Abstract
The mathematical concept of topology has brought about significant advantages that allow for a fundamental understanding of the underlying physics of a system. In magnetism, the topology of spin order manifests itself in the topological winding number which plays a pivotal role for the determination of the emergent properties of a system. However, the direct experimental determination of the topological winding number of a magnetically ordered system remains elusive. Here, we present a direct relationship between the topological winding number of the spin texture and the polarized resonant Xray scattering process. This relationship provides a onetoone correspondence between the measured scattering signal and the winding number. We demonstrate that the exact topological quantities of the skyrmion material Cu_{2}OSeO_{3} can be directly experimentally determined this way. This technique has the potential to be applicable to a wide range of materials, allowing for a direct determination of their topological properties.
Introduction
In a manybody system, a local order parameter can be assigned to individual entities, and by considering interactions among them, emergent phases and novel physical properties may evolve. The possible values of the order parameters constitute the order parameter space, which can be described in the framework of topology^{1,2}. In magnetism, the spins are the elementary entities, and the order parameter is the magnetization vector m. Its magnitude can be taken as a constant, that is, its three components satisfy , where M_{S} is the saturation magnetization. Therefore, the order parameter space is the surface of a threedimensional unit sphere, which is described by the homotopy group π_{2}(S^{2}) for a twodimensional physical space (x, y) (ref. 2). Different homotopy classes with distinct topological properties can be quantified based on the winding number N, which is an integer that counts the number of times the physical space fully covers the order parameter space^{3}. It is defined as
Recently, it has been demonstrated that noncentrosymmetric helimagnetic materials carry N=1 magnetic skyrmions^{4,5,6}, which leads to emergent phenomena, including novel magnetoelectrical transport effects (that is, the topological Hall effect^{7,8,9}, skyrmion motion induced by ultralow current densities^{10,11,12,13}, and emergent electromagnetic fields^{14,15,16}), as well as new spin dynamic properties^{17,18,19}. Utilizing this nontrivial topological order, advanced spintronics applications have been devised^{20,21,22,23,24}. More recently, several candidates with N=2 have also been discovered^{25,26}, suggesting that other elements from the π_{2}(S^{2}) group may exist in nature as well.
While the significance of the topological properties of ordered systems is being recognized more and more, the experimental determination of the winding number for spinordered media remains challenging. Commonly, the winding number is determined by comparing a microscopic image of the magnetization state with theoretical model calculations, making it a rather indirect process that has no unique answer^{21,22,23,24,27,28,29}. Most importantly, the established magnetic imaging techniques only give a partial picture of the local magnetization vector, as they are both limited in threedimensional sensitivity and lateral resolution. For example, Lorentz transmission electron microscopy (LTEM) has been the most common technique which is used to infer the topological winding number from a magnetization map^{27}. In most of the LTEM experiments, the magnetization configuration is obtained via an indirect transportofintensity equation simulation process, and, most importantly, the information of the outofplane spin component is missing^{27}. Consequently, LTEM is not a direct experimental method^{30} to determine the topological winding number (see Supplementary Note 3 for a detailed discussion). On the other hand, the presence of skyrmions leads to measurable signals in electric transport, that is, the topological Hall effect^{9}. Nevertheless, other noncollinear magnetic structures, which are not related to skyrmions, can also give rise to a measurable topological Hall effect^{7,31}, rendering transport measurements less ideal for the unambiguous determination of topological properties.
Here, we show that the winding number N can be unambiguously identified by utilizing the sensitivity of the light polarization to the magnetic order at resonant elastic Xray scattering (REXS) condition, referred as polarizationdependent REXS.
Results
Representation of a skyrmion with winding number N
A general magnetic skyrmion structure can be obtained by mapping the order parameter space to the physical space, described in the twodimensional polar coordinates ρ (radial coordinate) and Ψ (azimuthal angle), in the following way^{4,32}:
where the boundary conditions are defined such that the magnetization points up in the centre of the twodimensional physical space and down at the boundary. The function Θ(ρ) describes the radial profile of the outofplane component of the magnetization, starting from the centre and extending to the boundary; χ is the helicity, defined in the range of (−π, π]; and λ takes the values of ±1, describing the polarity of the skyrmion^{4}. For example, an N=1 skyrmion appears as a vortexlike texture. If the core magnetization points up (that is, λ=1), a χ=−π/2 skyrmion has a clockwise rotation sense when viewing from the top. The entire texture thus carries negative chirality, defined by C=sgn(λχ). Analogously, a χ=π/2 skyrmion carries positive chirality. On the other hand, for χ=0 and χ=π skyrmions, socalled Néeltype skyrmions, there is no chirality present^{33}.
