Abstract
Supersolid, an exotic quantum state of matter that consists of particles forming an incompressible solid structure while simultaneously showing superfluidity of zero viscosity1, is one of the long-standing pursuits in fundamental research2,3. Although the initial report of 4He supersolid turned out to be an artefact4, this intriguing quantum matter has inspired enthusiastic investigations into ultracold quantum gases5,6,7,8. Nevertheless, the realization of supersolidity in condensed matter remains elusive. Here we find evidence for a quantum magnetic analogue of supersolid—the spin supersolid—in the recently synthesized triangular-lattice antiferromagnet Na2BaCo(PO4)2 (ref. 9). Notably, a giant magnetocaloric effect related to the spin supersolidity is observed in the demagnetization cooling process, manifesting itself as two prominent valley-like regimes, with the lowest temperature attaining below 100 mK. Not only is there an experimentally determined series of critical fields but the demagnetization cooling profile also shows excellent agreement with the theoretical simulations with an easy-axis Heisenberg model. Neutron diffractions also successfully locate the proposed spin supersolid phases by revealing the coexistence of three-sublattice spin solid order and interlayer incommensurability indicative of the spin superfluidity. Thus, our results reveal a strong entropic effect of the spin supersolid phase in a frustrated quantum magnet and open up a viable and promising avenue for applications in sub-kelvin refrigeration, especially in the context of persistent concerns about helium shortages10,11.
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Data availability
The datasets generated and analysed during the current study are available from the corresponding authors on reasonable request. Source data are provided with this paper.
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Acknowledgements
We are indebted to D. Yu, L. Zhang, S. Gao, T. Shi and X.-F. Zhang for helpful discussions, E. Ressouche, K. Beauvois and P. Fouilloux for technical supports at ILL, and X. Lu and Y. Li for the help with the orientation of the single-crystal sample using X-ray Laue diffraction. This work was supported by the National Natural Science Foundation of China (grant nos. 12222412, 12074023, 11974036, 12047503, 12074024, 11834014, 52088101 and 12141002), Strategic Priority Research Program and Scientific Instrument Developing Program of the Chinese Academy of Sciences (CAS) (grant nos. XDB28000000, XDB33000000 and ZDKYYQ20210003), the National Key R&D Program of China (grant no. 2022YFA1402200), CAS Project for Young Scientists in Basic Research (grant no. YSBR-057) and the Fundamental Research Funds for the Central Universities in China. We thank the HPC-ITP for the technical support and generous allocation of CPU time. This work was supported by the Synergetic Extreme Condition User Facility and the beamline 1W1A of the Beijing Synchrotron Radiation Facility. The Australian Center for Neutron Scattering and ILL are gratefully acknowledged for providing neutron beam time through proposal nos. P14250 and 5-41-1193.
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W.L., J.X. and G.S. initiated this work. C.Z., B.L. and W.J. prepared the samples. J.X. and P.S. designed and performed the magnetocaloric measurements. J.X. and G.L. conducted the low-temperature specific heat measurements. J.X., Z.C., J.S., H.J., P.S., W.L. and G.S. conducted the magnetocaloric and specific heat data analysis. W.S., K.S., C.-W.W., C.Z. and W.J. performed the neutron scattering experiments. C.Z., Y.G., W.L. and W.J. analysed the neutron data. Y.G., N.X., X.-Y.L., Y.Q., Y.W. and W.L. conducted the microscopic spin model analysis and performed the many-body calculations. W.L., J.X., W.J., Y.G., P.S. and G.S. wrote the manuscript with input from all coauthors. W.J., W.L., P.S. and G.S. supervised the project.
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Extended data figures and tables
Extended Data Fig. 1 Sample structure characterization.
a Powder X-ray diffraction pattern of NBCP measured at room temperature. The red line indicates the calculated diffraction pattern for comparison. b Single-crystal X-ray diffraction pattern recorded for the (0,0,L) planes at room temperature. The insets show the photo of the NBCP single crystals grown from the same batch, which are used for the specific heat, magnetocaloric and neutron measurements, and the rocking-curve scan of the (003) reflection for one representative crystal. The very sharp peak profile with a full width at half maximum (FWHM) of 0.027° indicates the high quality of the crystals. c, d Images and X-ray Laue patterns of the single-crystal samples used for the neutron diffraction experiments at WOMBAT (50 mg, c) and D23 (125 mg, d), respectively, mounted on the Cu plates. Sharp and bright diffraction spots confirm good quality of the crystals. a.u., arbitrary units.
