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

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.