Skyrmion Number Current in Magnetic Textures

• Skyrmion Number Current is the transport of topological charge in magnetic textures, defined by continuity equations linking spatial gradients and time evolution of magnetization.
• It encompasses diverse formulations including continuum current densities, collective drift mechanisms, and quantized adiabatic pumping, as demonstrated by theoretical models and simulations.
• Experimental studies reveal its role in current-driven skyrmion motion, topological Hall effects, and optically induced control, offering new avenues for advanced spintronic devices.

Skyrmion number current denotes the transport of the topological charge carried by a magnetic texture. For a two-dimensional unit magnetization field, the relevant invariant is the skyrmion number, while the corresponding local density can satisfy a continuity equation whose flux defines a topological current. In the literature, this notion appears in several closely related forms: as an explicit continuum current density, as the collective drift of skyrmions or skyrmion bundles carrying integer topological charge, and as an experimentally inferred transport channel detected through topological Hall signals or quantized magnetoresistance changes (Östberg et al., 2023, Stier et al., 2017, Liang et al., 2015, Du et al., 2015). In three-dimensional magnets the concept extends to skyrmion strings, where conserved surface skyrmion number is coupled to bulk monopole charge (Koshibae et al., 2020).

1. Topological definition and local continuity

The standard two-dimensional skyrmion number is

$$Q=\frac{1}{4\pi}\int d^2r\,\mathbf{n}\cdot(\partial_x\mathbf{n}\times\partial_y\mathbf{n}),$$

with $\mathbf{n}(\mathbf{r},t)$ the normalized magnetization. Closely related notation appears throughout the literature. For magnetic bubbles, the skyrmion number is written as

$$\mathcal{N}_{\rm S}=\frac{1}{4\pi}\int dx\,dy\left(\mathbf{m}\cdot\partial_x\mathbf{m}\times\partial_y\mathbf{m}\right),$$

and, for axisymmetric bubbles, $\mathcal{N}_{\rm S}=QS$ with polarity $Q=\pm1$ and winding number $S\in\mathbb{Z}$, so the bubble skyrmion number can take any integer value (Yamane et al., 2015).

A local formulation introduces the skyrmion density $q(\mathbf{r},t)$ and a current density $\mathbf{j}_Q$. In a microscopic transport treatment,

$$\partial_t q(\mathbf{r},t)+\nabla\cdot\mathbf{j}_Q(\mathbf{r},t)=0,$$

with

$$[\mathbf{j}_Q]_k=\frac{1}{8\pi}\epsilon_{ijk}\epsilon_{\mu\nu}\,\partial_\mu n_i\,\partial_\nu n_j\,\partial_t n_k.$$

This makes the skyrmion number current a genuine topological flux built from spatial gradients of the spin texture and its time evolution, not an electrical current in the ordinary sense (Östberg et al., 2023).

An equivalent continuity-equation form was derived from the extended Landau-Lifshitz-Gilbert equation for in-plane spin-polarized currents:

$\mathbf{n}(\mathbf{r},t)$0

where $\mathbf{n}(\mathbf{r},t)$1, with a convective contribution

$\mathbf{n}(\mathbf{r},t)$2

and a second contribution proportional to

$\mathbf{n}(\mathbf{r},t)$3

In that formulation, the second term is the ingredient responsible for transverse separation of positive and negative skyrmion density under current drive (Stier et al., 2017).

2. Current-driven transport in ferromagnets, confined geometries, and bubbles

Current-driven skyrmion motion is commonly described by an LLG equation augmented by adiabatic and nonadiabatic spin-transfer torques or by damping-like spin-orbit torque. In effective collective-coordinate descriptions, the topological charge enters through a gyrovector or Magnus term. In the Thiele formulation for a rigid skyrmion, the gyrocoupling is proportional to the topological charge, $\mathbf{n}(\mathbf{r},t)$4 in one notation and $\mathbf{n}(\mathbf{r},t)$5 in the magnetic-bubble notation (Iwasaki et al., 2013, Yamane et al., 2015).

