Functional-integral approach to Gaussian fluctuations in Eliashberg theory

The Eliashberg theory of superconductivity is based on a dynamical electron-phonon interaction as opposed to a static interaction present in BCS theory. The standard derivation of Eliashberg theory is based on an equation of motion approach, which incorporates certain assumptions such as Migdal's approximation for the pairing vertex. In this paper we provide a functional-integral-based derivation of Eliashberg theory and we also consider its Gaussian-fluctuation extension. The functional approach enables a self-consistent method of computing the mean-field equations, which arise as saddle-point conditions, and here we observe that the conventional Eliashberg self-energy and pairing function both appear as Hubbard-Stratonovich auxiliary fields. An important consequence of this fact is that it provides a systematic derivation of the Cooper and density-channel interactions in the Gaussian-fluctuation response. We also investigate the fluctuation contribution to the diamagnetic susceptibility near the critical temperature.