High-Order Anti-Skyrmions in Spintronics

• High-order anti-skyrmions are topologically protected 2D spin textures with integer charges exceeding one and a reversed winding compared to conventional skyrmions.
• They are generated via the coalescence of multiple vertical Bloch lines and stabilized by a balance of exchange, anisotropy, and dipolar interactions.
• Their unique multi-lobed symmetry underpins applications in multilevel memory, spintronic oscillators, and structured optical devices.

High-order anti-skyrmions are topologically protected two-dimensional spin or polarization textures characterized by integer topological charge $\left|Q\right|>1$ and an orientation (or "winding") opposite to conventional skyrmions. In condensed matter, they arise as higher-winding-number solutions of the micromagnetic equations and are stabilized through a mixture of exchange, anisotropy, dipolar, and, in select contexts, higher-order exchange or optical field couplings. Analogous structures—engineered in polarization and angular momentum space—extend the definition of high-order anti-skyrmions to optical and SU($N$) fields. These textures exhibit unique nucleation pathways, symmetry properties, and stability criteria, and are central to proposals for multistate information encoding and advanced spintronic and photonic device architectures.

1. Mathematical Definition and Topological Classification

High-order anti-skyrmions are defined by their topological charge, or skyrmion number,

$$Q = \frac{1}{4\pi} \int_{\mathbb{R}^2} \mathbf{n}(x, y)\cdot \left(\partial_x \mathbf{n} \times \partial_y \mathbf{n}\right)dx\,dy$$

with $\mathbf{n}$ being either a real-space magnetization unit vector field $\mathbf{m}$ or, in optical contexts, the normalized Stokes vector $\mathbf{S}$. For fundamental (first-order) anti-skyrmions, $Q=+1$ (in the standard sign convention), whereas higher-order anti-skyrmions have even larger positive $Q$, e.g., $Q=+2, +3, +5$ (Koraltan et al., 8 Jan 2025, Fang et al., 2023, Tian et al., 13 Aug 2025). The "anti-" prefix denotes the orientation opposite to standard skyrmions (e.g., $Q>0$ vs $Q<0$ for the same field configuration, sign depending on convention and background orientation).

Distinct from their first-order analogs, higher-order anti-skyrmions exhibit multi-lobed symmetry and feature a Stokes or magnetization vector covering the sphere $|Q|$ times, resulting in robust, energetically protected quasi-particles. Their topology is linked to both the vorticity $m$ (the number of $2\pi$ windings in the in-plane component) and polarity $q$ (the boundary-to-center inversion), with $Q=q \times m$ (Fang et al., 2023).

2. Nucleation and Stabilization Mechanisms in Magnetic Systems

The creation of high-order anti-skyrmions in ultrathin magnetic multilayers proceeds via the pinching off of multiple vertical Bloch lines (vBLs), which are topological point defects in chiral or Bloch-type domain walls. Each vBL contributes a topological charge of $\pm 1$. When the collapse of a multidomain region under an increasing out-of-plane magnetic field leaves behind $n$ vBLs of the same sign, these can coalesce into a zero-dimensional texture with $Q=n$ (Koraltan et al., 8 Jan 2025):

• $Q=0$ bubbles occur if two opposite-signed vBLs annihilate.
• $|Q|=1$ skyrmions or antiskyrmions result from a single surviving vBL.
• $|Q|>1$ higher-order (anti)skyrmions arise if two or more equally signed vBLs are present.

Stabilization does not require large Dzyaloshinskii–Moriya interaction (DMI). Instead, the interplay of dipolar interaction, exchange stiffness $A$, and perpendicular anisotropy $K_u$—especially when $K_u \approx \mu_0 M_s^2/2$—enables wide Bloch walls and supports multiple vBLs (Heigl et al., 2020). In Fe/Gd and Co/Ni multilayers, higher-order anti-skyrmions are metastable in parameter domains where $$100\,\mathrm{kA/m} \lesssim M_s \lesssim 300\,\mathrm{kA/m}$$ and $$10\,\mathrm{kJ/m^3} \lesssim K_u \lesssim 30\,\mathrm{kJ/m^3}$$ (Heigl et al., 2020, Koraltan et al., 8 Jan 2025).

Higher-order exchange interactions, particularly four-site four-spin couplings, can further increase the energy barrier for collapse and soften saddle-point modes, extending anti-skyrmion lifetimes by orders of magnitude (Schrautzer et al., 7 Nov 2025). This stabilization is effective even in centrosymmetric systems lacking DMI, facilitating antiskyrmion formation in a broader range of materials.

3. Visualization and Experimental Signatures

High-resolution magnetic force microscopy (MFM) and Lorentz transmission electron microscopy (LTEM) enable direct imaging of high-order anti-skyrmions:

• MFM phase-shift maps, which are sensitive to the stray-field gradient $\partial_z B_z$, reveal distinct lobe patterns tied to $Q$ (Koraltan et al., 8 Jan 2025). For instance, a $Q=1$ anti-skyrmion displays two prominent lobes, whereas $Q=3$ and $Q=5$ anti-skyrmions exhibit four and five lobes, respectively.
• In LTEM, the pattern of bright and dark fringes reflects the alternating Bloch and Néel wall segments, with higher-order anti-skyrmions (e.g., $Q=–2$) manifesting three-fold symmetry (Heigl et al., 2020).
• Micromagnetic simulations reproduce these MFM and LTEM signatures, confirming the correspondence between the number of lobes and the topological charge $Q$. Table 1 summarizes key experimental lobe counts and symmetries.

