Dirac and Nodal Line Magnons in Three-Dimensional Antiferromagnets

We study the topological properties of magnon excitations in three-dimensional antiferromagnets, where the ground state configuration is invariant under time reversal followed by space inversion (PT symmetry). We prove that Dirac points and nodal lines, the former being the limiting case of the latter, are the generic forms of symmetry-protected band crossings between magnon branches. As a concrete example, we study a Heisenberg spin model for a "spin-web" compound, Cu_{3}TeO_{6}, and show the presence of the magnon Dirac points assuming a collinear magnetic structure. Upon turning on symmetry-allowed Dzyaloshinsky-Moriya interactions, which introduce a small noncollinearity in the ground state configuration, we find that the Dirac points expand into nodal lines with nontrivial Z_{2}-topological charge, a new type of nodal line not predicted in any materials so far.