Extended Data Fig. 2 Energy level diagram of the Co2+ 4F state.
Under the effects of octahedral crystal electric field and spin-orbit coupling, the lowest lying ground state is a j = 1/2 Kramers doublet separated from the first excited j = 3/2 level by a gap of about Δ ≈ 70 meV31.
Extended Data Fig. 3 Photos of PPMS-based quasi-adiabatic demagnetization cooling measurement setups.
a Setup for the measurements of the NBCP single-crystal samples and the paramagnetic coolants FAA and CPA, with two gold-plated thermal shields. To enhance thermal insulation, we employed a Vespel straw to support the sliver stage. We connected the field-calibrated RuO2 thermometer using two pairs of twisted manganin wires (25 μm in diameter and approximately 30 cm in length) to minimize heat leakage. Additionally, two thermal shields were used to protect the sample column from thermal radiation and other parasitic heat loads from the PPMS chamber. These shields were connected using PEEK tubes. b shows the setup for a relatively large mass (50 g) GGG sample with a gold-plated thermal shield.
Extended Data Fig. 4 Analysis of the specific heat and magnetic entropy results measured under magnetic fields.
a The measured specific heat data (Cp) of NBCP under magnetic fields of 0.75 T and 2 T, both of which turn up at low temperature T ≲ 150 mK due to the nuclear spin Schottky anomaly. b The open symbols are estimated parameter αn of nuclear spin contributions for different fields, based on the fitted αn under 0.75 T and 2 T, and supposed to be linear vs. field B. The provided error bars are estimated based on the fitting uncertainty in the B = 0.75 T case. c-g show the total specific heat Cp (open symbols) under different fields, which contains the fitted nuclear spin contribution (dotted-dashed line) and phonon specific heat (dotted line). After subtracting these contributions, we show the resulting magnetic specific heat Cm with solid symbols. h-j The magnetic entropy can be obtained by taking the integration of the specific heat, and both measured data (Exp.) and model calculation (Th.) are shown. The red arrows indicate an ideal adiabatic demagnetization cooling process, which starts from the initial conditions T0 = 2 K and B0 = 4 T and end with lowest temperature Tcorr shown in Fig. 2a of the main text. The estimated error bars, which are based on the fitting uncertainty of nuclear contribution, are smaller than the size of symbols.
Extended Data Fig. 5 The magnon dispersion by the linear spin-wave theory.
a Magnon dispersion Ek calculated by the linear spin-wave theory at the supersolid transition \({B}_{{\rm{c3}}}^{* }\simeq 1.69\)(6) T, and the dashed box in a is expanded in b. The easy-axis TLAF model simulated here use the effective parameters, i.e., Jz/Jxy = 1.68, with two gapless quadratic modes at the K point. They enhance the low-energy density of states and give rise to notable entropic effect at low temperature. The results indicate a quantum critical point (QCP) with dynamical exponent z = 2. Since two order parameters, the solid order associated with lattice rotational \({{\mathbb{Z}}}_{3}\) symmetry breaking and the superfluid order with U(1) spin symmetry breaking, both vanish at the same QCP (c.f., Fig. 3b in the main text), we argue that there may exist an emergent O(4) symmetry at the QCP that include the \({{\mathbb{Z}}}_{3}\times \) U(1) as its subgroup.
Extended Data Fig. 6 Comparisons on cooling performances.
a Quasi-adiabatic demagnetization cooling curves of NBCP (solid line) and GGG (dashed line) from the initial field of B0 = 4 T and different base temperatures T0 = 2-4 K. b compares the demagnetization cooling of NBCP with paramagnetic coolants CPA and FAA. The volumes of different materials are roughly equal (about 0.4 cm3). NBCP cools down much faster in the high-field regime, i.e., \({B}_{{\rm{c2}}}^{* }\le B\le {B}_{{\rm{c3}}}^{* }\) (V-type spin supersolid phase). We find NBCP cools down to 94 mK at about 1.5 T, much lower than that of paramagnets by ΔT ≃ 660 mK. The insets in a and b show single-crystal samples of NBCP (1.65 g, 0.395 cm3), GGG (50 g), FAA (0.7 g, 0.409 cm3), and CPA (0.76 g, 0.421 cm3) used in practical MCE measurements. c shows the volumetric magnetic entropy densities of NBCP at Bf = 0 T (solid line) and B0 = 4 T (dashed line). The heat absorption of NBCP ΔQc ≃ 19 mJ ⋅ cm−3 at zero field (indicated by the purple shaded area), which turns out to be much higher than those of FAA (3.0 mJ ⋅ cm−3) and CPA (3.1 mJ ⋅ cm−3) as reported in ref. 33. Therefore, NBCP is able to maintain the reached low temperatures for a longer period of time (hold time). As shown in d, the NBCP remains below 1 K for about 1.5 hours, while the temperatures of CPA and FAA quickly rise up and reach thermal equilibrium with 2 K environment within 30–40 mins. e tabulates the lowest cooling temperatures in the quasi-adiabatic demagnetization process from the same initial field B0 = 4 T and various initial temperature T0.
