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


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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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 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.


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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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Authors and Affiliations



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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Correspondence to Emanuel Gull or Andrew J. Millis.

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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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Supplementary Figs. 1–8 and discussion.

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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. (2022).

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