Spatiotemporal Skyrmionic Textures

• Spatiotemporal skyrmionic textures are topological structures exhibiting spatial winding and temporal evolution, found in magnetic systems as skyrmions, skyrmioniums, and skyrmion bags, and in optics as x–t skyrmions and related pulses.
• They are characterized by invariant metrics such as the skyrmion number, with advanced methods like deep neural networks and quantitative reconstruction enabling precise classification.
• Their control via geometric confinement, anisotropy, and engineered optical fields underpins functional innovations in spintronics, photonics, and ultrafast topological metrology.

Searching arXiv for recent and foundational papers on spatiotemporal skyrmionic textures across magnetic and optical systems. Spatiotemporal skyrmionic textures are topological field configurations in which a vector order parameter realizes skyrmion-like winding across spatial coordinates together with either explicit temporal structure or experimentally resolved time evolution. In current literature, the term spans two closely connected domains. In magnetism, it refers to skyrmions and higher-order textures whose stabilization, transformation, transport response, and reversal are governed by confinement, anisotropy, interfaces, and dynamical processes such as Bloch-line or Bloch-point motion. In optics, it denotes polarization or Stokes-vector textures formed directly in space-time, including $x\!-\!t$ skyrmions, sagittal skyrmionic pulses, and 4D topological textures in $3+1$ dimensions (Sadi et al., 15 Apr 2026, Teng et al., 10 Mar 2025, Vo et al., 2024, Marco et al., 2022).

1. Topological descriptors and classification

The common invariant across magnetic and optical realizations is a skyrmion number defined by the wrapping of a normalized vector field over an effective order-parameter sphere. For magnetic textures, the standard topological charge is written as

$$Q = \frac{1}{4\pi} \int \mathbf{m} \cdot \left( \frac{\partial \mathbf{m}}{\partial x} \times \frac{\partial \mathbf{m}}{\partial y} \right) dx\,dy ,$$

and is used to distinguish skyrmions, skyrmioniums, bags, sacks, and related composites. In Pt/Co/W multilayers, the charge assignments reported are $Q=\pm1$ for a skyrmion, $Q=0$ for a skyrmionium, and $Q=n$ for a skyrmion bag, where $n$ is the net number of internal skyrmions of given polarity (Sadi et al., 15 Apr 2026). In Fe$_{3-x}$GaTe$_2$, the same formula is used to classify skyrmions, skyrmionium, skyrmion bags, and skyrmion sacks, with sacks carrying $Q \le -1$ and bags $3+1$0 depending on the embedded structure (Li et al., 8 May 2025).

In optical spatiotemporal skyrmions, the invariant is defined on the normalized Stokes vector $3+1$1 over a space-time plane: $3+1$2 This measures how the polarization state covers the Poincaré sphere in the $3+1$3 domain. In a picosecond pulse wavepacket, the observed optical spatiotemporal skyrmion number was reported as $3+1$4, indicating topology preservation despite experimental imperfections (Teng et al., 10 Mar 2025).

Several works broaden the descriptor space beyond a single integer. In centrosymmetric frustrated magnets, vector vorticity $3+1$5 generalizes conventional scalar vorticity by encoding the orientation of the spin-rotation plane, with the vorticity angle $3+1$6 and helicity angle $3+1$7 together specifying the full rotation freedom of the skyrmionic texture (Yao et al., 2022). In nonparaxial optical lattices, the relevant order-parameter spaces are the complex projective plane $3+1$8 and a 4-sphere, leading to 4D Skyrme numbers defined over $3+1$9 (Marco et al., 2022).

