  The melting of quasi-long-range superconductivity in two spatial dimensions
occurs through the proliferation and unbinding of vortex-antivortex pairs -- a
phenomenon known as the Berezinskii-Kosterlitz-Thouless (BKT) transition.
Although signatures of this transition have been observed in bulk measurements,
these experiments are often complicated, ambiguous, and unable to resolve the
rich physics of the vortex unbinding transition. Here we show that local noise
magnetometry is a sensitive, noninvasive probe that can provide direct
information about the scale-dependent vortex dynamics. In particular, by
resolving the distance and temperature dependence of the magnetic noise, it may
be possible to experimentally study the renormalization group flow equations of
the vortex gas and track the onset of vortex unbinding in situ. Specifically,
we predict i) a nonmonotonic dependence of the noise on temperature and ii) the
local noise is almost independent of the sample-probe distance at the BKT
transition. We also show that noise magnetometry can distinguish Gaussian
superconducting order-parameter fluctuations from topological vortex
fluctuations and can detect the emergence of unbound vortices. The weak
distance dependence at the BKT transition can also be used to distinguish it
from quasiparticle background noise. Our predictions may be within experimental
reach for a number of unconventional superconductors.
