# Two-Cooper-pair problem and the Pauli exclusion principle

## TL;DR

This paper analyzes the two-Cooper pair problem to understand how Pauli exclusion influences pair energies and the superconducting gap, revealing that pair binding energy and gap behavior differ from common assumptions.

## Contribution

It provides the first exact derivation of the two-Cooper pair energy considering Pauli blocking, clarifying its impact on pair binding and the superconducting gap.

## Key findings

- Pauli blocking increases the free energy component of the two-pair ground state.
- Pauli blocking decreases the binding energy of pairs compared to isolated pairs.
- The excitation gap increases when going from one to two pairs.

## Abstract

While the one-Cooper pair problem is now a textbook exercise, the energy of two pairs of electrons with opposite spins and zero total momentum has not been derived yet, the exact handling of Pauli blocking between bound pairs being not that easy for N=2 already. The two-Cooper pair problem however is quite enlightening to understand the very peculiar role played by the Pauli exclusion principle in superconductivity. Pauli blocking is known to drive the change from 1 to $N$ pairs, but no precise description of this continuous change has been given so far. Using Richardson procedure, we here show that Pauli blocking increases the free part of the two-pair ground state energy, but decreases the binding part when compared to two isolated pairs - the excitation gap to break a pair however increasing from one to two pairs. When extrapolated to the dense BCS regime, the decrease of the pair binding while the gap increases strongly indicates that, at odd with common belief, the average pair binding energy cannot be of the order of the gap.