Fractional Dynamics and Modulational Instability in Long-Range Heisenberg Chains
Abstract
We study the effective dynamics of ferromagnetic spin chains in presence of long-range interactions. We consider the Heisenberg Hamiltonian in one dimension for which the spins are coupled through power-law long-range exchange interactions with exponent . We add to the Hamiltonian an anisotropy in the -direction. In the framework of a semiclassical approach, we use the Holstein-Primakoff transformation to derive an effective long-range discrete nonlinear Schr\"odinger equation. We then perform the continuum limit and we obtain a fractional nonlinear Schr\"odinger-like equation. Finally, we study the modulational instability of plane-waves in the continuum limit and we prove that, at variance with the short-range case, plane waves are modulationally unstable for . We also study the dependence of the modulation instability growth rate and critical wave-number on the parameters of the Hamiltonian and on the exponent .
Cite
@article{arxiv.2210.13201,
title = {Fractional Dynamics and Modulational Instability in Long-Range Heisenberg Chains},
author = {Laetitia Mbetkwe Youwa and Jean Pierre Nguenang and Paul André Paglan and Thierry Dauxois and Andrea Trombettoni and Stefano Ruffo},
journal= {arXiv preprint arXiv:2210.13201},
year = {2022}
}