Small Superconducting Grain in the Canonical Ensemble

By means of the Lanczos method we analyze superconducting correlations in ultrasmall grains at fixed particle number. We compute the ground-state properties and the excitation gap of the pairing Hamiltonian as a function of the level spacing $\ensuremath{\delta}$. Both quantities turn out to be parity dependent and universal functions of the ratio $\ensuremath{\delta}/\ensuremath{\Delta}$ ( $\ensuremath{\Delta}$ is the BCS gap). We then characterize superconductivity in the canonical ensemble from the scaling behavior of correlation functions in energy space.