Energy of N Cooper pair by analytically solving Richardson-Gaudin equations

TL;DR

This paper analytically solves the Richardson-Gaudin equations to determine the ground state energy of N Cooper pairs, confirming the BCS result and revealing that pair interactions depend on N as N(N-1), indicating Pauli blocking as the interaction mechanism.

Contribution

It provides the first exact analytical solution to the energy of an arbitrary number of Cooper pairs, advancing understanding of pair interactions in superconductors.

Findings

Supports the BCS result for pair energy at half-filling

Shows interaction energy depends on N as N(N-1)

Reveals Pauli blocking as the sole interaction mechanism

Abstract

This Letter provides the solution to a yet unsolved basic problem of Solid State Physics: the ground state energy of an arbitrary number of Cooper pairs interacting via the Bardeen-Cooper-Schrieffer potential. We here break a 50 year old math problem by analytically solving Richardson-Gaudin equations which give the exact energy of these $N$ pairs via $N$ parameters coupled through $N$ non-linear equations. Our result fully supports the standard BCS result obtained for a pair number equal to half the number of states feeling the potential. More importantly, it shows that the interaction part of the $N$-pair energy depends on $N$ as $N(N-1)$ only from N=1 to the dense regime, a result which evidences that Cooper pairs interact via Pauli blocking only.