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[Max Hirschberger](https://orcid.org/0000-0002-1780-1619), Bertalan G. Szigeti, Mamoun Hemmida, Moritz M. Hirschmann, [Sebastian Esser](https://orcid.org/0000-0001-8372-3551), [Hiroyuki Ohsumi](https://orcid.org/0000-0002-6418-8984), Yoshikazu Tanaka, [Leonie Spitz](https://orcid.org/0000-0002-1028-3548), Shang Gao, [Kamil K. Kolincio](https://orcid.org/0000-0002-9757-6764), Hajime Sagayama, [Hironori Nakao](https://orcid.org/0000-0003-4020-537X), [Yuichi Yamasaki](https://orcid.org/0000-0002-8560-3462), László Forró, Hans-Albrecht Krug von Nidda, Istvan Kezsmarki, [Taka-hisa Arima](https://orcid.org/0000-0002-6959-0454), [Yoshinori Tokura](https://orcid.org/0000-0002-2732-4983)

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[Lattice-commensurate skyrmion texture in a centrosymmetric breathing kagome magnet](https://mdr.nims.go.jp/datasets/efbfbc30-c81f-4c48-aec0-305e04e53013)

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Lattice-commensurate skyrmion texture in a centrosymmetric breathing kagome magnetnpj | quantummaterials ArticlePublished in partnership with Nanjing Universityhttps://doi.org/10.1038/s41535-024-00654-2Lattice-commensurate skyrmion texturein a centrosymmetric breathingkagome magnetCheck for updatesMax Hirschberger 1,2 , Bertalan G. Szigeti3, Mamoun Hemmida3, Moritz M. Hirschmann2,Sebastian Esser 1, Hiroyuki Ohsumi 4, Yoshikazu Tanaka4, Leonie Spitz 2,11, Shang Gao2,12,Kamil K. Kolincio 2,5, Hajime Sagayama6, Hironori Nakao 6, Yuichi Yamasaki 7, László Forró8,Hans-Albrecht Krug von Nidda3, Istvan Kezsmarki3, Taka-hisa Arima 2,9 & Yoshinori Tokura 1,2,10Skyrmion lattices (SkL) in centrosymmetric materials typically have a magnetic period on thenanometer-scale, so that the coupling between magnetic superstructures and the underlying crystallattice cannot be neglected. We reveal the commensurate locking of a SkL to the atomic lattice inGd3Ru4Al12 via high-resolution resonant elastic x-ray scattering (REXS). Weak easy-plane magneticanisotropy, demonstrated here by a combination of ferromagnetic resonance and REXS, penalizesplacing a skyrmion core on a site of the atomic lattice. Under these conditions, a commensurate SkL,locked to the crystal lattice, is stable at finite temperatures – but gives way to a competingincommensurate ground state upon cooling.Wediscuss the role ofUmklapp-terms in theHamiltonianfor the formation of this lattice-locked state, its magnetic space group, and the role of slightdiscommensurations, or (line) defects in the magnetic texture. We also contrast our findings with thecase of SkLs in noncentrosymmetric material platforms.Magnetic skyrmion lattices (SkLs) are periodic arrays of vortex-like spinstructures. In SkLs, magnetic moments are twisted into a knot, covering alldirections of a sphere as we traverse a single magnetic unit cell (UC)(Fig. 1a)1–3. These vortices were first described as topological solitons usingthe concepts of field theory, and such continuummodels are most suitablewhen the magnetic UC is at least two orders of magnitude larger than theunderlying crystallographic UC3–5. With a focus on frustrated, i.e. com-peting, interactions, recent theoretical work6–10 has proposed SkL formationin a high-symmetry context without spin-orbit driven Dzyaloshinskii-Moriya interactions, paving theway for the experimental observation of SkLphases with magnetic period on the nanometer-scale in centrosymmetricinsulators and metals3,11–14. These quasi-discrete SkLs have raised hopes ofenhanced functional responses, especially those related to the interplay ofemergent electromagnetic fields with conduction electron (Bloch) waves, orwith incident light waves3,15–20.Evidence for coupling between the atomic lattice and skyrmion tex-tures with lattice spacing 2− 3 nanometers has emerged in tetragonalmagnets: Centrosymmetric alloys host square and rhombic skyrmionlattices3,21, andnon-centrosymmetric EuNiGe3 exhibits a fascinating helicityreversal upon entering the SkL phase, where themagnetic texture breaks thesense of rotationprescribedby its polar structure13,14. A keyopen challenge isthe demonstration of a commensurate locking (C-locking) transition of theSkL’s spin superstructure to the underlying lattice potential in such a cen-trosymmetric bulk material. This phenomenon is conceptually related toinstabilities of the skyrmion vortex core anticipated for spin-1 systems22 andits observation would provide a bridge between – usually – large-scale SkL1Department of Applied Physics and Quantum-Phase Electronics Center, The University of Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan. 2RIKEN Center forEmergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan. 3Experimental Physics V, Center for Electronic Correlations and Magnetism, University ofAugsburg, 86135 Augsburg, Germany. 4RIKEN SPring-8 Center, 1-1-1 Kouto, Sayo, Hyogo 679-5148, Japan. 5Faculty of Applied Physics and Mathematics,Gdańsk University of Technology, Narutowicza 11/12, 80-233 Gdańsk, Poland. 6Institute of Materials Structure Science, High Energy Accelerator ResearchOrganization, Tsukuba, Ibaraki 305-0801, Japan. 7Research andServicesDivision ofMaterials Data and Integrated System (MaDIS), National Institute forMaterialsScience (NIMS), Tsukuba 305-0047, Japan. 8StavropoulosCenter forComplexQuantumMatter, Department of Physics andAstronomy, University of Notre Dame,Notre Dame, IN 46556, USA. 9Department of Advanced Materials Science, The University of Tokyo, Kashiwa 277-8561, Japan. 10Tokyo College, The University ofTokyo, Bunkyo-ku, Tokyo 113-8656, Japan. 11Present address: Paul Scherrer Institut, Forschungsstrasse 111, 5232 Villigen PSI, Switzerland, 12Present address:Department of Physics, University of Science and Technology of China, Hefei 230026, China. e-mail: hirschberger@ap.t.u-tokyo.ac.jpnpj Quantum Materials |            (2024) 9:45 11234567890():,;1234567890():,;http://crossmark.crossref.org/dialog/?doi=10.1038/s41535-024-00654-2&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41535-024-00654-2&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41535-024-00654-2&domain=pdfhttp://orcid.org/0000-0002-1780-1619http://orcid.org/0000-0002-1780-1619http://orcid.org/0000-0002-1780-1619http://orcid.org/0000-0002-1780-1619http://orcid.org/0000-0002-1780-1619http://orcid.org/0000-0001-8372-3551http://orcid.org/0000-0001-8372-3551http://orcid.org/0000-0001-8372-3551http://orcid.org/0000-0001-8372-3551http://orcid.org/0000-0001-8372-3551http://orcid.org/0000-0002-6418-8984http://orcid.org/0000-0002-6418-8984http://orcid.org/0000-0002-6418-8984http://orcid.org/0000-0002-6418-8984http://orcid.org/0000-0002-6418-8984http://orcid.org/0000-0002-1028-3548http://orcid.org/0000-0002-1028-3548http://orcid.org/0000-0002-1028-3548http://orcid.org/0000-0002-1028-3548http://orcid.org/0000-0002-1028-3548http://orcid.org/0000-0002-9757-6764http://orcid.org/0000-0002-9757-6764http://orcid.org/0000-0002-9757-6764http://orcid.org/0000-0002-9757-6764http://orcid.org/0000-0002-9757-6764http://orcid.org/0000-0003-4020-537Xhttp://orcid.org/0000-0003-4020-537Xhttp://orcid.org/0000-0003-4020-537Xhttp://orcid.org/0000-0003-4020-537Xhttp://orcid.org/0000-0003-4020-537Xhttp://orcid.org/0000-0002-8560-3462http://orcid.org/0000-0002-8560-3462http://orcid.org/0000-0002-8560-3462http://orcid.org/0000-0002-8560-3462http://orcid.org/0000-0002-8560-3462http://orcid.org/0000-0002-6959-0454http://orcid.org/0000-0002-6959-0454http://orcid.org/0000-0002-6959-0454http://orcid.org/0000-0002-6959-0454http://orcid.org/0000-0002-6959-0454http://orcid.org/0000-0002-2732-4983http://orcid.org/0000-0002-2732-4983http://orcid.org/0000-0002-2732-4983http://orcid.org/0000-0002-2732-4983http://orcid.org/0000-0002-2732-4983mailto:hirschberger@ap.t.u-tokyo.ac.jpspin textures in materials with broken inversion symmetry, and cantedantiferromagnetism on the scale of a single unit cell. Indeed, theory predictssuch C-locking based on Ruderman-Kittel-Kasuya-Yosida (RKKY) inter-actions andmagnetic anisotropy, when the length scale ofmagnetic texturesapproaches the size of a crystallographic UC23. Among inversion breakingmaterial platforms, the hcp-Fe/Ir(111) interface has been reported to exhibitC-locking using imaging techniques, although fcc-Fe/Ir(111) forms alattice-incommensurate structure16,24–26. However, such locking between theperiodicity of a magnetic skyrmion lattice and the underlying crystalstructure has never been observed in a bulk material.Using precise resonant x-ray measurements, we report a commensu-rate skyrmion lattice (C-SkL) surrounded by incommensurate (IC) phasesin bulk samples of the centrosymmetric intermetallic Gd3Ru4Al12, locked tothe distorted kagomenetwork ofmagnetic gadolinium ions.We discuss thisstate based on (weak) single-ion anisotropy K1, as supported by electron-spin resonance experiments.For large classical spins, calculations on both triangular and breathingkagome lattices show that, if the single-ion anisotropy is of easy-plane type(easy-axis type), a commensurate skyrmion vortex may gain energy bylocating its core at an interstitial site (on a lattice site)23,27; an incommen-surate skyrmion lattice does not benefit from this type of energy gain(Supplementary Fig. 5). We illustrate this point in the lower part of Fig. 1a,depicting a realistic magnetic structuremodel for the C-SkL of Gd3Ru4Al12,described by normalized vectors n(x, y), which is mapped onto a sphereusing a stereographic projection, seeMethods.Here,magneticmoments areconspicuously absent at the poles (Fig. 1a, upper). The sparsity of magneticmoments at the poles becomes more apparent when unfolding the sphereusing a cartographic projection (Fig. 1b and Supplementary Note 1).ResultsObservation of a commensurate skyrmion lattice (C-SkL)We use elastic x-ray scattering in resonance with the L2,3 absorptionedge of gadolinium, setting the sample in reflection geometry toprecisely detect the magnetic period of each magnetic phase (Fig. 1candMethods). Reporting data from a synchrotron light source, Fig. 1ddepicts the core observation of this work: At moderate temperaturesT = 7− 8 K, a magnetic field B applied along the c-axis drives theincommensurate proper screw ground state (IC-PS) into a commen-surate skyrmion lattice phase (C-SkL), and again to incommensuratefan-like order (IC-Fan). The C-locking of the SkL was not observed in aprevious