Nikolaev and Solovyev (Phys. Rev. B 89, 064428 (2014)) applied the modern theory of orbital magnetization to prototype Mott-insulating perovskite oxides with a net ferromagnetic moment (orthorhombic YTiO3, LaMnO3, and YVO3, and monoclinic YVO3). They found that the net itinerant correction to the orbital moment is small, but only because large, strongly wave-vector-dependent contributions from different parts of the Brillouin zone — tracking the Chern invariants — cancel almost completely; in a real-space Wannier representation these nonlocal (site-off-diagonal) contributions exceed the local ones. They left open two linked questions: whether these large nonlocal contributions can be measured or separated experimentally, or contribute measurably to other physical properties; and whether this essentially nonlocal, current-dependent part of the orbital magnetization enters the exchange-correlation energy — for example in current-spin-density-functional theory, whose explicit current-dependent functional form remains unknown — and whether including it would improve density-functional descriptions of orbital magnetism. No resolution of either question is known in the literature to date.

Background

Orbital magnetization is the contribution to a magnet's magnetic moment from the electrons' orbital motion, complementing the spin moment. In periodic crystals it is subtle because the position operator is ill-defined for Bloch states, and circulating currents in the bulk cannot be assigned to individual atoms. The 'modern theory of orbital magnetization' (Xiao, Shi, and Niu; Thonhauser, Ceresoli, Vanderbilt, and Resta, 2005) solved this by expressing the total orbital magnetization of a crystal as a Brillouin-zone (BZ) average over Berry-phase-like quantities, including an explicit 'itinerant' term that depends on the wave-vector (k\mathbf{k}) dependence of the Bloch-state expansion coefficients over localized Wannier orbitals (localized basis functions constructed from the occupied bands). Nikolaev and Solovyev (2014) applied this theory to prototype Mott insulators — materials whose insulating character arises from strong electron-electron repulsion rather than band filling — in the distorted perovskite structure: orthorhombic YTiO3 (a rare example of a ferromagnetic Mott insulator, ordering below about 29 K), LaMnO3 and YVO3 (weak ferromagnets produced by antisymmetric Dzyaloshinskii–Moriya exchange, i.e., spin-canting interactions allowed by the broken inversion symmetry of the distorted lattice), and monoclinic YVO3 (imperfect cancellation between two crystallographic sublattices). Using effective Hubbard-type models (multi-orbital lattice models retaining the dominant on-site Coulomb repulsion of the magnetically active bands) derived from first-principles calculations and solved with spin-orbit coupling in mean-field Hartree-Fock approximation, they recovered the local orbital moments from the site-diagonal density matrix via μ0=−μBTr{L^D^}\boldsymbol{\mu}^{0}=-\mu_{\mathrm{B}}\mathrm{Tr}\{\hat{\mathbf{L}}\hat{\mathcal{D}}\} (L^\hat{\mathbf{L}} the angular momentum operator, μB\mu_{\mathrm{B}} the Bohr magneton) and evaluated the itinerant correction ΔM\Delta\mathbf{M} from the modern theory. The net ΔM\Delta\mathbf{M} turned out to be negligible for all compounds, but its BZ integrand M(k)\mathbf{M}(\mathbf{k}) is large and follows the behavior of the Chern invariants (integer topological indices computed from the Berry curvature of the occupied bands); equivalently, in real space the nonlocal (inter-site, off-diagonal) circulations are substantially larger than the local ones, with the small total resulting from strong cancellation. This sits in tension with the view — supported by Bianco and Resta's proof that a gauge-invariant local orbital-magnetization density exists in the thermodynamic limit — that orbital magnetization is an essentially local quantity, and with the practice of spin-density-functional theory (SDFT, whose local spin density approximation LSDA is built from the homogeneous electron gas and contains no current dependence). In principle, current-spin-density-functional theory (current-SDFT, Vignale and Rasolt, 1987) uses the orbital current density as an additional variational variable, but the explicit form of its exchange-correlation energy in terms of the orbital currents is largely unknown, and practical approximations remain undeveloped for strongly correlated, spin-orbit-coupled insulators.

