\subsection{Adiabatic construction of itinerant solitons}
\sectionn{Moving solitons}
As the connection between the stationary solitons of the two models does not guarantee the existence of moving soliton solutions of the Heisenberg chain, we next use our adiabatic procedure to extend  the `existence diagram' in Fig~\ref{fig:solitons}(a) to finite $k$. We consider a resultant state a soliton of the Heisenberg chain if the following conditions are satisfied: (i) there is, for all times, a unique local minimum of $z_i(t)$; (ii) for $t > t_A$, the torsion $\tau = \sum_i \tau_i$ is constant in time; (iii) the unique local minimum propagates with a constant velocity. These conditions are examined in more detail in \cite{supplemental}. An example of a thus constructed moving soliton is shown in Fig.~\ref{fig:solitons}(c), and compared to the original Ishimori soliton.


The resulting existence diagram (Fig.~\ref{fig:solitons}(a)), shows  that the solitons are stable in the Heisenberg model over a remarkably large range of parameters $(R,k)$, with the narrow solitons apparently becoming unstable first with increasing velocity ($\sim k$). 

We find no indication of a finite lifetime of the single soliton states which are stable under this adiabatic procedure. Moreover, the torsion -- generally not a conserved quantity of the Heisenberg chain -- is conserved in these states. Whilst stationary solutions of non-linear classical equations of motion are well known (see e.g. \cite{FLACH20081}), stable moving solitons are not generally expected to exist. 

Next we consider how the properties of Ishimori solitons are modified in the Heisenberg model. Fig.~\ref{fig:renormalisation} shows that the internal frequency (the frequency with which the in-plane spin-components rotate) and velocity of a Heisenberg soliton are suppressed. The energy (measured in both cases by the Heisenberg Hamiltonian) is only slightly reduced -- whilst the torsion is very slightly higher for the Heisenberg solitons (see also Fig.~\ref{fig:stability}, \cite{supplemental}). Overall, we note a remarkable similarity between the one-soliton properties in the Ishimori and Heisenberg chain.

\sectionn{Two-soliton scattering}
We now turn to interactions between the solitons.
To set the stage, we  briefly recall scattering in the Ishimori chain.
As a fully integrable model, interactions are completely described by the two-soliton phase-shifts, even for thermal multi-soliton states \cite{theodorakopoulos1995nontopological,theodorakopoulos1988semiclassical}. When two
solitons collide, the asymptotic result (compared to two separate one-soliton solutions) is unchanged, except that the solitons are displaced by a so-called phase-shift \cite{theodorakopoulos1995nontopological}:
\eqa{
\Delta(&R, k; R', k') = \mathrm{sgn}(v(R, k) - v(R', k')) \nonumber \\ 
&\times \frac{1}{2R} \log \left[\frac{\cosh(2(R + R')) - \cos(2(k - k'))}{\cosh(2(R - R')) - \cos(2(k - k'))}\right].
\label{phase_shift}
}
experienced by the soliton $(R, k)$, due to a collision with the soliton $(R', k')$. 


\begin{figure}
    \centering
    \includegraphics{figs/plot_3.pdf}
    \caption{Soliton scattering in the Heisenberg chain. Colour-scale shows the $z$-components, and is the same for (a), (b), and (c). (a) Single scattering event between two solitons with parameters $(R, k) = (0.1, 0.1)$ and $(R', k') = (0.1, -0.15)$. (b) Screening of the magnetisation transported by a narrower soliton as it moves through a wider soliton. (c) Repeated scattering of two solitons [$(R, k) = (0.1, 0.1)$ and $(R', k') = (0.1, 0)$] under periodic boundary conditions. (d) Comparison of the scattering phase-shift $\Delta(R, k = 0; 0.1, 0.1)$ in the Ishimori chain (solid line) and in the Heisenberg chain, for stationary target solitons. Phase-shifts in the Heisenberg chain obtained by averaging over 10 scattering events, cf. (c), and over the relative phases of the solitons -- the error bars are the standard deviation w.r.t. the relative phases.}
    \label{fig:scattering}
\end{figure}


Fig.~\ref{fig:scattering} displays the scattering of two Heisenberg solitons. The solitons survive scattering essentially unchanged (Fig.~\ref{fig:scattering}(a)), akin to the fully integrable model. Whilst the collisions do leave the solitons unchanged asymptotically, the magnetisation of a moving soliton is
`screened' during the collision with a larger soliton as seen in Fig.~\ref{fig:scattering}(b). Importantly, solitons survive multiple collisions (Fig.~\ref{fig:scattering}(c)), with the change to their trajectories apparently given by simple consecutive phase-shifts. 

There exist, nonetheless, some important differences between the Heisenberg and Ishimori cases. First, absent integrability, scattering in the Heisenberg chain is not expected to be perfectly lossless. Indeed, there is a very small amount of radiation emitted during the collision (approx. $\delta S^z \sim 10^{-6}$ in magnitude) in Fig.~\ref{fig:scattering}(a). Second, scattering from narrow solitons at small $k$ (where the existence diagram is wider in $R$) can emit significant amounts of radiation, although, curiously, the modified solitons that emerge appear to be stable to subsequent collisions. 

In addition, the phase-shift $\Delta$ appears not to depend only on the soliton parameters $R, k, R', k'$. We extract the phase-shifts in Fig.~\ref{fig:scattering}(d) by averaging over 10 scattering events. They are also averaged over the relative phase (azimuthal angles) of the solitons at the moment of collision. In the integrable case, this has no effect -- in the Heisenberg case, however, in particular for larger $R$, the phase-shift depends on the relative phase (see Fig.~\ref{fig:scatter3}).  We discuss these points in more detail in \cite{supplemental}.

