Skyrmion-Antiskyrmion Lattice in Chiral Magnets

• Skyrmion–antiskyrmion lattice is a net-zero magnetic state where equal numbers of Q=-1 skyrmions and Q=+1 antiskyrmions coexist within each unit cell.
• The lattice stability is achieved through symmetry-enforced anisotropic Dzyaloshinskii–Moriya interactions and exchange anisotropy in low-symmetry chiral magnets.
• Field-induced transitions demonstrate a sequence from cycloidal spirals to the net-zero lattice, then to conical spirals and ferromagnetism in Fe heterostructures.

Searching arXiv for the specified paper and closely related work on skyrmion–antiskyrmion lattices, antiskyrmions, and net-zero topological phases. Skyrmion–antiskyrmion lattice denotes a periodic magnetic texture in which skyrmions and antiskyrmions coexist with equal population inside the magnetic unit cell, producing a vanishing global topological charge even though each constituent soliton remains individually topological. In low-symmetry frustrated chiral magnets, this state has been reported as a thermodynamically stable field-induced phase in two-dimensional Fe heterostructures on $C_{1v}$-symmetric $(110)$ semiconductor surfaces, where it appears between cycloidal and conical spin spirals and circumvents the ordinarily anticipated annihilation of oppositely charged solitons (Banik et al., 17 Sep 2025).

1. Topological definition and net-zero character

The local topological charge density of a smooth magnetization field $\mathbf n(x,y)$ is defined by

$$q(x,y)=\frac{1}{4\pi}\,\mathbf n\cdot\bigl(\partial_x\mathbf n\times\partial_y\mathbf n\bigr),$$

and the total topological charge in a unit cell $\Omega$ is

$Q_{\mathrm{UC}}=\int_{\Omega}q(x,y)\,dx\,dy.$

Within this convention, skyrmions carry $Q=-1$ and antiskyrmions carry $Q=+1$. In a skyrmion–antiskyrmion lattice with equal numbers $N_{\mathrm{Sk}}=N_{\mathrm{ASk}}$,

$$Q_{\mathrm{UC}}=N_{\mathrm{ASk}}(+1)+N_{\mathrm{Sk}}(-1)=0,$$

so the crystal is globally topologically neutral (Banik et al., 17 Sep 2025).

This neutrality does not mean that the texture is locally trivial. The net-zero state is built from individually nontrivial solitons of opposite sign, and its defining feature is therefore cancellation at the unit-cell level rather than absence of topological structure. A recurring misconception is to equate $(110)$0 with a topologically featureless phase; in the skyrmion–antiskyrmion lattice, the relevant distinction is between vanishing global charge and nonvanishing local topological charge density.

The coexistence of skyrmions and antiskyrmions is especially notable because isotropic chiral magnets generally favor lattices of one sign only. The central problem is thus not merely how to create opposite charges simultaneously, but how to prevent their mutual annihilation while retaining long-range order. In the low-symmetry frustrated case, the reported answer is a symmetry-selected energetic mechanism rather than metastable trapping (Banik et al., 17 Sep 2025).

2. Microscopic model and the role of $(110)$1 symmetry

The Fe/$(110)$2-zincblende heterostructures are modeled by an extended Heisenberg Hamiltonian that combines exchange, Dzyaloshinskii–Moriya interaction (DMI), out-of-plane uniaxial anisotropy, and Zeeman coupling: $(110)$3 In the full two-layer formulation, $(110)$4 and $(110)$5 include both intra- and interlayer couplings, with $(110)$6 the atomic spin moment and $(110)$7 the applied perpendicular field (Banik et al., 17 Sep 2025).

The decisive ingredient is the reduced interfacial symmetry. On the $(110)$8 surface of zincblende semiconductors, the only surviving point symmetry is a single mirror $(110)$9 along the [001] or $\mathbf n(x,y)$0-axis, while $\mathbf n(x,y)$1 is absent. This forces exchange couplings within a coordination shell to become inequivalent along $\mathbf n(x,y)$2 and $\mathbf n(x,y)$3, and it makes the DMI strongly anisotropic in both magnitude and orientation. In particular, intralayer nearest-neighbor DMI lies mainly in the $\mathbf n(x,y)$4-plane, is not orthogonal to the bond, and acquires a small out-of-plane component; intralayer next-nearest-neighbor and interlayer nearest-neighbor DMI are restricted by $\mathbf n(x,y)$5 to lie strictly along $\mathbf n(x,y)$6; and some interlayer $\mathbf n(x,y)$7 DM vectors contain both in-plane and out-of-plane parts (Banik et al., 17 Sep 2025).

