Skyrmion-String Lattice in Chiral Magnets

• Skyrmion-string lattice is a 3D magnetic phase characterized by a hexagonal array of line-like skyrmion tubes with nontrivial topology in chiral magnets.
• It is stabilized by thermal fluctuations near the ordering temperature and described by a Ginzburg–Landau model incorporating exchange, DMI, and Zeeman interactions.
• The phase exhibits emergent electrodynamics with quantized flux and low depinning currents, confirmed by techniques like SANS, Lorentz TEM, and magnetization measurements.

A skyrmion-string lattice is a three-dimensional magnetic phase in which the two-dimensional skyrmion texture in a plane normal to an applied field is repeated along the field direction, so that each skyrmion becomes a line-like topological object extending through the sample. In bulk cubic chiral magnets, this is not a distinct phase from the conventional skyrmion lattice but the same phase viewed from its 3D extension: a regular hexagonal array of topologically nontrivial spin whirls in cross section and a lattice of nearly straight, parallel skyrmion strings in the full bulk. The best-established realization is the small equilibrium phase pocket that appears at intermediate field just below the helimagnetic ordering temperature in non-centrosymmetric chiral magnets such as MnSi, FeGe, Fe$_{1-x}$Co$_x$Si, Mn$_{1-x}$Fe$_x$Si, and Cu$_2$OSeO$_3$ (Bauer et al., 2016).

1. Geometric structure and topological content

In bulk cubic chiral magnets, the skyrmion-string lattice is a hexagonal arrangement of skyrmion cores in the plane perpendicular to the magnetic field. In reciprocal space it is a phase-locked triple-$Q$ state: three helices with wavevectors at $120^\circ$ to each other in the plane normal to the field, supplemented by a uniform magnetization component along the field. In real space, this produces a triangular or hexagonal lattice of skyrmion cores. Because the texture is nearly unchanged along the field direction, each skyrmion is effectively a line-like topological tube through the sample thickness (Bauer et al., 2016).

The corresponding topological density is

$$\phi_i = \frac{1}{8\pi}\,\epsilon_{ijk}\,\hat{\psi}\cdot \left(\partial_j \hat{\psi}\times \partial_k \hat{\psi}\right),$$

with $$\hat{\psi}=\boldsymbol M(\boldsymbol r)/M(\boldsymbol r)$$. Its integral over a magnetic unit cell gives an integer winding number $_x$0, which is the sense in which the state is topologically nontrivial and cannot be continuously deformed into a trivial ferro-, para-, or simple helimagnetic state. In MnSi, the observed texture is anti-skyrmionic in sign convention, with core magnetization antiparallel to the external field and $_x$1, while remaining the standard bulk skyrmion-lattice phase of chiral magnets (Bauer et al., 2016).

The in-plane ordering vector has the same magnitude as in the helical state, $_x$2, so the skyrmion-core spacing exceeds the helix pitch by the factor $_x$3. Small-angle neutron scattering therefore shows a sixfold diffraction pattern normal to the field, while real-space imaging shows a trigonal array of circular or nearly circular objects. This cross-sectional description is the two-dimensional face of the string-lattice picture (Bauer et al., 2016).

A distinct geometric variant appears in the polar magnetic semiconductor GaV$_x$4S$_x$5, where the bulk SkL is also a bulk skyrmion-string lattice, but the string axis is confined to a rhombohedral easy axis rather than being selected by the external field. A regular triangular array of vortex lines along $_x$6 appears as a regular triangular lattice in the $_x$7 plane and as a distorted triangular lattice on a $_x$8 surface, making the string interpretation explicit at the level of projection geometry (Kézsmárki et al., 2015).

2. Microscopic setting and continuum description

The canonical setting is a non-centrosymmetric bulk magnet with long-wavelength helimagnetism generated by a hierarchy of interactions. Strong ferromagnetic exchange favors collinear order, the Dzyaloshinskii–Moriya interaction (DMI) favors twisted textures and fixes the handedness set by crystal chirality, and much weaker higher-order spin-orbit terms select preferred crystallographic directions and orient the helix or skyrmion lattice. In cubic chiral magnets with space group $_x$9, these ingredients yield the generic field sequence helical $_{1-x}$0 conical $_{1-x}$1 field-polarized, with the skyrmion-string lattice appearing as a separate equilibrium pocket just below $_{1-x}$2 (Bauer et al., 2016).

