  Interaction driven topological phases can significantly enrich the class of
topological materials and thus are of great importance. Here, we study the
phase diagram of interacting spinless fermions filling the two-dimensional
checkerboard lattice with a quadratic band touching (QBT) point. By developing
new diagnosis based on the state-of-the-art density-matrix renormalization
group and exact diagonalization, we determine accurate quantum phase diagram
for such a system at half-filling with three distinct phases. For weak
nearest-neighboring interactions, we demonstrate the instablity of the QBT
towards an interaction-driven spontaneous quantum anomalous Hall (QAH) effect.
For strong interactions, the system breaks the rotational symmetry realizing a
nematic charge-density-wave (CDW) phase. Interestingly, for intermediate
interactions we discover a symmetry-broken bond-ordered critical phase
sandwiched in between the QAH and CDW phases, which splits the QBT into two
Dirac points driven by interaction. Instead of the direct transition between
QAH and CDW phases, our identification of an intermediate phase shed new light
on the theoretical understanding of the interaction-driven phases in QBT
systems.
