Hopfions: 3D Topological Solitons

Updated 11 January 2026

Hopfions are three-dimensional topological solitons defined by a quantized Hopf number that measures the linking of field preimages.
They are modeled via micromagnetic and field-theoretic energy functionals that incorporate exchange, DMI, and anisotropy to ensure stability.
Experimental and numerical methods such as target skyrmion closure and finite-element topology tracking enable precise creation and control of hopfions.

A hopfion is a three-dimensional topological soliton characterized by a nontrivial linking of field preimages in real space, defined by a quantized topological invariant known as the Hopf number. Originally emerging in field theory as finite-energy maps from space compactified to the 3-sphere onto the 2-sphere, hopfions have seen realization and theoretical modeling in magnetic, polar, quantum fluid, photonic, and gauge field contexts. Their unique topological properties, robust particle-like stability, and dynamical control promise a new paradigm for three-dimensional structured matter and information storage.

1. Topological Definition, Classification, and Mathematical Characterizations

A hopfion is a soliton whose underlying vector field—such as a magnetization field $\mathbf{m}(\mathbf{r})$ , polarization, or complex order parameter—constitutes a map from compactified three-space $S^3$ to a target two-sphere $S^2$ , classified by the homotopy group $\pi_3(S^2)=\mathbb Z$ . The topological invariant is the Hopf number $Q_H$ , which counts the total linking number between any two distinct preimages of points on $S^2$ .

For magnetic hopfions (and many generalizations), $Q_H$ is computed using the Chern–Simons-type integral

$Q_H = \frac{1}{(4\pi)^2} \int_{\mathbb{R}^3} \mathbf{F}(\mathbf{r}) \cdot \mathbf{A}(\mathbf{r}) \; d^3 r,$

where

$F_i(\mathbf{r}) = \frac{1}{2} \varepsilon_{ijk} \mathbf{m} \cdot \left( \partial_j \mathbf{m} \times \partial_k \mathbf{m} \right)$

and $\mathbf{A}$ is a vector potential satisfying $\nabla \times \mathbf{A} = \mathbf{F}$ .

Different physical systems may realize hopfions where $Q_H$ is encoded as product or combination of additional winding numbers (e.g., $(P,Q)$ for toroidal domain walls, $M S$ for quantum droplets), but the generic property is that the integer invariant measures the linking of closed field lines or preimage loops in real space (Kent et al., 2020, Kobayashi et al., 2013, Zhao et al., 15 Jul 2025).

2. Micromagnetic and Field-Theoretic Models: Energy Functionals and Topological Stability

The stabilization of hopfions relies on energetic competition between local field stiffness and nonlocal twisting or anisotropy. In magnetism, a minimal continuum functional includes exchange, Dzyaloshinskii–Moriya interaction (DMI), and perpendicular anisotropy: $E[\mathbf m] = \int d^3x\, \left[ A(\nabla\mathbf m)^2 + D\mathbf m\cdot(\nabla\times\mathbf m) - K m_z^2 - \mu_0 M_s \mathbf H \cdot \mathbf m \right] + E_{\rm dip}$ (Kent et al., 2020, Sutcliffe, 2018). In bulk magnets, higher-order derivatives arising from frustrated competing exchange are essential for stability: $E[\mathbf{n}] = \int \left[ \mathcal{A} (\partial_\alpha n_\beta)^2 + \mathcal{B} (\partial_\alpha^2 n_\beta - \partial_\gamma^2 n_\beta )^2 + \mathcal{C}(\partial_\alpha^2 n_\beta \partial_\gamma^2 n_\beta) \right] d^3r$ with positive-definite coefficients (Rybakov et al., 2019, Sallermann et al., 2022). Derrick–Hobart and Vakulenko–Kapitanski scaling arguments prove the existence of finite-energy, stable, three-dimensional solitons for $Q_H \neq 0$ .

Hopfions in quantum fluids are constructed as stationary solutions to nonlinear Gross–Pitaevskii equations, stabilized by the interplay of mean-field nonlinearities and quantum fluctuation corrections (Lee–Huang–Yang term) in toroidal traps (Zhao et al., 15 Jul 2025).

3. Creation, Numerical Construction, and Experimental Observation

Several pathways exist for hopfion nucleation:

From 2D Target Skyrmions: In chiral magnets, a target skyrmion string is closed by modulating perpendicular magnetic anisotropy, producing a 3D Hopfion (Kent et al., 2020, Sutcliffe, 2018). Also, experimental Hopfion creation in Ir/Co/Pt multilayer nanodisks followed this strategy, confirmed by cross-referenced X-PEEM and MTXM imaging (Kent et al., 2020).
Direct Numerical Construction: Analytic ansätze based on rational maps or explicit stereographic projections are commonly minimized by gradient flow or conjugate-gradient methods in atomistic and mesoscale simulations (Hou et al., 4 Apr 2025, Sutcliffe, 2018, Rybakov et al., 2019).
Finite-Element Topology Tracking: Variants of the Chern–Simons integral, translated to discrete spatial elements, permit robust identification and mesh-invariant quantification of $Q_H$ in large-scale numerical micromagnetic simulations (Gallard et al., 12 May 2025).
Photonic and Field-Theoretic Realizations: Hopfions are synthesized in structured light by spatially programming superpositions of Laguerre–Gaussian modes, with polarization mapping the $S^2$ target, and topological invariance monitored through Stokes vector field tomography (Shen et al., 2022). In gauge field theory, Hopfions of massive vector fields are constructed in FLRW backgrounds via special helicity-mode expansions (Bousder et al., 2024).

