Quantum Skyrmions in Chiral Magnets

• Quantum skyrmions are quantum-mechanical spin textures defined by topological invariants and many-body wavefunctions rather than classical soliton approximations.
• They emerge in systems like chiral ferromagnets and frustrated magnets, showcasing quantized dynamics as seen in distinctive band splittings and quantum phase transitions.
• Their study informs quantum nucleation processes, magnetic phase control, and topological Hall effects, providing actionable insights for experimental and theoretical research.

Quantum skyrmions are topological spin textures whose description requires quantum mechanics at a structural level rather than as a small correction to a classical soliton. In the literature, the term encompasses quantized collective-coordinate dynamics of a skyrmion in a chiral magnet, fully quantum many-body states of microscopic spins whose correlations remain skyrmionic, and dense quantum liquids or crystals in which lattice discreteness, Berry phases, and entanglement remain essential (Takashima et al., 2016, Ochoa et al., 2018). For conventional magnetic textures the starting point is the continuum skyrmion number

$$\mathcal N=\frac{1}{4\pi}\int d^2r\,\mathbf n(\mathbf r)\cdot\big[\partial_x\mathbf n(\mathbf r)\times \partial_y\mathbf n(\mathbf r)\big],$$

but quantum formulations replace or supplement this invariant by operator algebras, scalar chirality, discretized order parameters, or explicit many-body wavefunctions.

1. Conceptual scope and definitions

A first notion of a quantum skyrmion treats a classical texture as a quasiparticle whose center coordinate is quantized. In that setting, the skyrmion remains spatially extended, but its translational dynamics, band structure, and condensation are quantum-mechanical objects (Takashima et al., 2016). A second notion identifies quantum skyrmions directly as many-body states of microscopic spin-$1/2$ or spin-1 Hamiltonians, without assuming a smooth classical magnetization field. There the characteristic signatures are multiple-$q$ structure factors, scalar chirality, dominant skyrmion-like basis states, or entanglement patterns (Sotnikov et al., 2018, Sotnikov et al., 2020). A third notion appears in dense phases, where the relevant objects are quantum skyrmion lattices or liquids rather than isolated defects (Haller et al., 2021, Mæland et al., 2022).

The standard classical skyrmion is a CP$^1$ texture because the target manifold of the magnetization field is $S^2\simeq \mathrm{CP}^1$. In spin-1 magnets the local coherent-state manifold becomes $\mathrm{CP}^2$, so the texture can mix dipolar and quadrupolar components. This enlarges the skyrmion concept: CP$^2$ skyrmions can exist not only in magnetically ordered phases but also as metastable textures of quantum paramagnets (Zhang et al., 2022).

The phrase “quantum origin” is also used in a multiscale sense. In Cu$_2$OSeO$_3$, skyrmions that look classical at long wavelengths arise from strongly fluctuating Cu$_4$ tetrahedra with an entangled triplet ground state; the continuum skyrmion texture is then an emergent object built from quantum cluster degrees of freedom rather than from rigid microscopic spins (Janson et al., 2014).

2. Microscopic settings and topological descriptors

One canonical microscopic setting is the square-lattice quantum chiral ferromagnet

$1/2$0

with nearest-neighbour ferromagnetic exchange, Dzyaloshinskii–Moriya interaction, out-of-plane field, and easy-axis anisotropy (Takashima et al., 2016). Other quantum-skyrmion platforms replace DMI by frustration, as in the spin-$1/2$1 triangular-lattice $1/2$2–$1/2$3 ferromagnet with easy-axis anisotropy and Zeeman field (Lohani et al., 2019), or use antiferromagnetic exchange with in-plane DMI on the triangular lattice (Corte et al., 26 Jan 2026). In spin-1 systems, a realistic triangular-lattice model with $1/2$4, $1/2$5, exchange anisotropy $1/2$6, field $1/2$7, and single-ion anisotropy $1/2$8 yields CP$1/2$9 skyrmions and skyrmion crystals (Zhang et al., 2022).

Because a local spin direction is not sharply defined in a generic quantum state, the continuum winding number is often replaced by lattice observables. One useful operator is the scalar chirality on an oriented triangle,

$q$0

whose expectation value vanishes for coplanar or fully polarized states and is nonzero for genuinely three-dimensional textures (Sotnikov et al., 2020). Another is the quantum Berg–Lüscher order parameter

$q$1

constructed from two-spin and three-spin correlators and designed to reduce to the classical discretized skyrmion number in the semiclassical limit (Mæland et al., 2022).

These descriptors play different roles. Scalar chirality is especially natural as a local operator and as a bridge to Hall-type responses. The Berg–Lüscher construction is closer to a lattice topological count. This suggests that “quantum skyrmion number” is not unique: different questions emphasize different observables.