The standard continuum framework uses a Ginzburg–Landau free-energy density with an isotropic part

$_{1-x}$3

and magnetization density $_{1-x}$4. Here $_{1-x}$5 controls proximity to ordering, $_{1-x}$6 is the exchange stiffness, $_{1-x}$7 the DMI coupling, $_{1-x}$8 the quartic mode-coupling term, and the last term the Zeeman coupling. Cubic anisotropy enters through much smaller terms such as

$_{1-x}$9

with $_x$0. These anisotropies choose easy directions such as $_x$1 or $_x$2 and fix the azimuthal orientation of the lattice, but they are not the primary origin of the phase (Bauer et al., 2016).

In polar $_x$3 magnets the same string-lattice concept survives with different local twisting symmetry. In GaV$_x$4S$_x$5, the polar rhombohedral $_x$6 distortion allows only Néel-type skyrmions; the state is describable as a superposition of spin cycloids rather than spin helices, and the easy-axis anisotropy is strong enough to dominate over the Zeeman tendency to align the strings with the field. The experimentally inferred exchange scales are $_x$7 K and $_x$8 K, underscoring that the easy axis is not a weak perturbation in that material class (Kézsmárki et al., 2015).

3. Stabilization mechanism and phase-diagram topology

A central result for bulk cubic chiral magnets is that the skyrmion-string lattice is generically stabilized by thermal fluctuations rather than primarily by anisotropy. Mean-field theory alone favors the conical state, but Gaussian fluctuation corrections lower the free energy of the skyrmion lattice at intermediate fields just below $_x$9. Both short-range and long-range fluctuations favor the phase in that regime, linking the stabilization mechanism to Brazovskii-type fluctuation physics near the paramagnetic-to-helimagnetic transition. In this window, strong chiral fluctuations suppress the mean-field second-order transition, create a fluctuation-disordered regime above $_2$0, drive a fluctuation-induced first-order transition, and stabilize the skyrmion-string lattice over a finite field interval (Bauer et al., 2016).

The resulting equilibrium phase diagram contains helical, conical, skyrmion lattice, fluctuation-disordered, paramagnetic, and field-polarized regimes, with a tricritical point at higher field where the fluctuation-induced first-order transition evolves into a second-order one. The skyrmion-string lattice appears specifically just below $_2$1, in an intermediate field window, in crystals with broken inversion symmetry and homochiral DMI, and with only weak dependence on cubic anisotropy. This underlies the paper’s universalist interpretation: despite large variations in transition temperature, ordered moment, critical fields, and helix wavelength across MnSi, FeGe, Fe$_2$2Co$_2$3Si, Mn$_2$4Fe$_2$5Si, and Cu$_2$6OSeO$_2$7, the phase-diagram topology is qualitatively the same (Bauer et al., 2016).

This universality also frames an important controversy. The generic picture argues against explanations that invoke special material-dependent anisotropies as the essential stabilizing mechanism, and it also argues against multiple complex precursor pockets as the generic interpretation of FeGe. The broader implication is that the bulk skyrmion-string lattice is governed mainly by symmetry, the exchange–DMI–Zeeman hierarchy, fluctuations, and weak anisotropy rather than by detailed electronic structure (Bauer et al., 2016).

The broadest known thermodynamic stability window among bulk hosts is found not in cubic chiral magnets but in GaV$_2$8S$_2$9. There, the Néel-type SkL survives down to about $_3$0, far below the narrow $_3$1–$_3$2 of $_3$3 typical of many bulk chiral hosts. The low-field phase is cycloidal rather than helical, there is no reported conical phase, and the orientation-dependent phase boundaries collapse when the applied field is projected onto the easy axis, showing that the relevant control parameter is $_3$4 rather than the total field magnitude (Kézsmárki et al., 2015).

4. Emergent electrodynamics, elasticity, and driven motion

For carriers moving adiabatically through the texture, the real-space Berry curvature acts as an emergent magnetic field

$_3$5

while a time-dependent texture generates an emergent electric field

$_3$6

Each skyrmion carries one quantum of emergent flux, $_3$7. In string language, each skyrmion string is therefore an extended tube of quantized emergent flux. In MnSi, the effective emergent field associated with the lattice is quoted as $_3$8, and the topological Hall contribution is estimated as $_3$9 (Bauer et al., 2016).

Under in-plane currents, the skyrmion lattice in cubic chiral magnets depins at exceptionally low current densities, $Q$0, and then drifts through the sample. In the string interpretation these are moving skyrmion lines. Their collective motion is described with Landau–Lifshitz–Gilbert dynamics and the Thiele approach, combining a Magnus force transverse to current, dissipative drag, defect pinning, and elastic deformations of individual skyrmions and of the lattice. The flexibility of the line array is already physically important even though the generic bulk theory does not formulate a separate string-tension functional (Bauer et al., 2016).