Another, more illustrative way to interpret equation (2), is to construct an Nskyrmion texture by assigning a onedimensional helix spin profile to a radial chain in physical space, and by repeating the process for all azimuthal angles Ψ, in the range from 0° to 360°, thereby mapping out the entire twodimensional physical space. This concept is illustrated in Fig. 1a–c. Starting from the line for Ψ=0° that is parallel to x axis, a standard harmonic helix structure is assigned. Subsequent Ψ angles get a helix assigned that is rotated by NΨ from Ψ=0°. As as result, when the twodimensional physical space is fully sampled, the order parameter space will have been mapped out N times. The exact structure of such harmonic helices does not affect the topological properties of the system, nor our measurement principle, as will be shown below. Using this onedimensional helix approximation, an analytical solution for the polarizationdependent REXS process can be obtained that is explicit in N.
Topology determination principle
The measurement geometry for determining N is illustrated in Fig. 1d. The incident and scattered Xray wavevectors are denoted as k_{i} and k_{s}, with the incident angle α, which satisfies the diffraction condition Q=k_{s}−k_{i}. The incident Xrays can be linearly polarized with the polarization angle β. We define β=0° corresponding to σpolarization, while β=90° corresponds to πpolarization. Alternatively, the light can be circularly polarized. Here, we define the circular dichroism (CD) signal as the difference of the scattering crosssections for leftcircularly and rightcircularly polarized incident light (at the same diffraction condition).
We demonstrate our new experimental principle for the determination of N on the magnetic skyrmion system Cu_{2}OSeO_{3}. This material carries an incommensurate, hexagonal lattice with an N≠0 topological entity motif^{34,35,36}. The modulation wavevector is ∼0.0158, r.l.u., and the motif’s periodic lattice lies in the x–yplane when the required magnetic field is along the z direction^{37} (see Fig. 1). Using the onedimensional helix approximation construction, an analytical form of the resonant magnetic diffraction crosssection I and the CD crosssection I_{CD} can be derived as a function of N (see Methods for the derivation)
and
where and Y are constants. The arbitrary phase parameters Φ_{1} and Φ_{2} can be chosen to adapt to other spin configurations with the same winding number, however, which deviate from the ‘standard’ configuration as constructed in Fig. 1a. These two relationships can be interpreted in the following way, which form the core of our experimental method for the determination of the winding number: In case of an odd winding number, N equals to the periodicity of the CD signal, while Ψ covers the full range from 0° to 360°. N is also equal to half the number of peaks in the polarizationazimuthal map (PAM; see below). For an even winding number, no CD signal is observed. The case of noninteger winding numbers is discussed in Supplementary Note 2.
Numerical results
Figure 2e–h shows the numerical calculation results for the CD crosssection for different topological spin motifs. Equation (4) can be represented by the CD amplitude as a function of a closedloop in reciprocal space. Each reciprocal space point (, ) on the loop (shown in red) corresponds to one azimuthal angle at which the diffraction condition for the modulation wavevector for Ψ is met. Therefore, according to equation (4), the CD intensity varies as a function of Ψ, with a periodicity that is equal to N if the spin winding is an odd number. For even winding numbers, such as N=2, there is no CD. This can be understood by treating a N=2 skyrmion as two N=1 skyrmions pulled together^{2} (see Fig. 2b). These two N=1 skyrmions have opposite chirality, thus there is no global chirality of this spin configuration. As CD is sensitive to chiral structures^{38}, the total CD for this state is zero. This also applies for other even winding number systems.
On the other hand, by varying the linear polarization β of the incident light from 0° to 180°, one can measure the polarizationdependent scattering intensity at each Ψ. By covering Ψ in the range from 0° to 360°, a PAM is obtained. The PAM plot in Fig. 2i–l shows humplike, twodimensional peaks of equal height. The peaks appear around β≈90°, and modulate along Ψ. The periodicity of the PAM signal is twice that of the winding number. This PAM feature is consistent with the analytical solution described by equation (3).
If both the CD and PAM data can be fitted by the two equations (3) and (4) simultaneously, the winding number can be unambiguously determined. If the detailed spin structure of the motif varies, such that the exact mapping from physical space to order parameter space changes within the same homotopy class, the corresponding shapes of the CD and PAM signals only undergo a linear shift, while the periodicities do not change (see Supplementary Note 1). Therefore, the topological robustness is also reflected in this type of Xray scattering measurement.
