  We analyze the theoretical derivation of the beyond-mean-field equation of
state for a two-dimensional gas of dilute, ultracold alkali-metal atoms in the
Bardeen-Cooper-Schrieffer (BCS) to Bose-Einstein condensate (BEC) crossover. We
show that at zero temperature our theory -- considering Gaussian fluctuations
on top of the mean-field equation of state -- is in very good agreement with
experimental data. Subsequently, we investigate the superfluid density at
finite temperature and its renormalization due to the proliferation of
vortex-antivortex pairs. By doing so, we determine the
Berezinskii-Kosterlitz-Thouless (BKT) critical temperature -- at which the
renormalized superfluid density jumps to zero -- as a function of the
inter-atomic potential strength. We find that the Nelson-Kosterlitz criterion
overestimates the BKT temperature with respect to the renormalization group
equations, this effect being particularly relevant in the intermediate regime
of the crossover.
