  The interaction between the Fermi sea of conduction electrons and a
non-adiabatic attractive impurity potential can lead to a power-law divergence
in the tunneling probability of charge through the impurity. The resulting
effect, known as the Fermi edge singularity (FES), constitutes one of the most
fundamental many-body phenomena in quantum solid state physics. Here we report
the first observation of FES for Dirac Fermions in graphene driven by isolated
Coulomb impurities in the conduction channel. In high-mobility graphene devices
on hexagonal boron nitride substrates, the FES manifests in abrupt changes in
conductance with a large magnitude $\approx e^{2}/h$ at resonance, indicating
total many-body screening of a local Coulomb impurity with fluctuating charge
occupancy. Furthermore, we exploit the extreme sensitivity of graphene to
individual Coulomb impurities, and demonstrate a new defect-spectroscopy tool
to investigate strongly correlated phases in graphene in the quantum Hall
regime.
