Abstract
Spin nematic is a magnetic analogue of classical liquid crystals, a fourth state of matter exhibiting characteristics of both liquid and solid1,2. Particularly intriguing is a valence-bond spin nematic3,4,5, in which spins are quantum entangled to form a multipolar order without breaking time-reversal symmetry, but its unambiguous experimental realization remains elusive. Here we establish a spin nematic phase in the square-lattice iridate Sr2IrO4, which approximately realizes a pseudospin one-half Heisenberg antiferromagnet in the strong spin–orbit coupling limit6,7,8,9. Upon cooling, the transition into the spin nematic phase at TC ≈ 263 K is marked by a divergence in the static spin quadrupole susceptibility extracted from our Raman spectra and concomitant emergence of a collective mode associated with the spontaneous breaking of rotational symmetries. The quadrupolar order persists in the antiferromagnetic phase below TN ≈ 230 K and becomes directly observable through its interference with the antiferromagnetic order in resonant X-ray diffraction, which allows us to uniquely determine its spatial structure. Further, we find using resonant inelastic X-ray scattering a complete breakdown of coherent magnon excitations at short-wavelength scales, suggesting a many-body quantum entanglement in the antiferromagnetic state10,11. Taken together, our results reveal a quantum order underlying the Néel antiferromagnet that is widely believed to be intimately connected to the mechanism of high-temperature superconductivity12,13.
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References
Blume, M. & Hsieh, Y. Y. Biquadratic exchange and quadrupolar ordering. J. Appl. Phys. 40, 1249–1249 (1969).
Andreev, A. F. & Grishchuk, I. A. Spin nematics. Sov. Phys. JETP 60, 267–271 (1984).
Läuchli, A., Domenge, J. C., Lhuillier, C., Sindzingre, P. & Troyer, M. Two-step restoration of SU(2) symmetry in a frustrated ring-exchange magnet. Phys. Rev. Lett. 95, 137206 (2005).
Shannon, N., Momoi, T. & Sindzingre, P. Nematic order in square lattice frustrated ferromagnets. Phys. Rev. Lett. 96, 027213 (2006).
Sato, M., Hikihara, T. & Momoi, T. Spin-nematic and spin-density-wave orders in spatially anisotropic frustrated magnets in a magnetic field. Phys. Rev. Lett. 110, 077206 (2013).
Kim, B. J. et al. Novel Jeff = 1/2 Mott state induced by relativistic spin-orbit coupling in Sr2IrO4. Phys. Rev. Lett. 101, 076402 (2008).
Kim, B. J. et al. Phase-sensitive observation of a spin-orbital mott state in Sr2IrO4. Science 323, 1329–1332 (2009).
Jackeli, G. & Khaliullin, G. Mott insulators in the strong spin-orbit coupling limit: from Heisenberg to a quantum compass and Kitaev models. Phys. Rev. Lett. 102, 017205 (2009).
Bertinshaw, J., Kim, Y. K., Khaliullin, G. & Kim, B. J. Square lattice iridates. Annu. Rev. Condens. Matter Phys. 10, 315–336 (2019).
Dalla Piazza, B. et al. Fractional excitations in the square-lattice quantum antiferromagnet. Nat. Phys. 11, 62–68 (2015).
Shao, H. et al. Nearly deconfined spinon excitations in the square-lattice spin-1/2 Heisenberg antiferromagnet. Phys. Rev. X 7, 041072 (2017).
Anderson, P. W. The resonating valence bond state in La2CuO4 and superconductivity. Science 235, 1196–1198 (1987).
Lee, P. A., Nagaosa, N. & Wen, X.-G. Doping a Mott insulator: physics of high-temperature superconductivity. Rev. Mod. Phys. 78, 17–85 (2006).
Scalapino, D. J. A common thread: the pairing interaction for unconventional superconductors. Rev. Mod. Phys. 84, 1383–1417 (2012).
Vaknin, D. et al. Antiferromagnetism in La2CuO4−y. Phys. Rev. Lett. 58, 2802–2805 (1987).
Singh, R. R. P. Thermodynamic parameters of the T = 0, spin-1/2 square-lattice Heisenberg antiferromagnet. Phys. Rev. B 39, 9760–9763 (1989).
Smerald, A. & Shannon, N. Theory of spin excitations in a quantum spin-nematic state. Phys. Rev. B 88, 184430 (2013).
Savary, L. & Senthil, T. Probing hidden orders with resonant inelastic X-ray scattering. Preprint at arxiv.org/abs/1506.04752 (2015).
Kohama, Y. et al. Possible observation of quantum spin-nematic phase in a frustrated magnet. Proc. Natl Acad. Sci. USA 116, 10686–10690 (2019).
