Exact Study of the Effect of Level Statistics in Ultrasmall Superconducting Grains

TL;DR

This paper uses Richardson's exact solution of the reduced BCS model to analyze how level statistics influence pairing correlations in ultrasmall superconducting grains, revealing differences based on level uniformity and electron parity.

Contribution

It provides an exact analysis of level statistics effects on superconducting grains, comparing uniform and non-uniform level spacings using Richardson's solution.

Findings

Ground state energies are lower for non-uniform level spacings.

Both uniform and non-uniform systems show a smooth crossover from bulk to few-electron regimes.

Ground state energy fluctuations depend strongly on electron number parity.

Abstract

The reduced BCS model that is commonly used for ultrasmall superconducting grains has an exact solution worked out long ago by Richardson in the context of nuclear physics. We use it to check the quality of previous treatments of this model, and to investigate the effect of level statistics on pairing correlations. We find that the ground state energies are on average somewhat lower for systems with non-uniform than uniform level spacings, but both have an equally smooth crossover from the bulk to the few-electron regime. In the latter, statistical fluctuations in ground state energies strongly depend on the grain's electron number parity.