Magnon Spin-Momentum Locking: Various Spin Vortices and Dirac magnons in Noncollinear Antiferromagnets

We generalize the concept of the spin-momentum locking to magnonic systems and derive the formula to calculate the spin expectation value for one-magnon states of general two-body spin Hamiltonians. We give no-go conditions for magnon spin to be independent of momentum. As examples of the magnon spin-momentum locking, we analyze a one-dimensional antiferromagnet with the Néel order and two-dimensional kagome lattice antiferromagnets with the 120° structure. We find that the magnon spin depends on its momentum even when the Hamiltonian has the $z$-axis spin rotational symmetry, which can be explained in the context of a singular band point or a $U(1)$ symmetry breaking. A spin vortex in momentum space generated in a kagome lattice antiferromagnet has the winding number $Q=-2$, while the typical one observed in topological insulator surface states is characterized by $Q=+1$. A magnonic analogue of the surface states, the Dirac magnon with $Q=+1$, is found in another kagome lattice antiferromagnet. We also derive the sum rule for $Q$ by using the Poincaré-Hopf index theorem.

Figure 1

(a) Schematics of the magnon spin-momentum locking with the winding number $Q=-2$ in a magnon band of a kagome lattice antiferromagnet with the Dzyaloshinskii-Moriya interaction denoted by $\pm D\widehat{z}$. The magnetic unit cell, shown by the green region, is the same as the chemical unit cell. Precise vector plots of magnon spin are shown in Fig. 3. (b) Upper: real space illustration of spin-momentum-locked magnon modes with $\mathit{k}=(k_x,0)$ (red) and $\mathit{k}=(0,-k_y)$ (blue). Lower: schematics of magnon spin flip device using the spin-momentum locking with $Q=-2$.

Figure 2

(a) Schematics of the ground state of a one-dimensional antiferromagnet described by Eq. (11). The ground state is the Néel state with two sublattices ($A$, $B$), and their spins are parallel to the DM vector $D\widehat{z}$. (b) Magnon band dispersions for $J=1$, $D=0.1$, and $K=-0.05$. The solid and dotted lines denote the magnon states with $S^z=-1$ and 1, respectively. (c) The contributions from the $A$ and $B$ sublattices to the $z$-component spin in the upper and lower bands are plotted for the momentum $k$. The total $S^z$ is quantized, and its sign is changed across the band crossing points.

Figure 3

Magnon band dispersions and $\mathit{k}$-dependent spin in kagome lattice antiferromagnets described by Eq. (14) with (a) $J^x=J^y=J^z=1$, $D=0.1$ and (b) $J^x=J^y=1$, $J^z=D=0$. The lattice constant $a=1$. The hexagonal region surrounded by the red dotted lines is the first Brillouin zone.