Very often the skyrmions form a triangular crystal in chiral magnets. Here we study the effect of itinerant electrons on the structure of skyrmion crystal (SkX) on triangular lattice using Kondo lattice model in the large coupling limit and treating the localized spins as classical vectors. To simulate the system, we employ hybrid Markov Chain Monte Carlo method (hMCMC) which includes electron diagonalization in each MCMC update for classical spins. We present the low temperature results for $12\times 12$ system at electron density $n=1/3$ which show a sudden jump in skyrmion number when we increase the hopping strength of the itinerant electrons. We find that this high skyrmion number SkX phase is stabilized by combined effects: lowering of density of states at electron filling $n=1/3$ and also pushing the bottom energy states further down. We show that these results hold for larger system using travelling cluster variation of hMCMC. We expect that itinerant triangular magnets might exhibit the possible transition between low density to high density SkX phases by applying external pressure.
Magnetic structures and their stability of the multiple helical spin density waves (MHSDW) and related magnetic skyrmions in itinerant electron system have been investigated on the basis of both the phenomenological and microscopic theories. It is shown by using the Ginzburg-Landau (GL) theory and the Application Visualization System (AVS) that the magnetic skyrmion structures emerge as a MHSDW and can be stabilized in itinerant electron system on the fcc lattice even if there is no Dzyaloshinskii-Moriya interaction. Moreover, by using the generalized Hartree-Fock theory for the Hubbard model,…
Magnetic skyrmions are swirling spin textures with topologically protected noncoplanarity. Recently, skyrmions with the topological number of unity have been extensively studied in both experiment and theory. We here show that a skyrmion crystal with an unusually high topological number of two is stabilized in itinerant magnets at zero magnetic field. The results are obtained for a minimal Kondo lattice model on a triangular lattice by an unrestricted large-scale numerical simulation and variational calculations. We find that the topological number can be switched by a magnetic field as $2\lef…
A skyrmion crystal (SkX) expressed as a multiple number of spiral modulations manifests itself not only in its peculiar magnetic texture but also in nontrivial transport properties originating from an emergent magnetic field. We here report our numerical results for multiple SkXs in a centrosymmetric tetragonal crystal system. By performing simulated annealing for an effective spin model for itinerant magnets, we find that three types of the SkXs with the skyrmion numbers of one and two, which are characterized by different superpositions of helices, are stabilized in the ground state, and are…
We theoretically investigate the origin of the square-type skyrmion crystal in centrosymmetric itinerant magnets, motivated from the recent experimental finding in GdRu$_2$Si$_2$ [N. D. Khanh $\mathit{et}$ $\mathit{al.}$, Nat. Nanotech. $\mathbf{15}$, 444 (2020)]. By simulated annealing for an effective spin model derived from the Kondo lattice model on a square lattice, we find that a square skyrmion crystal composed of a superposition of two spin helices is stabilized in a magnetic field by synergy between the positive biquadratic, bond-dependent anisotropic, and easy-axis anisotropic intera…
Magnetic skyrmions are promising spin textures for building next-generation magnetic memories and spintronic devices. Nevertheless, one of the major challenges in realizing skyrmion-based devices is the stabilization of ordered arrays of these spin textures in different geometries. Here we numerically study the skyrmion-skyrmion interaction potential that arises due to the dynamics of itinerant electrons coupled to the magnetic texture in a ferromagnetic background with racetrack geometry. We consider different topological textures (ferromagnetic (FM) and antiferromagnetic (AFM)), namely: skyr…
Topological spin textures, such as a skyrmion crystal, are a source of unusual physical phenomena owing to the interplay between magnetism and topology. Since physical phenomena depend on the topological property and the symmetry of underlying spin structures, the search for new topological spin textures and emergent phenomena is one of the challenges in condensed matter physics. In this letter, we theoretically explore new topological spin textures arising from the synergy between spin, charge, and sublattice degrees of freedom in an itinerant magnet. By performing simulated annealing for an …
