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.
The spectrum of collective (pair) excitations in the ground state of a dilute twodimensional (2D) attractive Fermi-gas is studied within the functional integral formalism. The linearized equations for the fluctuations about the non-trivial saddle point are analyzed for all coupling regimes, which are characterized by the ratio ε 0 /ε F , where ε 0 is the two-fermion binding energy and ε F is the Fermi energy. The approximation takes into account propagation of the fluctuations and their interaction with the condensate. In the strong-coupling, or ‘;Bosegas’, regime (ε 0 /ε F > 1) the spectrum is continuous and has the Bogolubov form, but in the weak-coupling limit (ε 0 /ε F «1) there are two types of excitations (different from the two-fermion scattering states): (i) long-wavelength sound-like excitations with the cut-off at momentum q c ≃ l/ξ 0 (where ξ 0 is the Cooper pair size), and (ii) pair excitations with q ≃ (8 m μ) ½ , where μ is the chemical potential and m is the fermion mass. The crossover between weakand strong-coupling behavior of the excitation spectrum is found to occur at the value of the coupling parameter ε 0 /ε F ≃¼. Introduction Since the discovery of high- T c superconductivity there has been growing interest in studying 2D Fermi gases with attractive interaction, especially in the crossover regime, when the pair size ξ 0 is of the order of the interparticle distance. The importance of such a model, which can be regarded as a semiphenomenological model of a 2D superconductor, is highlighted by experimental evidence that the high- T c superconductors, most of which have a layered structure, have a short coherence length, so that k F ξ 0 ≃1, where k F is the Fermi momentum.
We derive an expression for the superfluid density of a uniform two-component Fermi gas through the BCS-BEC crossover in terms of the thermodynamic potential in the presence of an imposed superfluid flow. Treating the pairing fluctuations in a Gaussian approximation following the approach of Nozi\`eres and Schmitt-Rink, we use this definition of ${\ensuremath{\rho}}_{s}$ to obtain an explicit result which is valid at finite temperatures and over the full BCS-BEC crossover. It is crucial that the BCS gap $\ensuremath{\Delta}$, the chemical potential $\ensuremath{\mu}$, and ${\ensuremath{\rho}}_{s}$ all include the effect of fluctuations at the same level in a self-consistent manner. We show that the normal fluid density ${\ensuremath{\rho}}_{n}\ensuremath{\equiv}n\ensuremath{-}{\ensuremath{\rho}}_{s}$ naturally separates into a sum of contributions from Fermi BCS quasiparticles $({\ensuremath{\rho}}_{n}^{F})$ and Bose collective modes $({\ensuremath{\rho}}_{n}^{B})$. The expression for ${\ensuremath{\rho}}_{n}^{F}$ is just Landau's formula for a BCS Fermi superfluid but now calculated over the BCS-BEC crossover. The expression for the Bose contribution ${\ensuremath{\rho}}_{n}^{B}$ is more complicated and only reduces to Landau's formula for a Bose superfluid in the extreme BEC limit, where all the fermions have formed stable Bose pairs and the Bogoliubov excitations of the associated molecular Bose condensate are undamped. In a companion paper, we present numerical calculations of ${\ensuremath{\rho}}_{s}$ using an expression equivalent to the one derived in this paper, over the BCS-BEC crossover, including unitarity, and at finite temperatures.
We develop a strong-coupling theory for the superfluidity of fermion pairing phase in a Bose-Fermi mixture. Dynamical screening, self-energy renormalization, and a pairing gap function are included self-consistently within the adiabatic limit (i.e., the phonon velocity is much smaller than the Fermi velocity). An analytical solution for the transition temperature (T(c)) is derived within reasonable approximations. Using typical parameters of a 40K-87Rb mixture, we find that the calculated T(c) is several times larger than that obtained in the weak coupling theory, and can be up to several percent of the Fermi temperature.
The self-consistent equations, which have been derived recently as a microscopic model for the crossover between BCS superconductivity and Bose-Einstein condensation in a three-dimensional interacting Fermi system [R. Haussmann, Z. Phys. B 91, 291 (1993)], are solved numerically by repeated Fourier transformation. We find a superfluid transition temperature
Tc which increases monotonically with increasing attractive coupling strength. Furthermore, we determine the chemical potential \ensuremath{\mu}, the fermion distribution function n(k), and the complex effective mass 2
m∗ of the fermion pairs at T=
Tc. The bound fermion pairs cause a power-law tail \ensuremath{\sim}${\mathit{k}}^{\mathrm{\ensuremath{-}}4}$ in n(k) for large k and behave as short-living quasiparticles in the crossover region, which is indicated by a large imaginary part of 2
m∗.