  The adhesion of graphene on slightly lattice-mismatched surfaces, for
instance of hexagonal boron nitride (hBN) or Ir(111), gives rise to a complex
landscape of sublattice symmetry-breaking potentials for the Dirac fermions.
Whereas a gap at the Dirac point opens for perfectly lattice-matched graphene
on hBN, we show that the small lattice incommensurability prevents the opening
of this gap and rather leads to a renormalized Dirac dispersion with a trigonal
warping. This warping breaks the effective time reversal symmetry in a single
valley. On top of this a new set of massless Dirac fermions is generated, which
are characterized by a group velocity that is half the one of pristine
graphene.
