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Hybridized quadrupolar excitations in the spin-anisotropic frustrated magnet FeI2

Abstract

Magnetic order is usually associated with well-defined magnon excitations. Exotic magnetic fluctuations with fractional, topological or multipolar character have been proposed for unconventional forms of magnetic matter such as spin liquids1. As a result, considerable effort has been expended to search for, and uncover, low-spin materials with suppressed dipolar order at low temperatures2,3. However, long-range order of magnetic dipoles is much more common. Here we report neutron-scattering experiments and quantitative theoretical modelling of a spin-1 system—the uniaxial triangular magnet FeI2 (ref. 4)—where a dispersive band of mixed dipolar–quadrupolar fluctuations with large spectral weight emerges just above a dipolar ordered ground state. This excitation arises from anisotropic exchange interactions that hybridize overlapping modes carrying fundamentally different quantum numbers. A generalization of spin–wave theory to local SU(3) degrees of freedom5 accounts for all details of the low-energy dynamical response of FeI2, without going beyond quadratic order. Our work highlights that quantum excitations without classical counterparts can be realized, even in the presence of fully developed magnetic order.

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Fig. 1: Elementary magnetic excitations of a ferromagnetic easy-axis spin-1 chain and their hybridization through anisotropic exchange interactions.
Fig. 2: Microscopic origin of the magnetic properties of FeI2.
Fig. 3: Low-energy magnetic excitations of FeI2 and matching anisotropic exchange model for SU(3) spins.

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Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding authors upon request.

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The source codes used in this study are available from the corresponding authors upon request.

References

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

    Article  ADS  Google Scholar 

  2. Nakatsuji, S. et al. Spin disorder on a triangular lattice. Science 309, 1697–1700 (2005).

    Article  ADS  Google Scholar 

  3. Broholm, C. L. et al. Quantum spin liquids. Science 367, eaay0668 (2020).

    Article  Google Scholar 

  4. Bertrand, Y., Fert, A. R. & Gelard, J. Susceptibilité magnétique des halogénures ferreux FeCl2, FeBr2, Fel2. J. Phys. 35, 385–391 (1974).

    Article  Google Scholar 

  5. Batista, C. D. & Ortiz, G. Algebraic approach to interacting quantum systems. Adv. Phys. 53, 1–82 (2004).

    Article  ADS  Google Scholar 

  6. Takagi, H., Takayama, T., Jackeli, G., Khaliullin, G. & Nagler, S. E. Concept and realization of Kitaev quantum spin liquids. Nat. Rev. Phys. 1, 264–280 (2019).

    Article  Google Scholar 

  7. Suzuki, M. T., Ikeda, H. & Oppeneer, P. M. First-principles theory of magnetic multipoles in condensed matter systems. J. Phys. Soc. Jpn 87, 041008 (2018).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  10. Baek, S. H. et al. Orbital-driven nematicity in FeSe. Nat. Mater. 14, 210–214 (2015).

    Article  ADS  Google Scholar 

  11. Chubukov, A. V. Chiral, nematic and dimer states in quantum spin chains. Phys. Rev. B 44, 4693–4696 (1991).

    Article  ADS  Google Scholar 

  12. Shannon, N., Momoi, T. & Sindzingre, P. Nematic order in square lattice frustrated ferromagnets. Phys. Rev. Lett. 96, 027213 (2006).

    Article  ADS  Google Scholar 

  13. Matsumoto, M. & Koga, M. Longitudinal spin–wave mode near quantum critical point due to uniaxial anisotropy. J. Phys. Soc. Jpn 76, 073709 (2007).

    Article  ADS  Google Scholar 

  14. Romhányi, J. & Penc, K. Multiboson spin–wave theory for Ba2CoGe2O7: a spin-3/2 easy-plane Néel antiferromagnet with strong single-ion anisotropy. Phys. Rev. B 86, 174428 (2012).

