Multi-step topological transitions among meron and skyrmion crystals in a centrosymmetric magnet

The study addresses whether a centrosymmetric magnet can host controllable, multi-step transformations among distinct topological spin textures (elliptic skyrmions, meron/anti-meron pairs, circular skyrmions) and what microscopic mechanism stabilizes them. Prior work predominantly realized skyrmion phases in non-centrosymmetric (Dzyaloshinskii-Moriya interaction-driven) materials, typically yielding skyrmion diameters of tens to hundreds of nanometers. Recent theory proposed alternative stabilization in centrosymmetric systems via itinerant-electron-mediated interactions, and experiments in rare-earth intermetallics uncovered nanometric skyrmions. The purpose here is to uncover and characterize multiple topological phases and their transitions in the centrosymmetric tetragonal compound GdRu₂Ge₂ under B || [001], correlate them with transport signatures (Hall effect), and identify a general mechanism based on competing RKKY interactions at inequivalent wave vectors.

The paper reviews skyrmions, anti-skyrmions, merons, and hopfions in systems with broken inversion symmetry (magnets, ferroelectrics, chiral liquid crystals), where DM interaction fixes the helicity and stabilizes skyrmion lattices (chiral for skyrmions, D2d/S4 for anti-skyrmions). In such DM systems, multi-Q states emerge under magnetic field and skyrmion diameters are typically tens to hundreds of nm. The authors discuss centrosymmetric alternatives: theoretical works indicating skyrmions without inversion breaking through RKKY or competing interactions; experimental discoveries of ultra-small (<3 nm) skyrmions in Gd2PdSi3 and GdRu2Si2; and a rare observed meron–skyrmion transition in the non-centrosymmetric Co8Zn9Mn3. They highlight the open question of achieving diverse quasi-particles beyond circular skyrmions in centrosymmetric magnets and the need for mechanisms enabling controlled interconversion.

Materials and sample preparation: Polycrystalline GdRu₂Ge₂ rods were synthesized by arc melting, and single crystals were grown by floating zone. Orientation was determined by Laue X-ray; phase purity by powder XRD. Typical sample thickness ~1 mm; multiple crystals were tested. Magnetization and transport: Magnetization M(B,T) measured by SQUID MPMS; electrical transport via five-terminal AC method in PPMS on the same sample for Fig. 1 (I || [100], B || [001]). Hall resistivity ρyx(B) was antisymmetrized; longitudinal ρxx(B) is even in B. Field-sweep hysteresis examined. Phase boundaries determined from M(B) anomalies; a B–T diagram was constructed. Neutron scattering: Conducted at HRC spectrometer BL12 (J-PARC MLF). Single crystal (6.5×8.0×0.4 mm) with (100) face, B || [001]. To mitigate Gd absorption, incident energy Ei = 153.5 meV; elastic energy resolution ~4%. Data collected over ω rotations and processed with DAVE to map reciprocal space around (-1,1,0) and (0,-1,-1). Limited Q-resolution noted owing to high-energy neutrons. Resonant X-ray scattering (RXS): Performed at BL-3A (Photon Factory, KEK) near Gd L2 edge (~7.935 keV) with polarization analysis using PG(006) analyzer (2θ ~ 88.7°). Single crystal (0.45×3.2×4.6 mm), B || [001]. Incident beam π-polarized within scattering plane (plane normal to [001]). Measured satellite reflections near (4,2,0) and (4,0,0) with defined line scans A and B; additionally scanned higher-order satellites (line scans C, D). Separated scattered π′ and σ′ components to resolve spin components via I ∝ |(ei×ef)·m(Q)|², decomposing m(Q) into components along c, Q, and c×Q. Rocking curves and Gaussian fits provided peak positions and integrated intensities; polarization selection rules identified screw vs sinusoidal modulations and domain contributions. Magnetic structure reconstruction: Using measured amplitudes of modulated spin components m̃α(Q) at QA, QA′, QB, QB′ and weak harmonics QC, QC′ (from RXS), and uniform magnetization mc^0 from M(B), the real-space spin texture m(r) was reconstructed as m(r) = ec mc^0 + Σα,Q eα m̃α(Q) sin(Q·r + θα). Phases θα, not directly accessible to RXS, were chosen to minimize spatial variation of |m(r)|, respecting localized Gd3+ moments. Domain populations (α, β) inferred for symmetry-broken phases. Topological transport analysis: Hall resistivity decomposed as ρyx = R0 B + Rs M + P R0 Bem. By comparing measured ρyx(B) with expectations for intrinsic (Rs ∝ ρxx²) and extrinsic (Rs ∝ ρxx) anomalous mechanisms, the residual peak-like features were attributed to topological Hall effect in phases with nonzero Nsk. Theory and simulation: An effective spin Hamiltonian derived from the Kondo lattice model was used: H = −2J Σν,α,β Γαβ^ν mα(Qν) mβ(Qν) − Σi B·m(ri), with |m(ri)|=1, capturing RKKY interactions at inequivalent wave vectors QA = (q,0), QA′ = (0,q), and QB = (q/2,q/2), QB′ = (−q/2,q/2) with tetragonal symmetry, weak easy-axis anisotropy, and χ(QA)=χ(QA′)>χ(QB)=χ(QB′). Simulated annealing Monte Carlo on a 100×100 square lattice with decreasing temperature (α between 0.99999–0.999999), 10^5–10^6 MCS per step. Scalar spin chirality Nsk computed as a lattice measure of noncoplanarity. Additional simulations contrasted models with and without QB contributions and with optional four-spin term K to assess mechanism minimality.

