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Neutron scattering and thermodynamic evidence for emergent photons and fractionalization in a pyrochlore spin ice

Nature Physics volume 21, pages 1203–1210 (2025)Cite this article

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Abstract

The three-dimensional pyrochlore lattice of corner-sharing tetrahedra can host a quantum spin ice, a quantum analogue of the classical spin ice found in other pyrochlore compounds. This state can manifest a quantum spin liquid, and indeed, these compounds are predicted to have emergent gauge fields that produce linearly dispersing collective magnetic excitations near zero energy, in addition to the presence of higher-energy spinon excitations. Here we use polarized neutron scattering experiments on single crystals of the Ce2Zr2O7 pyrochlore. We find evidence for magnetic excitations near zero energy, in addition to signatures of spinons at higher energies. Furthermore, we perform heat capacity measurements and find behaviour consistent with the cubic-in-temperature dependence expected for linearly dispersing gapless bosonic modes. Comparing the observed magnetic excitations with theoretical calculations, we argue that Ce2Zr2O7 is a strong candidate for a dipolar–octupolar quantum spin ice with dominant dipolar Ising interactions.

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Fig. 1: Summary of crystal structure, Brillouin zone, phase diagram, expected experimental signatures and excitations of Ce2Zr2O7.
Fig. 2: Energy scans of the polarized neutron scattering cross-sections at different momentum positions for Ce2Zr2O7.
Fig. 3: Wavevector and polarization dependencies of magnetic scattering at E = 0 ± 0.03 meV for Ce2Zr2O7.
Fig. 4: Wavevector and polarization dependencies of magnetic scattering at E = 0.1 ± 0.03 meV for Ce2Zr2O7.
Fig. 5: Specific heat of Ce2Zr2O7.

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The data presented in the article are available from the corresponding authors upon request.

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Acknowledgements

We thank H. Yan, A. Nevidomskyy, A. Scheie, P. Laurell and M. Gingras for helpful discussions and M. Taupin for initial support in setting up the specific heat measurements. The neutron scattering work at Rice is supported by the United States Department of Energy, Basic Energy Sciences, DE-SC0012311 (P.D.). The single-crystal growth work at Rice is supported by the Robert A. Welch Foundation under grant no. C-1839 (P.D.). Crystal growth by B.G. and S.-W.C. at Rutgers was supported by the visitor program at the Center for Quantum Materials Synthesis (cQMS), funded by the Gordon and Betty Moore Foundation’s EPiQS initiative through grant no. GBMF6402 and by Rutgers University. F.D. and Y.B.K. are supported by the Natural Science and Engineering Research Council of Canada and the Center for Quantum Materials at the University of Toronto. F.D. is further supported by the Vanier Canada Graduate Scholarship (CGV—186886). D.W.T. acknowledges funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement no. 884104 (PSI-FELLOW-III-3i). Neutron data (https://doi.org/10.5291/ILL-DATA.4-05-851; https://doi.org/10.5291/ILL-DATA.4-05-904) were obtained using OrientExpress and ThALES instruments at the Institut Laue Langevin with support from proposal nos 4-05-851 and 4-05-904. The specific heat measurements and analyses in Vienna were supported by the European Research Council (ERC Advanced Grant 101055088—CorMeTop) and the Austrian Science Fund (FWF project nos I 5868-N, F 86 and 10.55776/COE1).

Author information

Author notes

  1. These authors contributed equally: Bin Gao, Félix Desrochers, David W. Tam, Diana M. Kirschbaum.

Authors and Affiliations

  1. Department of Physics and Astronomy, Rice University, Houston, TX, USA

    Bin Gao & Pengcheng Dai

  2. Department of Physics, University of Toronto, Toronto, Ontario, Canada

    Félix Desrochers & Yong Baek Kim

  3. PSI Center for Neutron and Muon Sciences, Paul Scherrer Institut, Villigen PSI, Switzerland

    David W. Tam

  4. Institute of Solid State Physics, Vienna University of Technology, Vienna, Austria

    Diana M. Kirschbaum, Duy Ha Nguyen & Silke Paschen

  5. Institut Laue-Langevin, Grenoble Cedex 9, France

    Paul Steffens & Arno Hiess

  6. European Spallation Source ERIC, Lund, Sweden

    Arno Hiess

  7. Jülich Centre for Neutron Science (JCNS) at Heinz Maier-Leibnitz Zentrum (MLZ), Forschungszentrum Jülich, Garching, Germany

    Yixi Su

  8. Keck Center for Quantum Magnetism and Department of Physics and Astronomy, Rutgers University, Piscataway, NJ, USA

    Sang-Wook Cheong

  9. Smalley-Curl Institute, Rice University, Houston, TX, USA

    Pengcheng Dai

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Contributions

P.D. and B.G. conceived of the project. B.G. and S.-W.C. prepared the samples. The neutron scattering experiments were carried out and analysed by D.W.T., P.S., A.H., Y.S., B.G. and P.D. The specific heat measurements were performed by D.M.K., D.H.N. and S.P. Theoretical analysis was supervised by Y.B.K. and performed by F.D. and Y.B.K. The entire project was supervised by P.D. The paper is written by P.D., Y.B.K., B.G. and F.D., with contributions from D.M.K. and S.P. All authors made comments.

