Recently topological aspects of magnon band structure have attracted much interest, and especially, the Dirac magnons in Cu3TeO6 have been observed experimentally. In this work, we calculate the magnetic exchange interactions Js using the first-principles linear-response approach and find that these Js are short-range and negligible for the Cu-Cu atomic pair apart by longer than 7 Angstrom. Moreover there are only 5 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 (about 1.3{deg}) of the magnetic non-collinearity based on the competition between DMIs and Js, which is consistent with the experiment. The small canting angle agrees with that the 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 [Nat. Phys. 14, 1011 (2018)] holds but only up to the 11th nearest neighbour.
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