Souza and Vanderbilt (Phys. Rev. B 77, 054438 (2008)) proved an exact dichroic f-sum rule: the frequency-integrated magnetic circular dichroism (MCD) spectrum of an insulating crystal equalsMSR(I)\mathbf{M}_{\mathrm{SR}}^{(\mathrm{I})}, the gauge-invariant part of the self-rotation of the occupied Wannier orbitals, which is one of two individually gauge-invariant pieces of the total orbital magnetization,Morb=MSR(I)+ΔM\mathbf{M}_{\mathrm{orb}} = \mathbf{M}_{\mathrm{SR}}^{(\mathrm{I})} + \Delta\mathbf{M}. The itinerant remainderΔM\Delta\mathbf{M}is computable from the electronic ground state and is in principle measurable by combining a wide-frequency magneto-optical measurement with an independent gyromagnetic determination ofMorb\mathbf{M}_{\mathrm{orb}}. The open problem is to exploit this decomposition for real materials: first-principles evaluation of both terms in real crystals (including metals) to quantify the itinerant contribution and to test whether orbital magnetism in solids is atomic-like and adequately described by standard spin-density functional theory, and experimental extraction ofΔM\Delta\mathbf{M}from convergent MCD spectra combined with gyromagnetic measurements, including identification of materials where the itinerant contribution is appreciable.

Background

The orbital magnetizationMorb\mathbf{M}_{\mathrm{orb}}— the magnetic moment per unit volume carried by the electrons' orbital motion, as opposed to their spin — is subtle to define in a crystal, because the circulating currents that generate it are not confined to atoms and can flow over the whole sample. The "modern theory" of orbital magnetization (Thonhauser et al. 2005; Ceresoli et al. 2006; Xiao et al. 2005; Shi et al. 2007) solved this long-standing problem by writingMorb\mathbf{M}_{\mathrm{orb}}as a bulk, gauge-invariant property of the occupied Bloch states, expressed in terms of Berry-phase quantities. It is valid for conventional insulators, for Chern insulators (insulators whose Hall conductivity is quantized because the occupied bands carry a nonzero Chern number, a topological invariant of the band structure), and — in the formulation of Shi et al. (2007), which works from the outset with a periodic crystal and also covers interacting electrons — for metals. For insulators an intuitive real-space picture exists: the occupied bands are represented by localized Wannier functions (localized orbitals that span the occupied bands of the periodic crystal), andMorb\mathbf{M}_{\mathrm{orb}}then decomposes into the self-rotation of each Wannier orbital about its own center plus an "itinerant" circulation associated with the center-of-mass motion of overlapping neighboring orbitals, both in the interior and at the surface of the sample.

Magnetic circular dichroism (MCD) is the difference in absorption of right- and left-circularly-polarized light by a medium with broken time-reversal symmetry; it is measured by the antisymmetric part of the optical conductivity, whose absorptive spectrum is given by the Kubo–Greenwood formula (a standard expression for optical transition rates between electronic states). Souza and Vanderbilt (2008) proved an exact "dichroic f-sum rule" for the frequency integral of this spectrum, recasting it as a ground-state property: in insulators it equals the gauge-invariant part of the Wannier self-rotation, denotedMSR(I)\mathbf{M}_{\mathrm{SR}}^{(\mathrm{I})}. The total orbital magnetization then splits exactly as

Morb=MSR(I)+ΔM,\mathbf{M}_{\mathrm{orb}} = \mathbf{M}_{\mathrm{SR}}^{(\mathrm{I})} + \Delta\mathbf{M},

where both terms are individually gauge-invariant (i.e., independent of the arbitrary unitary rotation chosen among occupied states), while the further division ofΔM\Delta\mathbf{M}into a self-rotation remainder and interior/surface itinerant-circulation pieces is gauge-dependent and hence not separately observable. A key consequence is that the integrated MCD spectrum does not measureMorb\mathbf{M}_{\mathrm{orb}}itself — contrary to what had been assumed since the atomic sum rule of Hasegawa and Howard (1961) — and that combining a magneto-optical measurement over a sufficiently wide frequency range (givingMSR(I)\mathbf{M}_{\mathrm{SR}}^{(\mathrm{I})}via the sum rule) with an independent gyromagnetic determination ofMorb\mathbf{M}_{\mathrm{orb}}(an Einstein–de Haas-type experiment relating changes of magnetization to changes of angular momentum) would separately determine the localized and itinerant contributions.

