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Combined effect of thermal and quantum fluctuations in superconducting nanostructures: a path integral approach

We study the combined effect of thermal and quantum fluctuations in a zero dimensional superconductor. By using path integral techniques, we obtain novel expressions for the partition function and the superconducting order parameter which include both types of fluctuations. Our results are valid for any temperature and to leading order in δ/Δ_{0} where δis the mean level spacing and Δ_{0} is the bulk energy gap. We avoid divergences at low temperatures, previously reported in the literature, by identifying and treating non-perturbatively a low-energy collective mode. In the low and high temperature limit our results agrees with those from the random phase (RPA) and the static path approximation (SPA) respectively.