Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

A quantum liquid of magnetic octupoles on the pyrochlore lattice


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.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

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.


  1. 1.

    Savary, L. & Balents, L. Quantum spin liquids: a review. Rep. Prog. Phys. 80, 016502 (2016).

    ADS  Article  Google Scholar 

  2. 2.

    Hermele, M., Fisher, M. P. A. & Balents, L. Pyrochlore photons: The U(1) spin liquid in a S = ½ three-dimensional frustrated magnet. Phys. Rev. B 69, 064404 (2004).

    ADS  Article  Google Scholar 

  3. 3.

    Benton, O., Sikora, O. & Shannon, N. Seeing the light: experimental signatures of emergent electromagnetism in a quantum spin ice. Phys. Rev. B 86, 075154 (2012).

    ADS  Article  Google Scholar 

  4. 4.

    Gingras, M. J. P. & McClarty, P. A. Quantum spin ice: a search for gapless quantum spin liquids in pyrochlore magnets. Rep. Prog. Phys. 77, 056501 (2014).

    ADS  Article  Google Scholar 

  5. 5.

    Kuramoto, Y., Kusunose, H. & Kiss, A. Multipole orders and fluctuations in strongly correlated electron systems. J. Phys. Soc. Jpn 78, 072001 (2009).

    ADS  Article  Google Scholar 

  6. 6.

    Santini, P. et al. Multipolar interactions in f-electron systems: the paradigm of actinide dioxides. Rev. Mod. Phys. 81, 807–863 (2009).

    ADS  Article  Google Scholar 

  7. 7.

    Huang, Y.-P., Chen, G. & Hermele, M. Quantum spin ices and topological phases from dipolar–octupolar doublets on the pyrochlore lattice. Phys. Rev. Lett. 112, 167203 (2014).

    ADS  Article  Google Scholar 

  8. 8.

    Li, Y.-D. & Chen, G. Symmetry enriched U(1) topological orders for dipole–octupole doublets on a pyrochlore lattice. Phys. Rev. B 95, 041106 (2017).

    ADS  Article  Google Scholar 

  9. 9.

    Ikeda, H. et al. Emergent rank-5 nematic order in URu2Si2. Nat. Phys. 8, 528–533 (2012).

    Article  Google Scholar 

  10. 10.

    Paddison, J. A. M. et al. Hidden order in spin liquid Gd3Ga5O12. Science 350, 179–181 (2015).

    ADS  Article  Google Scholar 

  11. 11.

    Lovesey, S. W., Khalyavin, D. D. & Staub, U. Ferro-type order of magneto-electric quadrupoles as an order-parameter for the pseudo-gap phase of a cuprate superconductor. J. Phys. Condens. Matter 27, 292201 (2015).

    Article  Google Scholar 

  12. 12.

    Fechner, M., Fierz, M. J. A., Thöle, F., Staub, U. & Spaldin, N. A. Quasistatic magnetoelectric multipoles as order parameter for pseudogap phase in cuprate superconductors. Phys. Rev. B 93, 174419 (2016).

    ADS  Article  Google Scholar 

  13. 13.

    Zhao, L. et al. Evidence of an odd-parity hidden order in a spin–orbit coupled correlated iridate. Nat. Phys. 12, 32–36 (2016).

    Article  Google Scholar 

  14. 14.

    Lovesey, S. W. & Khalyavin, D. D. Dirac multipoles in diffraction by the layered room-temperature antiferromagnets BaMn2P2 and BaMn2As2. Phys. Rev. B 98, 054434 (2018).

    ADS  Article  Google Scholar 

  15. 15.

    Hayami, S., Yanagi, Y., Kusunose, H. & Motome, Y. Electric toroidal quadrupoles in the spin–orbit-coupled metal Cd2Re2O7. Phys. Rev. Lett. 122, 147602 (2019).

    ADS  Article  Google Scholar 

  16. 16.

    Spaldin, N. A., Fiebig, M. & Mostovoy, M. The toroidal moment in condensed-matter physics and its relation to the magnetoelectric effect. J. Phys. Condens. Matter 20, 434203 (2008).

    ADS  Article  Google Scholar 

  17. 17.

    Di Matteo, S. & Norman, M. R. Orbital currents, anapoles and magnetic quadrupoles in CuO. Phys. Rev. B 85, 235143 (2012).

