Abstract
The phase of the quantummechanical wave function can encode a topological structure with wideranging physical consequences, such as anomalous transport effects and the existence of edge states robust against perturbations. While this has been exhaustively demonstrated for electrons, properties associated with the elementary quasiparticles in magnetic materials are still underexplored. Here, we show theoretically and via inelastic neutron scattering experiments that the bulk ferromagnet Mn_{5}Ge_{3} hosts gapped topological Dirac magnons. Although inversion symmetry prohibits a net DzyaloshinskiiMoriya interaction in the unit cell, it is locally allowed and is responsible for the gap opening in the magnon spectrum. This gap is predicted and experimentally verified to close by rotating the magnetization away from the caxis with an applied magnetic field. Hence, Mn_{5}Ge_{3} realizes a gapped Dirac magnon material in three dimensions. Its tunability by chemical doping or by thin film nanostructuring defines an exciting new platform to explore and design topological magnons. More generally, our experimental route to verify and control the topological character of the magnons is applicable to bulk centrosymmetric hexagonal materials, which calls for systematic investigation.
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Introduction
Recent breakthroughs in the physics of electrons in solids resulted from the application of topological concepts to the quantummechanical wave function, highlighting the role of the Berry phase^{1}. For instance, the modern understanding of the integer quantum Hall effect in a twodimensional (2D) system is that of a gapped bulk with nonzero Chern numbers which imply the existence of chiral edge states responsible for the quantized conduction^{2}. Threedimensional (3D) topological insulators are gapped by the spinorbit interaction, leading to Diraclike surface states with a linear dispersion and spinmomentum locking that underpin the quantum spin Hall effect^{3}. There is now a systematic classification of the possible topological phases in electronic systems, encompassing also gapless systems such as Weyl semimetals, where the dimensionality but also spatial and magnetic symmetries are prominent^{4,5,6}.
Topology is agnostic to whether the (quasi)particles are fermions or bosons, so that magnons can also be responsible for novel physical effects^{7}. Perhaps the first example is the Hall effect experienced by a thermallyinduced magnon current in ferromagnetic (FM) Lu_{2}V_{2}O_{7}, with the DzyaloshinskiiMoriya interaction (DMI) playing a similar role to the one of spinorbit coupling (SOC) for electronic systems^{8,9,10}. Magnetic materials with an hexagonal lattice generally exhibit a Diraclike magnon dispersion at the Kpoint in the Brillouin zone, which if gapped signals the existence of nontrivial topology^{11,12,13,14,15}. Such topological magnon insulators were experimentally identified in 2D FM materials^{16,17,18}, while gapless Dirac magnons were characterized in 3D magnetic materials such as antiferromagnetic (AFM) Cu_{3}TeO_{6}^{19} and CoTiO_{3}^{20,21}, and the elemental FM Gd^{22}. While a symmetrybased approach has been proposed to predict materials hosting topological magnons^{23}, experimentally confirming the topological character is challenging, as the magnon Hall effect is difficult to measure and other signatures such as a characteristic winding of the scattering intensity have only recently been detected^{21}.
In this article, we study the 3D centrosymmetric ferromagnet Mn_{5}Ge_{3}^{24,25}. This material exhibits significant anomalous Hall^{26} and Nernst^{27} effects which are signatures of large electronic Berry phases. Its Curie temperature (T_{C}) can be enhanced by carbon doping^{28} as successfully described by a previous theoretical work^{29}, but its magnonic properties remain unexplored. Here, we theoretically predict and experimentally confirm the existence of gapped Dirac magnons at the Kpoint due to the DMI. This gap can be closed by rotating the magnetization direction with an external magnetic field, thus validating the proposed gap mechanism and confirming the topological character of the magnons at the Kpoint. Our experimental route to verify and control the topological character of the magnons is not limited to Mn_{5}Ge_{3} and should also be applicable to other bulk centrosymmetric hexagonal materials.
Results
Basic properties of Mn_{5}Ge_{3}
A highquality Mn_{5}Ge_{3} single crystal of about 10 g has been grown using the Czochralski method. The space group is P6_{3}/mcm and the unit cell contains 10 Mn atoms (and 6 Ge atoms), with Mn1 and Mn2 occupying the Wyckoff positions 4d and 6g, respectively^{24}. In the abplane, Mn1 adopts a honeycomb lattice while Mn2 adopts an hexagonal arrangement, Fig. 1a. Along the caxis, Mn1 forms chains and Mn2 columns of facesharing octahedra, Fig. 1b. The Curie temperature (T_{C}) is around 300 K and the magnetic moments of the Mnl and Mn2 atoms are 1.96(3) μ_{B} and 3.23(2) μ_{B}, respectively, aligned along the caxis^{24,25}.