According to the reported mechanism, this anisotropy has two crucial consequences. First, it frustrates the usual DMI energy gain of a uniformly rotating texture such as a homogeneous skyrmion lattice or a conventional spin spiral. Second, it vanishes on average for textures that alternate chirality along $\mathbf n(x,y)$8, namely the skyrmion–antiskyrmion lattice. As a result, the alternating texture pays little DMI penalty from the reduced components, whereas homogeneous-chirality phases lose more DMI energy when they sample the weaker direction. Exchange anisotropy similarly penalizes the conical spiral more strongly than the alternating lattice. The reported stability of the net-zero phase is therefore not accidental coexistence but a consequence of symmetry-enforced anisotropic magnetic interactions (Banik et al., 17 Sep 2025).

3. Field-induced phase sequence in Fe/GaAs(110) and Fe/CdTe(110)

Zero-temperature spin-lattice simulations for two-layer Fe films on $\mathbf n(x,y)$9 GaAs and CdTe yield a four-stage sequence under increasing perpendicular field: $$q(x,y)=\frac{1}{4\pi}\,\mathbf n\cdot\bigl(\partial_x\mathbf n\times\partial_y\mathbf n\bigr),$$0 At $$q(x,y)=\frac{1}{4\pi}\,\mathbf n\cdot\bigl(\partial_x\mathbf n\times\partial_y\mathbf n\bigr),$$1, the ground state is a cyclic cycloidal spin spiral (CySS). Increasing field drives a first-order transition to the skyrmion–antiskyrmion lattice, followed by a second first-order transition to a conical spin spiral (CoSS), and finally a continuous evolution to the uniformly magnetized ferromagnet (FM) (Banik et al., 17 Sep 2025).

The critical fields reported for the strained 2Fe/GaAs and 2Fe/CdTe systems are as follows.

Transition	2Fe/GaAs	2Fe/CdTe
$$q(x,y)=\frac{1}{4\pi}\,\mathbf n\cdot\bigl(\partial_x\mathbf n\times\partial_y\mathbf n\bigr),$$2: CySS $$q(x,y)=\frac{1}{4\pi}\,\mathbf n\cdot\bigl(\partial_x\mathbf n\times\partial_y\mathbf n\bigr),$$3 Sk-ASkL	$$q(x,y)=\frac{1}{4\pi}\,\mathbf n\cdot\bigl(\partial_x\mathbf n\times\partial_y\mathbf n\bigr),$$4 T	$$q(x,y)=\frac{1}{4\pi}\,\mathbf n\cdot\bigl(\partial_x\mathbf n\times\partial_y\mathbf n\bigr),$$5 T
$$q(x,y)=\frac{1}{4\pi}\,\mathbf n\cdot\bigl(\partial_x\mathbf n\times\partial_y\mathbf n\bigr),$$6: Sk-ASkL $$q(x,y)=\frac{1}{4\pi}\,\mathbf n\cdot\bigl(\partial_x\mathbf n\times\partial_y\mathbf n\bigr),$$7 CoSS	$$q(x,y)=\frac{1}{4\pi}\,\mathbf n\cdot\bigl(\partial_x\mathbf n\times\partial_y\mathbf n\bigr),$$8 T	$$q(x,y)=\frac{1}{4\pi}\,\mathbf n\cdot\bigl(\partial_x\mathbf n\times\partial_y\mathbf n\bigr),$$9 T
$\Omega$0: CoSS $\Omega$1 FM	$\Omega$2 T	$\Omega$3 T

These transition fields were identified from intersections of total-energy versus $\Omega$4 curves and from abrupt jumps in $\Omega$5 at the first-order boundaries (Banik et al., 17 Sep 2025).

The resulting phase diagram places the skyrmion–antiskyrmion lattice as an intermediate-field equilibrium phase rather than a transient nonequilibrium texture. This point is central: the reported state is thermodynamically stable within a finite field window, not merely a metastable arrangement of defects.

