  The Berezinskii-Kosterlitz-Thouless (BKT) transition in ultra-thin NbN films
is investigated in the presence of weak perpendicular magnetic fields. A jump
in the phase stiffness at the BKT transition is detected up to 5 G, while the
BKT features are smeared between 5 G and 50 G, disappearing altogether at 100
G, where conventional current-voltage behaviour is observed. Our findings
demonstrate that weak magnetic fields, insignificant in bulk systems, deeply
affect our ultra-thin system, promoting a crossover from Halperin-Nelson
fluctuations to a BCS-like state with Ginzburg-Landau fluctuations, as the
field increases. This behavior is related to field-induced free vortices that
screen the vortex-antivortex interaction and smear the BKT transition.
