Direct experimental determination of the topological winding number of skyrmions in Cu2OSeO3

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 X-ray scattering process. This relationship provides a one-to-one correspondence between the measured scattering signal and the winding number. We demonstrate that the exact topological quantities of the skyrmion material Cu2OSeO3 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.

. Robustness of the measurement principle for varying radial profile. a, Three different Θ(ρ) profiles that govern different radial spin distributions, labelled as (i), (ii), and (iii), are used for the subsequent numerical calculations. Note that profile (i) represents a linear relationship, which is equivalent to the one-dimensional helix modulation case. b-d, Circular dichroism (CD) profiles, and, e-g, polarisation-azimuthal maps (PAMs) calculated based on the three different radial functions. It can be seen that both CD and PAM are independent of the radial profile, confirming the robustness of the measurement principle. On the other hand, we use the one-dimensional helix approximation method to perform the numerical calculations for the same object, in order to confirm the equivalence of both methods.
In summary, three theoretical methods are used for calculating the CD and PAM as a function of the topological winding number, in order to demonstrate the consistency of the results: (I) Analytical solution based on Eq.
(3) for PAM and Eq. (4) for CD in the main text.
(II) Construction of the skyrmion configuration using the one-dimensional helix approximation model. For the azimuthal angle Ψ, the diffractive x-rays are sensitive to the structure factor of the spin helix, obtained by rotating the helix N Ψ from the base position. The CD and PAM are subsequently calculated numerically.
(III) Generation of a two-dimensional skyrmion lattice using the rigorous solution given by Eq.
(2) in the main text. The CD and PAM are then numerically obtained using Eq.
We first demonstrate the consistency between the three calculation methods, which fur- Second, we discuss the influence of χ and λ on the CD and PAM patterns. Supplementary Figure 4a shows another N = 3 topological object, which is essentially a continuous transformation from the object shown in Fig. 2c (see main text). This homotopic transformation can be achieved by adjusting χ. As shown in Supplementary Figures 4b and 4c, 8 compared to Fig. 2g and 2k in the main text, the CD and PAM patterns have identical periodicities, and the only difference is a linear phase shift. This is valid for all cases in our numerical studies. Moreover, as shown in Supplementary Figure 4d-f, flipping the polarity of the topological object does not alter the PAM, however, it imposes a phase shift on the CD profile. Therefore, the use of the phase parameters Φ 1 and Φ 2 in Eqs. (3) and (4) (see main text) can generalise the principle to all homotopies arising from variations in χ and λ.
To briefly summarise, our polarisation-dependent REXS method, represented by the circular dichroism plots and the polarisation-azimuthal maps, is only sensitive to the winding number and has a one-to-one correspondence to this topological quantity. Any homotopy change will not affect the outcome of the measurement. In other words, the method itself can be seen as 'topologically protected'.

Supplementary Note 2. NON-INTEGER WINDING NUMBERS
Here we discuss the case of non-integer winding numbers. Note that non-integer winding numbers correspond to energetically unstable states, due to the appearance of the singularities within their spin structures. As shown in Supplementary Figures 5a and 5d, the abrupt change of the spins across the red lines will cost extremely high energy, leading to the unstable states. However, we will calculate the corresponding CD and PAM in order to demonstrate that our new technique is only sensitive to spin configurations with integer topological winding numbers. Supplementary Figures 5a-c show the magnetisation distribution, CD and PAM for a N = 1.7 motif lattice. First, the CD shape is largely distorted from a well-defined sinusoidal curve shape. Second, in PAM, the humps are no longer of equal height due to the non-integer topology. These features can also be found for the N = 3.3 case, shown in Supplementary Figures 5d-f. The asymmetry is even more pronounced in CD, in which the periodically modulated peaks do not have equal height. This is also clearly shown in their PAM relationship. On the other hand, as expressed by Eq. (7) (see main text), polarisation-dependent REXS is sensitive to all three magnetisation components. As a consequence, the calculated CD signal as shown in Supplementary Figure 6d will be suppressed for N = 0 type of vortices, while PAM shows two humps. Combining the CD and PAM results, one can unambiguously conclude that the motif is a N = 0 vortex. This is in stark contrast to the CD and PAM results for an N = 1 skyrmion, as shown in Supplementary Figure 6i. Therefore, our method is a direct experimental technique that can accurately measure N . [2] S. Zhang, A. A. Baker, S. Komineas, and T. Hesjedal, "Topological computation based on direct magnetic logic communication," Sci. Rep. 5, 15773 (2015).