Magnetic manifestation of discrete scaling symmetry in Dirac semimetals

Two-dimensional and three-dimensional massless Dirac fermions can form a\nsequence of quasibound states with an attractive charged impurity. These\nquasibound states exhibit a discrete scaling symmetry, i.e., the energy ratio\nbetween two successive states is a constant. Through the calculation of the\nenergy spectrum directly, we find that in two dimension an applied magnetic\nfield can shift or even destroy the quasibound states around the Dirac point\nand their discrete scaling symmetry disappears. However, as the magnetic field\nincreases, the remaining quasibound states are pushed up to the Dirac point.\nWhen one quasibound state is close to the Dirac point, the spectrum is modified\nsignificantly, due to the resonant scattering. The magnetic oscillation of the\nspectrum displays the same discrete scaling symmetry as the quasibound state\ndoes in zero magnetic field. This phenomenon also occurs in the quantum limit\nof three-dimensional Dirac semimetals, where the system becomes quasi\none-dimensional essentially. Our theoretical analysis are in good agreement\nwith the recent experimental observations.\n