Abstract
Topological magnons are emergent quantum spin excitations featured by magnon bands crossing linearly at the points dubbed nodes, analogous to fermions in topological electronic systems. Experimental realisation of topological magnons in three dimensions has not been reported so far. Here, by measuring spin excitations (magnons) of a threedimensional antiferromagnet Cu_{3}TeO_{6} with inelastic neutron scattering, we provide direct spectroscopic evidence for the coexistence of symmetryprotected Dirac and triply degenerate nodes, the latter involving threecomponent magnons beyond the Dirac–Weyl framework. Our theoretical calculations show that the observed topological magnon band structure can be well described by the linearspinwave theory based on a Hamiltonian dominated by the nearestneighbour exchange interaction J_{1}. As such, we showcase Cu_{3}TeO_{6} as an example system where Dirac and triply degenerate magnonic nodal excitations coexist, demonstrate an exotic topological state of matter, and provide a fresh ground to explore the topological properties in quantum materials.
Introduction
By introducing the concept of topology into electronic bands, plenty of novel quantum materials, such as topological insulators^{1,2} with the edge state existing in the bulk gap, and Dirac^{3,4,5,6} and Weyl semimetals^{7,8,9} featured by linearband crossings at the Dirac and Weyl nodes, respectively, have been discovered. Excitations associated with these topological states are fermions described by the Dirac–Weyl equations^{10,11}. Recently, exotic new fermions^{12,13,14,15}, such as the triply degenerate ones beyond such a classification, have emerged^{16,17}, enriching the family of topological materials and advancing the understanding on band topology. Since topological band structure is independent of the statistics of the constituent quasiparticles, many efforts have been devoted to seeking for nontrivial topological analogues of fermions in bosonic systems, e.g., phononic^{18,19,20,21,22} and photonic crystals^{23,24,25,26,27,28,29}. In two dimensions, various topological states for magnons (also bosons), which are spin excitations in magnetically ordered systems, have also been proposed. These include topological magnon insulators^{30,31,32,33,34}, and magnonic Dirac^{35,36,37,38} and Weyl semimetals^{39,40,41,42,43}. Following the successful examples in fermionic systems^{13,14,16}, triply degenerate nodal excitations have been predicted for the magnonic case^{44,45}, extending the topological classification in bosonic systems. Topological magnonic systems exhibit: (i) nonzero Berry curvature which gives rise to the anomalous Hall effect of the heat current carried by the chargeneutral spin excitations^{33,46,47,48,49,50,51}; (ii) edge or surface state that is topologically protected^{30,32,52,53}. These exotic properties make the materials appealing in developing highefficiency and lowcost spintronic devices^{47,53,54,55}. However, candidate materials to realise topological magnons are scarce. Especially for topological magnons in three dimensions, there has been no experimental report so far. In this regard, Li et al. ^{56} have predicted Cu_{3}TeO_{6} to host Dirac magnons, offering an excellent opportunity for experimental investigations into the topological properties of magnons.
In this work, we measure the spin excitations in Cu_{3}TeO_{6} with inelastic neutron scattering (INS), and compare the INS data with the linearspinwave calculations performed based on a Hamiltonian dominated by the nearestneighbour (NN) exchange interaction J_{1}. From the results, we discover symmetryprotected threedimensional Dirac and triply degenerate magnons in Cu_{3}TeO_{6}.
Results
Sample characterisations
The crystal structure of Cu_{3}TeO_{6} with the Ia3 space group (no. 206) is illustrated in Fig. 1a (ref. ^{57}). Six Cu^{2+} ions form an almost coplanar hexagon, and each ion is vertexshared by two hexagons, constituting a threedimensional spinweb structure^{57,58,59,60}. Neutron powder diffraction has shown that Cu_{3}TeO_{6} develops a longrange collinear antiferromagnetic order below the transition temperature T_{N} of 61 K, with spins aligned along the [111] direction^{57,58}. Such a magnetic state as illustrated in Fig. 1a, with each up spin related to a down spin by centroinversion, belongs to a magnetic group with the PT symmetry^{56}, where P and T are spaceinversion and timereversal operations, respectively. Under the protection of this symmetry, magnons are expected to exhibit nontrivial topological properties^{56}. INS is a direct approach to visualise magnon bands in the momentum and energy space, which acts as angleresolvedphotoemission spectroscopy in characterising electronic band structures^{5,8,17}. Below, we present results from INS measurements on wellcharacterised highquality single crystals of Cu_{3}TeO_{6} (see Supplementary Figs. 1 and 2 for details).