Orlova, A. et al. Nuclear magnetic resonance signature of the spin-nematic phase in LiCuVO4 at high magnetic fields. Phys. Rev. Lett. 118, 247201 (2017).
Ye, F. et al. Magnetic and crystal structures of Sr2IrO4: a neutron diffraction study. Phys. Rev. B 87, 140406 (2013).
Porras, J. et al. Pseudospin-lattice coupling in the spin-orbit Mott insulator Sr2IrO4. Phys. Rev. B 99, 085125 (2019).
Cetin, M. F. et al. Crossover from coherent to incoherent scattering in spin-orbit dominated Sr2IrO4. Phys. Rev. B 85, 195148 (2012).
Gim, Y. et al. Isotropic and anisotropic regimes of the field-dependent spin dynamics in Sr2IrO4: Raman scattering studies. Phys. Rev. B 93, 024405 (2016).
Gretarsson, H. et al. Two-magnon Raman scattering and pseudospin-lattice interactions in Sr2IrO4 and Sr3Ir2O7. Phys. Rev. Lett. 116, 136401 (2016).
Aeppli, G. et al. Magnetic dynamics of La2CuO4 and La2−xBaxCuO4. Phys. Rev. Lett. 62, 2052–2055 (1989).
Christensen, N. B. et al. Quantum dynamics and entanglement of spins on a square lattice. Proc. Natl Acad. Sci. USA 104, 15264–15269 (2007).
Headings, N. S., Hayden, S. M., Coldea, R. & Perring, T. G. Anomalous high-energy spin excitations in the high-Tc superconductor-parent antiferromagnet La2CuO4. Phys. Rev. Lett. 105, 247001 (2010).
Martinelli, L. et al. Fractional spin excitations in the infinite-layer cuprate CaCuO2. Phys. Rev. X 12, 021041 (2022).
Tsyrulin, N. et al. Quantum effects in a weakly frustrated S = 1/2 two-dimensional Heisenberg antiferromagnet in an applied magnetic field. Phys. Rev. Lett. 102, 197201 (2009).
Powalski, M., Schmidt, K. P. & Uhrig, G. S. Mutually attracting spin waves in the square-lattice quantum antiferromagnet. SciPost Phys. 4, 001 (2018).
Chen, H. H. & Levy, P. M. Quadrupole phase transitions in magnetic solids. Phys. Rev. Lett. 27, 1383–1385 (1971).
Chandra, P. & Coleman, P. Quantum spin nematics: moment-free magnetism. Phys. Rev. Lett. 66, 100–103 (1991).
Läuchli, A., Mila, F. & Penc, K. Quadrupolar phases of the S = 1 bilinear-biquadratic Heisenberg model on the triangular lattice. Phys. Rev. Lett. 97, 087205 (2006).
Hikihara, T., Kecke, L., Momoi, T. & Furusaki, A. Vector chiral and multipolar orders in the spin-\(\frac{1}{2}\) frustrated ferromagnetic chain in magnetic field. Phys. Rev. B 78, 144404 (2008).
Ueda, H. T. & Totsuka, K. Magnon bose-einstein condensation and various phases of three-dimensonal quantum helimagnets under high magnetic field. Phys. Rev. B 80, 014417 (2009).
Zhitomirsky, M. E. & Tsunetsugu, H. Magnon pairing in quantum spin nematic. EPL 92, 37001 (2010).
Kim, J. et al. Magnetic excitation spectra of Sr2IrO4 probed by resonant inelastic X-ray scattering: establishing links to cuprate superconductors. Phys. Rev. Lett. 108, 177003 (2012).
Mourigal, M. et al. Evidence of a bond-nematic phase in LiCuVO4. Phys. Rev. Lett. 109, 027203 (2012).
Seyler, K. L. et al. Spin-orbit-enhanced magnetic surface second-harmonic generation in Sr2IrO4. Phys. Rev. B 102, 201113 (2020).
Jeong, J., Sidis, Y., Louat, A., Brouet, V. & Bourges, P. Time-reversal symmetry breaking hidden order in Sr2(Ir,Rh)O4. Nat. Commun. 8, 15119 (2017).
Murayama, H. et al. Bond directional anapole order in a spin-orbit coupled Mott insulator Sr2(Ir1−xRhx)O4. Phys. Rev. X 11, 011021 (2021).
Kim, J. et al. Single crystal growth of iridates without platinum impurities. Phys. Rev. Mater. 6, 103401 (2022).
