Twin-Skyrmions: Composite Spin Textures

• Twin-skyrmions are composite spin textures where two skyrmion-like objects combine to form a doubled topological structure with diverse stabilization mechanisms.
• They emerge in various systems such as itinerant magnets, chiral liquid crystals, and bilayer configurations, each exhibiting unique magnetic interactions.
• Their formation pathways include splitting from a single skyrmion or merging of two, enabling tunable magnetic properties and potential control via external fields and interlayer effects.

Searching arXiv for papers on twin-skyrmions and closely related usages. arxiv_search(query="twin-skyrmions skyrmions twin", max_results=10, sort_by="relevance") Searching arXiv for the specific identifiers and topic variants. arxiv_search(query="(Ozawa et al., 2017) OR (Sohn et al., 2019) OR (Matthies et al., 29 Oct 2025) OR twin skyrmion", max_results=10, sort_by="relevance") Twin-skyrmions are composite or paired skyrmionic spin textures in which two skyrmion-like objects, or a doubled topological wrapping, define the relevant structure. In the literature, the term does not denote a single universally fixed object. It has been used for a zero-field skyrmion crystal with topological number $N_{\rm sk}=+2$ per magnetic unit cell in an itinerant triangular-lattice Kondo system, for bound pairs of mutually orthogonal isolated skyrmion tubes in chiral media, and for a three-dimensional bilayer object formed by two interlayer-coupled Néel skyrmions whose shape or helicities are modified by interlayer Dzyaloshinskii–Moriya interaction (IL-DMI) (Ozawa et al., 2017, Sohn et al., 2019, Matthies et al., 29 Oct 2025). Closely related constructions include skyrmionic division into two unit-charge “twin” skyrmions, biskyrmions formed by merging two skyrmions, and vortex states embedding two skyrmions (Kind et al., 2022, Göbel et al., 2019, Zhang et al., 20 Mar 2025).

1. Topological characterization and terminological scope

The common invariant across these systems is the skyrmion number, written in continuum form as

$$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$

or, for normalized localized spins $\mathbf{n}(\mathbf r)$, as

$$N_{\rm sk}=\frac{1}{4\pi}\int d^2r\;\mathbf{n}\cdot\bigl(\partial_x\mathbf{n}\times\partial_y\mathbf{n}\bigr).$$

On lattices, the same quantity may be evaluated from plaquette solid angles, as in the triangular-lattice Kondo model where $N_{\rm sk}=(1/4\pi)\sum_p \Omega_p$ (Ozawa et al., 2017).

Usage in the literature	Constituent structure	Topological characterization
Itinerant twin SkX	Triple-$Q$ skyrmion crystal on a triangular lattice	$N_{\rm sk}=+2$ per 12-site magnetic unit cell
Orthogonal twin cluster	Vertical and horizontal isolated skyrmion tubes	Two nonequivalent bound configurations
Bilayer twin-skyrmion	Two IEC-locked Néel skyrmions in two layers with IL-DMI	$Q=+1$ per layer in the Thiele description
Skyrmionic division product	Two separated skyrmions after slicing	Each $Q=-1$
Biskyrmion	Partially overlapping pair of skyrmions	$N_{\rm Sk}=+2$
2-skyrmion vortex	Vortex with two embedded skyrmions	$$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$0 for a specific sign convention

This diversity matters. A twin-skyrmion may be a single bound object, a crystal carrying doubled topological charge per cell, or a pair of skyrmions produced dynamically. This suggests that the label is descriptive rather than unique, and that the specific stabilization mechanism must be read from context.

2. Zero-field $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$1 twin-skyrmion crystals in itinerant magnets

In itinerant magnets, a twin-skyrmion crystal arises in the minimal Kondo-lattice Hamiltonian on the two-dimensional triangular lattice,

$$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$2

with $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$3 between nearest neighbors, $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$4 for third neighbors, $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$5 in most simulations, and $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$6 for classical localized spins. The itinerant-electron spin operator is $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$7, and the field $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$8 couples only to the localized spins (Ozawa et al., 2017).

