Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Quantifying the role of antiferromagnetic fluctuations in the superconductivity of the doped Hubbard model

Abstract

Superconductivity arises from the pairing of charge-e electrons into charge-2e bosons—called Cooper pairs—and their condensation into a coherent quantum state. The exact mechanism by which electrons pair up into Cooper pairs in high-temperature superconductors is still not understood. One of the plausible candidates is that spin fluctuations can provide an attractive effective interaction that enables this1,2,3. Here we study the contribution of the electron–spin-fluctuation coupling to the superconducting state of the two-dimensional Hubbard model within dynamical cluster approximation4 using a numerically exact continuous-time Monte Carlo solver5. We show that only about half of the superconductivity can be attributed to a pairing mechanism arising from treating spin fluctuations as a pairing boson in the standard one-loop theory. The rest of the pairing interaction must come from as-yet unidentified higher-energy processes.

This is a preview of subscription content, access via your institution

Access options

Buy article

Get time limited or full article access on ReadCube.

$32.00

All prices are NET prices.

Fig. 1: Spin-fluctuation diagrams for normal and anomalous self-energy.
Fig. 2: Self-energies.
Fig. 3: Total measured anomalous self-energy \({{{\varSigma }}}_{K}^{A}\) and estimated spin-fluctuation contribution \({{{\varSigma }}}_{K}^{\mathrm{SF};A}\) of K = (0, π) at U = 6.0t, βt = 50 and μ = −1.0t (n ≈ 0.90).
Fig. 4: Comparison of true gap function, gap function from spin fluctuation, antiferromagnetic susceptibility \({{{\rm{Im}}}}{\chi }_{\mathrm{spin},(\uppi ,\uppi )}^{\omega +{{{\varDelta }}}_{0}}\) and FM susceptibility \({{{\rm{Im}}}}{\chi }_{\mathrm{spin},(0,0)}^{\omega +{{{\varDelta }}}_{0}}\) shifted by Δ0 = ReΔ(ω = 0) = 0.057 at U = 6.0t, βt = 50 and μ = − 1.0t (n ≈ 0.90).

Data availability

The datasets analysed during the current study are available via GitHub at https://github.com/CQMP/SCgap. Simulation data are available from the corresponding authors on request.

Code availability

Computer codes for data analysis are available from the corresponding authors upon request.

References

  1. Miyake, K., Schmitt-Rink, S. & C. M., Varma Spin-fluctuation-mediated even-parity pairing in heavy-fermion superconductors. Phys. Rev. B 34, 6554–6556 (1986).

    ADS  Article  Google Scholar 

  2. D. J., Scalapino Superconductivity and spin fluctuations. J. Low Temp. Phys. 117, 179–188 (1999).

    ADS  Article  Google Scholar 

  3. T. A., Maier, Poilblanc, D. & D. J., Scalapino Dynamics of the pairing interaction in the Hubbard and tJ models of high-temperature superconductors. Phys. Rev. Lett. 100, 237001 (2008).

    ADS  Article  Google Scholar 

  4. Maier, T., Jarrell, M., Pruschke, T. & M. H., Hettler Quantum cluster theories. Rev. Mod. Phys. 77, 1027–1080 (2005).

    ADS  Article  Google Scholar 

  5. Gull, E., Werner, P., Parcollet, O. & Troyer, M. Continuous-time auxiliary-field Monte Carlo for quantum impurity models. EPL 82, 57003 (2008).

    ADS  Article  Google Scholar 

  6. D. J., Scalapino, J. R., Schrieffer & J. W., Wilkins Strong-coupling superconductivity. I. Phys. Rev. 148, 263–279 (1966).

    ADS  Article  Google Scholar 

  7. W. L., McMillan Transition temperature of strong-coupled superconductors. Phys. Rev. 167, 331–344 (1968).

    ADS  Article  Google Scholar 

  8. Steglich, F. et al. Superconductivity in the presence of strong Pauli paramagnetism: CeCu2Si2. Phys. Rev. Lett. 43, 1892–1896 (1979).

    ADS  Article  Google Scholar 

  9. J. G., Bednorz & K. A., Müller Possible high Tc superconductivity in the Ba–La–Cu–O system. Z. Physik B—Condens. Matter 64, 189–193 (1986).

  10. Maeno, Y. et al. Superconductivity in a layered perovskite without copper. Nature 372, 532–534 (1994).

    ADS  Article  Google Scholar 

  11. Kamihara, Y. et al. Iron-based layered superconductor: LaOFeP. J. Am. Chem. Soc. 128, 10012–10013 (2006).

    Article  Google Scholar 

  12. Castellani, C., Di Castro, C. & Grilli, M. Non-Fermi-liquid behavior and d-wave superconductivity near the charge-density-wave quantum critical point. Z. Physik B—Condens. Matter 103, 137–144 (1996).

  13. C. M., Varma Non-Fermi-liquid states and pairing instability of a general model of copper oxide metals. Phys. Rev. B 55, 14554–14580 (1997).

    ADS  Article  Google Scholar 

  14. Capone, M., Fabrizio, M., Castellani, C. & Tosatti, E. Strongly correlated superconductivity and pseudogap phase near a multiband Mott insulator. Phys. Rev. Lett. 93, 047001 (2004).

