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Spin fluctuations associated with the collapse of the pseudogap in a cuprate superconductor


Theories of the origin of superconductivity in cuprates depend on an understanding of their normal state, which exhibits various competing orders. Transport and thermodynamic measurements on La2 − xSrxCuO4 show signatures of a quantum critical point and the associated fluctuations, including a peak in the electronic specific heat versus doping, near the doping p* where the pseudogap collapses. The fundamental nature of these quantum fluctuations is unclear. Here we use inelastic neutron scattering to show that, close to the superconducting critical temperature and near p*, there are very-low-energy collective spin excitations with characteristic energies of ~5 meV. Cooling and applying a magnetic field creates a mixed state with a stronger magnetic response below 10 meV. We conclude that the low-energy spin fluctuations are due to the collapse of the pseudogap combined with an underlying tendency to magnetic order. We show that the large specific heat near p* can be understood in terms of collective spin fluctuations. The spin fluctuations we measure exist across the superconducting phase diagram and may be related to the strange metal behaviour observed in overdoped cuprates.

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Fig. 1: Entropy and electronic specific heat in La2 − xSrxCuO4.
Fig. 2: Wavevector-dependent maps of low-energy spin fluctuations in La2 − xSrxCuO4 (x = 0.22).
Fig. 3: Fits of spin-fluctuation model to magnetic excitations.
Fig. 4: Magnetic excitations with a low-energy scale in the normal and mixed states of LSCO (p = 0.22).

Data availability

Data collected at ISIS on LET and MERLIN are available at and Data collected at the ILL on IN12 are available at Source data are provided with this paper.

Code availability

The Mathematica computer code used to evaluate γSF in Table 1 is available in the Supplementary Information.


  1. Stewart, G. R. Heavy-fermion systems. Rev. Mod. Phys. 56, 755–787 (1984).

    Article  ADS  Google Scholar 

  2. Coleman, P. Introduction to Many-Body Physics (Cambridge Univ. Press, 2015)

  3. Walter, U., Wohlleben, D. & Fisk, Z. Dynamics of the magnetization in the heavy fermion system CeCu6. Z. Phys. B 62, 325–330 (1986).

    Article  ADS  Google Scholar 

  4. Rourke, P. M. C. et al. A detailed de Haas-van Alphen effect study of the overdoped cuprate Tl2Ba2CuO6 + δ. New J. Phys. 12, 105009 (2010).

    Article  ADS  Google Scholar 

  5. Ramshaw, B. J. et al. Quasiparticle mass enhancement approaching optimal doping in a high-Tc superconductor. Science 348, 317–320 (2015).

    Article  ADS  Google Scholar 

  6. Yoshida, T. et al. Low-energy electronic structure of the high-Tc cuprates La2 − xSrxCuO4 studied by angle-resolved photoemission spectroscopy. J. Phys. Cond. Matter 19, 125209 (2007).

    Article  ADS  Google Scholar 

  7. Horio, M. et al. Three-dimensional Fermi surface of overdoped La-based cuprates. Phys. Rev. Lett. 121, 077004 (2018).

    Article  ADS  Google Scholar 

  8. Markiewicz, R. S., Sahrakorpi, S., Lindroos, M., Lin, H. & Bansil, A. One-band tight-binding model parametrization of the high-Tc cuprates including the effect of kz dispersion. Phys. Rev. B 72, 054519 (2005).

    Article  ADS  Google Scholar 

  9. Keimer, B., Kivelson, S. A., Norman, M. R., Uchida, S. & Zaanen, J. From quantum matter to high-temperature superconductivity in copper oxides. Nature 518, 179–186 (2015).

    Article  ADS  Google Scholar 

  10. Scalapino, D. J. A common thread: the pairing interaction for unconventional superconductors. Rev. Mod. Phys. 84, 1383–1417 (2012).

    Article  ADS  Google Scholar 

  11. Timusk, T. & Statt, B. The pseudogap in high-temperature superconductors: an experimental survey. Rep. Prog. Phys. 62, 61–122 (1999).

    Article  ADS  Google Scholar 

  12. Proust, C. & Taillefer, L. The remarkable underlying ground states of cuprate superconductors. Ann. Rev. Condens. Matter Phys. 10, 409–429 (2019).

    Article  ADS  Google Scholar 

  13. Cooper, R. A. et al. Anomalous criticality in the electrical resistivity of La2 − xSrxCuO4. Science 323, 603–607 (2009).

  14. Michon, B. et al. Thermodynamic signatures of quantum criticality in cuprate superconductors. Nature 567, 218–222 (2019).

