Toroidal Skyrmionic Domains

• Toroidal skyrmionic domains are spatially confined regions characterized by toroidal geometry and nontrivial topological charges (skyrmion or Hopf), ensuring robust field configurations.
• They are explored through diverse models, including Maxwell solutions and self-dual Skyrme-type frameworks, which validate their stability via space–time nonseparability and conformal invariance.
• These domains promise technological advances in telecom, remote sensing, and metamaterials by leveraging topological protection for programmable and resilient system designs.

Toroidal skyrmionic domains denote spatially localized regions, typically with toroidal geometry, where the order parameter field exhibits nontrivial topological structure described by a skyrmion number or, in more specialized contexts, a Hopf charge. These domains arise in a broad range of physical systems, including electromagnetic fields in free space, multicomponent condensed matter models, and nonlinear field theories with topological solitons. The toroidal topology both constrains and protects the topological charge, leading to distinctive field and energy configurations, robust excitation dynamics, and technologically relevant properties.

1. Maxwell Theory: Toroidal Electromagnetic Skyrmionic Domains

The canonical toroidal electromagnetic pulse is a non-transverse, space–time nonseparable solution to Maxwell’s equations in free space (Hellwarth–Nouchi pulse) (Wang et al., 2023). In cylindrical coordinates $(r,\phi,z)$, the field components are:

• $E_r(r,z,t)$ and $E_z(r,z,t)$ exhibit radial and longitudinal structure with explicit space–time nonseparability.
• $H_\phi(r,z,t)$ is purely azimuthal, yielding circulation around the toroidal axis.

The explicit Maxwell solution ensures a continuous, singularity-free evolution of the pulse with strong space-time coupling. The vector potential formulation, using $A_\phi(r,z,t)$, underlines the link between nontrivial topology and field configurations.

Each transverse slice in such a pulse is a vector skyrmion: the orientation of in-plane electric vectors ‘wraps’ the Poincaré sphere as a function of the radius, such that the skyrmion number $N_{\mathrm{sk}} = \pm 1$ remains topologically protected. Singularities include saddle points on-axis and vortex rings off-axis.

Experimentally, these domains are synthesized by launching a toroidal pulse with a broadband conical horn antenna (1.3–10 GHz bandwidth). Mapping the full skyrmion texture proceeds by point-by-point vector electrometry and reconstruction of the local unit vector, whose covering of the sphere yields a robust integer topological charge even as the pulse propagates.

Space–time nonseparability is quantified via quantum-tomographic metrics (concurrence $C$, entanglement of formation $E_{\mathrm{FoF}}$), showing that propagation ‘purifies’ the pulse toward the ideal toroidal form: for propagation distances up to $z \sim 1\,\mathrm{m}$, measured $C$ and $E_{\mathrm{FoF}}$ increase from $<0.2$ near the aperture to $>0.9$ far-field, with canonical pulse fidelity $F > 0.7$.

2. Field-Theoretic Models of Toroidal Skyrmions

Exact self-dual toroidal skyrmion solutions exist in Skyrme-type models with target space $S^3$ and space-time-dependent couplings (Ferreira et al., 2017). The model’s action includes quadratic and quartic derivative terms, with their strengths $m(x), e(x)$ chosen to depend on coordinates via a common conformal factor $f(x)$, yielding conformal invariance in three dimensions:

• The toroidal ansatz $$Z=(Z_1, Z_2)^{\mathrm{T}} = (\sqrt{F(z)}e^{in\phi}, \sqrt{1-F(z)}e^{im\xi})$$ leads to solutions labeled by integers $m,n$ (windings along toroidal angles).
• The Bogomolny (self-duality) equations are reduced to ODEs for $F(z)$ and $f(z,\xi)$, admitting exact solutions with topological charge $Q = -mn$.
• The energy density is spherically symmetric when $|m| = |n|$ and takes prolate or oblate toroidal shape otherwise.

These toroidal skyrmions saturate the Bogomolny bound ($E \geq 4\pi^2(m_0/e_0)|Q|$) and derive their existence from the conformal symmetry and judicious choice of spatially varying couplings that evade conventional no-go theorems for finite-energy, force-free configurations.

3. Skyrmion Domain Walls: Stable Toroidal Embedding

Domain-wall Skyrmions arise when the effective field theory admits wall solutions in one direction and baby Skyrmion solutions confined within the wall (Gudnason et al., 2014). In the O(4) sigma model with Skyrme term and suitable potential:

• The domain wall is stabilized by a kink solution ($f(x)$), with the planar baby-Skyrme model induced on the wall’s center.
• Axially-symmetric ring (toroidal) solutions with topological charge $B$ emerge, especially for quadratic mass term potentials.
• Numerical solutions show well-defined ring-shaped energy and baryon-density profiles. For the linear mass term, charge-2 rings become energetically favored, and large-$B$ ground states conjecturally form charge-2 ring lattices.

These wall-confined toroidal Skyrmions are relevant for physical systems with layered structures (e.g., magnetic multilayers, QCD domain walls) and potentially enable close-packed lattices of topological objects.

4. Twisted Vortex Rings and Handles

Twisted vortex-ring (toroidal) Skyrmions, as studied in (Gudnason et al., 2014, Gudnason et al., 2018), represent a physical realization where a U(1) modulus is wound along a closed toroidal defect in bulk or near a wall:

• The ansatz uses constrained two-component fields, with phase windings determining baryon charge and energy.
• Stability is governed by the competition between ring tension and twist-induced pressure, with explicit stabilization via sixth-order (BPS-Skyrme) terms.
• For BEC-inspired potentials, vortex handles bound to domain walls yield ‘toroidal braided string-junctions’ (specifically at $B=2$), which exhibit lower energy than isolated handles or bulk rings.

Major and minor radii of the torus scale with the mass parameter ($M^{-1}$), and energetics favor configurations tightly bound to the wall, with calculated binding energy differences ($\Delta E \approx 6.6$) confirming preference for the braided toroidal junction.

5. Hopfion Construction: Toroidal Domain Walls with Multiple Twists

In the Faddeev–Skyrme model with an Ising-type potential and two discrete vacua, domain walls can assume toroidal shapes with doubly-twisted internal structure (Kobayashi et al., 2013):

• The fields map $S^3 \rightarrow S^2$ via a Hopf map, embedding toroidal geometry.
• The ansatz introduces two winding numbers $(P,Q)$: $P$ for toroidal (longitudinal) phase twist, $Q$ for poloidal (meridional) phase twist.
• The topological Hopf charge is $H = PQ$, but distinct pairs $(P,Q)$ with equal product are topologically protected and cannot be smoothly deformed into one another in the presence of the Ising potential.

Numerical energy relaxation yields stable, doubly-twisted toroidal domain walls for $(P,Q) = 1,2,3$, with localized energy density on the tube and no decay into lower-charge states due to potential-induced barriers.

6. Topological Protection, Robustness, and Application Prospects

The toroidal topology confers strong topological protection on skyrmionic domains across electromagnetic, condensed-matter, and field-theoretic realizations. The quantized charge ($N_{\mathrm{sk}}$ or Hopf charge $H$) remains invariant unless the order parameter vanishes over a finite region, ensuring robustness against perturbations (e.g., mild misalignment, turbulence).

Potential application domains include:

• High-capacity telecom links, where the discrete charge could encode information beyond orbital angular momentum channels (Wang et al., 2023).
• Remote sensing and microscopy, leveraging subwavelength focal features and isodiffraction.
• Light-matter coupling to anapole and toroidal dipole resonances in metamaterials.
• Multilayer magnetic and cold-atom systems where wall-confined Skyrmion lattices may be engineered for programmable topological architectures.

A plausible implication is that the diverse realizations and universal topological robustness of toroidal skyrmionic domains render them a versatile platform for both fundamental studies of topology in field theories and emergent photonic or condensed-matter technologies.