In unconstrained geometries, a rigid skyrmion can follow the spin velocity, while confinement changes the current-velocity relation. For skyrmions in a narrow channel, the steady-state drift along the channel is

$\mathbf{n}(\mathbf{r},t)$6

and the corresponding skyrmion number current can be written as $\mathbf{n}(\mathbf{r},t)$7, with $\mathbf{n}(\mathbf{r},t)$8 the skyrmion density. The same analysis shows threshold behavior from pinning and boundary potentials, and it identifies current-driven creation and annihilation at notches or sample edges as mechanisms that change the total topological charge (Iwasaki et al., 2013).

For chiral ferromagnets under spin-transfer torque, the contrast between topological and non-topological textures is explicit. A $\mathbf{n}(\mathbf{r},t)$9 skyrmion converges to a steady state with constant velocity at an angle to the current flow, while a $$\mathcal{N}_{\rm S}=\frac{1}{4\pi}\int dx\,dy\left(\mathbf{m}\cdot\partial_x\mathbf{m}\times\partial_y\mathbf{m}\right),$$0 skyrmionium is accelerated in the direction of current flow and either reaches a steady state with constant velocity or is elongated to infinity. When the spin current stops, the $$\mathcal{N}_{\rm S}=\frac{1}{4\pi}\int dx\,dy\left(\mathbf{m}\cdot\partial_x\mathbf{m}\times\partial_y\mathbf{m}\right),$$1 skyrmion is spontaneously pinned, whereas the $$\mathcal{N}_{\rm S}=\frac{1}{4\pi}\int dx\,dy\left(\mathbf{m}\cdot\partial_x\mathbf{m}\times\partial_y\mathbf{m}\right),$$2 skyrmionium continues propagation (Komineas et al., 2015).

Yamane and Sinova’s treatment of magnetic bubbles generalizes this dependence beyond $$\mathcal{N}_{\rm S}=\frac{1}{4\pi}\int dx\,dy\left(\mathbf{m}\cdot\partial_x\mathbf{m}\times\partial_y\mathbf{m}\right),$$3. For a rigid circular bubble,

$$\mathcal{N}_{\rm S}=\frac{1}{4\pi}\int dx\,dy\left(\mathbf{m}\cdot\partial_x\mathbf{m}\times\partial_y\mathbf{m}\right),$$4

with $$\mathcal{N}_{\rm S}=\frac{1}{4\pi}\int dx\,dy\left(\mathbf{m}\cdot\partial_x\mathbf{m}\times\partial_y\mathbf{m}\right),$$5 and

$$\mathcal{N}_{\rm S}=\frac{1}{4\pi}\int dx\,dy\left(\mathbf{m}\cdot\partial_x\mathbf{m}\times\partial_y\mathbf{m}\right),$$6

They found that transverse motion is greatly suppressed as $$\mathcal{N}_{\rm S}=\frac{1}{4\pi}\int dx\,dy\left(\mathbf{m}\cdot\partial_x\mathbf{m}\times\partial_y\mathbf{m}\right),$$7 departs from unity, whereas the longitudinal motion is less sensitive. This identifies a specific regime in which skyrmion-number transport remains predominantly longitudinal even when the object carries nonzero integer charge (Yamane et al., 2015).

3. Compensation, cancellation, and multicharge transport

A major theme in the field is the suppression of transverse drift by cancellation of net topological charge. This occurs in skyrmionium textures, synthetic antiferromagnets, and some multilayer skyrmion tubes. It also provides a useful contrast with multicharge objects whose Hall response remains large.

