- Physical Review Journals
All JournalsPhysics Magazine

Physical Review B

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

Fractionalized superconductivity and spin stiffnesses in the SU(2) gauge theory of the two-dimensional Hubbard model

Demetrio Vilardi1 and Pietro M. Bonetti1,2

PDF

Phys. Rev. B 114, 065108 – Published 7 July, 2026

DOI: https://doi.org/10.1103/xcm5-c81y

Abstract

We consider the SU(2) gauge theory for spin fluctuations in the two-dimensional Hubbard model, where the electron field is fractionalized in terms of spinons and chargons. In this theory, spinons are described by a nonlinear sigma model, while chargons are treated as fermions at a mean-field level. We investigate the instability to a superconducting state (SC*), arising from a fractionalized Fermi liquid (FL*) where pairing between chargons occurs. Consistent with previous studies, our analysis reveals a coexisting phase characterized by both magnetic and superconducting order for the chargons. The central contribution of this work is the calculation of the feedback of superconductivity on spatial and temporal spin stiffnesses, thereby quantifying its impact on spin fluctuations. Our key finding is that superconductivity significantly suppresses these spin stiffnesses, enhancing quantum spin fluctuations. This enhancement suggests that superconductivity can play a crucial role in stabilizing quantum disorder against long-range magnetic ordering.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (42)

