There continues to be enormous interest in the BCS to BEC transition as there still is no exact theory. We recently reported a revealing reinterpretation of the condensed phase Boson-Fermion Model (BFM) by comparing it to a cold atom formulation [1]. While the ground and singly excited states appear to remain continuous in all models we have examined, the collective modes contain a singularity due to a Feshbach resonance (tuned by doping) causing a breakdown of the Migdal theorem. As a result of vertex corrections, there is a fundamental change in the nature of the superconductivity due to the formation of preformed pairs as the previously suggested location [1] of a quantum critical point in the fulleride phase diagram is passed. The result is a quantum phase transition (QPT) between BCS and BEC-like (or Feshbach resonance) superconductivity (SC). We discuss features of the resonance and the role of the experimentally observed preformed pair formation in fullerides, essential to the Boson-Fermion Model (BFM), and often speculated since the work of Nozieres and Schmitt-Rink [17]. Here, we present arguments to establish a model of the preformed pair which can be favorably compared to a circular charge density wave (CDW) isolated on a fulleride molecule. The binding is much larger than a Cooper pair. The CDW seems to be stabalized by splitting of the Jahn-Teller active vibrational modes to reduce Coulomb repulsions. Our conclusions are: 1) the doping of two electrons into triply degenerate orbitals results in the experimentally observed singlet state (CDW); and 2) this CDW (preformed pair) has a dual role as doping is varied: suppression of BCS SC and enabling a Feshbach resonance form of SC.
There continues to be enormous interest in the BCS to BEC transition as there still is no exact theory. We recently reported a revealing reinterpretation of the condensed phase Boson-Fermion Model (BFM) by comparing it to a cold atom formulation [1]. While the ground and singly excited states appear to remain continuous in all models we have examined, the collective modes contain a singularity due to a Feshbach resonance (tuned by doping) causing a breakdown of the Migdal theorem. As a result of vertex corrections, there is a fundamental change in the nature of the superconductivity due to the formation of preformed pairs as the previously suggested location [1] of a quantum critical point in the fulleride phase diagram is passed. The result is a quantum phase transition (QPT) between BCS and BEC-like (or Feshbach resonance) superconductivity (SC). We discuss features of the resonance and the role of the experimentally observed preformed pair formation in fullerides, essential to the Boson-Fermion Model (BFM), and often speculated since the work of Nozieres and Schmitt-Rink [17]. Here, we present arguments to establish a model of the preformed pair which can be favorably compared to a circular charge density wave (CDW) isolated on a fulleride molecule. The binding is much larger than a Cooper pair. The CDW seems to be stabalized by splitting of the Jahn-Teller active vibrational modes to reduce Coulomb repulsions. Our conclusions are: 1) the doping of two electrons into triply degenerate orbitals results in the experimentally observed singlet state (CDW); and 2) this CDW (preformed pair) has a dual role as doping is varied: suppression of BCS SC and enabling a Feshbach resonance form of SC.
The pioneering works by Eagles, Leggett, and Nozieres/Schmitt-Rink (reviewed and augmented by Randeria) emphasize that in the limits of the models studied both at T = 0 and, the cross-over from a BCS-type to a BEC-type superconductor is continuous. The BCS and BEC end points seem to be well-established. However, in the intermediate region, home to fulleride and high temperature superconductors, considerable extrapolation of the models must be done as there still is no exact theory. Yet, considerable current literature is devoted to what appears to be more singular-type phenomena such as quantum critical points, stripe formation, insulator to superconductor phase transitions, loss of validity of the Fermi liquid theory, etc. Using a connection we have made with cold atom Fermion-Boson crossover theory, we can establish that the resonance previously discussed is a result of the crossing of the Fermion band by the Boson band. While the ground and singly excited states appear to remain continuous, the collective modes due to the resonance transform the nature of the superconductivity. We discuss features of the resonance and the experimentally observed pre-formed BEC Cooper pair formation in fullerides, essential to the Boson-Fermion resonance theory. In addition some of the various singular phenomena discussed above can be put into perspective. PACS numbers: 61.48.+c, 74.20.-z, 81.05Tp
