  We show that the new quantum oscillations of the magnetization can occur when
the Fermi surface consists of points (massless Dirac points) or even when the
chemical potential is in a energy gap by studying the tight-binding electrons
on a honeycomb lattice in a uniform magnetic field. The quantum oscillations of
the magnetization as a function of the inverse magnetic field are known as the
de Haas-van Alphen (dHvA) oscillations and the frequency is proportional to the
area of the Fermi surface. The dominant period of the new oscillations
corresponds to the area of the first Brillouin zone and its phase is zero. The
origin of the new quantum oscillations is the characteristic magnetic field
dependence of the energy known as the Hofstadter butterfly and the Harper
broadening of Landau levels. The new oscillations are not caused by the
crossing of the chemical potential and Landau levels, which is the case in the
dHvA oscillations. The new oscillations can be observed experimentally in
systems with large supercell such as graphene antidot lattice or ultra cold
atoms in optical lattice at an external magnetic field of a few Tesla when the
area of the supercell is ten thousand times larger than that of graphene.
