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

BCS--Bose-Einstein condensation (BEC) crossover is effected by increasing pairing strength between fermions from weak to strong in the particle-particle channel. Here we study the effect of the particle-hole channel on the zero $T$ gap $Δ(0)$, superfluid transition temperature $T_{\text{c}}$ and the pseudogap at $T_{\text{c}}$, as well as the mean-field ratio $2Δ(0)/T_{\text{c}}^{\text{MF}}$, from BCS through BEC regimes, in the framework of a pairing fluctuation theory which includes self-consistently the contributions of finite-momentum pairs. These pairs necessarily lead to a pseudogap in single particle excitation spectrum above and below $T_{\text{c}}$. We sum over the infinite particle-hole ladder diagrams so that the particle-particle and particle-hole $T$-matrices are entangled with each other. We find that the particle-hole susceptibility has a complex dynamical structure, with strong momentum and frequency dependencies, and is sensitive to temperature, gap size and interaction strength. We conclude that neglecting the self-energy feedback causes a serious over-estimate of the particle-hole susceptibility. In the BCS limit, the particle-hole channel effect may be approximated by the same reduction in the overall pairing strength so that the ratio $2Δ(0)/T_{\text{c}}$ is unaffected, in agreement with Gor'kov \textit{et al.} to the leading order. However, the effect becomes more complex and pronounced in the crossover regime, where the particle-hole susceptibility 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 propose that precision measurements of the magnetic field for Feshbach resonance at low temperatures as a function of density can be used to quantify the particle-hole susceptibility and test different theories.

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