  Recent experiments on ultracold Bose gases in two dimensions have provided
evidence for the existence of the Berezinskii-Kosterlitz-Thouless (BKT) phase
via analysis of the interference between two independent systems. In this work
we study the two-dimensional quantum degenerate Bose gas at finite temperature
using the projected Gross-Pitaevskii equation classical field method. While
this describes the highly occupied modes of the gas below a momentum cutoff, we
have developed a method to incorporate the higher momentum states in our model.
We concentrate on finite-sized homogeneous systems in order to simplify the
analysis of the vortex pairing. We determine the dependence of the condensate
fraction on temperature and compare this to the calculated superfluid fraction.
By measuring the first order correlation function we determine the boundary of
the Bose-Einstein condensate and BKT phases, and find it is consistent with the
superfluid fraction decreasing to zero. We reveal the characteristic unbinding
of vortex pairs above the BKT transition via a coarse-graining procedure.
Finally, we model the procedure used in experiments to infer system
correlations [Hadzibabic et al., Nature 441, 1118 (2006)], and quantify its
level of agreement with directly calculated in situ correlation functions.
