  We investigate the impacts of combination of fermion-fermion interactions and
impurity scatterings on the low-energy stabilities of two-dimensional
asymmetric materials with a quadratic band crossing point by virtue of the
renormalization group that allows us to treat distinct sorts of physical
ingredients on the same footing. The coupled flow evolutions of all interaction
parameters which carry the central physical information are derived by taking
into account one-loop corrections. Several intriguing results are manifestly
extracted from these entangled evolutions. At first, we realize that the
quadratic band touching structure is particularly robust once the fermionic
couplings flow toward the Gaussian fixed point. Otherwise, it can either be
stable or broken down against the impurity scattering in the vicinity of
nontrivial fixed points. In addition, we figure out two parameters $\eta$ and
$\lambda$ that measure rotational and particle-hole asymmetries are closely
energy-dependent and exhibit considerably abundant behaviors depending upon the
fates of fermion-fermion couplings and different types of impurities.
Incidentally, as both $\eta$ and $\lambda$ can be remarkably increased or
heavily reduced in the low-energy regime, an asymmetric system under certain
restricted conditions exhibits an interesting phenomenon in which transitions
either from rotational or particle-hole asymmetry to symmetric situation would
be activated.