study11. The present and previous results are compared andsummarized in Supplementary Fig. 4. Let a* and c* be the reciprocalspace lattice constants for our target material Gd3Ru4Al12 in the hex-agonal P63/mmc space group, where magnetic gadolinium ions form akagome (star of David) lattice with a breathing distortion, corre-sponding to alternating bond distances. In zero magnetic field, themagnetic modulation wavevector is q = (q, 0, 0) with q ≈ 0.272 a* orwavelength λ ≈ 3.7 Luc, where Luc is the dimension of the crystal-lographic UC projected parallel to q (Supplementary Fig. 1). Thewavevector’s length q decreases with B in IC-PS, approaching a dis-continuous jump at the first-order transition11 towards q = 0.25 a* inC-SkL. The magnetic period λ in C-SkL is very close to 4 Luc, with aslight offset indicated by two dashed horizontal lines in Fig. 1d. Therole of this slight offset, or discommensuration λdisc, is discussed below.To further support the C-locking at q = 0.25 a*, we demonstrate inSupplementary Fig. 7 that the peak profiles are within less than onestandard deviation from the commensurate value, and in Supple-mentary Fig. 10 that q has weak temperature dependence in C-SkL ascompared to IC-PS.In the regime labeled as C-SkL in Fig. 1, previous real-space imagingexperiments have observed vortex structures11, and precise Hall effectmeasurements demonstrate that the noncoplanar magnetic state that iseasily destroyed by a slight in-plane magnetic field28. However, neither thisprior work nor the present REXS technique are able to determine whetherthe skyrmion core is located on a crystallographic lattice site, or on aninterstitial site. This question of the relative phase shift between magneticFig. 1 | Commensurate locking of the skyrmion lattice phase inGd3Ru4Al12 (GRA). aMagneticmoments of a skyrmionwinding a sphere (top), andcorresponding two-dimensional magnetic moment texture in real space (bottom).Each arrow corresponds to a site of themagnetic sublattice (Supplementary Note 1).bMap projection showing a hemisphere of (a) unfolded, with white dots indicatingdirections of magnetic moments. The colour code indicates the distance of eachpoint on the sphere from the nearest moment on the sphere, in radians.c Experimental geometry of resonant elastic x-ray scattering (REXS) at the Gd-L2absorption edge, where the pink line is the trajectory of the x-ray beam. d Scatteringintensity in REXS as a function ofmagnetic field (B) andmomentum transfer q fromq = (q, 0, 0); q = 0.25 corresponds to λ = 2π/q = 4 Luc, four times the projection of thelattice constant parallel to q (Supplementary Fig. 1). Incommensurate proper screw(IC-PS), commensurate skyrmion lattice (C-SkL), and IC fan phases are illustratedby insets. In C-SkL, the periodicity of the magnetic texture is locked to the crystallattice up to a weak discommensuration (offset) of Δq = 0.0018 r.l.u. B denotesmagnetic induction after demagnetization correction (Supplementary Fig. 4). Sup-plementary Fig. 6 and Supplementary Fig. 7 show raw data used to create thiscolormap and the results of Gaussian fitting to the data, respectively.https://doi.org/10.1038/s41535-024-00654-2 Articlenpj Quantum Materials |            (2024) 9:45 2texture and crystal lattice can be addressed by measurements of single-ionanisotropy, combined with theoretical modeling: We turn to the ferro-magnetic resonance (FMR) technique in the following section.Single-ion anisotropy in Gd3Ru4Al12We prepared a cylindrical disk-shaped sample for ferromagnetic resonance(FMR) experiments, spanned by the crystallographic a and c directions(Fig. 2a, right inset). This highly symmetric geometry allows for simple dataanalysis when rotating the magnetic field in the plane of the disk, seeMethods. In the experiment, we drive the crystal into the field-alignedferromagnetic (FA-FM) state with a large magnetic field, irradiate it withmicrowaves of frequency ν = 210 or 314 GHz, and observe a change of itsreflectivity when the microwaves excite a resonance between moment-upand -down states (Fig. 2a, left inset).In Fig. 2a, the anisotropic part of the free energy Fanis ¼ K1cos2ðθÞis deduced in FA-FM from a fit to the angular dependence of the FMRresonance field Bres; we disregard anisotropy constants beyond the firstorder, see Methods. Using the saturation magnetization MS = 7 μB /Gd3+, our fit yields easy-plane anisotropy withK1 =− 0.13 meV / Gd3+.Therefore, the anisotropy field Bani can be calculated asμ0∣2K1∣=MS ¼ 0:74 Tesla. As compared to magnetization measure-ments in Supplementary Fig. 11, the present FMR study yieldsenhanced precision; field-dependent FMR experiments separateg-factor anisotropy and exchange anisotropy from the single-ion term,see Methods. Figure 2b, c illustrates the resulting iso-energy surfacesFanis(θ, φ) in zero (finite) magnetic field along the c-axis, respectively,where the spherical coordinates refer to the direction of the sample’sbulk magnetization M.Anisotropy and anharmonic distortion of proper screw andskyrmion phasesEasy-plane anisotropy (K1 < 0) can also be verified semi-quantitativelyby resonant elastic x-ray scattering in the ordered phases with periodiclong-range order. Figure 1c shows the