The functional-integral approach to Gaussian fluctuations in Eliashberg theory

TL;DR

This paper presents a functional-integral derivation of Eliashberg theory for superconductivity, extending it to include Gaussian fluctuations and analyzing their effects on physical properties like diamagnetic susceptibility.

Contribution

It introduces a self-consistent functional-integral approach to Eliashberg theory and systematically derives Gaussian-fluctuation corrections, including Cooper and density-channel interactions.

Findings

Derivation of Eliashberg equations as saddle-point conditions

Systematic inclusion of Gaussian fluctuations in the theory

Analysis of fluctuation diamagnetic susceptibility near Tc

Abstract

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 approximations 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 transformations. 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 strong-coupling fluctuation diamagnetic susceptibility near the critical temperature.