Multi-Order Polar Skyrmions

• Multi-order polar skyrmions are topologically nontrivial solitonic textures characterized by systematically varying order parameters and quantized charges.
• They are stabilized through competing energetic interactions in ferroelectric, magnetic, and optical systems, enabling controllable device functionalities.
• External tuning via temperature, strain, doping, and fields enables dynamic control for high-density memory, reconfigurable logic, and active metamaterials.

A multi-order polar skyrmion is a solitonic, topologically nontrivial polarization or spin texture characterized by nonuniform spatial winding of the polarization, magnetization, or optical field, where the order parameter (e.g., handedness, vorticity, or winding number) varies systematically either in real space (e.g., along a superlattice stacking or within a skyrmion core) or as a function of system parameters, stabilizing distinct skyrmion orders (integer, half-integer, or higher) at equilibrium. Multi-order polar skyrmions are distinct from conventional (Néel or Bloch) skyrmions by exhibiting richer order parameter manifolds, multiple quantized topological charges, or controllably varying inner texture (e.g., Bloch–Néel hybridization, nested ring structures, or layerwise order) and are observed in a wide variety of ferroelectric, magnetic, and optical systems with strong polar, chiral, or strain–order coupling.

1. Topological Structure and Mathematical Definition

Multi-order polar skyrmions are defined via topological charge (or skyrmion number) in systems with a continuous vector order parameter. For a 2D spin or polarization field $\,\mathbf{n}(\mathbf{r})\,$ (normalized vector), the skyrmion number is

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

For generic “multi-order” textures, $Q$ can be integer, half-integer, or even higher depending on the spatial winding.

• In axisymmetric $k\pi$ polar skyrmions, the in-plane component winds by $k\pi$ as a function of radius, giving $Q = \pm 1$ for $k=1$, $Q=0$ for $k=2$ (even $k$), and generally $$Q = \frac{1}{4\pi} \int \mathbf{n} \cdot (\partial_x \mathbf{n} \times \partial_y \mathbf{n})\, d^2r.$$0 (odd $$Q = \frac{1}{4\pi} \int \mathbf{n} \cdot (\partial_x \mathbf{n} \times \partial_y \mathbf{n})\, d^2r.$$1) (Liu et al., 9 Jan 2026).
• In meron-like textures, the charge $$Q = \frac{1}{4\pi} \int \mathbf{n} \cdot (\partial_x \mathbf{n} \times \partial_y \mathbf{n})\, d^2r.$$2 arises when the out-of-plane component vanishes at the periphery (Shao et al., 2021).
• In layered (multilayer or superlattice) systems, the skyrmion order may vary from layer to layer (e.g., Néel-type to Bloch-type) while maintaining the same $$Q = \frac{1}{4\pi} \int \mathbf{n} \cdot (\partial_x \mathbf{n} \times \partial_y \mathbf{n})\, d^2r.$$3 per layer, defining a “multi-order” structure in stacking (Sen et al., 2024).
• In vortex-embedded states, the total topological charge is $$Q = \frac{1}{4\pi} \int \mathbf{n} \cdot (\partial_x \mathbf{n} \times \partial_y \mathbf{n})\, d^2r.$$4 (Zhang et al., 20 Mar 2025).

These textures are fully characterized by the spatial dependence of the vector field, the path it traces on the order-parameter sphere (e.g., Poincaré or Bloch spheres), and their quantized topological invariants.

2. Microscopic Mechanisms and Model Hamiltonians

The stabilization of multi-order polar skyrmions requires a combination of competing energetic terms. In ferroelectric and magnetic systems, the energy functional is commonly written as

$$Q = \frac{1}{4\pi} \int \mathbf{n} \cdot (\partial_x \mathbf{n} \times \partial_y \mathbf{n})\, d^2r.$$5

where each term may couple polarization/spin, strain, electromagnetic field, and octahedral tilts. Key mechanisms depend on the host system:

