Interaction-driven quantum criticality in two-dimensional quadratic band crossing semimetals with time-reversal symmetry breaking

Abstract

We present a systematic investigation of all sixteen marginally relevant fermion-fermion interactions in two-dimensional time-reversal symmetry-breaking kagomé 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 kagomé QBCP and related materials.

Figures (18)

• Figure 1: (Color online) Energy-dependent evolutions of fermion-fermion interactions at fixed structural parameters $d_1 = 0.4$, $d_2 = 0.06$, and $d_3 = 0.7$ (the behavior is insensitive to choices of structural parameters).
• Figure 2: (Color online) Energy-dependent rescaled evolutions of eight nondegenerate fermion-fermion interactions for the same structural parameters in Fig. \ref{['Fig_evolution']}.
• Figure 3: (Color online) Critical values of fermion-fermion interactions at $l=l_c$ with variations of the structural parameters $d_1$ and $d_3$ at a fixed $d_0=0.1$ (the basic results are insensitive to concrete value of $d_0$): (a) $\lambda_{00}$, (b) $\lambda_{01}$, (c) $\lambda_{03}$, (d) $\lambda_{10}$, (e) $\lambda_{11}$, and (f) $\lambda_{13}$.
• Figure 4: (Color online) Critical values of nondegenerate fermion-fermion interactions at $l=l_c$ with variation of the structural parameter $d_3$ at fixed $d_1=1$ and $d_2=2$. The right (light cyan) and left (light yellow) stable regions correspond to $\mathrm{FP}_1$ and $\mathrm{FP}_2$ in Case $\mathrm{I}$, respectively.
• Figure 5: (Color online) Critical values of fermion-fermion interaction $\lambda_{03}$ at $l=l_c$ as functions of the structural parameters $d_1$, $d_2$, and $d_3$ at a fixed $d_0=0.1$ (the basic results are insensitive to concrete value of $d_0$) rotated clockwise by (a) $0^\circ$, (b) $120^\circ$, and (c) $240^\circ$. The tendencies of the other seven nondegenerate fermion-fermion interaction parameters ($\lambda_{00}$, $\lambda_{01}$, $\lambda_{02}$, $\lambda_{10}$, $\lambda_{11}$, $\lambda_{12}$, and $\lambda_{13}$) are qualitatively similar and provided in Appendix \ref{['Sec_Appendix_BC_flows']}.