We investigate strong-coupling superfluidity in a uniform two-component gas of ultracold Fermi atoms attractively interacting via quasimolecular bosons associated with a Feshbach resonance. This interaction is tunable by the threshold energy $2\ensuremath{\nu}$ of the Feshbach resonance, becoming large as $2\ensuremath{\nu}$ is decreased (relative to $2{\ensuremath{\varepsilon}}_{\mathrm{F}},$ where ${\ensuremath{\varepsilon}}_{\mathrm{F}}$ is the Fermi energy of one component). In recent work, we showed that the enhancement of this tunable pairing interaction naturally leads to the BCS-BEC (Bose-Einstein condensation) crossover, where the character of the superfluid phase transition changes from the BCS type to a BEC of composite bosons consisting of preformed Cooper-pairs and Feshbach-induced molecules. In this paper, we extend our previous work and study both the quasiparticles and the collective dynamics of the superfluid phase below the phase-transition temperature Tc, limiting ourselves to a uniform gas. We show how the superfluid order parameter changes from the Cooper-pair amplitude $\ensuremath{\Delta}$ to the square root of the number of condensed molecules $({\ensuremath{\varphi}}_{\mathrm{m}})$ associated with the Feshbach resonance, as the threshold energy $2\ensuremath{\nu}$ is lowered. In the intermediate coupling regime, the superfluidity is shown to be characterized by an order parameter consisting of a superposition of $\ensuremath{\Delta}$ and ${\ensuremath{\varphi}}_{\mathrm{m}}.$ We also discuss the Goldstone mode associated with superfluidity, and show how its character smoothly changes from the Anderson-Bogoliubov phonon in the BCS regime to the Bogoliubov phonon in the BEC regime in the BCS-BEC crossover. The velocity of this Goldstone phonon mode is shown to strongly depend on the value of $2\ensuremath{\nu}.$ We also show that this Goldstone mode appears as a resonance in the spectrum of the density-density correlation function, which is experimentally accessible.
We discuss the BCS-BEC crossover in a degenerate Fermi gas of two hyperfine states interacting close to a Feshbach resonance. We show that, by including fluctuation contributions to the free energy similar to that considered by Nozières and Schmitt-Rink, the character of the superfluid phase transition continuously changes from the BCS-type to the BEC-type, as the threshold of the quasimolecular band is lowered. In the BEC regime, the superfluid phase transition is interpreted in terms of molecules associated with both the Feshbach resonance and Cooper pairing.
The BCS-to-BEC crossover, as well as the nature of Cooper pairs, in superconducting and Fermi superfluid media is studied from the exact ground state wave function of the reduced BCS Hamiltonian. As the strength of the interaction increases, the ground state continuously evolves from a mixed system of quasifree fermions and pair resonances (BCS), to pair resonances and quasibound molecules (pseudogap), and finally to a system of quasibound molecules (BEC). A single unified scenario arises where the Cooper-pair wave function has a unique functional form. Several exact analytic expressions such as the binding energy and condensate fraction are derived. We compare our results with recent experiments in ultracold atomic Fermi gases.
We investigate strong-coupling effects on the superfluid phase transition in a gas of Fermi atoms with a Feshbach resonance. The Feshbach resonance describes a composite quasiboson that can give rise to an additional pairing interaction between the Fermi atoms. This attractive interaction becomes stronger as the threshold energy $2\ensuremath{\nu}$ of the Feshbach resonance two-particle bound state is lowered. In a recent paper, we showed that in the uniform Fermi gas, this tunable pairing interaction naturally leads to a crossover from a BCS state to a Bose-Einstein condensate (BEC) of the Nozi\`eres and Schmitt-Rink kind, in which the BCS-type superfluid phase transition continuously changes into the BEC type as the threshold energy is decreased. In this paper, we extend our previous work by including the effect of a harmonic trap potential, treated within the local-density approximation. We also give results for both weak and strong coupling to the Feshbach resonance. We show that the BCS-BEC crossover phenomenon strongly modifies the shape of the atomic density profile at the superfluid phase-transition temperature Tc, reflecting the change of the dominant particles going from Fermi atoms to composite bosons. In the BEC regime, these composite bosons are shown to first appear well above Tc. We also discuss the ``phase diagram'' above Tc as a function of the tunable threshold energy $2\ensuremath{\nu}.$ We introduce a characteristic temperature ${T}^{*}(2\ensuremath{\nu})$ describing the effective crossover in the normal phase from a Fermi gas of atoms to a gas of stable molecules.