- Physical Review Journals
All JournalsPhysics Magazine

Physical Review X

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Inducing Ferroquadrupolar Order with Applied Magnetic Field in TbPO4

Paola Caterina Forino1,2,*, Jens Jensen3, Jian Rui Soh2,4,5, Sonia Francoual6, Oksana Zaharko7, Steffen Sloth1, Alexandra Turrini2,7, Ralf Feyerherm8, Karel Prokes8 et al.

Sofie Holm-Janas1, Ivica Zivkovic2, Yong Liu2, Arnaud Magrez2, Niels Bech Christensen1, Henrik M. Rønnow2,3, and Rasmus Toft-Petersen1,9,†

  • *Contact author: forino@fysik.dtu.dk
  • †Contact author: rasp@fysik.dtu.dk
PDF

Phys. Rev. X 16, 011021 – Published 6 February, 2026

DOI: https://doi.org/10.1103/bhgt-jkj7

Abstract

In this work, we demonstrate a remarkable field-induced ferroquadrupolar phase emanating from the magnetoelastic coupling. Using synchrotron x-ray and neutron diffraction, heat capacity measurements, as well as mean-field modeling, we show that above the critical magnetic field, ferroquadrupolar O2−2 order is stabilized as the main order parameter, in competition with the field-induced magnetic polarization. This is revealed by a severe lattice distortion and the suppression of antiferromagnetic ordering of the dipole moments. Even in zero magnetic field, the previously reported antiferromagnetic order is a mixed quadrupolar-dipolar phase, further highlighting the crucial role of spin-lattice interactions. This complete understanding of the magnetoelastic phase diagram establishes TbPO4 as a model system for studying nematic-antiferromagnetic transitions and provides key Hamiltonian parameters for its description.

View figure in article

Physics Subject Headings (PhySH)

Popular Summary

Rare-earth magnets are central to modern technology, yet they often harbor complex hidden orders where electronic arrangements do not produce detectable magnetic dipoles. We investigated this elusive phenomenon by studying how an applied magnetic field interacts with hidden electronic states in a crystalline material at temperatures near absolute zero. Using intense x-ray beams, we demonstrated that a magnetic field can unexpectedly induce hidden order rather than suppressing it, as previously assumed. Our research shows that electronic order can be switched on and off with magnetic fields by tracking how these states influence the crystal lattice. These results reveal a surprising level of control over elusive electronic phases and deepen our fundamental understanding of the behavior of complex rare-earth magnets in extreme conditions.

Article Text

References (53)