4. Energy landscape, annihilation barrier, and computational framework

A minimal square-lattice $\Omega$6–$\Omega$7–$\Omega$8 model clarifies the energetic mechanism. For moderate DMI, $\Omega$9, and frustrated antiferromagnetic exchange, $Q_{\mathrm{UC}}=\int_{\Omega}q(x,y)\,dx\,dy.$0, both a skyrmion lattice and a skyrmion–antiskyrmion lattice appear as local minima. When exchange anisotropy $Q_{\mathrm{UC}}=\int_{\Omega}q(x,y)\,dx\,dy.$1 and DMI anisotropy $Q_{\mathrm{UC}}=\int_{\Omega}q(x,y)\,dx\,dy.$2 are introduced, the energy of homogeneous-chirality phases such as the cycloidal spiral and skyrmion lattice increases, while the skyrmion–antiskyrmion lattice remains nearly unchanged; exchange anisotropy further raises the conical spiral. This places the alternating lattice in the favorable part of the anisotropic energy landscape (Banik et al., 17 Sep 2025).

The same study reports a finite local annihilation barrier. Atomistic Monte Carlo “nudge” calculations give an energy barrier of order a few meV per skyrmion–antiskyrmion pair, originating from the need to reverse the sign of alternating Bloch-type walls and overcome both exchange frustration and anisotropic DMI penalties. In the ab initio-parameterized simulations, the zero-field barrier exceeds $Q_{\mathrm{UC}}=\int_{\Omega}q(x,y)\,dx\,dy.$3 meV per soliton, which was reported as sufficient to suppress thermal annihilation well above room temperature (Banik et al., 17 Sep 2025).

The computational pipeline combines first-principles and spin-dynamics methods. Relaxed geometries were obtained with VASP using LDA–VWN, PAW, a $Q_{\mathrm{UC}}=\int_{\Omega}q(x,y)\,dx\,dy.$4 eV cutoff, and a $Q_{\mathrm{UC}}=\int_{\Omega}q(x,y)\,dx\,dy.$5 $Q_{\mathrm{UC}}=\int_{\Omega}q(x,y)\,dx\,dy.$6-mesh; exchange, DMI, and anisotropy were extracted with JuKKR using ASA, spin–orbit coupling, and a $Q_{\mathrm{UC}}=\int_{\Omega}q(x,y)\,dx\,dy.$7 $Q_{\mathrm{UC}}=\int_{\Omega}q(x,y)\,dx\,dy.$8-mesh. Atomistic spin dynamics employed the SPIRIT code with simulated annealing from $Q_{\mathrm{UC}}=\int_{\Omega}q(x,y)\,dx\,dy.$9 K to $Q=-1$0 K under both OBC and PBC in an $Q=-1$1 cell with interactions up to 12 shells. Minimal-model phase boundaries were mapped by classical Monte Carlo on the square-lattice $Q=-1$2–$Q=-1$3–$Q=-1$4 model. Complementary micromagnetic simulations in MuMax3 used a $Q=-1$5 nm track, mesh $Q=-1$6, and an energy functional containing $Q=-1$7, fourth-order gradient terms, DMI, anisotropy, and Zeeman coupling; these simulations confirmed the absence of a skyrmion Hall effect for the net-zero pair (Banik et al., 17 Sep 2025).

A related later study generalized the same motif to anisotropic frustrated chiral magnets and identified 2Fe/InSb(110) as a candidate material, again finding a cycloidal spin spiral $Q=-1$8 skyrmion–antiskyrmion lattice $Q=-1$9 conical spin spiral $Q=+1$0 ferromagnet sequence under field (Banik et al., 26 Oct 2025). This suggests that the mechanism is not confined to a single substrate family.

5. Relation to antiskyrmion crystals, antiferro skyrmion crystals, and skyrmion–antiskyrmion-like arrays

Several neighboring phases can resemble a skyrmion–antiskyrmion lattice but differ in topology, symmetry, or the mechanism of cancellation.

System	Cancellation mechanism	Distinctive feature
D$Q=+1$1 Heuslers (Nayak et al., 2017)	None	Field-stabilized antiskyrmion lattice only
Synthetic bilayer AF-SkX (Hayami, 2023)	Interlayer $Q=+1$2, $Q=+1$3	Net-zero topology across layers
Frustrated triangular-lattice AF-SkL/ASkL (Mohylna et al., 2022)	Neutrality after summing sublattices	Three interpenetrated sublattices
Magnetoelastic array (Go et al., 19 Sep 2025)	Alternating scalar chirality	Nonquantized charges; not topologically protected

The best-established experimental antecedent is the antiskyrmion lattice in tetragonal Heusler materials with $Q=+1$4 symmetry, where anisotropic Lifshitz invariants stabilize an antiskyrmion lattice over a broad field and temperature window; this is a single-sign crystal rather than a mixed-sign net-zero lattice (Nayak et al., 2017). Lattice Monte Carlo studies of $Q=+1$5-type DM interactions likewise found a stable antiskyrmion lattice pocket, again consisting of one topological sign only (Criado et al., 2021).

