  A clean noninteracting parabolic semimetal is characterized by quadratic band
touching between the conduction and the valence bands at isolated diabolic
points in the Brillouin zone and describes a fermionic quantum critical system
with dynamic exponent z=2. We consider the stability of such a semimetal
against electronic interaction and quenched disorder using a perturbative
renormalization group analysis for two and three spatial dimensions. For the
noninteracting problem infinitesimally weak disorder leads to an Anderson
insulator and a diffusive metal respectively in two and three dimensions. On
the other hand, the long range Coulomb interaction causes an excitonic
instability for the clean interacting problem towards a broken symmetry ground
state in both dimensions. Our weak coupling analysis of the combined effects of
disorder and interaction suggests the competition between a broken symmetry and
a disorder controlled metallic or insulating states, but is inadequate for
describing the quantum phase transitions among them. We discuss the relevance
of our results for bilayer graphene and some 227 iridate compounds, and
identify these materials as promising candidates for exploring novel disorder
and interaction controlled quantum critical phenomena.
