Interplay between 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 $T*gTg{T}_{c}$ and a true superconducting gap for $Tl{T}_{c}$ and (b) exhibits a considerable transfer of spectral weight between the incoherent background and the narrow coherent peak(s) signifying long-lived quasiparticle features. The Cooperon spectral function consists of a $\ensuremath{\delta}$-function peak, centered at the renormalized boson energy $\ensuremath{\omega}={\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{E}}_{\mathbf{q}}$ and a surrounding incoherent background which is spread over a wide energy range. When the temperature approaches ${T}_{c}$ from above, this peak for $\mathbf{q}=\mathbf{0}$ moves to $\ensuremath{\omega}=0$, 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 ${E}_{\mathbf{q}}\ensuremath{\propto}\ensuremath{\mid}\mathbf{q}\ensuremath{\mid}$ in the small-momentum region and simultaneously splits off from the incoherent background states which are expelled to the high-energy sector $\ensuremath{\mid}\ensuremath{\omega}\ensuremath{\mid}\ensuremath{\geqslant}2{\ensuremath{\Delta}}_{sc}(T)$. 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.