    ADS  Article  Google Scholar 

  15. P. W., Anderson Is there glue in cuprate superconductors? Science 316, 1705–1707 (2007).

    ADS  Article  Google Scholar 

  16. T. D., Stanescu, Galitski, V. & Das Sarma, S. Orbital fluctuation mechanism for superconductivity in iron-based compounds. Phys. Rev. B 78, 195114 (2008).

    ADS  Article  Google Scholar 

  17. Saito, T., Yamakawa, Y., Onari, S. & Kontani, H. Revisiting orbital-fluctuation-mediated superconductivity in LiFeAs: nontrivial spin-orbit interaction effects on the band structure and superconducting gap function. Phys. Rev. B 92, 134522 (2015).

    ADS  Article  Google Scholar 

  18. J. P. F., LeBlanc et al. Solutions of the two-dimensional Hubbard model: benchmarks and results from a wide range of numerical algorithms. Phys. Rev. X 5, 041041 (2015).

    Google Scholar 

  19. B.-X., Zheng et al. Stripe order in the underdoped region of the two-dimensional Hubbard model. Science 358, 1155–1160 (2017).

    ADS  MathSciNet  Article  Google Scholar 

  20. P. W., Anderson The resonating valence bond state in La2CuO4 and superconductivity. Science 235, 1196–1198 (1987).

    ADS  Article  Google Scholar 

  21. Gull, E., Ferrero, M., Parcollet, O., Georges, A. & A. J., M. Momentum-space anisotropy and pseudogaps: a comparative cluster dynamical mean-field analysis of the doping-driven metal-insulator transition in the two-dimensional Hubbard model. Phys. Rev. B 82, 155101 (2010).

    ADS  Article  Google Scholar 

  22. Gull, E., Parcollet, O. & A. J., Millis Superconductivity and the pseudogap in the two-dimensional Hubbard model. Phys. Rev. Lett. 110, 216405 (2013).

    ADS  Article  Google Scholar 

  23. Qin, M. et al. Absence of superconductivity in the pure two-dimensional Hubbard model. Phys. Rev. X 10, 031016 (2020).

    Google Scholar 

  24. Chen, X., J. P. F., LeBlanc & Gull, E. Superconducting fluctuations in the normal state of the two-dimensional Hubbard model. Phys. Rev. Lett. 115, 116402 (2015).

    ADS  Article  Google Scholar 

  25. P. W., Anderson & W. F., Brinkman Anisotropic superfluidity in 3He: a possible interpretation of its stability as a spin-fluctuation effect. Phys. Rev. Lett. 30, 1108–1111 (1973).

    ADS  Article  Google Scholar 

  26. Poilblanc, D. & D. J., Scalapino Calculation of Δ(k, ω) for a two-dimensional tJ cluster. Phys. Rev. B 66, 052513 (2002).

    ADS  Article  Google Scholar 

  27. Gull, E. & A. J., Millis Pairing glue in the two-dimensional Hubbard model. Phys. Rev. B 90, 041110 (2014).

    ADS  Article  Google Scholar 

  28. E. W., Huang, C. B., Mendl, H.-C., Jiang, Moritz, B. & T. P., Devereaux Stripe order from the perspective of the Hubbard model. npj Quantum Mater. 3, 22 (2018).

    ADS  Article  Google Scholar 

  29. Wietek, A., Y.-Y., He, S. R., White, Georges, A. & E. M., Stoudenmire Stripes, antiferromagnetism, and the pseudogap in the doped Hubbard model at finite temperature. Phys. Rev. X 11, 031007 (2021).

  30. Mai, P., Karakuzu, S., Balduzzi, G., Johnston, S. & T. A., Maier Intertwined spin, charge, and pair correlations in the two-dimensional Hubbard model in the thermodynamic limit. Proc. Natl Acad. Sci. USA 119, e2112806119 (2022).

    MathSciNet  Article  Google Scholar 

  31. Shen, J., Tang, T. & Wang, L.-L. Spectral Methods: Algorithms, Analysis and Applications Vol. 41 (Springer, 2011).

  32. Gull, E., Iskakov, S., Krivenko, I., A. A., Rusakov & Zgid, D. Chebyshev polynomial representation of imaginary-time response functions. Phys. Rev. B 98, 075127 (2018).

    ADS  Article  Google Scholar 

  33. Gull, E. et al. Continuous-time Monte Carlo methods for quantum impurity models. Rev. Mod. Phys. 83, 349–404 (2011).

    ADS  Article  Google Scholar 

Download references

Acknowledgements

X.D. is supported by NSF DMR 2001465. E.G. is supported by NSF DMR 2001465. The Flatiron Institute is a division of the Simons Foundation.

Author information

Authors and Affiliations

Authors

Contributions

X.D. participated in designing the project, writing the simulation and post-processing code, running the simulations, analysing the data and writing the paper. E.G. participated in designing the project, writing the simulation code, analysing the data and writing the paper. A.J.M. participated in designing the project, analysing the data and writing the paper.

Corresponding authors

Correspondence to Emanuel Gull or Andrew J. Millis.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Physics thanks Yao Wang, Matthias Eschrig and Ilya Eremin for their contribution to the peer review of this work.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Supplementary Figs. 1–8 and discussion.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Dong, X., Gull, E. & Millis, A.J. Quantifying the role of antiferromagnetic fluctuations in the superconductivity of the doped Hubbard model. Nat. Phys. (2022). https://doi.org/10.1038/s41567-022-01710-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1038/s41567-022-01710-z

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing