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 Sap of the pairing Hamiltonian as a function of the level spacing delta, Both quantities turn out to be parity dependent and universal functions of the ratio delta/Delta (Delta is the BCS gap). We then characterize superconductivity in the canonical ensemble from the scaling behavior of correlation functions in energy space.