Topological Charge $Q$	MFM/LTEM Lobe Pattern	Symmetry
+1	2 pronounced lobes	Elliptical, 2-fold
+3	4 strong lobes	4-fold
+5	5 lobes (large diameter)	5-fold
–2	3 alternating lobes	3-fold

4. High-Order Anti-Skyrmions in Optical and SU($N$) Systems

Beyond magnetism, high-order anti-skyrmions arise in structured light fields and in the polarization–OAM Hilbert space of photons:

• In high harmonic generation (HHG) using two full Poincaré beams, the EUV Stokes texture's skyrmion number is $Q = q \times m$, tunable by beam geometry, OAM difference $\ell_1-\ell_2$, and photon absorption channel. Anti-skyrmions of arbitrary order ($Q=+M$) are generated for $\ell_1-\ell_2=-M$, with spatially resolved spin inversion between the center and edge (Fang et al., 2023).
• Angular-spectrum engineering of Bessel beams enables the creation of optical anti-skyrmions with prescribed $Q$ and customized intensity/trajectory profiles. For instance, superposing $\ell_1=0$ and $\ell_2=2$ with a $\pi$ phase yields a $Q=-2$ anti-skyrmion (Tian et al., 13 Aug 2025).
• SU(6) structured-light frameworks map skyrmion and anti-skyrmion subalgebras onto higher-dimensional Poincaré spheres ($\mathbb{S}^2$ and beyond). High-order anti-skyrmions can be manipulated by compositions of rotations in the SU(6) algebra, implemented physically via Mach–Zehnder interferometry with waveplates and vortex lenses (Saito, 19 Mar 2025).

5. Micromagnetic and Optical Simulation Methodologies

Calculating and confirming high-order anti-skyrmion properties utilizes:

• Large-scale micromagnetic codes (e.g., magnum.np, magnum.af, magnum.fe) implementing Landau–Lifshitz–Gilbert (LLG) relaxation across discretized film geometries, with parameter sweeps in $M_s$, $K_u$, $D$, and applied field $H$ (Koraltan et al., 8 Jan 2025, Heigl et al., 2020).
• Atomistic spin simulations including up to fourth-order exchange, with harmonic transition-state theory (HTST) to compute collapse energy barriers and lifetimes (Schrautzer et al., 7 Nov 2025).
• Angular-spectrum/Fourier methods for generating and steering optical skyrmions—processing arbitrary field modulations and holographic patterns to encode $Q$ and spatial intensity/trajectory (Tian et al., 13 Aug 2025).
• In SU($N$) photonics, representations via generalized Euler decompositions and projections onto higher-order Poincaré spheres for direct mapping of spin and OAM coupled states (Saito, 19 Mar 2025).

6. Functional Implications and Prospects for Devices

High-order anti-skyrmions underpin several proposed functionalities:

• Multilevel memory: Each $Q$ encodes a distinct metastable state, enabling nonbinary logic and multistate racetrack memory (Koraltan et al., 8 Jan 2025).
• Reservoir computing: Distinct nonlinearity for each $Q$ broadens the phase space for neuromorphic processing (Koraltan et al., 8 Jan 2025).
• Spintronic oscillators and logic: Coexistence of multiple $Q$ objects supports reconfigurable, low-power, high-speed circuits, especially in ferrimagnetic films where skyrmion Hall angles and dynamics can be tuned (Heigl et al., 2020).
• Ultrafast spintronics and chiral photochemistry: EUV anti-skyrmions (particularly those engineered by HHG) enable femtosecond-scale manipulation of spin textures or excitation of three-dimensionally chiral responses for readout and separation protocols (Fang et al., 2023).
• Particle manipulation and encoding: Optical high-order anti-skyrmions can exert stronger, highly structured forces, and multiplex OAM/spin-based information throughput (Tian et al., 13 Aug 2025, Saito, 19 Mar 2025).

Limitations include the narrow parameter stability regions for higher-order anti-skyrmions, their tendency to fragment into lower-order objects without precise material control, and increased sensitivity to perturbations in optical implementations. Robust engineering of material parameters (e.g., enhancement of four-spin couplings, precision tuning of $K_u$ and $A$) and advanced optical field shaping are active avenues to overcome these challenges.

7. Topological Manipulation and Higher-Order Generalization

High-order anti-skyrmions are not limited to fixed $Q$ values: continuous deformation between skyrmions and anti-skyrmions is accessible through group-theoretic pathways in photonic SU($N$) spaces, notably via the skyrmionic torus in SU(3)—an explicit, toroidal trajectory parametrized by mixing different OAM and spin eigenstates (Saito, 19 Mar 2025). The ability to embed these states into higher-order Poincaré spheres or to modulate the spatial topology in light, matter, and hybrid systems is central to current research, with broader implications for topological phases, defect physics, and information transport in structured media.