Texture class	Net topological charge	Current-driven signature
Skyrmionium	$$\mathcal{N}_{\rm S}=\frac{1}{4\pi}\int dx\,dy\left(\mathbf{m}\cdot\partial_x\mathbf{m}\times\partial_y\mathbf{m}\right),$$8	No skyrmion Hall effect; longitudinal motion
Compensated SAF skyrmion	$$\mathcal{N}_{\rm S}=\frac{1}{4\pi}\int dx\,dy\left(\mathbf{m}\cdot\partial_x\mathbf{m}\times\partial_y\mathbf{m}\right),$$9	Along-current motion; Hall angle near zero
SyAF tube with even identical layers	$\mathcal{N}_{\rm S}=QS$0	No transverse motion in the rigid-tube limit
Skyrmion bundle with $\mathcal{N}_{\rm S}=QS$1	Integer $\mathcal{N}_{\rm S}=QS$2	Large Hall deflection
Skyrmion bundle with $\mathcal{N}_{\rm S}=QS$3	$\mathcal{N}_{\rm S}=QS$4	Collinear motion

In a frustrated magnetic monolayer, a skyrmionium carries $\mathcal{N}_{\rm S}=QS$5 and shows no skyrmion Hall effect because the transverse forces from the two constituent skyrmions cancel. Under damping-like spin-orbit torque it moves along the current direction, while small current or magnetic field can transform an unstable Néel-type skyrmionium into a metastable Bloch-type skyrmionium, and large current may distort and collapse the Bloch-type state (Xia et al., 2020).

In antiferromagnetic skyrmionium, a uniform spin current induces propagation without a Magnus-force deflection. The collective-coordinate model gives a steady-state velocity linear in the spin-current magnitude,

$\mathcal{N}_{\rm S}=QS$6

which matches micromagnetic simulations within the stated accuracy for the parameter range studied (Obadero et al., 2019).

The most direct experimental realization of compensated skyrmion transport so far is in synthetic antiferromagnets. In Pt/Co/Ru/Pt/Co/Ru multilayers, opposite topological charges in the two ferromagnetic layers cancel the net gyrovector, leading to motion strictly along the current direction at velocities of up to $\mathcal{N}_{\rm S}=QS$7. The measured skyrmion Hall angle is $\mathcal{N}_{\rm S}=QS$8, independent of velocity, in contrast to ferromagnetic or synthetic-ferromagnetic stacks, where Hall angles saturate to approximately $\mathcal{N}_{\rm S}=QS$9–$Q=\pm1$0 at high velocity (Pham et al., 2024).

The same compensation principle extends to three-dimensional synthetic antiferromagnetic multilayers. For even numbers of identical ferromagnetic layers, the total skyrmion number of the tube satisfies $Q=\pm1$1, the Magnus and interlayer forces cancel in the summed Thiele equation, and the rigid-tube result is purely longitudinal,

$Q=\pm1$2

Layer-resolved skyrmion number is still transported in each sublayer, even though the net topological charge vanishes (Xia et al., 2021).

By contrast, multicharge skyrmion bundles in FeGe thin plates show that nonzero $Q=\pm1$3 can retain a robust Hall response even when $Q=\pm1$4 is large. Bundles with arbitrarily integer values from negative up to at least $Q=\pm1$5 were reported. For $Q=\pm1$6, the Hall angle is almost constant at approximately $Q=\pm1$7 and the sign is determined by the sign of $Q=\pm1$8; for $Q=\pm1$9, the bundle propagates collinearly with the current flow without the skyrmion Hall effect (Tang et al., 2021).

4. Electrical probes and experimentally inferred skyrmion-number transport

A central experimental route to skyrmion number current is through emergent electrodynamics. In MnSi nanowires, the topological Hall resistivity decomposes as

$S\in\mathbb{Z}$0

and the current-dependent topological component is

$S\in\mathbb{Z}$1

When skyrmions move, they generate an emergent electric field

$S\in\mathbb{Z}$2

which reduces the measured topological Hall voltage. The transport of topological charge is then represented by

$S\in\mathbb{Z}$3

Experimentally, the topological Hall effect decreases with increasing current density in the range $S\in\mathbb{Z}$4–$S\in\mathbb{Z}$5, and at $S\in\mathbb{Z}$6 the inferred drift velocity is approximately $S\in\mathbb{Z}$7 (Liang et al., 2015).

