Evolution with Magnetic Field of Discrete Scale Invariant Supercritical States in Graphene

We investigate the quasi-bound states of a Coulomb impurity in graphene in the presence of a magnetic field. \ These states exhibit the dramatic and rather rare property of discrete scale invariance when the Coulomb potential is supercritical. \ We show using both Wentzel-Kramers-Brillouin (WKB) approximation and numerical studies that the supercritical states are converted to subcritical states as the field is increased. \ The local density of states is calculated and it shows direct signatures of discrete scale invariance. \ In a magnetic field, these signatures are gradually destroyed in a systematic way. \ Hence the effect that we propose can be detected via scanning tunneling microscope experiments. \ The range of magnetic field and energy resolution required are compatible with existing experimental setups. \ These experiments can be performed in a single sample by changing the field; they do not involve changing the nuclear charge.