A specifically three-dimensional instability appears when the spin current is driven parallel to the strings rather than across them. For a longitudinal current $Q$1, the static string is not directly translated because $Q$2 vanishes on a $Q$3-independent background, but finite-$Q$4 fluctuations are Doppler shifted. The first branch to destabilize is the translational Goldstone mode of the string or of the string lattice, yielding a Goldstone spin-wave instability rather than rigid drift. In a clean infinite system there is no finite threshold current for this instability; finite thickness and disorder restore one. For MnSi, a weak-collective-pinning estimate gives $Q$5, compared with the much smaller transverse depinning scale (Okumura et al., 2023).

This instability leads to a dynamical melting scenario for the skyrmion-string lattice. The sequence described in the simulations is longitudinal current $Q$6 exponential growth of helical line distortions $Q$7 large-amplitude deformation $Q$8 eventual string breaking. The final topological cutting mechanism is not derived microscopically in full detail, but the line-based mode structure is a genuinely 3D feature absent in strictly 2D skyrmion theories (Okumura et al., 2023).

5. Experimental identification, nucleation, and phase conversion

The standard experimental identification of a skyrmion-string lattice combines microscopic, thermodynamic, and transport probes. In bulk cubic chiral magnets, small-angle neutron scattering established the long-range ordered phase through its sixfold Bragg pattern perpendicular to the field, while higher-order peaks confirmed the phase-locked multi-$Q$9 character responsible for nontrivial topology. Lorentz transmission electron microscopy provided direct real-space images of periodic spin whirls in thinned samples, magnetic force microscopy detected the corresponding bulk-surface stray fields, and electron holography reconstructed the spin arrangement in three dimensions. Magnetization, $120^\circ$0, ac susceptibility, and specific heat map the phase pocket and show that the transitions into and out of the skyrmion lattice are first-order in the relevant regime near $120^\circ$1; in metals, the topological Hall effect supplies an additional probe of the winding number (Bauer et al., 2016).

In GaV$120^\circ$2S$120^\circ$3, AFM and SANS together established the bulk Néel-type string lattice. AFM on the $120^\circ$4 surface resolved a cycloidal modulation with period $120^\circ$5 at $120^\circ$6 K and $120^\circ$7 mT, and a distorted triangular vortex lattice with corrected lattice constant $120^\circ$8 at $120^\circ$9 K and $$\phi_i = \frac{1}{8\pi}\,\epsilon_{ijk}\,\hat{\psi}\cdot \left(\partial_j \hat{\psi}\times \partial_k \hat{\psi}\right),$$0 mT. SANS confirmed that these are bulk phases rather than surface artifacts and showed sixfold or four-spot reciprocal-space patterns depending on domain geometry and field orientation (Kézsmárki et al., 2015).

The formation of the bulk MnSi string lattice is itself topologically activated. High-precision SANS minor hysteresis loops show that the equilibrium SkL is separated from the conical phase by a finite activation barrier. The integrated SkL Bragg intensity tracks the SkL volume fraction, and the hysteresis can be modeled with an adapted Preisach description of bistable microscopic domains. The barrier corresponds to skyrmion-lattice formation from the conical phase in finite domains consisting of hundreds of skyrmions, with an activation barrier of several eV (Leishman et al., 2020).

Atomistic minimum-energy-path calculations refine this picture by showing that the transition does not proceed through spatially uniform conversion of the whole sample. Instead, skyrmion tubes are created progressively, one after another, in local domains; layer-resolved topological charge builds up through the sample thickness in a way consistent with Bloch-point-mediated nucleation. This directly supports the interpretation of the equilibrium bulk SkL as a skyrmion-string lattice whose creation and destruction require genuine topological events (Leishman et al., 2020).

6. Related variants and broader generalizations

A large fraction of the theoretical literature relevant to skyrmion-string lattices is explicitly two-dimensional and should be read as a theory of transverse cross sections rather than of full line objects. In a 2D chiral-magnet model, the field-induced sequence helical $$\phi_i = \frac{1}{8\pi}\,\epsilon_{ijk}\,\hat{\psi}\cdot \left(\partial_j \hat{\psi}\times \partial_k \hat{\psi}\right),$$1 skyrmion crystal $$\phi_i = \frac{1}{8\pi}\,\epsilon_{ijk}\,\hat{\psi}\cdot \left(\partial_j \hat{\psi}\times \partial_k \hat{\psi}\right),$$2 ferromagnet can be derived from a Ginzburg–Landau functional recast in CP$$\phi_i = \frac{1}{8\pi}\,\epsilon_{ijk}\,\hat{\psi}\cdot \left(\partial_j \hat{\psi}\times \partial_k \hat{\psi}\right),$$3 form, and the skyrmion crystal can be constructed by an Abrikosov-like Landau-level superposition. This does not solve the 3D string problem, but it provides the natural cross-sectional building block for a straight skyrmion-string lattice (Han et al., 2010).