A different route to global cancellation appears in synthetic bilayer antiferromagnets. In the bilayer triangular-lattice model under in-plane field, one layer carries skyrmion number $Q=+1$6 and the other $Q=+1$7, so the total skyrmion number and total scalar chirality vanish. This is a net-zero state, but the cancellation is interlayer rather than within a single mixed lattice (Hayami, 2023).

Centrosymmetric frustrated magnets provide yet another setting. On the triangular lattice with isotropic $Q=+1$8–$Q=+1$9–$N_{\mathrm{Sk}}=N_{\mathrm{ASk}}$0 antiferromagnetic exchange, Monte Carlo simulations found a spontaneous AF-SkL/ASkL at intermediate field and temperature. Real-space snapshots reveal three interpenetrated $N_{\mathrm{Sk}}=N_{\mathrm{ASk}}$1 sublattices, each hosting a ferromagnetic skyrmion or antiskyrmion crystal; summed over all three sublattices, the full antiferromagnetic texture is neutral (Mohylna et al., 2022). In a uniaxially distorted triangular-lattice antiferromagnet, related tuning of bond-dependent anisotropic exchange produces either a Bloch-type skyrmion crystal with $N_{\mathrm{Sk}}=N_{\mathrm{ASk}}$2 or an anti-type skyrmion crystal with $N_{\mathrm{Sk}}=N_{\mathrm{ASk}}$3, but not a coexisting equal-population lattice within one unit cell (Hayami, 2022).

The term “skyrmion–antiskyrmion-like” also requires care. A magnetoelastic-coupling-driven two-$N_{\mathrm{Sk}}=N_{\mathrm{ASk}}$4 spin-density-wave checkerboard can generate alternating scalar spin chirality and a checkerboard of skyrmion-like and antiskyrmion-like defects even without DMI, but the cores carry nonquantized “charges” and are not topologically protected (Go et al., 19 Sep 2025). This marks a strict distinction between a true net-zero topological soliton lattice and a chirality-alternating texture that only resembles one.

6. Experimental signatures and spintronic implications

For Fe/$N_{\mathrm{Sk}}=N_{\mathrm{ASk}}$5 semiconductor heterostructures, the reported signatures are modality-specific. Lorentz transmission electron microscopy is expected to show a hexagonal lattice with alternating bright and dark contrast corresponding to opposite topological charges. Spin-polarized scanning tunneling microscopy should resolve alternating up/down spin cores on an approximately $N_{\mathrm{Sk}}=N_{\mathrm{ASk}}$6–$N_{\mathrm{Sk}}=N_{\mathrm{ASk}}$7 nm periodic grid, with skyrmion cores down and antiskyrmion cores up. Magneto-optical Kerr effect microscopy under in-plane current should show one-dimensional motion of a skyrmion–antiskyrmion pair without transverse deflection, i.e. no Hall angle (Banik et al., 17 Sep 2025).

The dynamical implication emphasized in the simulations is suppression of the skyrmion Hall effect by net-zero topology. In the MuMax3 calculations, the net-zero pair showed no skyrmion Hall effect, and the proposed racetrack-memory operation uses straight-line motion at current densities of order $N_{\mathrm{Sk}}=N_{\mathrm{ASk}}$8 (Banik et al., 17 Sep 2025). A plausible implication is that such textures combine some of the transport advantages usually associated with antiferromagnetic or compensated topological states with the direct real-space addressability of individual solitons.

The proposed functionality remains broader than memory alone. Field-tunable creation and annihilation of balanced $N_{\mathrm{Sk}}=N_{\mathrm{ASk}}$9 textures were identified as a route to topological logic elements in which the global topology remains zero while local excitations encode information (Banik et al., 17 Sep 2025). More generally, the net-zero lattice stands apart from conventional hexagonal skyrmion lattices and from synthetic bilayer cancellation schemes because its compensation is realized inside a thermodynamically stable, long-range-ordered soliton crystal generated by low symmetry and frustration in a chiral magnet (Banik et al., 17 Sep 2025).