  The discovery of Weyl and Dirac semimetals has produced a number of dramatic
physical effects, including the chiral anomaly and topological Fermi arc
surface states. We point out that a very different but no less dramatic
physical effect is also to be found in these materials: discrete scale
invariance. This invariance leads to bound state spectra for Coulomb impurities
that repeat when the binding energy is changed by a fixed factor, reminiscent
of fractal behavior. We show that this effect follows from the peculiar
dispersion relation in Weyl and Dirac semimetals. It is observed when such a
material is placed in very strong magnetic field B: there are oscillations in
the magnetoresistivity somewhat similar to Shubnikov-de Haas oscillations but
with a periodicity in ln B rather than 1/B. These oscillations should be
present in other thermodynamic and transport properties. The oscillations have
now been seen in three topological semimetals: ZrTe$_{5}$, TaAs, and Bi.