    Article  ADS  Google Scholar 

  15. Hartnoll, S. A. & Mackenzie, A. P. Plankian dissipation in metals. Rev. Mod. Phys. Preprint at (2021).

  16. Loram, J. W., Luo, J., Cooper, J. R., Liang, W. Y. & Tallon, J. L. Evidence on the pseudogap and condensate from the electronic specific heat. J. Phys. Chem. Sol. 62, 59–64 (2001).

    Article  ADS  Google Scholar 

  17. Matsuzaki, T., Momono, N., Oda, M. & Ido, M. Electronic specific heat of La2 − xSrxCuO4: pseudogap formation and reduction of the superconducting condensation energy. J. Phys. Soc. Jpn. 73, 2232–2238 (2004).

  18. Momono, N. et al. Low-temperature electronic specific heat of La2 − xSrxCuO4 and La2 − xSrxCuO1 − yZnyO4, evidence for a d wave superconductor. Phys. C Superconduct. 233, 395–401 (1994).

  19. Girod, C. et al. Normal state specific heat in the cuprate superconductors La2 − xSrxCuO4 and Bi2 + ySr2 − x − yLaxCuO6 + δ near the critical point of the pseudogap phase. Phys. Rev. B 103, 214506 (2021).

  20. Shibauchi, T., Carrington, A. & Matsuda, Y. A quantum critical point lying beneath the superconducting dome in iron pnictides. Ann. Rev. Cond. Matter Phys. 5, 113–135 (2014).

    Article  ADS  Google Scholar 

  21. Lester, C. et al. Magnetic-field-controlled spin fluctuations and quantum criticality in Sr3Ru2O7. Nat. Commun. 12, 5798 (2021).

    Article  ADS  Google Scholar 

  22. Thurston, T. R. et al. Neutron scattering study of the magnetic excitations in metallic and superconducting La2 − xSrxCuO4. Phys. Rev. B 40, 4585–4595 (1989).

    Article  ADS  Google Scholar 

  23. Mason, T. E., Aeppli, G., Hayden, S. M., Ramirez, A. P. & Mook, H. A. Low energy excitations in superconducting La1.86Sr0.14CuO4. Phys. Rev. Lett. 71, 919–922 (1993).

    Article  ADS  Google Scholar 

  24. Aeppli, G., Mason, T. E., Hayden, S. M., Mook, H. A. & Kulda, J. Nearly singular magnetic fluctuations in the normal state of a high-Tc cuprate superconductor. Science 278, 1432–1435 (1997).

    Article  ADS  Google Scholar 

  25. Mook, H. A. et al. Spin fluctuations in YBa2Cu3O6.6. Nature 395, 580–582 (1998).

    Article  ADS  Google Scholar 

  26. Hinkov, V. et al. Two-dimensional geometry of spin excitations in the high-transition-temperature superconductor YBa2Cu3O6 + x. Nature 430, 650–654 (2004).

  27. Wakimoto, S. et al. Direct relation between the low-energy spin excitations and superconductivity of overdoped high-Tc superconductors. Phys. Rev. Lett. 92, 217004 (2004).

    Article  ADS  Google Scholar 

  28. Lipscombe, O. J., Hayden, S. M., Vignolle, B., McMorrow, D. F. & Perring, T. G. Persistence of high-frequency spin fluctuations in overdoped superconducting La2 − xSrxCuO4 (x = 0.22). Phys. Rev. Lett. 99, 067002 (2007).

  29. Li, Y. et al. Low-energy antiferromagnetic spin fluctuations limit the coherent superconducting gap in cuprates. Phys. Rev. B 98, 224508 (2018).

    Article  ADS  Google Scholar 

  30. Ikeuchi, K. et al. Detailed study of the structure of the low-energy magnetic excitations in overdoped La1.75Sr0.25CuO4. Physica B 536, 717–719 (2018).

  31. Yamada, K. et al. Doping dependence of the spatially modulated dynamical spin correlations and the superconducting-transition temperature in La2 − xSrxCuO4. Phys. Rev. B 57, 6165–6172 (1998).

    Article  ADS  Google Scholar 

  32. Headings, N. S., Hayden, S. M., Kulda, J., Babu, N. H. & Cardwell, D. A. Spin anisotropy of the magnetic excitations in the normal and superconducting states of optimally doped YBa2Cu3O6.9 studied by polarized neutron spectroscopy. Phys. Rev. B 84, 104513 (2011).

