Pairing in the Bogoliubov-de Gennes equations

TL;DR

This paper discusses how the Bogoliubov-de Gennes equations describe electron pairing in superconductors, emphasizing the need for BCS-like reduction and Anderson's pairing condition to accurately determine the vacuum state and account for impurity effects.

Contribution

It clarifies the pairing mechanisms within the Bogoliubov-de Gennes framework, highlighting the necessity of BCS reduction and Anderson's pairing condition for correct state determination.

Findings

The kernel of the self-consistency equation must be constrained by BCS pairing.

Impurities require Anderson's pairing condition for proper vacuum state.

Localization effects modify phonon-mediated interactions.

Abstract

It is shown that the Bogoliubov-de Gennes equations pair the electrons in states which are linear combinations of the normal states. Accordingly, the BCS-like reduction procedure is required to choose a correct pairing. For a homogeneous system, we point out that the kernel of the self-consistency equation derived from the Bogoliubov-de Gennes equations needs to be constrained by the BCS pairing condition. In the presence of ordinary impurities, on the other hand, the Bogoliubov-de Gennes equations should be supplemented by Anderson's pairing condition to obtain the correct vacuum state by the corresponding unitary transformation. This results in localization correction to the phonon-mediated interaction.