Classification is increasingly automated. A deep neural network trained according to transfer learning was used to distinguish nine skyrmionic textures with 98% accuracy, while related models extracted $$Q = \frac{1}{4\pi} \int \mathbf{m} \cdot \left( \frac{\partial \mathbf{m}}{\partial x} \times \frac{\partial \mathbf{m}}{\partial y} \right) dx\,dy ,$$0, $$Q = \frac{1}{4\pi} \int \mathbf{m} \cdot \left( \frac{\partial \mathbf{m}}{\partial x} \times \frac{\partial \mathbf{m}}{\partial y} \right) dx\,dy ,$$1, and $$Q = \frac{1}{4\pi} \int \mathbf{m} \cdot \left( \frac{\partial \mathbf{m}}{\partial x} \times \frac{\partial \mathbf{m}}{\partial y} \right) dx\,dy ,$$2 from spin-texture images with 90% accuracy for a MISO deep learning model and 80% accuracy for an SVR model (Feng et al., 2023). This suggests that the taxonomy of spatiotemporal skyrmionic textures is now being formalized not only topologically but also algorithmically.

2. Magnetic routes to complex textures

A central result in recent magnetic work is that confinement can act as a deterministic selector of higher-order textures. In $$Q = \frac{1}{4\pi} \int \mathbf{m} \cdot \left( \frac{\partial \mathbf{m}}{\partial x} \times \frac{\partial \mathbf{m}}{\partial y} \right) dx\,dy ,$$3 multilayers patterned into microtracks of $$Q = \frac{1}{4\pi} \int \mathbf{m} \cdot \left( \frac{\partial \mathbf{m}}{\partial x} \times \frac{\partial \mathbf{m}}{\partial y} \right) dx\,dy ,$$4, $$Q = \frac{1}{4\pi} \int \mathbf{m} \cdot \left( \frac{\partial \mathbf{m}}{\partial x} \times \frac{\partial \mathbf{m}}{\partial y} \right) dx\,dy ,$$5, and $$Q = \frac{1}{4\pi} \int \mathbf{m} \cdot \left( \frac{\partial \mathbf{m}}{\partial x} \times \frac{\partial \mathbf{m}}{\partial y} \right) dx\,dy ,$$6, increasing confinement drives a hierarchical transformation pathway: labyrinth domains fragment into isolated skyrmions, skyrmion pairs are progressively suppressed, skyrmioniums are enhanced by recombination, and skyrmion bags become dominant in the narrowest tracks (Sadi et al., 15 Apr 2026). The reported phase sequence is summarized as magnetic domains $$Q = \frac{1}{4\pi} \int \mathbf{m} \cdot \left( \frac{\partial \mathbf{m}}{\partial x} \times \frac{\partial \mathbf{m}}{\partial y} \right) dx\,dy ,$$7 skyrmion pairs $$Q = \frac{1}{4\pi} \int \mathbf{m} \cdot \left( \frac{\partial \mathbf{m}}{\partial x} \times \frac{\partial \mathbf{m}}{\partial y} \right) dx\,dy ,$$8 skyrmioniums $$Q = \frac{1}{4\pi} \int \mathbf{m} \cdot \left( \frac{\partial \mathbf{m}}{\partial x} \times \frac{\partial \mathbf{m}}{\partial y} \right) dx\,dy ,$$9 skyrmion bags. This is significant because the control parameter is geometric rather than a finely tuned field or current protocol.

Micromagnetic simulations in the same system identify a separation-dependent recombination regime for skyrmion pairs. When the initial core-to-core separation satisfies $Q=\pm1$0, the pair annihilates rapidly; for $Q=\pm1$1, recombination into a stable skyrmionium occurs; and for $Q=\pm1$2, the pair remains stable and non-interacting (Sadi et al., 15 Apr 2026). The dynamics are modeled by the Landau-Lifshitz-Gilbert equation

$Q=\pm1$3

with $Q=\pm1$4 including exchange, anisotropy, external field, and DMI.

