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A quantum liquid of magnetic octupoles on the pyrochlore lattice

Abstract

Spin liquids are highly correlated yet disordered states formed by the entanglement of magnetic dipoles1. Theories define such states using gauge fields and deconfined quasiparticle excitations that emerge from a local constraint governing the ground state of a frustrated magnet. For example, the ‘2-in–2-out’ ice rule for dipole moments on a tetrahedron can lead to a quantum spin ice2,3,4 in rare-earth pyrochlores. However, f-electron ions often carry multipole degrees of freedom of higher rank than dipoles, leading to intriguing behaviours and ‘hidden’ orders5,6. Here we show that the correlated ground state of a Ce3+-based pyrochlore, Ce2Sn2O7, is a quantum liquid of magnetic octupoles. Our neutron scattering results are consistent with a fluid-like state where degrees of freedom have a more complex magnetization density than that of magnetic dipoles. The nature and strength of the octupole–octupole couplings, together with the existence of a continuum of excitations attributed to spinons, provides further evidence for a quantum ice of octupoles governed by a ‘2-plus–2-minus’ rule7,8. Our work identifies Ce2Sn2O7 as a unique example of frustrated multipoles forming a ‘hidden’ topological order, thus generalizing observations on quantum spin liquids to multipolar phases that can support novel types of emergent fields and excitations.

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Fig. 1: Dipole–octupole degrees of freedom and their thermal evolution under the effect of magnetic interactions.
Fig. 2: Energy-integrated powder-averaged neutron diffuse scattering data and elastic cross-section calculations for octupole ice and spin ice correlations.
Fig. 3: Bulk properties and their modelling using a ‘dipole–octupole’ Hamiltonian.
Fig. 4: Low-energy neutron spectroscopy data evidencing a continuum of excitations attributed to spinons of an octupolar quantum spin ice.

Data availability

Source data for Figs. 1–4 are provided with the paper. The data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request (R.S. for all of the experimental data, and S.P. for the mean-field fits of the bulk properties and Monte Carlo simulations). The datasets for the inelastic neutron scattering experiment on MAPS and MERLIN are available from the ISIS Neutron and Muon Source Data Catalogue49,50. The datasets for the polarized neutron scattering experiment on D7 (see Supplementary Information), the thermal-neutron inelastic measurements on IN4 (see Supplementary Information) and the diffraction experiment on D20 are available from the Institute Laue–Langevin data portal51,52,53,54.

Code availability

The code and numerical methods described in the Supplementary Information are available upon request to the authors.

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Acknowledgements

This work is based on experiments performed at the Swiss spallation neutron source SINQ, Paul Scherrer Institute, Switzerland. Experiments at the ISIS Neutron and Muon Source were supported by a beamtime allocation from the Science and Technology Facilities Council. Additional neutron scattering experiments were also carried out at the Institut Laue–Langevin, France. We thank M. Kenzelmann, O. Zaharko, Y.-P. Huang, M. Müller and C. Mudry for useful discussions. We thank C. Paulsen for use of his magnetometers, P. Lachkar for help with the PPMS, B. Fåk for assisting with IN4c, E. Pomjakushina and D. Gawryluk for providing help and access to the solid-state chemistry laboratory and M. Zolliker and M. Bartkowiak for dedicated work running the dilution refrigerators at SINQ. R.S. thanks N. Shannon for fruitful exchanges and great hospitality at OIST, as well as H. Yan for enlightening conversations. We acknowledge funding from the Swiss National Science Foundation (grant no. 200021_179150 and fellowship no. P2EZP2 178542).