Despite these differences, however, we emphasise that collisions over a large parameter regime in the Heisenberg model strongly resemble the scattering in the Ishimori case. Importantly, whilst the phase-shift $\Delta$ acquires some fluctuations, the velocities of the solitons remain unaffected by the collisions. 




\sectionn{Solitons in thermal states}
Whilst the Heisenberg chain supports solitons as stable solitary waves, which suffer only very weak dissipation in scattering events, the imperfect nature of the scattering
implies the existence of a thermal timescale on which they eventually decay. The question arises, then, as to what extent these solitons exert their influence on the hydrodynamics and transport properties: thermal states are not in any sense a dilute soliton gas, and it would not be unreasonable to expect solitons to experience so many scattering events that, unprotected by integrability, they collapse too swiftly to generate a discernible superdiffusive contribution to the transport of spin or energy.

We find that the torsion (\ref{tau}) allows us to track the trajectories of solitons through a thermal background: Figs.~\ref{fig:thermal_solitons}(a) and \ref{fig:thermal_solitons}(b) show the spacetime profile of the torsion $\tau(j, t)$ for a low-temperature thermal state of both the Heisenberg and Ishimori chains. The expected ballistic trajectories of the solitons are clearly observed in the Ishimori chain. Remarkably, very long-lived ballistic trajectories are also observed in the Heisenberg chain. These trajectories can also be seen in the $z$-spin component $S^z(j, t)$ (Figs.~\ref{fig:thermal_solitons}(c) and \ref{fig:thermal_solitons}(d)) -- though, since the magnetisation changes as they propagate, spin is not transported ballistically.
In a complementary approach, for both the Ishimori and the Heisenberg chain the thermal solitons can also be isolated by surrounding an initial thermal state with a fully-polarised state $\boldsymbol{S} = \hat{\boldsymbol{z}}$, and allowing the thermal state to expand into this vacuum during the subsequent dynamics(see Fig.~\ref{fig:inverse_scattering}, \cite{supplemental}).

\begin{figure}
    \centering
   
    \includegraphics{figs/plot_4_interp=gauss.pdf}
    \caption{Solitons in the thermal state at $T = 0.1 J$ for the Heisenberg (left) and Ishimori (right) chains. Upper and lower panels show  torsion $\tau_j(t)$, and $S^z_j(t)$ respectively. The ballistic trajectories in the torsion indicate the presence of long-lived solitons in the Heisenberg chain. These ballistic trajectories can also be seen in the plots of $S^z$, where the magnetisation carried by a soliton changes as it moves through the chain -- the mechanism preventing ballistic spin transport.
   
    }
    \label{fig:thermal_solitons}
\end{figure}

Having established the existence and nature of the almost integrable behaviour of the Heisenberg chain, we now address the observed KPZ scaling \cite{mcroberts2022anomalous}. In the Ishimori chain, KPZ -- rather than ballistic -- spin transport emerges as follows. As smaller, faster solitons move through larger, slower solitons, they rotate to the `local vacuum'  within the larger soliton. Thus, in any thermal state the magnetisation carried by the smaller solitons is screened on a timsecale set by the rate at which they encounter larger solitons. This argument is qualitatively the same as that for KPZ scaling in the quantum $S = 1/2$ Heisenberg chain that appears in \cite{gopalakrishnan2019kinetic,de2020superdiffusion,DeNardis_2021,Ilievski_2021}. Since the same behaviour is present in the Heisenberg chain at low temperature, this provides a qualitative picture of and explanation for the KPZ regime in spin transport.%



\sectionn{Conclusions}
Our work clearly establishes the existence of a family of solitons in the non-integrable ferromagnetic Heisenberg chain in terms of those known to exist in the integrable Ishimori chain.  
Furthermore, these solitons are shown to exist and be relevant for the dynamics of thermal low-temperature states of the Heisenberg model.
This then allows us to explain the observation of KPZ scaling as a direct consequence of the nearly integrable scattering behaviour of long-lived solitons and the screening of magnetisation during collisions.

The fact that these solitons actually survive and determine the hydrodynamic behaviour at finite temperatures, where correlation lengths are only a few lattice sites, seems truly remarkable, in particular considering that, in that case, the \textit{adiabatically stable solitons are larger than the correlation lengths}. The crossover from this situation to the increasingly normal diffusive transport at higher temperatures is an obvious object for future studies. 

Our work contributes to the broader study of the role of approximate integrability in many-body systems, see, e.g.,\ \cite{Lange_2017}. For this, the Heisenberg chain  provides a suitable setting as, besides the proximity to the Ishimori chain exploited here, it can also be thought of a lattice version of the integrable continuum Landau-Lifshitz model,
via which route a family of approximate mobile solitons can be obtained by discretisation \cite{Schmidt_2011}, while
in the limit of low temperatures, magnon-type excitations exhibit the usual `emergent integrability' of weakly interacting quasiparticles.  
Our work in particular raises the question about a wider applicability of these ideas about anomalous transport to classical spin models with $SO(3)$ spin-symmetry. It certainly illustrates the point that, even in the very simplest settings, many-body dynamics  still holds many surprises awaiting discovery.


\begin{acknowledgments}
This work was in part supported by the Deutsche Forschungsgemeinschaft under grants SFB 1143 (project-id 247310070) and the cluster of excellence ct.qmat (EXC 2147, project-id 390858490). 
\end{acknowledgments}




