- Open Access
Real-space determination of orbital states driving successive phase transitions in
Phys. Rev. Research 8, 043058 – Published 16 October, 2026
DOI: https://doi.org/10.1103/j864-1gpm
Abstract
Direct experimental access to orbital states in strongly correlated materials remains a major challenge, despite their central role in driving coupled structural and magnetic phase transitions. In systems where electronic correlations, electron-lattice coupling, and relativistic spin-orbit interactions compete on comparable energy scales, first-principles calculations often yield multiple metastable solutions, hindering the unambiguous identification of the ground state. Here, we demonstrate that the orbital states of the spinel oxide , which possesses active orbital degrees of freedom on both Fe and V ions, are uniquely resolved by combining valence electron density (VED) analysis based on state-of-the-art synchrotron x-ray diffraction with spin-polarized density-functional-theory calculations. Our results reveal that temperature-dependent rearrangements of orbital occupations drive successive structural transitions that accompany collinear and noncoplanar ferrimagnetic ordering, establishing a direct correspondence between orbital anisotropy and spin structure. This work shows that experimentally determined VED provides a decisive real-space constraint on competing theoretical solutions, offering a powerful and broadly applicable framework for elucidating the microscopic mechanisms of complex phase transitions in strongly correlated electron systems.
Physics Subject Headings (PhySH)
Article Text
Supplemental Material
References (60)
- A. J. Millis, Lattice effects in magnetoresistive manganese perovskites, Nature (London) 392, 147 (1998).
- P. G. Radaelli, Y. Horibe, M. J. Gutmann, H. Ishibashi, C. H. Chen, R. M. Ibberson, Y. Koyama, Y. S. Hor, V. Kiryukhin, and S.-W. Cheong, Formation of isomorphic and octamers and spin dimerization in the spinel , Nature (London) 416, 155 (2002).
- M. S. Senn, J. P. Wright, and J. P. Attfield, Charge order and three-site distortions in the Verwey structure of magnetite, Nature (London) 481, 173 (2012).
- L. Camerano, A. O. Fumega, J. L. Lado, A. Stroppa, and G. Profeta, Multiferroic nematic -wave altermagnetism driven by orbital-order on the honeycomb lattice, npj 2D Mater. Appl. 9, 75 (2025).
- M. Neupane, P. Richard, Z.-H. Pan, Y.-M. Xu, R. Jin, D. Mandrus, X. Dai, Z. Fang, Z. Wang, and H. Ding, Observation of a novel orbital selective Mott transition in , Phys. Rev. Lett. 103, 097001 (2009).
- G. M. Luke, Y. Fudamoto, K. M. Kojima, M. I. Larkin, J. Merrin, B. Nachumi, Y. J. Uemura, Y. Maeno, Z. Q. Mao, Y. Mori, H. Nakamura, and M. Sigrist, Time-reversal symmetry-breaking superconductivity in , Nature (London) 394, 558 (1998).
- Á. A. Carrasco Álvarez, S. Petit, W. Prellier, M. Bibes, and J. Varignon, Orbital-order as the driving mechanism for superconductivity in ruthenates, Nat. Commun. 16, 1458 (2025).
- H. Takagi, T. Takayama, G. Jackeli, G. Khaliullin, and S. E. Nagler, Concept and realization of Kitaev quantum spin liquids, Nat. Rev. Phys. 1, 264 (2019).
- D. Hirai, H. Sagayama, S. Gao, H. Ohsumi, G. Chen, T.-H. Arima, and Z. Hiroi, Detection of multipolar orders in the spin-orbit-coupled Mott insulator , Phys. Rev. Res. 2, 022063(R) (2020).
- S.-H. Lee, D. Louca, H. Ueda, S. Park, T. J. Sato, M. Isobe, Y. Ueda, S. Rosenkranz, P. Zschack, J. Íñiguez, Y. Qiu, and R. Osborn, Orbital and spin chains in , Phys. Rev. Lett. 93, 156407 (2004).
