In a quantum spin liquid, the magnetic moments of the constituent electron spins evade classical long-range order to form an exotic state that is quantum entangled and coherent over macroscopic length scales1,2. Such phases offer promising perspectives for device applications in quantum information technologies, and their study can reveal new physics in quantum matter. Quantum spin ice is an appealing proposal of one such state, in which the fundamental ground state properties and excitations are described by an emergent U(1) lattice gauge theory3,4,5,6,7. This quantum-coherent regime has quasiparticles that are predicted to behave like magnetic and electric monopoles, along with a gauge boson playing the role of an artificial photon. However, this emergent lattice quantum electrodynamics has proved elusive in experiments. Here we report neutron scattering measurements of the rare-earth pyrochlore magnet Pr2Hf2O7 that provide evidence for a quantum spin ice ground state. We find a quasi-elastic structure factor with pinch points—a signature of a classical spin ice—that are partially suppressed, as expected in the quantum-coherent regime of the lattice field theory at finite temperature. Our result allows an estimate for the speed of light associated with magnetic photon excitations. We also reveal a continuum of inelastic spin excitations, which resemble predictions for the fractionalized, topological excitations of a quantum spin ice. Taken together, these two signatures suggest that the low-energy physics of Pr2Hf2O7 can be described by emergent quantum electrodynamics. If confirmed, the observation of a quantum spin ice ground state would constitute a concrete example of a three-dimensional quantum spin liquid—a topical state of matter that has so far mostly been explored in lower dimensionalities.
Access optionsAccess options
Subscribe to Journal
Get full journal access for 1 year
only $15.58 per issue
All prices are NET prices.
VAT will be added later in the checkout.
Rent or Buy article
Get time limited or full article access on ReadCube.
All prices are NET prices.
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Balents, L. Spin liquids in frustrated magnets. Nature 464, 199–208 (2010).
Savary, L. & Balents, L. Quantum spin liquids: a review. Rep. Prog. Phys. 80, 016502 (2016).
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).
Banerjee, A., Isakov, S. V., Damle, K. & Kim, Y.-B. Unusual liquid state of hard-core bosons on the pyrochlore lattice. Phys. Rev. Lett. 100, 047208 (2008).
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).
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).
Shannon, N., Sikora, O., Pollmann, F., Penc, K. & Fulde, P. Quantum ice: a quantum Monte Carlo study. Phys. Rev. Lett. 108, 067204 (2012).
Anderson, P. W. Resonating valence bonds: a new kind of insulator? Mat. Res. Bull. 8, 153–160 (1973).
Wen, X.-G. Quantum orders and symmetric spin liquids. Phys. Rev. B 65, 165113 (2002).
Moessner, R. & Sondhi, S. L. Resonating valence bond liquid physics on the triangular lattice. Prog. Theor. Phys. 145 (Suppl.), 37–42 (2002).
Tennant, D. A., Perring, T. G., Cowley, R. A. & Nagler, S. E. Unbound spinons in the spin-1/2 antiferromagnetic chain KCuF3. Phys. Rev. Lett. 70, 4003–4006 (1993).
Han, T.-H. et al. Fractionalized excitations in the spin-liquid state of a Kagome-lattice antiferromagnet. Nature 492, 406–410 (2012).
Banerjee, A. et al. Proximate Kitaev quantum spin liquid behaviour in a honeycomb magnet. Nat. Mater. 15, 733–740 (2016).
Shen, Y. et al. Evidence for a spinon Fermi surface in a triangular-lattice quantum-spin-liquid candidate. Nature 540, 559–562 (2016).
Castelnovo, C., Moessner, R. & Sondhi, S. L. Spin ice, fractionalization, and topological order. Annu. Rev. Condens. Matter Phys. 3, 35–55 (2012).
Bramwell, S. T. et al. Spin correlations in Ho2Ti2O7: a dipolar spin ice system. Phys. Rev. Lett. 87, 047205 (2001).
Fennell, T. et al. Magnetic Coulomb phase in the spin ice Ho2Ti2O7. Science 326, 415–417 (2009).
Henley, C. L. The “Coulomb phase” in frustrated systems. Annu. Rev. Condens. Matter Phys. 1, 179–210 (2010).
Castelnovo, C., Moessner, R. & Sondhi, S. L. Magnetic monopoles in spin ice. Nature 451, 42–45 (2008).
Curnoe, S. H. Structural distortion and the spin liquid state in Tb2Ti2O7. Phys. Rev. B 78, 094418 (2008).
Ross, K. A., Savary, L., Gaulin, B. D. & Balents, L. Quantum excitations in quantum spin ice. Phys. Rev. X 1, 021002 (2011).