    Article  ADS  Google Scholar 

  15. Michaud, F., Vernay, F. & Mila, F. Theory of inelastic light scattering in spin-1 systems: resonant regimes and detection of quadrupolar order. Phys. Rev. B 84, 184424 (2011).

    Article  ADS  Google Scholar 

  16. Fert, A. R. et al. Excitation of two spin deviations by far infrared absorption in FeI2. Solid State Commun. 26, 693–696 (1978).

    Article  ADS  Google Scholar 

  17. Petitgrand, D., Brun, A. & Meyer, P. Magnetic field dependence of spin waves and two magnon bound states in FeI2. J. Magn. Magn. Mater. 15, 381–382 (1980).

    Article  ADS  Google Scholar 

  18. Silberglitt, R. & Torrance, J. B. Jr. Effect of single-ion anisotropy on two-spin-wave bound state in a Heisenberg ferromagnet. Phys. Rev. B 2, 772–778 (1970).

    Article  ADS  Google Scholar 

  19. Oguchi, T. Theory of two-magnon bound states in the Heisenberg ferro- and antiferromagnet. J. Phys. Soc. Jpn 31, 394–402 (1971).

    Article  ADS  Google Scholar 

  20. Petitgrand, D., Hennion, B. & Escribe, C. Neutron inelastic scattering from magnetic excitations of FeI2. J. Magn. Magn. Mater. 14, 275–276 (1979).

    Article  ADS  Google Scholar 

  21. Katsumata, K. et al. Single-ion magnon bound states in an antiferromagnet with strong uniaxial anisotropy. Phys. Rev. B 61, 11632 (2000).

    Article  ADS  Google Scholar 

  22. Balucani, U. & Stasch, A. Hybrid excitations in layered iron halides. Phys. Rev. B 32, 182–193 (1985).

    Article  ADS  Google Scholar 

  23. Fujita, T., Ito, A. & Ôno, K. The Mössbauer study of the ferrous ion in FeI2. J. Phys. Soc. Jpn 21, 1734–1736 (1966).

    Article  ADS  Google Scholar 

  24. Gelard, J., Fert, A. R., Meriel, P. & Allain, Y. Magnetic structure of FeI2 by neutron diffraction experiments. Solid State Commun. 14, 187–189 (1974).

    Article  ADS  Google Scholar 

  25. Wiedenmann, A. et al. A neutron scattering investigation of the magnetic phase diagram of FeI2. J. Magn. Magn. Mater. 74, 7–21 (1988).

    Article  ADS  Google Scholar 

  26. Trooster, J. M. & de Valk, W. Spin ordering in FeBr2 and FeI2. Evidence for first order phase transition in FeI2. Hyperfine Interact. 4, 457–459 (1978).

    Article  ADS  Google Scholar 

  27. Tanaka, Y. & Uryû, N. Ground state spin configurations of the triangular Ising net with the nearest and next nearest neighbor interactions. J. Phys. Soc. Jpn 39, 825–826 (1975).

    Article  ADS  Google Scholar 

  28. Conlon, P. H. & Chalker, J. T. Absent pinch points and emergent clusters: further neighbor interactions in the pyrochlore Heisenberg antiferromagnet. Phys. Rev. B 81, 224413 (2010).

    Article  ADS  Google Scholar 

  29. 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 

  30. Bai, X. et al. Magnetic excitations of the classical spin liquid MgCr2O4. Phys. Rev. Lett. 122, 097201 (2019).

    Article  ADS  Google Scholar 

  31. Wu, X. et al. Magnetic ordering and multiferroicity in MnI2. Phys. Rev. B 86, 134413 (2012).

    Article  ADS  Google Scholar 

  32. Muniz, R. A., Kato, Y. & Batista, C. D. Generalized spin-wave theory: application to the bilinear-biquadratic model. Prog. Theor. Exp. Phys. 2014, 083101 (2014).

    Article  MATH  Google Scholar 

  33. 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).

