We show that the natural variable to follow the crossover from Cooper-pair-based superconductivity to Bose–Einstein condensation within the model of Noziéres and Schmitt-Rink is the product k F ξ, where k F is the Fermi wave vector and ξ, is the coherence length for two-electron correlation. In terms of this product, the results of the model do not depend on the detailed form of the (separable) pairing potential, and the crossover turns out to be restricted to the universal region π −1 ≲k F ξ ≲ 2π. Experimental estimates indicate that k F ξ ≈10 (> 2π) for high-T c superconductors. Evolution from weak to strong coupling superconductivity has been considered by Nozières and Schmitt-Rink [1] (hereafter referred to as NSR) following the pioneering work by Leggett [2]. After the discovery of high- T c superconductivity, the interest in this problem has grown, and many papers on this subject have appeared [3]. In the present work, we show that working within the simplified treatment by NSR, it is already possible to isolate the essential features of the crossover. Central to the work of NSR and Leggett is the argument [4] that the BCS wave function has the Bose–Einstein condensation (BEC) built in as a limiting case. (See the review by Randeria in this volume.) NSR study the evolution from BCS to BEC through the increase of the coupling strength associated to an effective fermionic attractive potential, and conclude that the evolution is “smooth”.
We show that the natural variable to follow the crossover from Cooper-pair-based superconductivity to Bose–Einstein condensation within the model of Noziéres and Schmitt-Rink is the product k F ξ, where k F is the Fermi wave vector and ξ, is the coherence length for two-electron correlation. In terms of this product, the results of the model do not depend on the detailed form of the (separable) pairing potential, and the crossover turns out to be restricted to the universal region π −1 ≲k F ξ ≲ 2π. Experimental estimates indicate that k F ξ ≈10 (> 2π) for high-T c superconductors. Evolution from weak to strong coupling superconductivity has been considered by Nozières and Schmitt-Rink [1] (hereafter referred to as NSR) following the pioneering work by Leggett [2]. After the discovery of high- T c superconductivity, the interest in this problem has grown, and many papers on this subject have appeared [3]. In the present work, we show that working within the simplified treatment by NSR, it is already possible to isolate the essential features of the crossover. Central to the work of NSR and Leggett is the argument [4] that the BCS wave function has the Bose–Einstein condensation (BEC) built in as a limiting case. (See the review by Randeria in this volume.) NSR study the evolution from BCS to BEC through the increase of the coupling strength associated to an effective fermionic attractive potential, and conclude that the evolution is “smooth”.
Comment on "Evolution from BCS Superconductivity to Bose-Einstein Condensation: Role of the Parameter kFξ in Interpreting the Experimental Plot by Uemura et al.''}
The problem of crossover from BCS to Bose-Einstein condensation has recently attracted considerable interest owing to the discovery of high-Tc cuprates. Their short coherence length, ξ, places these materials in the interesting region between BCS and Bose-Einstein condensation. In the paper of F. Pistolesi and G.C. Strinati (Phys. Rev. B {\bf 49}, 6356 (1994)), the Nozi\`eres and Schmitt-Rink approach (NSR) is taken, which is valid for the weak coupling regime. They derive a relation between Tc and TF, which they insist is valid for any coupling strength. We present arguments that their assumptions are incorrect by using our fully self-consistent T-matrix formalism in two dimensions, and show that the NSR approach produces unphysical results in this case.
The crossover between the Cooper-pair condensation and the Bose-Einstein condensation of ``di-electronic'' molecules in two-dimensional superconductors is investigated in detail on the basis of the Nozi\`eres and Schmitt-Rink formalism. It is shown that temperature dependence of the chemical potential \ensuremath{\mu} so calculated is classified into two classes as decreasing temperatures; i.e., class (a) where \ensuremath{\mu} approaches the point of Bose-Einstein condensation of two-dimensional ideal Bose gas of ``di-electronic'' molecules, and class (b) where \ensuremath{\mu} 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 V is fixed, class (a) is realized for electron number density N smaller than Ncr, which is an increasing function of V, and class (b) is realized for N larger than Ncr. If N\ensuremath{\gg}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 \ensuremath{\mu}(T), at temperatures far below the Fermi temperature, shows a discontinuous jump at V=Vcr(N) 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 Nozi\`eres and Schmitt-Rink formalism itself especially in two dimensions. A preliminary approach beyond their formalism is briefly discussed.
