BCS-BEC Crossover: From High Temperature Superconductors to Ultracold Superfluids

We review the BCS to Bose Einstein condensation (BEC) crossover scenario which is based on the well known crossover generalization of the BCS ground state wavefunction $\Psi_0$. While this ground state has been summarized extensively in the literature, this Review is devoted to less widely discussed issues: understanding the effects of finite temperature, primarily below $T_c$, in a manner consistent with $\Psi_0$. Our emphasis is on the intersection of two important problems: high $T_c$ superconductivity and superfluidity in ultracold fermionic atomic gases. We present the evidence in support of a crossover scenario for the "pseudogap state" in the copper oxide superconductors. We argue that current experiments in atomic gases are most likely in the counterpart pseudogap regime. That is, superconductivity takes place out of a non-Fermi liquid phase where preformed, metastable fermion pairs are present at the onset of their Bose condensation. As a microscopic basis for this work, we summarize a variety of T-matrix approaches, and assess their theoretical consistency. A close connection with conventional superconducting fluctuation theories is emphasized and exploited.