Domain Wall Dynamics in Classical Spin Chains: Free Propagation, Subdiffusive Spreading, and Soliton Emission

The nonequilibrium dynamics of domain wall initial states in a classical anisotropic Heisenberg chain exhibits a striking coexistence of apparently linear and nonlinear behaviors: the propagation and spreading of the domain wall can be captured quantitatively by linear, i.e., noninteracting, spin wave theory absent its usual justifications; while, simultaneously, for a wide range of easy-plane anisotropies, emission can take the place of stable solitons—a process and objects intrinsically associated with interactions and nonlinearities. The easy-axis domain wall only has transient dynamics, the isotropic one broadens diffusively, while the easy-plane one yields a pair of ballistically counterpropagating domain walls which, unusually, broaden subdiffusively, their width scaling as $t^{1/3}$.

Figure 1

(a) Schematic of the initial conditions (3), in the $XY$ plane, shown for $\phi =\pi /2$. (b) Boundaries between the different dynamical regimes as a function of the anisotropy $\mathrm{\Delta }$ and amplitude $\phi$. I.A and I.B are the two linear regimes (distinguished by whether the oscillations are behind or ahead of the counter-propagating domain walls, respectively), where all of the dynamical features are well described by linear spin wave theory; II is the easy-plane nonlinear regime, where solitons coexist with the spreading domain walls. III is the easy-axis regime, with a single, static domain wall. The isotropic point $\mathrm{\Delta }=1$ corresponds to the transition between I.B and III, and is well described by linear spin wave theory with a single, diffusively broadening domain wall. The vertical bars denote the uncertainty in determining the transition between I.B and II from the simulations. (c) Overview of the domain wall dynamics shown for the easy-plane $\mathrm{\Delta }=0.3$ (inverted shaded triangle), the isotropic point $\mathrm{\Delta }=1$ (shaded square) and the easy-axis $\mathrm{\Delta }=1.2$ (shaded triangle), respectively. Note the different ranges of the $x$ axes. $\phi =\pi /2$ for the easy-plane and the isotropic case, where $S^y$ is plotted. $S^z$ is plotted for the easy-axis case, with $\phi =\pi$. Ballistic counterpropagation is observed in the easy-plane, diffusive melting of the original domain wall is seen at the isotropic point, while the easy-axis approaches a very narrow static soliton. Note that, since the subdiffusive domain wall spreading in the easy plane is parametrically slower than the ballistic propagation, and the emitted solitons move only very slightly slower than the domain walls and so are not well separated over the timescales shown, both of these are difficult to see on this overview plot—they are seen more readily in Fig. 2 instead.

Figure 2

Dynamics in the easy-plane case. Dotted lines show spin wave predictions, where relevant. Only the left-moving domain walls are shown, and lighter colors indicate later times. (a) Linear regime I.A ($\mathrm{\Delta }=0$, $\phi =\pi /2$), showing the ballistic propagation and subdiffusive spreading of the domain wall, with the oscillations trailing. (b) Linear regime I.B ($\mathrm{\Delta }=0.3$, $\phi =3\pi /10$), where the oscillations are now ahead of the domain wall. (c) Nonlinear regime II ($\mathrm{\Delta }=0.25$, $\phi =\pi /2$), showing that the domain wall decays by emitting a soliton, though its speed and width scaling are unaffected; the soliton moves at a slower velocity $v<c$, and the separation increases with time. (d) Same parameters and times as (c), but in the comoving frame of the soliton.

Figure 3

Soliton emission in the easy-plane dynamics. (a),(b) Dependence of the soliton energy $E_S$ and soliton amplitude ${\phi }_S$, respectively, on the initial amplitude $\phi$. We observe that the amplitude of the emitted solitons is almost constant. (c),(d) Two- and three-soliton emission, respectively, when $\phi \gg {\phi }_S$.

Figure 4

Dynamics (a) at the isotropic point, and (b) in the easy-axis regime ($\mathrm{\Delta }=1.2$). There are no propagating domain walls in either case: the initial state spreads diffusively at the isotropic point, whilst it approaches the static soliton in the easy-axis case.