thies and K. Urlichs, Phys. Rev. D 67, 125015 (2003).
  21. O. Schnetz, M. Thies, and K. Urlichs, Ann. Phys. (Amsterdam) 314, 425 (2004).
  22. R. F. Dashen, B. Hasslacher, and A. Neveu, Phys. Rev. D 12, 2443 (1975).
  23. R. Narayanan, Phys. Rev. D 101, 096001 (2020).
  24. J. Lenz, L. Pannullo, M. Wagner, B. Wellegehausen, and A. Wipf, Phys. Rev. D 101, 094512 (2020).
  25. A. Koenigstein, L. Pannullo, S. Rechenberger, M. J. Steil, and M. Winstel, J. Phys. A 55, 375402 (2022).
  26. V. Melin, Y. Sekiguchi, P. Wiegmann, and K. Zarembo, Phys. Rev. Lett. 133, 101601 (2024).
  27. G. Başar, G. V. Dunne, and M. Thies, Phys. Rev. D 79, 105012 (2009).
  28. R. Ciccone, L. Di Pietro, and M. Serone, Phys. Rev. Lett. 129, 071603 (2022).
  29. R. Ciccone, L. Di Pietro, and M. Serone, J. High Energy Phys. 02 (2024) 211.
  30. N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 (1966).
  31. S. R. Coleman, Commun. Math. Phys. 31, 259 (1973).
  32. F. Benini, O. Mamroud, T. Reis, and M. Serone (to be published).

  33. There are two more diagrams that contribute to general four-point correlators, but they vanish for our particular choice of momenta.

  34. Γ develops large logs also at q→∞: this leads to the UV β-function of [4] and to Λ in (2).

  35. Without this test, one would be ambiguously led to other slightly different scales depending on the component of Γ one starts with.

  36. M. Thies and K. Urlichs, Phys. Rev. D 72, 105008 (2005).
  37. M. Thies, Phys. Rev. D 69, 067703 (2004).
  38. O. Schnetz, M. Thies, and K. Urlichs, Ann. Phys. (Amsterdam) 321, 2604 (2006).

  39. The phase transition is due to the large N limit. In any finite volume L there is a first-order phase transition which becomes smoother as L grows, leading to a second-order transition in infinite volume.

  40. P. Fendley, J. High Energy Phys. 05 (2001) 050.
  41. T. Reis, On the resurgence of renormalons in integrable theories, Ph.D. thesis, University of Geneva, 2022, arXiv:2209.15386.
  42. M. Mariño and R. Miravitllas, SciPost Phys. 18, 101 (2025).
  43. C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, NY, USA, 1978), p. 375, 10.1007/978-1-4757-3069-2.
  44. K. Fukushima and T. Hatsuda, Rep. Prog. Phys. 74, 014001 (2011).
  45. F. Benini, O. Mamroud, T. Reis, and M. Serone, Datapoints for “Condensates, crystals, and renormalons in the Gross-Neveu model at finite density” [Data set], (2025), 10.5281/zenodo.17157990.

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Phys. Rev. Lett. 135, 141601– Published 29 September, 2025
  • Received 11 June 2025
  • Revised 24 July 2025
  • Accepted 6 August 2025
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