We study pseudogap behaviors of ultracold Fermi gases in the BCS-BEC crossover region. We calculate the density of states (DOS), as well as the single-particle spectral weight, above the superfluid transition temperature $T_{\rm c}$ including pairing fluctuations within a $T$-matrix approximation. We find that DOS exhibits a pseudogap structure in the BCS-BEC crossover region, which is most remarkable near the unitarity limit. We determine the pseudogap temperature $T^*$ at which the pseudogap structure in DOS disappears. We also introduce another temperature $T^{**}$ at which the BCS-like double-peak structure disappears in the spectral weight. While one finds $T^*>T^{**}$ in the BCS regime, $T^{**}$ becomes higher than $T^*$ in the crossover and BEC regime. We also determine the pseudogap region in the phase diagram in terms of temperature and pairing interaction.
Recently, the BCS-BEC crossover has been realized in ultracold Fermi gases [1]. In this phenomenon, using a tunable pairing interaction associated with a Feshbach resonance, one can study Fermi superfluids from the weak-coupling BCS regime to the strong coupling BEC regime in a unified manner. Because of this advantage, superfluid Fermi gases would be also useful for the study of high-Tc cuprates with a strong pairing interaction.
In the under-doped regime of high-Tc cuprates, the so-called pseudogap structure has been observed in the density of states (DOS) [2]. As the origin of the pseudogap, strong pairing fluctuations has been proposed [3,4,5,6]. However, because of the complexity of this system due to strongly correlated electrons, other possibilities, such as antiferromagnetic spin fluctuations and a hidden ordered state, have been also discussed. Thus, to confirm the pairing fluctuation scenario, another simple system only having superfluid fluctuations would be useful.
The cold Fermi gas system meets this demand. It is much simpler than high-Tc cuprates, and pairing fluctuations dominate over the BCS-BEC crossover physics. Indeed, the pseudogap phenomenon in this system has been recently predicted [3,6,7]. Although the s-wave pairing symmetry of superfluid Fermi gas is different from the d-wave one in high-Tc cuprates, we can still expect that the study of pseudogap phenomenon in cold Fermi gases would be helpful in understanding the under-doped regime of high-Tc cuprates. Since a photoemission-type experiment has recently become possible in cold Fermi gases [8], observation of strong-coupling effects on single-particle excitations is now possible within the current technology.
In this paper, we investigate pseudogap behaviors of atomic Fermi gases above the superfluid transition temperature Tc. Including pairing fluctuations within a T -matrix approximation, we calculate DOS and single-particle spectral weight. we examine how pairing fluctuations affect them over the entire BCS-BEC crossover region. We also determine the pseudogap regime in the phase diagram in terms of temperature and the strength of pairing interaction.
We consider a uniform two-component Fermi gas described by pseudospin σ =↑, ↓. For a broad Feshbach resonance (which all the current experiments are using), it is known that one can safely study the interesting BCS-BEC crossover physics by using the ordinary BCS model [1], given by
Here, cpσ is an annihilation operator of a Fermi atom with pseudospin σ =↑, ↓. ξp ≡ εp -µ = p 2 /2m -µ is the kinetic energy, measured from the chemical potential µ, where m is an atomic mass. The pairing interaction (U > 0) is assumed to be tunable by a Feshbach resonance. In cold atom physics, the strength of pairing interaction is conveniently described in terms of the parameter (k F as) -1 , where as is the s-wave scattering length and k F the Fermi momentum. In this scale, the BCS limit and BEC limit are, respectively, given by (k F as)
∼ + 1 is referred to as the crossover region. The relation between U and as is given by [9]
]. The single-particle thermal Green’s function is given by Gp
Here, ωn is the fermion Matsubara frequency, and G 0 p (iωn) = 1/(iωn -ξp) is the non-interacting Fermi Green’s function. The self-energy Σ(p, iωn) involves effects of pairing fluctuations. In this paper, we include strong-coupling corrections within the T -matrix approximation [3,4,6]. The resulting self-energy has the form
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where νn is the boson Matsubara frequency. The particle-particle scattering matrix Γ (describing pairing fluctuations) is given by Γ (q, iνn) = -U/[1 -U Π(q, iνn)], where Π(q, iνn) = T p,ωn G 0 p+q/2 (iνn + iωn)G 0 -p+q/2 (-iωn) is a pair-propagator. To discuss the pseudogap phenomenon above Tc, we need to determine Tc. Following Ref. [10], we employ the Thouless criterion [11], Γ (q = 0, iνn = 0, T = Tc) -1 = 0, and solve this equation, together with the equation for the number of fermions, N = 2T p,ωn e iωnδ Gp(iωn).
(
The above treatment can describe the smooth crossover behavior of Tc and µ in the BCS-BEC crossover [6,10]. Namely, starting from the weak-coupling BCS regime, Tc gradually deviates from the mean-field result to approach Tc = 0.218ε F of an N/2 ideal molecular Bose gas (where ε F is the Fermi energy). The chemical potential monotonically decreases from ε F in the crossover regime to be negative in the BEC regime ((k F as) -1 > 0.35). The negative µ indicates the formation of two-body bound states, so that the BEC regime is well described by a molecular Bose gas, as expected.
Above Tc, we solve Eq. ( 3) to determine µ. DOS ρ(ω) and the spectral weight A(p, ω) are, respectively, evaluated from the analytic continued Green’s function, as
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Fig. 2 Temperature dependence of the density of states ρ(ω). To clearly show how the pseudogap disappears, we have offset results for T > Tc. The horizontal line in the left end of each curve indicates z
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