Fixed-NSuperconductivity: The Crossover from the Bulk to the Few-Electron Limit

We present a truly canonical theory of superconductivity in ultrasmall metallic grains by variationally optimizing fixed- $N$ projected BCS wave functions, which yields the first full description of the entire crossover from the bulk BCS regime (mean level spacing $d\ll \mathrm{bulk}\mathrm{gap}\tilde{\Delta }$) to the “fluctuation-dominated” few-electron regime $\left(d\gg \tilde{\Delta }\right)$. A wave-function analysis shows in detail how the BCS limit is recovered for $d\ll \tilde{\Delta }$, and how for $d\gg \tilde{\Delta }$ pairing correlations become delocalized in energy space. An earlier grand-canonical prediction for an observable parity effect in the spectral gaps is found to survive the fixed- $N$ projection.