Magnetic bubble refraction and quasibreathers in inhomogeneous antiferromagnets

We study the dynamics of magnetic bubble solitons in a two-dimensional isotropic antiferromagnetic spin lattice in the case where the exchange integral $J(x,y)$ is position dependent. In the near-continuum regime, this system is described by the relativistic $O(3)$ sigma model on a space–time with a spatially inhomogeneous metric determined by $J$. We use the geodesic approximation to describe the low-energy soliton dynamics in this system: the $n$-soliton motion is approximated by geodesic motion in the moduli space $\mathsf M_n$ of static $n$-solitons equipped with the $L^2$ metric $\gamma$. We obtain explicit formulas for $\gamma$ for various natural choices of $J(x,y)$. Based on these, we show that single soliton trajectories are refracted with $J^{-1}$ being analogous to the refractive index and that this refraction effect allows constructing simple bubble lenses and bubble guides. We consider the case where $J$ has a disk inhomogeneity (with the value $J_+$ outside a disk and $J_-<J_+$ inside) in detail. We argue that for sufficiently large $J_+/J_-$, this type of antiferromagnet supports approximate quasibreathers: two or more coincident bubbles confined within the disk spin internally while their shape oscillates with a generically incommensurate period.