Magnetic properties from first principles
Density functional theory (DFT) calculations were performed prior to the inelastic neutron scattering measurements in order to provide an initial picture of the expected magnon excitations and to identify interesting regions in (Q,E) space. The theoretical magnetic moments from juKKR (2.11 μ_{B} and 3.14 μ_{B} for Mn1 and Mn2, respectively) and the computed Heisenberg exchange interactions are comparable to the ones previously reported^{29}, as seen in Table 1. Spinorbit coupling leads to significant DMI (c.f. Table 1), much weaker symmetric anisotropic exchange (not included in Eq. (1)), and a small uniaxial magnetic anisotropy energy (K ≈ − 0.1 meV), so that the relevant spin Hamiltonian (with ∣S_{i}∣ = 1) reads:
Here J_{ij} are the Heisenberg exchange interactions and D_{ij} are the DMI vectors which mostly align along the caxis, with the strongest shown in Fig. 1b. We discovered that some of the magnetic interactions, namely the AFM ones, are quite sensitive to small changes in the unit cell parameter and the atomic positions. The impact of this can be seen in the theoretical magnon dispersions shown in Fig. 1c, where we compare the results obtained with the experimental crystal structure parameters and with the theoretically optimized ones. In both cases there are two magnon bands in the energy range of experimental interest, with an energy gap at the Kpoint where otherwise a Diraclike crossing of the bands would be expected by symmetry. This is straightforwardly verified to arise from the DMI, as omitting it from the magnon calculation leads to the closing of the gap.
DzyaloshinskiiMoriya interaction in centrosymmetric systems
The DMI is the key magnetic interaction for the subsequent interpretation of our experimental findings, so before we continue we wish to clarify how it can be present and have an effect in a centrosymmetric material. In his seminal paper^{30}, Moriya established the symmetry rules that the interaction named after Dzyaloshinskii and himself must obey. The most famous of these rules is that if two spins are connected by an inversion center then the respective DMI must identically vanish. This pointedly explains why we find finite DMI in our calculations for Mn_{5}Ge_{3}: it is finite for those spin pairs that do not contain an inversion center in the midpoint of the corresponding bond, such as those illustrated in Fig. 1b.
Centrosymmetry does ensure that the net DMI of the unit cell is zero, which is also verified in our simulations. This means that the ferromagnetic domain walls are not chiral and that magnetic skyrmions cannot form, in agreement with Neumann’s principle. In contrast, magnons can still be influenced by the local DMI. In a semiclassical picture, the spins at different sites precess with different phases and/or amplitudes, so that certain pairs of spins are noncollinear and can be affected by the torque arising from the DMI. This is a strong effect at the Kpoint, where two magnon modes with opposite chirality cross and the degeneracy is lifted in a nonperturbative way by the DMI. We now report the experimental observation of this effect and its implications.
Magnons from inelastic neutron scattering
Inspired by these theoretical predictions, the experimental magnon spectrum of Mn_{5}Ge_{3} was investigated by INS. Several constantQ and constantE scans have been performed at T = 10 K along the reciprocal space directions (h, 0, 0), (h, 0, 2), (h, h, 0), and (0, 0, l), with representative examples shown in Fig. 2a. The measured scattering intensity (circles) was fitted with Gaussian line shapes on top of a sloping background (lines).
The magnetic nature of the excitations has been confirmed through their temperature dependence (see Supplementary Figs. 4–8 and Fig. 3a below), and the obtained q and E position is given as red symbols in Fig. 2b. For the inplane directions (Γ − M − K − Γ) one can distinguish the presence of three modes: two acousticlike modes dispersing upwards in energy away from Γ and additionally a third higherenergy mode with a steep dispersion along Γ − M and weakly dispersive along K − M. In contrast, in the same investigated E range a single stiff spinwave mode is observed for the outofplane direction (Γ − A).