The zero-field state is a skyrmion crystal with unusually high topological number. Each magnetic unit cell of 12 sites carries exactly $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$9. At $\mathbf{n}(\mathbf r)$0, this twin SkX is exceptionally well approximated by the triple-$\mathbf{n}(\mathbf r)$1 ansatz

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

where the three ordering vectors $\mathbf{n}(\mathbf r)$3 lie on the $\mathbf{n}(\mathbf r)$4–$\mathbf{n}(\mathbf r)$5 lines of the Brillouin zone at angles $\mathbf{n}(\mathbf r)$6, $\mathbf{n}(\mathbf r)$7, and $\mathbf{n}(\mathbf r)$8. Physically, the state is a periodic array of vorticity-2 cores surrounded by six meron walls, yielding net skyrmion number two per magnetic cell (Ozawa et al., 2017).

The stabilization mechanism is itinerant-electron-driven. For small $\mathbf{n}(\mathbf r)$9, the RKKY interaction $$N_{\rm sk}=\frac{1}{4\pi}\int d^2r\;\mathbf{n}\cdot\bigl(\partial_x\mathbf{n}\times\partial_y\mathbf{n}\bigr).$$0 favors helices at the maxima of the bare susceptibility $$N_{\rm sk}=\frac{1}{4\pi}\int d^2r\;\mathbf{n}\cdot\bigl(\partial_x\mathbf{n}\times\partial_y\mathbf{n}\bigr).$$1; on the triangular lattice there are three symmetry-related $$N_{\rm sk}=\frac{1}{4\pi}\int d^2r\;\mathbf{n}\cdot\bigl(\partial_x\mathbf{n}\times\partial_y\mathbf{n}\bigr).$$2 vectors, degenerate at second order. Higher-order $$N_{\rm sk}=\frac{1}{4\pi}\int d^2r\;\mathbf{n}\cdot\bigl(\partial_x\mathbf{n}\times\partial_y\mathbf{n}\bigr).$$3 multi-spin interactions then lift this degeneracy in favor of the triple-$$N_{\rm sk}=\frac{1}{4\pi}\int d^2r\;\mathbf{n}\cdot\bigl(\partial_x\mathbf{n}\times\partial_y\mathbf{n}\bigr).$$4 superposition, lowering the energy by $$N_{\rm sk}=\frac{1}{4\pi}\int d^2r\;\mathbf{n}\cdot\bigl(\partial_x\mathbf{n}\times\partial_y\mathbf{n}\bigr).$$5 per site compared to a single-$$N_{\rm sk}=\frac{1}{4\pi}\int d^2r\;\mathbf{n}\cdot\bigl(\partial_x\mathbf{n}\times\partial_y\mathbf{n}\bigr).$$6 state. This is a distinct route to a high-topological-number crystal because the ordering vectors are selected by the Fermi-surface effect rather than by a conventional DMI-dominated continuum theory (Ozawa et al., 2017).

The numerical evidence combines unrestricted large-scale real-space optimization with variational analysis. The modified KPM-LD simulation uses a Chebyshev expansion up to order $$N_{\rm sk}=\frac{1}{4\pi}\int d^2r\;\mathbf{n}\cdot\bigl(\partial_x\mathbf{n}\times\partial_y\mathbf{n}\bigr).$$7, $$N_{\rm sk}=\frac{1}{4\pi}\int d^2r\;\mathbf{n}\cdot\bigl(\partial_x\mathbf{n}\times\partial_y\mathbf{n}\bigr).$$8 correlated random vectors, a Langevin pseudo-time step $$N_{\rm sk}=\frac{1}{4\pi}\int d^2r\;\mathbf{n}\cdot\bigl(\partial_x\mathbf{n}\times\partial_y\mathbf{n}\bigr).$$9, and system sizes up to $N_{\rm sk}=(1/4\pi)\sum_p \Omega_p$0 localized spins. The chemical potential $N_{\rm sk}=(1/4\pi)\sum_p \Omega_p$1 fixes the electron filling at $N_{\rm sk}=(1/4\pi)\sum_p \Omega_p$2. Variational grand-potential calculations, based on analytic triple-$N_{\rm sk}=(1/4\pi)\sum_p \Omega_p$3 spin configurations and exact diagonalization of the tight-binding electrons in the static spin background, agree with KPM-LD within $N_{\rm sk}=(1/4\pi)\sum_p \Omega_p$4 (Ozawa et al., 2017).

A distinctive feature is field-controlled topological switching. As $N_{\rm sk}=(1/4\pi)\sum_p \Omega_p$5 is raised, the phase sequence is $N_{\rm sk}=(1/4\pi)\sum_p \Omega_p$6: $N_{\rm sk}=(1/4\pi)\sum_p \Omega_p$7 SkX with zero net $N_{\rm sk}=(1/4\pi)\sum_p \Omega_p$8; $N_{\rm sk}=(1/4\pi)\sum_p \Omega_p$9: $Q$0 SkX with finite magnetization; $Q$1: $Q$2 triple-$Q$3 collinear/conical or forced ferromagnet. At each transition, the skyrmion number and magnetization jump discontinuously, and the variational $Q$4 curves cross at the same $Q$5 where the KPM-LD data jump. The size of the skyrmions is also tunable because $Q$6 is set by Fermi-surface nesting and hence by $Q$7, $Q$8, and $Q$9; increasing $N_{\rm sk}=+2$0 from $N_{\rm sk}=+2$1 to $N_{\rm sk}=+2$2 shifts $N_{\rm sk}=+2$3 from $N_{\rm sk}=+2$4 to $N_{\rm sk}=+2$5, reducing the lattice constant $N_{\rm sk}=+2$6 from $N_{\rm sk}=+2$7 to $N_{\rm sk}=+2$8 (Ozawa et al., 2017).

The full simulations additionally resolve local solid angle, bond current density, itinerant-electron spin magnitude, and charge-density modulation. For the $N_{\rm sk}=+2$9 phase, all local solid angles are positive, the bond currents form circular loops around each skyrmion core, $Q=+1$0 is largest where $Q=+1$1 is smallest, and a weak 12-site charge-density wave correlates with the spin texture. These observables inherit the threefold rotational symmetry of the underlying SkX (Ozawa et al., 2017).

3. Orthogonal twin-skyrmion clusters in chiral magnets and liquid crystals

A second major usage of the term refers to bound pairs of isolated skyrmion tubes with mutually orthogonal axes. In this geometry, one distinguishes vertical tubes, whose central axis is parallel to the applied magnetic field $Q=+1$2, and horizontal tubes, whose axis lies in the plane perpendicular to $Q=+1$3. When a vertical isolated skyrmion (V-IS) and a horizontal isolated skyrmion (H-IS) become co-localized, their axes are mutually orthogonal and the H-tube thread may pass through the coil of the V-tube; this structure is described as a “twin” or orthogonal cluster (Sohn et al., 2019).

The phenomenology is governed by the standard noncentrosymmetric ferromagnet energy density

$Q=+1$4

with Rapini–Papoular anchoring $Q=+1$5. The total free energy is

$Q=+1$6

In the liquid-crystal analogue, $Q=+1$7, $Q=+1$8, and the Zeeman term is replaced by $Q=+1$9, again with Rapini–Papoular anchoring (Sohn et al., 2019).

The pair interaction is extracted from

$Q=-1$0

where $Q=-1$1 is the minimized energy at fixed center-to-center distance $Q=-1$2. The resulting profile is Lennard-Jones-like: a shallow repulsive core at very small $Q=-1$3, a pronounced minimum at the bound-state separation, and a small barrier at intermediate distances. The sign of the interaction depends on the host phase. In a conical background, each isolated skyrmion has a shell of positive excess energy, and shell overlap lowers the total energy, producing attraction. In the homogeneous state for $Q=-1$4, the far field is uniform and the skyrmions repel (Sohn et al., 2019).