- Discovery of multi-step topological transitions among distinct multi-Q magnetic crystal states in centrosymmetric GdRu₂Ge₂ under B || [001]. Magnetization shows intermediate steps near 1.0 T, 1.2 T, and 1.35 T before saturation above ~4.5 T at 6 K, delineating Phases I–VI and FM. - RXS and neutron scattering identify in-plane magnetic modulation vectors QA ≈ (q,0,0) or (0,q,0) in all non-FM phases; additional satellites at QB ≈ (q/2,q/2,0) or (−q/2,q/2,0) occur only in Phases II–IV, evidencing multi-Q order. At B = 0 (Phase I), single-Q screw with q ≈ 0.213 r.l.u. - Fourfold symmetry breaking in Phases II and III: line scan A splits into two peaks q1<q2, reflecting coexisting rotational domains α and β; symmetry restored in Phase IV. - Polarization-resolved RXS shows screw-type modulations for QA/QA′ and QB/QB′ in Phase IV. Similar selection rules in Phase II; Phase III exhibits slight tilt of spiral planes toward QB, implying small additional spin components. - Real-space reconstruction yields: Phase I single-Q screw (Nsk=0); Phase II square lattice of elliptic skyrmions (Nsk = −1 per particle); Phase III square lattice of meron/anti-meron pairs (Nsk = 0 net, composed of Nsk=−1/2 and +1/2); Phase IV square lattice of circular skyrmions (Nsk = −1); Phase V square vortex lattice (Nsk = 0, not genuine meron lattice due to uniform out-of-plane component). The characteristic lattice constant/skyrmion diameter is ~2.7 nm. - Hall transport: ρyx(B) shows peak-like enhancements in Phases II and IV consistent with a topological Hall contribution ρyx^T from nonzero Nsk; anomalous Hall contributions proportional to Mρxx or Mρxx² cannot account for the sharp peaks. Using R0 ≈ 7.5 mΩ·cm/T and estimated P ~ 0.01 gives ρyx^T ~ 0.04 μΩ·cm, matching observed amplitudes. - Theory reproduces sequence Phase I → II → III → IV → V → FM and scattering intensity distributions. Crucially, competition between RKKY interactions at inequivalent wave vectors QA and QB stabilizes the elliptic skyrmion and meron/anti-meron phases without requiring a strong four-spin interaction, unlike models considering only QA interactions. - The nanometric scale (≈2.7 nm) of observed textures is one to two orders smaller than in typical DM-based non-centrosymmetric magnets.

The work demonstrates that centrosymmetric magnets can host a rich set of topological spin crystals and multi-step transformations between them, addressing the key question of realizing and controlling diverse magnetic quasi-particles without DM interaction. The combined diffraction and transport analyses connect nontrivial topology (nonzero Nsk) to measurable Hall responses (topological Hall effect), validating the identification of skyrmion phases. Theoretical modeling shows that the competition of itinerant-electron-mediated RKKY interactions at inequivalent nesting-related wave vectors QA and QB suffices to stabilize elliptic skyrmions, meron/anti-meron pairs, and circular skyrmions, explaining the observed domain behavior, multi-Q character, and field-evolution of wave numbers. This mechanism contrasts with DM-driven skyrmions and differs from earlier models requiring substantial four-spin terms, offering a general route to engineer nanometric topological textures via Fermi-surface tuning. These findings are significant for spintronics: the coexistence and field-tunable interconversion among particle-like textures with distinct Nsk suggests multi-valued memory or logic functionality. The extremely small lattice constant implies very high information density and potentially large emergent-field-driven responses. The observed hysteresis indicates sizable energy barriers between topological states, beneficial for metastability and device operation. The comparative theoretical analysis with GdRu₂Si₂ highlights how adding QB interactions transforms a single skyrmion phase into multiple distinct topological phases, guiding materials design.

This study uncovers multi-step topological transitions in GdRu₂Ge₂ between elliptic skyrmion, meron/anti-meron pair, and circular skyrmion crystal phases, all within a centrosymmetric lattice. Resonant X-ray and neutron scattering establish the multi-Q nature and symmetry evolution across phases, while Hall transport reveals topological Hall signatures specifically in the skyrmion phases. A minimal theoretical model with competing RKKY interactions at QA and QB quantitatively reproduces the phase sequence and scattering, showing that four-spin interactions are not essential. The textures are nanometric (~2.7 nm), far smaller than typical DM-based systems, positioning centrosymmetric rare-earth intermetallics as platforms for dense, tunable topological spin crystals. Future work should aim to: (1) discover and engineer materials with tailored χ(Q) landscapes to realize broader families of topological textures (including higher-order skyrmions with |Nsk| ≥ 2); (2) directly image these nanometric textures in real space and monitor field-driven transformations; (3) raise the ordering temperatures via compositional/design strategies (e.g., higher magnetic ion density or 3d–4f hybrids); and (4) explore and harness associated topological transport and optical responses for multi-level memory/logic devices.

- Real-space imaging at the ~2.7 nm period is challenging; LTEM resolution is near its limit and was not conclusive here. Spin textures were reconstructed from reciprocal-space data using phase choices minimizing |m(r)| variation, introducing model dependence (though cross-validated by theory). - High-energy neutron scattering used to mitigate Gd absorption limited Q-resolution, merging close satellites (q1, q2) into broader peaks; RXS provided more reliable q-values. - Slight discrepancies in critical fields between samples arise from shape anisotropy/demagnetizing effects due to different specimen geometries. - The Hall response outside skyrmion phases (e.g., at transitions from Phase V) may involve additional mechanisms (e.g., chiral Hall effect or scattering processes), not fully resolved here. - Phase V exhibits vortex-like textures with local fractional topology but is not a genuine meron lattice due to uniform out-of-plane magnetization, complicating topological quantification.