    Article  ADS  Google Scholar 

  33. Lee, C. H., Yamada, K., Hiraka, H., Venkateswara Rao, C. R. & Endoh, Y. Spin pseudogap in La2 − xSrxCuO4 studied by neutron scattering. Phys. Rev. B 67, 134521 (2003).

    Article  ADS  Google Scholar 

  34. Birgeneau, R. J. et al. Soft-phonon behavior and transport in single-crystal La2CuO4. Phys. Rev. Lett. 59, 1329–1332 (1987).

    Article  ADS  Google Scholar 

  35. Frachet, M. et al. Hidden magnetism at the pseudogap critical point of a cuprate superconductor. Nat. Phys. 16, 1064–1068 (2020).

    Article  Google Scholar 

  36. Lake, B. et al. Spins in the vortices of a high-temperature superconductor. Science 291, 1759–1762 (2001).

    Article  ADS  Google Scholar 

  37. Millis, A. J., Monien, H. & Pines, D. Phenomenological model of nuclear relaxation in the normal state of YBa2Cu3O7. Phys. Rev. B 42, 167–177 (1990).

    Article  ADS  Google Scholar 

  38. Chaikin, P. M. & Lubensky, T. C. Principles of Condensed Matter Physics (Cambridge Univ. Press, 1995).

  39. Béal-Monod, M. T., Ma, S.-K. & Fredkin, D. R. Temperature dependence of the spin susceptibility of a nearly ferromagnetic Fermi liquid. Phys. Rev. Lett. 20, 929–932 (1968).

    Article  ADS  Google Scholar 

  40. Brinkman, W. F. & Engelsberg, S. Spin-fluctuation contributions to the specific heat. Phys. Rev. 169, 417–431 (1968).

    Article  ADS  Google Scholar 

  41. Lonzarich, G. G. The magnetic equation of state and heat capacity in weak itinerant ferromagnets. J. Magn. Magn. Mater. 54–57, 612–616 (1986).

    Article  ADS  Google Scholar 

  42. Edwards, D. M. & Lonzarich, G. G. The entropy of fluctuating moments at low temperatures. Phil. Mag. B 65, 1185–1189 (1992).

    Article  ADS  Google Scholar 

  43. Ishigaki, A. & Moriya, T. On the spin fluctuation-enhanced specific heat around the magnetic instabilities. J. Phys. Soc. Jpn. 68, 3673–3676 (1999).

    Article  ADS  Google Scholar 

  44. Moriya, T. & Ueda, K. Antiferromagnetic spin fluctuation and superconductivity. Rep. Prog. Phys. 66, 1299–1341 (2003).

    Article  ADS  Google Scholar 

  45. Hayden, S. M., Doubble, R., Aeppli, G., Perring, T. G. & Fawcett, E. Strongly enhanced magnetic excitations near the quantum critical point of Cr1 − xVx and why strong exchange enhancement need not imply heavy fermion behavior. Phys. Rev. Lett. 84, 999–1002 (2000).

  46. Miao, H. et al. Charge density waves in cuprate superconductors beyond the critical doping. npj Quant. Mater. 6, 31 (2021).

    Article  ADS  Google Scholar 

  47. Wu, T. et al. Incipient charge order observed by NMR in the normal state of YBa2Cu3Oy. Nat. Commun. 6, 6438 (2015).

    Article  ADS  Google Scholar 

  48. Dai, P. et al. The magnetic excitation spectrum and thermodynamics of high-Tc superconductors. Science 284, 1344–1347 (1999).

    Article  ADS  Google Scholar 

  49. Takagi, H. et al. Superconductor-to-nonsuperconductor transition in La2 − xSrxCuO4 as investigated by transport and magnetic measurements. Phys. Rev. B 40, 2254–2261 (1989).

  50. Nakamae, S. et al. Electronic ground state of heavily overdoped nonsuperconducting La2 − xSrxCuO4. Phys. Rev. B 68, 100502 (2003).

    Article  ADS  Google Scholar 

  51. Ewings, R. A. et al. Horace: software for the analysis of data from single crystal spectroscopy experiments at time-of-flight neutron instruments. Nucl. Instrum. Methods Phys. Res. A 834, 132–142 (2016).

    Article  ADS  Google Scholar 

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We are grateful to J. R. Stewart for running the LET experiment. We acknowledge useful discussions with A. Carrington and N. E. Hussey. M.Z. and S.M.H. acknowledge funding and support from the Engineering and Physical Sciences Research Council (EPSRC) under grant no. EP/R011141/1. We acknowledge the ISIS Facility for instrument time at beamline LET under proposal RB1920542, MERLIN under proposal RB2010576 and Institut Laue-Langevin for time at IN12 under proposal no. 4-02-561.

