We present a systematic investigation of all 16 marginally relevant fermion-fermion interactions in two-dimensional (2D) time-reversal symmetry-breaking kagome semimetals hosting a quadratic band crossing point (QBCP). Employing a momentum-shell renormalization-group approach that treats every interaction on an 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 evolve towards divergence at a critical energy scale, signaling quantum critical behavior governed by a certain fixed point (FP). The character of this FP depends intimately on structural parameters $d_{0,1,2,3}$ which classify the microscopic model into rotationally symmetric and asymmetric cases. Then, we identify two stable FPs in the rotationally symmetric case and nine additional FPs in the asymmetric case dubbed ${\mathrm{FP}}_{1-10}$. 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 near 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 ${\mathrm{FP}}_{2,4,5,6,8}$ and ${\mathrm{FP}}_{1,9,10}$, while the $x$-current and bond density prevail at ${\mathrm{FP}}_3$ and ${\mathrm{FP}}_7$, 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 a 2D kagome QBCP and related materials.

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• Renormalization group
• Two-dimensional electron system
• Effective field theory
• Feynman diagrams