Interplay between itinerant electrons and localized spins in itinerant magnets gives rise to a variety of noncoplanar multiple-$Q$ spin textures, such as the skyrmion, hedgehog, meron, and vortex. We elucidate that another type of multiple-$Q$ state consisting of collinear sinusoidal waves, a magnetic bubble crystal, appears at finite temperatures in a centrosymmetric itinerant electron system. The results are obtained for the classical Kondo lattice model with easy-axis single-ion anisotropy on a triangular lattice by a large-scale numerical simulation. We find that a finite-temperature topol…
We report our numerical results for the effect of magnetic anisotropy on a Skyrmion crystal with a high topological number of two, which was recently discovered in an itinerant electron model [R. Ozawa, S. Hayami, and Y. Motome, Phys. Rev. Lett. 118, 147205 (2017)]. By performing numerical simulations based on the kernel polynomial method and the Langevin dynamics for the Kondo lattice model on a triangular lattice, we find that the topological property remains robust against the single-ion anisotropy, while the magnetic texture is deformed continuously. The resultant spin structure is charact…
A mutual interplay between the charge and spin degrees of freedom in itinerant magnets leads to a plethora of topological spin textures, such as magnetic skyrmion and vortex crystals, in both centrosymmetric and noncentrosymmetric hosts. Meanwhile, their stabilization has been extensively studied in the system including the classical localized spins. We here study a realization of the skyrmion crystal in the centrosymmetric triangular-lattice Hubbard model, where the itinerant nature of electrons plays a more significant role. By performing the self-consistent mean-field calculations, we find …
Magnetic skyrmions were thought to be stabilised only in inversion-symmetry breaking structures, but skyrmion lattices were recently discovered in inversion symmetric Gd-based compounds, spurring questions of the stabilisationmechanism. A natural consequence of a recent theoretical proposal, a coupling between itinerant electrons and localised magnetic moments, is that the skyrmions are amenable to detection using even non-magnetic probes such as spectroscopic-imaging scanning tunnellingmicroscopy (SI-STM). Here SI-STM observations of GdRu$_2$Si$_2$ reveal patterns in the local density of stat…
Spin textures with nontrivial topology, such as vortices and skyrmions, have attracted attention as a source of unconventional magnetic, transport, and optical phenomena. Recently, a new generation of topological spin textures has been extensively studied in itinerant magnets; in contrast to the conventional ones induced, e.g., by the Dzyaloshinskii-Moriya interaction in noncentrosymmetric systems, they are characterized by extremely short magnetic periods and stable even in centrosymmetric systems. Here we review such new types of topological spin textures with particular emphasis on their st…
We propose a skyrmion liquid state for the non-Fermi liquid (NFL) phase in Ge-substituted YbRh$_{2}$Si$_{2}$, where skyrmions form their Fermi surface, argued to result from the strongly coupled nature between skyrmions and itinerant electrons. The fermionized skyrmion theory identifies the antiferromagnetic (AF) transition with the Lifshitz transition, where the quantum critical point (QCP) is characterized by the dynamical critical exponent $z = 2$. Nonlocal interactions between skyrmions allow a critical line above the AF QCP, which originates from the Kondo-coupling effect with itinerant e…
Hertz-Moriya-Millis theory with dynamical critical exponent $z = 2$ has been proposed to describe antiferromagnetic quantum criticality in itinerant electron systems. In this study we show that the dynamical critical exponent changes from $z = 2$ to $z = 3$ at low temperatures, which results from effective long-range interactions between spin fluctuations, generated by Fermi-surface fluctuations beyond the Eliashberg framework. We claim that the underlying physics for the $z = 3$ antiferromagnetic quantum criticality is the emergence of fermionized skyrmion excitations at low energies, which f…
Multiple-$Q$ magnetic states often induce nontrivial topological spin textures, such as a skyrmion and a hedgehog. We theoretically investigate yet another multiple-$Q$ state with topological defects, a meron-antimeron crystal (MAX), represented by a periodic array of the meron and antimeron with a half-integer skyrmion number. Performing simulated annealing for an effective spin model of noncentrosymmetric itinerant magnets on a triangular lattice, we show that rectangular-shaped and triangular-shaped MAXs are stabilized by the interplay between the biquadratic interaction arising from the sp…
We theoretically demonstrate microwave-induced dynamical switching of magnetic topology in centrosymmetric itinerant magnets by taking the Kondo-lattice model on a triangular lattice, which is known to exhibit two types of skyrmion lattices with different magnetic topological charges of |Nsk| = 1 and 2. Our numerical simulations reveal that intense excitation of a resonance mode with circularly polarized microwave field can switch the magnetic topology, i.e., from the skyrmion lattice with |Nsk| = 1 to another skyrmion lattice with |Nsk| = 2 or to a nontopological magnetic order with |Nsk| = 0…
We theoretically investigate multiple-$Q$ spin textures, which are composed of superpositions of spin density waves with different wave numbers, for an effective spin model of centrosymmetric itinerant magnets. Our focus is on the interplay between biquadratic interactions arising from the spin-charge coupling and magnetic anisotropy caused by the spin-orbit coupling. Taking into account two types of the magnetic anisotropy, single-ion anisotropy and bond-dependent anisotropy, we elucidate magnetic phase diagrams for an archetypal triangular-lattice system in the absence and presence of an ext…
Multiple spin density waves and related vortex-type magnetic structures in the itinerant electron system with inversion symmetry have been investigated on the basis of the Hubbard model and the generalized Hartree-Fock approximation combined with the recursion method. Possible magnetic phase diagram on the fcc lattice is presented in the space of the Coulomb interaction strength and the electron occupation number. It is found that the 2Q multiple helical spin density waves (SDW) with half-skyrmion half-antiskyrmion paired structure, the 3Q multiple helical SDW with 3 dimensional type vortex st…
We theoretically study the stability of magnetic skyrmion crystals (SkXs) in chiral cubic antiferromagnets with the EuPtSi hosting unconventional nanometric SkXs in mind. By performing numerical simulations for a minimal effective spin model with the long-range Dzyaloshinskii-Moriya interaction and the multiple-spin interaction arising from itinerant nature of electrons, we clarify two key ingredients for the emergence of the SkXs in EuPtSi: One is the low-symmetric ordering vectors that lead to the field-direction sensitive SkXs in an external magnetic field, while the other is the synergy be…
Motivated by recent experiments in Gd$_3$Ru$_4$Al$_{12}$, we demonstrate the emergence of atomic scale Skyrmions in interacting spins on a breathing kagome lattice with competing nearest neighbor ferromagnetic and next nearest neighbor antiferromagnetic exchange interactions. In the presence of an applied longitudinal magnetic field, the ground state magnetic order evolves from a helical phase at low fields to a Skyrmion phase at intermediate fields before finally entering a polarized phase at high fields. The size of each Skyrmion spans only two unit cells of the lattice, in contrast to tens …
We propose an experiment to explore the magnetic phase transitions in interacting fermionic Hubbard systems, and describe how to obtain the ferromagnetic phase diagram of itinerant electron systems from these observations. In addition signatures of ferromagnetic correlations in the observed ground states are found: for large trap radii (trap radius $R_T > 4$, in units of coherence length $ξ$), ground states are topological in nature -- a "skyrmion" in 2D, and a "hedgehog" in 3D.
We theoretically study spin and charge excitations of skyrmion crystals stabilized by conduction-electron-mediated magnetic interactions via spin-charge coupling in a centrosymmetric Kondo-lattice model by large-scale spin-dynamics simulations combined with the kernel polynomial method. We reveal clear segregation of spin and charge excitation channels and nonreciprocal nature of the spin excitations governed by the Fermi-surface geometry, which are unique to the skyrmion crystals in centrosymmetric itinerant hosts and can be a source of novel physical phenomena.
A skyrmion crystal has internal degrees of freedom in terms of vorticity and helicity, which gives rise to fascinating types of skyrmion phases (Néel, Bloch, and anti-skyrmion). We show how the degeneracy lifting in tetragonal skyrmion crystals occurs in centrosymmetric itinerant magnets by focusing on microscopic interactions through the Fermi surface instability. We demonstrate that four types of skyrmion crystals can be realized according to the signs of the anisotropic Ruderman-Kittel-Kasuya-Yosida interaction, which are sensitive to band structures, electron fillings, atomic orbitals, and…
Noncollinear and noncoplanar magnetic textures including skyrmions and vortices act as emergent electromagnetic fields and give rise to novel electronic and transport properties. We here report a unified understanding of noncoplanar magnetic orderings emergent from the spin-charge coupling in itinerant magnets. The mechanism has its roots in effective multiple spin interactions beyond the conventional Ruderman-Kittel-Kasuya-Yosida (RKKY) mechanism, which are ubiquitously generated in itinerant electron systems with local magnetic moments. Carefully examining the higher-order perturbations in t…
Magnetic materials with competing magnetocrystalline anisotropy and dipolar energies can develop a wide range of domain patterns, including classical stripe domains, domain branching, as well as topologically trivial and non-trivial (skyrmionic) bubbles. We image the magnetic domain pattern of Fe$_3$Sn$_2$ by magnetic force microscopy (MFM) and study its evolution due to geometric confinement, magnetic fields, and their combination. In Fe$_3$Sn$_2$ lamellae thinner than 3 $μ$m, we observe stripe domains whose size scales with the square root of the lamella thickness, exhibiting classical Kitte…
We performed a systematic study of the temperature- and field-dependence of magnetization and resistivity of Gd2PdSi3, which is a centrosymmetric skyrmion crystal. While the magnetization behavior is consistent with the reported phase diagram based on susceptibility, we show that a phase diagram can also be constructed based on the anomalous magnetoresistance with one-to-one correspondence among all the features. In addition, the crossover boundary into the field-induced ferromagnetic state is also identified. Our results suggest that the ferromagnetic spin fluctuations above the Néel temperat…
Analytic properties of the multiple spin density waves showing vortex structures, which have recently been found in the fcc Hubbard model using the generalized Hartree-Fock (GHF) approximation, have been investigated. It is shown that the 2Q multiple helical spin density waves (SDW) consist of two types of half-skyrmion vortices and two types of antivortices. In the 3Q multiple helical SDW, the half-skyrmion vortex structures with the "westerlies" are shown to be twisted along the axis perpendicular to the vortex plane. The 12Q multiple SDW (12QMSDW) are analyzed as a superposition of the 2-4Q…
Skyrmions represent topologically stable field configurations with particle-like properties. We used neutron scattering to observe the spontaneous formation of a two-dimensional lattice of skyrmion lines, a type of magnetic vortices, in the chiral itinerant-electron magnet MnSi. The skyrmion lattice stabilizes at the border between paramagnetism and long-range helimagnetic order perpendicular to a small applied magnetic field regardless of the direction of the magnetic field relative to the atomic lattice. Our study experimentally establishes magnetic materials lacking inversion symmetry as an…
Skyrmions are small magnetic quasiparticles, which are uniquely characterized by their topological charge and their helicity. In this Rapid Communication, we show via calculations how both properties can be determined without relying on real-space imaging. The orbital magnetization and topological Hall conductivity measure the arising magnetization due to the circulation of electrons in the bulk and the occurrence of topologically protected edge channels due to the emergent field of a skyrmion crystal. Both observables quantify the topological Hall effect and distinguish skyrmions from antisky…
The tensile strain is a promising tool for creation and manipulation of magnetic solitonic textures in the chiral helimagnets via tunable control of magnetic anisotropy and Dzyaloshinskii-Moriya interaction. Here, by using the in-situ resonant small-angle x-ray scattering we demonstrate that the skyrmion and chiral soliton lattices can be achieved as metastable states in FeGe lamella as distinct states or even simultaneously by combining the tensile strain and magnetic fields in various orientations with respect to the deformation. The small-angle scattering data are discussed in the frame of …
Comment on "Theory of helimagnons in itinerant quantum systems" by D.Belitz, T.K.Kirpatrick and A.Rosch and "Cubic magnets with Dzyaloshinskii-Moriya interaction at low temperatures" by S.V.Maleyev