  1. P. Santini, S. Carretta, G. Amoretti, R. Caciuffo, N. Magnani, and G. H. Lander, Multipolar interactions in f-electron systems: The paradigm of actinide dioxides, Rev. Mod. Phys. 81, 807 (2009).
  2. D. F. McMorrow, K. A. McEwen, U. Steigenberger, H. M. Rønnow, and F. Yakhou, X-ray resonant scattering study of the quadrupolar order in UPd3, Phys. Rev. Lett. 87, 057201 (2001).
  3. Y. Kuramoto, H. Kusunose, and A. Kiss, Multipole orders and fluctuations in strongly correlated electron systems, J. Phys. Soc. Jpn. 78, 072001 (2009).
  4. R. Toft-Petersen, T. B. S. Jensen, J. Jensen, M. von Zimmermann, S. Sloth, F. W. Isaksen, N. B. Christensen, Y. Chen, K. Siemensmeyer, H. Kawano-Furukawa, H. Takeya, A. B. Abrahamsen, and N. H. Andersen, Magnetoelastic phase diagram of TbNi2B2C, Phys. Rev. B 97, 224417 (2018).
  5. R. Sibille, N. Gauthier, E. Lhotel, V. Porée, V. Pomjakushin, R. Ewings, T. Perring, J. Ollivier, A. Wildes, C. Ritter, T. C. Hansen, D. A. Keen, G. Nilsen, L. Keller, S. Petit, and T. Fennell, A quantum liquid of magnetic octupoles on the pyrochlore lattice, Nat. Phys. 16, 546 (2020).
  6. C. Pfleiderer, Superconducting phases off-electron compounds, Rev. Mod. Phys. 81, 1551 (2009).
  7. A. Sakai, K. Kuga, and S. Nakatsuji, Superconductivity in the ferroquadrupolar state in the quadrupolar Kondo lattice PrTi2Al20, J. Phys. Soc. Jpn. 81, 083702 (2012).
  8. E. Rosenberg, M. Ikeda, and I. Fisher, The nematic susceptibility of the ferroquadrupolar metal TmAg2 measured via the elastocaloric effect, npj Quantum Mater. 9 (2024).
  9. P. Massat, J. Wen, J. Jiang, A. Hristov, Y. Liu, R. Smaha, R. S. Feigelson, Y. Lee, R. M. Fernandes, and I. Fisher, Field-tuned ferroquadrupolar quantum phase transition in the insulator TmVO4, Proc. Natl. Acad. Sci. U.S.A. 119, e2119942119 (2022).
  10. N. Kolmakova, L. Takunov, and O. Shishkina, Magnetoelastic effects in orthorhombic and tetragonal rare-earth crystals, Physica (Amsterdam) 352B, 259 (2004).
  11. P. Morin, J. Rouchy, and D. Schmitt, Susceptibility formalism for magnetic and quadrupolar interactions in hexagonal and tetragonal rare-earth compounds, Phys. Rev. B 37, 5401 (1988).
  12. K. Kishimoto, T. Ishikura, H. Nakamura, Y. Wakabayashi, and T. Kimura, Antiferroelectric lattice distortion induced by ferroquadrupolar order in DyVO4, Phys. Rev. B 82, 012103 (2010).
  13. J. Kjems, W. Hayes, and S. Smith, Wave-vector dependence of the Jahn-Teller interactions in TmVO4, Phys. Rev. Lett. 35, 1089 (1975).
  14. I. Vinograd, K. Shirer, P. Massat, Z. Wang, T. Kissikov, D. Garcia, M. Bachmann, M. Horvatić, I. Fisher, and N. Curro, Second order Zeeman interaction and ferroquadrupolar order in TmVO4, npj Quantum Mater. 7, 1 (2022).
  15. D. Ranaut and K. Mukherjee, Evolution of field induced magnetic phase attributed to higher order magnetic moments in TbVO4, Sci. Rep. 13, 487 (2023).
  16. J. Rivera, A short review of the magnetoelectric effect and related experimental techniques on single phase (multi-) ferroics, Eur. Phys. J. B 71, 299 (2009).
  17. S. Bluck and H. G. Kahle, Measurement and interpretation of the magnetoelectric effect in TbPO4, J. Phys. 21, 5193 (1988).
  18. C. P. Grams, M. Grüninger, and J. Hemberger, Magnetic field induced critical dynamics in magnetoelectric TbPO4, Phys. Rev. B 110, 041106 (2024).
