Topological Orbital Magnetization (TOM)

• Topological orbital magnetization (TOM) is orbital magnetization arising from nonlocal geometric effects, such as Berry curvature, emergent fields, and Chern-Simons forms.
• It manifests in various settings including quantum spin Hall nanoislands, skyrmion textures, and Chern insulators, highlighting its role in topological and magnetic materials.
• TOM influences transport properties and optical responses, offering novel insights for spintronics, orbitronics, and topological device engineering.

Searching arXiv for recent and foundational papers on topological orbital magnetization and closely related formulations. Topological orbital magnetization (TOM) denotes orbital magnetization whose origin is tied to topology rather than to a purely local, atom-centered orbital moment. In the literature, the term is used for several closely related settings: orbital edge currents of quantum spin Hall nanoislands, chirality-driven orbital moments of skyrmions and noncoplanar magnets, Berry-curvature-driven orbital magnetization in Chern insulators and topological metals, and adiabatically pumped orbital magnetization described by Chern-Simons forms (Potasz et al., 2015, Dias et al., 2016, Pershoguba et al., 2021, Ren et al., 9 Jan 2025). A common thread is that the relevant orbital moment is controlled by geometric objects—Berry curvature, emergent magnetic fields, or Chern-Simons forms—and therefore is not, in general, captured by an atom-centered expectation value of $\mathbf{L}$ (Hanke et al., 2016, Hanke et al., 2016).

1. Terminology and scope

Orbital magnetization in a solid is the magnetic moment arising from circulating charge currents. In isolated atoms, orbital moments are protected by rotational symmetry, whereas in crystals the crystal field typically quenches atomic orbital moments, so that magnetization in most magnetic solids is predominantly spin in origin (Potasz et al., 2015). TOM designates the regime in which this expectation fails because topology stabilizes orbital currents that are robust, amplified, or even dominant (Dias et al., 2016, Hanke et al., 2016).

The literature does not use the term in a single universal way. In quantum spin Hall nanoislands, TOM refers to orbital magnetization carried by topological edge states, with a singly occupied Kramers doublet converting a helical spin current into a net charge current along the boundary (Potasz et al., 2015). In skyrmions and related noncoplanar textures, TOM denotes the chirality-driven contribution to orbital magnetization generated by the emergent magnetic field of the spin texture, and it can persist even when spin–orbit coupling is switched off (Dias et al., 2016). In Chern insulators, TOM denotes orbital magnetization arising from Berry curvature, valley polarization, and a nonzero Chern number (Pershoguba et al., 2021). In altermagnets, the term is used for the topological component of the modern orbital magnetization that is tied to the intrinsic anomalous Hall conductivity by a generalized Středa relation (Zhao et al., 24 Jun 2026).

A recurring misconception is that TOM is simply an unusually large atomic orbital moment. The modern literature instead treats it as an itinerant, geometric, or emergent-field response. This is explicit in the proposal of “topological orbital ferromagnets,” defined as systems whose macroscopic magnetization is dominated by orbital magnetization originating from nontrivial topology in real space rather than from spin–orbit interaction (Hanke et al., 2016, Hanke et al., 2016).

2. Geometric and thermodynamic formulations

A representative modern-theory expression used for orbital magnetization is

$$\mathbf{M} = \frac{e}{\hbar}\int_{\mathrm{BZ}} \frac{d^3k}{(2\pi)^3}\left[ f_{n\mathbf{k}}\,\mathbf{m}_{n}(\mathbf{k}) + \big(E_{n\mathbf{k}}-\mu\big)\,\mathbf{\Omega}_{n}(\mathbf{k}) \right],$$

where $\mathbf{\Omega}_{n}(\mathbf{k})$ is Berry curvature and $\mathbf{m}_{n}(\mathbf{k})$ is the orbital magnetic moment of Bloch wave packets (Zhang et al., 2023). In correlated systems, the same quantity can be written in terms of the full Green’s function and vertex functions; within dynamical mean-field theory for the Kane–Mele–Hubbard model, the orbital magnetization of the quantum spin Hall insulating phase with inversion symmetry is renormalized by the bulk quasiparticle weight, while it vanishes for the in-plane antiferromagnetic phase with trivial topology (Nourafkan et al., 2014).