  1. B. Keimer, S. A. Kivelson, M. R. Norman, S. Uchida, and J. Zaanen, From quantum matter to high-temperature superconductivity in copper oxides, Nature (London) 518, 179 (2015).
  2. C. Collignon, S. Badoux, S. A. A. Afshar, B. Michon, F. Laliberté, O. Cyr-Choinière, J.-S. Zhou, S. Licciardello, S. Wiedmann, N. Doiron-Leyraud, and L. Taillefer, Fermi-surface transformation across the pseudogap critical point of the cuprate superconductor La1.6−xNd0.4SrxCuO4, Phys. Rev. B 95, 224517 (2017).
  3. Y. Fang, G. Grissonnanche, A. Legros, S. Verret, F. Laliberté, C. Collignon, A. Ataei, M. Dion, J. Zhou, D. Graf, M. J. Lawler, P. A. Goddard, L. Taillefer, and B. J. Ramshaw, Fermi surface transformation at the pseudogap critical point of a cuprate superconductor, Nat. Phys. 18, 558 (2022).
  4. M. Frachet, I. Vinograd, R. Zhou, S. Benhabib, S. Wu, H. Mayaffre, S. Krämer, S. K. Ramakrishna, A. P. Reyes, J. Debray, T. Kurosawa, N. Momono, M. Oda, S. Komiya, S. Ono, M. Horio, J. Chang, C. Proust, D. LeBoeuf, and M.-H. Julien, Hidden magnetism at the pseudogap critical point of a cuprate superconductor, Nat. Phys. 16, 1064 (2020).
  5. I. Vinograd, R. Zhou, H. Mayaffre, S. Krämer, S. K. Ramakrishna, A. P. Reyes, T. Kurosawa, N. Momono, M. Oda, S. Komiya, S. Ono, M. Horio, J. Chang, and M.-H. Julien, Competition between spin ordering and superconductivity near the pseudogap boundary in La2−xSrxCuO4: Insights from NMR, Phys. Rev. B 106, 054522 (2022).
  6. S. Sachdev and D. Chowdhury, The novel metallic states of the cuprates: Topological Fermi liquids and strange metals, Prog. Theor. Exp. Phys. 2016, 12C102 (2016).
  7. S. Chatterjee, S. Sachdev, and M. S. Scheurer, Intertwining topological order and broken symmetry in a theory of fluctuating spin-density waves, Phys. Rev. Lett. 119, 227002 (2017).
  8. M. S. Scheurer, S. Chatterjee, W. Wu, M. Ferrero, A. Georges, and S. Sachdev, Topological order in the pseudogap metal, Proc. Natl. Acad. Sci. USA 115, E3665 (2018).
  9. S. Sachdev, H. D. Scammell, M. S. Scheurer, and G. Tarnopolsky, Gauge theory for the cuprates near optimal doping, Phys. Rev. B 99, 054516 (2019).
  10. S. Sachdev, Topological order, emergent gauge fields, and Fermi surface reconstruction, Rep. Prog. Phys. 82, 014001 (2019).
  11. M. Oshikawa, Topological approach to Luttinger's theorem and the Fermi surface of a Kondo lattice, Phys. Rev. Lett. 84, 3370 (2000).
  12. T. Senthil, S. Sachdev, and M. Vojta, Fractionalized Fermi liquids, Phys. Rev. Lett. 90, 216403 (2003).
  13. T. Senthil, M. Vojta, and S. Sachdev, Weak magnetism and non-Fermi liquids near heavy-fermion critical points, Phys. Rev. B 69, 035111 (2004).
  14. M. Christos, Z.-X. Luo, H. Shackleton, Y.-H. Zhang, M. S. Scheurer, and S. Sachdev, A model of d-wave superconductivity, antiferromagnetism, and charge order on the square lattice, Proc. Natl. Acad. Sci. USA 120, e2302701120 (2023).
  15. M. Christos and S. Sachdev, Emergence of nodal Bogoliubov quasiparticles across the transition from the pseudogap metal to the d-wave superconductor, npj Quantum Mater. 9, 4 (2024).
  16. M. Bunney, U. F. P. Seifert, S. Rachel, and M. Vojta, Fractionalized superconductivity mediated by Majorana fermions in the Kitaev-Kondo lattice, Phys. Rev. Lett. 134, 206602 (2025).
  17. S. D. Lundemo and A. Sudbø, Topological superconductivity induced by a Kitaev spin liquid, Phys. Rev. B 109, 184508 (2024).
  18. K. Kurokawa, S. Isono, Y. Kohama, S. Kunisada, S. Sakai, R. Sekine, M. Okubo, M. D. Watson, T. K. Kim, C. Cacho, S. Shin, T. Tohyama, K. Tokiwa, and T. Kondo, Unveiling phase diagram of the lightly doped high-tc cuprate superconductors with disorder removed, Nat. Commun. 14, 4064 (2023).
  19. W. Metzner and H. Yamase, Phase stiffness in an antiferromagnetic superconductor, Phys. Rev. B 100, 014504 (2019).
  20. M. Qin, T. Schäfer, S. Andergassen, P. Corboz, and E. Gull, The Hubbard model: A computational perspective, Annu. Rev. Condens. Matter Phys. 13, 275 (2022).
  21. E. Gull, O. Parcollet, and A. J. Millis, Superconductivity and the pseudogap in the two-dimensional Hubbard model, Phys. Rev. Lett. 110, 216405 (2013).
  22. C. J. Halboth and W. Metzner, d-wave superconductivity and Pomeranchuk instability in the two-dimensional Hubbard model, Phys. Rev. Lett. 85, 5162 (2000).
  23. D. Vilardi, C. Taranto, and W. Metzner, Dynamically enhanced magnetic incommensurability: Effects of local dynamics on nonlocal spin correlations in a strongly correlated metal, Phys. Rev. B 97, 235110 (2018).
  24. O. Gunnarsson, T. Schäfer, J. P. F. LeBlanc, E. Gull, J. Merino, G. Sangiovanni, G. Rohringer, and A. Toschi, Fluctuation diagnostics of the electron self-energy: Origin of the pseudogap physics, Phys. Rev. Lett. 114, 236402 (2015).
  25. P. M. Bonetti and W. Metzner, SU(2) gauge theory of the pseudogap phase in the two-dimensional Hubbard model, Phys. Rev. B 106, 205152 (2022).
  26. N. Dupuis, L. Canet, A. Eichhorn, W. Metzner, J. Pawlowski, M. Tissier, and N. Wschebor, The nonperturbative functional renormalization group and its applications, Phys. Rep. 910, 1 (2021).
  27. J. Wang, A. Eberlein, and W. Metzner, Competing order in correlated electron systems made simple: Consistent fusion of functional renormalization and mean-field theory, Phys. Rev. B 89, 121116(R) (2014).
  28. H. Yamase, A. Eberlein, and W. Metzner, Coexistence of incommensurate magnetism and superconductivity in the two-dimensional Hubbard model, Phys. Rev. Lett. 116, 096402 (2016).
  29. D. Vilardi, P. M. Bonetti, and W. Metzner, Dynamical functional renormalization group computation of order parameters and critical temperatures in the two-dimensional Hubbard model, Phys. Rev. B 102, 245128 (2020).
  30. V. L. Berezinskii, Destruction of long-range order in one-dimensional and two-dimensional systems having a continuous symmetry group. I. Classical systems, Zh. Eksp. Teor. Fiz. 59, 907 (1971) [Sov. Phys. JETP 32, 493 (1971)].
  31. J. M. Kosterlitz and D. J. Thouless, Ordering, metastability and phase transitions in two-dimensional systems, J. Phys. C 6, 1181 (1973).
  32. A. Auerbach, Interacting Electrons and Quantum Magnetism (Springer-Verlag, Berlin, 1994).
  33. N. Dupuis, Spin fluctuations and pseudogap in the two-dimensional half-filled Hubbard model at weak coupling, Phys. Rev. B 65, 245118 (2002).
  34. F. D. M. Haldane, Nonlinear field theory of large-spin Heisenberg antiferromagnets: Semiclassically quantized solitons of the one-dimensional easy-axis Néel state, Phys. Rev. Lett. 50, 1153 (1983).
  35. F. Haldane, Continuum dynamics of the 1-d Heisenberg antiferromagnet: Identification with the O(3) nonlinear sigma model, Phys. Lett. A 93, 464 (1983).
  36. P. M. Bonetti, Local Ward identities for collective excitations in fermionic systems with spontaneously broken symmetries, Phys. Rev. B 106, 155105 (2022); Erratum: Local Ward identities for collective excitations in fermionic systems with spontaneously broken symmetries, 110, 079902(E) (2024).
  37. I. A. Goremykin and A. A. Katanin, Antiferromagnetic and spin spiral correlations in the doped two-dimensional Hubbard model: Gauge symmetry, Ward identities, and dynamical mean-field theory analysis, Phys. Rev. B 110, 085153 (2024).
  38. N. Verma, T. Hazra, and M. Randeria, Optical spectral weight, phase stiffness, and Tc bounds for trivial and topological flat band superconductors, Proc. Natl. Acad. Sci. USA 118, e2106744118 (2021).
  39. A. P. Kampf, Collective excitations in itinerant spiral magnets, Phys. Rev. B 53, 747 (1996).
  40. P. M. Bonetti and W. Metzner, Spin stiffness, spectral weight, and Landau damping of magnons in metallic spiral magnets, Phys. Rev. B 105, 134426 (2022).
  41. C. Honerkamp and M. Salmhofer, Temperature-flow renormalization group and the competition between superconductivity and ferromagnetism, Phys. Rev. B 64, 184516 (2001).
  42. D. Vilardi, P. M. Bonetti, and W. Metzner, Spin stiffnesses and stability of magnetic order in the lightly doped two-dimensional Hubbard model, Phys. Rev. B 112, 245149 (2025).

Outline

Information

Phys. Rev. B 114, 065108– Published 7 July, 2026
  • Received 11 November 2025
  • Revised 4 March 2026
  • Accepted 2 June 2026
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. Open access publication funded by Max Planck Society.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation