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Signatures of two gaps in the spin susceptibility of a cuprate superconductor

Nature Physics volume 21, pages 97–103 (2025)Cite this article

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Abstract

A fundamental obstacle to understanding high-temperature superconducting cuprates is that the occurrence of superconductivity hinders the observation of the normal-state properties at low temperature. One important property illustrating this issue is the spin susceptibility: its decrease upon cooling in the normal state is considered as evidence of pseudogap behaviour. However, unambiguous interpretation of this decrease has been impossible, as the crucial low-temperature data inevitably reflect the superconducting pairing rather than the normal state. Here we measure the spin susceptibility of YBa2Cu3Oy at low temperature while suppressing superconductivity in high magnetic field. We found that there are two thermally activated contributions, each of which comes from a different gap, alongside a residual component due to gapless excitations. We relate these two distinct gaps to short-range charge density waves and to the formation of singlets, as occurs in certain quantum spin systems. Both phenomena contribute to the pseudogap at low temperature, supplementing the short-lived antiferromagnetism that initiates pseudogap behaviour at high temperatures. We, therefore, propose that the pseudogap should be regarded as a composite property and that, when not undergoing spin-stripe ordering, underdoped cuprates tend to form short-range spin singlets.

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Fig. 1: Examples of systems with gaps or pseudogaps in the spin susceptibility.
Fig. 2: Temperature and doping dependence of the spin susceptibility.
Fig. 3: Two-gap phenomenology and phase diagram of underdoped YBa2Cu3Oy.

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Source data are provided with this paper. All other data are available from the corresponding author upon reasonable request.

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Acknowledgements

We thank K. Behnia, C. Bernhard, P. Bourges, A. V. Chubukov, A. Georges, S. A. Kivelson, D. LeBoeuf, P. Lee, P. Mendels, D. Orgad, C. Proust, M. Punk, S. Sachdev, A. Sacuto, J. Schmalian, Y. Sidis, J. Tallon, J.M. Tranquada, A.-M. Tremblay, S. R. White and J. Zaanen for discussions. This work was performed at the LNCMI, a member of the European Magnetic Field Laboratory. R.Z., I.V., M.H., T.W., H.M., S.K. and M.-H.J. were supported by the Laboratoire d’Excellence LANEF (Grant No. ANR-10-LABX-51-01) and by the French National Agency for Research (Grant No. ANR-19-CE30-0019, Neptun).

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Author notes

  1. Rui Zhou

    Present address: Institute of Physics, Chinese Academy of Sciences and Beijing National Laboratory for Condensed Matter Physics, Beijing, China

  2. Michihiro Hirata

    Present address: Materials Physics and Applications – Quantum, Los Alamos National Laboratory, Los Alamos, NM, USA

  3. Tao Wu

    Present address: Hefei National Research Center for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei, China

Authors and Affiliations

  1. Laboratoire National des Champs Magnétiques Intenses, CNRS – Université Grenoble Alpes – Université Paul Sabatier – Institut National des Sciences Appliquées – European Magnetic Field Laboratory, Grenoble, France

    Rui Zhou, Igor Vinograd, Michihiro Hirata, Tao Wu, Hadrien Mayaffre, Steffen Krämer & Marc-Henri Julien

  2. Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia, Canada

    W. N. Hardy, Ruixing Liang & D. A. Bonn

  3. Canadian Institute for Advanced Research, Toronto, Ontario, Canada

    W. N. Hardy, Ruixing Liang & D. A. Bonn

  4. Max-Planck-Institut für Festkörperforschung, Stuttgart, Germany

    Toshinao Loew, Juan Porras & Bernhard Keimer

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Contributions

R.Z., I.V., M.H. and T.W. performed the NMR experiments with the help of H.M., S.K. and M.-H.J. R.Z. performed all the data analysis with preliminary input from I.V., M.H. and T.W. and guidance from M.-H.J. W.N.H., R.L., D.A.B., T.L., J.P. and B.K. contributed to the synthesis and the characterization of the samples. M.-H.J. supervised the project and wrote the paper with constant feedback from R.Z. and input from all co-authors.

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Correspondence to Marc-Henri Julien.

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Extended data

Extended Data Fig. 1 Evidence of activated behavior.

a-f: Knight shift (from which Kres has been subtracted) in vertical logarithmic scale vs. inverse temperature for the different doping levels. The straight line indicates exponential behaviour. g-l: same data vs. temperature. The nonlinear behaviour speaks against a power-law dependence. The p = 0.135 data undulates around the straight line (panel f) because of the larger difficulty in determining the exact line position for this sample: the line shapes are more complex due to ortho-III chain-oxygen order being both more complex and shorter ranged.

Extended Data Fig. 2 Alternative fits.

a–f, Kspin for p = 0.064 (a), p = 0.072 (b), p = 0.090 (c), p = 0.109 (d), p = 0.125 (e) and p = 0.135 (f) (same data as in Fig. 2). The solid lines are fits with the two-gap function Kspin = Kres + KL (1-tanh2(ΔL/2kBT)) + KH (1-tanh2(ΔH/2kBT)). Fit results are shown in Extended Data Fig. 3. Error bars represent the added uncertainties in the line positions, in the field reference and in the 2nd-order quadrupole shift correction, all calculated from one standard deviation in the Gaussian peak fits of the relevant NMR lines.

