Condensation energy in Eliashberg theory -- from weak to strong coupling

TL;DR

This paper analyzes the condensation energy in Eliashberg superconductors, deriving a correction to BCS theory and showing its dependence on the pairing interaction's form across different coupling strengths.

Contribution

It provides a first-order correction to the BCS condensation energy formula and demonstrates the impact of the pairing interaction's functional form on the energy.

Findings

Derived a correction to BCS condensation energy at first order in coupling.

Showed the strong dependence of condensation energy on the pairing interaction's form.

Analyzed the effects across weak to strong coupling regimes.

Abstract

We consider two issues related to the condensation energy in superconductors described by the Eliashberg theory for various forms of the pairing interaction, associated either with phonon or electronic mechanisms of superconductivity. First, we derive a leading correction to the BCS formula for the condensation energy to first order in the coupling $\lambda$. Second, we show that at a given $\lambda$, the value of the condensation energy strongly depends on the functional form of the effective pairing interaction $\Gamma (\omega)$.