  Two-dimensional and three-dimensional massless Dirac fermions can form a
sequence of quasibound states with an attractive charged impurity. These
quasibound states exhibit a discrete scaling symmetry, i.e., the energy ratio
between two successive states is a constant. Through the calculation of the
energy spectrum directly, we find that in two dimension an applied magnetic
field can shift or even destroy the quasibound states around the Dirac point
and their discrete scaling symmetry disappears. However, as the magnetic field
increases, the remaining quasibound states are pushed up to the Dirac point.
When one quasibound state is close to the Dirac point, the spectrum is modified
significantly, due to the resonant scattering. The magnetic oscillation of the
spectrum displays the same discrete scaling symmetry as the quasibound state
does in zero magnetic field. This phenomenon also occurs in the quantum limit
of three-dimensional Dirac semimetals, where the system becomes quasi
one-dimensional essentially. Our theoretical analysis are in good agreement
with the recent experimental observations.