We report that in a $β$-Mn-type chiral magnet Co$_9$Zn$_9$Mn$_2$, skyrmions are realized as a metastable state over a wide temperature range, including room temperature, via field-cooling through the thermodynamic equilibrium skyrmion phase that exists below a transition temperature $T_\mathrm{c}$ $\sim$ 400 K. The once-created metastable skyrmions survive at zero magnetic field both at and above room temperature. Such robust skyrmions in a wide temperature and magnetic field region demonstrate the key role of topology, and provide a significant step toward technological applications of skyrmi…
New mechanism of magnetoresistivity in itinerant metamagnets with a structural disorder is introduced basing on analysis of experimental results on magnetoresistivity, susceptibility, and magnetization of structurally disordered alloys (Y$_{1-x}$Gd$_{x}$)Co$_{2}$. In this series, YCo$_{2}$ is an enhanced Pauli paramagnet, whereas GdCo$_{2}$ is a ferrimagnet (T$_{\rm c}$=400 K) with Gd sublattice coupled antiferromagnetically to the itinerant Co-3d electrons. The alloys are paramagnetic for $x < 0.12$. Large positive magnetoresistivity has been observed in the alloys with magnetic ground state …
Compared with the traditional magnetic bubble, skyrmion has smaller size, better stability therefore is considered as a very promising candidate for future memory devices. When skyrmions are manipulated, erased and created, the density of skyrmions can be varied, however the relationship between the radii and the densities of skyrmions needs more exploration. In this paper, we study this problem both theoretically and by using the lattice simulation. The average radius of skyrmions as a function of material parameters, the strength of external magnetic field and the density of skyrmions is obt…
Skyrmions are nanometric spin whirls that can be stabilized in magnets lacking inversion symmetry. The properties of isolated skyrmions embedded in a ferromagnetic background have been intensively studied. We show that single skyrmions and clusters of skyrmions can also form in the helical phase and investigate theoretically their energetics and dynamics. The helical background provides natural one-dimensional channels along which a skyrmion can move rapidly. In contrast to skyrmions in ferromagnets, the skymion-skyrmion interaction has a strong attractive component and thus skyrmions tend to …
Dielectric properties were investigated under various magnitudes and directions of magnetic field (H) for a chiral magnetic insulator Cu2OSeO3. We found that the skyrmion crystal induces electric polarization (P) along either in-plane or out-of-plane direction of the spin vortices depending on the applied H-direction. The observed H-dependence of P in ferrimagnetic, helimagnetic, and skyrmion crystal state can be consistently described by the d-p hybridization model, highlighting an important role of relativistic spin-orbit interaction in the magnetoelectric coupling in Cu2OSeO3. Our analysis …
Chiral nematic liquid crystals sometimes form blue phases characterized by spirals twisting in different directions. By combining model calculations with neutron-scattering experiments, we show that the magnetic analogue of blue phases does form in the chiral itinerant magnet MnSi in a large part of the phase diagram. The properties of this blue phase explain a number of previously reported puzzling features of MnSi such as partial magnetic order and a two-component specific-heat and thermal-expansion anomaly at the magnetic transition.
Magnetic skyrmions are vortex-like topological spin textures often observed to form a triangular-lattice skyrmion crystal in structurally chiral magnets with Dzyaloshinskii-Moriya interaction. Recently $β$-Mn structure-type Co-Zn-Mn alloys were identified as a new class of chiral magnet to host such skyrmion crystal phases, while $β$-Mn itself is known as hosting an elemental geometrically frustrated spin liquid. Here we report the intermediate composition system Co$_7$Zn$_7$Mn$_6$ to be a unique host of two disconnected, thermal-equilibrium topological skyrmion phases; one is a conventional s…
The microscopic magnetism in the helical, the conical and the ferro-magnetically polarized phases in an itinerant helical magnet, MnSi, has been studied by an extended 29Si NMR at zero field and under external magnetic fields. The temperature dependence of staggered moment, M_Q(T), determined by the 29Si NMR frequency, nu(T), and nuclear relaxation rate, 1/T_1(T) is in general accord with the SCR theory for weak itinerant ferromagnetic metals and its extension. The external field dependence of resonance frequency, nu(H), follows a vector sum of the contributions from atomic hyperfine and macro…
We study the double-exchange model at half-filling with competing superexchange interactions on a triangular lattice, combining exact diagonalization and Monte-Carlo methods. We find that in between the expected itinerant ferromagnetic and $120^{\circ}$ Yafet-Kittel phases a robust scalar-chiral, insulating spin state emerges. At finite temperatures the ferromagnet - scalar-chiral quantum critical point is characterized by anomalous bad-metal behavior in charge transport as observed in frustrated itinerant magnets R$_2$Mo$_{2}$O$_7$.
