  We study the quantum phases driven by interaction in a semimetal with a
quadratic band touching at the Fermi level. By combining the density matrix
renormalization group (DMRG), analytical power expanded Gibbs potential method,
and the weak coupling renormalization group, we study a spinless fermion system
on a checkerboard lattice at half-filling, which has a quadratic band touching
in the absence of interaction. In the presence of strong nearest-neighbor
($V_1$) and next-nearest-neighbor ($V_2$) interactions, we identify a site
nematic insulator phase, a stripe insulator phase, and a phase separation
region, in agreement with the phase diagram obtained analytically in the strong
coupling limit (i.e. in the absence of fermion hopping). In the intermediate
interaction regime, we establish a quantum anomalous Hall phase in the DMRG as
evidenced by the spontaneous time-reversal symmetry breaking and the appearance
of a quantized Chern number $C = 1$. For weak interaction, we utilize the power
expanded Gibbs potential method that treats $V_1$ and $V_2$ on equal footing,
as well as the weak coupling renormalization group. Our analytical results
reveal that not only the repulsive $V_1$ interaction, but also the $V_2$
interaction (both repulsive and attractive), can drive the quantum anomalous
Hall phase. We also determine the phase boundary in the $V_1$-$V_2$ plane that
separates the semimetal from the quantum anomalous Hall state. Finally, we show
that the nematic semimetal, which was proposed for $|V_2| \gg V_1$ at weak
coupling in a previous study, is absent, and the quantum anomalous Hall state
is the only weak coupling instability of the spinless quadratic band touching
semimetal.