geometry of our experiment withpolarization analysis: the purple scattering plane is spanned by thewavevectors ki and kf of the incoming and outgoing x-ray beams, withbeam polarization εi and εf, respectively. We choose the incoming beampolarization εi to lie within the scattering plane for all our experiments(π-polarization). While the data in Fig. 1d represents a sum of scatteredphotons with all possible εf, we now add an analyser plate before thex-ray detector to separate two components of the scattered beam: Iπ�π0and Iπ�σ 0 with εf within or perpendicular to the scattering plane,respectively. From the scattered intensities at various magnetic reflec-tions, we extract the ratio, see Methods,R sin2ð2θÞ ¼ Iπ�σ 0 sin2ð2θÞ=Iπ�π0 ¼ ki �mabðqÞ=mcðqÞ� �2 ð1Þwith mab the component of the modulated magnetic moment in the scat-tering plane. The expectedbehavior of IC-PS (spin plane a-c) and IC cycloid(spin plane a*-c) is indicated in Fig. 3b by black and green dashed lines,respectively; a line with positive slope indicates IC-PS character.Supplementary Fig. 3 shows representative raw data, as used to createthis panel.Beyond identifying the PS character of the spin modulation, thisanalysis allows us to estimate the effect of single-ion anisotropy in IC-PS.Specifically,weobserve an elliptic distortion, i.e., a deviation fromtheproperscrew model. Figure 3b, c displays maximum values of Rsin2ð2θÞ around1.5 (IC-PS) and 7 (C-SkL), so that Eq. (1) delivers mab(q)/mc(q) around1.2 (IC-PS) and 2.6 (C-SkL). Figure 3b, inset, illustrates the proposedanharmonic distortion in IC-PS, where magnetic moments prefer to tilttowards the basal plane to gain anisotropy energy. In Fig. 3d, a simulation ofREXS anisotropy as a function of K1 for a spin model in both IC-PS andC-SkL (Supplementary Note 1) shows that the experimentally observedellipticity is consistent with the results of FMR in Fig. 2. Given the robustobservation ofK1 < 0 via various experimental techniques, we conclude thata C-SkL in magnetic space group P632020 and with skyrmion core betweenlattice sites is stable in Gd3Ru4Al12 (Supplementary Note 2).Competition of commensurate and incommensurate phasesFigure 4a shows a magnetic field scan of x-ray scattering intensity at thelowest accessible temperature, T = 1.5 K; see Supplementary Fig. 4 fordetailed phase diagrams and comparison to prior work. In contrast to Fig.1d, the C-SkL phase is now absent, being replaced by an incommensuratetransverse conical (IC-TC) state11.We compare the experimental and theoretical results for Gd3Ru4Al12to earlier studies of magnetism in elemental rare earth metals. Variousscenarios for the interplay of commensurate (C) and IC phases have beenadvanced29,30: One typical observation in systems with strong single-ionFig. 2 | Magnetic anisotropy in field-polarizedferromagnetic (FA-FM), incommensurate properscrew (IC-PS), and skyrmion lattice (C-SkL) pha-ses of Gd3Ru4Al12. a Resonance field Bres ¼ μ0Hresin ferromagnetic resonance (FMR) experiments inFA-FM at 210 and 314 GHz (grey and green mar-kers). Red lines: Fit to the data according to Smit-Beliers-Suhl, corresponding to weak in-plane ani-sotropy, see Methods. Left inset: principle of FMR,where transitions between up and down momentsare induced by microwave radiation in a cavity.Right inset: geometry of magnetic field B withrespect to the principal axes of a disk-shaped singlecrystal. b, c Anisotropic energy landscape Fanis ¼K1cos2ðθÞ � EZ cosðθÞ in zero field (Zeeman termEZ = 0) and finite magnetic field (EZ/K1 = 0.3), as afunction of the direction of the bulk magnetizationvector M. d Crystal structure of Gd3Ru4Al12 withmagnetic breathing kagome sublattice of gadoli-nium, shown in magenta.https://doi.org/10.1038/s41535-024-00654-2 Articlenpj Quantum Materials |            (2024) 9:45 3anisotropy is the appearance of spin-slip structures in the IC order, andeventual squaring up at low temperatures. In this scenario, thermal fluc-tuations at higher T locally reduce the ordered magnetic moment, allowingfor the formation of ICmagnetism close to the Néel point, e.g. in elementalHolmium29. Secondly, when single-ion anisotropy is much weaker thanexchange interactions, C orders are favored at higher temperatures whereshort-range correlations dominate. However, they can give way to ICground states as further-neighbor exchange gains importance upon cooling(e.g. ref. 31). Thirdly, in inversion breaking systems with strongDzyaloshinskii-Moriya interactions, IC-C transitions appear at low tem-perature when solitonic distortions are introduced into an IC order via amagnetic field32–35, especially if a large charge gap is opened due to nesting35.The present study generalizes the second scenario to the case of complex,twisted magnetic textures, such as the C-SkL, with an important role for(weak) magnetic anisotropy.We consider the observation of C-locking induced by magnetic fields,in our experiment, based on the single-ion contribution to the spinHamiltonian (Supplementary Note 3)Hanis / �K1XGXq2BZSzðqÞSzðG� qÞXd2u:c:eiG�d" #ð2Þwhere r = R+ d is the position of a magnetic ion, decomposed into aunit cell coordinate R and an intra-cell coordinate; q and G are themomentum in the first Brillouin zone and a reciprocal lattice vector,respectively. Typically, a small number of Fourier modes q = qν andG = 0 are selected in models of incommensurate helimagneticordering12,21,28. Meanwhile, C-locking is favored by the Umklapp termsG ≠ 036,37, representing a coupling between the primary Fourier modeof a helimagnet and its higher harmonics. Specifically for the q = 0.25 a*C-SkL in Gd3Ru4Al12, Sð3qÞ � SðqÞ and Sð2qÞ2 are the leadingcontributors, adding to G = (1, 0, 0) and coupling the helimagneticorder to the lattice. Application of a magnetic field to IC-PS enhancesthe elliptic distortion as demonstrated in Fig. 3, amplifies anharmo-nicity, shifts q away from the value preferred by the exchangeinteraction and towards commensurability, and ultimately inducesC-locking between the spin texture and the underlying lattice. InSupplementary Note 3, we thus derive an expression for the energeticcontribution that depends on the position of the skyrmion core.Nevertheless, a full numerical treatment of Eq. (2) with large numbersof Fourier modes, to capture changes in the optimal q – as well aschanges in the cycloidal / screw character of the modulation – as afunction of magnetic field and temperature, remains a challenge atpresent (Supplementary Note 1).To explain the effect of single-ion anisotropy more intuitively, weinitialize C-SkL andC-PS on the lattice and allow themagneticmomentsto relax in the combined potential of exchange interaction, magneticanisotropy, and external field (white/green arrows in Fig. 4b, c). The redarrows show a significant distortion of the textures, especially around thesouth pole. This density of strongly distortedmoments ismuch larger forthe C-PS state with quasi one-dimensional spin texture, which has anextended south pole region. In particular, the south pole direction of themagnetic moment sphere in Fig. 1a corresponds to a point (to a line) incase of a SkL (of a PS). We also calculated the z-projection of magneticmoments for a spin model of C-PS, C-SkL, and other orders, using ourexperimental K1 from FMR, thus confirming numerically the favorablepinning of the C-SkL due to stronger higher harmonics (SupplementaryFig. 5).DiscussionFigure 1d reveals a discommensuration of the magnetic lattice, i.e., a slightoffset from q/a* = 0.25 that indicates occasional magnetic defects. The dis-commensuration effect is well understood in one-dimensional chainsystems38: For example, the introduction of small amounts of chemicaldisorder causes proliferation of discommensurations in spin-Peierls chains,Fig. 3 | Magnetic anisotropy in ordered states ofGd3Ru4Al12. a Experimental geometry of resonantelastic x-ray scattering (REXS) with (in contrast toFig. 1c) polarization analysis. π and σ indicate x-rays(photons) polarized parallel and perpendicular to thescattering plane (purple), respectively. b Elliptic dis-tortion of IC-PS determined by polarization analysis inREXS. Iπ�σ 0 , Iπ�π0 , ki, and e⊥ are integrated intensitiesin the polarization-flipping and polarization-conserving scattering channels, the incoming beam’swavevector, and the in-plane directionperpendicular tothe propagation vector q of IC-PS, as in Eq. (1). Green,black, and magenta dashed lines indicate model pre-dictions for spherical cycloid, spherical proper screw,andellipticallydistorted screw, the latter being shown inthe inset cartoon. The value of Rsin2ð2θÞ at the rightboundary of the panel corresponds to the elliptic dis-tortion ðmab=mcÞ2 as defined in Eq. (1). c Analogousdata for the commensurate skyrmion (C-SkL) state; seeSupplementary Table 1 for more details. Note the dif-ferent y-axis range as compared to (b). d Spin-modelcalculation of elliptic distortion in IC-PS andC-SkL as afunction of anisotropy constant K1, normalized to thecritical field Bc= 4T for transition to the field-alignedferromagnetic (FA-FM) state. Dashed horizontal lines,dashed vertical line, and highlighted area indicate theexperimental values for the ellipticity, the value of K1from FMR, and and the range of K1 consistent withREXS experiments, respectively. Error bars in (b, c) areobtained from Gaussian fits to line profiles as in Sup-plementary Fig. 3, with quadratic error propagation forthe intensity ratio.https://doi.org/10.1038/s41535-024-00654-2 Articlenpj Quantum Materials |            (2024) 9:45 4as manifested in a drop of λdisc that ultimately destroys the commensurate(C) order39–42. Discommensurations in two dimensions, mostly line defects,also appear for surface-adsorbed atoms and in themulti-directional charge-density wave state of transition metal dichalcogenides such as 2H-TaSe2,where C-IC transitions have been extensively studied36,37. Among materialswith two-dimensional spin textures, we compare to the C-SkL observed inan interfacial system24–26: For inversion-breaking hcp-Fe/Ir(111) (for cen-trosymmetric Gd3Ru4Al12), the spin structure is found to be commensuratewithin less than 10% (within 0.7 %) of the magnetic period, and there is no(there is) evidence of C-IC transitions by cooling or application of a mag-netic field. Based on measurements of magnetic anisotropy and modeling,wearguehere thatGd3Ru4Al12has skyrmioncoreson interstitial lattice sites,as