We now turn to the theoretical interpretation of these measurements. Comparing the experimental results of Fig. 2b with Fig. 1c, we find that the spin model derived from the DFT calculations qualitatively reproduces several features. The absence of a clearly visible gap in the spinwave spectrum near zero energy transfer (Γpoint) agrees with the expected weakness of the uniaxial magnetic anisotropy^{25}, as also computed from DFT. The theoretical dispersion along Γ − A is slightly stiffer than the experimental one, and a simulation of the INS intensity reveals that the second mode which is higher in energy should be invisible (see Supplementary Fig. 14). We find rather poor quantitative agreement between theory and experiment in the Γ − K − M plane, which is probably related to the alreadyidentified strong dependence of the magnetic interactions computed from DFT on small changes of the structural parameters. We verified that this dependence is systematic by considering various deformations of the unit cell (see Supplementary Figs. 11 and 12). However, the most crucial feature is observed both in theory and in experiment, that is the existence of a gap at the Kpoint where two magnon bands should otherwise cross.
In order to provide a more quantitative description of the experimentally obtained magnon bands for the inplane directions, we constructed a simplified effective spin model (additional details are given in the Supplementary Information, Section II.E). We replace each Mn2 triangle at a constant height in the unit cell with a single effective spin S = 9/2 (S = 1 for Mn1), so that the column of Mn2 octahedra is replaced by a spin chain. Seen from the caxis, the unit cell for this model thus contains two Mn1 chains and one effective Mn2 chain, and we determine the model parameters using the measured magnon energies at the highsymmetry points. The lines in Fig. 2b show that the results of this model approach indeed provide a realistic band dispersion, and confirm once more that the gap at the Kpoint is a consequence of a finite DMI. The model results also highlight a peculiarity of the measured magnon energies in the vicitiny of Γ, to which we shall briefly return in the Discussion.
Closing the topological magnon gap
Although there are strong theoretical arguments in favor of the topological character of the magnon gap at the Kpoint, a convincing experimental demonstration is in order. The gap is expected to arise from the DMI, but such a microscopic interaction cannot easy be manipulated experimentally. However, it is known that the impact of the DMI on the FM magnon spectrum depends on the relative alignment between the vectors that characterize this interaction and the FM magnetization^{31,32,33}. Adapting these arguments to the hexagonal symmetry of Mn_{5}Ge_{3} leads to the prediction that the magnitude of the gap should depend on the relative alignment of the FM magnetization and the crystalline caxis, and in particular should vanish if the two are perpendicular. We have verified that the magnon gap closes both in the DFTbased spin model and in the one fitted to the experimental measurements when the magnetisation is perpendicular to the caxis (as it does when disregarding the DMI). The magnon dispersions computed with DMI and setting the magnetization along the a^{*}axis are identical to the dashed lines shown in Fig. 1c and Fig. 2b, which were obtained by excluding the DMI from the calculations.
This hypothesis can be experimentally tested by applying an external magnetic field. First we rule out the possibility of a phonon contribution to the inelastic peaks at the Kpoint, by heating the sample above its T_{C}, as seen in Fig. 3a, and verifying that the peaks disappear conforming their magnetic origin. The measurements reported so far in this work were performed in zero field, for which the magnetization is parallel to the caxis due to the uniaxial magnetic anisotropy. Applying a magnetic field along the caxis should lead to a very small Zeeman shift of the magnon energies, and this is indeed what we observed, as seen in Fig. 3b. If a magnetic field of similar magnitude is applied along the a^{*}axis it overcomes the magnetic anisotropy energy and saturates the magnetization^{25}. The results obtained in this way are shown in Fig. 3c. Now the effect of the field cannot be explained by a simple Zeeman shift, and instead we find the anticipated closing of the magnon gap. The two peaks observed in zero field coalesce into a single one with an integrated intensity approximately matching the sum of intensities for the two peaks in zero field (see also Supplementary Fig. 10), although shifted to slightly higher energy than anticipated. The distinct response of the magnon excitation to a magnetic field applied to orthogonal crystal directions is consistent with the DMI mechanism, and so the gapped Dirac magnons at the Kpoint should consequently have a topological nature.
Ruling out alternative explanations for a gap at the Kpoint
Next we rule out potential alternative mechanisms to the DMI that could lead to a magnon gap opening at the Kpoint, such as dipolar, Kitaev and magnonphonon interactions:

(i)
Dipolar interactions are longranged but much weaker than the magnetic exchange interactions, so their effect is usually seen for rather small wave vectors in the vicinity of the Γpoint. Even if they did lift the magnon degeneracy at the Kpoint, their intrinsic weakness could not account for the observed magnitude of the gap.

