  Kohn's theorem and the $f$-sum rule are powerful theorems, the first applying
to translationally invariant single-band electronic systems with parabolic
electronic dispersion relations and the second applying to materials in
general, that impose restrictions on the effects of electron-electron
interactions on electrical conductivity and on dielectric response,
respectively. We show rigorously that Kohn's theorem does not hold for
intrinsic Dirac-Weyl materials with filled Fermi seas where the chemical
potential is pinned at the band touching points. We also demonstrate that the
low-energy effective "relativistic" theories used in many-body studies of these
materials violate the $f$-sum rule due to the neglect of the full band
structure of the materials in the effective low-energy relativistic
approximations.
