Instability of in-plane vortices in two-dimensional easy-plane ferromagnets.

An analysis of the core region of an in-plane vortex in the two-dimensional Heisenberg model with easy-plane anisotropy λ = Jz/Jxy leads to a clear understanding of the instability towards transformation into an out-of-plane vortex as a function of anisotropy. The anisotropy parameter λc at which the in-plane vortex becomes unstable and develops into an out-of-plane vortex is determined with an accuracy comparable to computer simulations for square, hexagonal, and triangular lattices. For λ < λc, the in-plane vortex is stable but exhibits a normal mode whose frequency goes to zero as ω ∝ (λc − λ) as λ approaches λc. For λ > λc, the static nonzero out-of-plane spin components grow as (λ − λc). The lattice dependence of λc is determined strongly by the number of spins in the core plaquette, is fundamentally a discreteness effect, and cannot be obtained in a continuum theory.