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Quantum oscillations of magnetization in the tight-binding electrons on honeycomb lattice

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.