Domain-wall dynamics in the Landau-Lifshitz magnet and the classical-quantum correspondence for spin transport

We investigate the dynamics of spin in the axially anisotropic Landau-Lifshitz field theory with a magnetic domain-wall initial condition. Employing the analytic scattering technique, we obtain the exact scattering data and reconstruct the time-evolved profile. We identify three qualitatively distinct regimes of spin transport, ranging from ballistic expansion in the easy-plane regime, absence of transport in the easy-axis regime, and logarithmically enhanced diffusion for the isotropic interaction. Our results are in perfect qualitative agreement with those found in the anisotropic quantum Heisenberg spin-$1/2$ chain, indicating a remarkable classical-quantum correspondence for macroscopic spin transport.

Figure 1

Schematic representation of the integration protocol: The forward “nonlinear Fourier transform” $\mathcal{F}$ maps the initial spin field $\vec{S}(x,0)$ to the spectral data $\left\{a\right(\lambda ),b(\lambda \left)\right\}$. The latter satisfies simple time evolution (8). The inverse transform ${\mathcal{F}}^{-1}$ amounts to solving an appropriate linear integral equation from where one reconstructs the time-evolved spin field $\vec{S}(x,t)$.

Figure 2

Time-dependent density profiles of $S^z$ component in the easy-plane $\delta =-1$ (left), isotropic $\delta =0$ (middle), and easy-axis $\delta =9$ (right) regimes, displaying ballistic spin transport, logarithmically enhanced diffusion, and absence of transport, respectively. The dashed lines show $|S^z|=\{0.2,0.4,0.8\}$.

Figure 3

Spin diffusion constant $\mathfrak{D}\left(\gamma \right)$ as a function of the twisting angle $\gamma$, shown for the self-similarity solutions (open circles) and numerical integration up to $t=2000$ (red crosses). The blue dashed line shows the leading term in the large-$E$ asymptotic expansion of the self-similar solution. Inset: Numerical solution to the inverse scattering transform of the untwisted domain-wall profile (2).