Applying BCS-BEC Crossover Theory To High Temperature Superconductors and Ultracold Atomic Fermi Gases

This review is written at the time of the twentieth anniversary of the discovery of high temperature superconductors, which, nearly coincides with the important discovery of the superfluid phases of ultracold trapped fermionic atoms. We show how these two subjects have much in common. Both have been addressed from the perspective of the BCS-Bose Einstein condensation (BEC) crossover scenario, which is designed to treat short coherence length superfluids with transition temperatures which are "high", with respect to the Fermi energy. A generalized mean field treatment of BCS-BEC crossover at general temperatures $T$, based on the BCS-Leggett ground state, has met with remarkable success in the fermionic atomic systems. Here we summarize this success in the context of four different cold atom experiments, all of which provide indications, direct or indirect, for the existence of a pseudogap. This scenario also provides a physical picture of the pseudogap phase in the underdoped cuprates which is a central focus of high $T_c$ research. We summarize successful applications of BCS-BEC crossover to key experiments in high $T_c$ systems including the phase diagram, specific heat, and vortex core STM data, along with the Nernst effect, and exciting recent data on the superfluid density in very underdoped samples,