The discovery of Weyl and Dirac semimetals has produced a number of dramatic physical effects, including the chiral anomaly and Fermi arc surface states. A very different but no less dramatic physical effect is also to be found in these materials: discrete scale invariance. This invariance leads to quasibound-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 $lnB$ 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: ${\mathrm{ZrTe}}_5$, ${\mathrm{HfTe}}_5$, and TaAs.

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• Electrical conductivity
• Transport phenomena
• Topological materials
• Green's function methods