(ii)
Kitaev interactions were proposed for instance in ref. ^{34} to explain measurements on CrI_{3} but are ruled out for Mn_{5}Ge_{3} both by our simulations and by general considerations. The interactions extracted from our DFT calculations include both the Heisenberg exchange, the DMI and the symmetric anisotropic exchange (SAE), which includes the Kitaev interaction. The SAE was found to be rather weak and unable to open the observed magnon gap at the Kpoint. The weakness of the SAE (and so of potential Kitaev interactions) could be anticipated from the weak magnetic anisotropy measured for this system. This reflects the lack of heavy elements in the material that could supply a strong spinorbit interaction, which is a key ingredient for obtaining a sizeable Kitaev interaction. We are also not aware of any Kitaev material candidates containing only elements from the first four rows of the periodic table (i.e., Z < 36), likely due to the preceding reason. To make this argument more quantitative, we employ the theory of magnetic exchange interactions for systems with weak SOC presented by Moriya^{30}, Eqs. 2.3 and 2.4. The DMI is firstorder in the weak SOC, while the Kitaev interaction is part of the SAE which is secondorder in SOC and so is much weaker than the DMI. The Kitaev interaction, if present, would contribute to the magnetic anisotropy energy, which is about 1 meV/unit cell for Mn_{5}Ge_{3} (the DMI does not contribute to the magnetic anisotropy energy of the ferromagnetic state). To give an estimate of the potential magnitude of the Kitaev interaction using only experimental input, we distribute the magnetic anisotropy energy on one of the set of bonds for which we identified the DMI, bond #2 indicated in Fig. 1b. This set of bonds occurs four times in the unit cell, as it connects each Mn1 site to its six Mn2 neighbours, and so could have 1 meV/24 = 0.04 meV maximum Kitaev strength. The SAE obtained directly from the DFT calculations is about 0.02 meV in magnitude, which is in line with this estimate, and is one order of magnitude smaller than the values found for the DMI (0.57 meV for the set of bonds #2), as expected from the theory of the magnetic exchange interactions for systems with weak SOC.

(iii)
Magnonphonon interactions can result in gaps at the crossing points between the magnon and phonon branches. However, our INS data ruled out this possibility. The measured excitations at different Q vectors around and at the Kpoint are solely of magnetic origin. This has been verified through their temperature dependence. All the peaks observed in the energy range from 10 to 20 meV at T = 10 K are replaced by a broad quasielastic signal (centered at 0 meV) above the ordering temperature as shown in Fig. 3a. Hence no phonon modes were detected in the vicinity of the Kpoint with an energy compatible with that of the magnons, which is a requirement for the gap opening mechanism through magnonphonon interactions.
Therefore, we can assert that the only reasonable mechanism for the gap opening at the Kpoint is the DMI.
Simulation of magnon surface states
Here we explore the expectation that if the bulk magnon band structure has some topologically nontrivial character it should be accompanied by magnon surface states. To do so, we compute the magnon band structure of slabs which are finite in one direction and periodic (i.e., infinite) in the other two directions. Comparing the simulations performed for the same slab with periodic and open boundary conditions along the chosen surface normal enables us to identify the energy range corresponding to the surface projection of the bulk magnon bands. Surface magnons are then expected to appear in the regions of (E, q) where bulk magnon bands are absent.
To illustrate this point, we performed simulations using the simplified spin model depicted in Fig. 4a with parameters fitted to the experimental measurements in Fig. 2b. We created a rectangular supercell and extended it by 20 unit cells along the \((01\bar{1}0)\) direction of the original hexagonal lattice. The chosen path in reciprocal space is perpendicular to the surface normal and shown in Fig. 4b. Figure 4c shows that indeed there is a pocket in the K–M–K path and with energies between 10 and 12 meV from which bulk magnons are absent, with or without DMI. Figure 4d shows the modified magnon band structure upon truncating the crystal along the \((01\bar{1}0)\) direction, i.e., making a horizontal cut through the lattice shown in Fig. 4a. We indeed find that magnon surface states do appear in the identified region where bulk magnon bands were absent. Without the DMI, these surface magnons are disconnected and gapped from each other. The DMI restructures the band connectivity and leads to a crossing that resembles a distorted Weyllike crossing. The crossing is not located at the Mpoint as the slab loses the acmirror plane, retaining instead a twofold rotation around the caxis. There are other surface magnons at around 3 meV and 18 meV that are only weakly affected by the DMI, and result from the reduced coordination number introduced by creating the surface. Lastly, the identified surface magnons were found to extend only a couple of unit cells towards the interior of the slab, confirming their localization at the surface.