Author information

Authors and Affiliations



M.Z. and O.J.L. prepared the samples. M.Z., D.J.V., S.R., C.C.T. and S.M.H. acquired neutron scattering measurements. M.Z. and S.M.H. analysed the data and wrote the initial manuscript. All authors contributed to the discussion and provided feedback on the manuscript.

Corresponding authors

Correspondence to M. Zhu or S. M. Hayden.

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Nature Physics thanks Igor Zaliznyak and Pengcheng Dai for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 Spin fluctuations and phonons in La2-xSrxCuO4 (x = 0.22) near Qδ.

S(Q, ω) as a function of energy and wavevector along a trajectory through two incommensurate wave vectors Qδ = (0.5-δ, 0.5, L) and (0.5, 0.5-δ, L) (see inset to panel a). Integration ranges are a L [ − 1, 1] and b L [3.8, 4.2]. Strong phonons are observed (panel b) for L ≈ 4, but these are not visible near L = 0 (panel a) where spin fluctuations are seen. Data were collected on LET (panel a) and MERLIN (panel b).

Extended Data Fig. 2 Phonons in La2−xSrxCuO4 (x = 0.22) near Q = (1.5, 1.5, 2).

a S(Q, ω) as a function of energy and wavevector across Q = (1.5, 1.5, 2) with L [1.8, 2.2] at T = 26 K. b Energy dependence of S(Q, ω) at (1.5, 1.5, 2). The arrow denotes a phonon at approximately 3 meV corresponding to the rotation of the CuO6 octahedra. These features are quite different from the scattering we observe near (0.5, 0.5, 0) identified as magnetic scattering. Data were collected on MERLIN.

Source data

Extended Data Fig. 3 No evidence for field-induced spin density wave (SDW) order in La2−xSrxCuO4 (x = 0.22).

Elastic scans through the Qδ = (0.5 − δ, 0.5, 0) position collected on IN12 with Ef = 4.7 meV, T = 1.5 K and B = 10 T. a No evidence of SDW order is seen in the La2−xSrxCuO4 (x = 0.22) sample studied here. The dashed line shows the position and width (due to instrumental resolution) that a SDW peak at Qδ would have. b For comparison, we show a SDW peak measured on an underdoped La2−xSrxCuO4 (x = 0.132) sample of similar size (4.9 g) with IN12 using similar spectrometer conditions and scaled to the same time as a. The values of ordered moments are for a single Qδ and have been determined by comparison with scattering from the (110) Bragg peak.

Source data

Extended Data Fig. 4 Low-energy spin fluctuations measured by IN12 cold neutron triple-axis spectrometer.

a-b Measurements made at Qδ = (0.5, 0.37, 0) (closed symbols) and a background estimated from the average of (0.56, 0.31, 0) and (0.44, 0.43, 0) (open symbols). c-d Signal isolated from the data in a-b and corrected by a bose factor. Data are consistent with the LET data and show low-energy spin fluctuations in the normal state (panel c) and a field-induced signal in the superconducting state (panel d). e-f Constant-energy scans across Qδ at T = 1.5 K, B = 0 and 10 T.

Source data

Extended Data Fig. 5 Fits of low-energy spin fluctuations in the normal state at Tc.

a-c Constant-energy cuts of S(Q, ω). Integration range perpendicular to the trajectory is shown in Fig. 2h by dashed lines with L [ − 1, 1]. Solid lines are fitted curves using Eqn. (7) convoluted with the instrumental resolution. d-f Energy dependence of χ(Qδ, ω), \({\kappa }_{1}^{2}(\omega )\), and δ in Eqn. (7). The solid lines in d, e and f are fits of Eqn. (8), Eqn. (9) and a constant respectively.

Source data

Supplementary information

Supplementary Information

A discussion of the high-frequency cut-off in the spin fluctuation model used in the paper.

Supplementary Software 1

Mathematica script (gamma_calculation.wls) used to evaluate the linear heat capacity (equation (12)) with pdf of output.

Source data

Source Data Fig. 1

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Source Data Fig. 3

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Source Data Fig. 4

Source data text file

Source Data Extended Data Fig. 2

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Source Data Extended Data Fig. 3

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Source Data Extended Data Fig. 4

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Source Data Extended Data Fig. 5

Source data text file

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Zhu, M., Voneshen, D.J., Raymond, S. et al. Spin fluctuations associated with the collapse of the pseudogap in a cuprate superconductor. Nat. Phys. (2022).

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