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arXiv:cond-mat/0406022 · Physical Review B

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

T. Domański, J. Ranninger

Published Nov 9, 2004• 21 citations
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Abstract

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 $T<{T}_{c}$ 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.

DOI
10.1103/physrevb.70.184503

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We study the interplay between the single-particle and fermion-pair features in the boson fermion model, both above and below the transition temperature Tc{T}_{c}Tc​, 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>TcT*>T>{T}_{c}T∗>T>Tc​ and a true superconducting gap for T<TcT<{T}_{c}T<Tc​ 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 \ensuremathδ\ensuremath{\delta}\ensuremathδ-function peak, centered at the renormalized boson energy \ensuremathω=E\ifmmode~\else˜\fiq\ensuremath{\omega}={\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{E}}_{\mathbf{q}}\ensuremathω=E\ifmmode~\else˜\fi​q​ and a surrounding incoherent background which is spread over a wide energy range. When the temperature approaches Tc{T}_{c}Tc​ from above, this peak for q=0\mathbf{q}=\mathbf{0}q=0 moves to \ensuremathω=0\ensuremath{\omega}=0\ensuremathω=0, so that the static pair susceptibility diverges (Thouless criterion for the broken symmetry phase transition). Upon decreasing the temperature below Tc{T}_{c}Tc​ the Cooperon peak becomes the collective (Goldstone) mode Eq\ensuremath∝\ensuremath∣q\ensuremath∣{E}_{\mathbf{q}}\ensuremath{\propto}\ensuremath{\mid}\mathbf{q}\ensuremath{\mid}Eq​\ensuremath∝\ensuremath∣q\ensuremath∣ in the small-momentum region and simultaneously splits off from the incoherent background states which are expelled to the high-energy sector \ensuremath∣\ensuremathω\ensuremath∣\ensuremath⩾2\ensuremathΔsc(T)\ensuremath{\mid}\ensuremath{\omega}\ensuremath{\mid}\ensuremath{\geqslant}2{\ensuremath{\Delta}}_{sc}(T)\ensuremath∣\ensuremathω\ensuremath∣\ensuremath⩾2\ensuremathΔsc​(T). We discuss the smooth evolution of these features upon approaching Tc{T}_{c}Tc​ from above and consider its feedback on the single-particle spectrum where a gradual formation of damped Bogoliubov modes (above Tc{T}_{c}Tc​) is observed.
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We study a pseudogap region of the mixed boson fermion system using a recent formulation of the renormalization group technique through the set of infinitesimal unitary transformations. Renormalization of fermion energies gives rise to a depletion of the low energy states (pseudogap) for temperatures T∗>T>TcT^{*}>T>T_{c}T∗>T>Tc​ which foreshadows appearance of the pairwise correlations with their long range phase coherence being missed. With a help of the flow equations for boson and fermion operators we analyze spectral weights and finite life times of these quasiparticles caused by interactions.
We have developed a semi-analytical framework formulated in the canonical fermion representation to investigate strongly correlated electron systems. We consider the U=∞\infty∞ Hubbard model and used the equation of motion method to calculate the fermion self-energy which has two parts: single and two-boson exchange processes. The emergent bosons here are self-generated local charge and spin-density fluctuations which become strongly time-dependent due to extreme correlations. The computed boson spectral density is a diffusive damped mode with a long tail. The electron self-energy at d=∞d=\inftyd=∞ is computed self-consistently. The corresponding fermionic spectral density displays a pronounced coherence peak at ω=0ω=0ω=0, while its frequency derivative develops a two-peak structure at finite ωωω. The resistivity shows a linear temperature dependence over a broad range, crossing over to coherent Fermi-liquid behavior at extremely low temperatures.
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We study the normal state spectral properties of the fermionic excitations in the Boson-Fermion model. The fermionic single particle excitations show a flattening of the dispersion as the Fermi vector kF is approached from below, forshadowing a Bogoliubov spectrum of a superconducting ground state. The width of the quasiparticle excitations near k increases monotonically as the F temperature is lowered. In the fermionic distribution function this temperature dependence is manifest in a strong modification of n(k) in a small region below k, but a nearly T independant n(k).