- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review C

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Novel way of recasting the Bardeen-Cooper-Schrieffer gap equations

Georgios Palkanoglou1,2 and Alexandros Gezerlis2

  • 1TRIUMF, 4004 Wesbrook Mall, Vancouver, British Columbia V6T 2A3, Canada
  • 2Department of Physics, University of Guelph, Guelph, Ontario N1G 2W1, Canada

PDF

Phys. Rev. C 114, 024340 – Published 27 August, 2026

DOI: https://doi.org/10.1103/bbff-kxpy

Abstract

The gap equations lie at the core of the Bardeen-Cooper-Schrieffer (BCS) theory, a standard tool in the description of superfluidity. As a set of non-inear integral equations, the gap equations' inherent difficulties oftentimes hinder even the crudest descriptions of superfluid states. Hard-core potentials, high-density superfluids, and coupled-channel pairing are all reasons that have historically required one to provide special treatment to the gap equations to get a solution. In this paper we present a new method for solving the gap equations that holds the promise of being an efficient universal solver that requires the minimum amount of a priori knowledge of the targeted solutions. With theoretical evidence of exotic nuclear superfluidity posing new questions to our understanding of this fundamental property of nuclear systems, the presented method can be a valuable tool when exploring new pairing states, finite-temperature properties, or the development of sophisticated descriptions of nuclear superfludity.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (41)