    ADS  Article  Google Scholar 

  18. 18.

    Wen, X.-G. Quantum orders and symmetric spin liquids. Phys. Rev. B 65, 165113 (2002).

    ADS  Article  Google Scholar 

  19. 19.

    Knolle, J. & Moessner, R. A field guide to spin liquids. Annu. Rev. Condens. Matter Phys. 10, 451–472 (2019).

    ADS  Article  Google Scholar 

  20. 20.

    Han, T.-H. et al. Fractionalized excitations in the spin-liquid state of a kagome-lattice antiferromagnet. Nature 492, 406–410 (2012).

    ADS  Article  Google Scholar 

  21. 21.

    Plumb, K. W. et al. Continuum of quantum fluctuations in a three-dimensional S = 1 Heisenberg magnet. Nat. Phys. 15, 54–59 (2019).

    Article  Google Scholar 

  22. 22.

    Kitagawa, K. et al. A spin–orbital-entangled quantum liquid on a honeycomb lattice. Nature 554, 341–345 (2018).

    ADS  Article  Google Scholar 

  23. 23.

    Paddison, J. A. M. et al. Continuous excitations of the triangular-lattice quantum spin liquid YbMgGaO4. Nat. Phys. 13, 117–122 (2017).

    Article  Google Scholar 

  24. 24.

    Rau, J. G. & Gingras, M. J. P. Frustrated quantum rare-earth pyrochlores. Annu. Rev. Condens. Matter Phys. 10, 357–386 (2019).

    ADS  Article  Google Scholar 

  25. 25.

    Castelnovo, C., Moessner, R. & Sondhi, S. L. Spin ice, fractionalization and topological order. Annu. Rev. Condens. Matter Phys. 3, 35–55 (2012).

    Article  Google Scholar 

  26. 26.

    Rau, J. G. & Gingras, M. J. P. Magnitude of quantum effects in classical spin ices. Phys. Rev. B 92, 144417 (2015).

    ADS  Article  Google Scholar 

  27. 27.

    Onoda, S. & Tanaka, Y. Quantum melting of spin ice: emergent cooperative quadrupole and chirality. Phys. Rev. Lett. 105, 047201 (2010).

    ADS  Article  Google Scholar 

  28. 28.

    Sibille, R. et al. Candidate quantum spin liquid in the Ce3+ pyrochlore stannate Ce2Sn2O7. Phys. Rev. Lett. 115, 097202 (2015).

    ADS  Article  Google Scholar 

  29. 29.

    Curnoe, S. H. Structural distortion and the spin liquid state in Tb2Ti2O7. Phys. Rev. B 78, 094418 (2008).

    ADS  Article  Google Scholar 

  30. 30.

    Lee, S., Onoda, S. & Balents, L. Generic quantum spin ice. Phys. Rev. B 86, 104412 (2012).

    ADS  Article  Google Scholar 

  31. 31.

    Benton, O., Jaubert, L. D. C., Singh, R. R. P., Oitmaa, J. & Shannon, N. Quantum spin ice with frustrated transverse exchange: from a π-flux phase to a nematic quantum spin liquid. Phys. Rev. Lett. 121, 067201 (2018).

    ADS  Article  Google Scholar 

  32. 32.

    Chen, G. Spectral periodicity of the spinon continuum in quantum spin ice. Phys. Rev. B 96, 085136 (2017).

    ADS  Article  Google Scholar 

  33. 33.

    Taillefumier, M., Benton, O., Yan, H., Jaubert, L. D. C. & Shannon, N. Competing spin liquids and hidden spin-nematic order in spin ice with frustrated transverse exchange. Phys. Rev. X 7, 041057 (2017).

    Google Scholar 

  34. 34.

    Yao, X.-P., Li, Y.-D. & Chen, G. Pyrochlore U(1) spin liquid of mixed-symmetry enrichments in magnetic fields. Phys. Rev. Res. 2, 013334 (2020).

    Article  Google Scholar 

  35. 35.

    Huang, C.-J., Deng, Y., Wan, Y. & Meng, Z.-Y. Dynamics of topological excitations in a model quantum spin ice. Phys. Rev. Lett. 120, 167202 (2018).

    ADS  Article  Google Scholar 

  36. 36.

