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

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