A different electrical signature appears in ultra-narrow MnSi nanowires supporting skyrmion cluster states. There, quantized jumps in magnetoresistance are directly associated with one-by-one creation or annihilation of skyrmions in the cross-section. In a $S\in\mathbb{Z}$8 nanowire, at most two skyrmions can exist and two jumps are observed; in a $S\in\mathbb{Z}$9 nanowire, four jumps are observed. The magnetoresistive step associated with a single skyrmion is modeled as

$q(\mathbf{r},t)$0

within an anisotropic magnetoresistance description (Du et al., 2015).

Electrical control of integer skyrmion number has also been demonstrated in confined Fe$q(\mathbf{r},t)$1Sn$q(\mathbf{r},t)$2 nanostripes. Single current pulses of duration $q(\mathbf{r},t)$3–$q(\mathbf{r},t)$4 and current densities up to approximately $q(\mathbf{r},t)$5 drive reversible transitions between stripe states and skyrmion chains. The reported control sequence includes states with $q(\mathbf{r},t)$6, and micromagnetic simulations attribute the switching to the combined effect of current-induced Joule heating and magnetic hysteresis (Jiang et al., 2023). These results concern current-controlled topological state conversion rather than rigid translation, but they establish that electrical protocols can write and erase discrete skyrmion numbers in nanostructures.

5. Sources, sinks, and topology-changing events

A recurring misconception is that skyrmion number is always conserved under current drive. The literature is more precise: it is conserved under smooth continuum dynamics, but it can change through singular events, lattice discretization, boundaries, and bulk monopoles.

For in-plane currents, the explicit continuity equation for the skyrmion density allows pair creation without violating continuum conservation of total $q(\mathbf{r},t)$7. Small fluctuations of positive and negative skyrmion density are separated by the transverse contribution to $q(\mathbf{r},t)$8, generating skyrmion–antiskyrmion pairs. In lattice simulations, the antiskyrmion is unstable and annihilates below the lattice scale, so the surviving skyrmion changes the net topological charge by $q(\mathbf{r},t)$9. The creation time scales as $\mathbf{j}_Q$0, and pair creation is suppressed for $\mathbf{j}_Q$1 (Stier et al., 2017).

Spatially varying spin currents provide another route to nonconservation. In a heavy-metal/ferromagnet bilayer with counter-propagating spin currents,

$\mathbf{j}_Q$2

the skyrmion elongates, may rotate back to a circular state, or may split into two skyrmions. The paper explicitly states that it does not present a continuity equation or an explicit formula for the skyrmion number current, but the simulations show that the splitting event changes the total skyrmion number from $\mathbf{j}_Q$3 to $\mathbf{j}_Q$4 (Duijn et al., 2021).

High-topological-number textures show the same principle in a more singular form. A $\mathbf{j}_Q$5 skyrmion can be created and stabilized only as a nonequilibrium dynamic object under vertical spin-polarized current, and its nucleation and destruction proceed through abrupt topological transitions associated with Bloch-point-like events. In that setting, a continuity equation for skyrmion number is meaningful only away from the singular events; at creation and destruction the topological charge changes in integer steps (Zhang et al., 2015).

In three-dimensional magnets the source and sink structure is explicit. For skyrmion strings, the relevant indices are the skyrmion number on a developed surface and the monopole charge in the bulk. As long as the configuration remains slowly varying, the surface skyrmion number is conserved; when it changes, the change is associated with nonzero monopole charge. The topological bookkeeping is summarized by relations such as

$\mathbf{j}_Q$6

and

$\mathbf{j}_Q$7

This makes monopoles the sources and sinks of skyrmion number current in three dimensions (Koshibae et al., 2020).

