Superconductivity in Ultrasmall Grains: Introduction to Richardson's Exact Solution

TL;DR

This paper introduces Richardson's exact solution for pairing correlations in ultrasmall metallic grains, providing a tutorial and evaluating previous approaches to this reduced BCS-model.

Contribution

It offers a clear tutorial on Richardson's exact solution and assesses the accuracy of earlier methods used for this model.

Findings

Richardson's solution accurately describes pairing in ultrasmall grains

Previous treatments vary in their approximation quality

The tutorial clarifies the application of the exact solution

Abstract

Studies of pairing correlations in ultrasmall metallic grains have commonly been based on a simple reduced BCS-model describing the scattering of pairs of electrons between discrete energy levels that come in time-reversed pairs. This model has an exact solution, worked out by Richardson in the context of nuclear physics in the 1960s. Here we give a tutorial introduction to his solution, and use it to check the quality of various previous treatments of this model.