Other 2D works clarify how such cross sections fluctuate or distort. A collective-coordinate theory of the 2D skyrmion lattice identifies phasons as the low-energy canonical coordinates, finds one quadratic gapless mode and one massive mode, and introduces a phase-sum mode whose fluctuations screen the effective topological charge of the lattice. A cautious extrapolation is that analogous phase-locking variables should matter for the transverse elastic sector of a string lattice as well (Tatara et al., 2014). Likewise, perturbative solutions of the exchange-plus-DM model around Belavin–Polyakov seeds predict noncircular and anisotropic skyrmions, implying that string cross sections need not be circular even when the lattice as a whole is hexagonal (Kundu, 2015). In 2D thin films with in-plane anisotropy, anisotropic deformation can even induce weak or strong directional attractions, elongated triangular SkX states, and domain-wall-bound bimeron lattices; this suggests that anisotropic cross-sectional forces may substantially reshape string arrays in 3D, although that extension is not proved in the paper itself (Kameda et al., 2021).

Microscopic 2D mechanisms beyond DMI also matter for broader skyrmion-string thinking. In the triangular Kondo lattice model, skyrmion-crystal wavelengths are dictated by the Fermi wave-vector and enhanced by higher-order effective spin interactions beyond RKKY, so the in-plane lattice constant of a weakly stacked 3D analogue would plausibly be electronically tunable (Wang et al., 2021). In frustrated antiferromagnets with DMI on the triangular lattice, the AF-SkX consists of three interpenetrating sublattice skyrmion crystals and seems to survive to very low temperature, providing a multi-sublattice 2D analogue rather than a true string lattice (Rosales et al., 2015). In an $$\phi_i = \frac{1}{8\pi}\,\epsilon_{ijk}\,\hat{\psi}\cdot \left(\partial_j \hat{\psi}\times \partial_k \hat{\psi}\right),$$4-symmetric triangular-lattice model with scalar chirality, a field-free SkX-2 with two topological charges per unit cell and zero $$\phi_i = \frac{1}{8\pi}\,\epsilon_{ijk}\,\hat{\psi}\cdot \left(\partial_j \hat{\psi}\times \partial_k \hat{\psi}\right),$$5-point magnetization supplies another possible transverse building block, but again only at the level of a strict 2D theory (Bocquet et al., 20 May 2026).

More direct layered 3D analogues also exist. In a nonsymmorphic lattice with a threefold screw axis, simulated annealing finds a layered skyrmion crystal stabilized even without threefold rotational symmetry in each plane, together with fractional phases in which only one or two layers in a three-layer unit cell carry skyrmion number $$\phi_i = \frac{1}{8\pi}\,\epsilon_{ijk}\,\hat{\psi}\cdot \left(\partial_j \hat{\psi}\times \partial_k \hat{\psi}\right),$$6. This is best interpreted as a screw-modulated, layered skyrmion-crystal or skyrmion-string-lattice precursor rather than a continuum theory of smooth strings (Hayami, 2022).

Finally, there are broader generalizations in which “string lattice” is no longer a mere 3D extension of an $$\phi_i = \frac{1}{8\pi}\,\epsilon_{ijk}\,\hat{\psi}\cdot \left(\partial_j \hat{\psi}\times \partial_k \hat{\psi}\right),$$7 skyrmion crystal. A recent $$\phi_i = \frac{1}{8\pi}\,\epsilon_{ijk}\,\hat{\psi}\cdot \left(\partial_j \hat{\psi}\times \partial_k \hat{\psi}\right),$$8 sigma-model study with generalized DMI finds genuinely three-dimensional magnetic-string-lattice phases, negative string tension, anti-confinement, endpoint half-Skyrmions, and a mixed-topology regime with fractional $$\phi_i = \frac{1}{8\pi}\,\epsilon_{ijk}\,\hat{\psi}\cdot \left(\partial_j \hat{\psi}\times \partial_k \hat{\psi}\right),$$9 charge at string bends. These phases are conceptually related but topologically distinct from the ordinary skyrmion-string lattice of chiral magnets (Francini et al., 5 Jun 2026).

In this broader literature, “skyrmion-string lattice” therefore has two principal meanings. In the established bulk chiral-magnet sense, it is the three-dimensional realization of the familiar skyrmion lattice: a lattice of nearly straight, field-oriented topological tubes stabilized in a narrow but robust equilibrium pocket. In a wider theoretical sense, it also serves as a framework for layered, modulated, or generalized line-lattice phases whose transverse sections are skyrmionic even when the full 3D topology, stacking, or order-parameter manifold differs from the conventional cubic-chiral case (Bauer et al., 2016).