Interaction and disorder effects on Cooper instability in two-dimensional fractional Dirac semimetals

Employing a renormalization group analysis that allows for an unbiased treatment of competing physical ingredients, we systematically trace how the interplay between Cooper pairing and disorder scatterings governs the emergence or suppression of Cooper instability in the low-energy regime of fractional Dirac semimetals.In the clean limit, we find that the emergence of Cooper instability requires surpassing a finite interaction threshold $|\lambda_c|$, and depends sensitively on both the fractional exponent $\alpha$ and the transfer momentum $\mathbf{Q}=(Q,\phi)$. Specifically, bigger values of $\alpha$ enhance the tendency toward BCS instability. For $\alpha\in(0.001,0.61)$, the $(Q,\phi)$ parameter space separates into two distinct regions: Zone-\uppercase\expandafter{\romannumeral1}, where Cooper instability is suppressed, and Zone-\uppercase\expandafter{\romannumeral2}, where it is allowed. In the presence of disorders, we demonstrate that they can either promote or suppress Cooper instability. Disorder of type $\Delta_1$ or $\Delta_2$ enhances superconductivity by reducing the critical interaction threshold $|\lambda_c|$ and expanding the superconducting phase space (Zone-\uppercase\expandafter{\romannumeral2}). In sharp contrast, either $\Delta_0$ or $\Delta_3$ suppresses Cooper pairing by increasing $|\lambda_c|$ and shrinking the available phase space (Zone-\uppercase\expandafter{\romannumeral1}). Although Cooper instability can be enhanced when promotive disorders ($\Delta_1$, $\Delta_2$) coexist with a single suppressive disorder ($\Delta_0$ or $\Delta_3$), the suppressive influence of $\Delta_{0,3}$ generally dominates the promotive effects of $\Delta_{1,2}$ in the presence of all sorts of disorders. These results would be helpful for further studies of fractional Dirac semimetals and alike materials.