- T. Maitra and R. Valentí, Orbital order in , Phys. Rev. Lett. 99, 126401 (2007).
- E. M. Wheeler, B. Lake, A. T. M. N. Islam, M. Reehuis, P. Steffens, T. Guidi, and A. H. Hill, Spin and orbital order in the vanadium spinel , Phys. Rev. B 82, 140406(R) (2010).
- A. Kiswandhi, J. Ma, J. S. Brooks, and H. D. Zhou, Effects of inter-vanadium distance and -site magnetism in ( = Cd, Mg, Zn) spinels near the itinerant electron limit, Phys. Rev. B 90, 155132 (2014).
- T. Suzuki, M. Katsumura, K. Taniguchi, T. Arima, and T. Katsufuji, Orbital ordering and magnetic field effect in , Phys. Rev. Lett. 98, 127203 (2007).
- Y. Nii, H. Sagayama, T. Arima, S. Aoyagi, R. Sakai, S. Maki, E. Nishibori, H. Sawa, K. Sugimoto, H. Ohsumi, and M. Takata, Orbital structures in spinel vanadates ( = Fe, Mn), Phys. Rev. B 86, 125142 (2012).
- T. Katsufuji, T. Suzuki, H. Takei, M. Shingu, K. Kato, K. Osaka, M. Takata, H. Sagayama, and T.-H. Arima, Structural and magnetic properties of spinel with two ions having orbital degrees of freedom, J. Phys. Soc. Jpn. 77, 053708 (2008).
- G. J. MacDougall, V. O. Garlea, A. A. Aczel, H. D. Zhou, and S. E. Nagler, Magnetic order and ice rules in the multiferroic spinel , Phys. Rev. B 86, 060414(R) (2012).
- S. Kawaguchi, H. Ishibashi, S. Nishihara, S. Mori, J. Campo, F. Porcher, O. Fabelo, K. Sugimoto, J. Kim, K. Kato, M. Takata, H. Nakao, and Y. Kubota, Orthorhombic distortion and orbital order in the vanadium spinel , Phys. Rev. B 93, 024108 (2016).
- Q. Zhang, K. Singh, F. Guillou, C. Simon, Y. Bréard, V. Caignaert, and V. Hardy, Ordering process and ferroelectricity in a spinel derived from , Phys. Rev. B 85, 054405 (2012).
- J.-S. Kang, J. Hwang, D. H. Kim, E. Lee, W. C. Kim, C. S. Kim, S. Kwon, S. Lee, J.-Y. Kim, T. Ueno, M. Sawada, B. Kim, B. H. Kim, and B. I. Min, Valence states and spin structure of spinel with different orbital degrees of freedom, Phys. Rev. B 85, 165136 (2012).
- G. J. MacDougall, I. Brodsky, A. A. Aczel, V. O. Garlea, G. E. Granroth, A. D. Christianson, T. Hong, H. D. Zhou, and S. E. Nagler, Magnons and a two-component spin gap in , Phys. Rev. B 89, 224404 (2014).
- Y. Ishitsuka, T. Ishikawa, R. Koborinai, T. Omura, and T. Katsufuji, Comparative studies of the thermal conductivity of spinel oxides with orbital degrees of freedom, Phys. Rev. B 90, 224411 (2014).
- K. Matsuura, H. Sagayama, Y. Nii, N. D. Khanh, N. Abe, and T. Arima, X-ray magnetic circular dichroism study of an orbital ordered state in the spinel-type vanadium oxide ( = Mn, Fe), Phys. Rev. B 92, 035133 (2015).
- J. Okabayashi, S. Miyasaka, M. Takahashi, and S. Tajima, Local electronic and magnetic properties of ferro-orbital-ordered , Jpn. J. Appl. Phys. 57, 0902BD (2018).