    Udagawa, M. & Moessner, R. Spectrum of itinerant fractional excitations in quantum spin ice. Phys. Rev. Lett. 122, 117201 (2019).

    ADS  Article  Google Scholar 

  37. 37.

    Morampudi, S. D., Wilczek, F. & Laumann, C. R. Spectroscopy of spinons in Coulomb quantum spin liquids. Phys. Rev. Lett. 124, 097204 (2019).

    ADS  Article  Google Scholar 

  38. 38.

    Gaudet, J. et al. Quantum spin ice dynamics in the dipole–octupole pyrochlore magnet Ce2Zr2O7. Phys. Rev. Lett. 122, 187201 (2019).

    ADS  Article  Google Scholar 

  39. 39.

    Gao, B. et al. Experimental signatures of a three-dimensional quantum spin liquid in effective spin-1/2 Ce2Zr2O7 pyrochlore. Nat. Phys. 15, 1052–1057 (2019).

    Article  Google Scholar 

  40. 40.

    Tolla, B. et al. Oxygen exchange properties in the new pyrochlore solid solution Ce2Sn2O7–Ce2Sn2O8. CR Acad. Sci. 2, 139–146 (1999).

    Google Scholar 

  41. 41.

    Ewings, R. A. et al. Horace: upgrade to the MAPS neutron time-of-flight chopper spectrometer. Rev. Sci. Instrum. 90, 035110 (2019).

    ADS  Article  Google Scholar 

  42. 42.

    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).

    ADS  Article  Google Scholar 

  43. 43.

    Princep, A. J., Prabhakaran, D., Boothroyd, A. T. & Adroja, D. T. Crystal-field states of Pr3+ in the candidate quantum spin ice Pr2Sn2O7. Phys. Rev. B 88, 104421 (2013).

    ADS  Article  Google Scholar 

  44. 44.

    Boothroyd, A. T. SPECTRE, a program for calculating spectroscopic properties of rare earth ions in crystals (1990–2018).

  45. 45.

    Fischer, P. et al. High-resolution powder diffractometer HRPT for thermal neutrons at SINQ. Phys. B 146, 276–278 (2000).

    Google Scholar 

  46. 46.

    Hansen, T. C., Henry, P. H., Fischer, H. E., Torregrossa, J. & Convert, P. The D20 instrument at the ILL: a versatile high-intensity two-axis neutron diffractometer. Meas. Sci. Technol. 19, 034001 (2008).

    ADS  Article  Google Scholar 

  47. 47.

    Ollivier, J. & Mutka, H. IN5 cold neutron time-of-flight spectrometer, prepared to tackle single crystal spectroscopy. J. Phys. Soc. Jpn 80, SB003 (2011).

    Article  Google Scholar 

  48. 48.

    Kusunose, H. Description of multipole in f-electron systems. J. Phys. Soc. Jpn 77, 64710 (2008).

    Article  Google Scholar 

  49. 49.

    Sibille R. et al. Crystal electric field excitations of pyrochlore quantum antiferromagnet Ce2Sn2O7 (STFC ISIS Neutron and Muon Source, 2015);

  50. 50.

    Sibille R. et al. Hidden correlations in Ce2Sn2O7 (STFC ISIS Neutron and Muon Source, 2019);

  51. 51.

    Sibille R. et al. Investigation of the exotic low temperature groundstate of the S eff = 1/2 pyrochlore antiferromagnet Ce2Sn2O7 (Institut Laue–Langevin, 2015);

  52. 52.

    Sibille R. et al. Investigation of the exotic low temperature groundstate of the S eff = 1/2 pyrochlore antiferromagnet Ce2Sn2O7—continuation (Institut Laue–Langevin, 2016);

  53. 53.

    Sibille R. et al. Crystal electronic field excitations of pyrochlore antiferromagnet Ce2Sn2O7 (Institut Laue–Langevin, 2015);

  54. 54.

    Sibille R. et al. Hidden correlations in a quantum liquid of magnetic octupoles (Institut Laue–Langevin, 2019);

Download references


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




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.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

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.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Sibille, R., Gauthier, N., Lhotel, E. et al. A quantum liquid of magnetic octupoles on the pyrochlore lattice. Nat. Phys. 16, 546–552 (2020).

Download citation

Further reading


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing