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Kohn's theorem and the $f$-sum rule are powerful theorems, the first applyingto translationally invariant single-band electronic systems with parabolicelectronic dispersion relations and the second applying to materials ingeneral, that impose restrictions on the effects of electron-electroninteractions on electrical conductivity and on dielectric response,respectively. We show rigorously that Kohn's theorem does not hold forintrinsic Dirac-Weyl materials with filled Fermi seas where the chemicalpotential is pinned at the band touching points. We also demonstrate that thelow-energy effective "relativistic" theories used in many-body studies of thesematerials violate the $f$-sum rule due to the neglect of the full bandstructure of the materials in the effective low-energy relativisticapproximations.

Failure of Kohn's theorem and the apparent failure of the $f$-sum rule in intrinsic Dirac-Weyl materials in the presence of a filled Fermi sea