Self-Consistent Approximations for Superconductivity beyond the Bardeen-Cooper-Schrieffer Theory

TL;DR

This paper introduces a self-consistent perturbation framework for superconductivity that simplifies the inclusion of pair processes, automatically satisfies conservation laws, and extends existing theories to study complex superconducting properties.

Contribution

It develops a symmetric interaction vertex formalism that unifies various superconducting theories and derives the FLEX-S approximation with comprehensive pair process inclusion.

Findings

Reproduces BCS theory at lowest order

Includes all pair processes in a unified formalism

Derives closed equations for superconducting properties

Abstract

We develop a concise self-consistent perturbation expansion for superconductivity where all the pair processes are naturally incorporated without drawing "anomalous" Feynman diagrams. This simplification results from introducing an interaction vertex that is symmetric in the particle-hole indices besides the ordinary space-spin coordinates. The formalism automatically satisfies conservation laws, includes the Luttinger-Ward theory as the normal-state limit, and reproduces the Bardeen-Cooper-Schrieffer theory as the lowest-order approximation. It enables us to study the thermodynamic, single-particle, two-particle, and dynamical properties of superconductors with competing fluctuations based on a single functional $\Phi[\hat{G}]$ of Green's function $\hat{G}$ in the Nambu space. Specifically, we derive closed equations in the FLEX-S approximation, i.e., the fluctuation exchange approximation for superconductivity with all the pair processes, which contains extra terms besides those in the standard FLEX approximation.