Onoda, S. & Tanaka, Y. Quantum melting of spin ice: emergent cooperative quadrupole and chirality. Phys. Rev. Lett. 105, 047201 (2010).
Kimura, K. et al. Quantum fluctuations in spin-ice-like Pr2Zr2O7. Nat. Commun. 4, 1934 (2013).
Petit, S. et al. Antiferroquadrupolar correlations in the quantum spin ice candidate Pr2Zr2O7. Phys. Rev. B 94, 165153 (2016).
Wen, J.-J. et al. Disordered route to the Coulomb quantum spin liquid: random transverse fields on spin ice in Pr2Zr2O7. Phys. Rev. Lett. 118, 107206 (2017).
Martin, N. et al. Disorder and quantum spin ice. Phys. Rev. X 7, 041028 (2017).
Benton, O. From quantum spin liquid to paramagnetic ground states in disordered non-Kramers pyrochlores. Preprint at http://arXiv.org/abs/1706.09238 (2017).
Sibille, R. et al. Candidate quantum spin ice in the pyrochlore Pr2Hf2O7. Phys. Rev. B 94, 024436 (2016).
Wan, Y., Carrasquilla, J. & Melko, R. G. Spinon walk in quantum spin ice. Phys. Rev. Lett. 116, 167202 (2016).
Hao, Z., Day, A. G. R. & Gingras, M. J. P. Bosonic many-body theory of quantum spin ice. Phys. Rev. B 90, 214430 (2014).
Huang, C.-J., Deng, Y., Wan Y. and Meng Z.-Y. Dynamics of topological excitations in a model quantum spin ice. Preprint at http://arXiv.org/abs/1707.00099 (2017).
Savary, L. & Balents, L. Disorder-induced quantum spin liquid in spin ice pyrochlores. Phys. Rev. Lett. 118, 087203 (2017).
Chen, G. Dirac’s ‘magnetic monopoles’ in pyrochlore ice U(1) spin liquids: spectrum and classification. Phys. Rev. B 96, 195127 (2017).
Petrova, O., Moessner, R. & Sondhi, S. L. Hydrogenic states of monopoles in diluted quantum spin ice. Phys. Rev. B 92, 100401(R) (2015).
Kato, S. & Onoda, S. Numerical evidence of quantum melting of spin ice: quantum-to-classical crossover. Phys. Rev. Lett. 115, 077202 (2015).
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).
Ciomaga Hatnean, M. et al. Single crystal growth, structure and magnetic properties of Pr2Hf2O7 pyrochlore. J. Phys. Condens. Matter 29, 075902 (2017).
Ollivier, J. & Mutka, H. IN5 cold neutron time-of-flight spectrometer, prepared to tackle single crystal spectroscopy. J. Phys. Soc. Jpn 80, SB003 (2011).
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. 834, 132–142 (2016).
Winn, B. et al. Recent progress on HYSPEC, and its polarization analysis capabilities. EPJ Web Conf. 83, 03017 (2015).
Stewart, J. R. et al. Disordered materials studied using neutron polarization analysis on the multi-detector spectrometer, D7. J. Appl. Cryst. 42, 69–84 (2009).
Isakov, S. V., Gregor, K., Moessner, R. & Sondhi, S. L. Dipolar spin correlations in classical pyrochlore magnets. Phys. Rev. Lett. 93, 167204 (2004).
Henley, C. L. Power-law spin correlations in pyrochlore antiferromagnets. Phys. Rev. B 71, 014424 (2005).
Sibille, R., Fennell, T., Gauthier, N., Kenzelmann, M. & Ollivier, J. Time-of-flight Inelastic Neutron Scattering Investigation of the Quantum Spin Ice and Quantum Kagome Ice Phases in Pr 2 Hf 2 O 7 (Institut Laue-Langevin, 2016); https://doi.org/10.5291/ILL-DATA.4-05-641
We acknowledge the Institut Laue-Langevin (Grenoble, France) for the allocated beamtime. We acknowledge funding from the Swiss National Science Foundation (grant nos 200021_140862; 206021_139082; and 200021_138018). This research used resources at the Spallation Neutron Source, a DOE Office of Science User Facility operated by the Oak Ridge National Laboratory. The work at ORNL was supported by the US DOE, Office of Science, Office of Basic Energy Sciences, under contract number DE-AC05-00OR22725. The work at the University of Warwick was supported by the EPSRC, UK, through grant EP/M028771/1. Additional neutron scattering experiments were carried out at the continuous spallation neutron source SINQ at the Paul Scherrer Institut at Villigen PSI in Switzerland.
Supplementary Figures 1–4, Supplementary References