Magnetic excitation spectra
We have obtained a rich INS dataset which covers up to eight Brillouin zones in the whole energy range of interest at various temperatures. In Fig. 2a–c, we present the excitation spectra obtained at T = 5 K along three highsymmetry directions [001], [101], and [111], respectively. These directions are illustrated in Fig. 1b. The INS spectra show clear excitations dispersing up from the magnetic Bragg peaks^{57}. More magnetic peaks are shown in Supplementary Fig. 4. These peaks can be almost perfectly indexed with the collinear magnetic structure which respects the PT symmetry^{57,58}. We have performed measurements at higher temperatures up to T = 70 K, above the T_{N} of 61 K, and the results are shown in Supplementary Figs. 3 and 4. We find that the welldefined dispersions at T = 5 K become almost featureless at 70 K (Supplementary Fig. 3g–i), along with the disappearance of the magnetic Bragg peaks (Supplementary Fig. 4e, f). The wavevector and temperature dependences of the excitations are clearly evidencing that they are spinwave excitations.
Turning back to the data at 5 K, we can see that the acoustic bands extend up to about 15 meV, and the optical bands are present roughly between 15 and 20 meV. As there are 12 Cu^{2+} atoms in a primitive unit cell^{57}, there are six doubly degenerate magnon bands due to the PT symmetry^{56}. Since these six bands coexist in such a narrow energy window, band crossings are expected. In fact, by bare visual inspection of Fig. 2a–c, we can already identify various highsymmetry points at which the bands cross each other. Taking Fig. 2a as an example, Γ points at E ≈ 15 meV, and H point at E ≈ 16 and 18.5 meV exhibit as hot spots in the dispersion. At the H point, the interval between the two spots is clearly visible. To better characterise these points, we perform theoretical calculations as described below.
Comparison with linearspinwave calculations
The welldefined acoustic modes and quick disappearance of the magnetic order and excitations when approaching T_{N} (Supplementary Fig. 4) indicate that Cu_{3}TeO_{6} is a threedimensional antiferromagnet without much frustration, consistent with a small frustration index of f = Θ_{CW}/T_{N} ≈ 2.9 in our sample, where Θ_{CW} = −175 K is the Curie–Weiss temperature (Supplementary Fig. 2a). Therefore, we carry out the linearspinwave calculation to fit the experimental data. We find that a J_{1}–J_{2} model with only NN and nextnearestneighbour (NNN) exchange interactions cannot fit the data, given the apparent discrepancies especially on the optical branches (Supplementary Fig. 3a–f). We have added longerrange exchange interactions and found that at least up to sixthNN (J_{6}) can we fit the data satisfactorily. The necessity for using terms up to J_{6} may lie in the highly interconnected threedimensional spin network and the large number of Cu^{2+} ions in the unit cell, such that differences between different exchange paths can be small. However, J_{1} is the only dominant term, which is compatible with the modest frustration of the system as there is no comparable interaction to compete with J_{1}. The calculated spinwave spectra using these parameters along [001], [101], and [111] directions are presented in Fig. 2d–f, respectively, and the corresponding dispersions are plotted on top of the experimental data in Fig. 2a–c.
The calculated magnetic excitation spectra capture most of the features in the experimental results, as shown in Fig. 2. We note that we can include more longerrange interactions to improve the fittings, mostly for the acoustic branches in the lowenergy range where no band crossings occur. But given the present agreement between the theoretical and experimental results, we believe that our Hamiltonian up to J_{6} is appropriate, since the main purpose for the calculations is to guide our characterisations on the crossing points, which are in the highenergy range. The comparison between the calculated dispersions and experimental data assures that we have observed multiple nodal points along different trajectories in the Brillouin zone in the energy range of 15–18.5 meV. Again, we remind that these nodes are symmetry protected^{56}. The presence of nodes along all these directions indicates that the associated nodal excitations are of threedimensional nature, similar to those in the fermionic systems^{5,8}. We first identify the fourfold degenerate Dirac nodes, at which two doubly degenerate magnon bands cross each other, for example, the Γ(Γ′) and P points.
