Cooper pairing reexamined

TL;DR

This paper reexamines Cooper pairing by solving the Bethe-Salpeter equation, revealing both real and imaginary solutions for the pair energy, and questions the common interpretation of the energy gap as a binding energy.

Contribution

It provides a detailed analysis of Cooper-pair energies, showing the existence of real solutions coinciding with the BCS gap and clarifying the nature of imaginary solutions.

Findings

Real solutions match the BCS energy gap in weak coupling.

Imaginary solutions are confirmed as the only solutions in certain regimes.

Challenges the interpretation of the energy gap as a binding energy.

Abstract

When both two-electron \textit{and} two-hole Cooper-pairing are treated on an equal footing in the ladder approximation to the Bethe-Salpeter (BS) equation, the zero-total-momentum Cooper-pair energy is found to have two \textit{real} solutions $\mathcal{E}_{0}^{BS}=\pm 2\hbar \omega_{{D}%}/\sqrt{{e}^{2/\lambda }+{1}}$ which coincide with the zero-temperature BCS energy gap $\Delta =\hbar \omega_{D}/\sinh (1/\lambda) $ in the weak coupling limit. Here, $\hbar \omega_{D}$ is the Debye energy and $\lambda \geq 0$ the BCS model interaction coupling parameter. The interpretation of the BCS energy gap as the binding energy of a Cooper-pair is often claimed in the literature but, to our knowledge, never substantiated even in weak-coupling as we find here. In addition, we confirm the two purely-\textit{imaginary} solutions assumed since at least the late 1950s as the \textit{only} solutions, namely, $\mathcal{E}_{0}^{BS}=\pm i2\hbar \omega_{{D}}/\sqrt{{e}^{2/\lambda}{-1}}.$