3. Quantized collective dynamics of isolated skyrmions

Near the skyrmion-crystal–ferromagnet boundary in two-dimensional chiral ferromagnets, the low-energy excitation can be a skyrmion rather than a magnon. Restricting the spin field to a rigidly translated skyrmion profile $q$2 with $q$3 gives the effective action

$q$4

where the Berry/Magnus term implies

$q$5

The periodic potential turns the problem into Harper’s Hamiltonian, and the single-skyrmion spectrum splits into $q$6 magnetic subbands. For $q$7, the band minimum lies at $q$8 for even $q$9 and at $^1$0 for odd $^1$1; the same parity governs the many-body onset of skyrmions, yielding a 3D Ising quantum phase transition for odd $^1$2 and only a crossover for even $^1$3 (Takashima et al., 2016).

In that framework, condensation of the lowest skyrmion band produces a quantum liquid phase between the ferromagnet and the classical skyrmion crystal. The liquid has delocalized skyrmion centers and preserves translational symmetry, whereas the crystal is a density wave of localized skyrmions (Takashima et al., 2016). The bandwidth is set by the exponentially small lattice pinning scale $^1$4, so the skyrmion bands are much flatter than magnon bands.

A more recent field-theoretic reformulation identifies the natural coordinate not as the center of magnetization but as the topological dipole

$^1$5

with momentum

$^1$6

This yields a dipole conservation law,

$^1$7

the commutator

$^1$8

and a long-wavelength Girvin–MacDonald–Platzman algebra for the topological charge density. In the translationally invariant spin-only continuum theory, the effective action of the conserved coordinate is purely Berry-phase, the skyrmion mass vanishes, and an isolated skyrmion is fracton-like in the sense of intrinsic immobility (Sorn et al., 2024).

These formulations are not identical. This suggests that the choice of collective coordinate, lattice pinning, and coupling to additional degrees of freedom are central to the skyrmion-mass debate rather than secondary modeling details.

4. Quantum many-body skyrmion phases

In finite spin-$^1$9 clusters, quantum skyrmions can be defined directly from the many-body ground state. On a 12-site plaquette with DMI and Zeeman field, exact diagonalization identifies a quantum nanoskyrmion through simultaneous structure-factor peaks at $S^2\simeq \mathrm{CP}^1$0 and $S^2\simeq \mathrm{CP}^1$1 with $S^2\simeq \mathrm{CP}^1$2, together with dominant basis states corresponding to off-site, bond, and on-site classical nanoskyrmions. In that model the multiple-$S^2\simeq \mathrm{CP}^1$3 signature remains visible up to at least $S^2\simeq \mathrm{CP}^1$4, although the classical skyrmion phase ends near $S^2\simeq \mathrm{CP}^1$5, and it survives up to $S^2\simeq \mathrm{CP}^1$6 in units of $S^2\simeq \mathrm{CP}^1$7 (Sotnikov et al., 2018).

In frustrated ferromagnets on the triangular lattice, quantum skyrmions and antiskyrmions appear as stable many-magnon bound states. Exact diagonalization yields the selection rule

$S^2\simeq \mathrm{CP}^1$8

for skyrmions and

$S^2\simeq \mathrm{CP}^1$9

for antiskyrmions, an exponentially small bandwidth generated by skyrmion–antiskyrmion tunneling, and a band structure that changes qualitatively when a single spin is added or removed. In the weak-force regime the wavepacket accelerates parallel to the force, whereas for stronger forces it moves perpendicular to it (Lohani et al., 2019).

For extended two-dimensional lattices, DMRG establishes a broad zero-temperature quantum-skyrmion-lattice region in chiral Heisenberg ferromagnets with spin-$\mathrm{CP}^2$0 moments. Single centered skyrmions appear for disk diameters $\mathrm{CP}^2$1, larger flakes support chains and then lattices of skyrmions, and the skyrmion state remains stable in parts of parameter space where the corresponding classical model is already field-polarized (Haller et al., 2021).

Quantum antiferromagnetic skyrmions arise in a different way. In the spin-$\mathrm{CP}^2$2 Heisenberg antiferromagnet with in-plane DMI on the triangular lattice, DMRG finds three-sublattice quantum antiferromagnetic skyrmion textures stabilized over $\mathrm{CP}^2$3 at $\mathrm{CP}^2$4. Each sublattice hosts a ferromagnetic-skyrmion-like texture, and the full state carries nonzero quantum chirality while remaining entangled (Corte et al., 26 Jan 2026).

