Analysis of the Strong Coupling Limit of the Richardson Hamiltonian using the Dyson Mapping

TL;DR

This paper applies the Dyson mapping to analyze the strong coupling limit of the Richardson Hamiltonian, revealing new insights into superconducting correlations in small metallic grains without relying on Bethe Ansatz solutions.

Contribution

It introduces a boson expansion method using Dyson mapping for perturbative analysis of the Richardson Hamiltonian in the strong pairing limit, avoiding Bethe Ansatz reliance.

Findings

Uncovers a selection rule simplifying perturbation expansions.

Highlights subtleties in modeling superconductivity as a Bose-Einstein condensate.

Provides recursive formulas for high-order perturbation corrections.

Abstract

The Richardson Hamiltonian describes superconducting correlations in a metallic nanograin. We do a perturbative analysis of this and related Hamiltonians, around the strong pairing limit, without having to invoke Bethe Ansatz solvability. Rather we make use of a boson expansion method known as the Dyson mapping. Thus we uncover a selection rule that facilitates both time-independent and time-dependent perturbation expansions. In principle the model we analise is realised in a very small metalic grain of a very regular shape. The results we obtain point to subtleties sometimes neglected when thinking of the superconducting state as a Bose-Einstein condensate. An appendix contains a general presentation of time-independent perturbation theory for operators with degenerate spectra, with recursive formulas for corrections of arbitrarily high orders.