  Fractional Dirac materials (FDMs) feature a fractional energy-momentum
relation $E(\vec{k}) \sim |\vec{k}|^{\alpha}$, where $\alpha \; (<1)$ is a real
noninteger number, in contrast to that in conventional Dirac materials with
$\alpha=1$. Here we analyze the effects of short- and long-range Coulomb
repulsions in two- and three-dimensional FDMs. Only a strong short-range
interaction causes nucleation of a correlated insulator that takes place
through a quantum critical point. The universality class of the associated
quantum phase transition is determined by the correlation length exponent
$\nu^{-1}=d-\alpha$ and dynamic scaling exponent $z=\alpha$, set by the band
curvature. On the other hand, the fractional dispersion is protected against
long-range interaction due to its nonanalytic structure. Rather, a linear Dirac
dispersion gets generated under coarse graining, and the associated Fermi
velocity increases logarithmically in the infrared regime, thereby yielding a
two-fluid system. Altogether, correlated FDMs unfold a rich landscape
accommodating unconventional emergent many-body phenomena.
