Fixed-N Superconductivity: The Exact Crossover from the Bulk to the Few-Electron Limit

TL;DR

This paper investigates the crossover from bulk superconductivity to few-electron regimes in ultrasmall metallic grains using canonical approaches, revealing how pairing correlations evolve with system size.

Contribution

It introduces a combined variational and exact solution approach to describe superconductivity across the entire size spectrum of metallic grains.

Findings

Recovery of BCS limit in large grains

Delocalization of pairing correlations in energy space for small grains

Detailed wave-function analysis of the crossover

Abstract

We use two truly canonical approaches to describe superconductivity in ultrasmall metallic grains: (a) a variational fixed-N projected BCS-like theory and (b) an exact solution of the model Hamiltonian developed by Richardson in context with Nuclear Physics. Thereby we obtain a description of the entire crossover from the bulk BCS regime (mean level spacing $d\ll$ bulk gap $\tilde\Delta$) to the `fluctuation-dominated' few-electron regime ($d\gg\tilde\Delta$). A wave-function analysis shows in detail how the BCS limit is recovered and how for $d\gg\tilde\Delta$ pairing correlations become delocalized in energy space.