Long-lived solitons and their signatures in the classical Heisenberg chain

Motivated by the Kardar-Parisi-Zhang (KPZ) scaling recently observed in the classical ferromagnetic Heisenberg chain, we investigate the role of solitonic excitations in this model. We find that the Heisenberg chain, although well known to be nonintegrable, supports a two-parameter family of long-lived solitons. We connect these to the exact soliton solutions of the integrable Ishimori chain with $ln(1+S_i·S_j)$ interactions. We explicitly construct infinitely long-lived stationary solitons, and provide an adiabatic construction procedure for moving soliton solutions, which shows that Ishimori solitons have a long-lived Heisenberg counterpart when they are not too narrow and not too fast moving. Finally, we demonstrate their presence in thermal states of the Heisenberg chain, even when the typical soliton width is larger than the spin correlation length, and argue that these excitations likely underlie the KPZ scaling.

Figure 1

Solitons in the classical Heisenberg chain. (a) Parameter space of Ishimori solitons for which we found an adiabatically connected soliton in the Heisenberg chain. (b),(c) Comparisons between adiabatically connected solitons (only $z$ component shown) in the Heisenberg ($H$, $•$) and Ishimori ($I$, $\times$) chains. The solitons in (b) are stationary $(R=0.25,k=0)$; and in (c), they move $(R=0.1,k=0.15)$. The Heisenberg soliton moves at a slower velocity, but both preserve their initial profile.

Figure 2

Physical properties of the Ishimori solitons (solid lines) compared to Heisenberg solitons [dotted lines: up to the boundary of the existence diagram; Fig. 1], shown as a function of inverse-width $R$, for various values of the wave number $k$. (a)–(d) The internal frequency $\omega$, the velocity $v$, the energy $E$ (measured by the Heisenberg Hamiltonian), and the torsion $\tau$, respectively.

Figure 3

Soliton scattering in the Heisenberg chain. Color scale shows the $z$ components and is the same for (a)–(c). (a) Single scattering event between two solitons with parameters $(R,k)=(0.1,0.1)$ and $(R^{\prime },k^{\prime })=(0.1,-0.15)$. (b) Screening of the magnetization 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 $(R^{\prime },k^{\prime })=(0.1,0)$] under periodic boundary conditions. (d) Comparison of the scattering phase shift $\mathrm{\Delta }(R,k=0;0.1,0.1)$ 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 with respect to the relative phases.

Figure 4

Solitons in the thermal state at $T=0.1J$ for the Heisenberg (left) and Ishimori (right) chains. Upper and lower panels show torsion ${\tau }_j\left(t\right)$, and $S_j^z\left(t\right)$, respectively. Ballistic trajectories are clearly visible in the torsion plots, both in the Ishimori and in the Heisenberg chains. These ballistic trajectories can also be seen in the plots of $S^z$, where the magnetization carried by a soliton changes as it moves through the chain—the mechanism preventing ballistic spin transport.