Annals of Physics 275, 126 (1999) Fluctuations and Odd-Even Effects in Small Superfluid Systems R. Rossignoli and N. Canosa Departamento de F@sica, Universidad Nacional de La Plata, c.c. 67, 1900 La Plata, Argentina and P. Ring Physik-Department der Technischen Universitat Munchen, D-85748 Garching, Germany Received June 26, 1998; revised November 9, 1998 Fluctuations and odd-even effects in small superfluid systems at finite temperature are investigated by means of the static path plus RPA approximation. A general derivation of this method is presented, which allows a straightforward implementation in statistical ensembles with fixed number parity (NP). A significant smoothing of the superconducting to normal transition is obtained in systems where the gap is comparable to the level spacing, as well as a decrease of pairing correlations in the odd case. Results are shown for a schematic model, where an excellent agreement with exact canonical results is obtained with the present method, and then for a heavy nucleus. Comparison with NP projected BCS and BCS+RPA results is made. An effective BCS+RPA approach which remains smooth in transitional regions is also derived.  1999 Academic Press I. INTRODUCTION Small correlated quantum systems exhibit important fluctuation phenomena, which lead to significant deviations from the predictions of conventional mean field approximations (MFA). In particular, in finite systems at finite temperature, the sharp phase transitions displayed by the mean field become increasingly washed out as the size of the system decreases, which can be attributed to the effects of growing fluctuations in the relevant order parameters [1, 2]. On the other hand, in small systems with fixed particle number, one can also expect important deviations from the predictions of standard grand canonical (GC) statistics due to canonical correc- tions. These facts have recently acquired special relevance in solid state physics with relation to the development of diverse mesoscopic structures, as for instance super- conducting islands [3] and ultrasmall superconducting metallic grains [4], where odd-even effects, i.e., differences between systems with odd and even particle number, are non-negligible and play an important role [511]. Most theoretical studies at finite temperature have been based, however, on the extension of the BCS approxi- mation to an ensemble with fixed number parity [5], rigorously derived in [11] Article ID aphy.1999.5914, available online at http:www.idealibrary.com on 1 0003-491699 30.00 Copyright  1999 by Academic Press All rights of reproduction in any form reserved.