Torchinsky, D. et al. Structural distortion-induced magnetoelastic locking in Sr2IrO4 revealed through nonlinear optical harmonic generation. Phys. Rev. Lett. 114, 096404 (2015).
Zhu, X. D., Ullah, R. & Taufour, V. Oblique-incidence Sagnac interferometric scanning microscope for studying magneto-optic effects of materials at low temperatures. Rev. Sci. Instrum. 92, 043706 (2021).
Varma, C. M. Non-Fermi-liquid states and pairing instability of a general model of copper oxide metals. Phys. Rev. B 55, 14554–14580 (1997).
Zhao, L. et al. Evidence of an odd-parity hidden order in a spin-orbit coupled correlated iridate. Nat. Phys. 12, 32–36 (2016).
Acknowledgements
We thank N. Shannon, G. Khaliullin and Y. B. Kim for helpful discussions. This project is supported by the Institute for Basic Science (Project IBS-R014-A2) and the Samsung Science and Technology Foundation (Project SSTF-BA2201-04). Experiments at the PLS-II 1C beamline were supported in part by the Ministry of Science and ICT of Korea. The use of the Advanced Photon Source at the Argonne National Laboratory was supported by the US Department of Energy (Contract DE-AC02-06CH11357). We acknowledge the European Synchrotron Radiation Facility for provision of synchrotron radiation facilities (Proposal HC-5311), and we also thank F. Gerbon for assistance and support in using beamline ID20. J.-K.K. was supported by the Global PhD Fellowship Program by the National Research Foundation of Korea (Grant 2018H1A2A1059958). G.Y.C. is supported by the National Research Foundation of Korea (Grants 2020R1C1C1006048, RS-2023-00208291, 2023M3K5A1094810 and 2023M3K5A1094813) funded by the Korean Government (Ministry of Science and ICT), the Institute of Basic Science (Project IBS-R014-D1), the Air Force Office of Scientific Research (Award FA2386-22-1-4061) and the Samsung Science and Technology Foundation (Project SSTF-BA2002-05). H.H. and J.J. are supported by the National Research Foundation of Korea (Grant 2020R1A5A1016518) and the Creative-Pioneering Researchers Program through Seoul National University.
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B.J.K. conceived and managed the project. H.K., K.K. and Jonghwan Kim performed Raman experiments. H.K., J. Kwon and S.H. performed resonant X-ray diffraction experiments with help from J.S., G.F., Y.C., D.H. and J.W.K.; H.K., J.-K.K., J. Kwon and H.-W.J.K. performed resonant inelastic X-ray scattering experiments and analysed the data with help from C.J.S., A.L. and Jungho Kim. Jimin Kim grew single crystals. H.H. and J.J. performed Kerr measurements. H.K. and B.J.K. performed representation analysis. W.L. and G.Y.C. assisted in the interpretation of the data. H.K., J.-K.K. and B.J.K. wrote the manuscript with inputs from all authors.
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Extended data figures and tables
Extended Data Fig. 1 Temperature evolution of the low-energy Raman modes.
The Raman spectra of the A1g and B2g modes shown in Fig. 2c,e are displayed with vertical offset for clarity. The A1g (red) and B2g (black) spectra were measured on the same crystal under the same experimental conditions including laser power and acquisition time, and the spectra are plotted in the same arbitrary unit.
Extended Data Fig. 2 Magneto-optical Kerr measurement.
a, Relative Kerr angle in the 0.35 T in-plane magnetic field (red line) and ambient magnetic field near 0 T (black line). The relative Kerr angle (left axis) is converted to magnetization (right axis) using a conversion factor of 7.7 × 10−4 (μB/Ir ion)/(μ rad). b, Magnified plot of the 0 T data in a. Dashed line indicates the standard deviation for the T = 230 ~ 300 K range. The Kerr signal at B = 0 T in the range of 230 K < T < 300 K shows that no net magnetization is present within our experimental resolution, 3.4 × 10−5 μB/ion (dashed line), thus confirming the preservation of time-reversal symmetry above TN ~ 230 K.
Extended Data Fig. 3 Polarziation-resolved RIXS spectra.
a–f, RIXS spectra along the zone boundary from (π/2, π/2) to (π, 0) with the spin components transverse (T) and longitudinal (L) to the ordered AF moments resolved. The spectra were acquired with the same experimental setup in Fig. 4. The error bars are derived using error propagation based on the standard deviations of the raw spectra.
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Kim, H., Kim, JK., Kwon, J. et al. Quantum spin nematic phase in a square-lattice iridate. Nature 625, 264–269 (2024). https://doi.org/10.1038/s41586-023-06829-4
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DOI: https://doi.org/10.1038/s41586-023-06829-4
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