Effect of the particle-hole channel on BCS-Bose-Einstein condensation crossover in atomic Fermi gases

Abstract

BCS-Bose-Einstein condensation (BEC) crossover is effected by increasing pairing strength between fermions from weak to strong in the particle-particle channel, and has attracted a lot of attention since the experimental realization of quantum degenerate atomic Fermi gases. Here we study the effect of the (often dropped) particle-hole channel on the zero T gap Δ(0), superfluid transition temperature Tc, the pseudogap at Tc, and the mean-field ratio 2Δ(0)/, from BCS through BEC regimes, using a pairing fluctuation theory which includes self-consistently the contributions of finite-momentum pairs and features a pseudogap in single particle excitation spectrum. Summing over the infinite particle-hole ladder diagrams, we find a complex dynamical structure for the particle-hole susceptibility χph, and conclude that neglecting the self-energy feedback causes a serious over-estimate of χph. While our result in the BCS limit agrees with Gor'kov et al., the particle-hole channel effect becomes more complex and pronounced in the crossover regime, where χph is reduced by both a smaller Fermi surface and a big (pseudo)gap. Deep in the BEC regime, the particle-hole channel contributions drop to zero. We predict a density dependence of the magnetic field at the Feshbach resonance, which can be used to quantify χph and test different theories.

Figure 1. Feynman diagrams for the particle-particle channel T-matrix t_pg and the self energy Σ(K).

The dotted lines represent the bare pairing interaction U. The dashed line, t_sc, represents the superfluid condensate.

Figure 2. Feynman diagrams for the particle-hole susceptibility χ_ph in the presence of self-energy feedback effect.

Panel (b) is identical to panel (a), with twisted external legs. The total particle-hole momentum P in (a) is equal to K + K′ − Q in (b), with Q being the particle-particle pair momentum.

Figure 3. Strong momentum dependence of the real part of the particle-hole susceptibility at zero frequency ν = 0 in the unitarity limit, with (black curve) and without (blue dashed curve) self-energy feedback, calculated at T = 0.3T_c, where T_c = 0.256E_F.

While the undressed shows a simple monotonic behavior, the dressed susceptibility has a nonmonotonic p dependence, and a substantially reduced value at p = 0. This reduction derives from the gap effect in the Green’s function G(K). Namely, seriously over-estimated particle-hole fluctuations.

Figure 4. Feynman diagrams showing the particle-hole channel effect on fermion pairing, in the presence of self-energy feedback.

(a) Particle-particle T matrix t₁(Q), with external four momenta labeled. (b) Particle-hole T matrix t_ph(P), with P = K + K′ − Q being the total particle-hole 4-momentum. (c) An effective, composite particle-particle T-matrix, t₂(Q), with the contribution from the particle-hole channel included. Here different shadings represent different T matrices.

Figure 5. Angular average of the on-shell particle-hole susceptibility, 〈χ_ph(0, p = |k + k′|)〉 at ν = 0 as a function of momentum k/k_F, under the condition k = k′, calculated at unitarity for different temperatures T = 0.1T_c (black solid curve) and T = T_c (green dot-dashed curve), in units of .

Also plotted is its undressed counterpart, , which shows a serious over-estimate due to the neglect of the self-energy feedback. Here T_c = 0.256E_F and associated gap and μ values are calculated without the particle-hole channel effect. The open circles on each curve denote level 1 average, i.e., k = k_μ. The vertical axis readings of the horizontal short bars indicate the corresponding values of level 2 average. The thick section of each curve indicates the range of k used for level 2 averaging. Clearly, there are strong temperature and k dependencies in both 〈χ_ph(0, p)〉 and . The (absolute) values of Level 2 average are substantially smaller than their level 1 counterpart.

Figure 6. Effect of the particle-hole channel contributions on the zero temperature gap in BCS-BEC crossover.

In (a), the black solid curve is the gap without the particle-hole effect. The rest curves are calculated with the particle-hole channel effect but at different levels, i.e., using undressed particle-hole susceptibility with level 1 averaging (red dotted line), dressed particle-hole susceptibility 〈χ_ph〉 with level 1 (green dot-dashed curve) and level 2 (blue dashed line) averaging, respectively. The corresponding values of the average particle-hole susceptibility with a minus sign are plotted in (b), in units of . The particle-hole channel effect can be essentially neglected beyond 1/k_Fa > 1.5.

Figure 7. Effect of the particle-hole channel contributions on T_c and the pseudogap Δ at T_c in BCS-BEC crossover.

In (a,b), the black solid curves are calculated without the particle-hole effect. The rest curves are calculated with the particle-hole channel effect but at different levels, using undressed particle-hole susceptibility with level 1 averaging (red dotted line), dressed particle-hole susceptibility 〈χ_ph〉 with level 1 (green dot-dashed curve) and level 2 (blue dashed line) averaging, respectively. The corresponding values of the average particle-hole susceptibility with a minus sign are plotted in (c), in units of . The particle-hole channel effect can be essentially neglected beyond 1/k_Fa > 1.5.

Figure 8. Effect of the particle-hole channel contributions on the ratio in BCS-BEC crossover.

Shown is the mean-field ratio calculated with (black solid curve) and without (blue dashed curve) the particle-hole channel contributions. Here the particle-hole susceptibility 〈χ_ph〉 is calculated with level 2 averaging.

Figure 9. Higher order T-matrix, t₃, obtained by repeating the T-matrix t₂.