We investigate the superfluid phase transition in a gas of Fermi atoms loaded on a three-dimensional optical lattice. When the lattice potential is strong, this system can be well described by an attractive Hubbard model. In this model, we calculate the superfluid phase transition temperature Tc, including both superfluid and (spin and charge) density fluctuations within the self-consistent t-matrix theory and fluctuation exchange approximation, respectively. Since we treat these fluctuations in a consistent manner, our theory satisfies the required particle-hole symmetry over the entire BCS-BEC crossover region. We show that charge density fluctuations compete against superfluid fluctuations near the half-filling, leading to the suppression of Tc. As a result, the maximum Tc is obtained away the half-filling. Since the strong density fluctuations originate from the nesting property of the Fermi surface at the half filling (which is absent in a uniform gas with no lattice potential), our results would be useful in considering lattice effects on strong-coupling superfluidity.
We model fermions with an attractive interaction in an optical lattice with a single-band Hubbard model away from half-filling with on-site attraction
U and nearest neighbor hopping
t. Our goal is to understand the crossover from BCS superfluidity in the weak attraction limit to the BEC of molecules in the strong attraction limit, with particular emphasis on how this crossover in an optical lattice differs from the much better studied continuum problem. We use a large-
N theory with Sp(2N) symmetry to study the fluctuations beyond mean field theory. At T=0, we calculate across the crossover various observables, including chemical potential, gap, ground state energy, speed of sound and compressibility. The superfluid density
ns is found to have non-trivial
U/t dependence in this lattice system. We show that the transition temperature
Tc scales with the energy gap in the weak coupling limit but crosses over to a
t2/U scaling in the BEC limit, where phase fluctuations controlled by
ns determine
Tc. We also find, quite contrary to our expectations, that in the strong coupling limit, the large-
N theory gives qualitatively wrong trends for compressibility. A comparison with a simple Hartree shifted BCS theory, which takes into account both pairing and Hartree shifts, and correctly recovers the atomic limit and the right qualitative trend for compressibility, reveals that the large-
N theory on the lattice, although considers a larger number of diagrams, is in fact inferior to the simpler Hartree shifted BCS theory. The failure of the large-
N approach is explained by noting (i) the importance of Hartree shift in lattice problems, and (ii) inability of the large-
N approach to treat particle-particle and particle-hole channels at equal footing at the saddle point level.
The superfluid density is a fundamental quantity describing the response to a rotation as well as in two-fluid collisional hydrodynamics. We present extensive calculations of the superfluid density ${\ensuremath{\rho}}_{s}$ in the BCS-BEC crossover regime of a uniform superfluid Fermi gas at finite temperatures. We include strong-coupling or fluctuation effects on these quantities within a Gaussian approximation. We also incorporate the same fluctuation effects into the BCS single-particle excitations described by the superfluid order parameter $\ensuremath{\Delta}$ and Fermi chemical potential $\ensuremath{\mu}$, using the Nozi\`eres--Schmitt-Rink approximation. This treatment is shown to be necessary for consistent treatment of ${\ensuremath{\rho}}_{s}$ over the entire BCS-BEC crossover. We also calculate the condensate fraction
Nc as a function of the temperature, a quantity which is quite different from the superfluid density ${\ensuremath{\rho}}_{s}$. We show that the mean-field expression for the condensate fraction
Nc is a good approximation even in the strong-coupling BEC regime. Our numerical results show how ${\ensuremath{\rho}}_{s}$ and
Nc depend on temperature, from the weak-coupling BCS region to the BEC region of tightly bound Cooper pair molecules. In a companion paper [Phys. Rev. A 74, 063626 (2006)], we derive an equivalent expression for ${\ensuremath{\rho}}_{s}$ from the thermodynamic potential, which exhibits the role of the pairing fluctuations in a more explicit manner.
We investigate single-particle excitations and strong-coupling effects in the BCS-BEC crossover regime of a superfluid Fermi gas. Including phase and amplitude fluctuations of the superfluid order parameter within a
T-matrix theory, we calculate the superfluid density of states (DOS), as well as single-particle spectral weight, over the entire BCS-BEC crossover region below the superfluid transition temperature
Tc. We clarify how the pseudogap in the normal state evolves into the superfluid gap, as one passes through
Tc. While the pseudogap in DOS continuously evolves into the superfluid gap in the weak-coupling BCS regime, the superfluid gap in the crossover region is shown to appear in DOS after the pseudogap disappears below
Tc. In the phase diagram with respect to the temperature and interaction strength, we determine the region where strong pairing fluctuations dominate over single-particle properties of the system. Our results would be useful for the study of strong-coupling phenomena in the BCS-BEC crossover regime of a superfluid Fermi gas.
In this paper we address the behavior of the superfluid transition temperature
Tc in the attractive Hubbard model. The attractive Hubbard model can be regarded as the generalization of the BCS to Bose-Einstein condensation (BEC) crossover to a lattice and, as such, may have implications for future optical lattice studies. Nevertheless, the BEC limit of the Hubbard model is very different from that of the better-studied Fermi gases, owing to the strong intersite repulsion between pairs. Here we address systematically the effects of pairing fluctuations for all filling fractions over the entire range of attractive interaction strength. A central conclusion of our work is that in a lattice, around half filling, the smooth evolution from the BCS to the BEC limits is interrupted:
Tc vanishes when the system approaches the bosonic regime with increasing interaction strength. We suggest that this interruption of crossover may signal a quantum critical transition to another form of superfluid not continuously connected to a BCS-like phase. A simple variational ansatz for an alternate ground state in this more strongly coupled superfluid is presented. A generalization of the (
s-wave) Hubbard model to
d-wave pairing allows us to address issues of relevance to high-
Tc superconductivity. The phase diagram (representing the pairing or pseudogap onset temperature
T∗ and
Tc) shows that here too one observes a vanishing of
Tc when
T∗ becomes sufficiently large. Given this predicted breakdown of the crossover, and given the striking similarity to features of the cuprate phase diagram, we suggest that future experiments on ultracold fermions in optical lattices should not be exclusively limited to the repulsive Hubbard model, but should address the attractive model in order to elucidate features of high-temperature superconductivity.
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.