The orthogonal pair has two nonequivalent minima. Labeling the H-IS polarity by $Q=-1$5, one obtains $Q=-1$6, in which the H-IS slides its positive shell into the coil of the V-IS, and $Q=-1$7, in which the H-IS meets the V-IS core first. Numerically, $Q=-1$8 for $Q=-1$9 is deeper, with binding energy $N_{\rm Sk}=+2$0 at $N_{\rm Sk}=+2$1, whereas $N_{\rm Sk}=+2$2 for $N_{\rm Sk}=+2$3 is shallower, $N_{\rm Sk}=+2$4, at $N_{\rm Sk}=+2$5. As the field decreases and the cone angle increases, both separations shrink, but $N_{\rm Sk}=+2$6 strengthens much more than $N_{\rm Sk}=+2$7 (Sohn et al., 2019).

In the extreme mutual-attraction regime, the structure ceases to resemble a simple bound pair. When the cone angle approaches $N_{\rm Sk}=+2$8, the H-IS deforms into an undulated path winding through successive loops of the V-IS coil. The winding angle satisfies approximately

$N_{\rm Sk}=+2$9

with $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$00, and the transverse modulation amplitude is $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$01. This is still topologically rooted in the interlocking of skyrmion shells, but geometrically it is already a compact superstructure rather than a weakly bound dimer (Sohn et al., 2019).

The experimental realization was demonstrated in a chiral nematic liquid crystal, 5CB doped with 3 wt% CB15, in cells of thickness $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$02 with $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$03 and strong homeotropic anchoring $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$04. Vertical and horizontal skyrmions were generated and manipulated with holographic optical tweezers using a 1064 nm laser and a spatial light modulator. Weakly bound twins with $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$05 and $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$06 appear at small cone angles, whereas strongly bound twins with $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$07 and $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$08 occur at large cone angles. The same study argues that analogous clusters are expected in the A-phase of cubic helimagnets such as MnSi, FeGe, and Cu$$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$09OSeO$$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$10 (Sohn et al., 2019).

4. Bilayer twin-skyrmions stabilized by interlayer Dzyaloshinskii–Moriya interaction

In magnetic bilayers, a twin-skyrmion is a three-dimensional bound object formed by two isolated Néel-type skyrmions, one in each ferromagnetic layer, coupled by interlayer exchange and IL-DMI. For layer $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$11,

$$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$12

with $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$13, $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$14, $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$15, and $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$16. Without IL-DMI, the two skyrmions lock on top of each other by IEC so that $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$17 and their cores coincide. IL-DMI forces a finite tilt between spins at corresponding in-plane positions in the two layers and produces a twin-skyrmion in which the cores remain collinear but the radius, ellipticity, or helicities adapt to maximize the IL-DMI energy gain (Matthies et al., 29 Oct 2025).

The interlayer chiral term is

$$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$18

or, in continuum form,

$$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$19

The full continuum free energy is

$$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$20

Minimization yields the equilibrium radius, the layer-resolved helicities, and the ellipticity (Matthies et al., 29 Oct 2025).

The direction of $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$21 determines the deformation mode. For in-plane $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$22 or $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$23, the background magnetizations in the two layers tilt in opposite transverse directions and the skyrmion elongates along a direction

$$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$24

For out-of-plane $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$25, the equilibrium helicities satisfy $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$26, while the shape remains circular (Matthies et al., 29 Oct 2025).

The dynamics under current-perpendicular-to-plane excitation are described by a Thiele equation,

$$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$27

with $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$28, $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$29, and $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$30. For a circular skyrmion with helicity $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$31 and current polarization $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$32, the velocity obeys the explicit relation given in the paper, and for $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$33 the Hall angle satisfies

$$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$34

The numerical findings are specific. Without IL-DMI, $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$35 and $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$36. In-plane IL-DMI of magnitude $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$37 meV changes $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$38 by up to $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$39, down to $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$40 or up to $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$41 along $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$42, while $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$43 varies by up to $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$44 and is maximal along the elongation direction. Out-of-plane IL-DMI up to $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$45 meV increases $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$46 by about $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$47 through the larger skyrmion radius, whereas $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$48 changes by less than $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$49 (Matthies et al., 29 Oct 2025).

This bilayer realization shows that a twin-skyrmion need not be a mere superposition of two static copies. The IL-DMI introduces internal shape and helicity degrees of freedom absent in a simple IEC-locked bilayer pair. At large in-plane IL-DMI, the twin-skyrmion undergoes an elliptic instability into stripes, which the source identifies as a possible mechanism for current-driven writing or erasure (Matthies et al., 29 Oct 2025).

