Renormalization-group approach to strong-coupled superconductors

We develop an asymptotically exact renormalization-group approach that treats electron-electron and electron-phonon interactions on an equal footing. The approach allows an unbiased study of the instabilities of Fermi liquids without the assumption of a broken symmetry. We apply our method to the problem of strongly coupled superconductors and find the temperature $T^{*}$ below which the high-temperature Fermi liquid state becomes unstable towards Cooper pairing. We show that $T^{*}$ is the same as the critical temperature $T_c$ obtained in Eliashberg’s strong coupling theory starting from the low-temperature superconducting phase. A $1∕N$-expansion shows that the method is asymptotically exact and Migdal’s theorem follows as a consequence. Finally, our results lead to a novel way to calculate numerically, from microscopic parameters, the transition temperature of superconductors.

Figure 1

(a) The retarded interaction $\tilde{u}$. (b) The self-energy correction. (c) The interaction vertex.

Figure 2

(a) The electron-phonon vertex; (b)–(d) are self-energy corrections.

Figure 3

The numerical solution of (15) (continuous line) for ${\mu }^{*}=0$, 0.17, and 0.24 (from top to bottom) as a function of $\lambda$. Dashed line: $T^{*}∕{\omega }_E=1.1\exp [-(1+\lambda )∕(\lambda +{\mu }^{*}(1+\lambda )\phantom{)}]$ for ${\mu }^{*}=0.24$; dotted line: $T^{*}∕{\omega }_E=0.11\sqrt{\lambda -1.49}$ for ${\mu }^{*}=0.24$. The inset shows the determinant in (15) as a function of $T$ for ${\mu }^{*}=0$ and $\lambda =0.3$, 0.4, 0.5, 0.7, and 1 (from top to bottom).