Topological orbital ferromagnetism in γ-FeMn beyond the virtual-crystal approximation: disorder-inclusive theory, chirality expansion of the Hall response, inverse effects, and experimental detection

The noncoplanar 3Q antiferromagnet γ-FeMn was predicted from first principles (Hanke et al., Sci. Rep. 7, 41078, 2017) to be a prototypical topological orbital ferromagnet: a spin-compensated alloy whose macroscopic magnetization and anomalous Hall conductivity (AHC) arise from the scalar spin chirality of its spin texture rather than from spin-orbit coupling, with orbital magnetization of order 0.1 μB per four-atom cell and AHC up to about 1000 S/cm under [111] strain. That prediction rested on a virtual-crystal (VCA) treatment of the chemical disorder, left the higher-order chirality contributions to the AHC analytically undecomposed, conjectured but did not quantify inverse effects (topological orbital magnetostriction and a purely topology-driven spin Hall effect), deferred thin-film and interface studies, and awaited experimental confirmation. The problem is to place chirality-driven topological orbital magnetism in γ-FeMn on the same footing as conventional orbital magnetism: a disorder-inclusive quantitative theory, a microscopic chirality decomposition of the Hall response, quantitative inverse-effect predictions, an ab initio account of how reduced dimensionality (surfaces, interfaces, and finite-size effects) modifies these responses, and a detection-and-attribution scheme for the chirality-driven response that isolates it from residual spin-orbit and extrinsic mechanisms. A 2025 experiment reporting a surface topological Hall effect in γ-FeMn films partially advances the detection facet, but the core remains open.

Background

In a solid, the orbital magnetization (OM) is the magnetic moment generated by the electrons' orbital motion, as distinct from their spin moments; it is measured, for example, by x-ray magnetic circular dichroism (XMCD), and its rigorous ground-state theory — the Berry-phase or 'modern' theory of orbital magnetization — expresses the OM as a bulk property of the Bloch eigenstates and their Berry curvature in reciprocal space. In ordinary ferromagnets, spontaneous OM is a manifestation of the spin-orbit interaction (SOC), which partially lifts the quenching of orbital angular momentum by the crystal field. However, in magnets whose neighboring spins are noncoplanar, an effective 'emergent' magnetic field appears that can replace SOC as the degeneracy-breaking mechanism. Its strength is quantified by the scalar spin chirality $\kappa=\mathbf{S}_i\cdot(\mathbf{S}_j\times\mathbf{S}_k)$ between three neighboring spins — the same quantity that endows skyrmions — nanoscale whirl-like spin textures — with their topological charge. A finite chirality-driven OM, existing even when SOC is switched off, is called topological orbital magnetization (TOM), and it is microscopically and symmetry-wise related to the anomalous Hall conductivity (AHC, the transverse conductivity at zero external field); in this context the chirality-driven part of the AHC is often called the topological Hall effect (THE). Disordered face-centered-cubic (fcc, the γ phase) alloys Fe$_x$Mn$_{1-x}$ realize, for $0.35<x<0.80$, a noncoplanar so-called 3Q spin structure — a superposition of three spin-density waves whose moments point toward the center of a tetrahedron, with maximal scalar chirality at the tetrahedral polar angle $\theta\approx54.7°$ and zero net spin moment. On the ideal fcc lattice the chirality contributions from the four equivalent (111) faces cancel, so strain along [111] is needed to unbalance them and expose a net chirality-driven OM and AHC along that axis. Hanke, Freimuth, Blügel and Mokrousov (Sci. Rep. 7, 41078, 2017; arXiv:1610.07573) computed, from density functional theory in the virtual crystal approximation (VCA — replacing the mixed Fe/Mn sublattice by an averaged nuclear charge, which avoids spurious periodic ordering of Fe and Mn atoms), that strained γ-FeMn carries a TOM of order $0.05$–$0.1\,\mu_\mathrm{B}$ per four-atom magnetic cell that reverses sign between tensile and compressive [111] strain and changes sign with composition near $x\approx0.5$–$0.6$, together with a gigantic AHC reaching about $1000\,\mathrm{S/cm}$ — both essentially unchanged when SOC is included. They called such a spin-compensated material a topological orbital ferromagnet. The calculation also uncovered an asymmetry that defines one part of the open problem: as the texture angle $\theta$ is varied, the OM tracks the scalar chirality $\kappa(\theta)$ almost perfectly (a 'topological orbital susceptibility' proportionality), whereas the AHC deviates strongly from $\kappa(\theta)$ — including a local minimum at the 3Q angle — behavior attributed to higher-order, non-adiabatic chirality terms that were not analytically decomposed. For atomic-scale textures like 3Q there is no small parameter controlling such an expansion, unlike the adiabatic limit of slowly varying textures studied in the skyrmion and domain-wall orbital-magnetism literature (e.g., Lux et al., Commun. Phys. 2018; dos Santos Dias et al., Nat. Commun. 2016). The authors were explicit about the limits of their treatment: for the anomalous Hall effect a coherent ab initio description of disordered alloys including extrinsic contributions (skew scattering and side jump, the two impurity-scattering mechanisms that produce the anomalous Hall effect in disordered alloys) existed (Lowitzer, Ködderitzsch and Ebert, PRL 105, 266604, 2010), whereas 'a coherent description of orbital magnetism in the presence of disorder has not been developed so far' — hence their recourse to VCA. They further conjectured an inverse phenomenon, topological orbital magnetostriction — strain of the crystal driven by the interaction of the TOM with an external field, so that reversing the field direction along [111] would switch the crystal between expansion and contraction — and an analogous purely topology-driven spin Hall effect in noncoplanar magnets, neither of which they quantified. They also noted that TOM magnitudes should be enhancable in thin films with noncollinear textures, and motivated experimental detection of the predicted large AHC and TOM. Subsequent work has built relevant machinery without closing these questions: model-level studies of disorder effects on the OM (Wang et al., arXiv:2007.10548), a real-space spectral method for OM in large disordered systems (Vidarte et al., arXiv:2512.01575), a reformulation of band-resolved OM valid for degenerate bands (Mitscherling et al., arXiv:2512.09051), second-order chirality expansions for magneto-optical responses of γ-FeMn (Yang et al., PRB 106, 174427, 2022), scanning-tunneling and ab initio studies of topological orbital moments in atomic-scale skyrmion and 3Q lattices on surfaces (Nickel et al., npj Spintronics 2025 and Phys. Rev. B 108, L180411 (2023); Beyer et al., arXiv:2506.05091), model theories of a topological spin Hall effect in antiferromagnets (Nakazawa et al., arXiv:2304.02850), and γ-FeMn thin-film growth and magnetotransport experiments (Shiomi, 2018; Li et al., 2023, 2025).