Problem Statement

For ferromagnetic insulating perovskite transition-metal oxides such as orthorhombic YTiO3, LaMnO3, and YVO3 and monoclinic YVO3, the modern theory of orbital magnetization predicts that the itinerant correction to the net orbital moment is small only through near-complete cancellation of large, strongly wave-vector-dependent contributions whose BZ distribution tracks the Chern invariants, so that in a Wannier representation the nonlocal (site-off-diagonal) circulations substantially exceed the local ones. The problem, as posed by Nikolaev and Solovyev, comprises two scientifically linked questions about this nonlocal part: (1) Physical significance — can these large nonlocal contributions to the orbital magnetization be measured or experimentally separated from the local moments (which are accessible, e.g., by x-ray magnetic circular dichroism), or do they contribute measurably to other physical properties or response functions where the cancellation does not occur? (2) Density-functional significance — does this large, essentially nonlocal, current-dependent part of the orbital magnetization contribute to the exchange-correlation energy, for example within current-spin-density-functional theory; can an explicit, computationally tractable exchange-correlation functional of the orbital current density be constructed that captures these contributions; and would including them improve practical density-functional descriptions of orbital magnetism relative to purely local (site-diagonal) treatments — in other words, how general is the assumption that the orbital magnetization is a purely site-diagonal (on-site) quantity in a Wannier representation? The two questions are linked, and a resolution of the stated problem is expected to address both; the enumerated sub-parts of each question are concrete aspects of that question rather than separately required deliverables. For question (1), an acceptable resolution must either name a specific observable or response function and argue or estimate its sensitivity to the off-diagonal circulations, or establish that no gauge-invariant observable isolates them. For question (2), an acceptable resolution must state whether the current-dependent part enters the exchange-correlation energy and, if so, exhibit an explicit approximate functional form, or show why no such functional can capture it. The common objective is to determine the physical reality and the theoretical weight of the nonlocal itinerant orbital magnetization in these prototypical Mott ferromagnets.

References

  1. S. A. Nikolaev and I. V. Solovyev, 'Orbital magnetization of insulating perovskite transition-metal oxides with a net ferromagnetic moment in the ground state', Phys. Rev. B 89, 064428 (2014). DOI: 10.1103/PhysRevB.89.064428; arXiv:1311.4263; https://arxiv.org/abs/1311.4263
  2. Di Xiao, Junren Shi, and Qian Niu, 'Berry Phase Correction to Electron Density of States in Solids', Phys. Rev. Lett. 95, 137204 (2005). DOI: 10.1103/PhysRevLett.95.137204; https://doi.org/10.1103/PhysRevLett.95.137204
  3. T. Thonhauser, Davide Ceresoli, David Vanderbilt, and R. Resta, 'Orbital Magnetization in Periodic Insulators', Phys. Rev. Lett. 95, 137205 (2005). DOI: 10.1103/PhysRevLett.95.137205; https://doi.org/10.1103/PhysRevLett.95.137205
  4. G. Vignale and Mark Rasolt, 'Density-functional theory in strong magnetic fields', Phys. Rev. Lett. 59, 2360 (1987). DOI: 10.1103/PhysRevLett.59.2360; https://doi.org/10.1103/PhysRevLett.59.2360
  5. Raffaello Bianco and Raffaele Resta, 'Orbital Magnetization as a Local Property', Phys. Rev. Lett. 110, 087202 (2013). DOI: 10.1103/PhysRevLett.110.087202; https://doi.org/10.1103/PhysRevLett.110.087202
  6. F. Aryasetiawan and K. Karlsson, 'Modern theory of orbital magnetic moment in solids', J. Phys. Chem. Solids 128, 87 (2019). DOI: 10.1016/j.jpcs.2017.12.004; https://doi.org/10.1016/j.jpcs.2017.12.004
  7. T. J. Sjöstrand, K. Karlsson, and F. Aryasetiawan, 'Influence of correlations on the orbital magnetization of the spin-1/2 Haldane-Hubbard model', Phys. Rev. B 100, 054427 (2019). DOI: 10.1103/PhysRevB.100.054427; https://doi.org/10.1103/PhysRevB.100.054427
  8. Zhimou Zhou, Shishir Kumar Pandey, and Ji Feng, 'Dynamical correlation enhanced orbital magnetization in VI3', Phys. Rev. B 103, 035137 (2021). DOI: 10.1103/PhysRevB.103.035137; https://doi.org/10.1103/PhysRevB.103.035137
  9. Tarik P. Cysne, Sayantika Bhowal, Giovanni Vignale, and Tatiana G. Rappoport, 'Orbital Hall effect in bilayer transition metal dichalcogenides: From the intra-atomic approximation to the Bloch states orbital magnetic moment approach', Phys. Rev. B 105, 195421 (2022). DOI: 10.1103/PhysRevB.105.195421; https://doi.org/10.1103/PhysRevB.105.195421
  10. Yannick J. Franzke and Christof Holzer, 'Current density functional framework for spin-orbit coupling: Extension to periodic systems', J. Chem. Phys. 160, 184101 (2024). DOI: 10.1063/5.0209704; arXiv:2403.14420; https://doi.org/10.1063/5.0209704
  11. Giovanni Vignale, Junren Shi, Di Xiao, and Qian Niu, 'Ward identities and orbital magnetization in current density functional theory', Phys. Rev. B 114, 115113 (2026). DOI: 10.1103/bdmq-3139; arXiv:2605.14336; https://doi.org/10.1103/bdmq-3139
  12. Igor Solovyev, Sergey Nikolaev, and Akihiro Tanaka, 'Altermagnetism and weak ferromagnetism', npj Quantum Materials 11, 54 (2026). DOI: 10.1038/s41535-026-00900-9; https://doi.org/10.1038/s41535-026-00900-9