Dynamics of magnon fluid in Dzyaloshinskii-Moriya magnet and its manifestation in magnon-Skyrmion scattering

We construct a Holstein-Primakoff Hamiltonian for magnons in an arbitrary slowly varying spin background for a microscopic spin Hamiltonian consisting of ferromagnetic spin exchange, Dzyaloshinskii-Moriya exchange, and the Zeeman term. The Gross-Pitaevskii-type equation for magnon dynamics contains several background gauge fields pertaining to local spin chirality, inhomogeneous potential, and anomalous scattering that violates the boson number conservation. Nontrivial corrections to previous formulas derived in the literature are given. Subsequent mapping to hydrodynamic fields yields the continuity equation and the Euler equation of the magnon fluid dynamics. Magnon wave scattering off a localized Skyrmion is examined numerically based on our Gross-Pitaevskii formulation. The dependence of the effective flux experienced by the impinging magnon on the Skyrmion radius is pointed out, and compared to an analysis of the same problem using the Landau-Lifshitz-Gilbert equation.

Figure 1

Snapshots of scattered magnon waves for $\tilde{\kappa }=\kappa R$ equal to (a) 0.5, (b) 1.0, and (c) 1.5. Shown are color plots of the magnon density $\rho ={\left|b\right|}^2$ normalized by the incoming magnon density ${\rho }_0$. Skyrmion is indicated by a red circle of radius $R$. The magnon trajectory changes direction at $\tilde{\kappa }=1$ as anticipated from the total flux consideration in the HP theory. Arrows indicate the direction of incoming spin wave.

Figure 2

Snapshots of scattered spin waves in the LLG calculation for $\kappa R$ equal to (a,d) 0.9, (b,e) 1.5, and (c,f) 2.0. In the top row are plots of the spin wave density $\rho =\sqrt{{\left(\delta n_x\right)}^2+{\left(\delta n_y\right)}^2}$. See the main text for the definition of $\delta \mathbf{n}=(\delta n_x,\delta n_y)$. Bottom row shows the phase $\varphi =arctan(\delta n_y/\delta n_x)$. Skyrmion positions and their sizes are indicated by red circles. A clear sign of skew scattering at $\kappa R=0.9$ is completely gone at $\kappa R=2.0$. Arrows indicate the direction of incoming spin wave.