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=${\mathit{J}}_{\mathit{z}}$/${\mathit{J}}_{\mathit{x}\mathit{y}}$ leads to a clear understanding of the instability towards transformation into an out-of-plane vortex as a function of anisotropy. The anisotropy parameter ${\lambda }_{\mathit{c}}$ 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<${\lambda }_{\mathit{c}}$, the in-plane vortex is stable but exhibits a normal mode whose frequency goes to zero as ω∝(${\lambda }_{\mathit{c}}$-λ${)}^{1/2}$ as λ approaches ${\lambda }_{\mathit{c}}$. For λ>${\lambda }_{\mathit{c}}$, the static nonzero out-of-plane spin components grow as (λ-${\lambda }_{\mathit{c}}$${)}^{1/2}$. The lattice dependence of ${\lambda }_{\mathit{c}}$ 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.