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Spin–orbital liquid state and liquid–gas metamagnetic transition on a pyrochlore lattice

Abstract

Crystal structures with degenerate electronic orbitals are unstable towards lattice distortions that lift the degeneracy. Although these Jahn–Teller distortions have profound effects on magnetism, they are typically unaffected by the onset of magnetic ordering because of a separation in energy scales. Here we show the contrary case in Pr2Zr2O7, where orbital degeneracy remains down to the millikelvin range due to an interplay between spins and orbitals. Pr2Zr2O7 is a multipolar spin ice with strongly localized 4f electrons in an even-number configuration, giving rise to a non-Kramers doublet that carries transverse quadrupolar and longitudinal dipolar moments. Our study of ultrapure single crystals of Pr2Zr2O7 finds comprehensive evidence for enhanced spin–orbital quantum dynamics of the non-Kramers doublet. This dynamical Jahn–Teller effect is encapsulated by the liquid–gas metamagnetic transition that is characteristic of spin ice being accompanied by strong lattice softening. This behaviour suggests that a spin–orbital liquid state forms on the pyrochlore lattice at low temperatures and low magnetic fields.

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Fig. 1: Spin–orbital interlocking in the non-Kramers doublet of the multipolar spin ice Pr2Zr2O7 and magnetic-field tuning of its ground state with spin–orbital quantum dynamics.
Fig. 2: First-order liquid–gas metamagnetic transition and strong lattice anomalies in Pr2Zr2O7.
Fig. 3: Low-lying CEP of the liquid–gas metamagnetic transition in Pr2Zr2O7.
Fig. 4: Evidence for magnetodistortive dynamics: dielectric and length-change anomalies and strong lattice softening in Pr2Zr2O7.

Data availability

Source data are provided with this paper. All other data that support the plots within this paper are available from the corresponding authors upon reasonable request.

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Acknowledgements

We thank Y. Motome, Y. Kato, T. A. Bojeson, S. Nakamura, N. Kawashima, A. A. Nugroho, H. Kusunose and S. Onoda for helpful discussion. This work is partially supported by JST-CREST (JPMJCR18T3), JST-PRESTO (JPMJPR15N5) and by Grants-in-Aid for Scientific Research (JP19H00650 and 20K03829) from the Japan Society for the Promotion of Science (JSPS). Institute for Quantum Matter, an Energy Frontier Research Center, was funded by the Department of Energy, Office of Science, Basic Energy Sciences, under award no. DE-SC0019331. The use of the facilities of the Materials Design and Characterization Laboratory at the ISSP, The University of Tokyo, is acknowledged. S.B. acknowledges funding from a Max Planck Partner group Grant at ICTS; hospitality of visitors program at the Max Planck Institute for the Physics of Complex Systems, Dresden; and ISSP, The University of Tokyo; Science and Engineering Research Board—Department of Science and Technology (India) for funding through project grant no. ECR/2017/000504 and the support of the Department of Atomic Energy, Government of India, under project nos. 12-R&D-TFR-5.10-1100 and RTI4001. We acknowledge support of the HLD at HZDR, member of the European Magnetic Field Laboratory (EMFL). This work was in part supported by the Deutsche Forschungsgemeinschaft under grants SFB 1143 (Project ID 247310070) and the Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter—ct.qmat (EXC 2147, Project ID 390858490).

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Authors and Affiliations

Authors

Contributions

S.N. conceived the project and planned the experiments. S.N., J.Wosnitza. and C.B. supervised the experiments. K.K. and S.N. grew the single crystals. N.T. and A.S. performed the thermal expansion/magnetostriction measurements and N.T., M.F. and A.S. analysed the data. Y.G. and S.Z. performed the ultrasound measurements and analysed the data. K.K. and Y.M. performed the dielectric measurements and K.K. analysed the data. Y.M., K.K. and J.Wen. performed the specific heat measurements and N.T. and Y.M. analysed the data. T.S., Y.S., K.K. and N.T. performed the magnetization measurements, and K.K. and N.T. analysed the data. K.S. and H.S. collected and analysed the synchrotron X-ray diffraction data. H.T. and M.T. carried out the NQR measurements and analysed the data. S.B. and R.M. wrote the theoretical notes in the Supplementary Information. S.N., N.T., S.B., M.F., R.M., Y.G. and K.K. wrote the main text and prepared the figures. N.T., M.F., K.K., S.N., Y.G. and H.M. wrote the Methods section. All the authors discussed the results and commented on the manuscript.

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Correspondence to Satoru Nakatsuji.

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Extended data

Extended Data Fig. 1 Crystal structure analysis by single crystal synchrotron X-ray diffraction of Pr2Zr2O7.

Contour plot of the reliability factor R1 as a function of x and y in (Pr1−xZrx)2(Zr1−yPry)2O7 is shown.