In addition to the Dirac nodes predicted in ref. ^{56}, we also observe some doubletriply (triply degenerate, hereafter) nodal points at some highsymmetry positions, e.g., the H(H′) points in Fig. 2a, c. At the Γ point, there exist both Dirac and triply degenerate nodes very close in energy. In fact, the triply degenerate nodes are generically expected for a system with a PT plus some pointgroup symmetry, such as C_{3} (refs. ^{13,17,44,45}), which is the case in Cu_{3}TeO_{6} (refs. ^{56,57}).
The existence of abovementioned nodal points is guaranteed by the symmetry, independent of the Hamiltonian^{44,45,56}. As a demonstration, we show the calculated results using a J_{1}–J_{2} model in Supplementary Fig. 3a–c—given the apparent failure of this model in describing the experimental data, this model still gives Dirac nodes at P and triply degenerate nodes at H. This further strengthens our conclusion that the unprecedented magnon band structure with coexisting Dirac and triplydegenerate nodes has been discovered in Cu_{3}TeO_{6}. Since the Γ point hosts two types of nodes too close in energy to be resolved experimentally, we pick the P and H points for further elaborations.
Triply degenerate and Dirac nodes
In the dispersions shown in Fig. 2a, we observe two triply degenerate points at H at two energies of E ≈ 16 and 18.5 meV. We first perform a constantenergy (E) cut in the dispersions at an energy interval of ΔE = 18.5 ± 0.5 meV through the H point. The results are plotted in Fig. 3a, where it clearly shows a circle centring at the H point. A cut along the [001] direction through this point yields a peak exactly centring at the H point, as shown in Fig. 3c. Peaks at L = ±2 rlu correspond to H points in the next Brillouin zones. We have also analysed the triply degenerate point at the lower energy of E = 16 meV in Supplementary Fig. 5, which confirms our conclusion on the observation of two triply degenerate nodal points at H.
Results from similar practices for the P points in Fig. 2c at an energy interval of ΔE = 15 ± 0.5 meV are presented in Fig. 3b, from which we observe two Dirac nodes at the P points on top of the ring centring (−2, 0, 1), i.e., the H′ point. Two P and Γ points in other directions are also present in this constantE contour. We perform a cut through the two P points along the [111] direction, and the results are shown in Fig. 3d, with two peaks centring at the P points.
We have also performed a series of constantQ (wavevector) cuts of the dispersions at the Q positions indicated in Fig. 4a, b through the H and P points, respectively. Results of these cuts are shown in Fig. 4c–e. From the linear cuts, we identify two energies around 15 and 18 meV corresponding to these points at q = 0 (Fig. 4c) and ∓0.5 (Fig. 4d, e), evidenced by the sharpest peaks at these Dirac and triply degenerate nodes. We also illustrate the linear dispersions near these nodes in Fig. 4c–e.
To further characterise the triply degenerate points at H, we perform simulations using the effective Hamiltonian derived from our J_{1}, J_{2}…J_{6} model (Eq. (1)). The results are plotted in Supplementary Fig. 6, which shows that, near each of the H points, there are two linear bands and one flat band. Thus, the magnons can be regarded as threecomponent bosons^{44,45}, similar to the new fermions^{13,14,15,16}. The two linear bands have a Chern number of 2 and −2, respectively, and the flat band has a Chern number of 0 (refs.^{13,14,15,22,44,45}).
