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

## Abstract

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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## Data availability

Data collected at ISIS on LET and MERLIN are available at https://doi.org/10.5286/ISIS.E.RB1920542 and https://doi.org/10.5286/ISIS.E.RB2010576. Data collected at the ILL on IN12 are available at https://doi.org/10.5291/ILL-DATA.4-02-583. 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.

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## Acknowledgements

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

### Contributions

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.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

## Peer review

### Peer review information

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.

### 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.

### 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.

### 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.

## 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

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### 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

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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). https://doi.org/10.1038/s41567-022-01825-3

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• DOI: https://doi.org/10.1038/s41567-022-01825-3