Pairing in the Bogoliubov–de Gennes Equations

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.