Interacting Dirac Fermions in (3+1) Dimensions

Abstract

The long-range electron-electron interaction between Dirac electrons in $(3+1)$ dimensions is considered by the perturbative renormalization group analysis. When the chemical potential in Dirac energy dispersion is tuned at the band crossing point, the density of states vanishes and hence the Coulomb interaction is not screened to become a long-range force. We analyze that long-range interaction relativistically, taking into consideration of the retardation effect of the propagation of the interaction. Those massless Dirac electrons in $(3+1)$ dimensions emerge, for example, at the quantum critical point of topological phase transitions and in Weyl/Dirac semimetals. They receive logarithmic corrections in the nonrelativistic regime, where the electron’s speed v is much larger than the speed of light c; v increases logarithmically depending on the renormalization scale. In the low energies, however, the Dirac systems attain a relativistic regime $v \approx c$, and finally the Lorentz invariance is recovered ($v=c$).