Interaction-driven topological phases 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 instability 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 sheds new light on the theoretical understanding of the interaction-driven phases in QBT systems.

相互作用驱动的拓扑阶段显着丰富了拓扑材料的种类，因此非常重要。在这里，我们研究了具有二维带状接触（QBT）点的填充二维棋盘格的相互作用的无旋转费米子的相图。通过基于最新的密度矩阵重归一化组和精确的对角线化开发新的诊断，我们为这种系统在半填充三个不同的相的情况下确定了准确的量子相图。对于较弱的最近邻相互作用，我们证明了QBT对于相互作用驱动的自发量子异常霍尔（QAH）效应的不稳定性。对于强相互作用，系统破坏了旋转对称性，实现了向列电荷密度波（CDW）相。有趣的是，对于中间相互作用，我们发现了一个夹在QAH和CDW相之间的对称断裂的键序临界相，该相将QBT分为两个Dirac点，这些点由相互作用驱动。代替在QAH和CDW阶段之间的直接过渡，我们对中间阶段的识别为QBT系统中相互作用驱动的阶段的理论理解提供了新的思路。