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

We show that 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 an energy gap by studying 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 de Haas-van Alphen (dHvA) oscillations and the frequency is proportional to the area of the Fermi surface. The dominant period of the oscillations shown in this paper corresponds to the area of the first Brillouin zone and its phase is zero. The origin of these quantum oscillations is the characteristic magnetic field dependence of the energy known as the Hofstadter butterfly and the Harper broadening of Landau levels. These oscillations are not caused by the crossing of the chemical potential and Landau levels, which is the case in dHvA oscillations. These oscillations can be observed experimentally in systems with a large supercell such as a graphene antidot lattice or ultracold atoms in an optical lattice at an external magnetic field of a few Tesla when the area of the supercell is 10(4) times larger than that of graphene.