  1. D. J. Dean and M. Hjorth-Jensen, Pairing in nuclear systems: From neutron stars to finite nuclei, Rev. Mod. Phys. 75, 607 (2003).
  2. A. Sedrakian and J. W. Clark, Superfluidity in nuclear systems and neutron stars, Eur. Phys. J. A 55, 167 (2019).
  3. Y. Fujimoto, Renormalization-group approach to Kohn-Luttinger superconductivity: Amplification of the pairing gap from ℓ4 to ℓ, Phys. Rev. B 111, 184510 (2025).
  4. M. Kumamoto and S. Reddy, Kohn-Luttinger effect in dense matter and its implications for neutron stars, Phys. Rev. C 110, 025804 (2024).
  5. E. Krotscheck, P. Papakonstantinou, and J. Wang, Triplet pairing in neutron matter, Astrophys. J. 955, 76 (2023).
  6. D. Ding, A. Rios, H. Dussan, W. H. Dickhoff, S. J. Witte, A. Carbone, and A. Polls, Pairing in high-density neutron matter including short- and long-range correlations, Phys. Rev. C 94, 025802 (2016).
  7. G. F. Burgio, H. J. Schulze, I. Vidaña, and J. B. Wei, Neutron stars and the nuclear equation of state, Prog. Part. Nucl. Phys. 120, 103879 (2021).
  8. V. Allard and M. Chamel, Evidence of gapless superfluidity in MXB 1659-29 with and without late time cooling, Universe, 11, 140 (2025).
  9. A. Tichai, P. Arthuis, T. Duguet, H. Hergert, V. Somá, and R. Roth, Bogoliubov many-body perturbation theory for open-shell nuclei, Phys. Lett. B 786, 195 (2018).
  10. M. Drissi, A. Rios, and C. Barbieri, Nambu-covariant many-body theory I: Perturbative approximations, Ann. Phys. 469, 169729 (2024).
  11. Fifty Years of Nuclear BCS: Pairing in Finite Systems, edited by R. A. Broglia and V. Zelevinsky (World Scientific, Singapore, 2013).
  12. J. F. Annett, Superconductivity, Superfluids and Condensates (Oxford University Press, Oxford, 2004).
  13. G. Almirante and M. urban, Superfluid density in linear response theory: Pulsar glitches from the inner crust of neutron stars, Phys. Rev. Lett. 135, 132701 (2025).
  14. A. Gezerlis and J. Carlson, Low-density neutron matter, Phys. Rev. C 81, 025803 (2010).
  15. S. Gandolfi, G. Palkanoglou, J. Carlson, A. Gezerlis, and K. E. Schmidt, The S01 pairing gap in neutron matter, Condens. Matter 7, 19 (2022).
  16. N. Martin and M. Urban, Superfluid hydrodynamics in the inner crust of neutron stars, Phys. Rev. C 94, 065801 (2016).
  17. V. A. Khodel, V. V. Khodel, and J. W. Clark, Universalities of triplet pairing in neutron matter, Phys. Rev. Lett. 81, 3828 (1998).
  18. V. V. Khodel, V. A. Khodel, and J. W. Clark, Triplet pairing in neutron matter, Nucl. Phys. A 679, 827 (2001).
  19. E. Krotscheck, Superfluidity in neutron matter, Z. Phys. 251, 135 (1972).
  20. S. Ramanan, S. Bogner, and R. J. Furnstahl, Weinberg eigenvalues and pairing with low-momentum potentials, Nucl. Phys. A 797, 81 (2007).
  21. A. L. Goodman, Generalized gap equations and the coherence of the α−α pair field, Nucl. Phys. A 186, 475 (1972).
  22. A. L. Goodman, T=0 and T=1 pairing in rotational states of the N=Z nucleus Zr80, Phys. Rev. C 63, 044325 (2001).
  23. A. Gezerlis, G. F. Bertsch, and Y. L. Luo, Mixed-spin pairing condensates in heavy nuclei, Phys. Rev. Lett. 106, 252502 (2011).
  24. T. Takatsuka and R. Tamagaki, Chapter II. Superfluidity in neutron star matter and symmetric nuclear matter, Prog. Theor. Phys. Suppl. 112, 27 (1993).
  25. Y. Guo, T. Naito, H. Tajima, and H. Liang, Quartet correlations near the surface of N=Z nuclei, Phys. Rev. C 112, 024310 (2025).
  26. A. Bulgac, M. McNeil Forbes, and A. Schwenk, Induced P-wave superfluidity in asymmetric Fermi gases, Phys. Rev. Lett. 97, 020402 (2006).
  27. G. Palkanoglou, M. Stuck, and A. Gezerlis, Spin-triplet pairing in heavy nuclei is stable against deformation, Phys. Rev. Lett. 134, 032501 (2025).
  28. S. Frauendorf and A. O. Macchiavelli, Overview of neutron–proton pairing, Prog. Part. Nucl. Phys. 78, 24 (2014).
  29. K. Atkinson, A survey of numerical methods for solving nonlinear integral equations, J. Integral Equ. Appl. 4, 15 (1992).
  30. C. Drischler, T. Krüger, K. Hebeler, and A. Schwenk, Pairing in neutron matter: New uncertainty estimates and three-body forces, Phys. Rev. C 95, 024302 (2017).
  31. D. D. Johnson, Modified Broyden's method for accelerating convergence in self-consistent calculations, Phys. Rev. B 38, 12807 (1988).
  32. S. Kumar, Superconvergence of a collocation-type method for Hummerstein equations, IMA J. Numer. Anal. 7, 313 (1987).
  33. S. Kumar and I. Sloan, A new collocation-type method for Hammerstein integral equations, Math. Comp. 48, 585 (1987).
  34. J.-P. Berrut and L. N. Trefethen, Barycentric Lagrange interpolation, SIAM Rev. 46, 501 (2004).
  35. A. Gezerlis, Numerical Methods in Physics with Python, 2nd ed. (Cambridge University Press, Cambridge, 2023).
  36. J. Pera and J. Boronat, Low-energy scattering parameters: A theoretical derivation of the effective range and scattering length for arbitrary angular momentum, Am J. Phys. 91, 90 (2023).
  37. A. Gezerlis, I. Tews, E. Epelbaum, S. Gandolfi, K. Hebeler, A. Nogga, and A. Schwenk, Quantum Monte Carlo calculations with chiral effective field theory interactions, Phys. Rev. Lett. 111, 032501 (2013).
  38. D. R. Entem and R. Machleidt, Accurate charge-dependent nucleon-nucleon potential at fourth order of chiral perturbation theory, Phys. Rev. C 68, 041001(R) (2003).
  39. K. Hebeler, A. Schwenk, and B. Friman, Dependence of the BCS S01 superfluid pairing gap on nuclear interactions, Phys. Lett. B 648, 176 (2007).
  40. N. Hinohara, T. Oishi, and K. Yoshida, Triplet-odd pairing in finite nuclear systems: Even-even singly closed nuclei, Phys. Rev. C 109, 034302 (2024).
  41. X. Lian, C. L. Bai, H. Sagawa, and H. Q. Zhang, Proton-neutron pairing correlations in N = Z nuclei with deformed Skyrme+pnBCS model, arXiv:2507.18124.

Outline

Information

Phys. Rev. C 114, 024340– Published 27 August, 2026
  • Received 10 October 2025
  • Accepted 11 June 2026
Crossmark - Check for updates button

©2026 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation