  Nanoscale defects such as Nitrogen Vacancy (NV) centers can serve as
sensitive and non-invasive probes of electromagnetic fields and fluctuations
from materials, which in turn can be used to characterize these systems. Here
we specifically discuss how NV centers can directly probe time-reversal
symmetry breaking (TRSB) phenomena in low-dimensional conductors and magnetic
insulators. We argue that the relaxation rate of NV centers can vary
dramatically depending on whether its magnetic dipole points towards or away
from the TRSB material. This effect arises from the difference in the
fluctuation spectrum of left and right-polarized magnetic fields emanating from
such materials. It is perhaps most dramatic in the quantum Hall setting where
the NV center may experience no additional contribution to its relaxation due
to the presence of the material when initialized in a particular spin state but
a large decay rate when initialized in the opposite spin state. More generally,
we show that the NV center relaxation rate is sensitive to the imaginary part
of the wave-vector dependent Hall conductivity of a TRSB material. We argue
that this can be used to determine the Hall viscosity, which can potentially
distinguish candidate fractional quantum Hall states and pairing angular
momentum in TRSB chiral superconductors. We also consider Wigner crystals
realized in systems with large Berry curvature and discuss how the latter may
be extracted from NV center relaxometry.
