Magnetic Excitations in Non-Collinear Antiferromagnetic Weyl Semimetal $\mathsf{Mn_{3}Sn}$

$\mathsf{Mn_{3}Sn}$ has recently attracted considerable attention as a magnetic Weyl semimetal exhibiting concomitant transport anomalies at room temperature. The topology of the electronic bands, their relation to the magnetic ground state and their nonzero Berry curvature lie at the heart of the problem. The examination of the full magnetic Hamiltonian reveals otherwise hidden aspects of these unusual physical properties. Here, we report the full spin wave spectra of $\mathsf{Mn_{3}Sn}$ measured over a wide momentum - energy range by the inelastic neutron scattering technique. Using a linear spin wave theory, we determine a suitable magnetic Hamiltonian which not only explains the experimental results but also stabilizes the low-temperature helical phase, consistent with our DFT calculations. The effect of this helical ordering on topological band structures is further examined using a tight-binding method, which confirms the elimination of Weyl points in the helical phase. Our work provides a rare example of the intimate coupling between the electronic and spin degrees of freedom for a magnetic Weyl semimetal system.


Introduction
It is an intriguing question how certain electronic band structures give rise to nontrivial topological properties. Initial work focused on how unique edge states form for particular symmetries, leading to the now broad and popular field of topological insulators 1 . More recently, attention has concentrated on magnetic systems, where Dirac points at band crossings are broken into two sets of so-called Weyl points because of the broken time-reversal symmetry. Systems with Weyl points exhibit a nontrivial Berry's phase, which can be characterized by measuring transport anomalies such as the anomalous Hall effect (AHE). Mn 3 Sn, a non-collinear metallic antiferromagnet whose magnetic ions sit on a kagome lattice, has recently drawn growing interest for its rather remarkable transport anomalies. Density functional calculations examining the Weyl points 2−3 , and transport measurements 4−8 of Mn 3 Sn show that this antiferromagnetic metal exhibits significant transport anomalies at room temperature. Subsequently, the presence of the Weyl points in the electronic band structure have been confirmed by angle-resolved photoemission spectroscopy (ARPES) and magnetoresistance measurements 9 . More recently, it was suggested that it may also have spin polarized current, which would make it an interesting candidate for spintronics applications 10 .
According to studies done in the 1980's and 1990's the magnetic structure of Mn 3 Sn has a 120 • structure with a negative vector chirality at room temperature. This ground state indicates the presence of significant Dzyaloshinskii-Moriya (DM) interactions mediated by a spin-orbit coupling 11−14 , which, together with an easy-axis anisotropy, stabilizes a weak ferromagnetic moment 13,15 . On the other hand, the low temperature phase of Mn 3 Sn is known to depend on the precise ratio between Mn and Sn contents (ref. 16). Some earlier results show that Mn 3 Sn displays an additional incommensurate helical ordering along the c-axis (k z 0.09) below 220∼270 K 17,18 (we call it as the A-type Mn 3 Sn), while others found a glass phase below 50 K, instead of the helical ordering 19,20 (we call it as the B-type Mn 3 Sn). While both types show the same transport anomalies at room temperature 4,7 , the A-type Mn 3 Sn does not show the transport anomaly at the low temperature phase with the helical ordering 21 .
It is usually beneficial to carry out inelastic neutron scattering experiments at lower temperature because of thermal fluctuations. However, the complex glass phase of the B-type Mn 3 Sn makes it difficult to use the 2 inelastic neutron scattering technique at low temperature. Meanwhile, we note that both types are identical in terms of the crystal and magnetic structures at room temperature 9,18,20,21 , and it is also known that the phase transition at 220∼270 K in the A-type Mn 3 Sn is not accompanied by any structural transition 22 .
Thus, we can gain insight into the magnetic Hamiltonian of Mn 3 Sn at room temperature by measuring the A-type Mn 3 Sn at low temperature, since there should be, a priori, no significant difference in the intrinsic magnetic Hamiltonian for both types of Mn 3 Sn samples with the same crystal structure.
The magnetic excitation spectra of the A-type, helically ordered, Mn 3 Sn were previously studied using inelastic neutron scattering (INS) in the early 1990's with limited success 23−25 . Here, we report the full spin wave spectra of the A-type Mn 3 Sn over a wide momentum-energy range measured by inelastic neutron scattering. The full magnetic Hamiltonian based on a local moment model is proposed in order to explain the data completely using a linear spin wave theory. Supported by density functional theory (DFT) calculations, we also present a low temperature magnetic structure which involves a helical ordering with some important details that have so far been neglected in previous studies, including its effect on the Weyl points of Mn 3 Sn.

Results
Magnetic Structure A 120 • magnetic structure with negative vector chirality (Γ 5 ) is used within each single a-b plane, which has been confirmed by both theoretical calculations and experiments 11−15 (Fig 1b).
However, because our inelastic neutron scattering experiment was done at 5 K, extra effects from the low temperature phase must be considered rather than just using the room temperature phase. As we noted in the introduction, there are two types of Mn 3 Sn : A and B-type. By measuring the magnetization of our sample, we observed a clear phase transition near 260 K but failed to observe any increasing magnetization below 50 K (see Supplementary Fig. 1). This result confirms that our sample has the helical order of the A-type at low temperature as shown in Fig 1a. There are a few other noteworthy features of this phase. First, the spin directions of two Mn atoms connected by inversion symmetry are no longer parallel in the helical phase. Instead, one of them is rotated by as much as half a turn ( 16.3 • ) in the a-b plane by the helical ordering (Fig. 1a). This helical ordering has 3 significant effects on Weyl fermions and the corresponding Hall conductivity of Mn 3 Sn, which will be dealt with in the discussion section. The helical ordering of the A-type Mn 3 Sn is rather complicated : a previous neutron diffraction result shows that there are multiple satellite peaks at (1, 0, τ 1 ), (1, 0, τ 2 ), and (1, 0, 3τ 2 ), with τ 1 0.07 and τ 2 0.09 (ref. 17). The first two peaks correspond to two propagating helices, and the third one is due to the anharmonicity of the second helix. We can safely ignore the contribution by τ 1 for our spin waves calculations, since the diffraction peak of τ 1 is much weaker than the other one 26 . The existence of anharmonicity can also be well explained by our Hamiltonian, which will be shown later.

INS data and magnetic Hamiltonian
Our inelastic neutron scattering data show the spin wave spectra over the full Brillouin zone, which extends to almost 0.1 eV (Fig 2a). These coherent spin wave spectra prove to have a well-defined magnetic structure at 5 K. To explain the data, we use the following where ∆k is the bond vector corresponding to the k th nearest neighbor. The first two terms denote the usual isotropic exchange interaction and the DM interaction between Mn atom and its k th nearest neighbor, while the last two terms denote a single ion anisotropy related to the local easy axes on the a-b plane and the 6 th order anisotropy term, respectively. The last term originates from crystal field effects, described in terms of Stevens operator equivalents 27 . The necessity of this last term will be discussed in further detail later in this paper. With this Hamiltonian, we fitted the measured dispersion curves using a linear spin wave theory. Indicated by solid lines layered on Fig 2a, the fitted result from the Hamiltonian with coupling constants stated in Table 1 is consistent with the data. For our fitting, we used seven isotropic exchange parameters : up to the 2 nd in-plane nearest neighbor coupling and up to the 3 rd inter-plane nearest neighbor coupling.
We have also taken into account the effects of magnon damping and twins for our spin wave calculations.
The calculated spin wave spectra from a linear spin theory yield very large spectral weight above 80 meV (See Supplementary Fig. 3), whereas our data show quite a uniform intensity throughout almost the whole energy range. To resolve this discrepancy, energy-dependent magnon damping is included to the calculation, which is often necessary to explain the spin waves for metallic magnets 28,29 (see Supplementary Note 3 for theoretical background). The damping effect we considered is similar to that of refs 28, 29. By analyzing the elastic peak positions of the INS data, we found two kinds of twins rotated about ±3.5 • from the original lattice orientation. The calculated spin wave spectra with both twins and damping effect included are quite consistent with the data, supporting our Hamiltonian (Fig 2b & Fig 3).
When fitting the magnetic Hamiltonian, we also performed Monte Carlo simulations before the spin wave calculations to check whether our Hamiltonian stabilizes the magnetic structure mentioned before. Indeed, our Hamiltonian successfully demonstrates a helical ordering as the ground state, with k z of 0.0904. This stabilization is found in our model calculations to arise from exchange frustration, particularly due to the positive J 6 and the negative J 4 and J 5 . Note that the optimized k z value is found to be narrowly focused around 0.0904±0.01, since the resulting k z value varies sharply even if the J values are adjusted only slightly.