What the source paper left open is the exploitation of this decomposition. First-principles calculations of the two terms for real materials are needed to learn how large the itinerant part is and whether orbital magnetism in solids is "atomic-like" — an assumption made routinely, for instance, in interpreting x-ray MCD (XMCD) sum rules (which extract site-projected orbital moments from core-level absorption spectra) and in constructing orbital-polarization corrections to spin-density functional theory (SDFT, the standard Kohn-Sham framework in which the basic variable is the spin-resolved density). Such calculations would clarify whether standard SDFT can describe orbital magnetism in solids or whether extended frameworks — LDA+U with orbital-polarization terms, or current-density functional theory, in which the orbital current is an explicit basic variable — are required. Experimentally, candidate materials whereΔM\Delta\mathbf{M}is appreciable must be identified; the source suggests wide-bandss-ppparamagnets and diamagnets, "zero-magnetization ferromagnets" whose orbital moment cancels the spin moment, and proposes testing Yafet's 1963 conjecture that the anomalously largegg-factors of bismuth arise from itinerant circulations of precisely the kind captured byΔM\Delta\mathbf{M}. Subsequent literature has computed total orbital magnetizations of real metals and realized the adjacent Hall-sum-rule connection in ultracold atoms, while the dichroic-sum-rule decomposition for bulk crystals and its experimental implementation remain unrealized.

Problem Statement

Determine the localized-versus-itinerant decomposition of the orbital magnetization of real crystalline materials, exploiting the exact sum-rule splitMorb=MSR(I)+ΔM\mathbf{M}_{\mathrm{orb}} = \mathbf{M}_{\mathrm{SR}}^{(\mathrm{I})} + \Delta\mathbf{M}of Souza and Vanderbilt (2008). Concretely: (1) compute, from first principles, the two individually gauge-invariant terms for real crystals with appreciable orbital magnetization — including metals and other systems beyond conventional Wannier-representable insulators, for which a fully general derivation and practical computational scheme remain to be settled — establishing the magnitude and material dependence of the itinerant remainderΔM\Delta\mathbf{M}relative to the self-rotation term, and assess, by confronting spin-density-functional calculations that include the itinerant terms with experimental orbital magnetizations, whether orbital magnetism in solids is atomic-like or requires beyond-SDFT treatments (orbital-polarization corrections, LDA+U, or current-density functional theory); and (2) realize the decomposition experimentally, by measuring the magnetic circular dichroism spectrum over a frequency range wide enough to converge the dichroic f-sum rule and combining it with an independent gyromagnetic determination of the total orbital magnetization, so thatΔM=Morb−MSR(I)\Delta\mathbf{M} = \mathbf{M}_{\mathrm{orb}} - \mathbf{M}_{\mathrm{SR}}^{(\mathrm{I})}is extracted with quantified uncertainty, and by identifying materials where the itinerant contribution is appreciable — for example wide-bandss-ppparamagnets or diamagnets, zero-magnetization ferromagnets, or bismuth as a test of the Yafet conjecture on its anomalously largegg-factors. The two routes are complementary rather than cumulative: a completed first-principles determination (1) or a completed experimental extraction (2) each resolves the localized-versus-itinerant decomposition for the materials studied, and each settles the accompanying question of whether standard SDFT suffices by confronting the resulting decomposition with spin-density-functional calculations; a combined determination, the strongest form, cross-validates the two. Either way the objective is the same: a quantified, experimentally anchored answer to how atomic-like orbital magnetism in solids actually is, and whether standard spin-density functional theory suffices to describe it.

