  We examine a basic lattice model of interacting fermions that exhibits
quadratic band crossing points (QBCPs) in the non-interacting limit. In
particular, we consider spinless fermions on the honeycomb lattice with nearest
neighbor hopping $t$ and third-nearest neighbor hopping $t''$, which exhibits
fine-tuned QBCPs at the corners of the Brillouin zone for ${t'' = t/2}$. In
this situation, the density of states remains finite at the Fermi level of the
half-filled band and repulsive nearest-neighbor interactions $V$ lead to a
charge-density-wave (CDW) instability at infinitesimally small $V$ in the
random-phase approximation or mean-field theory. We examine the fragility of
the QBCPs against dispersion renormalizations in the ${t\mbox{-}t''\mbox{-}V}$
model using perturbation theory, and find that the $t''$-value needed for the
QBCPs increases with $V$ due to the hopping renormalization. However, the
instability toward CDW formation always requires a nonzero threshold
interaction strength, i.e., one cannot fine-tune $t''$ to recover the QBCPs in
the interacting system. These perturbative arguments are supported by quantum
Monte Carlo simulations for which we carefully compare the corresponding
threshold scales at and beyond the QBCP fine-tuning point. From this analysis,
we thus gain a quantitative microscopic understanding of the fragility of the
QBCPs in this basic interacting fermion system.