• Ferroelectric superlattices: Polar skyrmions are stabilized not by DMI but by elastic, electrostatic, and gradient energies and their competition, especially under temperature- or strain-induced phase transitions (Shao et al., 2021). The coupling of polarization and strain at the layer level can play the orienting role of DMI (even in the absence of intrinsic chirality) (Sen et al., 2024).
• Magnetic multilayers: Layer-dependent Dzyaloshinskii–Moriya interaction with symmetry tuning (e.g., in trilayers) creates twisted surface states and high-topological-number skyrmions (Hayami, 2022).
• Anisotropic exchange: Symmetry-allowed polar DMI vectors and symmetric anisotropy yield nontrivial hybrid or rhombic multi-q states, including hybridized Néel–Bloch–anti-skyrmion lattices (Hayami, 2024).
• Optical counterparts: Spin–orbit coupling arising from field configuration can generate multi-order hybrid skyrmions in structured light, with the order set by orbital angular momentum or polarization degree (Yao et al., 2024, Saito, 19 Mar 2025).

3. Classification and Realizations

Multi-order polar skyrmions can be systematically classified by:

Texture Type	Defining Feature	Example System
$$Q = \frac{1}{4\pi} \int \mathbf{n} \cdot (\partial_x \mathbf{n} \times \partial_y \mathbf{n})\, d^2r.$$6-order skyrmion	$$Q = \frac{1}{4\pi} \int \mathbf{n} \cdot (\partial_x \mathbf{n} \times \partial_y \mathbf{n})\, d^2r.$$7 half-turns of in-plane polarization; $$Q = \frac{1}{4\pi} \int \mathbf{n} \cdot (\partial_x \mathbf{n} \times \partial_y \mathbf{n})\, d^2r.$$8 concentric rings	BiFeO$$Q = \frac{1}{4\pi} \int \mathbf{n} \cdot (\partial_x \mathbf{n} \times \partial_y \mathbf{n})\, d^2r.$$9 superlattices (Liu et al., 9 Jan 2026)
Half-integer meron/antimeron	$Q$0, vanishing P$Q$1 at periphery, checkerboard order	PTO/STO membranes (Shao et al., 2021)
Layer-twisted/stacked	Bloch $Q$2 Néel variation along stack	PTO/STO superlattice (Sen et al., 2024)
High-$Q$3 multi-q	Multiple windings, $Q$4, $Q$5, or non-integer	Trilayer triangular magnets (Hayami, 2022)
Hybrid optical/multi-DOF	Order set by OAM, spin, or SU(N) mixing of polarization	Structured light (Yao et al., 2024, Saito, 19 Mar 2025)

In chiral-nematics and liquid crystals, schools of polar skyrmions with tunable order and collective fluid dynamics emerge due to elastic dipolar interactions (Sohn et al., 2019).

4. Control and Tunability of Skyrmion Order

The multi-order character can be tuned through external fields, temperature, mechanical boundary conditions, or composition:

• Temperature cycling: Drives transitions from solitons ($Q$6) $Q$7 1$Q$8-, 2$Q$9-, 3$k\pi$0-, 4$k\pi$1-skyrmions in BFO/STO, with each order occupying distinctive stability windows; odd $k\pi$2 orders have nonzero $k\pi$3, even $k\pi$4 orders are topologically trivial (Liu et al., 9 Jan 2026).
• Strain modulation: In PTO/STO freestanding films, in-plane strain is set by thermal expansion mismatch, producing tensile-strain-stabilized skyrmion stripes, isotropic bubbles, or tetratic meron lattices with alternating chirality (Shao et al., 2021).
• Doping: Sm substitution in BFO lowers transition temperatures, making multi-order skyrmions accessible at room temperature (Liu et al., 9 Jan 2026).
• External magnetic field and interlayer coupling: High-order ($k\pi$5) skyrmions are stabilized in trilayer triangular models by tuning interlayer exchange $k\pi$6, DMI $k\pi$7, and field $k\pi$8 (Hayami, 2022).
• Optical field design: Skyrmion order in structured light is set by the orbital angular momentum index $k\pi$9 and polarization control (e.g., via vortex beams and waveplates) (Yao et al., 2024, Saito, 19 Mar 2025).

