Zero-energy modes and valley asymmetry in the Hofstadter spectrum of   bilayer graphene van der Waals heterostructures with hBN
  We investigate the magnetic minibands of a heterostructure consisting of bilayer graphene (BLG) and hexagonal boron nitride (hBN) by numerically diagonalizing a two-band Hamiltonian that describes electrons in BLG in the presence of a moire potential. Due to inversion-symmetry breaking characteristic for the moire potential, the valley symmetry of the spectrum is broken, but despite this, the zero-energy Landau level in BLG survives, albeit with reduced degeneracy. In addition, we derive effective models for the low-energy features in the magnetic minibands and demonstrate the appearance of secondary Dirac points in the valence band, which we confirm by numerical simulations. Then, we analyze how single-particle gaps in the fractal energy spectrum produce a sequence of incompressible states observable under a variation of carrier density and magnetic field. 
Xi Chen, J. R. Wallbank, M. Mucha-Kruczynski, E. McCann, and V. I.   Fal'ko
cond-mat.mes-hall