Existence of breathers in classical ferromagnetic lattices

Abstract

In this paper, we study the dynamics of classical spins interacting via Heisenberg exchange in the presence of a single-ion anisotropy on spatial d-dimensional lattices. We focus on easy-plane ferromagnets and easy-axis ferromagnets. We present a rigorous proof for the existence of ‘out of plane’ breathers, which have no analogue in continuum theory. In this configuration, one or several spins precess around an axis out of a plane where all the other spins lie. Such discrete breathers represent excitations with a tilted magnetization and possess an energy threshold. The travelling waves of the linearized system possess frequencies in an acoustic band, but we shall see that with a suitable restriction of the functional spaces, the anticontinuous method introduced by MacKay and Aubry still works. This analysis is independent of the lattice dimension, and we present the proof in the one-dimensional case. On the other hand, we prove the existence of small amplitude breathers precessing around an axis (‘easy’ axis) using centre manifold reduction for quasilinear mappings. In this case, the motion of precession is small and decreases to 0 far from the lattice centre.