Corresponding authors

Correspondence to Silke Paschen, Yong Baek Kim or Pengcheng Dai.

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Extended data

Extended Data Fig. 1 Pictures of samples.

a, One piece of single-crystalline Ce2Zr2O7 was mounted on a copper holder. b, The X-ray Laue pattern in the [0, 0, 1] direction. The sample is mounted inside a dilution refrigerator maintained at \(T=50\) mK for the entire experiment. The sample is tied inside copper foils to ensure good thermalization at 50 mK (sample 1). c, d show pictures of sample 2 and its Laue pattern, respectively.

Extended Data Fig. 2 Raw data of energy scans.

a-c, The energy scan of \({\sigma }_{x}^{{NSF}}({\boldsymbol{Q}},E)\) channel at \({\boldsymbol{Q}}=\left(\mathrm{0,0,1}\right),(3/4,3/\mathrm{4,0})\) and \((\mathrm{1,1,0})\) using \({E}_{f}=3.23\), 3.23, and 2.52 meV, respectively, measured on sample 1. We use the Gaussian fit to determine the energy resolution to be 0.076 meV, 0.062 meV, and 0.042 meV in FWHM, respectively. d, The energy scan of \({\sigma }_{x}^{{NSF}}({\boldsymbol{Q}},E)\) channel at \({\boldsymbol{Q}}=\left(\mathrm{0,0,1}\right)\) using \({E}_{f}=2.52\) meV on sample 2, which has an energy resolution of 0.035 meV in FWHM. The curve is shown as the solid black line in Fig. 2n. e, Similar energy scan using \({E}_{f}=3.23\) meV on sample 2, which gives an energy resolution of 0.052 meV in FWHM. The vertical error bars in a–e are statistical errors of 1 standard deviation.

Extended Data Fig. 3 Raw data of energy scans.

a-f, Comparison of the polarized NSF neutron scattering cross sections \({\sigma }_{x}^{{NSF}}({\boldsymbol{Q}},E)\) and SF neutron scattering cross sections \({\sigma }_{x,y,z}^{{SF}}({\boldsymbol{Q}},E)\) (a-c) and \({\sigma }_{x,y}^{{SF}}({\boldsymbol{Q}},E)\) (d-f) at E = 0 ± 0.03 meV along the [0, 0, l], [h, h, 1], [h, h, 0], [h, h, 0.25], [h, h, 0.5] and [h, h, 0.75] directions. \({\sigma }_{x}^{{NSF}}\left({\boldsymbol{Q}},E\right) > {\sigma }_{x,y,z}^{{SF}}({\boldsymbol{Q}},E)\) at all \({\boldsymbol{Q}}\) points in the scattering plane. g-h, Comparison of the polarized \({\sigma }_{x}^{{NSF}}({\boldsymbol{Q}},E)\) and \({\sigma }_{x,y,z}^{{SF}}({\boldsymbol{Q}},E)\) at E = 0.1 ± 0.03 meV along the [0, 0, l] and [h, h, 0] directions. \({\sigma }_{x}^{{NSF}}\left({\boldsymbol{Q}},E\right) < {\sigma }_{x,y,z}^{{SF}}({\boldsymbol{Q}},E)\) at most \({\boldsymbol{Q}}\) points in the scattering plane. Gray windows in panels b, c & h indicate nuclear Bragg peaks at (1, 1, 1) and (2, 2, 0) points, respectively. Data are obtained with \({E}_{f}=3.23\) meV. The vertical error bars in a–h are statistical errors of 1 standard deviation.

Extended Data Fig. 4 Unpolarized neutron scattering data.

a,b, The raw scattering intensity at 35 mK and 12 K using Ei = 1.55 meV at the elastic line (E = 0 ± 0.03 meV) from our previous unpolarized neutron scattering experiment at CNCS17. c,d, The comparison of raw scattering intensity at 35 mK and 12 K along the [0, 0, l] and [h, h, 0] directions from cuts using panel a. As one can see, the scattering is highly structured and the scattering has higher intensity at 12 K at almost all \({\boldsymbol{Q}}\) space probed. e,f, Inelastic scattering signals obtained at 35 mK by subtracting 12 K as background along the (h, h, 0) and (0, 0, l) directions from our previous unpolarized INS experiment at CNCS17 and the corresponding Gaussian fits. The vertical error bars in c-f are statistical errors of 1 standard deviation.

Extended Data Fig. 5 Theoretical calculations.

Predictions from GMFT for the width of the two-spinon continuum as a function of transverse coupling for a, 0-flux QSI and b, \({\rm{\pi }}\)-flux QSI. We compare these with QMC results of Ref. 47 and the 32-site ED results of Ref. 29 extracted from the transverse dynamical spin structure factor \({S}^{\pm }\left({\boldsymbol{Q}},E\right)\) for 0- and \({\rm{\pi }}\)-flux QSI, respectively. The dashed and dashed-dotted lines denote the parameter sets obtained in Refs. 28,20.