- M. Nakano, T. Kobayashi, and T. Katsufuji, Coupling between orbital and spin degrees of freedom in Jahn-Teller ions for , Phys. Rev. Lett. 135, 186701 (2025).
- H. Tsunetsugu and Y. Motome, Magnetic transition and orbital degrees of freedom in vanadium spinels, Phys. Rev. B 68, 060405(R) (2003).
- O. Tchernyshyov, Structural, orbital, and magnetic order in vanadium spinels, Phys. Rev. Lett. 93, 157206 (2004).
- S. Di Matteo, G. Jackeli, and N. B. Perkins, Orbital order in vanadium spinels, Phys. Rev. B 72, 020408(R) (2005).
- Y. Motome and H. Tsunetsugu, Theory of successive transitions in vanadium spinels and order of orbitals and spins, Prog. Theor. Phys. Suppl. 160, 203 (2005).
- S. Sarkar and T. Saha-Dasgupta, Orbital ordering in : Spinel with two orbitally active sites, Phys. Rev. B 84, 235112 (2011).
- D. Dey, T. Maitra, and A. Taraphder, Orbital ordering in : A first-principles study, Phys. Rev. B 93, 195133 (2016).
- J. Krishna, N. Singh, S. Shallcross, J. K. Dewhurst, E. K. U. Gross, T. Maitra, and S. Sharma, Complete description of the magnetic ground state in spinel vanadates, Phys. Rev. B 100, 081102(R) (2019).
- K. I. Kugel’ and D. I. Khomskii, The Jahn-Teller effect and magnetism: Transition metal compounds, Sov. Phys. Usp. 25, 231 (1982).
- D. I. Khomskii and T. Mizokawa, Orbitally induced Peierls state in spinels, Phys. Rev. Lett. 94, 156402 (2005).
- T. Manjo, S. Kitou, N. Katayama, S. Nakamura, T. Katsufuji, Y. Nii, T.-H. Arima, J. Nasu, T. Hasegawa, K. Sugimoto, D. Ishikawa, A. Q. R. Baron, and H. Sawa, Do electron distributions with orbital degree of freedom exhibit anisotropy? Mater. Adv. 3, 3192 (2022).
- S. Kitou, T. Fujii, T. Kawamoto, N. Katayama, S. Maki, E. Nishibori, K. Sugimoto, M. Takata, T. Nakamura, and H. Sawa, Successive dimensional transition in revealed by synchrotron x-ray diffraction, Phys. Rev. Lett. 119, 065701 (2017).
- S. Kitou, T. Manjo, N. Katayama, T. Shishidou, T.-h. Arima, Y. Taguchi, Y. Tokura, T. Nakamura, T. Yokoyama, K. Sugimoto, and H. Sawa, Collapse of the simple localized orbital picture in Mott insulator, Phys. Rev. Res. 2, 033503 (2020).
- See Supplemental Material at http://link.aps.org/supplemental/10.1103/j864-1gpm for details of the single-crystal structural analysis, -mode analysis, orbital-state determination, and spin-polarized DFT++SOC calculations, which includes Refs. [15, 39].
- S. Kitou, M. Gen, Y. Nakamura, K. Sugimoto, Y. Tokunaga, S. Ishiwata, and T.-H. Arima, Real-space observation of ligand hole state in cubic perovskite , Adv. Sci. 10, 2302839 (2023).
- K. Sugimoto, H. Ohsumi, S. Aoyagi, E. Nishibori, C. Moriyoshi, Y. Kuroiwa, H. Sawa, and M. Takata, Extremely high resolution single crystal diffractometory for orbital resolution using high energy synchrotron radiation at SPring-8, AIP Conf. Proc. 1234, 887 (2010).
- L. Krause, K. Tolborg, T. B. E. Grønbech, K. Sugimoto, B. B. Iversen, and J. Overgaard, Accurate high-resolution single-crystal diffraction data from a Pilatus 3X CdTe detector, J. Appl. Crystallogr. 53, 635 (2020).