A persistent misconception is that skyrmions are necessarily circular. The literature on stripe skyrmions argues that stripy magnetic textures previously termed spiral, helical, or cycloidal spin orders are in fact skyrmions, with morphology controlled by skyrmion-number density. The reported progression is ramified stripe $Q=\pm1$5 rectangular stripe $Q=\pm1$6 circular skyrmion $Q=\pm1$7 skyrmion crystal, and at low density the natural stripe width is proportional to $Q=\pm1$8 (Wang et al., 2021). In anisotropic environments, periodic modulations of exchange and anisotropy break rotational symmetry, generate elongated $Q=\pm1$9 skyrmion-like textures, and create skyrmionic tracks (Hagemeister et al., 2016).

Van der Waals ferromagnets add a real-space engineering route. In Fe$Q=0$0GaTe$Q=0$1, skyrmion lattices can be realized by field cooling and then controllably erased and painted via delicate manipulation of tip stray field, including deterministic creation of skyrmions with opposite topological charges $Q=0$2 and $Q=0$3 (Mi et al., 2024). In non-stoichiometric Fe$Q=0$4GaTe$Q=0$5, room-temperature stable stripe, striped skyrmionium, and striped skyrmion sack states were observed, and field-cooling was reported to suppress trivial transitions and favor the stabilization of composite skyrmion textures including skyrmionium, skyrmion bags, and sacks (Li et al., 8 May 2025).

Frustrated kagome magnets provide a related but distinct setting. In Fe$Q=0$6Sn$Q=0$7, in-situ Lorentz transmission electron microscopy at room temperature showed stripe domains evolving into dumbbell-shaped domains, bubbles, and finally skyrmionic bubbles under increasing out-of-plane field, with the transformation between different bubbles and domains occurring via Bloch-line motion driven by the applied field (Hou et al., 2017). This establishes that the spatial complexity of skyrmionic matter is often inseparable from its temporal conversion pathway.

3. Three-dimensionality and dynamical evolution

Magnetic skyrmionic textures are intrinsically three-dimensional in many confined systems. In helimagnetic FeGe nanodisks, full 3D finite-element micromagnetics including demagnetisation effects showed that incomplete skyrmions and isolated skyrmions can be ground states at zero external magnetic field and in the absence of magnetocrystalline anisotropy, whereas neglecting demagnetisation or imposing translational invariance along $Q=0$8 suppresses the skyrmion ground state in favor of helical states (Beg et al., 2013). This directly ties stability to three-dimensional modeling rather than to a purely two-dimensional effective picture.

The same study reported hysteretic behavior between two energetically equivalent skyrmionic states with different core orientation and identified the core-reversal mechanism as Bloch-point occurrence and propagation through the thickness (Beg et al., 2013). Such results are genuinely spatiotemporal: the topology is spatially localized, but the experimentally relevant operation is a time-resolved singular process.

Direct 3D imaging has made these features observable rather than inferred. Holographic vector field electron tomography provided the first quantitative reconstruction of the 3D spin texture of skyrmion tubes in FeGe with sub-10 nanometer resolution, revealing local deviations from homogeneous Bloch character, collapse of the texture at surfaces, and correlated modulation of the tubes along their axes (Wolf et al., 2021). The reconstructed magnetic induction further allowed spatially resolved maps of exchange and Dzyaloshinskii-Moriya energy densities,

$Q=0$9

showing that the DM energy density is maximal and negative in the tube core.

A complementary multilayer realization is provided by skyrmionic cocoons in aperiodic Pt/Co/Al. HERALDO-based vector tomography reconstructed the full 3D magnetization vector field with a spatial resolution of approximately $Q=n$0, determined by Fourier shell correlation, and directly visualized cocoons as localized tube-like textures confined to fractions of the multilayer thickness rather than through the whole stack (Chiliquinga-Jacome et al., 21 Jan 2026). The reconstructions showed vertical misalignment and overall chirality, while the limited angular coverage and missing wedge were identified as the principal imaging constraints.