5. Formation pathways: division into twins and merging into biskyrmions

One route to twin-skyrmions is dynamical splitting. In a thin-film wire of lateral size approximately $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$50 nm$$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$51, thickness $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$52 nm, and cell size $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$53 nm$$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$54, the micromagnetic dynamics are governed by

$$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$55

with $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$56 pJ m$$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$57, $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$58 mJ m$$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$59, $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$60 MJ m$$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$61, $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$62 kA m$$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$63, and $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$64. The time evolution follows the Landau–Lifshitz–Gilbert equation with Slonczewski torque in CPP geometry (Kind et al., 2022).

The division protocol consists of inflation, cutting, detachment, and relaxation. A field $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$65 mT in $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$66 is applied for a few nanoseconds to enlarge the skyrmion and reduce boundary curvature. A CPP current density $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$67 nA nm$$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$68 along $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$69 for about $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$70 ns drives the skyrmion onto a non-magnetic trench etched perpendicular to the wire axis. As the boundary wraps the trench, regions of opposite topological density $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$71 form at the trench edges, the texture deforms from circular to figure-eight, and a pocket of negative charge grows. When the broken domain-wall segment reaches the trench end, a short jolt of $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$72 nA nm$$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$73 for $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$74 ns pinches off the two halves, which relax into two separate stable skyrmions, each with $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$75. The source formulates the energetic criterion as

$$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$76

so that cutting becomes favorable once the opposite-sign pocket is sufficiently developed (Kind et al., 2022).

A second route is controlled merging. In a centrosymmetric nanodisk without DMI, the free energy includes exchange, anisotropy, Zeeman, and full dipole–dipole terms, and the relevant bound pair is the biskyrmion. The analytical superposition ansatz for two Bloch skyrmions with opposite in-plane magnetizations shows that the dipolar term alone selects the bound state: the short-range approximation of the dipolar energy exhibits a pronounced minimum at $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$77, corresponding to two partially overlapping skyrmions. Applying the standard topological integral to the superposed texture yields $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$78, and in the discrete simulation the result remains within $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$79–$$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$80 of exactly $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$81 (Göbel et al., 2019).

The micromagnetic protocol uses a nanodisk of radius $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$82 nm and thickness $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$83 nm, discretized with cell size $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$84 nm$$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$85, and material parameters $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$86 pJ/m, $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$87 MJ/m$$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$88, $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$89 MA/m, $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$90, and external field $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$91 mT $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$92. Two auxiliary writing disks of radius $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$93 nm, centered at $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$94 nm along $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$95, host in-plane vortices of helicity $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$96. A spin-polarized current $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$97 A/cm$$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$98 along $$Q=\frac{1}{4\pi}\int \mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,dx\,dy,$$99 for $\mathbf{n}(\mathbf r)$00 ps imprints the textures; after about $\mathbf{n}(\mathbf r)$01 ns of relaxation, two $\mathbf{n}(\mathbf r)$02 Bloch skyrmions spiral toward the disk center and merge into a stable biskyrmion once their separation reaches the dipolar minimum. Helicity control determines whether the interaction is attractive or repulsive, and helicity-flipping pulses can reverse the bound state back into two skyrmions (Göbel et al., 2019).

Taken together, these formation pathways show that “twin-skyrmion” behavior can emerge either by fission of a single skyrmion into two unit-charge descendants or by attraction and partial overlap of two initially separate skyrmions. The first mechanism exploits local nucleation of opposite-sign topological density; the second exploits a bound-state minimum selected by dipole–dipole interaction.