The experimental detection of the predicted surface magnons is quite challenging, as the scattering volume is too small for a straightforward detection using INS. Other techniques such as Brillouin light scattering (for magnons near the Γpoint) or electron energy loss spectroscopy could be considered for this purpose^{35}.
Discussion
We now briefly discuss some outstanding points. Firstly, we return to the quantitative disagreement between theory and experiment concerning the magnon bands. The main issue seems to be an overestimation of the magnitude of the magnetic interactions in the DFT calculations, which has also been found in other studies. A recent example from the literature is Co_{3}Sn_{2}S_{2}^{36}, where two different DFT approaches are compared with experiment, with disagreements also in the Γ − M − K − Γ plane but not in the Γ − A direction. Another issue is that the simplified spin model does not adequately capture the relative intensities of the two peaks around the Kpoint, despite providing a reasonable description of the experimental magnon energies. This is likely due to the model assumptions, namely treating the Mn2 sites as a single effective spin and neglecting the atomic magnetic form factor, as well as employing a simple Lorentzian broadening for the peaks. This shows the need for further developments on the theoretical side. We also noticed that the ratio of the intensities of the two peaks varies slightly in different experimental setups, see Figs. 3b, c. This might be due to changes in the domain state of the sample arising from the measurement history.
Our INS measurements for Mn_{5}Ge_{3} revealed a peculiar dispersion for the second mode in the Γ − K direction. Such a steep linearlike dispersion close to the Γpoint resembles an AFM magnon or a phonon mode. We restate that the material is FM and the measured excitations are of magnetic nature as verified by measurements above T_{C} (Fig. 3a and Supplementary Figs. 4–8), so that both simple explanations are ruled out. However, it is possible to have formation of hybrid collective modes. INS can identify the magnetic and phononic component of such excitations, which are referred as magnonpolarons (magnetoelastic modes), and are reported in several magnetic materials^{37,38}. Avoided crossings are the common signature of magnetoelastic interactions, but they also underpin the magnetovibrational scattering term in the INS scattering cross section^{39,40}. Additional discussion is given in the Supplementary Information, Section I.C. We propose that Mn_{5}Ge_{3} is also an interesting 3D FM candidate material for detailed investigation of magnetoelastic effects^{41,42}.
The key interaction responsible for opening the magnon gap at the Kpoint and thus endowing the Dirac magnons with a topological character is the DMI. There are several possible routes by which this interaction could be engineered, so that the magnon properties can be tuned for specific purposes for magnonic devices operating in a broad temperature range. Firstly, chemical substitution of Ge by Si has been explored in the literature to connect to the multifunctional AFM Mn_{5}Si_{3}^{43,44,45,46}. Substituting Ge by Si reduces T_{C}^{47}, but the impact on the magnons and on the DMI is unknown. On the other hand, carbon implantation is demonstrated to enhance T_{C}^{27,48} for which the imprint on the Heisenberg exchange interactions has been theoretically established^{29}, but once again the effect on the magnons and the DMI remains unexplored. Lastly, Mn_{5}Ge_{3} can also be grown in thin film form^{27,48}. This could modify the DMI by epitaxial strain, which would also be interesting in connection to potential magnetoelastic effects, by interfaces to other materials, or by quantum confinement effects if the thickness is just a few nanometers.
