We consider a system of classical spins on the sphere ( ), interacting on a 2D square lattice. The Hamiltonian is
where the sum is over nearest-neighbor lattice sites, J > 0 determines ferromagnetic coupling and introduces easy-plane anisotropy.
There are two types of static vortices which arise in this model--- out-of-plane and in-plane ones, depending on the presence or absence respectively of nonzero out-of-plane spin components (), with identical in-plane spin structure. A study of their static properties,, shows that their stability depends on the anisotropy parameter . Below a specific value ( for square lattice) only the in-plane vortex is stable while for only the out-of-plane one is stable. Because the out-of-plane structure may influence the vortex interactions and therefore their lifetime, we consider both and .
Assuming an ideal gas of free vortices with infinite lifetime, Mertens et al. obtained the asymptotic behavior for the in-plane correlation function :
The characteristic time implied by this equation is , which is approximately the time for a vortex to move one correlation length. Thus, the theory is reasonable provided the vortex lifetime is at least that long. For an order of magnitude result the characteristic time is approximately from the estimates and at temperature (length is in lattice constant units, T in , and time in ). We use this value to compare with the lifetime we obtain from our simulation at this particular temperature.
The space-time Fourier transformation of Eq. (2.2) leads to a squared Lorentzian central peak form
where , with a wavevector-dependent characteristic frequency width
However, a finite vortex lifetime will lead to fluctuations in the number of free vortices on a length scale of of the order of , with frequency higher than the inverse vortex lifetime, . Thus, will represent a cutoff frequency below which the ideal vortex gas theory for cannot be a valid description.
As mentioned above, the two types of static vortices have the same xy behavior of their spin components but different components. For , there is no contribution to from static in-plane vortices and the vortex contribution can only be from moving vortices. On the other hand, for , the main contribution can be from static (out-of-plane) vortex structures. In either case, the ideal gas theory predicts a central peak in Fourier space, but with a Gaussian shape for rather than the squared Lorentzian for the in-plane correlation function. Since for , the theory for may be built in first approximation by assuming that the out-of-plane structure of a moving vortex can be approximated by the static structure, it is important to compare the vortex lifetime in this case with the case , where the spin-wave peak is strongly softened and the central peak can be clearly attributed to the motion of vortices.