4. Dynamical Properties, Transport, and Topology Switching

The dynamical response of hopfions is governed by their vanishing net gyrovector, contrast to 2D skyrmions (Wang et al., 2019, Kasai et al., 28 Nov 2025).

Current-Driven Motion: Under spin-transfer torque (STT) or spin-Hall torque (SHT), Néel-type hopfions translate strictly along the current direction, while Bloch-type hopfions can move transversely or longitudinally depending on the drive (Wang et al., 2019). Absence of a Magnus force precludes any skyrmion Hall effect.
Spin–Orbit-Torque-Induced Topology Switching: The application of SOT can induce translational and precessional dynamics for $Q_H=1$ hopfions, or drive controlled splitting and recombination for $Q_H>1$ , allowing for dynamic manipulation and encoding of topological charge (Kasai et al., 28 Nov 2025).
Eigenmode Excitations: Sub-GHz breathing oscillations of both core diameter and shell width, with preserved $Q_H$ , have been quantitatively modeled via domain-wall interaction frameworks and confirmed in micromagnetic simulations (Tejo et al., 30 Aug 2025).

5. Extensions beyond Magnetism: Hopfions in Polar, Quantum, Photonic, and Gauge Fields

Ferroelectrics: Three-dimensional hopfion polarization textures emerge as ground states within spherical nanoparticles of weakly anisotropic ferroelectric compounds under strong screening and surface boundary conditions, offering new dielectric and negative capacitance functionalities (Luk'Yanchuk et al., 2019).
Quantum Fluids: Self-trapped hopfion droplets have been predicted in binary atomic Bose–Einstein condensates, stabilized by competition of mean-field attraction, LHY quantum correction, and toroidal confinement, with the Hopf index built from vorticity and intrinsic twist (Zhao et al., 15 Jul 2025).
Electromagnetism, Photonics, and Gravitation: Hopfions have exact formulations as null (type N) and radiative (type D, III) solutions to Maxwell and gravitational field equations, with topological lines of force corresponding to linked circles and all relevant Petrov classes constructed via Penrose contour integrals in twistor space (Thompson et al., 2014, Shen et al., 2022). Structured light experiments directly realize and manipulate $Q_H$ in optical beams.
Gauge Fields and Early Universe Cosmology: Gauge-hopfions in Proca-type U(1) fields in cosmological backgrounds induce Hubble parameter shifts and chiral gravitational-wave backgrounds via their nontrivial topology (Bousder et al., 2024).
Statistical Mechanics and Lattice Models: Hopfions are exact topological defects in three-dimensional bipartite dimer models, protected against local moves (plaquette flips), with classification and explicit lattice construction realized via duality with classical (and artificial) spin ices (Bednik, 2019).

6. Collective Phases, Crystalline Order, and Higher-Index Constructions

Crystals and Lattices: Systematic frameworks based on rational maps and wave superpositions in $\mathbb{R}^4$ permit the design of periodic arrays of hopfions with arbitrary cubic symmetry (SC, FCC, BCC), variable $Q_H$ , and explicit higher-winding links and knots per unit cell (Hou et al., 4 Apr 2025).
Phase Behavior under Confinement: In thin films or chiral liquid crystals, multiple metastable and topologically distinct hopfion species, "bags," and composite configurations can be engineered through anisotropy tuning and boundary manipulation, but periodic hopfion lattices generically relax to stripe or finger phases, indicating intrinsic instability in simple confined geometries (Leonov et al., 28 Nov 2025).

7. Detection, Application Prospects, and Device Architectures

Transport and Orbital Hall Effects: Hopfions in magnetic systems generate a three-dimensional orbital Hall effect, with finite response tensor components corresponding to each axis—constituting an unambiguous electronic signature of the Hopf index (Göbel et al., 13 Jun 2025). Conversely, the charge topological Hall effect is globally absent, but hopfion position can be locally read via asymmetric contacts exploiting uncompensated emergent fields (Göbel et al., 2020).
Spintronics and Memory: The robust topological character, multilevel encoding enabled by controlled $Q_H$ switching, and vanishing skyrmion Hall effect designate hopfions as promising carriers for dense, three-dimensional, nonvolatile memory and logic elements in spintronic architectures (Kasai et al., 28 Nov 2025, Kent et al., 2020, Göbel et al., 2020). Demonstrated lifetimes (energy barriers up to hundreds of meV), topological switching by tunable SOT, and direct electrical readout support feasibility (Sallermann et al., 2022, Tejo et al., 30 Aug 2025).
Metrology, Magnonics, and Photonics: Frequency-domain addressing via breathing modes, waveguide designs exploiting field topology, and photonic encoding of information in $Q_H$ -distinct optical beams collectively extend hopfion utility into metrology, reconfigurable magnonic devices, and topological communication channels (Shen et al., 2022, Tejo et al., 30 Aug 2025).
Other Realizations: Similar topological classification and stabilization arguments operate in polar liquid crystals, plasma, and even early-universe gauge fields, pointing to a broad relevance for both fundamental science and advanced technology (Luk'Yanchuk et al., 2019, Bousder et al., 2024).