Discussions
By now, we have unambiguously demonstrated the coexistence of Dirac and triply degenerate magnons in Cu_{3}TeO_{6}, and so this material is the first topological system where both Dirac and triply degenerate nodal excitations are present, enabling the investigations into the possible interplay between them and other topological properties of the material. Due to the presence of the nontrivial Berry curvature in topological magnons, the anomalous thermal Hall transport resulting from the spin current is expected^{33,46,47,48,49,50,51}. In Cu_{3}TeO_{6}, under an external magnetic field, a Dirac point should split into two Weyl points carrying a monopole charge of 1 and −1, respectively, which will give rise to the thermal Hall conductivity^{10,11}. Furthermore, the two bands with a Chern number of ±2 crossing with the flat band at the triply degenerate H point are also expected to show a thermal Hall effect under an external magnetic field that opens a gap^{33}. Another important feature of the topological magnons in Cu_{3}TeO_{6} is the topologicallyprotected surface arc state^{56}, which may be detected using surfacesensitive probes, such as highresolution electron energy loss spectroscopy^{61}, or helium atom energy loss spectroscopy^{62}. Recent developments in optical measurements of the spin excitations via the magnetooptical effect may also be helpful^{63,64,65}. Furthermore, spin current flowing on the surface may be directly measured^{53}. Further explorations of these topological properties should lend support to developing spintronics with outstanding performance^{47,53,54,55}. Finally, since the band topology does not rely on the constituent quasiparticles, both the electron and phonon bands of Cu_{3}TeO_{6} may exhibit topological properties^{13,14}, calling for future theoretical and experimental investigations.
After we finished this work, we became aware of a preprint reporting similar INS results^{66}.
Methods
Singlecrystal growth and characterisations
Highquality single crystals of Cu_{3}TeO_{6} were grown using PbCl_{2} (4 N) as the flux, following the procedures in ref.^{59}. Xray diffraction data were collected in an xray diffractometer (X′TRA, ARL) using the CuK_{α} edge with a wave length of 1.54 Å. Rietveld refinements on the data were run in the Fullprof. suite. A singlecrystal xray diffractometer was used to confirm the orientation of the single crystals. Susceptibility and heat capacity were measured in the physical property measurement system (PPMS9T) from Quantum Design.
INS experiment
Our INS experiment was performed on wide angularrange chopper spectrometer (ARCS) at Spallation Neutron Source (SNS) of Oak Ridge National Laboratory (ORNL). For the experiment, we coaligned 40 pieces of single crystals weighing about 3 g in total using a backscattering Laue xray diffractometer. The single crystals glued on an aluminum plate with a sample mosaic of 1.5° were loaded into a closedcycle refrigerator with the [010] direction aligned in the vertical direction. Data were collected by rotating the sample about the [010] axis with an incident energy E_{i} = 35 meV and a chopper frequency of 300 Hz resulting in an energy resolution of about 1.4 meV. We collected data at various temperatures. At 5 K, the data were collected by rotating the sample by 90° in a 1.25° step. For other high temperatures, data were collected with a 5° step. We used DAVE^{67} to analyse the data. The wave vector Q was expressed as Q = (2π/a, 2π/b, 2π/c) reciprocal lattice unit (rlu) with a = b = c = 9.537(3) Å. Data in Fig. 2a–c were obtained by integrating the experimental data along two other orthogonal directions, with a thickness of [H, 0, 0] = [−3.2, −2.8], [0, K, 0] = [−0.2, 0.2]; [−L, 0, L] = [0.8, 1.2], [0, K, 0] = [−0.2, 0.2]; and [−L, 0, L] = [1.3, 1.7], [−K, 2K, −K] = [0.1, 0.2], respectively. Data in Fig. 3a,b were integrated over [H, 0, 0] = [−3.2, −2.8] and [−L, 0, L] = [1.3, 1.7], respectively.
Linearspinwave theory
In order to fit the experimental data, we used the Heisenberg model involving exchange interactions up to the sixth nearest neighbour (NN),
where nNN indicates the nth NN bond and J_{n} is the magnitude of the nth NN Heisenberg term. Since this material is a collinear antiferromagnet, we performed the calculations with the linearspinwave theory. After performing standard Holstein–Primakoff transformation and diagonalizing the quadric Hamiltonian, we obtained the magnon dispersions.