Dynamics are not restricted to switching. In centrosymmetric frustrated magnets, rotating magnetic field can tune vector vorticity and generate straight motion, while the same internal degree of freedom modulates current-driven motion under spin-polarized current (Yao et al., 2022). In skyrmionic division, a skyrmion or skyrmion bag is driven across a “skyrmionic slicer” so that a transient region of opposite topological degree forms at the cut site and the object divides into two or more stable skyrmionic structures (Kind et al., 2022). In distorted skyrmionic textures with different topological charges, linearized LLG analysis showed that broken cylindrical symmetry hybridizes angular-momentum eigenmodes, produces avoided crossings in the magnon spectrum, and makes many more modes visible to homogeneous excitation (Rózsa et al., 2020). Taken together, these studies indicate that spatiotemporal skyrmionic behavior in magnets is governed as much by mode structure and singular events as by static topology.

4. Optical spatiotemporal skyrmions

In optics, spatiotemporal skyrmionic textures are literal structures in space-time rather than solely dynamical evolutions of spatial textures. A central experimental realization constructs skyrmions in the $Q=n$1 plane of a picosecond pulse wavepacket by vectorial sculpturing of spatiotemporal wavepackets. The governing scalar envelope satisfies

$Q=n$2

and the skyrmion is formed by superposing orthogonally polarized spatiotemporal Laguerre-Gaussian pulses (Teng et al., 10 Mar 2025). The reported key distinction is that these skyrmions are constructed upon transverse orbital angular momentum, not longitudinal OAM. As a result, they exhibit no helical twisting perpendicular to the skyrmion plane and therefore show enhanced stability to deformations or perturbations relative to spatial skyrmions (Teng et al., 10 Mar 2025).

A complementary analytic route is provided by exact nonparaxial solutions of Maxwell’s equations for spatiotemporal optical vortices and sagittal skyrmionic pulses. The scalar pulsed field is generated within a complex focus model, and differential operators then produce a STOV with transverse OAM and an electromagnetic vector solution whose polarization distribution in the sagittal plane covers nearly all possible full polarization states (Vo et al., 2024). The paper explicitly frames these pulses as skyrmionic polarization distributions in a longitudinal plane rather than in a purely transverse one.

A broader generalization appears in 4D topological textures in light. There, five plane waves with adiabatically varying relative amplitudes generate nonparaxial optical lattices that contain all possible polarization ellipses with every combination of ellipticity and orientation in 3D space, spanning the nonparaxial polarization space and a 4-sphere within specific spatiotemporal regions (Marco et al., 2022). The corresponding 4D Skyrme numbers are defined as

$Q=n$3

for the polarization-space and 4-sphere formulations, respectively (Marco et al., 2022). This moves the optical notion of a skyrmion from a 2D wrapping of the Poincaré sphere to a higher-dimensional mapping over $Q=n$4 variables.

Structured intense laser fields provide a related route in which skyrmionic optical textures are imprinted in a field experienced by matter. Superpositions of vortex and vector-beam potentials generate optical skyrmions, while structured-light Volkov states show how local phase, polarization, and field gradients enter the electron wavefunction and allow photoelectrons to image the field topology with subwavelength spatial sensitivity and attosecond temporal resolution (Wätzel et al., 2020). This suggests that, in optics, spatiotemporal skyrmionic textures are not only objects of field design but also targets of ultrafast topological metrology.

5. Observation, reconstruction, and inverse inference

The experimental study of spatiotemporal skyrmionic textures depends heavily on reconstruction methods that recover vector fields rather than scalar contrasts. In magnetic multilayers, HERALDO tomography uses two orthogonal slits as references, 58 projections over $Q=n$5, and iterative vector reconstruction to obtain $Q=n$6, with the NiFe region serving as a chemical marker for depth selectivity in cocoon imaging (Chiliquinga-Jacome et al., 21 Jan 2026). In FeGe, vector field electron tomography reconstructs $Q=n$7 and $Q=n$8 from magnetic phase shifts and obtains $Q=n$9 from $n$0, enabling quantitative 3D induction maps with sub-10 nm spatial resolution (Wolf et al., 2021).