Spin-model based on DFT calculations
In order to render further credence to the above study of the helical ordering in Mn 3 Sn in terms of equation (1), we performed ab-initio calculations of the exchange interactions. Indeed, the dominating interactions fall within a distance of 5-6Å (see Fig. 4a) as predicted by our model Hamiltonian (Table 1) DM interaction and single ion anisotropy As shown in Table 1, we applied the DM interaction on the nearest in-plane neighbor coupling (J 2 and J 3 ) with its DM vector being parallel to theẑ axis (with the direction from site i to site j defined counterclockwise in the triangle formed by J 2 ). The most important role of this term is to stabilize the magnetic structure with negative vector chirality (Γ 5 ) since this term lowers the energy of that structure 12−14 . It also serves as an effective easy-plane anisotropy, stabilizing the structure in which the spins lie in the a-b plane. For these reasons, we assumed that the easy-plane anisotropy effect purely comes from our DM interaction term, rather than a single-ion anisotropy term with its easy plane perpendicular to the c-axis, similar to a recent theoretical study 30 .
To explore the effect of DM interactions more quantitatively, we calculated the DM interactions in the system using relativistic DFT 31 . The dominating z-components of the DM vectors are smaller at least by two orders of magnitude than the leading isotropic interactions (Fig 4a), consistent with our fitting results (see Table 1). When including the DM interactions to the calculation of E(k z ), we observe that the curve is shifted downwards (at k z = 0 by about 3 meV) stabilizing the Γ 5 spin-state against the Γ 3 state with different chirality (red line in Fig 4b). As the DM energy E DM (k z ) monotonously increases with k z (inset of Fig 4b), the out-of-plane DM interactions somewhat destabilize the helical ordering but this effect is marginal. Surprisingly, E DM (k z ) shows a quadratic dependence for small k z , in contrast to the usual linear dependence of spin-spirals with a ferromagnetic order normal to the propagation. Analytic calculations with nearest neighbor out-of-plane DM vectors indeed show that E DM (k z ) cos(k z c/2), which is fully confirmed numerically (inset of Fig 4b).
When it comes to an analysis of the DM interaction in the data, the low energy dynamics of the spin wave is important since the DM interaction is relatively small and has little influence on the spectra in We also found single ion anisotropy to be as crucial as the DM interaction in explaining several micro-6 scopic effects like an energy gap (∼ 5 meV) observed at the magnetic zone center (Fig 5a). It is already known that neither the ordinary easy axis anisotropy term (the third term in equation (1)) nor the fourthorder anisotropy term can produce the energy gap since the effect of these terms coming from each Mn site cancel out 25,27 . Considering symmetrically allowed two-ion anisotropy is also unnecessary since this term would destroy the subtle spin canting observed experimentally 13 (see Supplementary Note 4 for more details.) Thus we applied the sixth-order anisotropy term (the fourth term in equation (1)), and calculated spin waves near the magnetic zone center (Fig 5c). For the details of this calculation, see Supplementary Note 4. The calculation result clearly reproduces the energy gap at the C point (Fig 5c).
Moreover, the single ion anisotropy term related to the easy-axes (the third term in equation (1)) can explain the anharmonicity of the helical order. As noted before, a magnetic Bragg peak also exists at (1,0,3τ 2 ) in addition to the primary satellite peak at (1,0,τ 2 ) 17 . In terms of Fourier components, this can be interpreted as the third order anharmonicity of the helical ordering; some part of which shows a tendency to rotate three times the normal angle per unit cell. However, finding a corresponding magnetic structure analytically is very difficult. Instead, we performed a Monte Carlo simulation with our Hamiltonian to obtain the energy-minimized magnetic structure, and calculated the structure factor along the (1 0 L) line (See Methods). Interestingly enough, the simulation results show that the easy-axis anisotropy term clearly produces the third order anharmonicity (Fig 5e), which is also consistent with the data from the previous study (ref. 17).

Discussion
It is worth pointing out the key difference between our data and the previous reported results, extending the to a different symmetry due to the different Sn environments, (See Supplementary Fig. 4) and the same 7 feature holds also for J 2 and J 3 . Our DFT calculation indeed demonstrates the difference between J 4 and J 5 as well, giving multiple values for the bonds with the same bond length (See supplementary Fig. 5).
Also, our fitting parameters show similar oscillatory behavior to that of the parameters obtained from DFT calculation of Mn 3 Ir (ref. 31,33), which has a structure comparable to that of Mn 3 Sn (See Supplementary   Fig. 5). However, it is to be noted that both the parameters in Table 1  Magnetization was also measured to check the bulk properties (MPMS-3 EverCool, Quantum Design USA). The results were consistent with the previously reported results (See Supplementary Fig. 1).
Inelastic neutron scattering experiment. Inelastic neutron scattering experiment was done on the single crystal using the MERLIN time-of-flight spectrometer at ISIS, UK 39,40 . We used highly pure Al sample holder with proper Cd shielding to minimize unnecessary background signals. The measurements were performed at 5 K. By using the repetition-rate-multiplication (RRM) technique 41 , we could collect data simultaneously with various incident neutron energies (23, 42 and 100 meV, with chopper frequency of 400 Hz) in order to obtain the wide energy-momentum spectra with high quality. To cover the full spin wave spectra, the measurement with an incident neutron energy of 200 meV was done using a sloppy chopper (with chopper frequency of 500 Hz) in a single E i mode. Following standard procedures, time-independent background was removed from the raw data using the data collected between 15000 µs and 18000 µs, with the data corrections to compensate for the detector efficiency 42 . All data were symmetrized while conserving the lattice symmetry to improve statistics. Then we properly combined all the data from different neutron incident energies to obtain uniformly fine quality over a wide energy and momentum range. After the experiment, we analyzed the data using the Horace software 43 .
Theoretical calculations. We used spinW to calculate theoretical spin wave spectra, which is based on a linear spin wave theory using the method of diagonalizing the magnetic Hamiltonian by applying the Holstein-Primakoff transformation after local coordinate rotation 44 . For the simulation showing the emergence of anharmonicity in the helical ordering, we set a supercell of [1 1 6600] size and executed the energy-minimizing algorithm built in spinW to obtain the corresponding magnetic structure. Omitting some constants, the following well-known equation is used to calculate the structure factor.
where F(q) and p λ k denotes the magnetic form factor and polarization, respectively. The damping effect is also