To compare with the experimental data, we calculated the neutron scattering cross section
where Q_{α = x,y,z} is the α component of Q and the S^{αβ}(Q, E) is the spin–spin correlation function defined by
Here, S_{i} is the effective spin at site i with the coordinate r_{i}.
Data availability
Data supporting the findings of this study are available from the corresponding author J.S.W. (Email: jwen@nju.edu.cn) upon reasonable request.
References
Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).
Qi, X.L. & Zhang, S.C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011).
Young, S. M. et al. Dirac semimetal in three dimensions. Phys. Rev. Lett. 108, 140405 (2012).
Wang, Z. et al. Dirac semimetal and topological phase transitions in A_{3}Bi (A = Na, K, Rb). Phys. Rev. B 85, 195320 (2012).
Liu, Z. K. et al. Discovery of a threedimensional topological Dirac semimetal, Na_{3}Bi. Science 343, 864–867 (2014).
Liu, Z. K. et al. A stable threedimensional topological Dirac semimetal Cd_{3}As_{2}. Nat. Mater. 13, 677–681 (2014).
Wan, X., Turner, A. M., Vishwanath, A. & Savrasov, S. Y. Topological semimetal and Fermiarc surface states in the electronic structure of pyrochlore iridates. Phys. Rev. B 83, 205101 (2011).
Xu, S.Y. et al. Discovery of a Weyl fermion semimetal and topological Fermi arcs. Science 349, 613–617 (2015).
Lv, B. Q. et al. Observation of Weyl nodes in TaAs. Nat. Phys. 11, 724–727 (2015).
Wehling, T., BlackSchaffer, A. M. & Balatsky, A. V. Dirac materials. Adv. Phys. 63, 1–76 (2014).
Bansil, A., Lin, H. & Das, T. Colloquium: topological band theory. Rev. Mod. Phys. 88, 021004 (2016).
Zhu, Z., Winkler, G. W., Wu, Q., Li, J. & Soluyanov, A. A. Triple point topological metals. Phys. Rev. X 6, 031003 (2016).
Bradlyn, B. et al. Beyond Dirac and Weyl fermions: unconventional quasiparticles in conventional crystals. Science 353, aaf5037 (2016).
Bradlyn, B. et al. Topological quantum chemistry. Nature 547, 298–305 (2017).
Tang, P., Zhou, Q. & Zhang, S.C. Multiple types of topological fermions in transition metal silicides. Phys. Rev. Lett. 119, 206402 (2017).
Ma, J. Z. et al. Threecomponent fermions with surface Fermi arcs in tungsten carbide. Nat. Phys. 14, 349–354 (2018).
Lv, B. Q. et al. Observation of threecomponent fermions in the topological semimetal molybdenum phosphide. Nature 546, 627–631 (2017).
Strohm, C., Rikken, G. L. J. A. & Wyder, P. Phenomenological evidence for the phonon Hall effect. Phys. Rev. Lett. 95, 155901 (2005).
Stenull, O., Kane, C. L. & Lubensky, T. C. Topological phonons and Weyl lines in three dimensions. Phys. Rev. Lett. 117, 068001 (2016).
SerraGarcia, M. et al. Observation of a phononic quadrupole topological insulator. Nature 555, 342 (2018).
Peterson, C. W., Benalcazar, W. A., Hughes, T. L. & Bahl, G. A quantized microwave quadrupole insulator with topologically protected corner states. Nature 555, 346 (2018).
Zhang, T. et al. DoubleWeyl phonons in transitionmetal monosilicides. Phys. Rev. Lett. 120, 016401 (2018).
Lu, L., Joannopoulos, J. D. & Soljacic, M. Topological photonics. Nat. Photon. 8, 821–829 (2014).
Lu, L. et al. Experimental observation of Weyl points. Science 349, 622–624 (2015).
Zilberberg, O. et al. Photonic topological boundary pumping as a probe of 4D quantum Hall physics. Nature 553, 59 (2018).
Zhou, H. et al. Observation of bulk Fermi arc and polarization half charge from paired exceptional points. Science 359, 1009–1012 (2018).