Spin-1 systems further enlarge the taxonomy. In realistic triangular-lattice models, CP$\mathrm{CP}^2$5 skyrmions can be metastable both in the field-polarized phase and in a quantum paramagnet dominated by the $\mathrm{CP}^2$6 single-ion state. The same models host field-induced CP$\mathrm{CP}^2$7 skyrmion crystals: SkX-I has magnetic cores in a nematic background, whereas SkX-II has nematic cores in a polarized background (Zhang et al., 2022).

5. Wavefunctions, entanglement, and quantum order

A multiscale example of the quantum origin of skyrmions is Cu$\mathrm{CP}^2$8OSeO$\mathrm{CP}^2$9. There the fundamental magnetic units are strongly fluctuating Cu$^2$0 tetrahedra with a triplet $^2$1 ground state separated from excited multiplets by $^2$2 K. Projecting onto these cluster states yields effective spin-1 degrees of freedom on a trillium lattice, from which a continuum chiral ferromagnet, a skyrmion texture, and even a fractionalized half-skyrmion phase emerge. The same construction explains reduced local moments, weakly dispersive high-energy tetrahedral excitations, and a weak antiferromagnetic modulation of the primary ferrimagnetic order (Janson et al., 2014).

At the microscopic many-body level, quantum skyrmions can be represented by explicit creation operators. A variational operator construction writes a skyrmion creator as a classical rotation field multiplying a superposition of local spin-flip configurations, so that the leading term reproduces the classical texture while subleading terms form a quantum cloud of local spin-flip excitations. Matrix product state simulations of adiabatic braiding show that, at large inter-skyrmion distance, these operators behave as bosonic quasiparticle creators (Haller et al., 2024).

Entanglement data sharpen the distinction between classical and quantum skyrmions. In chiral Heisenberg ferromagnets, the field-polarized phase is almost unentangled in the bulk, whereas the quantum skyrmion state has strong concurrence concentrated on the skyrmion rim and a reduced local spin norm there; the core is comparatively weakly entangled (Haller et al., 2021). This identifies the domain wall as the main location of the skyrmion’s quantum dressing.

Dense quantum skyrmion crystals provide a complementary perspective. In a triangular-lattice model with approximately 15 spins per skyrmion, Holstein–Primakoff theory around the classical SkX ground state yields a quantum Berg–Lüscher order parameter $^2$3 that remains close to one skyrmion per unit cell but jumps discontinuously at the transition between two distinct skyrmion-crystal phases, SkX1 and SkX2. The classical $^2$4 equals one in both phases, so the jump is a purely quantum distinction (Mæland et al., 2022).

For small spin-$^2$5 clusters, the scalar chirality

$^2$6

plays an analogous role. It vanishes for coplanar or fully polarized states, is nearly constant throughout the quantum skyrmion phase, and reduces to the classical lattice discretization of the continuum topological charge in the large-skyrmion limit (Sotnikov et al., 2020).

6. Observation, nucleation, and unresolved issues

Near the ferromagnetic side of the skyrmion-crystal boundary, quantum skyrmions were predicted to appear in inelastic neutron scattering as multiple low-energy branches with bandwidth $^2$7, rather than a single magnon branch. The dominant signal occurs near the band minima, with $^2$8 for even $^2$9 and $_2$0 for odd $_2$1. In the intermediate-momentum window $_2$2, the intensity behaves as

$_2$3

so small skyrmions and moderate $_2$4 are favored experimentally (Takashima et al., 2016).

Because local polarization measurements can miss the quantum texture, scalar-chirality-sensitive responses such as the topological Hall effect, topological orbital magnetization, and second-harmonic generation have been proposed as more faithful probes of the quantum analog of a classical skyrmion (Sotnikov et al., 2020). On the preparation side, zero-temperature quantum nucleation has been analyzed for ultrathin films with interfacial DMI by applying a local magnetic field opposite to a uniform stabilizing field. In a collective-coordinate instanton treatment, the ferromagnet tunnels to a single-skyrmion state with rate

$_2$5

where the exponent depends on the tip field and the circular spot radius (Diaz et al., 2016).

Several issues remain unsettled. A rigorous, universally accepted quantum topological invariant for microscopic spin-$_2$6 skyrmions is still lacking, so many constructions rely on scalar chirality, structure factors, or discretized order parameters instead of a strictly quantized charge (Sotnikov et al., 2018). The skyrmion-mass problem also remains formulation-dependent: one line of work derives flat lattice bands and condensation of mobile skyrmions (Takashima et al., 2016), whereas another finds that the conserved topological-dipole coordinate carries no inertial term and that an isolated skyrmion is intrinsically pinned in the translationally invariant spin-only continuum theory (Sorn et al., 2024). This suggests that the relevant coordinate, lattice effects, and coupling to electrons, phonons, disorder, or hedgehog defects are part of the definition of the quasiparticle itself, not merely corrections to it.