Superconductivity in the uniform electron gas: Irrelevance of the Kohn-Luttinger mechanism

We study the Cooper instability in jellium model in the controlled regime of small to intermediate values of the Coulomb parameter $r_s\le 2$. We confirm that superconductivity naturally emerges from purely repulsive interactions described by the Kukkonen-Overhauser vertex function. By employing the implicit renormalization approach and the discrete Lehmann representation we reveal that even in the small-$r_s$ limit, the dominant mechanism behind Cooper instability is based on dynamic screening of the Coulomb interaction—accurately captured by the random phase approximation, whereas the Kohn-Luttinger contribution is negligibly small and, thus, not relevant.

Figure 1

Second-order diagrams contributing to the particle-particle irreducible four-point vertex $\mathrm{\Gamma }$ in the KL analysis. Only the first bubble diagram is resummed within the RPA approach, while all three diagrams are resummed in the KO formulation. In the small-$r_s$ limit, the contributions of the diagrams (b) and (c) might prove appreciable only if the Kohn-Luttinger mechanism is the leading channel of Cooper instability; otherwise, the leading contribution is accurately captured by RPA.

Figure 2

Static KO vertex function on the Fermi surface $W_{\ell }(k_F,k_F,\omega =0)$ at $r_s=1.0$. Dotted red line with circles represents the original $W_{\ell }$ with clear KL oscillations decaying as ${\ell }^{-4}$ (dot-dash green and dashed blue fits). The solid black line with squares represents the regularized $W_{\ell }$.

Figure 3

Temperature dependence of the eigenvalues $\overline{\lambda }\left(T\right)$ for RPA (red circles) and KO (black triangles) vertex functions at $r_s=2.0$ and $\ell =3$. The linear fits of the RPA (red dotted line) and KO (black dashed line) data are almost identical. The extrapolated value of $T_c=2.71\times {10}^{-20}{E}_F$ is extremely small.

Figure 4

Amplitudes of the relative eigenvalue oscillations, $|{\eta }_{\ell +1}-{\eta }_{\ell }|$, induced by regularization of the KO vertex at $T={10}^{-5}{E}_F$ and $r_s=0.5$ (blue circles), $r_s=1$ (green squares), and $r_s=2$ (yellow triangles). Inset: relative eigenvalue changes ${\eta }_{\ell }$. Data in the main figure and inset involve scaling factors of ${10}^{-7}$ and ${10}^{-6}$, respectively.

Figure 5

Oscillation amplitudes as functions of ${log}_{10}(E_F/T)$ for $r_s=0.5$ and $\ell =21$ (blue squares), $\ell =23$ (green circles), and $\ell =25$ (yellow triangles). When $T/E_F$ is reduced by one order of magnitude, the amplitudes increase by less than $3\times {10}^{-8}$ for $\ell \ge 21$ at $T\sim {10}^{-6}{E}_F$.

Figure 6

Dominant superconducting channel ${\ell }_c$ (green squares connected by the dashed line) as a function of $r_s$ along with the critical channel of the KL mechanism ${\ell }_{KL}$ (blue circles connected by the dotted line).