The reduced BCS model that is commonly used for ultrasmall superconducting grains has an exact solution worked out long ago by Richardson in the context of nuclear physics. We use it to check the quality of previous treatments of this model, and to investigate the effect of level statistics on pairing correlations. We find that the ground state energies are on average somewhat lower for systems with non-uniform than uniform level spacings, but both have an equally smooth crossover from the bulk to the few-electron regime. In the latter, statistical fluctuations in ground state energies strongly depend on the grain's electron number parity. . Nevertheless, these two regimes are qualitatively very different [9,10]: the condensation energy, e.g., is an extensive function of volume in the former and almost intensive in the latter, and pairing correlations are quite strongly localized around the Fermi energy ε F , or more spread out in energy, respectively.After the appearance of all these works, we became aware that the reduced BCS Hamiltonian on which they are based actually has an exact solution. It was published by Richardson in the context of nuclear physics (where it is known as the "picket-fence model"), in a series of papers between 1963 and 1977 [12,13] which seem to have completely escaped the attention of the condensed matter community. The beauty of this solution, besides its mathematical elegance, is that it also works for the case of randomly-spaced levels. It thus presents us with two rare opportunities, which are the subject of this Letter: (i) to compare the results of various previously-used approximations against the benchmark set by the exact solution, in order to gauge their reliability for related problems for which no exact solutions exist; and very interestingly, (ii) to study the interplay of randomness and interactions in a non-trivial model exactly, by examining the effect of level statistics on the SC/FD crossover.There is a previous study of the latter question by Smith and Ambegaokar using the g.c. mean-field BCS approach [5], who concluded, interestingly, that randomness enhances pairing correlations: compared to the case of uniform spacings [2], they found that a random spacing of levels (distributed according to the gaussian orthogonal ensemble) on average lowers the condensation energy E C to more negative values and increases the critical level spacings at which E C vanishes abruptly, but these still are parity dependent ( d c 1 = 1.8∆, d c 0 ≃ 14∆). However, the abrupt vanishing of E C found by SA can be suspected to be an artifact of their g.c. mean-field treatment, as was the case in [2][3][4]. Indeed, our exact results for random levels show (1) that the SC/FD crossover is as smooth as for the case of uniformly-spaced levels; this means, remarkably, that (2) even in the presence of randomness pairing correlations never vanish, no matter how large d/∆ becomes; quite the opposite, (3) the randomness-induced lowering of E C is strongest in the FD regime; in the latter, moreover, (...