Problem Statement

Establish chirality-driven topological orbital magnetism and Hall responses in $\gamma$-Fe$_x$Mn$_{1-x}$ beyond the clean-limit virtual-crystal prediction of Hanke et al. (2017), jointly along the following linked fronts that together define completion of the objective — (i) Theory of disorder: develop and execute a quantitative first-principles description of the TOM and AHC of the 3Q state that treats chemical disorder beyond the virtual crystal approximation (e.g., supercell or coherent-potential-type (CPA, an effective-medium treatment of substitutional disorder) approaches), quantifying intrinsic and, where applicable, extrinsic (skew-scattering/side-jump) contributions, and comparing against the VCA baseline under the same strain, texture-angle, and composition conventions. (ii) Chirality decomposition: derive a microscopic decomposition of the AHC into powers/functions of the scalar spin chirality and texture angles, identifying the leading chirality-proportional term and the higher-order, non-adiabatic terms responsible for the AHC's local minimum at the 3Q angle, with its scaling with the polar and azimuthal texture angles θ and φ, the [111] strain δ, and the composition x. (iii) Inverse effects: provide quantitative, experimentally testable predictions for the conjectured topological orbital magnetostriction (including whether the predicted switch between expansion and compression along [111] upon field reversal occurs, and its magnitudes/field/temperature dependence) and for a purely topology-driven spin Hall effect in noncoplanar magnets, so that this conjectured inverse functionality can be established or refuted. (iv) Reduced dimensionality: determine by ab initio calculation how finite size, surfaces, and interfaces modify the TOM and its transport signatures in noncollinear γ-FeMn-class thin films, quantifying the conjectured enhancement. (v) Experimental confirmation: determine how the chirality-driven AHC and TOM in γ-FeMn samples can be detected with attribution controls that isolate the chirality mechanism from residual SOC and extrinsic effects — exploiting the predicted strain-sign reversal of the TOM, the composition-dependent sign change, and consistency with the chirality-decomposed response — the front being resolved by a specified, falsifiable detection-and-interpretation protocol (whether designed afresh or assessed against existing γ-FeMn data) rather than by an executed measurement. These fronts are inseparable parts of one objective: a claim that topological orbital ferromagnetism is established requires the disorder-robustness of the prediction (i), an understood microscopic response function (ii), its inverse-functionality consequences (iii), its tunability in reduced geometry (iv), and an experiment that discriminates the chirality mechanism (v). Each front is resolved by the specific deliverable named for it in (i)–(v); the objective is completed only when all five fronts are resolved, and partial progress on any subset is assessed front by front rather than counted as resolution of the objective.

References

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