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=J_z/J_xy leads to a clear understanding of the instability towards transformation into an out-of-plane vortex as a function of anisotropy. The anisotropy parameter λ_c 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<λ_c, the in-plane vortex is stable but exhibits a normal mode whose frequency goes to zero as ω∝(λ_c-λ)^1/2 as λ approaches λ_c. For λ>λ_c, the static nonzero out-of-plane spin components grow as (λ-λ_c)^1/2. The lattice dependence of λ_c 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.