does hcp-Fe/Ir(111); further that the discommensuration represents theappearance of line defects (characteristic spacing λdisc = 430 nm) in theformer – while phase-slip domain walls (spacing ~ 10 nm) and domains ofthe net magnetization (spacing ~ 30 nm) have been observed in the latter.We note that domain walls of the skyrmion helicity may appear in the bulkC-SkL of centrosymmetric materials6,43; these are forbidden in inversionbreaking platforms such as hcp-Fe/Ir(111).Finally, Fig. 4d summarizes recent progress onmaterial search andmagnetic structure studies of noncoplanar magnets with lattice-averaged net spin chirality, especially SkL host compounds3,13,14,44,45. Asthe dimension λ of the magnetic UC increases, magnetic momentscover all directions of the S2 sphere, and the maximum gap in angularcoverage (as defined in percent of 4π) approaches zero. For example,noncoplanar antiferromagnets on the left side of Fig. 4d leave largefractions of S2 out of reach of magnetic moments. Being located at thecenter of the plot, the C-SkL in the centrosymmetric, breathing kagomemagnet Gd3Ru4Al12 represents an essential link between long-period,incommensurate magnetic textures, stabilized e.g. by Dzyaloshinskii-Moriya interactions in MnSi3,46, and so-called topological antiferro-magnets with canted magnetic moments, such as pyrochlore systems44.Although the present C-SkL is locked to the crystal lattice by weak in-plane magnetic anisotropy, confirmed here by ferromagnetic reso-nance experiments, the magnetic unit cell is large enough so thatmoments densely cover S2, leaving only marginal gaps on the order of5% of 4π.MethodsCartographic projectionFor visualizing the magnetic texture in Fig. 1a, we mapped a magneticskyrmion onto a sphere in three dimensions, and subsequently mappedfrom the sphere onto a planar surface using the cartographic Nicolosiglobular projection. Let (θ, φ) denote a point where a line between (x, y)and the north pole of the sphere penetrates the sphere’s surface. Thesespherical coordinates are identified with (x, y) in Fig. 1a, upper side,while a point on the sphere is projected back onto the plane using theformalism in ref. 47.Fig. 4 | Stability of the commensurate skyrmion lattice (C-SkL) as compared toother magnetic phases. a Resonant x-ray scattering intensity, normalized tomonitor counts, at base temperature (T = 1.5 K). Three incommensurate (IC)magnetic phases are shown: from left to right, proper screw (IC-PS), transverseconical (IC-TC), and fan-like (IC-Fan) as illustrated by insets. Quantitative analysisof Gaussian line-scan profiles is given in Supplementary Fig. 8. b, c Distortion ofproper screw (left) and skyrmion textures in anisotropy potential and externalmagnetic field along the z-direction (Supplementary Note 1). The undistortedmoment directions (red) are superimposed on those moments that have beenrotated bymore than a critical angle. d Relationship of magnetic texture dimensions(λ) and coverage of directions on the sphere for various materials with noncoplanartextures and spin chirality3,11,13,14,44,45,51. At the center of the plot, the commensurateC-SkL state in Gd3Ru4Al12 is highlighted by a red box. On the y-axis, a continuousmagnetic texture has zero uncovered solid angle, i.e. we assign a value of 0 %.Note thestretched scale of the y-axis.https://doi.org/10.1038/s41535-024-00654-2 Articlenpj Quantum Materials |            (2024) 9:45 5Starting frompolar angles θ,φ on the sphere, the x and y coordinates ofthe planar projection are defined asx ¼ π2R M þ sgnðφ� φ0ÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 þ cos2ðθÞ1þ b2=d2s !ð3Þy ¼ π2R N � sgnðθÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiN2 � d2sin2ðθÞ=b2 þ d sinðθÞ � 11þ d2=b2s !ð4Þwith functionsM ¼ b sinðθÞ=d � b=21þ b2=d2; ð5ÞN ¼ d2 sinðθÞ=b2 þ d=21þ d2=b2; ð6Þb ¼ π2ðφ� φ0Þ� 2πðφ� φ0Þ; ð7Þc ¼ 2πθ; and ð8Þd ¼ 1� c2sinðθÞ � c: ð9ÞThe sphere’s radiusR and the reference point for the longitude φ0 can be setto 1 and 0, respectively.Sample preparation and characterizationWe prepared polycrstals of Gd3Ru4Al12 by arc melting of the constituentelements in argon atmosphere, carefully turning pellets at least three times.Subsequently, single crystals were grown from these polycrystals by thefloating zone technique under argon flow. Before the melting step, thehalogen-lamp based furnace was evacuated to a base pressure of 8 ⋅ 10−4 Pa,pumping for about three hours. The growth speed in the float zoning stepwas2− 4mm/hr.Wecrushedsingle crystallinepieces into afinepowder forx-ray diffraction and refined the data by the Rietveld method using theRIETAN software package48. In this analysis, we did not find impurityphases with volume fraction larger than 4%. Further, single-crystallinepieces were oriented using a Laue camera, cut with a diamond saw, andhand-polished to a high sheen (~ 1 μm grit) for single crystal x-ray dif-fraction measurements. Using a microscope with Nicolet prism, single-crystalline pieces with small amounts of RuAl3 impurity phase, which formstear-drop shaped inclusions of < 2% volume fraction that are hard to detectby laboratory x-rays, were excluded based on patterns in the surface-reflected light. Finally,magnetizationmeasurements (M-H curve atT = 2 K,M-T curve at μ0H = 0.1 T) were used to confirm a systematic evolution oflong-range magnetic order in these single crystals.Ferromagnetic resonance measurementsIn order to avoid unwanted shape anisotropy effects, we polished theGd3Ru4Al12 samples cut within the a-c plane into a cylindrical disc form,with ellipticity below 4%. The ratio of diameter to thickness was 12 for thefinal disc, with a thickness of 120 μm.High-field ferromagnetic resonance (FMR) measurements were per-formed at École Polytechnique Fédérale de Lausanne (EPFL, Switzerland)using a home built, high-sensitivity, quasi-optical spectrometer operating inthe range of 50− 420 GHz. This instrument covers a broad magnetic fieldregimeup to16 T, using a superconductingmagnet. Its variable temperatureinsert operates across the temperature range 1.5−300 K, using the dynamicflow of helium gas, or liquid helium, through a heat exchanger right belowthe sample space. For more details, see ref. 49.The overall temperature accuracy of the system is 0.1 K. The polisheddisc-like sample is mounted on a goniometer with the a-c plane coincidingwith plane of rotation for the static magnetic field, the angular position ofwhich is controllable and detectable via a potentiometer. Rotation pro-ceeded in 5− 10degree steps, at sample temperatureT = 2 K.The signal-to-noise ratio of the spectra is improved by recording the field-derivative of theabsorbed power dP/dH using a lock-in technique with magnetic fieldmodulation. The angular-dependent resonance data can be evaluated usingthe Smit-Beliers-Suhl formula as described in ref. 50. Using the saturationmagnetization MS = 7 μB / Gd3+, our fit yields the easy-plane anisotropyconstant K1 =− 1.944 ⋅ 106 erg/cm3 (− 0.13meV / Gd3+) and the tem-perature independent, isotropic g-factor 2.005.Resonant x-ray scattering experiments (Gd-L2,3)Resonant elastic x-ray scattering (REXS) experiments are carried out inreflection geometry at RIKEN beamline BL19LXU of SPring-8 and beam-line BL-3A of Photon Factory, KEK, with the sample mounted inside acryomagnet. The preparation of SampleA, used to obtain the data in Fig. 3b,is discussed in ref. 11. Sample B, which was used to obtain all other data inthis manuscript, has a surface perpendicular to the [110] axis, and waspolished to reduce loss of intensity bydiffuse scattering of x-rays.The energyof incident x-rays ismatched to the L2 or L3 absorption edge ofGadolinium,where magnetic scattering involves virtual excitations from the 2p to the 5datomic shells (Supplementary Fig. 2). The 5d shell is coupled to the domi-nant magnetic species, the half-filled 4f orbitals, by intra-atomic exchangecorrelations.For data in Figs. 1 and 4, which are collected at the Gd-L3 edge(Ex-ray = 7.243 keV) at BL19LXU of SPring-8, we do not carry out polar-ization analysis of the diffracted beam. For data in Fig. 3 and SupplementaryFig. 3, which were collected at the Gd-L2 edge (Ex-ray = 7.932 keV) at BL-3Aof Photon Factory, π � σ 0 and π � π0 components of the diffracted beamare separated using an analyser plate made from pyrolytic graphite(PG-006).The details of the polarization analysis in the latter case are as follows.Let ki (kf) be the wavevector of the incoming (outgoing) x-ray beam withpolarization vector εi (εf), where the scattering plane is spanned by the twowavevectors; c.f. purple plane in Fig. 1c. Let z be the direction perpendicularto the scatteringplane.The incident beam isπ-polarized, so that εi⊥ ez,ki. Inthe resonant elastic scattering process, the scattering cross-section f rescontains a term∝ (εi × εf) ⋅m(q), where q = kf− ki, and m(q) are themomentum transfer and the periodically modulated magnetic moment,respectively (Fig. 1c). In thepresent case,where εi∝ ki × ez and the scatteringplane is aligned with the crystal’s hexagonal ab plane, we separate scatteredx-rays with εf∥ez∥c* (i.e., π � σ 0) and εf∝ kf × ez (i.e., π � π0). Hencef π�σ0res / ki �mðqÞ and f π�π0res / ðki × kf Þ �mðqÞ / mzðqÞ sinð2θÞ, where 2θis the scattering angle between ki and kf. The observed scattered intensitiesare I / ∣f res∣2.We obtain integrated intensities Iπ�σ0 and Iπ�π0 from Gaussian fits toline-cuts in momentum space. As described, the intensity ratio R ¼Iπ�σ 0=Iπ�π0 at a givenmagnetic reflection is sensitive to the cycloidal (properscrew) character of themagnetic order by virtue of being large (small) whenki ⋅ (q−G′) is large (small). For proper screw order, magnetic momentsarrange themselves perpendicular to the propagation direction (q−G′),with a finite projection onto both ez (i.e., mz) and the in-plane vectore⊥ = (q−G′) × ez (i.e., mip). Here we introduce G′, the closest reciprocallattice vector to q, and ez, a unit vector along the c-direction. It followsRsin2ð2θÞ / ki � e?