# Quantum nature of Skyrmions

Oleg Janson,^{1,2} Ioannis Rousochatzakis,^{3} Alexander A. Tsirlin,^{1,2} Marilena Belesi^{3}, Andrei A. Leonov,^{3} Ulrich K. Rößler,^{3} Jeroen van den Brink,^{3,4} and Helge Rosner^{1}

^{1}Max Planck Institute for Chemical Physics of Solids, D-01087 Dresden, Germany. ^{2}National Institute of Chemical Physics and Biophysics, EE-12618 Tallinn, Estonia. ^{3}Leibniz Institute for Solid State and Materials Research, IFW D-01069 Dresden, Germany. ^{4}Department of Physics, TU Dresden, D-01062 Dresden, Germany

Originally, skyrmions were introduced over half a century ago in the context of dense nuclear matter. However, being at first hand mathematical objects — special types of topological solitons — they can emerge in much broader contexts. Recently skyrmions were observed in metallic helimagnets like MnSi and FeGe, forming nanoscale spin-textures. Extending over length scales much larger than the interatomic spacing, they behave as large, classical objects, yet deep inside they are of quantum nature. As a large surprise came the recent observation of skyrmions in the strongly correlated spin-½ quantum magnet Cu_{2}OSeO_{3}, which, in contrast to MnSi and FeGe, should exhibit few relevant short-range exchange interactions, only. The key aspects of helimagnetism and skyrmion formation in chiral magnetic systems such as Cu_{2}OSeO_{3}, are shown in Fig. 1.

**Figure 1** - Spin textures in chiral helimagnets. Besides flat helices, chiral helimagnets like Cu_{2}OSeO_{3} manifest radially symmetric topological solitons like **(a)** skyrmions or half- skyrmions. **(b)** Parallel skyrmions can form densely packed lattices in two spatial dimensions. **(c)** Quantitative first-principles calculations predict that the ferrimagnetic order in Cu_{2}OSeO_{3} is locally altered by the multi-sublattice structure, leading to weak antiferromagnetism (depicted by orange arrowheads). **(d)** The skyrmion texture is locally composed of these three-dimensional canted spin patterns. Thus, the weak antiferromagnetic order itself is modulated along with the primary ferrimagnetic twisting shown in **(a)**.

**Figure 1** - Spin textures in chiral helimagnets. Besides flat helices, chiral helimagnets like Cu_{2}OSeO_{3} manifest radially symmetric topological solitons like **(a)** skyrmions or half- skyrmions. **(b)** Parallel skyrmions can form densely packed lattices in two spatial dimensions. **(c)** Quantitative first-principles calculations predict that the ferrimagnetic order in Cu_{2}OSeO_{3} is locally altered by the multi-sublattice structure, leading to weak antiferromagnetism (depicted by orange arrowheads). **(d)** The skyrmion texture is locally composed of these three-dimensional canted spin patterns. Thus, the weak antiferromagnetic order itself is modulated along with the primary ferrimagnetic twisting shown in **(a)**.

Penetrating into the microscopic roots of helimagnetism and skyrmion formation requires a multi-scale approach (see Fig. 2), spanning the full quantum to classical domain. For the mentioned metallic systems such an approach is presently intractable due to the strong mixing of delocalized low energy electronic and magnetic degrees of freedom. However, since its band gap enforces a natural separation between electronic and magnetic energy scales, we achieve for the first time a complete multi-scale approach in the skyrmionic Mott insulator Cu_{2}OSeO_{3}. We find that its magnetic building blocks are strongly fluctuating Cu_{4} tetrahedra, spawning a continuum theory that culminates in 51nm large skyrmions, in striking agreement with experiment.

**Figure 2**- From crystal structure towards an effective magnetic model of Cu

_{2}OSeO

_{3}

**a)**The crystal structure is shaped by Cu(1)O4 plaquettes (yellow) and Cu(2)O5 bypyramids (orange), and covalent Se-O bonds (thick lines), forming a three-dimensional lattice. This lattice can be tiled into tetrahedra (dashed lines), each composed of one Cu(1) and three Cu(2) polyhedra.

**b)**The arrangement of the tetrahedra is topologically equivalent to a diamond lattice, where each green cube comprises a tetrahedron.

**c)**DFT calculations evince the presence of both types of magnetic interactions: antiferromagnetic (red) and ferromagnetic (blue), in agreement with the experimental magnetic structure (arrows). The strength of a certain coupling is indicated by the thickness of the respective line. The strongest couplings are found within the tetrahedra (shaded), while the couplings between the tetrahedra (dashed lines) are substantially weaker.

**d)**A quantum mechanical treatment of a single tetrahedron yields a magnetic S=1 ground state, separated from the lowest lying excitation by about 240 K. Due to this large energy scale, the tetrahedra behave as rigid S=1 entities at low temperatures.

**e)**Effective magnetic model of S=1 tetrahedra, where all effective interactions are ferromagnetic (see text). The composite nature of the effective S=1 moments is indicated by sectioned arrows.

