Magnetic Vortex Mass in 2d Easy-plane Magnets

Nonlinear vortex excitations in models for layered or two-dimensional (2D) magnets [1] have attracted much study not only for their role in the thermodynamics of the Berezinskii-Kosterlitz-Thouless vortex-unbinding transition [2,3], but more recently because their microscopic dynamic behavior is not fully understood. The dynamics of individual vortices continues under study for ferromagnets (FM) [4,5] and antiferromagnets (AFM) [6,7], with attention to external and gyrotopic force terms, dissipation terms, and mass terms in appropriate equations of motion. Specifically, a question of great importance that we consider here is whether vortices have dynamics that requires an effective mass. The value of the vortex effective mass, if present, will relate to vortex motion in thermal equilibrium and vortex rms velocity. The results may also be relevant for vortex fluctuations experiments in related systems such as high temperature superconductors [8]. We consider 2D magnetic models with easy-plane anisotropy, where the spins have an energetic preference to lie in the xy-plane, with smaller z-components, depending on the strength of the anisotropy. A vortex position is defined by the location of a local singularity in the in-plane (xy) components of the spin field. If the anisotropy is weak, there is an associated nonzero outof-plane (z) spin structure that peaks at the vortex center, and falls off away from the vortex center [9,10]. If one supposes that this “out-of-plane” vortex spin structure (in continuum theory) is fixed as the vortex moves with constant velocity ~ V = ~̇ X, then the response of the vortex position ~ X to an external force ~ F (due to applied field or other vortices) is associated with a topological charge of the vortex, or gyrovector, ~ G = Gêz, according to an equation of motion derived by Thiele [11] for domain walls and applied by Huber [12] to dynamics of vortices: