Short Coherence Length Superconductivity: A Generalization of BCS Theory for the Underdoped Cuprates

On the basis of the observed short coherence lengths in the cuprates we argue that a BCS-Bose-Einstein condensation (BEC) crossover approach is an appropriate starting point for correcting the mean field approach of BCS and, thereby, for addressing pseudogap phenomena in these materials. Our version of the BCS-BEC approach is based on a particular Greens' function decoupling scheme which should be differentiated from others in the literature, and which yields (i) the Leggett crossover ground state (for all coupling constants g, at T=0) and (ii) BCS theory (for all $ T \leq T_c$ over a range of small g). In this paper we provide a simple physical picture of the pseudogap phase above and below $T_c$, and review the quantitative and qualitative implications of this theory, which, for the most part have been published in a series of recent papers.