Yang, B. et al. Ideal Weyl points and helicoid surface states in artificial photonic crystal structures. Science 359, 1013–1016 (2018).
Harari, G. et al. Topological insulator laser: theory. Science 359, eaar4003 (2018).
Bandres, M. A. et al. Topological insulator laser: experiments. Science 359, eaar4005 (2018).
Zhang, L., Ren, J., Wang, J.S. & Li, B. Topological magnon insulator in insulating ferromagnet. Phys. Rev. B 87, 144101 (2013).
Chisnell, R. et al. Topological magnon bands in a Kagome lattice ferromagnet. Phys. Rev. Lett. 115, 147201 (2015).
Mook, A., Henk, J. & Mertig, I. Edge states in topological magnon insulators. Phys. Rev. B 90, 024412 (2014).
Romhnyi, J., Penc, K. & Ganesh, R. Hall effect of triplons in a dimerized quantum magnet. Nat. Commun. 6, 6805 (2015).
McClarty, P. A. et al. Topological triplon modes and bound states in a Shastry–Sutherland magnet. Nat. Phys. 13, 736 (2017).
Fransson, J., BlackSchaffer, A. M. & Balatsky, A. V. Magnon Dirac materials. Phys. Rev. B 94, 075401 (2016).
Okuma, N. Magnon spinmomentum locking: various spin vortices and Dirac magnons in noncollinear antiferromagnets. Phys. Rev. Lett. 119, 107205 (2017).
Owerre, S. A. Magnonic analogs of topological Dirac semimetals. J. Phys. Commun. 1, 025007 (2017).
Pershoguba, S. S. et al. Dirac magnons in honeycomb ferromagnets. Phys. Rev. X 8, 011010 (2018).
Mena, M. et al. Spinwave spectrum of the quantum ferromagnet on the pyrochlore lattice Lu_{2}V_{2}O_{7}. Phys. Rev. Lett. 113, 047202 (2014).
Li, F.Y. et al. Weyl magnons in breathing pyrochlore antiferromagnets. Nat. Commun. 7, 12691 (2016).
Mook, A., Henk, J. & Mertig, I. Tunable magnon Weyl points in ferromagnetic pyrochlores. Phys. Rev. Lett. 117, 157204 (2016).
Su, Y., Wang, X. S. & Wang, X. R. Magnonic Weyl semimetal and chiral anomaly in pyrochlore ferromagnets. Phys. Rev. B 95, 224403 (2017).
Ross, K. A., Savary, L., Gaulin, B. D. & Balents, L. Quantum excitations in quantum spin ice. Phys. Rev. X 1, 021002 (2011).
Owerre, S. A. Weyl magnons in noncoplanar stacked kagome antiferromagnets. Phys. Rev. B 97, 094412 (2018).
Owerre, S. A. Magnonic triplydegenerate nodal points. Eur. Phys. Lett. 120, 57002 (2017).
Katsura, H., Nagaosa, N. & Lee, P. A. Theory of the thermal Hall effect in quantum magnets. Phys. Rev. Lett. 104, 066403 (2010).
Onose, Y. et al. Observation of the magnon Hall effect. Science 329, 297–299 (2010).
Matsumoto, R. & Murakami, S. Theoretical prediction of a rotating magnon wave packet in ferromagnets. Phys. Rev. Lett. 106, 197202 (2011).
Ideue, T. et al. Effect of lattice geometry on magnon Hall effect in ferromagnetic insulators. Phys. Rev. B 85, 134411 (2012).
Zhang, L. Berry curvature and various thermal Hall effects. New J. Phys. 18, 103039 (2016).
Hirschberger, M., Chisnell, R., Lee, Y. S. & Ong, N. P. Thermal Hall effect of spin excitations in a Kagome magnet. Phys. Rev. Lett. 115, 106603 (2015).
Shindou, R., Matsumoto, R., Murakami, S. & Ohe, J.i Topological chiral magnonic edge mode in a magnonic crystal. Phys. Rev. B 87, 174427 (2013).
Rückriegel, A., Brataas, A. & Duine, R. A. Bulk and edge spin transport in topological magnon insulators. Phys. Rev. B 97, 081106 (2018).