Author information

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Authors

Contributions

The project and experiments were designed by R.S. Sample preparation and characterization were performed by R.S. and V. Porée. Neutron scattering experiments were carried out by R.S., E.L., V. Porée and T.F., with V. Pomjakushin, R.A.E., T.G.P., J.O., A.W., C.R., T.C.H., D.A.K., G.J.N. and L.K. as local contacts. Measurements of bulk properties were performed by E.L. Experimental data were analysed by R.S., N.G., E.L., V. Porée, V. Pomjakushin, S.P. and T.F. Calculations were performed by N.G. and S.P. Graphical representations of the magnetic charge densities shown in Figs. 1f–h and 2d,f were produced by N.G., while all the other elements of the main figures were produced by R.S. and V. Porée based on (1) neutron scattering data taken by R.S. and T.F. (Figs. 1a–c), R.S. and V. Porée (Figs. 2a–c and 3b) and V. Porée and E.L. (Fig. 4), (2) macroscopic measurements performed by E.L. (Figs. 1e and 3a–c) and (3) calculations performed by S.P. (Fig. 2c,e,g). The paper was written by R.S., with feedback from all authors, especially N.G., E.L., S.P. and T.F.

Corresponding authors

Correspondence to Romain Sibille or Sylvain Petit.

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The authors declare no competing interests.

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Peer review information Nature Physics thanks Gang Chen, Dmytro Inosov and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 Photograph of the sample.

The 28-grams powder sample of Ce2Sn2O7 used in this work is shown here, inside a silica tube for the purpose of the photography only.

Extended Data Fig. 2 Constant-wavelength high-resolution neutron powder diffraction data.

Rietveld refinements of data acquired at 1.5 K (a) and 300 K (b). Data (red points), calculated pattern (black line), Bragg positions (green markers) and difference pattern (blue line) are shown. The results of the refinements are presented in details in the Supplementary Information and indicate a phase-pure, stoichiometric sample of Ce2Sn2O7 free from any defects detectable using this technique.

Extended Data Fig. 3 Time-Of-Flight neutron powder diffraction data and pair distribution function.

Panel a shows the Rietveld refinement of ambient temperature Time-of-Flight neutron powder diffraction data from a medium-resolution detector bank centred around 2θ = 63.6. Data (red points), calculated pattern (black line), Bragg positions (green markers) and difference pattern (blue line) are shown. Panel b displays the experimental Pair Distribution Function (PDF) of Ce2Sn2O7 (blue open circles) obtained from all detector banks and after usual corrections and background subtraction (see Supplementary Information). The red line is a fit of the PDF using a perfect pyrochlore structure, and the green line shows the difference between the experiment and this calculation. The PDF provides a powerful tool to further check and confirm the structure of the sample. In particular, it is highly sensitive to the environment around Ce3+, and the analysis fails to provide evidence for any significant deviations of local structure relative to that in a perfect pyrochlore structure.

Extended Data Fig. 4 Thermogravimetric analysis of the sample.

The change of mass of approximately 50 mg of the Ce2Sn2O7 powder sample used in this work was recorded upon heating under an oxygen flow up to 1000 Celsius. This method provides a direct and precise measurement of the oxygen stoichiometry according to the reaction Ce2Sn2O7+δ + (1 − δ)/2 O2 → 2 CeO2 + 2 SnO2. The theoretical mass gain for this reaction is 2.54% (δ = 0). Experimentally, we have measured a mass gain of 2.54% ± 0.02, which translates into δ = 0.00 ± 0.01. The residue of the sample after the thermogravimetric analysis was analysed by X-ray powder diffraction, corroborating the above chemical reaction.

Supplementary information

Supplementary Information

Supplementary Figs. 1–17, Tables 1–6 and refs. 1–19.

Source data

Source Data Fig. 1

Inelastic neutron scattering cuts, experimental effective magnetic moment and crystal-electric field fit for the high-temperature part.

Source Data Fig. 2

Neutron scattering data and powder-averaged model calculations.

Source Data Fig. 3

Macroscopic measurements and model calculations and integrated octupolar scattering intensity.

Source Data Fig. 4

Low-energy inelastic neutron scattering data and derived imaginary susceptibility.

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Sibille, R., Gauthier, N., Lhotel, E. et al. A quantum liquid of magnetic octupoles on the pyrochlore lattice. Nat. Phys. 16, 546–552 (2020). https://doi.org/10.1038/s41567-020-0827-7

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