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arXiv:0709.2621 · Physical Review Letters

BCS-BEC Crossover in a Random External Potential

Giuliano Orso

Published Dec 17, 2007• 38 citations
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Abstract

We investigate the ground state properties of a disordered superfluid Fermi gas across the BCS-BEC (Bose-Einstein condensate) crossover. We show that, for weak disorder, both the depletion of the condensate fraction of pairs and the normal fluid density exhibit a nonmonotonic behavior as a function of the interaction parameter 1/k{F}a, reaching their minimum value near unitarity. We find that, moving away from the weak-coupling BCS regime, Anderson's theorem ceases to apply and the superfluid order parameter is more and more affected by the random potential.

DOI
10.1103/physrevlett.99.250402

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We investigate the ground state properties of a disordered superfluid Fermi gas across the BCS-BEC (Bose-Einstein condensate) crossover. We show that, for weak disorder, both the depletion of the condensate fraction of pairs and the normal fluid density exhibit a nonmonotonic behavior as a function of the interaction parameter 1/k{F}a, reaching their minimum value near unitarity. We find that, moving away from the weak-coupling BCS regime, Anderson's theorem ceases to apply and the superfluid order parameter is more and more affected by the random potential.
In this paper, we have investigated the effect of weak random disorder in the Bardeen–Cooper–Schrieffer to Bose–Einstein condensation (BCS–BEC) crossover region. The disorder is included in the mean field formalism through Nozières–Smith-Rink (NSR) theory of superconducting fluctuations. A self-consistent numerical solution of the coupled equations involving the superfluid gap parameter and density as a function of the disorder strength, albeit unaffected in the BCS phase, yields a depleted order parameter in the BEC regime and an interesting nonmonotonic behaviour of the condensate fraction in the vicinity of the unitary region, and a gradual depletion thereafter, as the pairing interaction is continuously tuned across the BCS–BEC crossover. The unitary regime thus demonstrates a robust paradigm of superfluidity even when the disorder is introduced. To support the above feature and shed light on a lingering controversial issue, we have computed the behaviour of the sound mode across the crossover that distinctly reveals a suppression of the sound velocity. We also find the Landau critical velocity that shows similar nonmonotonicity as that of the condensate fraction data, thereby supporting a stable superfluid scenario in the unitary limit.
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{N}_{c}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{N}_{c}Nc​ is a good approximation even in the strong-coupling BEC regime. Our numerical results show how ${\ensuremath{\rho}}_{s}$ and Nc{N}_{c}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.
PACS. 03.75.Hh – Static properties of condensates; thermodynamical, statistical and structural properties. PACS. 03.75.Ss – Degenerate Fermi gases. PACS. 05.30.Fk – Fermion systems and electron gas. Abstract. – We present a theory for a superfluid Fermi gas near the BCS-BEC crossover, including pairing fluctuation contributions to the free energy similar to that considered by Nozières and Schmitt-Rink for the normal phase. In the strong coupling limit, our theory is able to recover the Bogoliubov theory of a weakly interacting Bose gas with a molecular scattering length very close to the known exact result. We compare our results with recent Quantum Monte Carlo simulations both for the ground state and at finite temperature. Excellent agreement is found for all interaction strengths where simulation results are available. The recent experimental realization of strongly interacting Fermi gases of 6 Li and 40 K atoms near a Feshbach resonance has opened up the exciting possibility of investigating the crossover from a Bardeen-Cooper-Schrieffer (BCS) superfluid to a Bose-Einstein condensate (BEC) [1–5]. In these systems, the inter-atomic interaction strength can be varied by tuning
We study the impact of weak quenched disorder on the BCS-BEC crossover in the Hubbard model within a functional-integral approach. By deriving the thermodynamic potential up to second order in both the disorder potential and pairing fluctuations, we obtain self-consistent expressions for the number equation, condensate fraction, superfluid fraction and sound speed at zero temperature. In the dilute BEC limit, our results analytically reproduce the known continuum limits of weakly-interacting bosons, where weak disorder depletes the superfluid more strongly than the condensate due to broken translational symmetry, and enhances the sound speed through the overcompensation of the static compressibility. These findings establish a unified and controlled approach for describing the BCS-BEC crossover in disordered lattice models, and they provide a foundation for future extensions to finite temperatures and multiband Hubbard models.
We investigate the Bose-Einstein condensation of fermionic pairs in a uniform two-component Fermi gas, obtaining an explicit formula for the condensate density as a function of the chemical potential and the energy gap. We analyze the condensate fraction in the crossover from the Bardeen-Cooper-Schrieffer (BCS) state of weakly interacting Cooper pairs to the Bose-Einstein condensate of molecular dimers. By using the local-density approximation we study confined Fermi vapors of alkali-metal atoms for which there is experimental evidence of condensation also on the BCS side of the Feshbach resonance. Our theoretical results are in agreement with these experimental data and give the behavior of the condensate on both sides of the Feshbach resonance at zero temperature.
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 TTT-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{T}_{\mathrm{c}}Tc​. We clarify how the pseudogap in the normal state evolves into the superfluid gap, as one passes through Tc{T}_{\mathrm{c}}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{T}_{\mathrm{c}}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.