Experimental demonstration
To demonstrate the measurement principle, we performed experiments on singlecrystalline Cu_{2}OSeO_{3} with the setup sketched in Fig. 1d. At 57 K, and in an applied magnetic field of 32 mT, the skyrmion lattice phase emerges, manifesting itself as a hexagonal lattice of N=1 topological motifs. The lattice gives rise to the sixfoldsymmetric diffraction pattern in reciprocal space, shown in Fig. 3a. The six sharp firstorder magnetic peaks correspond to the ‘unit cell’ of the skyrmion lattice, with one of them locked along h, that is, the [100] crystallographic direction in real space. This is due to the higherorder magnetic anisotropy of the material^{34,36,37}. Note that the coordinates (q_{x}, q_{y}) used here, as defined in Fig. 1d, are independent of the crystallographic directions. Therefore, by rotating Ψ, the same modulation wavevectors rotate accordingly in the coordinate system (see Fig. 3b,c). The measured CD intensity as a function of Ψ (see Fig. 3d) shows exactly one period when the Xrays map the physical space once, suggesting that the skyrmion motif has a winding number of N=1. Moreover, as shown in Fig. 3e,f, the PAM is in excellent agreement with the theoretical calculations presented in Fig. 2i. The equal height of the two humps confirms the N=1 topology of this material.
Discussion
Another type of N=1 system, which does not carry chirality, is the socalled Néeltype skyrmion^{33} (see Fig. 4a). Its spin texture has a different appearance; however, it is topologically equivalent to the other skyrmion form. Consequently, the CD profile in Fig. 4b shows the same periodicity; however, a constant phase shift, as compared with Fig. 2e, appears. The phase shift, on the other hand, is due to the different mapping of the spin configuration under a continuous transformation. The analytical solution takes the value of Φ_{1}=π/2 for equation (4). The same behaviour is found for the PAM, as shown in Fig. 4c, for which the shape and height of the two humps are essentially the same as in Fig. 2i; however, the entire pattern undergoes a linear shift along Ψ. The analytical solution takes the value of Φ_{2}=π/6 for equation (3). Moreover, if the winding number is negative (see Fig. 4d), the CD signal still shows the same periodicity, that is, it does not distinguish between N and −N. However, the appearance of the humps is fundamentally different (compare Fig. 4f with Fig. 4c).
In summary, we have demonstrated that for a longwavelength magnetically ordered system, the topological winding number of the motif can be unambiguously determined by polarized Xrays. First, our polarizationdependent REXS strategy is a direct measurement method as the winding number is naturally encoded in the underlying physics of the light–matter interaction, and is explicit in the measurement principle, expressed in equation (3) and (4). Second, although we used resonant soft Xray diffraction for the demonstration of the measurement principle, the fundamentally same theory, with slight modifications, can be applied to the hard Xray wavelength regime as well. It can further be expanded to nonresonant magnetic Xray scattering by adding certain corrections. Third, this experimental technique can be applied to a wide range of magnetic systems, including both metallic and insulating materials, as well as other genres of materials that host topologically ordered spin systems, making it a general experimental principle.
Methods
Polarizationdependent resonant magnetic Xray scattering
For the derivation of the polarizationdependent REXS signal of an Nskyrmion system, we start from the basic resonant Xray scattering process from chiral magnets^{39,40,41,42}. For a single magnetic ion at site n carrying a moment m_{n} the resonant scattering form factor in the electricdipole approximation takes the form:
where and are the polarization unit vectors of the incident and outgoing Xrays, and the asterisks denotes the complex conjugate. The and terms are the charge and the linear magnetic part of the energydependent resonance amplitude, respectively. The term describes the anomalous charge scattering at resonance, which is added to the Thomson scattering part. The term describes resonant magnetic scattering, which can be of the same order of magnitude as charge scattering. In equation (5), we have neglected the term that is quadratic in the magnetization as it is much smaller than the leading terms, and which gives rise to higherorder effects.
In the first Born approximation, the diffraction intensity for the scattering vector Q=k_{s}−k_{i} from a periodic lattice with sites r_{n} can be written as
The complex amplitudes and are energydependent. Here we take them as constant, since in our numerical calculations the photon energy is not varied.