Baltz, V. et al. Antiferromagnetic spintronics. Rev. Mod. Phys. 90, 015005 (2018).
Chumak, A., Vasyuchka, V., Serga, A. & Hillebrands, B. Magnon spintronics. Nat. Phys. 11, 453–461 (2015).
Li, K., Li, C., Hu, J., Li, Y. & Fang, C. Dirac and nodal line magnons in threedimensional antiferromagnets. Phys. Rev. Lett. 119, 247202 (2017).
Herak, M. et al. Novel spin lattice in Cu_{3}TeO_{6}: an antiferromagnetic order and domain dynamics. J. Phys. Condens. Matter 17, 7667 (2005).
Månsson, M. et al. Magnetic order and transitions in the spinweb compound Cu_{3}TeO_{6}. Phys. Procedia 30, 142–145 (2012).
He, Z. & Itoh, M. Magnetic behaviors of Cu_{3}TeO_{6} with multiple spin lattices. J. Magn. Magn. Mater. 354, 146–150 (2014).
Norman, M. Copper tellurium oxides  a playground for magnetism. J. Magn. Magn. Mater. 452, 507–511 (2018).
Zhu, X. et al. High resolution electron energy loss spectroscopy with twodimensional energy and momentum mapping. Rev. Sci. Instrum. 86, 083902 (2015).
Harten, U. & Toennies, J. P. Surface phonons on GaAs(110) measured by inelastic helium atom scattering. Eur. Phys. Lett. 4, 833 (1987).
van Kampen, M. et al. Alloptical probe of coherent spin waves. Phys. Rev. Lett. 88, 227201 (2002).
Shen, K. & Bauer, G. E. W. Laserinduced spatiotemporal dynamics of magnetic films. Phys. Rev. Lett. 115, 197201 (2015).
Hashimoto, Y. et al. Alloptical observation and reconstruction of spin wave dispersion. Nat. Commun. 8, 15859 (2017).
Yao, W. et al. Topological spin excitations observed in a threedimensional antiferromagnet. Preprint at https://arxiv.org/abs/1711.00632 (2017).
Azuah, R. T. et al. DAVE: a comprehensive software suite for the reduction, visualization, and analysis of low energy neutron spectroscopic data. J. Res. NIST 114, 341 (2009).
Acknowledgements
Work at Nanjing University was supported by National Natural Science Foundation of China with Grant Nos. 11674157, 11774152, 11374138, 11674158 and 11525417, National Key Projects for Research & Development of the Ministry of Science and Technology of China with Grant No. 2016YFA0300401, and Fundamental Research Funds for the Central Universities with Grant No. 020414380105. The research at Oak Ridge National Laboratory’s Spallation Neutron Source was sponsored by the US Department of Energy, office of Basic Energy Sciences, Scientific User Facilities Division. We thank Jian Sun, S. A. Owerre, Yuan Li, and Ka Shen for stimulating discussions.
Author information
Authors and Affiliations
Contributions
J. S. W. and J. X. L. conceived the project. S. B. grew the crystals. S. B. and Z. M. carried out the neutron scattering experiments with help from D. L. A. W. W., S. L. Y., D. W. and X. G. W. performed the theoretical calculations. S. B., J. H. W., Z. W. C. and J. S. W. analysed the data. J. S. W., J. X. L. and S. B. wrote the paper with inputs from all authors.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Electronic supplementary material
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Bao, S., Wang, J., Wang, W. et al. Discovery of coexisting Dirac and triply degenerate magnons in a threedimensional antiferromagnet. Nat Commun 9, 2591 (2018). https://doi.org/10.1038/s41467018050542
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41467018050542
Further reading

Dipolar spinwaves and tunable band gap at the Dirac points in the 2D magnet ErBr3
Communications Physics (2022)

Orderbydisorder from bonddependent exchange and intensity signature of nodal quasiparticles in a honeycomb cobaltate
Nature Communications (2021)

Acoustic spin1 Weyl semimetal
Science China Physics, Mechanics & Astronomy (2020)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.