ðqÞ� �2for a proper screw-type order, as demonstratedfor Gd3Ru4Al12 in Fig. 3b, c. Moreover, R provides information on thedegree of elliptical distortion, capturing e.g. the ratio ðmy=mzÞ2 for aproper screw propagating along ex, written as ey �my sinð2πx=λÞþez �mz cosð2πx=λÞ, where ex, ey, ez areCartesian unit vectors aligned so thatex∥q. In real materials with large (classical) magnetic moments, where thelength of the moment is fixed to be spatially uniform in space, this type ofdeformed screw can be realized by anharmonic distortion of the properscrew texture, as shown in Fig. 3b (inset).https://doi.org/10.1038/s41535-024-00654-2 Articlenpj Quantum Materials |            (2024) 9:45 6Data availabilityThe data supporting the findings of this study are available from the cor-responding author upon reasonable request.Received: 17 September 2023; Accepted: 26 April 2024;References1. Bogdanov, A. & Yablonsky, D. Thermodynamically stable vortexes inmagnetically ordered crystals: mixed state of magnetics. J. Exp.Theor. Phys. 95, 178 (1989).2. Mühlbauer, S. et al. Skyrmion Lattice in a Chiral Magnet.Science 323,915 (2009).3. Tokura, Y. & Kanazawa, N. Magnetic SkyrmionMaterials.Chem. Rev.5, 2857 (2021).4. Binz, B., Vishwanath, A. & Aji, V. Theory of the Helical Spin Crystal: ACandidate for the Partially Ordered State of MnSi. 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Continuous-wave far-infraredESRspectrometer for high-pressuremeasurements.J.Magn. Reson. 195, 206 (2008).50. Ehlers, D. et al. Exchange anisotropy in the skyrmion host GaV4S8. J.Phys. Condens. Matter 29, 065803 (2017).51. Butykai, Á. et al. Squeezing the periodicity of Néel-type magneticmodulations by enhanced Dzyaloshinskii-Moriya interaction of 4delectrons. npj Quantum Mater. 7, 26 (2022).AcknowledgementsWe thank Taro Nakajima for support regarding resonant elastic x-ray scat-tering (REXS) experiments at beamline BL-3A of Photon Factory. AkikoKikkawa provided guidance regarding single crystal synthesis. Weacknowledge Ferenc Simon for fruitful discussions and technical support atEPFL, and Dieter Ehlers for providing his software to model the ferromag-netic resonance data. We are also grateful to Vladimir Tsurkan for polishingthe samples for FMR. REXS measurements at the Institute of MaterialStructure Science of the High Energy Accelerator Research Organization(KEK) and at SPring-8 BL19LXU were carried out under the approval of thePhoton Factory program advisory committee (Proposal No. 2020G665) andunder grant number 20210007, respectively. M.He. and H.-A. K.v.N.acknowledge funding within the joint RFBR-DFG research project contractNo. 19-51-45001 and KR2254/3-1. S.E. and Ma.Hi. benefited from JSPSKAKENHI Grant No. 22H04463 and 23H05431, while also acknowledginggrantsby theMurataScience Foundation, theYamadaScience Foundation,the Hattori Hokokai Foundation, the Iketani Science and TechnologyFoundation, the Mazda Foundation, the Casio Science Promotion Foun-dation, the Inamori Foundation, the Kenjiro Takayanagi Foundation, and theMarubun foundation through their Exchange Grant. This work is partiallyfunded by the Deutsche Forschungsgemeinschaft (DFG, GermanResearchFoundation) under project numbers 518238332 and TRR 360–492547816and by the Japan Science and Technology Agency via Core Research forEvolutional Science and Technology (CREST) Grant Nos. JPMJCR1874,JPMJCR20T1 (Japan), and by FOREST No. JPMJFR2238.Author contributionsMa.Hi., I.K., Y.To. and T.-h.A. conceived the project.Ma.Hi. synthesized andcharacterized the single crystals.M.He., B.G.S., H.-A.K.v.N. and L.F. carriedout electron spin-resonance experiments at EPFL. Ma.Hi., L.S., H.O. andY.Ta. performed REXS at SPring-8. Ma.Hi., L.S., S.G., K.K., H.S., H.N. andY.Y. carried out REXS measurements at BL-3A of Photon Factory (KEK).Ma.Hi. and S.E. were in charge of spin model calculations. Ma.Hi. wrote themanuscript in close collaboration with M.M.H. and S.E., and with contribu-tions and comments from all co-authors.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41535-024-00654-2.Correspondence and requests for materials should be addressed toMax Hirschberger.Reprints and permissions information is available athttp://www.nature.com/reprintsPublisher’s note Springer Nature remains neutral with regard tojurisdictional claims in published maps and institutional affiliations.Open Access This article is licensed under a Creative CommonsAttribution 4.0 International License, which permits use, sharing,adaptation, distribution and reproduction in anymedium or format, as longas you give appropriate credit to the original author(s) and the source,provide a link to the Creative Commons licence, and indicate if changeswere made. 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To view a copy of thislicence, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2024https://doi.org/10.1038/s41535-024-00654-2 Articlenpj Quantum Materials |            (2024) 9:45 8https://doi.org/10.1038/s41535-024-00654-2http://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/ Lattice-commensurate skyrmion texture in a centrosymmetric breathing kagome�magnet Results Observation of a commensurate skyrmion lattice (C-SkL) Single-ion anisotropy in Gd3Ru4Al12 Anisotropy and anharmonic distortion of proper screw and skyrmion�phases Competition of commensurate and incommensurate�phases Discussion Methods Cartographic projection Sample preparation and characterization Ferromagnetic resonance measurements Resonant x-ray scattering experiments (Gd-L2,3) Data availability References Acknowledgements Author contributions Competing interests Additional information