From one to $N$ Cooper pairs, step by step

Abstract

We extend the one-pair Cooper configuration towards Bardeen-Cooper-Schrieffer (BCS) model of superconductivity by adding one-by-one electron pairs to an energy layer where a small attraction acts. To do it, we solve Richardson's equations analytically in the dilute limit of pairs on the one-Cooper pair scale. We find, through only keeping the first order term in this expansion, that the $N$ correlated pair energy reads as the energy of $N$ isolated pairs within a $N(N-1)$ correction induced by the Pauli exclusion principle which tends to decrease the average pair binding energy when the pair number increases. Quite remarkably, extension of this first-order result to the dense regime gives the BCS condensation energy exactly. This leads us to suggest a different understanding of the BCS condensation energy with a pair number equal to the number of pairs feeling the potential and an average pair binding energy reduced by Pauli blocking to half the single Cooper pair energy - instead of the more standard but far larger superconducting gap.

14 pages, 2 figures

References in corpus (4)

• Many-Body Physics with Ultracold Gases
• Theory of ultracold Fermi gases
• Microscopic derivation of Frenkel excitons in second quantization
• Ground state energy of N Frenkel excitons

Cited by in corpus (9)

• Variational Principle of Bogoliubov and Generalized Mean Fields in Many-Particle Interacting Systems
• Neutrino Spectral Split in the Exact Many Body Formalism
• Richardson-Gaudin integrability in the contraction limit of the quasispin
• "Probabilistic" approach to Richardson equations
• BCS ansatz, Bogoliubov approach to superconductivity and Richardson-Gaudin exact wave function
• Ground-state energies of the open and closed $p+ip$-pairing models from the Bethe Ansatz
• Multi-particle instability in a spin-imbalanced Fermi gas
• BCS ansatz for superconductivity in the canonical ensemble and the Pauli exclusion principle
• Circuit Quantisation from First Principles