Cooper-pair and Bose-Einstein condensations in two dimensions: A critical analysis based on the Nozières and Schmitt-Rink formalism

Abstract

The crossover between the Cooper-pair condensation and the Bose-Einstein condensation of "dielectronic" molecules in two-dimensional superconductors is investigated in detail on the basis of the Nozieres and Schmitt-Rink formalism. It is shown that temperature dependence of the chemical potential p so calculated is classified into two classes as decreasing temperatures; i.e. , class (a) where p approaches the point of Bose-Einstein condensation of two-dimensional ideal Bose gas of "di-electronic" molecules, and class (b) where p diverges positively along the line of BCS-type mean-field pair condensation. This feature is rather universal irrespective of strength V of the attractive interaction of the s-wave type. While the former class (a) has been found by Schmitt-Rink, Varma, and Ruckenstein, existence of the latter class (b) is recognized here. In the case where Vis fixed, class (a) is realized for electron number density N smaller than N", which is an increasing function of V, and class (b) is realized for X larger than N". If N ))Ncr in particular, there exists a regime, where the Fermi-liquid-like description is valid, between the BCS-type mean-field transition temperature and the Fermi temperature. In the situation where V is changed with N being fixed, low-temperature states for the strong-coupling case belong to class (a) while those for the weak-coupling case belong to class (b). Therefore, with decreasing V, the chemical potential p( T), at temperatures far below the Fermi temperature, shows a discontinuous jump at V = V"(Ã) corresponding to the transition from class (a) to (b). However, this is in contradiction to a physical picture that the chemical potential should smoothly cross over between the above two limits unless the liquid-gas transition occurs. This shows in turn a necessity of improving the Nozieres and Schmitt-Rink formalism itself especially in two dimensions. A preliminary approach beyond their formalism is briefly discussed.

BCS/BEC crossover in quark matter and evolution of its static and dynamic properties from the atomic unitary gas to color superconductivity

“…is the number of "flavors" of boson and χ 0 ≡ χ µ=0,T =0 ; we have subtracted the "vacuum" contribution at µ = T = 0, which in general leads a G-dependent fermion mass and wavefunction renormalization through a Hartree and higher order terms [75]. Some notes are in order here.…”

BCS-BEC Crossover in a Gas of Fermi Atoms with a Feshbach Resonance

“…However, µ can deviate from ε F in a more fundamental way when the quasi-molecules with q = 0, superfluid fluctuations, and the pre-formed Cooper-pairs are all properly included. [6,7,10] This is the goal of the present letter. …”

“…[7,10] In the present case, when we plot T c with respect to the effective pairing interaction, U eff = U + g 2 /(2ν − 2µ), we obtain the inset in Fig. 3(b).…”

Pairing and radio-frequency spectroscopy in two-dimensional Fermi gases

“…The first one is the Noziéres-Schmitt-Rink (NSR) approximation [30][31][32]37] and the second method is based on the number density given by the dressed Green's function [13,38]. The motivation for studying the two approximations is the fact that the NSR scheme is numerically much more affordable but expected to be accurate only when self-energy corrections are small [38].…”

“…Furthermore, we have set = 1. Within the non-self-consistent ladder approximation [10,11,13,[30][31][32][33], the self-energy is given by the T -matrix and the bare Green's function,…”