LAUSR

Recently, topological aspects of magnon band structure have attracted much interest, and especially, the Dirac magnons in ${\mathrm{Cu}}_{3}{\mathrm{TeO}}_{6}$ have been observed experimentally. In this work, we calculate the magnetic exchange interactions $J$'s using the first-principles linear-response approach and find that these $J$'s are short range and negligible for the Cu-Cu atomic pair longer than 7 \AA{}. Moreover, there are only five sizable magnetic exchange interactions, and according to their signs and strengths, modest magnetic frustration is expected. Based on the obtained magnetic exchange couplings, we successfully reproduce the experimental spin-wave dispersions. The calculated neutron scattering cross section also agrees very well with the experiments. We also calculate Dzyaloshinskii-Moriya interactions (DMIs) and estimate the canting angle ($\ensuremath{\sim}1.3{}^{\ensuremath{\circ}}$) of the magnetic noncollinearity based on the competition between DMIs and $J$'s, which is consistent with the experiment. The small canting angle agrees with that current experiments cannot distinguish the DMI-induced nodal line from a Dirac point in the spin-wave spectrum. Finally, we analytically prove that the ``sum rule'' conjectured in W. Yao et al. [Nat. Phys. 14, 1011 (2018)] holds but only up to the 11th-nearest neighbor.