  Measuring degeneracy and broken-symmetry states of a system at nanoscale
requires extremely high energy and spatial resolution, which has so far eluded
direct observation. Here, we realize measurement of the degeneracy and subtle
broken-symmetry states of graphene at nanoscale for the first time. By using
edge-free graphene quantum dots, we are able to measure valley splitting and
valley-contrasting spin splitting of graphene at the single-electron level. Our
experiments detect large valley splitting around atomic defects of graphene due
to the coexistence of sublattice symmetry breaking and time reversal symmetry
breaking. Large valley-contrasting spin splitting induced by enhanced
spin-orbit coupling around the defects is also observed. These results reveal
unexplored exotic electronic states in graphene at nanoscale induced by the
atomic defects.