Extended Data Fig. 7 Low temperature magnetic Grüneisen ratio and location of quantum critical points.
a Magnetic Grüneisen ratio ΓB deduced from the isentropic T-B lines, \({\Gamma }_{B}=\frac{1}{T}{\left(\frac{\partial T}{\partial B}\right)}_{S}\), with the red (black) arrows indicating the dip (peak) locations. The quasi-adiabatic cooling starting from various low initial temperatures T0≤300 mK, and an initial field of B0 = 3 T. The lines have been shifted with a constant offset of 3.5 T−1 for the sake of clarity. b Characteristic (T, B) values, including those determined from \({T}_{{\rm{MCE}}}^{\min }\) (lowest temperature) and the peak (\({\Gamma }_{B}^{{\rm{peak}}}\)) and dip (\({\Gamma }_{B}^{{\rm{dip}}}\)) of ΓB. They naturally form quantum critical “fans” (orange shaded areas) that converge to the QCPs (\({B}_{{\rm{c1}}}^{* },{B}_{{\rm{c2}}}^{* },{B}_{{\rm{c3}}}^{* }\)) in the low temperature limit, denoted by the purple pentagons along with error bars estimated. c lists the field-induced quantum phase transitions (B // c axis) in NBCP, and compare our results to the previous experimental and theoretical works, where excellent agreements are seen.
Extended Data Fig. 8 Single-crystal neutron diffraction scans.
Reciprocal-space scans at 30 mK performed at WOMBAT in zero-field across (1/3, 1/3, 0.183), the strongest magnetic reflection, along a the in-plane (H, H, 0.183) direction and b out-of-plane (1/3, 1/3, L) direction, respectively. The solid and dashed lines represent the Gaussian fittings to the experimental data and the instrumental resolutions represented by the nuclear reflections, respectively. The fitted peak position in a is slightly off H = 1/3, due to the inaccuracy of the lattice constant a determined in the long-wavelength condition. Error bars represent the standard deviations in a and b. c-e Temperature dependencies of the representative magnetic reflections measured at D23, under different applied fields applied along the c axis. c and d show the rocking-curve scans of the (1/3, 1/3, 0.836) reflection appearing under B = 0 and the (2/3, 2/3, 0) reflection appearing under B = 0.8 T, respectively, while e represents the reciprocal-space scan for the (1/3, 1/3, 0.664) peak emerging under B = 1.3 T along the L direction. Int., Intensity; r.l.u., reciprocal lattice units.; a.u., arbitrary units.
Extended Data Fig. 9 Refinements to the magnetic diffraction intensities at 95 mK.
a and b tabulate the basis vectors of the irreducible representations (IRs) for the Co2+ ions on the 1b Wyckoff sites in NBCP, for k = (1/3, 1/3, 0.166) and k = (1/3, 1/3, 0), respectively, obtained from representation analysis. c and d show the comparisons between the observed and calculated integrated intensities of the non-equivalent magnetic reflections for B = 0 and B = 0.8 T, adopting the modulated and non-modulated UUD spin configurations with the irreducible representation Γ1 and Γ2, respectively. Error bars represent the standard deviations, and the R factors of the refinements are listed in both cases. a.u., arbitrary units.
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Xiang, J., Zhang, C., Gao, Y. et al. Giant magnetocaloric effect in spin supersolid candidate Na2BaCo(PO4)2. Nature 625, 270–275 (2024). https://doi.org/10.1038/s41586-023-06885-w
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DOI: https://doi.org/10.1038/s41586-023-06885-w
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