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.