For anomalous Hall systems, the thermodynamic connection is particularly direct. In MnTe-type altermagnets, the generalized Středa relation ties the intrinsic anomalous Hall conductivity to the orbital magnetization by

$\sigma_{xy}=-e\frac{\partial M_z}{\partial \mu},$

and the paper identifies the relevant $M_z$ with the topological component of the modern orbital magnetization (Zhao et al., 24 Jun 2026). This relation makes clear that TOM is not merely a static moment; it is also the thermodynamic quantity governing intrinsic anomalous transport.

In adiabatically driven topological insulators, the bulk contribution can instead be expressed through a Chern-Simons form. For coherent spin precession, the topological bulk contribution to orbital magnetization is

$M_z^t = -\frac{e}{T}\,\frac{\theta}{2\pi},$

where $T$ is the precession period and $\theta$ is a Chern-Simons 3-form over combined momentum–time parameter space (Ren et al., 9 Jan 2025). This places TOM within the same geometric framework as axion electrodynamics, Thouless pumping, and higher-Chern responses.

The modern Berry-phase formulation is therefore not a technical refinement but the essential language of TOM. In chemically or structurally inhomogeneous systems, in Chern insulators, and in noncollinear magnets, the atom-centered approximation fails qualitatively or quantitatively because it omits the nonlocal Berry-phase contributions that define the orbital response (Hanke et al., 2016).

3. Real-space topology: chirality, skyrmions, and topological orbital ferromagnets

For noncoplanar magnetic textures, the basic local invariant is scalar spin chirality,

$$C_{123}=\mathbf{S}_1\cdot(\mathbf{S}_2\times\mathbf{S}_3),$$

and in the continuum the corresponding topological charge density is

$$\mathbf{M} = \frac{e}{\hbar}\int_{\mathrm{BZ}} \frac{d^3k}{(2\pi)^3}\left[ f_{n\mathbf{k}}\,\mathbf{m}_{n}(\mathbf{k}) + \big(E_{n\mathbf{k}}-\mu\big)\,\mathbf{\Omega}_{n}(\mathbf{k}) \right],$$0

The emergent magnetic field generated by such textures acts on itinerant electrons and produces orbital magnetization even without spin–orbit coupling (Dias et al., 2016, Göbel et al., 2024). In the language of spin textures, TOM is the second-order gradient contribution to orbital magnetization, distinct from the first-order chiral orbital magnetization that already appears in one-dimensional textures with spin–orbit coupling (Lux et al., 2017).

The skyrmion case is the canonical real-space realization. In the absence of spin–orbit coupling, the local emergent field is

$$\mathbf{M} = \frac{e}{\hbar}\int_{\mathrm{BZ}} \frac{d^3k}{(2\pi)^3}\left[ f_{n\mathbf{k}}\,\mathbf{m}_{n}(\mathbf{k}) + \big(E_{n\mathbf{k}}-\mu\big)\,\mathbf{\Omega}_{n}(\mathbf{k}) \right],$$1

and the resulting TOM can be understood as a Landau–Peierls response to this emergent field (Lux et al., 2017). In the weak-spin–orbit, exchange-dominated regime, the integrated TOM of a skyrmion is quantized to a universal value of $$\mathbf{M} = \frac{e}{\hbar}\int_{\mathrm{BZ}} \frac{d^3k}{(2\pi)^3}\left[ f_{n\mathbf{k}}\,\mathbf{m}_{n}(\mathbf{k}) + \big(E_{n\mathbf{k}}-\mu\big)\,\mathbf{\Omega}_{n}(\mathbf{k}) \right],$$2 at $$\mathbf{M} = \frac{e}{\hbar}\int_{\mathrm{BZ}} \frac{d^3k}{(2\pi)^3}\left[ f_{n\mathbf{k}}\,\mathbf{m}_{n}(\mathbf{k}) + \big(E_{n\mathbf{k}}-\mu\big)\,\mathbf{\Omega}_{n}(\mathbf{k}) \right],$$3, while finite Rashba spin–orbit coupling can enhance TOM by more than an order of magnitude and make it strongly dependent on helicity and band filling (Lux et al., 2017, Lux et al., 2018). This is the basis of the proposed “chiral orbitronics” program, in which orbital degrees of freedom are engineered through spin texture and spin–orbit coupling (Lux et al., 2018).

