Real Space Imaging of the Quantum Hall Effect and Valley Polarization in Graphene

When a perpendicular magnetic field is applied to a graphene sheet, the resulting eigenenergies (Landau Levels or LLs) have a nonlinear energy distribution that includes a four-fold degenerate zero-energy state ($LL_{0}$). Maps of the energy-resolved local density of states (LDOS) acquired via cryogenic scanning tunneling spectroscopy (STS) provide atomic-scale images of the LL spatial distribution. Focusing on $LL_{0}$, we use STS maps to show the distribution of ``drift states'' and find unexpected atomic-scale spatial variations of the LDOS above a critical field of $B_{*} = 4 T$. We resolve an energy gap in $LL_{0}$ and show how it depends on the local A-B lattice symmetry and magnetic field. The gap is observed only within patches of at least a few magnetic lengths in size, which forces the splitting to ``turn off'' below the critical field. We attribute this behavior to a breaking of the local sublattice symmetry imposed by moire layer stacking.