Topological swirling spin textures, such as skyrmions and merons, are attracting significant attention for their potential in high-density magnetic information storage. These textures, along with hopfions, have been observed in various systems with broken space-inversion symmetry, including magnets, ferroelectrics, and chiral liquid crystals. Their formation mechanisms and responses to external stimuli remain key areas of research. Skyrmions, in particular, are promising candidates for high-density information carriers due to their small size and electric controllability. Their topological charge, defined by an integral over the spin configuration, quantifies the number of times the spin direction wraps around a sphere. Skyrmions (N_sk = ±1), anti-skyrmions (N_sk = ±1), merons (N_sk = ±1/2), and anti-merons (N_sk = ±1/2) are characterized by their distinct topological charges and spin textures. Skyrmions were initially discovered in non-centrosymmetric compounds where the Dzyaloshinskii-Moriya (DM) interaction is crucial for their formation. However, recent theoretical work suggests that skyrmions can also exist in centrosymmetric materials via different mechanisms, potentially leading to much smaller skyrmions (less than 3 nm). This paper focuses on the experimental identification of a wide variety of magnetic quasi-particles and the underlying mechanism driving their transformations within a single material system, a challenge in the field. Previous studies have primarily focused on circular skyrmion phases, while controlled transformations among different topological spin textures are less common.

The literature extensively details the study of topological spin textures in non-centrosymmetric magnets, particularly focusing on the role of the Dzyaloshinskii-Moriya (DM) interaction in skyrmion formation. Studies have shown that the DM interaction influences the size and stability of skyrmions and the associated topological Hall effect. Recent theoretical and experimental works have explored the possibility of stabilizing skyrmions in centrosymmetric magnets through alternative mechanisms, particularly focusing on itinerant-electron-mediated interactions. These studies have reported the observation of small-diameter skyrmions in centrosymmetric rare-earth intermetallic compounds such as Gd₂PdSi₃ and GdRu₂Si₂. However, most of these materials exhibit only a single, circular skyrmion phase. While some studies have shown B-induced transitions between different topological phases in non-centrosymmetric systems, the controlled transformation between various meron and skyrmion phases within a centrosymmetric material remained largely unexplored before this work.

The study utilized GdRu₂Ge₂, a centrosymmetric magnet with a ThCr₂Si₂-type crystal structure. The researchers employed several techniques to characterize the material's magnetic and transport properties. Magnetization measurements were performed using a superconducting quantum interference device (SQUID) magnetometer, revealing multiple metamagnetic phase transitions. Electrical transport measurements using a five-terminal AC method were conducted to probe longitudinal (ρ_xx) and Hall (ρ_yx) resistivities. Neutron scattering experiments, conducted at the high-resolution chopper (HRC) spectrometer at J-PARC's Materials and Life Science Facility (MLF), provided information on the magnetic structure by measuring intensity profiles around reciprocal lattice points. Resonant X-ray scattering (RXS) experiments at the Photon Factory, KEK, Japan, using energy tuned to the Gd-L₂ absorption edge, allowed for a more detailed investigation of the magnetic satellite peaks and their polarization analysis. Polarization analysis of the scattered X-ray beam was performed to identify the detailed spin orientations in each phase. Theoretical analysis was conducted using simulated annealing based on an effective magnetic Hamiltonian derived from a Kondo lattice model. This model considered the competition between RKKY interactions at inequivalent wave vectors Q_A (q,0,0) and Q_B (q/2,q/2,0) in a two-dimensional square lattice. The parameters of the model were adjusted to reproduce the experimental observations of magnetization and the topological Hall effect. The real-space spin textures were reconstructed based on the experimental data from the RXS and neutron scattering experiments and using the constraints imposed by the localized nature of Gd³⁺ moments and the requirement for a spatially uniform magnetization amplitude.

The magnetization measurements revealed three distinct intermediate steps in the magnetization curve at 1.0 T, 1.2 T, and 1.35 T, indicating successive metamagnetic phase transitions. Corresponding anomalies were observed in both longitudinal and Hall resistivities. The peak-like enhancements in Hall resistivity at Phases II and IV suggest the presence of topological spin textures. Neutron scattering experiments confirmed that all phases except the ferromagnetic phase exhibit in-plane magnetic modulations. The RXS measurements revealed a multi-step evolution of magnetic structure, characterized by different wave vectors. The polarization analysis of the scattered X-ray beam allowed the researchers to determine the detailed spin orientations. The experimentally-obtained data were used to reconstruct the real-space spin textures for each phase: Phase I (single-Q screw spin texture), Phase II (elliptic skyrmion lattice), Phase III (meron/anti-meron pair lattice), Phase IV (circular skyrmion lattice), and Phase V (vortex lattice). The diameter of the skyrmions and merons in Phases II, III, and IV was determined to be approximately 2.7 nm. The theoretical model successfully reproduced the experimental observations, demonstrating the importance of competing RKKY interactions at inequivalent wave vectors Q_A and Q_B in stabilizing the observed variety of topological phases. The four-spin interaction was found not to be significant for reproducing the magnetism in GdRu₂Ge₂. The calculated scalar spin chirality shows non-zero values for Phases II and IV, consistent with the experimental observation of topological Hall effect.

The findings demonstrate that a centrosymmetric magnet can exhibit complex multi-step topological transitions, contrasting with previous observations mainly in non-centrosymmetric systems relying on DM interaction. The small size of the observed skyrmions and merons suggests a mechanism distinct from DM interaction. The successful reproduction of experimental results using a theoretical model incorporating competing RKKY interactions highlights the crucial role of this interaction in the stabilization of multiple topological phases. The observed transitions between elliptic skyrmions, meron/anti-meron pairs, and circular skyrmions can be understood as a step-by-step transformation of spin textures, driven by the Zeeman energy gain upon application of the magnetic field. The observed topological Hall effect confirms the existence of nontrivial spin textures in Phases II and IV, and its magnitude is consistent with the estimated skyrmion density. The study opens avenues for exploring diverse topological magnetic structures in centrosymmetric systems by tuning the interplay of various magnetic interactions mediated by itinerant electrons.

This study successfully demonstrated the experimental observation and theoretical understanding of multi-step topological transitions among various nanometric magnetic quasi-particles in the centrosymmetric magnet GdRu₂Ge₂. The key findings highlight the importance of competing RKKY interactions at inequivalent wave vectors for stabilizing a rich variety of topological phases, offering a new paradigm for designing materials with tunable topological properties. Future research directions include exploring materials with higher magnetic ordering temperatures, achieving direct real-space observation of the spin textures, investigating skyrmions with even higher densities, and studying the responses of these quasi-particles to various external stimuli.

The limitations of the study primarily involve the challenges in directly visualizing the spin textures at the nanoscale. The achieved resolution using the experimental techniques, such as neutron and X-ray scattering, is limited. The theoretical model employed simplifying assumptions, such as the two-dimensional square lattice approximation, potentially overlooking some complexities of the real three-dimensional structure. The precise determination of the phase θ_α in the reconstruction of real-space spin textures relies on the assumption of a spatially uniform magnetization amplitude, which might not perfectly hold in reality.