Source data

Extended Data Fig. 2 Sample quality dependence of physical properties of Pr2Zr2O7.

a, Magnetic field (B) dependence of the magnetization measured along the crystallographic [111] axis and its B-derivative for three samples (sample #1, sample #2, and sample #3) taken from different single crystalline rods (rod 1, rod 2, and rod 3). Measurement temperatures are also indicated. b, NQR spectra and Lorentzian fits for the three samples measured at 4.2 K. Sample #1 and sample #2 that exhibit the metamagnetism produce sharper NQR spectra (smaller FWHM values as shown in each panel) than sample #3 that does not show any metamagnetism. c, Temperature (T) dependence of the magnetic specific heat CM for three samples of different quality. CM is obtained by subtracting CL, CCEF and CN from CP (See Extended Data Fig. 6). The data of sample #3 is taken from Ref. 26. The inset shows CM with respect to TThump(K) so that the hump is centred at 0 K. Thump is 1.4 K, 1.7 K, 2.6 K for sample #1, #2, #3, respectively. It is easily seen that the best sample exhibits the narrowest hump structure.

Source data

Extended Data Fig. 3 Strongly anisotropic magnetization curve of Pr2Zr2O7.

a, Magnetization (M) measured under field (B) along the crystallographic [100], [110], and [111] axes at T = 65(5) mK. Dashed lines denote the saturation M along each axis estimated using the effective moment (μeff ≈ 2.5 μB) deduced from the Curie-Weiss fit. The magnetization curve for Dy2Ti2O7 at 50 mK, adapted from Ref. 40, is also shown. M and B for Dy2Ti2O7 are scaled to those for Pr2Zr2O7 with respect to the value of μeff( 2.5 μB for Pr3+ and 10 μB for Dy3+) and the metamagnetic field Bc ( 2.2 T for Pr2Zr2O7 and 0.9 T for Dy2Ti2O7), respectively. b, B dependence of the field derivative of M in (a).

Source data

Extended Data Fig. 4 Van Vleck and other contributions to magnetization M of Pr2Zr2O7.

M (left axis) and its susceptibility dM /dB (right axis) of Pr2Zr2O7 obtained at T = 60 mK under B [111]. The Van Vleck term (red dashed line), calculated from the CEF scheme obtained from the neutron scattering results in Ref. 26, is subtracted from the experimental M (left axis, dark blue circles). The result is shown in light blue circles. Magnetic susceptibility dM /dB (right axis) corresponds to the subtracted data.

Source data

Extended Data Fig. 5 Splitting of non-Kramers ground doublet due to the linear coupling between transverse quadrupolar component Sx and elastic strain εx in magnetic field Hz.

a, Quadrupolar-strain coupling lifts the degeneracy of the ground state doublet by opening a gap of ∆. In this case, |Sx+〉state is energetically more favorable and physically selected. b, Evolution of the non-Kramers doublet with increasing elastic strain εx by distorting D3d CEF under a fixed strength of magnetic field Hz. |Sx+〉state is stretched along the local x direction and hence, energetically more favorable in this case. Each three wavefunctions of |Sx+〉(top) and |Sx〉(bottom) states correspond to the doublet gap of ∆ 0 K (Left), 1.3 K (middle), 5.4 K (Right). It is important to note that in real crystals of Pr2Zr2O7, there is no static strain or internal magnetic field. These pictures are only for an illustrative purpose.

Extended Data Fig. 6 Temperature (T) dependence of the specific heat for three Pr2Zr2O7 samples of different quality.

a, b, c, T dependence of the magnetic and nuclear part of the specific heat CMN (open circles) for three Pr2Zr2O7 samples of different quality after subtracting lattice (dull yellow line) and CEF (dark blue line) contributions from the measured specific heat Cp (open rhombus) in zero field (see Methods for details). CMN is defined as CMNCM + CN = CPCLCCEF. CM is the magnetic specific heat (solid circle). CN is the Schottky-like specific heat that fits CMN below T ≈ 0.2 K. A refined estimate CM = CMNCN was obtained by subtracting a scaled nuclear Schottky-like anomaly (cyan line). d, e, f, T dependence of the corresponding entropies obtained through integration from absolute zero. The fitting results CN is used for the experimentally inaccessible temperature range in order to perform the integration. Open circles denote the magnetic and nuclear entropy SMN, while the solid circles denote the magnetic entropy SM. The three dashed black lines denote ∆S for a spin ice system, a spin ice system with a nuclear contribution of I = 5/2, and a two-level electronic system with a nuclear contribution of I = 5/2, respectively.

Source data

Extended Data Fig. 7 Temperature (T) dependence of the magnetization M and the thermal expansion ∆L/L0 at high fields for Pr2Zr2O7.

a, b, T dependence of (a) M and (b) longitudinal ∆L/L0 at B = 3 and 5 T, where B is applied along the [111] direction. ∆L/L0 has a constant shift along the y axis for clarity.

Source data

Extended Data Table 1

Crystallographic parameters for Pr2Zr2O7 at 20 K.

Extended Data Table 2

Atomic positions for Pr2Zr2O7 at 20 K.

Extended Data Table 3

Sample dependence of Curie-Weiss temperature Tθ and metamagnetic behaviour in Pr2Zr2O7.

Supplementary information

Supplementary Information

Supplementary Sections 1–7, Fig. 1, Table 1 and references.

Source data

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Tang, N., Gritsenko, Y., Kimura, K. et al. Spin–orbital liquid state and liquid–gas metamagnetic transition on a pyrochlore lattice. Nat. Phys. 19, 92–98 (2023). https://doi.org/10.1038/s41567-022-01816-4

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