The same logic extends from skyrmions to bulk antiferromagnets with noncoplanar order. In $$\mathbf{M} = \frac{e}{\hbar}\int_{\mathrm{BZ}} \frac{d^3k}{(2\pi)^3}\left[ f_{n\mathbf{k}}\,\mathbf{m}_{n}(\mathbf{k}) + \big(E_{n\mathbf{k}}-\mu\big)\,\mathbf{\Omega}_{n}(\mathbf{k}) \right],$$4-FeMn, first-principles calculations predict an entirely topological orbital magnetization in the noncoplanar $$\mathbf{M} = \frac{e}{\hbar}\int_{\mathrm{BZ}} \frac{d^3k}{(2\pi)^3}\left[ f_{n\mathbf{k}}\,\mathbf{m}_{n}(\mathbf{k}) + \big(E_{n\mathbf{k}}-\mu\big)\,\mathbf{\Omega}_{n}(\mathbf{k}) \right],$$5 state, originating from scalar spin chirality and persisting even when spin–orbit coupling is set to zero (Hanke et al., 2016). Under suitable strain, the orbital magnetization reaches about $$\mathbf{M} = \frac{e}{\hbar}\int_{\mathrm{BZ}} \frac{d^3k}{(2\pi)^3}\left[ f_{n\mathbf{k}}\,\mathbf{m}_{n}(\mathbf{k}) + \big(E_{n\mathbf{k}}-\mu\big)\,\mathbf{\Omega}_{n}(\mathbf{k}) \right],$$6 per four-atom unit cell and the anomalous Hall conductivity reaches $$\mathbf{M} = \frac{e}{\hbar}\int_{\mathrm{BZ}} \frac{d^3k}{(2\pi)^3}\left[ f_{n\mathbf{k}}\,\mathbf{m}_{n}(\mathbf{k}) + \big(E_{n\mathbf{k}}-\mu\big)\,\mathbf{\Omega}_{n}(\mathbf{k}) \right],$$7, while the net spin magnetization vanishes (Hanke et al., 2016). This is the prototype of a topological orbital ferromagnet: a system that is antiferromagnetic in the spin channel yet ferromagnetic in the orbital channel.

Experimentally, the chirality-driven contribution can be isolated by soft x-ray spectroscopy. The proposed topological orbital magnetization ratio,

$$\mathbf{M} = \frac{e}{\hbar}\int_{\mathrm{BZ}} \frac{d^3k}{(2\pi)^3}\left[ f_{n\mathbf{k}}\,\mathbf{m}_{n}(\mathbf{k}) + \big(E_{n\mathbf{k}}-\mu\big)\,\mathbf{\Omega}_{n}(\mathbf{k}) \right],$$8

uses XMCD sum rules to separate the orbital moment induced by spin chirality from the ordinary spin–orbit-driven contribution (Dias et al., 2016).

4. Reciprocal-space topology and boundary-state realizations

A distinct but closely related usage of TOM appears in finite topological insulators. In Bi(111) bilayer nanoislands, the orbital moment of an eigenstate is defined by

$$\mathbf{M} = \frac{e}{\hbar}\int_{\mathrm{BZ}} \frac{d^3k}{(2\pi)^3}\left[ f_{n\mathbf{k}}\,\mathbf{m}_{n}(\mathbf{k}) + \big(E_{n\mathbf{k}}-\mu\big)\,\mathbf{\Omega}_{n}(\mathbf{k}) \right],$$9