Extended Data Fig. 6 Comparison of theory and data.

a-c, Total magnetic scattering \({M}_{z}+{M}_{y}\) as a function of energy and theoretical prediction for the spinon contribution using \({{\mathscr{J}}}_{\parallel }=0.06\) meV and \({{\mathscr{J}}}_{\pm }/{{\mathscr{J}}}_{\parallel }=-0.35\) at \({\boldsymbol{Q}}=(\mathrm{0,0,1})\) (X point), \({\boldsymbol{Q}}=(3/\mathrm{4,3}/\mathrm{4,0})\) (K point), and \({\boldsymbol{Q}}=(\mathrm{1,1,0}).\) The theoretical results are broadened using a Gaussian with a FWHM of 0.076 meV, 0.062 meV, and 0.042 meV at the X, K and \({\boldsymbol{Q}}=(\mathrm{1,1,0})\) point, respectively. d-f, Residual of the fit using only the spinons. The residual is fitted at all three momentum transfers using a Gaussian function centered close to the elastic line. The vertical error bars are propagating errors using Eq. (3).

Extended Data Fig. 7 Comparison of theory and data.

a-i, The energy scan of pure magnetic components \({M}_{z}+{M}_{y}\), \({M}_{z}\), and \({M}_{y}\) at \({\boldsymbol{Q}}=\left(\mathrm{0,0,1}\right),(3/4,3/\mathrm{4,0})\) and \((\mathrm{1,1,0})\). We use the Gaussian fit to determine the relative shift of the signals compared with the elastic line. The vertical error bars are propagating errors using Eq. (3).

Extended Data Fig. 8 Raw data of high resolution measurements.

a, The raw energy scan around the elastic line of the second sample at \({\boldsymbol{Q}}=\left(\mathrm{0,0,1}\right)\) using the same setup as the first sample, Ef = 3.23 meV. b, The raw data of the polarized \({\sigma }_{x}^{{NSF}}({\boldsymbol{Q}},E)\) and \({\sigma }_{x,y}^{{SF}}({\boldsymbol{Q}},E)\) at E = 0 ± 0.02 meV using Ef = 3.23 meV along the [0, 0, l] direction. The vertical error bars represent statistical errors of 1 standard deviation.

Extended Data Fig. 9 Summary of heat capacity measurements.

a Top view of the sample holder for specific heat measurements, showing the silver sample stage suspended by NbTi wires from the silver frame that was directly screwed to the mixing chamber of the cryostat, as well as the gold wire serving as thermal link to the bath. b Bottom view of the sample holder showing the thermometer (left) and heater (right) chips that are glued to the sample stage with GE varnish. c Single-crystalline sample of Ce2Zr2O7 used for specific heat measurements. d Comparison of the specific heat data of Ce2Zr2O7 obtained in this work with published results39,42,67. The grey line is a guide to the eyes. e Same data as in a, except for the data measured on powder down to only 0.4 K67, rescaled to the average of the data points of Smith et al.42 and Gao et al.39 at 100 mK, which is deemed to be the most precise estimate of the absolute magnitude of the specific heat of Ce2Zr2O7 at this temperature. The grey line is a cubic-in-temperature fit to data below 50 mK. f Magnetic specific heat data on Ce2Zr2O7 as a function of temperature on double-logarithmic scales, compared to power-law fits, \(A{T}^{\alpha }\), with fixed powers \(\alpha\) of 3 (grey), and 2.5 (red) and 3.5 (blue) for comparison, illustrating that \(\alpha =3\) describes the data best. A minimization procedure with open α yields the \({\chi }_{v}^{2}(\alpha)\) dependence shown in the inset, confirming that, within the error bars, α = 3 is the best description of the data. g Arrhenius plot of the magnetic specific heat together with a linear fit to the data (grey line), showing that the low-temperature specific heat of Ce2Zr2O7 could also be accounted for by a thermally activated behavior, with a gap of 0.1 K (inset). Both fits yield similar minimal \({\chi }_{v}^{2}\), preventing discrimination between the two on purely statistical grounds. h Magnetic entropy release as function of temperature obtained by integrating our Cmag/T data (full black symbols) together with previously published Cmag/T data39 (open symbols), both scaled as done in e. Within the error of the measurements, the full entropy of \(\bar{R}\mathrm{ln}2\) is reached at 10 K (grey shaded area). The vertical error bars in d-g are estimated to be at a maximum 10 % at 100 mK and up to 20 % at the lowest temperatures. Horizontal error bars represent maximal errors in the temperatures of the sample, which amount to 10 % on average (depending on whether a larger or smaller heater power was used).

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Gao, B., Desrochers, F., Tam, D.W. et al. Neutron scattering and thermodynamic evidence for emergent photons and fractionalization in a pyrochlore spin ice. Nat. Phys. 21, 1203–1210 (2025). https://doi.org/10.1038/s41567-025-02922-9

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