Chiral magnets support topological skyrmion textures due to the Dzyaloshinskii-Moriya spin-orbit interaction. In the presence of a sufficiently large applied magnetic field, such skyrmions are large amplitude excitations of the field-polarized magnetic state. We investigate analytically the interaction between such a skyrmion excitation and its small amplitude fluctuations, i.e., the magnons in a clean two-dimensional chiral magnet. The magnon spectrum is found to include two magnon-skyrmion bound states corresponding to a breathing mode and, for intermediate fields, a quadrupolar mode, which …
Magnetic skyrmions are vortex-like topological spin textures often observed in structurally chiral magnets with Dzyaloshinskii-Moriya interaction. Among them, Co-Zn-Mn alloys with a $β$-Mn-type chiral structure host skyrmions above room temperature. In this system, it has recently been found that skyrmions persist over a wide temperature and magnetic field region as a long-lived metastable state, and that the skyrmion lattice transforms from a triangular lattice to a square one. To obtain perspective on chiral magnetism in Co-Zn-Mn alloys and clarify how various properties related to the skyrm…
In a recent study the coalescence of magnetic skyrmions was observed in a metallic chiral magnet Fe0.5Co0.5Si when the skyrmion phase is destroyed, and numerical simulations demonstrated the existence of a monopole at the merging point of two skyrmion lines. The exchange interaction between such magnetic textures and the conduction electrons can be described by emergent electromagnetism. In this paper, we investigate the effect of a skyrmions-merging process on conduction electrons by calculating induced electric currents. Here, in addition to the exchange interactions, we consider the antisym…
The magnetic skyrmions generated in a centrosymmetric crystal were recently first discovered in Gd2PdSi3. In light of this, we observe the electronic structure by angle-resolved photoemission spectroscopy (ARPES) and unveil its direct relationship with the magnetism in this compound. The Fermi surface and band dispersions are demonstrated to have a good agreement with the density functional theory (DFT) calculations carried out with careful consideration of the crystal superstructure. Most importantly, we find that the three-dimensional Fermi surface has extended nesting which matches well the…
The Shastry-Sutherland model and its generalizations have been shown to capture emergent complex magnetic properties from geometric frustration in several quasi-two-dimensional quantum magnets. Using an $sd$ exchange model, we show here that metallic Shastry-Sutherland magnets can exhibit a topological Hall effect driven by magnetic skyrmions under realistic conditions. The magnetic properties are modeled with competing symmetric Heisenberg and asymmetric Dzyaloshinskii-Moriya exchange interactions, while a coupling between the spins of the itinerant electrons and the localized moments describ…
Effect of uniaxial stress on helimagnetic phases in the square-lattice itinerant magnet EuAl$_{4}$
Institute for Solid State Physics, University of Tokyo（东京大学固体物理研究所）；RIKEN Center for Emergent Matter Science (CEMS)（理化学研究所新兴物质科学中心）；Department of Physics, Tokyo Metropolitan University（东京都立大学物理学系）；Institute of Materials Structure Science, High Energy Accelerator Research Organization（高能加速器研究机构材料结构科学研究所）；Department of Advanced Materials Science, The University of Tokyo（东京大学先进材料科学系）；Research Center for Advanced Science and Technology, University of Tokyo（东京大学先进科学技术研究中心）；Department of Applied Physics, The University of Tokyo（东京大学应用物理学系）
Magnetic skyrmions in 2D chiral magnets are in general stabilized by a combination of Dzyaloshinskii-Moriya interaction and external magnetic field. Here, we show that skyrmions can also be stabilized in twisted moiré superlattices in the absence of an external magnetic field. Our setup consists of a 2D ferromagnetic layer twisted on top of an antiferromagnetic substrate. The coupling between the ferromagnetic layer and the substrate generates an effective alternating exchange field. We find a large region of skyrmion crystal phase when the length scales of the moiré periodicity and skyrmions …
We study the quantum mechanics of magnetic skyrmions in the vicinity of the skyrmion-crystal to ferromagnet phase boundary in two-dimensional magnets. We show that the skyrmion excitation has an energy dispersion that splits into multiple bands due to the combination of magnus force and the underlying lattice. Condensation of the skyrmions can give rise to an intermediate phase between the skyrmion crystal and ferromagnet: a quantum liquid, in which skyrmions are not spatially localized. We show that the critical behavior depends on the spin size $S$ and the topological number of the skyrmion.…
The centrosymmetric tetragonal itinerant magnet EuAl$_{4}$ exhibits an intricate magnetic phase diagram including rhombic and square skyrmion-lattice (SkL) phases in the external magnetic field. Here, we report a multi-axis dilatometric investigation of EuAl$_{4}$ by means of a newly designed fiber-Bragg-grating technique complemented by a resonant x-ray scattering experiment, revealing anisotropic magnetostriction and magnetovolume effect associated with successive phase transitions. The rhombic and square SkL phases are found to possess $\sim$0.10% and $\sim$0.03% orthorhombic structural dis…
