The subject of BCS - Bose Einstein condensation (BEC) crossover is particularly exciting because of its realization in ultracold Fermi gases and its possible relevance to high temperature superconductors. In the paper we review that body of theoretical work on this subject which represents a natural extension of the seminal papers by Leggett and by Nozieres and Schmitt-Rink (NSR). The former addressed only the ground state, now known as the "BCS-Leggett" wave-function and the key contributions of the latter pertain to calculations of the superfluid transition temperature $T_c$. These two papers have given rise to two main and, importantly, distinct, theoretical schools in the BCS-BEC crossover literature. The first of these extends the BCS-Leggett ground state to finite temperature and the second extends the NSR scheme away from $T_c$ both in the superfluid and normal phases. It is now rather widely accepted that these extensions of NSR produce a different ground state than that first introduced by Leggett. Our analysis shows how the NSR-based approach views the bosonic contributions more completely but it treats the fermions as "quasi-free". By contrast, the BCS-Leggett based approach treats the fermionic contributions more completely but it treats the bosons as "quasi-free". The NSR based schemes approach the crossover between BCS and BEC by starting from the BEC limit and the BCS-Leggett based scheme approaches this crossover by starting from the BCS limit. Ultimately, one would like to combine these two schemes. In this paper we review the strengths and weaknesses of both approaches. To reach a full understanding, it is important in the future to invest effort in investigating in more detail the T=0 aspects of NSR-based theory and the $T \neq 0$ aspects of BCS-Leggett theory.
Pairing fluctuations and the superfluid density through the BCS-BEC crossover
我们推导了一个表达式的超流密度的均匀的两个组件的费米气体通过BCS-BEC交叉的热力学势在存在一个强加的超流。按照Nozières和Schmitt-Rink的方法处理高斯近似中的配对涨落，我们使用$ρ_s$的这个定义来获得在有限温度下和在完整的BCS-BEC交叉上有效的明确结果。BCS能隙$Δ$、化学势$μ$和$ρ_s$都以自洽的方式包括相同水平的波动效应，这一点至关重要。我们发现，正常的流体密度$ρ_n \equiv n - ρ_s$自然分离成费米BCS准粒子（$ρ^F_{n}$）和玻色集体模（$ρ^B_{n}$）的贡献之和。$ρ^F_{n}$的表达式只是BCS费米超流体的朗道公式，但现在是在BCS-BEC交叉上计算的。玻色贡献的表达式$ρ^B_{n}$更复杂，并且仅在极端BEC极限下简化为玻色超流体的朗道公式，其中所有费米子都形成了稳定的玻色对，并且相关分子玻色凝聚体的Bogoliubov激发是无阻尼的。在一个配套文件中，我们提出了数值计算的$ρ_s$使用一个表达式相当于本文推导出的，在BCS-BEC交叉，包括么正性，并在有限的温度。