  19. P. Morin, J. Rouchy, and Z. Kazei, Magnetic and magnetoelastic properties in tetragonal TbPO4, Phys. Rev. B 50, 12625 (1994).
  20. C. K. Loong, L. Soderholm, G. L. Goodman, M. M. Abraham, and L. A. Boatner, Ground-state wave functions of Tb3+ ions in paramagnetic TbPO4: A neutron scattering study, Phys. Rev. B 48, 6124 (1993).
  21. J. Coing-Boyat, F. Sayetat, and A. Apostolov, Caractéristiques cristallographiques, propriétés et structure magnétiques de TbPO4 dans la gamme 1, 5 K-300 K, J. Phys. 36, 1165 (1975).
  22. A. U. Muller, J. Jakelski, and H. G. Kahle, Extensive studies on the low-temperature properties of TbPO4. III. Birefringence measurements and mean-field calculations, J. Phys. Condens. Matter 5, 955 (1993).
  23. M. Rotter, D. M. Le, J. Keller, L. G. Pascut, T. Hoffmann, M. Doerr, R. Schedler, P. F. né Hoffmann, S. Rotter, M. Banks, and N. Klüver, McPhase users manual (2013), https://www2.cpfs.mpg.de/rotter/homepage_mcphase/manual/manual.html.
  24. J. Jensen and A. R. Mackintosh, Rare Earth Magnetism: Structures and Excitations (Clarendon Press, Oxford, 1991), http://www.nbi.ku.dk/page40667.htm.
  25. S. Francoual, J. Strempfer, J. Warren, Y. Liu, A. Skaugen, S. Poli, J. Blume, F. Wolff-Fabris, P. C. Canfield, and T. Lograsso, Single-crystal X-ray diffraction and resonant X-ray magnetic scattering at helium-3 temperatures in high magnetic fields at beamline P09 at PETRA III, J. Synchrotron Radiat. 22, 1207 (2015).
  26. W. Nägele, D. Hohlwein, and G. Domann, Structural and magnetic phase transitions in TbPO4 studied by neutron diffraction, Z. Phys. B Condens. Matter 39, 305 (1980).
  27. G. Üffinger and A. Kasten, Specific heat and magnetocaloric effects in TbPO4 1. Experiments, Phys. Status Solidi (b) 128, 201 (1985).
  28. P. J. Becker, H. G. Kahle, and K. Keller, The phase diagram of TbPO4 measured by linear optical birefringence, Phys. Status Solidi B 130, 191 (1985).
  29. A. Maharaj, E. Rosenberg, A. T. Hristov, E. Berg, R. M. Fernandes, I. R. Fisher, and S. A. Kivelson, Transverse fields to tune an Ising-nematic quantum phase transition, Proc. Natl. Acad. Sci. U.S.A. 114, 13430 (2017).
  30. R. M. Fernandes, A. V. Chubukov, and J. Schmalian, What drives nematic order in iron-based superconductors?, Nat. Phys. 10, 97 (2014).
  31. T. Shibauchi, A. Carrington, and Y. Matsuda, A quantum critical point lying beneath the superconducting dome in iron pnictides, Annu. Rev. Condens. Matter Phys. 5, 113 (2014).
  32. A. E. Böhmer, F. Hardy, L. Wang, T. Wolf, P. Schweiss, and C. Meingast, Superconductivity-induced re-entrance of the orthorhombic distortion in Ba1−xKxFe2As2, Nat. Commun. 6, 7911 (2015).
  33. R. Grasset, K. Katsumi, P. Massat, H.-H. Wen, X.-H. Chen, Y. Gallais, and R. Shimano, Terahertz pulse-driven collective mode in the nematic superconducting state of Ba1−−xKxFe2As2, npj Quantum Mater. 7, 4 (2022).
  34. K. R. Islam and A. Chubukov, Unconventional superconductivity near a nematic instability in a multi-orbital system, npj Quantum Mater. 9, 28 (2024).
  35. A. Yamamoto, J. Takeda, T. Koyama, T. Mito, S. Wada, I. Shirotani, and C. Sekine, Evidence for an antiferroquadrupolar ordering in YbSb probed by Sb121 and Sb123 nuclear magnetic resonances, Phys. Rev. B 70, 220402 (2004).
