Superconducting correlations in small metallic grains

By applying the path-integral formulation and the Feynman theorem, we calculate the off-diagonal superconducting correlation in an ultrasmall metallic grain. Unlike its behavior in the bulk limit, we find that this correlation function is an intensive quantity in this case. Therefore, superconductivity is indeed completely suppressed by quantum fluctuations as the diameter of the grain shrinks to the nanometer scale. This conclusion confirms the previous results by numerical calculations. Furthermore, it also imposes a strong constraint on the delocalizing amplitude of the pair-mixing function, which was recently proposed to characterize superconductivity in the canonical ensemble.