Extended Data Fig. 3 Comparison of results for two different fitting functions.

(a) Onset temperature TCDW of short-range 2D CDW, from refs. 13,14,15. The thick trace is a parabolic function that represents the p dependence of TCDW and its experimental uncertainty. (b) Low-gap values ΔL extracted from the two different fits, and using one or two gap functions (see text), compared to the same parabola as in (a). (c–f) Other fit parameters. Error bars for ΔH, ΔL, KH, KL and Kres represent one standard deviation in the fit results.

Extended Data Fig. 4 89Y NMR evidence of spin freezing for p = 0.064 (YBa2Cu3O6.38).

A broad peak in the spin-lattice relaxation rate 1/T1 vs. T is the typical signature of spin fluctuations becoming as slow as the NMR frequency scale of 31 MHz (here in a field of 15 T). Error bars correspond to one standard deviation in fits of the time-dependence of the NMR signal to the theoretical law for relaxation by magnetic fluctuations. The line is a guide to the eye.

Extended Data Fig. 5 Comparison of the spin susceptibility of LSCO and YBCO.

Bulk magnetization data for La1.86Sr0.14CuO4 is from Nakano et al.36. 63Cu Knight shift data for YBa2Cu4O8 is from Curro et al.68. Unlike YBa2Cu3Oy, for which chain-oxygen atoms become mobile above room temperature, stoichiometric YBa2Cu4O8 does not suffer this problem. This allows one to see that the broad susceptibility maximum, inherited from the behaviour of the undoped square lattice, is present in the YBCO system as well.

Extended Data Fig. 6 Predicting Kres for YBa2Cu3O6.43.

The data at B = 15 T (measured with B | |c) and at 30 T (measured with B tilted by 16° off the c axis) correspond to the zero-temperature extrapolation of our K(T) measurements of O(2) sites at low T. The value at B = 0 is the zero-temperature extrapolation of our K(T) measurements of O(2) sites at low T, with B ⊥ c and B = 9 T « \({B}_{{\rm{c}}2}^{\perp }\). The thick blue line represents the expected B dependence on the basis of results at other doping levels8: linear increase up to Bc2(T = 0) = 45 T (value taken from an interpolation of the results in refs. 69,70) and saturation above Bc2. The value K(T = 0, B = Bc2) = Kres is predicted to be equal to about 0.036%. Within error bars, this agrees well with the fit result in Fig. 2b: Kres(fit) = 0.039 ± 0.01%. Error bars represent the added uncertainties in the line positions, in the field reference and in the 2nd-order quadrupole shift correction, all calculated from one standard deviation in the Gaussian peak fits of the relevant NMR lines.

Extended Data Fig. 7 Absence of direct correlation between residual spin susceptibility and disorder.

The linewidth values correspond to the width of the 17O(2) central line at room temperature. For p = 0.064, we used the ratio of 89Y NMR linewidth between p = 0.064 and p = 0.072 samples. While Kres=Kspin(T = 0) is nearly identical for p = 0.109 and p = 0.090, disorder (as quantified by the NMR line width) differs by a factor of five. p = 0.135 also has identical Kres, yet its linewidth is larger than p = 0.109 by a factor 2.3. Therefore, although both χres and disorder tend to increase at low doping, this likely occurs for different reasons: weakening of CDW correlations for the former, increased oxygen disorder for the latter (notice that the weakening of CDW correlations may, or may not, be partially caused by the increased disorder at low p). Error bars for Kres represent one standard deviation in the fit results. Lines are guides to the eye.

Extended Data Fig. 8 Determining the oxygen concentration with chain-Cu NMR.

63Cu(1E) (empty-chain site) low-frequency quadrupole satellites of the 63Cu NMR spectrum. The labels iV (i = 0, 1, 2, 3) indicate Cu(1E) sites having a number i of nearest-neighbor vacancies. Each line is fitted with a Gaussian function and the relative integrated intensities of the Cu(1E)0V and Cu(1E)1V sites are used to determine the actual oxygen concentration y (see ref. 65 for details about the method).

Extended Data Fig. 9 Orbital contribution to the 17O Knight shift (O(2) sites, B | |b).

The values (red dots) are obtained from the scaling of O(2) and O(3) Knight shifts (Supplementary Information). The data is shown in the same vertical scale as Fig. 2, in order to facilitate comparison. Clearly, Korb « Kspin at any T and can therefore be neglected. Error bars represent the estimated uncertainty in the scaling between O(2) and O(3) Knight shifts.

Extended Data Table 1 Sample properties
Full size table

Supplementary information

Supplementary Information

Two figures with additional Knight shift data.

Source data

Source Data Fig. 2

Knight data for all samples in Fig. 2a–f.

Source Data Fig. 3

Fitting parameters in Fig. 3a–d.

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Zhou, R., Vinograd, I., Hirata, M. et al. Signatures of two gaps in the spin susceptibility of a cuprate superconductor. Nat. Phys. 21, 97–103 (2025). https://doi.org/10.1038/s41567-024-02692-w

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