  36. L. Keller, W. Henggeler, and A. Furrer, Giant quadrupolar interaction in the heavy-fermion compound YbAs, Europhys. Lett. 26, 353 (1994).
  37. Y. Iizuka, T. Yamada, K. Hanzawa, and Y. Ōno, First-principles study of the RKKY interaction and the quadrupole order in the Pr 1-2-20 systems PrT2Al20 (T=Ti, V), J. Phys. Soc. Jpn. 91, 074708 (2022).
  38. M. Tsujimoto, Y. Matsumoto, T. Tomita, A. Sakai, and S. Nakatsuji, Heavy-fermion superconductivity in the quadrupole ordered state of PrTi2Al20, Phys. Rev. Lett. 113, 267001 (2014).
  39. E. W. Rosenberg, J.-H. Chu, J. P. C. Ruff, A. T. Hristov, and I. R. Fisher, Divergence of the quadrupole-strain susceptibility of the electronic nematic system YbRu2Ge2, Proc. Natl. Acad. Sci. U.S.A. 116, 7232 (2019).
  40. P. Morin and Z. Kazei, Stimulated cooperative jahn-teller effect in TmPO4, Phys. Rev. B 55, 8887 (1997).
  41. Z. A. Kazei and R. I. Chanieva, Magnetoelastic effects in rare-earth vanadates YbVO4 and HoVO4, J. Exp. Theor. Phys. 102, 266 (2006).
  42. Z. A. Kazei, Jahn-teller transitions in YbXO4 (X=V, P) stimulated by a strong magnetic field, Phys. Solid State 46, 2249 (2004).
  43. M. Jaime, A. Saul, M. Salamon, V. S. Zapf, N. Harrison, T. Durakiewicz, J. C. Lashley, D. A. Andersson, C. R. Stanek, J. L. Smith, and K. Gofryk, Piezomagnetism and magnetoelastic memory in uranium dioxide, Nat. Commun. 8, 99 (2017).
  44. K. Kimura, T. Katsuyoshi, Y. Sawada, S. Kimura, and T. Kimura, Imaging switchable magnetoelectric quadrupole domains via nonreciprocal linear dichroism, Commun. Mater. 1, 39 (2020).
  45. Introduction to Complex Mediums for Optics and Electromagnetics, edited by W. S. Weiglhofer and A. Lakhtakia, SPIE Press Monograph (SPIE Press, Bellingham, WA, 2003), Vol. 123, p. 757.
  46. P. C. Forino, J. Jensen, J. R. Soh, S. Francoual, O. Zaharko, S. Sloth, A. Turrini, R. Feyerherm, K. Prokes, S. Holm-Janas, I. Zivkovic, Y. Liu, A. Magrez, N. B. Christensen, H. M. Rønnow, and R. Toft-Petersen, Data for article “Inducing ferroquadrupolar order with applied magnetic field in TbPO4”, 10.5281/zenodo.16940462 (2025).
  47. E. du Trémolet de Lacheisserie, Les coefficients de magnétostriction, Ann. Phys. (Berlin) 14, 267 (1970).
  48. W. Böhm, H. G. Kahle, and W. Wüchner, Spectroscopic study of the crystal field splittings in TbPO4, Phys. Status Solidi B 126, 381 (1984).
  49. Y. Hikichi and T. Nomura, Melting temperatures of monazite and xenotime, J. Am. Ceram. Soc. 70, C-252 (1987).
  50. S. Ushakov, K. Helean, A. Navrotsky, and L. Boatner, Thermochemistry of rare-earth orthophosphates, J. Mater. Res. 16, 2623 (2001).
  51. R. S. Feigelson, Synthesis and single-crystal growth of rare-earth orthophosphates, J. Am. Ceram. Soc. 47, 257 (1964).
  52. B. Campbell, H. T. Stokes, D. E. Tanner, and D. M. Hatch, ISODISPLACE: A web-based tool for exploring structural distortions, J. Appl. Crystallogr. 39, 607 (2006).
  53. CrystalMaker Software Ltd., Single Crystal: A crystal and molecular structures program (2024), version 5, Oxford, UK.

Outline

Information

Phys. Rev. X 16, 011021– Published 6 February, 2026
  • Received 27 August 2025
  • Revised 5 December 2025
  • Accepted 2 January 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation