Renormalization-group approach to superconductivity: from weak to strong electron-phonon coupling

We present the numerical solution of the renormalization group (RG) equations derived in Tsai, Castro Neto, Shankar, et al. [Phys. Rev. B to appear], for the problem of superconductivity in the presence of both electron-electron and electron-phonon coupling at zero temperature. We study the instability of a Fermi liquid to a superconductor and the RG flow of the couplings in the presence of retardation effects and the crossover from weak to strong coupling. We show that our numerical results provide an ansatz for the analytic solution to the problem in the asymptotic limits of weak and strong coupling.