- M. Mueller, M. Wang, and C. Schulze-Briese, Optimal fine φ-slicing for single-photon-counting pixel detectors, Acta Crystallogr. D 68, 42 (2012).
- Agilent Technologies, CrysAlis PRO (Agilent Technologies Ltd., Yarnton, Oxfordshire, England, 2014).
- R. H. Blessing, DREADD—Data reduction and error analysis for single-crystal diffractometer data, J. Appl. Crystallogr. 22, 396 (1989).
- V. Petříček, M. Dušek, and L. Palatinus, Crystallographic computing system JANA2006: General features, Z. Kristallogr. 229, 345 (2014).
- K. Momma and F. Izumi, VESTA3 for three-dimensional visualization of crystal, volumetric and morphology data, J. Appl. Crystallogr. 44, 1272 (2011).
- P. Giannozzi et al., Advanced capabilities for materials modelling with quantum ESPRESSO, J. Phys.: Condens. Matter 29, 465901 (2017).
- J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77, 3865 (1996).
- M. J. van Setten, M. Giantomassi, E. Bousquet, M. J. Verstraete, D. R. Hamann, X. Gonze, and G.-M. Rignanese, The PseudoDojo: Training and grading a 85 element optimized norm-conserving pseudopotential table, Comput. Phys. Commun. 226, 39 (2018).
- A. I. Liechtenstein, V. I. Anisimov, and J. Zaanen, Density-functional theory and strong interactions: Orbital ordering in Mott-Hubbard insulators, Phys. Rev. B 52, R5467 (1995).
- N. Marzari and D. Vanderbilt, Maximally localized generalized Wannier functions for composite energy bands, Phys. Rev. B 56, 12847 (1997).
- I. Souza, N. Marzari, and D. Vanderbilt, Maximally localized Wannier functions for entangled energy bands, Phys. Rev. B 65, 035109 (2001).
- K. Nakamura, Y. Yoshimoto, Y. Nomura, T. Tadano, M. Kawamura, T. Kosugi, K. Yoshimi, T. Misawa, and Y. Motoyama, RESPACK: An ab initio tool for derivation of effective low-energy model of material, Comput. Phys. Commun. 261, 107781 (2021).
- M. Charlebois, J.-B. Morée, K. Nakamura, Y. Nomura, T. Tadano, Y. Yoshimoto, Y. Yamaji, T. Hasegawa, K. Matsuhira, and M. Imada, Ab initio derivation of low-energy Hamiltonians for systems with strong spin-orbit interaction: Application to , Phys. Rev. B 104, 075153 (2021).
- G. Pizzi et al., Wannier90 as a community code: New features and applications, J. Phys.: Condens. Matter 32, 165902 (2020).
- P. P. Ewald, Die Berechnung optischer und elektrostatischer Gitterpotentiale, Ann. Phys. 369, 253 (1921).
- S. Nakamura and A. Fuwa, Distinct evidence of orbital order in spinel oxide by Mössbauer spectroscopy, J. Phys. Soc. Jpn. 85, 014702 (2016).
- S. Ohtani, Y. Watanabe, M. Saito, N. Abe, K. Taniguchi, H. Sagayama, T.-H. Arima, M. Watanabe, and Y. Noda, Orbital dilution effect in ferrimagnetic : Competition between anharmonic lattice potential and spin-orbit coupling, J. Phys.: Condens. Matter 22, 176003 (2010).
- S. Bordács, D. Varjas, I. Kézsmárki, G. Mihály, L. Baldassarre, A. Abouelsayed, C. A. Kuntscher, K. Ohgushi, and Y. Tokura, Magnetic-order-induced crystal symmetry lowering in ferrimagnetic spinels, Phys. Rev. Lett. 103, 077205 (2009).
- H. Nakai and Y. Nomura, Spin-orbital exchange as a route to intertwined dipole-quadrupole orbital order in under strong trigonal crystal field, arXiv:2605.03028.