The coordinate system used to carry out the polarizationdependent study of the scattering crosssection is shown in Fig. 1b. The Cartesian coordinates are determined by the scattering plane (containing Q), that is, the x–zplane in this case. The y axis is perpendicular to this plane. This defines the three components for the magnetization vectors, as well as the reciprocal space coordinates (q_{x}, q_{y}, q_{z}). Thus, k_{i}=k(cos α, 0, sin α), k_{s}=k(cos α, 0, −sin α), k_{s} × k_{i}=k(0, −2 cos α sin α, 0), where the magnitude of the Xray wavevector, k, relates to the photon energy [k=2π/λ=E/(ħc)].
In the PoincaréStokes representation, the polarization of the incident Xrays is characterized by P=(P_{0}, P_{1}, P_{2}, P_{3}). For left and rightcircularly polarized light, P_{0}=1, P_{1}=P_{2}=0, P_{3}=±1. For linearly polarized light, P_{0}=1, P_{1}=cos 2β, P_{2}=sin 2β, P_{3}=0.
For a magnetic system carrying incommensurate magnetic modulations, magnetic diffraction occurs as satellites q surrounding the structure peak G, so that Q=G+q. Therefore, the charge and magnetic part of the diffraction can be separated, and no chargemagnetic interference term is expected. Consequently, at the diffraction condition for the periodic magnetic structure, the scattering intensity is described by the magnetic part (that is, second term of equation (5)). It is straightforward, but rather tedious, to derive the intensity for the magnetic scattering, which is given by^{41}:
where F_{1} is the energydependent resonant term, and M(Q) is the Fourier transform of the realspace magnetic moment modulation m(r) at Q. Note that k_{i} and k_{s} are tuned to fulfil the diffraction condition for Q.
Xray polarization dependence of the winding number
As shown in Fig. 1a, a onedimensional properscrew helix pitch^{43}, otherwise called ‘Blochtype’ helix, propagating along x, can be written as
where q_{h} is the helix propagation wavevector. We define a base position of the helix such that q_{h} is along the x axis, this also corresponds to Ψ=0°. While the Xrays probe the physical space at an finite angle Ψ, the order parameter space magnetization profile rotates the base helix position specified by equation (8) by NΨ within the q_{x}–q_{y}plane. By applying the rotation matrix = to equation (8), the rotated magnetic structure becomes
To meet the diffraction condition for Q=G+q_{h} at Ψ, one has to bring Q into the scattering plane. In a common fourcircle diffractometer, this is achieved by compensating the diffraction offset with the other two rotation axes, that is, the α axis, and the κ axis, which is perpendicular to both the α and Ψ axes. As a consequence, the components of the magnetic structure transform into:
where and are the rotation matrices corresponding to the κ and α rotation axes, and the combination of the two rotations brings Q into the scattering plane for the diffraction condition.
However, it is essential to note that this change would be negligible for most of the longwavelength modulated magnetic structures. For example, Cu_{2}OSeO_{3} has qa=0.0158 (ref. 37), where a is the lattice constant. Therefore, for G=(0, 0, 1), Q=G+q_{h}, the change of α is less than 0.9° for all Ψ angles. This makes and . Therefore, the longwavelength approximation suggests , as well as M(Q)≈M(q_{h}), and we can take one angle α for the diffraction condition of all Ψ positions.
Thus, the Fourier transform of equation (9) at the diffraction condition of q_{h} takes the form
Inserting equation (11) into equation (7), and evaluating the expressions described above, the CD profile is obtained as
where . However, note that the CD intensity is zero for even values of N. This condition is not captured by the analytical relationship of equation (12); however, it can be generalized from the numerical calculations. The reason why the CD vanishes is discussed in the main text, and is based on the assumption that, for an example, an N=2 skyrmion can be considered as two N=1 skyrmions pulled together^{2}. The ‘chirality cancelling’ effect does not occur for odd winding number motifs; however, it exists for all even winding numbers. Moreover, as will be discussed shortly, adding another phase factor Φ_{1} is also necessary for generalizing the CD relationship to Nskyrmions.
The linear polarization dependence can be derived as
Equations (12) and (13) are the foundation of our measurement principle, and are derived based on a standard onedimensional helix structure. Therefore, this analytical form is only valid for ‘standard’ types of spin configurations, for an example, N=1 chiral skyrmions with χ=±π/2. However, in principle, there is an infinite number of homotopies for a certain winding number, that is, the same topological property will always hold if continuous transformations are acting on a ‘standard’ skyrmion configuration, as we have used and derived so far. For example, if the onedimensional helix takes other forms, such as a cycloidal type structure^{33}, the overall spin texture will change while the winding number remains invariant. As shown in the Supplementary Note 1, this degree of freedom is dealt with by adding a phase factor to equations (12) and (13), which makes the measurement principle generally valid for all cases.
Numerical calculations
Numerical calculations were carried out using the materials parameters of Cu_{2}OSeO_{3}, that is, a helix pitch of 60 nm. This leads to a skyrmion coretocore distance of ∼69.28 nm, as well as a wavevector of ∼0.015 r.l.u. Resonant Xray scattering at the Cu L_{3} edge with a photon energy of 931.25 eV gives k=2π × 4.7187, nm^{−1}, with α≈48.24° for the (0, 0, 1) diffraction peak. In the calculation, F_{1} and M_{S} are kept constant as the CD profile and PAM are measured for the same photon energy and temperature. For detailed numerical results, please see Supplementary Note 1.
REXS
Resonant soft Xray scattering experiments were carried out in the RASOR diffractometer on beamline I10 at the Diamond Light Source (UK). Single crystals of Cu_{2}OSeO_{3} are precharacterized by Xray diffraction and electron backscattering diffraction to confirm the crystalline quality and single chirality. Magnetometry measurements were performed to map out the magnetic phase diagram. The polished crystal surface was (001)oriented for the subsequent resonant Xray scattering measurements.
The incident soft Xray beam with variable polarization was tuned to the Cu L_{3} edge. The experimental geometry is shown in Fig. 1d. The scattered beam is captured by either a CCD camera or a photodiode point detector. The modulated magnetic structure leads to satellites surrounding the structural peaks in reciprocal space. Further details about the experimental methods on resonant soft Xray scattering can be found in refs 37, 42. Polarizationdependent measurements are performed by varying the incident light polarization, while measuring the scattering intensities for different diffraction conditions for varying Ψ.
Data availability
The data that support the findings of this study are available from the corresponding author on request.
Additional information
How to cite this article: Zhang, S. L. et al. Direct experimental determination of the topological winding number of skyrmions in Cu_{2}OSeO_{3}. Nat. Commun. 8, 14619 doi: 10.1038/ncomms14619 (2017).
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Acknowledgements
The resonant soft Xray scattering experiments were carried out on beamline I10 at the Diamond Light Source, UK, under proposals SI11784 and SI12958. S.L.Z. and T.H. acknowledge financial support by the Semiconductor Research Corporation (SRC) and EPSRC (EP/N032128/1). We thank H. Berger for providing the crystal and A. Bauer and C. Pfleiderer for fruitful discussions.
Author information
Affiliations
Clarendon Laboratory, Department of Physics, University of Oxford, Parks Road, Oxford OX1 3PU, UK
 S. L. Zhang
 & T. Hesjedal
Magnetic Spectroscopy Group, Diamond Light Source, Didcot OX11 0DE, UK
 G. van der Laan
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Contributions
S.L.Z. conceived and derived polarizationdependent REXS for the determination of the winding number. S.L.Z., T.H. and G.v.d.L. designed and performed the experiments, as well as carried out the data analysis. S.L.Z. performed the numerical calculations. S.L.Z., G.v.d.L. and T.H. discussed all aspects of the work and wrote the manuscript.
Competing interests
The authors declare no competing financial interests.
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Correspondence to T. Hesjedal.
Supplementary information
PDF files
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Supplementary Information
Supplementary Figures, Supplementary Notes and Supplementary References
Videos
 1.
Supplementary Movie 1
Covering N=1.mov. Demonstration of the topological covering concept based on onedimensional helix chains for N = 1, 2, 3 and 6 skyrmions. The left panel shows the orderparameterspace coverage and the right panel the corresponding magnetisation configuration.
 2.
Supplementary Movie 2
Covering N=2.mov. Demonstration of the topological covering concept based on onedimensional helix chains for N = 1, 2, 3 and 6 skyrmions. The left panel shows the orderparameterspace coverage and the right panel the corresponding magnetisation configuration.
 3.
Supplementary Movie 3
Covering N=3.mov. Demonstration of the topological covering concept based on onedimensional helix chains for N = 1, 2, 3 and 6 skyrmions. The left panel shows the orderparameterspace coverage and the right panel the corresponding magnetisation configuration.
 4.
Supplementary Movie 4
Covering N=6.mov. Demonstration of the topological covering concept based on onedimensional helix chains for N = 1, 2, 3 and 6 skyrmions. The left panel shows the orderparameterspace coverage and the right panel the corresponding magnetisation configuration.
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