Long-Lived Solitons, anomalous dynamics and equilibration in the classical Heisenberg chain

The search for departures from standard hydrodynamics in many-body systems has yielded a number of promising leads, especially in low dimensions. Here, we study one of the simplest classical interacting lattice models, the nearest-neighbor Heisenberg chain, with temperature as the tuning parameter. Our numerics expose strikingly different spin dynamics between the antiferromagnet, where it is largely diffusive, and the ferromagnet, where we observe strong evidence either of spin superdiffusion or an extremely slow crossover to diffusion.

Motivated by this observed KPZ scaling in the classical ferromagnetic Heisenberg chain, we investigate the role of solitonic excitations. We find that the Heisenberg chain, although well-known to be non-integrable, supports a two-parameter family of long-lived solitons. We connect these to the exact soliton solutions of the integrable Ishimori chain with log(1+Si⋅Sj) 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.

[1] Phys. Rev. B 105, L100403

[2] arXiv:2207.08866