Emergence of Massless Dirac Fermions in Graphene's Hofstadter Butterfly at Switches of the Quantum Hall Phase Connectivity

The fractal spectrum of magnetic minibands (Hofstadter butterfly), induced by the moire superlattice of graphene on a hexagonal crystal substrate, is known to exhibit gapped Dirac cones. We show that the gap can be closed by slightly misaligning the substrate, producing a hierarchy of conical singularities (Dirac points) in the band structure at rational values phi= (p / q) (h / e) of the magnetic flux per supercell. Each Dirac point signals a switch of the topological quantum number in the connected component of the quantum Hall phase diagram. Model calculations reveal the scale-invariant conductivity sigma = 2qe(2) / pi h and Klein tunneling associated with massless Dirac fermions at these connectivity switches.