In conclusion, we presented a combined theoretical and experimental study of the magnons in the centrosymmetric 3D FM Mn_{5}Ge_{3}. Despite the inversion symmetry, significant DMI has been theoretically identified on MnMn bonds which do not contain an inversion center. This DMI is responsible for opening a gap in the magnon spectrum at the Kpoint, where otherwise symmetry would enforce a Diraclike crossing of the magnon bands. INS measurements of the magnon spectrum show qualitative agreement with the main points predicted by theory, and confirm the expected gap at the Kpoint. We experimentally observe the closing of the gap by rotating the magnetization from the caxis to the a^{*}axis with a magnetic field. This both validates the gap generation mechanism and the topological nature of the magnons at the Kpoint, thus establishing Mn_{5}Ge_{3} as a realization of a gapped Dirac magnon material in three dimensions. The ability to control the gap at the Kpoint with an external magnetic field will also impact topological magnon surface states, and deserves further study. As the macroscopic magnetic properties of Mn_{5}Ge_{3} can be tuned by chemical substitution of Ge with Si or by carbon implantation, and it can also be grown as thin films in spintronics heterostructures, we foresee that the features of the newlydiscovered topological magnons can be engineered and subsequently integrated in novel device concepts for magnonic applications. Looking beyond Mn_{5}Ge_{3}, the physical mechanism leading to the formation of topological magnons at the Kpoint should be present in many other bulk centrosymmetric hexagonal materials, which opens an exciting avenue for future investigations.
Methods
Experimental methods
Inelastic neutron scattering (INS) experiments have been carried out on a cold (IN12) and a thermal (IN22) triple axis spectrometer at the Institut LaueLangevin, in Grenoble, France. We use the hexagonal coordinate system and the scattering vector Q is given in reciprocal lattice units (r.l.u.). The wavevector q is related to Q through Q = q + G, where G is a Brillouin zone center. Inelastic scans were performed with constant ∣k_{f}∣, where k_{f} is the wavevector of the scattered neutron beam. Data were collected holding either the energy (constantE) or the scattering vector (constantQ) fixed. Further details on the experimental procedures and additional measurements can be found in the Supplementary Information, Section I).
Theoretical methods
The theoretical results were obtained with DFT calculations and the extracted spin Hamiltonian. The unit cell parameters and the atomic positions were optimized with the DFT code Quantum Espresso^{49}. The magnetic parameters were computed with the DFT code juKKR^{50,51}, which are then used to solve a spin Hamiltonian in the linear spin wave approximation^{52}. Further details on all these aspects can be found in the Supplementary Information, Section II).
Data availability
The authors declare that the data supporting the findings of this study are available within the paper, its supplementary information file and in the following repositories^{54,55,56,57,58}.
Code availability
The DFT simulation packages Quantum Espresso and juKKR are publicly available (see Methods). The code for the solution of the linear spin wave problem is available from the corresponding authors upon request.
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Acknowledgements
We thank S. Raymond for discussions and comments. The work of M.d.S.D. made use of computational support by CoSeC, the Computational Science Centre for Research Communities, through CCP9. N.B. acknowledges the support of JCNS through the Tasso Springer fellowship. F.J.d.S. acknowledges support of the European H2020 Intersect project (Grant no. 814487), and N.M. of the Swiss National Science Foundation (SNSF) through its National Centre of Competence in Research (NCCR) MARVEL. This work was also supported by the Brazilian funding agency CAPES under Project No. 13703/137 and the Deutsche Forschungsgemeinschaft (DFG) through SPP 2137 “Skyrmionics” (Project LO 1659/81). We gratefully acknowledge the computing time granted through JARA on the supercomputer JURECA^{53} at Forschungszentrum Jülich GmbH and by RWTH Aachen University. The neutron data collected at the Institut LaueLangevin are available at refs. ^{54,55,56,57}.
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M.d.S.D., N.B. and F.J.d.S. contributed equally to this work. M.d.S.D. and N.B. conceived the project together with T.B. and S.L. M.d.S.D. performed most DFT calculations and the spinwave modelling, with additional calculations performed by F.J.d.S. J.P. grew the Mn_{5}Ge_{3} single crystal. N.B. performed the experimental measurements and the corresponding data analysis. K.S. and F.B. were local contacts of IN12 and IN22 and provided instrument support. The theoretical aspects of the work were discussed with N.M., S.B. and S.L. All authors participated in the discussion of the results. M.d.S.D., N.B. and F.J.d.S. wrote the manuscript with input from all authors.
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dos Santos Dias, M., Biniskos, N., dos Santos, F.J. et al. Topological magnons driven by the DzyaloshinskiiMoriya interaction in the centrosymmetric ferromagnet Mn_{5}Ge_{3}. Nat Commun 14, 7321 (2023). https://doi.org/10.1038/s41467023430423
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DOI: https://doi.org/10.1038/s41467023430423
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