and the total orbital magnetization is

$\mathbf{\Omega}_{n}(\mathbf{k})$0

A perpendicular magnetic field splits each edge-state Kramers doublet, and when only one state of the highest doublet is occupied, the helical spin current is converted into a net charge current around the edge, producing a large orbital edge magnetization (Potasz et al., 2015). In this setting, TOM is literally carried by topological boundary states. Its key hallmarks are linear size scaling, $\mathbf{\Omega}_{n}(\mathbf{k})$1, and robustness to disorder, edge roughness, and shape changes; even for Anderson disorder $\mathbf{\Omega}_{n}(\mathbf{k})$2, the average total magnetization at $\mathbf{\Omega}_{n}(\mathbf{k})$3 drops by less than $\mathbf{\Omega}_{n}(\mathbf{k})$4 (Potasz et al., 2015).

In Chern insulators, TOM is instead a bulk Berry-curvature response. For twisted graphene multilayers and some transition-metal dichalcogenide heterostructures, spontaneous valley polarization selects one sign of Berry curvature and produces an orbital magnetization perpendicular to the layers (Pershoguba et al., 2021). Circularly polarized light couples to the antisymmetric part of the dynamical polarizability tensor through

$\mathbf{\Omega}_{n}(\mathbf{k})$5

and, for a two-band Chern insulator, $\mathbf{\Omega}_{n}(\mathbf{k})$6 is given by an integral of Berry curvature over the Brillouin zone (Pershoguba et al., 2021). This provides a purely optical handle on TOM and underlies proposals for topological memory based on orbital magnetization, as well as optically written domain walls carrying topologically protected chiral edge modes (Pershoguba et al., 2021).

The static Haldane model and its Floquet extensions show the same reciprocal-space logic in a more explicit band-topological form. In Chern gaps, the orbital magnetization varies linearly with chemical potential with slope fixed by the Chern number, while in anomalous Floquet phases the Chern number does not fully account for the topology and orbital magnetization remains large because the Berry curvature distribution and edge-mode structure remain nontrivial (Topp et al., 2022, Dag et al., 2022).

5. Metallic and unconventional materials

Metallic systems broaden TOM beyond insulating and finite-size settings. A central case is Fe$\mathbf{\Omega}_{n}(\mathbf{k})$7Sn$\mathbf{\Omega}_{n}(\mathbf{k})$8, a kagome-derived ferromagnetic metal in which the orbital moment measured by Fe $\mathbf{\Omega}_{n}(\mathbf{k})$9-edge XMCD is large and directly tied to spin–orbit coupling and Weyl-band topology (Zhang et al., 2023). The orbital-to-spin ratio is

$\mathbf{m}_{n}(\mathbf{k})$0

in contrast to $\mathbf{m}_{n}(\mathbf{k})$1 for pure iron, and the material hosts a very large number of Weyl nodes within $\mathbf{m}_{n}(\mathbf{k})$2 of the Fermi level (Zhang et al., 2023). The term TOM is not used there, but the paper explicitly identifies the large orbital contribution as a quantitative manifestation of the same spin–orbit coupling that makes Fe$\mathbf{m}_{n}(\mathbf{k})$3Sn$\mathbf{m}_{n}(\mathbf{k})$4 a topological material and suggests calibrating the spin–orbit coupling in DFT by matching the measured orbital-to-spin ratio (Zhang et al., 2023). This suggests a metallic version of TOM in which orbital ferromagnetism is amplified by magnetization-dependent Weyl-node rearrangements.

Altermagnets provide another metallic setting. In MnTe-type altermagnets, the modern orbital magnetization is decomposed as

$\mathbf{m}_{n}(\mathbf{k})$5

where $\mathbf{m}_{n}(\mathbf{k})$6 is the Berry-curvature contribution (Zhao et al., 24 Jun 2026). The same work argues that residual orbital and spin magnetization is an intrinsic thermodynamic property, not an irrelevant remanence, because the intrinsic anomalous Hall conductivity obeys $\mathbf{m}_{n}(\mathbf{k})$7 (Zhao et al., 24 Jun 2026). Microscopically, a local crystal field combined with spin–orbit coupling produces a net orbital moment with angular dependence

$\mathbf{m}_{n}(\mathbf{k})$8

without invoking Dzyaloshinskii–Moriya interaction (Zhao et al., 24 Jun 2026).

