We investigate the effects of long-ranged Coulomb interactions in a tiltedDirac semimetal in two dimensions by using the perturbativerenormalization-group method. Depending on the magnitude of the tiltingparameter, the undoped system can have either Fermi points (type-I) or Fermilines (type-II). Previous studies usually performed the renormalization-grouptransformations by integrating out the modes with large momenta. This isproblematic when the Fermi surface is open, like type-II Dirac fermions. Inthis work, we study the effects of Coulomb interactions, following the spiritof Shankar\cite{Shankar}, by introducing a cutoff in the energy scale aroundthe Fermi surface and integrating out the high-energy modes. For type-I Diracfermions, our result is consistent with that of the previous work. On the otherhand, we find that for type-II Dirac fermions, the magnitude of the tiltingparameter increases monotonically with lowering energies. This implies thestability of type-II Dirac fermions in the presence of Coulomb interactions, incontrast with previous results. Furthermore, for type-II Dirac fermions, thevelocities in different directions acquire different renormalization even ifthey have the same bare values. By taking into account the renormalization ofthe tilting parameter and the velocities due to the Coulomb interactions, weshow that while the presence of a charged impurity leads only to chargeredistribution around the impurity for type-I Dirac fermions, for type-II Diracfermions, the impurity charge is completely screened, albeit with a very longscreening length. The latter indicates that the temperature dependence ofphysical observables are essentially determined by the RG equations we derived.We illustrate this by calculating the temperature dependence of thecompressibility and specific heat of the interacting tilted Dirac fermions.
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