References

  1. I. Souza and D. Vanderbilt, "Dichroic f-sum rule and the orbital magnetization of crystals", Phys. Rev. B 77, 054438 (2008). DOI: 10.1103/PhysRevB.77.054438; arXiv:0709.2389. https://doi.org/10.1103/PhysRevB.77.054438
  2. T. Thonhauser, D. Ceresoli, D. 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
  3. D. Ceresoli, T. Thonhauser, D. Vanderbilt, and R. Resta, "Orbital magnetization in crystalline solids: Multi-band insulators, Chern insulators, and metals", Phys. Rev. B 74, 024408 (2006). DOI: 10.1103/PhysRevB.74.024408. https://doi.org/10.1103/PhysRevB.74.024408
  4. D. Xiao, J. Shi, and Q. 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
  5. J. Shi, G. Vignale, D. Xiao, and Q. Niu, "Quantum Theory of Orbital Magnetization and Its Generalization to Interacting Systems", Phys. Rev. Lett. 99, 197202 (2007). DOI: 10.1103/PhysRevLett.99.197202. https://doi.org/10.1103/PhysRevLett.99.197202
  6. M. G. Lopez, D. Vanderbilt, T. Thonhauser, and I. Souza, "Wannier-based calculation of the orbital magnetization in crystals", Phys. Rev. B 85, 014435 (2012). DOI: 10.1103/PhysRevB.85.014435; arXiv:1112.1938. https://doi.org/10.1103/PhysRevB.85.014435
  7. T. Thonhauser, "Theory of Orbital Magnetization in Solids", Int. J. Mod. Phys. B 25, 1429 (2011). DOI: 10.1142/S0217979211058912. https://doi.org/10.1142/S0217979211058912
  8. 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
  9. R. Resta, "Magnetic circular dichroism versus orbital magnetization", Phys. Rev. Research 2, 023139 (2020). DOI: 10.1103/PhysRevResearch.2.023139. https://doi.org/10.1103/PhysRevResearch.2.023139
  10. D. Seleznev and D. Vanderbilt, "Towards a theory of surface orbital magnetization", Phys. Rev. B 107, 115102 (2023). DOI: 10.1103/PhysRevB.107.115102; arXiv:2210.08736. https://doi.org/10.1103/PhysRevB.107.115102
  11. B. Ghosh, Y. Onishi, S.-Y. Xu, H. Lin, L. Fu, and A. Bansil, "Probing quantum geometry through optical conductivity and magnetic circular dichroism", Sci. Adv. 10, eado1761 (2024). DOI: 10.1126/sciadv.ado1761; arXiv:2401.09689. https://doi.org/10.1126/sciadv.ado1761
  12. P. Li, K. Jiang, Z. Wang, J. Kang, and Y. Zhang, "Contribution of remote bands to orbital magnetization in twisted bilayer graphene", arXiv:2603.03702 (2026). https://arxiv.org/abs/2603.03702
  13. B. Bermond, L. Peralta Gavensky, A. Defossez, and N. Goldman, "Dichroism from Chiral Thermoelectric Probes: Generalized Sum Rules for Orbital and Heat Magnetizations", Phys. Rev. B 113, 205127 (2026). DOI: 10.1103/y88d-tx4l; arXiv:2511.21599. https://doi.org/10.1103/y88d-tx4l
  14. D. T. Tran, A. Dauphin, A. G. Grushin, P. Zoller, and N. Goldman, "Probing topology by “heating”: Quantized circular dichroism in ultracold atoms", Sci. Adv. 3, e1701207 (2017). DOI: 10.1126/sciadv.1701207. https://doi.org/10.1126/sciadv.1701207
  15. L. Asteria, D. T. Tran, T. Ozawa, M. Tarnowski, B. S. Rem, N. Fläschner, K. Sengstock, N. Goldman, and C. Weitenberg, "Measuring quantized circular dichroism in ultracold topological matter", Nat. Phys. 15, 449 (2019). DOI: 10.1038/s41567-019-0417-8. https://doi.org/10.1038/s41567-019-0417-8
  16. M. dos Santos Dias and S. Lounis, "Insights into the orbital magnetism of noncollinear magnetic systems", Proc. SPIE 10357, Spintronics X (2017). DOI: 10.1117/12.2275305; arXiv:1707.04518. https://doi.org/10.1117/12.2275305
  17. S. Okamoto, A. Moreo, N. Nagaosa, and S. S. P. Parkin, "Berry curvature and field-induced intrinsic anomalous Hall effect in an antiferromagnet FeTe", arXiv:2604.11583 (2026). https://arxiv.org/abs/2604.11583