We argue that the natural variable to establish the crossover from Cooper-pair-based superconductivity to Bose-Einstein condensation of bound-electron pairs is the product kF\ensuremath{\xi} of Fermi electron wave vector times the coherence length for two-electron correlation, and that Cooper-pair-based superconductivity is stable against bosonization down to kF\ensuremath{\xi}\ensuremath{\simeq}2\ensuremath{\pi}. We also propose that the experimental plot by Uemura et al., relating the superconducting temperature Tc to the Fermi temperature TF for a variety of ``exotic'' as well as conventional superconductors, can be rationalized by correlating the ratio Tc/TF to kF\ensuremath{\xi}. It appears in this way that the high-Tc superconductors lie in the plot by Uemura et al. near the ``instability'' line kF\ensuremath{\xi}=2\ensuremath{\pi}.
This article presents a comparison of two finite-temperature BCS--Bose-Einstein condensation (BEC) crossover theories above the transition temperature: Nozieres--Schmitt-Rink (NSR) theory and finite-T extended BCS-Leggett theory. The comparison is cast in the form of numerical studies of the behavior of the fermionic spectral function both theoretically and as constrained by (primarily) radio frequency (rf) experiments. Both theories include pair fluctuations and exhibit pseudogap effects, although the nature of this pseudogap is very different. The pseudogap in finite-T extended BCS-Leggett theory is found to follow a BCS-like dispersion which, in turn, is associated with a broadened BCS-like self-energy, rather more similar to what is observed in high-temperature superconductors (albeit, for a d-wave case). The fermionic quasiparticle dispersion is different in NSR theory and the damping is considerably larger. We argue that the two theories are appropriate in different temperature regimes with the BCS-Leggett approach being more suitable nearer to condensation. There should, in effect, be little difference at higher T as the pseudogap becomes weaker and where the simplifying approximations used in the BCS-Leggett approach break down. On the basis of momentum-integrated rf studies of unpolarized gases, it would be difficult to distinguish which theory is the better one. A full comparison for polarized gases is not possible since it is claimed that there are inconsistencies in the NSR approach (not found in the BCS-Leggett scheme). Future experiments along the lines of momentum-resolved experiments look to be very promising in distinguishing the two theories.
We consider a fermionic system at zero temperature interacting through an effective nonretarded potential of the type introduced by Nozi\`eres and Schmitt-Rink, and calculate the phase coherence length ${\ensuremath{\xi}}_{\mathrm{phase}}$ (associated with the spatial fluctuations of the superconducting order parameter) by exploiting a functional-integral formulation for the correlation functions and the associated loop expansion. This formulation is especially suited to follow the evolution of the fermionic system from a BCS-type superconductor for weak coupling to a Bose-condensed system for strong coupling, since in the latter limit a direct mapping of the original fermionic system onto an effective system of bosons with a residual boson-boson interaction can be established. Explicit calculations are performed at the one-loop order. The phase coherence length ${\ensuremath{\xi}}_{\mathrm{phase}}$ is compared with the coherence length ${\ensuremath{\xi}}_{\mathrm{pair}}$ for two-electron correlation, which is relevant to distinguish the weak- (kF${\ensuremath{\xi}}_{\mathrm{pair}}$\ensuremath{\gg}1) from the strong- (kF${\ensuremath{\xi}}_{\mathrm{pair}}$\ensuremath{\ll}1) coupling limits (kF being the Fermi wave vector) as well as to follow the crossover in between. It is shown that ${\ensuremath{\xi}}_{\mathrm{phase}}$ coincides with ${\ensuremath{\xi}}_{\mathrm{pair}}$ down to kF${\ensuremath{\xi}}_{\mathrm{pair}}$\ensuremath{\simeq}10, ${\ensuremath{\xi}}_{\mathrm{pair}}$ in turn coinciding with the Pippard coherence length. In the strong-coupling limit we find instead that ${\ensuremath{\xi}}_{\mathrm{phase}}$\ensuremath{\gg}${\ensuremath{\xi}}_{\mathrm{pair}}$, with ${\ensuremath{\xi}}_{\mathrm{pair}}$ coinciding with the radius of the bound-electron pair. From the mapping onto an effective system of bosons in the strong-coupling limit we further relate ${\ensuremath{\xi}}_{\mathrm{pair}}$ with the ``range'' of the residual boson-boson interaction, which is physically the only significant length associated with the dynamics of the bosonic system. \textcopyright{} 1996 The American Physical Society.