5. Experimental Observation, Characterization, and Phase Diagrams

• Phase-field and micromagnetic simulations reveal precise equilibrium sequences and phase boundaries as a function of strain, temperature, field, and system geometry (Shao et al., 2021, Hayami, 2022, Liu et al., 9 Jan 2026, Zhang et al., 20 Mar 2025).
• 4D-STEM/SCBED methodologies enable quantitative high-resolution mapping of local polarization, strain, and chirality at the nanometer scale; extraction of $k\pi$0 is achieved by integrating local vector data (Shao et al., 2021).
• Polarization tomography and structured-light analysis confirm the mapping of optical skyrmion textures to higher-order Poincaré spheres, validating the control over multi-order states (Yao et al., 2024, Saito, 19 Mar 2025).
• Synchrotron X-ray reciprocal-space mapping identifies the emergence of satellite peaks and rotation of symmetry axes corresponding to changing skyrmion lattice order and structure (Shao et al., 2021).
• Multipole decomposition of local polarization in real materials (e.g., GaV$k\pi$1S$k\pi$2) discriminates between mono-, di-, and quadrupolar contributions, with $k\pi$3, $k\pi$4 multipoles directly observable in force or Kelvin-probe microscopy (Ruff et al., 2015).

6. Applications and Functional Implications

Multi-order polar skyrmions enable several unique device and material functionalities:

• High-density, multilevel nonvolatile memory: Each skyrmion order provides an additional state; robust odd-$k\pi$5 protection for data retention, even-$k\pi$6 as intermediate “erase” levels (Liu et al., 9 Jan 2026).
• Externally reconfigurable logic: Local control of strain, field, or boundary condition allows dynamic transformation between orders and chirality for logic-in-memory primitives (Shao et al., 2021).
• Spintronic and magnonic devices: Vortex–skyrmion hybrids provide fourfold (chirality, polarity) degree-of-freedom and topological protection, enabling high-density multi-bit state encoding, tunable magnonic band structures, and neuromorphic circuit design (Zhang et al., 20 Mar 2025).
• Topological optical information channels: High-order Stokes skyrmions in light enable large-capacity multiplexing and robust, orthogonal information representation, protected by topological invariance of $k\pi$7 (Yao et al., 2024, Saito, 19 Mar 2025).
• Active matter and soft-matter machines: Multi-order polar skyrmion clusters and schools (e.g., in chiral nematics) display programmable motility and collective fluidity, facilitating emergent metamaterials and micro-machines (Sohn et al., 2019).
• Metamaterials: Dipolar, quadrupolar, and higher multipole moments in multiferroics couple local spin and charge order for responsive, field-tunable metamaterial design (Ruff et al., 2015).

7. Outlook and Generalizations

Multi-order polar skyrmions exist across material classes (ferroelectric, multiferroic, magnetic, liquid crystal, and photonic) and symmetry types. Future directions include:

• Engineering of targeted interlayer coupling and DMI symmetry to stabilize arbitrary topological charge, twisted, or hybrid textures (Hayami, 2022, Hayami, 2024).
• Synthetic structuring of optical and electronic order via SU(N) symmetry manipulation (SU(6) in structured light) for continuous passage between skyrmion and antiskyrmion manifolds and higher-dimensional topological textures (Saito, 19 Mar 2025).
• Development of layer-resolved and 3D characterization tools (e.g., TEM holography, X-ray dichroism) for direct imaging of multi-order and twisted skyrmion profiles in superlattices (Sen et al., 2024).
• Thermodynamically reversible “order switching” protocols via thermal cycling, compositional grading, or field-tuned strain, critical for device integration of multi-level nonvolatile logic (Liu et al., 9 Jan 2026).
• Theory and experiment of active topological matter: long-range order, number fluctuations, and topological fluidity in living and inanimate skyrmion “schools” (Sohn et al., 2019).

In summary, multi-order polar skyrmions represent a unifying topological framework for solitonic textures with tunable winding, chirality, and internal structure, providing a paradigm for controlled, robust, and multifunctional nano-architectures in condensed matter, optics, and beyond (Shao et al., 2021, Liu et al., 9 Jan 2026, Hayami, 2022, Sen et al., 2024, Yao et al., 2024, Sohn et al., 2019, Ruff et al., 2015, Hayami, 2024, Saito, 19 Mar 2025, Zhang et al., 20 Mar 2025, Javed et al., 2021).