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Hidden orders in spin–orbit-entangled correlated insulators

Nature Reviews Materials volume 10, pages 674–696 (2025)Cite this article

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Abstract

In many materials, ordered phases and their order parameters are easily characterized by standard experimental methods. ‘Hidden order’ refers to a phase transition in which an ordered state emerges without such an easily detectable order parameter, despite clear thermodynamic evidence of the transition. The underlying mechanisms for these unconventional states of matter stem from spin–orbit coupling, which intertwines intersite exchange, classical electron–magnetic interactions and electron–lattice effects. This physics is elusive to experimental probes and beyond traditional theories of insulating magnetism, requiring sophisticated methodologies for its exploration. In this Review, we survey exotic hidden-order phases in correlated insulators, particularly focusing on the latest progress in material-specific theories and numerical approaches. The relevant degrees of freedom in these phases are local high-rank multipole moments of magnetic and charge density that emerge from spin–orbit-entangled correlated shells of heavy d and f electron ions and interact on the lattice via various mechanisms. We discuss approaches to modelling hidden orders in realistic systems via direct ab initio calculations or by constructing low-energy many-body effective Hamiltonian. We also describe how these new theoretical tools have helped to uncover driving mechanisms for recently discovered multipolar phases in double perovskites of heavy transition metals and how they have proved instrumental in disentangling the role of various interactions in ‘traditional’ f-electron multipolar materials such as actinide dioxides. In both cases, material-specific theories have played a key part in interpreting and predicting experimental signatures of hidden orders.

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Fig. 1: Spin–orbit-entangled phenomena.
Fig. 2: Levels structure, ground-state multiplets and hopping paths in d1 and d2 spin–orbit double perovskites.
Fig. 3: Intersite exchange interaction matrices for three Jeff = 3/2 spin–orbit insulators calculated by the force theorem in Hubbard-I method.
Fig. 4: Multipolar properties of d-electron systems computed with different methods.
Fig. 5: Multipolar orders of f-electron systems computed with different methods.

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Acknowledgements

The authors thank the Erwin Schrödinger Institute (ESI) for hosting the ESI-PsiK workshop ‘Spin–orbit entangled quantum magnetism’ and all participants for the many enlightening discussions. This research was funded in part by the Austrian Science Fund (FWF) projects I4506, I1490-N19 and J4698. For Open Access purposes, the authors have applied a CC BY public copyright license to any author accepted manuscript version arising from this submission. The work here presented is partly funded by the European Union’s Next Generation EU initiative, ‘PNRR — M4C2, investimento 1.1 — Fondo PRIN 2022’ and ‘Superlattices of relativistic oxides’ (ID No. 2022L28H97, CUP D53D23002260006). A.P. acknowledges support from the Natural Sciences Engineering Council of Canada. The authors sincerely thank F. Perpetuini for her professional assistance in the design and preparation of Fig. 1.

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Authors and Affiliations

  1. CPHT, CNRS, École polytechnique, Institut Polytechnique de Paris, Palaiseau, France

    Leonid V. Pourovskii & Dario Fiore Mosca

  2. Collège de France, Université PSL, Paris, France

    Leonid V. Pourovskii & Dario Fiore Mosca

  3. Faculty of Physics and Center for Computational Materials Science, University of Vienna, Vienna, Austria

    Dario Fiore Mosca & Lorenzo Celiberti

  4. Vienna Scientific Cluster Research Center, Vienna University of Technology, Vienna, Austria

    Sergii Khmelevskyi & Cesare Franchini

  5. Department of Physics, University of Toronto, Toronto, Ontario, Canada

    Arun Paramekanti

  6. Department of Physics and Astronomy ‘Augusto Righi’, Alma Mater Studiorum — Università di Bologna, Bologna, Italy

    Cesare Franchini

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Contributions

C.F. conceived the initial idea for the review and coordinated its writing. L.V.P., C.F. and A.P. outlined the content. The initial draft was written by C.F. (Introduction), L.V.P. (theories and methods), A.P. (d-electrons: double perovskites) and S.K. (f-electron systems). L.C. and D.F.M. contributed to drafting, compiled the tables and designed the major figures. All authors participated in the final editing of the text and figures.

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Pourovskii, L.V., Fiore Mosca, D., Celiberti, L. et al. Hidden orders in spin–orbit-entangled correlated insulators. Nat Rev Mater 10, 674–696 (2025). https://doi.org/10.1038/s41578-025-00824-z

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