    Article  ADS  Google Scholar 

  34. Paddison, J. A. M. Scattering signatures of bond-dependent magnetic interactions. Preprint at https://arxiv.org/pdf/2002.12894.pdf (2020).

  35. Li, Y. et al. Rare-earth triangular lattice spin liquid: a single-crystal study of YbMgGaO4. Phys. Rev. Lett. 115, 167203 (2015).

    Article  ADS  Google Scholar 

  36. Maksimov, P. A., Zhu, Z., White, S. R. & Chernyshev, A. L. Anisotropic-exchange magnets on a triangular lattice: spin waves, accidental degeneracies and dual spin liquids. Phys. Rev. X 9, 021017 (2019).

    Google Scholar 

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

    Article  Google Scholar 

  38. Carretta, S., Santini, P., Caciuffo, R. & Amoretti, G. Quadrupolar waves in uranium dioxide. Phys. Rev. Lett. 105, 167201 (2010).

    Article  ADS  Google Scholar 

  39. Penc, K. et al. Spin-stretching modes in anisotropic magnets: spin–wave excitations in the multiferroic Ba2CoGe2O7. Phys. Rev. Lett. 108, 257203 (2012).

    Article  ADS  Google Scholar 

  40. Akaki, M. et al. Direct observation of spin–quadrupolar excitations in Sr2CoGe2O7 by high-field electron spin resonance. Phys. Rev. B 96, 214406 (2017).

    Article  ADS  Google Scholar 

  41. Zvyagin, S. A. et al. Observation of two-magnon bound states in the spin-1 anisotropic Heisenberg antiferromagnetic chain system NiCl2-4SC(NH2)2. Physica B 403, 1497–1499 (2008).

    Article  ADS  Google Scholar 

  42. Yoshizawa, H., Kozukue, W. & Hirakawa, K. Neutron scattering study of magnetic excitations in pseudo-one-dimensional singlet ground state ferromagnets CsFeCl3 and RbFeCl3. J. Phys. Soc. Jpn 49, 144–153 (1980).

    Article  ADS  Google Scholar 

  43. Hayashida, S. et al. Novel excitations near quantum criticality in geometrically frustrated antiferromagnet CsFeCl3. Sci. Adv. 5, eaaw5639 (2019).

    Article  ADS  Google Scholar 

  44. Birgeneau, R. J., Yelon, W. B., Cohen, E. & Makovsky, J. Magnetic properties of FeCl2 in zero field. I. Excitations. Phys. Rev. B 5, 2607 (1972).

    Article  ADS  Google Scholar 

  45. Tartaglia, T. A. et al. Accessing new magnetic regimes by tuning the ligand spin-orbit coupling in van der Waals magnets. Sci. Adv. 6, eabb9379 (2020).

    Article  ADS  Google Scholar 

  46. Pasternak, M. P. et al. Pressure-induced magnetic and electronic transitions in the layered Mott insulator FeI2. Phys. Rev. B 65, 035106 (2001).

    Article  ADS  Google Scholar 

  47. Fert, A. R., Gelard, J. & Carrara, P. Phase transitions of Fel2 in high magnetic field parallel to the spin direction, static field up to 150 kOe, pulsed field up to 250 kOe. Solid State Commun. 13, 1219–1223 (1973).

    Article  ADS  Google Scholar 

  48. Dally, R. L. et al. Three-magnon bound state in the quasi-one-dimensional antiferromagnet α-NaMnO2. Phys. Rev. Lett. 124, 197203 (2020).

    Article  ADS  Google Scholar 

  49. Lockwood, D. J., Mischler, G. & Zwick, A. Raman scattering from magnons, electronic excitations and phonons in antiferromagnetic FeI2. J. Phys. Condens. Matter 6, 6515 (1994).

    Article  ADS  Google Scholar 

  50. Coleman, C. C. & Yamada, E. Optimization of the vapor reaction growth of single crystal FeI2. J. Cryst. Growth 132, 129–133 (1993).

