Strong-coupling limit of Eliashberg theory

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Abstract
We study the strong-coupling limit of the Eliashberg theory of superconductivity, where the coupling strength λ goes to infinity and the critical temperature gets large compared to a typical phonon energy. This limit is of interest because it is both universal and simple, and we may hope to obtain from this study a deeper understanding of the conventional strong-coupling regime of superconductivity. Our work on this problem is both analytical and numerical. At T=0, we find that the excitation spectrum is discrete. We interpret physically the excited states as bound states due to a type of polaronic effect. We show that one can solve the Eliashberg equations essentially analytically by working fully on the real frequency axis. At finite temperature we find a thermal smearing of the T=0 structure. Since the critical temperature is small compared to the zero-temperature gap, thermal effects can be treated as a kind of perturbation over almost all the temperature range. In this spirit, we give a simple approximate solution which reproduces almost quantitatively the exact numerical results.