The same broadening of scope appears in unconventional superconductors. In chiral superconductors, orbital magnetization cannot be reduced to quasiparticle currents because Bogoliubov quasiparticles do not carry a definite electric charge, and the microscopic expression instead involves a dressed photon vertex, interband coherence, and collective modes (Zhu et al., 18 Jan 2026). In rhombohedral tetralayer graphene, the superconducting contribution can enhance or suppress the normal-state orbital magnetization depending sensitively on band structure, while a generalized clapping mode contributes to the orbital response through its dressing of the photon vertex (Zhu et al., 18 Jan 2026). This extends TOM into Bogoliubov band topology.

6. Driven, Floquet, and transport extensions

Driven systems make explicit the distinction between geometric and topological pieces of orbital magnetization. A semiclassical theory of adiabatically induced orbital magnetization shows that, in general, the induced magnetization is gauge dependent because of the second Chern form of Berry curvatures; the only gauge-invariant instances are the orbital magnetoelectric effect and the periodic-evolution pumped orbital magnetization (Xiao et al., 2020). In two-dimensional metals and Chern insulators, the electric-field-induced intrinsic orbital magnetization is controlled by Berry connections and quantum metric dipoles, while in insulating pumps the time-averaged orbital magnetization is generated over one adiabatic cycle (Xiao et al., 2020).

Floquet systems add another layer. A general expression for the orbital magnetization of a Floquet system shows that it can be large not only for Chern insulators but also for anomalous phases where the Chern number does not fully account for the topology (Topp et al., 2022). In two-dimensional Floquet topological systems with flat bands, the drive can create anomalous Floquet phases with $\mathbf{m}_{n}(\mathbf{k})$9 and yet chiral edge modes exist, and the orbital magnetization is enhanced at half filling by the broken particle–hole symmetry of the Haldane model (Dag et al., 2022). In this sense, Floquet TOM is governed not only by band Chern numbers but also by the broader topology of the evolution operator and the edge-mode structure.

Spin dynamics provide a further extension. In antiferromagnetic topological insulators, coherent spin precession pumps orbital magnetization through a Chern-Simons form, with $\sigma_{xy}=-e\frac{\partial M_z}{\partial \mu},$0; for small cone angle the result is proportional to $\sigma_{xy}=-e\frac{\partial M_z}{\partial \mu},$1, while for large cone angle the magnetization can reach its natural unit $\sigma_{xy}=-e\frac{\partial M_z}{\partial \mu},$2 (Ren et al., 9 Jan 2025). When the pumped magnetization is spatially inhomogeneous, a dissipationless charge current is generated, and edge Thouless pumps contribute boundary terms that are tied to the gauge ambiguity of the Chern-Simons form (Ren et al., 9 Jan 2025). Closely related transport physics appears in skyrmion crystals: the topological Hall effect is accompanied by a topological orbital Hall effect even for $\sigma_{xy}=-e\frac{\partial M_z}{\partial \mu},$3 electrons without spin–orbit coupling, and antiferromagnetic skyrmions or bimerons can carry a topological orbital Hall conductivity without charge transport, in some cases orders of magnitude larger than the topological spin Hall conductivity (Göbel et al., 2024).

Taken together, these developments show that TOM is best understood as a family of geometric orbital responses unified by Berry curvature, emergent fields, or Chern-Simons structure. Its realizations range from edge-state currents and skyrmion textures to kagome metals, altermagnets, Floquet anomalous phases, and adiabatic pumps, but the central claim remains the same: orbital magnetization can become a primary, topology-governed degree of freedom rather than a small correction to spin magnetism (Potasz et al., 2015, Dias et al., 2016, Zhang et al., 2023).