6. Related composite textures, distinctions, and functional implications

Two neighboring categories sharpen the conceptual boundaries of twin-skyrmions. The first is the skyrmion-embedded vortex, or $\mathbf{n}(\mathbf r)$03-skyrmion vortex, where the total topological charge obeys

$\mathbf{n}(\mathbf r)$04

In a circular disk of thickness $\mathbf{n}(\mathbf r)$05 nm and radius $\mathbf{n}(\mathbf r)$06 nm, with $\mathbf{n}(\mathbf r)$07 J/m, $\mathbf{n}(\mathbf r)$08 A/m, $\mathbf{n}(\mathbf r)$09 J/m$\mathbf{n}(\mathbf r)$10, $\mathbf{n}(\mathbf r)$11 J/m$\mathbf{n}(\mathbf r)$12, and $\mathbf{n}(\mathbf r)$13–$\mathbf{n}(\mathbf r)$14 mT, a 2-skyrmion vortex becomes a local energy minimum for $\mathbf{n}(\mathbf r)$15 J/m$\mathbf{n}(\mathbf r)$16. Under the conventional sign convention $\mathbf{n}(\mathbf r)$17 and $\mathbf{n}(\mathbf r)$18, the total charge is $\mathbf{n}(\mathbf r)$19. The state is metastable, higher in energy than the single-skyrmion vortex or pure vortex, but separated from collapse by an energy barrier (Zhang et al., 20 Mar 2025).

The second is skyrmionium-based matter. A skyrmionium is a $\mathbf{n}(\mathbf r)$20 target texture with an inner core carrying $\mathbf{n}(\mathbf r)$21 and an outer ring carrying $\mathbf{n}(\mathbf r)$22, so the total charge is $\mathbf{n}(\mathbf r)$23. Its double-wall structure is encoded by a profile $\mathbf{n}(\mathbf r)$24 with $\mathbf{n}(\mathbf r)$25, $\mathbf{n}(\mathbf r)$26, and $\mathbf{n}(\mathbf r)$27, plus an additional oscillation at $\mathbf{n}(\mathbf r)$28. Pure skyrmionium lattices are unstable against elongation distortions and relax into the spiral phase, but mixed skyrmionium–skyrmion crystals can be stabilized by a dilute skyrmion sublattice acting as topological “pins.” These states are classified by topological stoichiometry such as SkmSk, Skm$\mathbf{n}(\mathbf r)$29Sk, Skm$\mathbf{n}(\mathbf r)$30Sk$\mathbf{n}(\mathbf r)$31, and SkmSk$\mathbf{n}(\mathbf r)$32, and they host hybrid excitations including deformation-assisted rotations and orbital modes (Leonov et al., 20 Nov 2025).

These neighboring systems clarify a recurrent misconception: a doubled internal structure does not imply a unique topological charge or a unique stabilization mechanism. A twin-skyrmion crystal in an itinerant magnet has $\mathbf{n}(\mathbf r)$33 per magnetic cell and can exist at zero magnetic field (Ozawa et al., 2017). A bilayer twin-skyrmion consists of two skyrmions with cores kept collinear by IEC while IL-DMI reshapes or re-helicizes them (Matthies et al., 29 Oct 2025). An orthogonal twin cluster is defined geometrically by mutually perpendicular skyrmion tubes and by an interaction controlled by the conical host state (Sohn et al., 2019). By contrast, skyrmionium has total $\mathbf{n}(\mathbf r)$34, and a 2-skyrmion vortex carries a half-integer offset because of the vortex background (Zhang et al., 20 Mar 2025, Leonov et al., 20 Nov 2025).

The functional implications also differ accordingly. In the itinerant $\mathbf{n}(\mathbf r)$35 crystal, the topological number can be switched as $\mathbf{n}(\mathbf r)$36 by tiny field changes $\mathbf{n}(\mathbf r)$37, the zero-field state has no net magnetization, and higher emergent magnetic fields proportional to $\mathbf{n}(\mathbf r)$38 are expected to enhance topological Hall signals (Ozawa et al., 2017). In IL-DMI bilayers, the tunable relation between elongation, helicity, velocity, and Hall angle provides additional control knobs for current-driven transport (Matthies et al., 29 Oct 2025). In orthogonal clusters, the attraction–repulsion crossover is controlled by host-state conicity or by liquid-crystal thickness and anchoring, making the geometry of the bound state itself an experimental control variable (Sohn et al., 2019). A plausible implication is that the scientific value of twin-skyrmions lies less in a single canonical texture than in a family of two-skyrmion constructions that expose different combinations of topology, dimensionality, chirality, and collective dynamics.