As a preliminary step we made a Monte Carlo
finite-size scaling[14] 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.*[15] [(planar rotator)], combined with the
MC calculations of Gupta * et al.*[16] [(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.*[8] 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 [15].

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 .

Fri Sep 1 16:26:33 CDT 1995