**f)**In moderate magnetic fields, the effective spins form a spiral.

**Figure 2**- From crystal structure towards an effective magnetic model of Cu

_{2}OSeO

_{3}

**a)**The crystal structure is shaped by Cu(1)O4 plaquettes (yellow) and Cu(2)O5 bypyramids (orange), and covalent Se-O bonds (thick lines), forming a three-dimensional lattice. This lattice can be tiled into tetrahedra (dashed lines), each composed of one Cu(1) and three Cu(2) polyhedra.

**b)**The arrangement of the tetrahedra is topologically equivalent to a diamond lattice, where each green cube comprises a tetrahedron.

**c)**DFT calculations evince the presence of both types of magnetic interactions: antiferromagnetic (red) and ferromagnetic (blue), in agreement with the experimental magnetic structure (arrows). The strength of a certain coupling is indicated by the thickness of the respective line. The strongest couplings are found within the tetrahedra (shaded), while the couplings between the tetrahedra (dashed lines) are substantially weaker.

**d)**A quantum mechanical treatment of a single tetrahedron yields a magnetic S=1 ground state, separated from the lowest lying excitation by about 240 K. Due to this large energy scale, the tetrahedra behave as rigid S=1 entities at low temperatures.

**e)**Effective magnetic model of S=1 tetrahedra, where all effective interactions are ferromagnetic (see text). The composite nature of the effective S=1 moments is indicated by sectioned arrows.

**f)**In moderate magnetic fields, the effective spins form a spiral.

In our approach, we start at the atomic level with density functional (DFT) band structure calculations to estimate the Heisenberg exchange parameters J_{ij} and Dzyaloshinskii-Moriya (DM) interactions. After deriving a tight-binding model as a first step to analyse the relevant orbitals and couplings, DFT+U calculations include the missing electronic correlations in a static mean-field approximation and yield specific values for the coupling parameters. The resulting J_{ij}'s are cross-checked for their accuracy by Quantum Monte Carlo (QMC) simulations. The calculated magnetic ordering temperature T_{C} and the T-dependence of magnetization and magnetic susceptibility are in very good agreement with the measurements (see Fig. 3).

**Figure 3**- Microscopic magnetic model of Cu

_{2}OSeO

_{3}- comparison with the experiments

**a)**QMC simulations of the magnetic model (see Fig. 2c) reproduce the experimental magnetic susceptibility in the paramagnetic regime above the Curie point at 70 K.

**b)**Similarly, in the ferrimagnetic phase (shaded), the bulk magnetization of QMC results agrees with the experimental magnetization data.

**c)**Cu(1) and Cu(2) occupy topologically different sites of the spin lattice, hence the effect of quantum fluctuations and thus the local magnetization of both sites is different. This key feature of our model is consistently reproduced in both QMC and our tetrahedral mean-field theory (TMF, see Fig. 2 d, e).

**Figure 3**- Microscopic magnetic model of Cu

_{2}OSeO

_{3}- comparison with the experiments

**a)**QMC simulations of the magnetic model (see Fig. 2c) reproduce the experimental magnetic susceptibility in the paramagnetic regime above the Curie point at 70 K.

**b)**Similarly, in the ferrimagnetic phase (shaded), the bulk magnetization of QMC results agrees with the experimental magnetization data.

**c)**Cu(1) and Cu(2) occupy topologically different sites of the spin lattice, hence the effect of quantum fluctuations and thus the local magnetization of both sites is different. This key feature of our model is consistently reproduced in both QMC and our tetrahedral mean-field theory (TMF, see Fig. 2 d, e).

The evaluated parameters define a clear separation of energy scales, leading to an effective model (with renormalized interaction parameters) of weakly interacting Cu_{4} tetrahedra in a spin-1 triplett state. Interestingly, although Cu_{2}OSeO_{3} exhibits a much complexer crystal structure, the obtained effective so-called trillium-lattice model is strongly reminiscent to the B20 helimagnets MnSi and FeGe. This effective model can be further extended to a mesoscopic scale by a long wavelength approximation, delivering two basic parameters for the exchange stiffness A and the twisting parameter D which control the skyrmion physics in Cu_{2}OSeO_{3}. Based on these parameters A and D, micromagnetic simulations enable a detailed investigation of the magnetic phase diagram.

In summary, our multi-scale approach successfully models the complex skyrmion formation in Cu_{2}OSeO_{3} on a basis of weakly coupled, strongly fluctuating Cu_{4} tetrahedra in surprisingly good agreement with the experiments.

O. Janson, I. Rousochatzakis, A. A. Tsirlin, M. Belesi, A. A. Leonov, U. K. Rößler, J. van den Brink, H. Rosner, The quantum nature of skyrmions and half-skyrmions in Cu_{2}OSeO_{3}. Nature Commun. **5**, 1-11 (2014) http://dx.doi.org/10.1038/ncomms6376