Interplay between the single particle and collective features in the boson fermion model

We study the interplay between the single particle and fermion pair features in the boson fermion model, both above and below the transition temperature $T_{c}$, using the flow equation method. Upon lowering the temperature the single particle fermionic spectral function: (a) gradually develops a depletion of the low energy states (pseudogap) for $T^{*}>T>T_{c}$ and a true superconducting gap for $T<T_{c}$, (b) exhibits a considerable transfer of spectral weight between the incoherent background and the narrow coherent peak(s) signifying long-lived quasi-particle features. The cooperon spectral function consists of a delta function peak, centered at the renormalized boson energy $\omega=\tilde{E}_{\bf q}$, and a surrounding incoherent background which is spread over a wide energy range. When the temperature approaches $T_{c}$ from above this peak for ${\bf q}={\bf 0}$ moves to $\omega=0$, so that the static pair susceptibility diverges (Thouless criterion for the broken symmetry phase transition). Upon decreasing the temperature below $T_{c}$ the cooperon peak becomes the collective (Goldstone) mode $E_{\bf q} \propto |{\bf q}|$ in the small momentum region and simultaneously splits off from the incoherent background states which are expelled to the high energy sector $|\omega| \geq 2 \Delta_{sc}(T)$. We discuss the smooth evolution of these features upon approaching $T_{c}$ from above and consider its feedback on the single particle spectrum where a gradual formation of damped Bogoliubov modes (above $T_{c}$) is observed.