As a preliminary step we made a Monte Carlo finite-size scaling study to determine accurately the KT-temperature, for , since the precise critical point for this model is not known. Using L= 10, 20, 40 and 80, and averaging over 160,000 states at each temperature, we calculated the reduced fourth-order cumulant,
where M is the total in-plane magnetic moment. This definition is appropriate for systems with XY symmetry, with approaching 0.5 in the low-temperature phase and 0.0 in the high-temperature phase. The result is shown in Fig. 1; the curves for different L cross at . This is consistent with the prediction of Menezes et al. [(planar rotator)], combined with the MC calculations of Gupta et al. [(planar rotator) ], although the Menezes et al. theory does not give the correct for either model. On this basis, to have enough vortices for lifetime measurements, we considered the temperature range with a step .
We obtained the free vortex lifetime for two values of the anisotropy parameter and , as shown in Figure 2 for system sizes L=32, 64. The lifetime decreases starting from and it is close to saturation when approaching . The data points for are higher than those for for all T studied. For and , we have , whereas (see Sec. ii). This implies a cutoff frequency several times greater than the characteristic frequency in the Mertens et al. theory. For comparison, some typical results for from an L=100 system are shown in Fig. 3, for . The observed CP is strong for higher temperatures, and for frequencies well below our measured values of . This implies that the ideal vortex gas description for frequencies below is inappropriate.
The lifetime does not show dependence on the size of the system for ; the data points for and overlap within their error bars. On the contrary, there is a pronounced finite size effect when approaching the critical region from above for . This indicates that the system size is too small to be used to study the bulk properties, unless some finite size scaling is applied in a domain close enough to the critical temperature. The presence of this finite size effect and its absence for is because the correlation length for is greater than that for for the corresponding temperatures, since .
In Fig. 4 we show the free vortex number---number correlation function for temperatures , , and , for . As expected, the correlation function decays faster and decorrelates at earlier times for larger temperature. The correlation function cannot be described by linear response theory, implying that is not governed by a single time scale. This is also confirmed by the tails of these curves. For example, for , a linear fit to from the first 4--5 points gives . However decorrelates at large times t>40, which contradicts the simple behavior. On the other hand, relaxation times determined from the small-t linear fits are of the same order of magnitude and have the same behavior with temperature as determined above. Similar results for show a faster decay of than for , completely in agreement with our measurements of for both values of , however, at large times decorrelates slightly slower than for the case .