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, ${\mathrm{Cu}}_3{\mathrm{TeO}}_6$, 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.

Figure 1

The ${\mathrm{Cu}}^{2+}$ sublattice of ${\mathrm{Cu}}_3{\mathrm{TeO}}_6$ in a cubic unit cell, with spin-up and -down ions represented by different colors. Nearest-neighbor ($J_1$) and next-nearest-neighbor ($J_2$) interactions are indicated.

Figure 2

(a) A typical band structure of the spin-wave dispersion along high-symmetry lines with $J_2=0.134J_1>0$, where the inset shows an enlarged region near $P$. (b) Positions of all Dirac points in the first Brillouin zone. Red and yellow indicate, respectively, a monopole charge of $+1$ or $-1$. For clarity, only one of the eight $D_3$ points is displayed in the three-dimensional Brillouin zone in (b). Above the Brillouin zone, we schematically show how, upon adding the DMIs, a Dirac point at $P$ expands either into a nodal ring around [111] or into a line along [111], preserving the threefold rotation along the [111] axis.

Figure 3

(a) The classical ground state configuration for $D/J_1=0.2$. Spin-up and -down ions are represented by different colors with nearest-neighbor ($J_1$) and next-nearest-neighbor ($J_2$) interactions indicated. (b) The same configuration projected on the (111) plane. (c) Table of the exact direction of each polarization, where angle $\theta$ and $\varphi$ are defined in the corner panels of (a),(b).