Optical platforms require different observables but comparable reconstruction logic. The experimental realization of spatiotemporal optical skyrmions used time-resolved off-axis interferometry with a synchronized probe pulse and polarization tomography to reconstruct the full spatiotemporal field and the Stokes distribution (Teng et al., 10 Mar 2025). In structured dielectric planar media, skyrmionic polarization textures were reconstructed over one spatial period by quantum process tomography based on supervised machine learning, using a neural network trained on $n$1 simulated examples to infer local SU(2) operator parameters from six polarimetric images per setting (Colandrea et al., 10 Oct 2025). The same framework yielded local Berry curvature and quantum metric maps in the synthetic optical lattice picture.

Automated recognition and parameter extraction are emerging as a general layer above imaging. The nine-class skyrmionic-texture classifier based on fine-tuned Inception-v3, together with the MISO and SVR parameter regressors, demonstrates that spin texture images can be mapped to phase labels and Hamiltonian parameters directly (Feng et al., 2023). A plausible implication is that future spatiotemporal datasets—videos, tomograms, or multidimensional polarization maps—will be treated as inverse problems in which topology, interactions, and dynamical pathways are inferred jointly rather than sequentially.

6. Functional implications and open questions

The device relevance of spatiotemporal skyrmionic textures derives from multistability, programmability, and transport or wave-dynamical signatures. In Pt/Co/W multilayers, the confinement-driven redistribution of populations toward skyrmion bags in narrow tracks was presented as a scalable materials strategy for multistate skyrmion-based spintronic and memory architectures (Sadi et al., 15 Apr 2026). In Fe$n$2GaTe$n$3, topological skyrmion junctions formed by adjoining $n$4 and $n$5 lattices showed enhanced longitudinal conductivity and reduced resistance because the opposite skyrmion Hall deflections cancel (Mi et al., 2024). In Fe$n$6GaTe$n$7, the pathway model based on DMI, field, and cooling history offers deterministic access to composite states at room temperature, which was explicitly connected to memory, logic, neuromorphic computing, and heterostructure spintronics (Li et al., 8 May 2025).

Transport signatures remain sensitive to sample quality. In disordered skyrmionic textures, the topological Hall effect and topological spin Hall effect originate from the emergent magnetic field

$n$8

yet both signals were found to be highly sensitive to disorder strength and momentum scattering, even though the adiabatic approximation remained valid down to very small skyrmions (N'diaye et al., 2016). This places a practical limit on how directly topological charge can be converted into electronic functionality.

In photonics, the applications emphasized are optical metrology, sensing, optical communication, and data storage for $n$9 skyrmions, as well as topological photonics and all-optical quantum Hall analogues for skyrmionic eigenpolarization textures in planar dielectric media (Teng et al., 10 Mar 2025, Colandrea et al., 10 Oct 2025). Because spatiotemporal optical skyrmions rely on transverse OAM and avoid propagation-induced helical twisting, they provide a route to robust polarization topology in ultrafast wavepackets (Teng et al., 10 Mar 2025).

Several unresolved issues recur across magnetic and optical settings. Real structures are distorted by surfaces, missing-wedge limitations, disorder, pinning at grains, and anisotropy variations rather than by ideal continuum symmetries alone (Chiliquinga-Jacome et al., 21 Jan 2026, Wolf et al., 2021, N'diaye et al., 2016). At the same time, those deviations are often productive: they enable confinement-controlled recombination, chirality changes, track formation, mode hybridization, or synthetic-lattice band topology (Sadi et al., 15 Apr 2026, Hagemeister et al., 2016, Rózsa et al., 2020, Colandrea et al., 10 Oct 2025). The field therefore treats spatiotemporal skyrmionic textures not as a single canonical object but as a family of topological configurations whose defining feature is robust winding under controlled spatial and temporal deformation.