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Abstract

We present a systematic investigation of all sixteen marginally relevant fermion-fermion interactions in two-dimensional time-reversal symmetry-breaking kagom\'{e} semimetals hosting a quadratic band crossing point. Employing a momentum-shell renormalization group approach that treats every interaction on equal footing, we derive energy-dependent flow equations that capture the hierarchical evolutions of interaction parameters. Our analysis begins by tracking the energy-dependent flows of fermion-fermion interactions. The interaction couplings go towards divergence at a critical energy scale, signaling quantum critical behavior. Such behavior is characterized by a certain fixed point (FP) whose characteristics depends intimately on structural parameters $d_{0,1,2,3}$ that cluster the microscopic model into rotationally symmetric and asymmetric cases. Then, we identify two stable FPs in the rotationally symmetric and nine additional FPs in asymmetric case dubbed FP$_{1-10}$. Their boundary conditions are approximately demarcated and established by linear and plane fitting techniques in the structural parameter space. Furthermore, we examine distinct interaction-driven instabilities nearby these FPs by incorporating the relevant external source terms and computing their susceptibilities. It indicates that the charge density wave and superconductivity become dominant at FP$_{2,4,5,6,8}$ and FP$_{1,9,10}$, while the $x$-current and bond density prevail at FP$_3$ and FP$_7$, respectively. In addition to these leading states, several underlying subordinate instabilities are presented as well. These results would be helpful to further study the low-energy critical behavior in 2D kagom\'{e} QBCP and related materials.