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.
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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
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).
D. J., Scalapino Superconductivity and spin fluctuations. J. Low Temp. Phys. 117, 179–188 (1999).
T. A., Maier, Poilblanc, D. & D. J., Scalapino Dynamics of the pairing interaction in the Hubbard and t−J models of high-temperature superconductors. Phys. Rev. Lett. 100, 237001 (2008).
Maier, T., Jarrell, M., Pruschke, T. & M. H., Hettler Quantum cluster theories. Rev. Mod. Phys. 77, 1027–1080 (2005).
Gull, E., Werner, P., Parcollet, O. & Troyer, M. Continuous-time auxiliary-field Monte Carlo for quantum impurity models. EPL 82, 57003 (2008).
D. J., Scalapino, J. R., Schrieffer & J. W., Wilkins Strong-coupling superconductivity. I. Phys. Rev. 148, 263–279 (1966).
W. L., McMillan Transition temperature of strong-coupled superconductors. Phys. Rev. 167, 331–344 (1968).
Steglich, F. et al. Superconductivity in the presence of strong Pauli paramagnetism: CeCu2Si2. Phys. Rev. Lett. 43, 1892–1896 (1979).
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).
Maeno, Y. et al. Superconductivity in a layered perovskite without copper. Nature 372, 532–534 (1994).
Kamihara, Y. et al. Iron-based layered superconductor: LaOFeP. J. Am. Chem. Soc. 128, 10012–10013 (2006).
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).
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).
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).
P. W., Anderson Is there glue in cuprate superconductors? Science 316, 1705–1707 (2007).
T. D., Stanescu, Galitski, V. & Das Sarma, S. Orbital fluctuation mechanism for superconductivity in iron-based compounds. Phys. Rev. B 78, 195114 (2008).
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).
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).
B.-X., Zheng et al. Stripe order in the underdoped region of the two-dimensional Hubbard model. Science 358, 1155–1160 (2017).
P. W., Anderson The resonating valence bond state in La2CuO4 and superconductivity. Science 235, 1196–1198 (1987).
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).
Gull, E., Parcollet, O. & A. J., Millis Superconductivity and the pseudogap in the two-dimensional Hubbard model. Phys. Rev. Lett. 110, 216405 (2013).
Qin, M. et al. Absence of superconductivity in the pure two-dimensional Hubbard model. Phys. Rev. X 10, 031016 (2020).
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).
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).
Poilblanc, D. & D. J., Scalapino Calculation of Δ(k, ω) for a two-dimensional t−J cluster. Phys. Rev. B 66, 052513 (2002).
Gull, E. & A. J., Millis Pairing glue in the two-dimensional Hubbard model. Phys. Rev. B 90, 041110 (2014).
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).
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).
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).
Shen, J., Tang, T. & Wang, L.-L. Spectral Methods: Algorithms, Analysis and Applications Vol. 41 (Springer, 2011).
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).
Gull, E. et al. Continuous-time Monte Carlo methods for quantum impurity models. Rev. Mod. Phys. 83, 349–404 (2011).
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.
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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.
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Nature Physics thanks Yao Wang, Matthias Eschrig and Ilya Eremin for their contribution to the peer review of this work.
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Dong, X., Gull, E. & Millis, A.J. Quantifying the role of antiferromagnetic fluctuations in the superconductivity of the doped Hubbard model. Nat. Phys. 18, 1293–1296 (2022). https://doi.org/10.1038/s41567-022-01710-z
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DOI: https://doi.org/10.1038/s41567-022-01710-z