Weak localization of magnons in chiral magnets

We report on the impact of the Dzyaloshinskii-Moriya interaction on the coherent backscattering of spin waves in a disordered magnetic material. This interaction breaks the inversion symmetry of the spin-wave dispersion relation, such that ${\omega }_{\mathbf{k}}={\omega }_{2{\mathbf{K}}^{\mathrm{I}}-\mathbf{k}}\ne {\omega }_{-\mathbf{k}}$, where ${\mathbf{K}}^{\mathrm{I}}$ is related to the Dzyaloshinskii-Moriya vectors. The nonequivalence of $\mathbf{k}$ and $-\mathbf{k}$ also means that time-reversal symmetry is broken. As a result of numerical investigations we find that the backscattering peak of a wave packet with initial wave vector ${\mathbf{k}}_0$ shifts from $-{\mathbf{k}}_0$ to $2{\mathbf{K}}^{\mathrm{I}}-{\mathbf{k}}_0$, such that the backscattering wave vector and the initial wave vector are in general no longer antiparallel. The shifted coherence condition is explained by a diagrammatic approach and opens up an avenue to measure sign and magnitude of the Dzyaloshinskii-Moriya interaction in weakly disordered chiral magnets. Surprisingly, although time-reversal symmetry is broken, our system shows coherent backscattering as a manifestation of weak localization, which is due to the fact that reciprocity is still preserved.

Figure 1

CBS of spin waves in chiral magnets with DM interaction. (a) Ensemble-averaged spin-wave intensity in $k$ space, $I_{\mathbf{k}}=\langle {\left|{\mathcal{S}}_{\mathbf{k}}\right|}^2\rangle$ at time $t=20t_J$. The initial wave packet [Eq. (6)] is centered at ${\mathbf{k}}_0$. A CBS peak rises at ${\mathbf{k}}_{\mathrm{CBS}}\ne -{\mathbf{k}}_0$ over the incoherent background. Both the center of the Brillouin zone $\mathrm{\Gamma }=(0,0)$ and the center of inversion ${\mathbf{K}}^{\mathrm{I}}$ are plotted. The CBS peak position is the conjugate of ${\mathbf{k}}_0$ with respect to ${\mathbf{K}}^{\mathrm{I}}$, namely, ${\mathbf{k}}_{\text{CBS}}-{\mathbf{K}}^{\mathrm{I}}={\mathbf{K}}^{\mathrm{I}}-{\mathbf{k}}_0$. (b) Time evolution of the CBS contrast with and without DM interaction. The dashed curve is the expected decay $\mathcal{C}\left(t\right)={(1+Dt/2{\sigma }_0^2)}^{-1}$ with spin-wave diffusion constant $D\approx 19a^2/t_J$ extracted from the real-space diffusive spread of the wave packet. Within the noise of the data, there is no observable difference between the cases with and without DM interaction.

Figure 2

(a) Diagrammatic representation for the ensemble-averaged Green's function, Eq. (7). (b) Incoherent background intensity kernel, a sum of ladder diagrams [Eq. (8)]. (c) CBS contribution given by the sum of maximally crossed diagrams [Eq. (9)]. Upon choosing $\mathbf{k}+{\mathbf{k}}_0=2{\mathbf{K}}^{\text{I}}$, this contribution equals the background of (b) and thus yields perfect interference contrast at the shifted position ${\mathbf{k}}_{\text{CBS}}=2{\mathbf{K}}^{\text{I}}-{\mathbf{k}}_0$.