    Article  ADS  Google Scholar 

  51. Petitgrand, D. & Meyer, P. Far infrared antiferromagnetic resonance in FeCl2, FeBr2 and FeI2. J. Phys. 37, 1417–1422 (1976).

    Article  Google Scholar 

  52. Friedt, J. M., Sanchez, J. P. & Shenoy, G. K. Electronic and magnetic properties of metal diiodides MI2 (M = V, Cr, Mn, Fe, Co, Ni and Cd) from 129I Mössbauer spectroscopy. J. Chem. Phys. 65, 5093–5102 (1976).

    Article  ADS  Google Scholar 

  53. Katsumata, K., Hagiwara, M., Tokunaga, M. & Yamaguchi, H. Observation of single-ion magnon bound states in the metamagnet FeI2. J. Appl. Phys. 87, 5085–5087 (2000).

    Article  ADS  Google Scholar 

  54. Rodríguez-Carvajal, J. Recent advances in magnetic structure determination by neutron powder diffraction. Physica B 192, 55–69 (1993).

    Article  ADS  Google Scholar 

  55. Ye, F., Liu, Y., Whitfield, R., Osborn, R. & Rosenkranz, S. Implementation of cross correlation for energy discrimination on the time-of-flight spectrometer CORELLI. J. Appl. Crystallogr. 51, 315–322 (2018).

    Article  Google Scholar 

  56. Granroth, G. E. et al. SEQUOIA: a newly operating chopper spectrometer at the SNS. J. Phys. Conf. Ser 251, 012058 (2010).

    Article  Google Scholar 

  57. Stone, M. B. et al. Comparison of four direct geometry time-of-flight spectrometers at the Spallation Neutron Source. Rev. Sci. Instrum. 85, 045113 (2014).

    Article  ADS  Google Scholar 

  58. Arnold, O. et al. Mantid—Data analysis and visualization package for neutron scattering and μ SR experiments. Nucl. Instrum. Methods Phys. Res. A 764, 156–166 (2014).

    Article  ADS  Google Scholar 

  59. 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).

    Article  ADS  Google Scholar 

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Acknowledgements

We thank C. Broholm, I. Kimchi, S. Nagler, O. Starykh and A. Tennant for valuable discussions. The work of X.B., Z.D. and M.M. at Georgia Tech was supported by the US Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division under award DE-SC-0018660. The work of S.-S.Z. and C.D.B. at the University of Tennessee was supported by the Lincoln Chair of Excellence in Physics and the work by H. Zhang was supported by the US Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division. The work of Q.H. and H. Zhou at the University of Tennessee was supported by the US Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division under award DE-SC-0020254. This research used resources at the High Flux Isotope Reactor and Spallation Neutron Source, a DOE Office of Science User Facility operated by the Oak Ridge National Laboratory.

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Contributions

X.B. and M.M. conceived the project, which was supervised by C.D.B. and M.M. Z.D. and X.B. grew the samples with the assistance of Q.H. and H. Zhou using the floating zone furnace. Z.D. aligned the sample for measurements. X.B., Z.D., M.B.S., A.I.K., F.Y. and M.M. performed the neutron-scattering measurements. X.B. analysed the data and performed fits. S.-S.Z., H. Zhang and C.D.B. carried out the GSWT calculations and assisted with the theoretical interpretation. S.-S.Z. made the GSWT code used to fit the inelastic spectra. X.B. and M.M. wrote the manuscript with input from all authors.

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Correspondence to Xiaojian Bai or Martin Mourigal.

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Supplementary Figs. 1–12, Discussion and Tables 1–3.

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Bai, X., Zhang, SS., Dun, Z. et al. Hybridized quadrupolar excitations in the spin-anisotropic frustrated magnet FeI2. Nat. Phys. 17, 467–472 (2021). https://doi.org/10.1038/s41567-020-01110-1

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