  Quadratic band touching (QBT) points are widely observed in 2D and 3D
materials, including bilayer graphene and Luttinger semimetals, and attract
significant attention from theory to experiment. However, even in its simplest
form, the 2D checkerboard lattice QBT model, the phase diagram characterized by
temperature and interaction strength still remains unknown beyond the
weak-coupling regime. Intense debates persist regarding the existence of
various interaction-driven insulating states in this system. To address these
uncertainties, we employ thermal tensor network simulations, specifically
exponential tensor renormalization group and tangent space tensor
renormalization group, along with density matrix renormalization group
calculations to provide a comprehensive finite-temperature phase diagram for
this model and shed light on previous ambiguities. Notably, our findings reveal
the emergence of a robust bond-nematic Dirac semimetal phase with distinct
thermodynamic properties that set it part from the nematic insulating state and
other symmetry broken states. This previously overlooked feature is found to be
ubiquitous in interacting QBT systems. We also discuss the implications of
these results for experimental systems such as bilayer graphene and iridate
compounds.