The experimental consequence is visible in metastable MnSi skyrmion-string lattices under pulsed current. Rather than being transported, the strings undergo topological unwinding, as evidenced by a reduction of the topological Hall signal. For pulse widths of $\mathbf{j}_Q$8–$\mathbf{j}_Q$9 at $$\partial_t q(\mathbf{r},t)+\nabla\cdot\mathbf{j}_Q(\mathbf{r},t)=0,$$0, the practical critical current density for clear unwinding is on the order of $$\partial_t q(\mathbf{r},t)+\nabla\cdot\mathbf{j}_Q(\mathbf{r},t)=0,$$1–$$\partial_t q(\mathbf{r},t)+\nabla\cdot\mathbf{j}_Q(\mathbf{r},t)=0,$$2, and the critical current increases for shorter pulses in a viscoelastic manner (Kagawa et al., 2017). This establishes that current-induced skyrmion-number transport in three-dimensional systems can be interrupted by defect-mediated annihilation instead of rigid motion.

6. Microscopic, quantized, and optical formulations

Beyond classical drift, skyrmion number current also appears as a topological pumping response. In two-dimensional topological chiral magnetic insulators, adiabatic deformation of an inhomogeneous spin texture generates a spin gauge flux and induces quantized charge and spin polarization currents. The induced currents are proportional to the product of a momentum-space Chern number and the real-space skyrmion density:

$$\partial_t q(\mathbf{r},t)+\nabla\cdot\mathbf{j}_Q(\mathbf{r},t)=0,$$3

$$\partial_t q(\mathbf{r},t)+\nabla\cdot\mathbf{j}_Q(\mathbf{r},t)=0,$$4

For a full adiabatic cycle, the pumped charge and spin are

$$\partial_t q(\mathbf{r},t)+\nabla\cdot\mathbf{j}_Q(\mathbf{r},t)=0,$$5

and the fully quantized topological response is endowed with the second Chern number (Yang et al., 2011).

A microscopic nonequilibrium theory of current-induced skyrmion transport combines nonequilibrium Green’s functions with the LLG equation in open systems. In that framework, the local skyrmion number current is computed from the time-dependent spin texture while the current-induced torque is obtained from electronic reservoirs and the nonequilibrium electron density. The results show that skyrmion dynamics in disordered spin textures are sensitive to the specific form of spin disorder, and different disorder configurations can lead to qualitatively different trajectories for the same applied bias (Östberg et al., 2023).

An optical formulation pushes the concept further by generating skyrmion number current without charge transport. With a time-dependent Hamiltonian

$$\partial_t q(\mathbf{r},t)+\nabla\cdot\mathbf{j}_Q(\mathbf{r},t)=0,$$6

where the Zeeman term is induced by circularly polarized light, the local current can be written as

$$\partial_t q(\mathbf{r},t)+\nabla\cdot\mathbf{j}_Q(\mathbf{r},t)=0,$$7

A first-order perturbative treatment based on a breathing Skyrmion ansatz shows that time-dependent deformation of the boundary generates an anisotropic breathing mode and hence a nonzero skyrmion number current. The resulting dynamics in momentum space form a limit cycle whose characteristics depend solely on the external magnetic field amplitude, the Heisenberg exchange coupling, and the Gilbert damping constant (Fadhilla et al., 15 Aug 2025).

Taken together, these works define skyrmion number current as the transport channel of topological charge across a broad hierarchy of descriptions: local continuity equations in continuum field theory, collective motion of textures carrying integer charge, source-sink processes mediated by antiskyrmions or monopoles, quantized adiabatic pumping, and optically induced low-dissipation motion. The unifying principle is that the current is fixed by the time evolution of the spin texture’s topology, even when the experimental drive and readout are electrical, thermal, or optical.