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, 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>Tc and a true superconducting gap for
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 $\ensuremath{\delta}$-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
Tc from above, this peak for
q=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
Tc 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
Tc from above and consider its feedback on the single-particle spectrum where a gradual formation of damped Bogoliubov modes (above
Tc) is observed.
Applying two-time Green-function techniques to the Friedberg-T.D. Lee phenomenological Hamiltonian of a many-fermion system, it is shown that positive-energy resonant bosonic pairs associated with four-fermion excitations above the Fermi sea are energetically lower in a ground-state that is a mixture of two coexisting and dynamically interacting many-particle subsystems: a) unpaired fermions and b) composite bosons. It is argued that an interaction between free fermions and bosons excited above the Fermi sea in the mixture, namely, the continuous processes of pair-formation from, and disintegration into, two unpaired electrons, results in a substantially lowering the total system energy. The positive-energy composite bosons begin to appear incoherently below a depairing temperature T* as their coupling- and temperature-dependent number density gradually increases from zero. This leads quite naturally to the pseudogap phenomenon observed in high-Tccuprates.
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>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.
The change-over from phase- to amplitude-fluctuation-driven superconductivity is examined for a composite system of free electrons (fermions with concentration
nF) and localized electron pairs (hard-core bosons with concentration
nB) as a function of doping---changing the total concentration of charge carriers
(ntot=nF+2nB). The coupling together of these two subsystems via a charge exchange term induces electron pairing and ultimately superconductivity in the fermionic subsystem. The difference in statistics of the two species of charge carriers has important consequences on the doping mechanism, showing an onset temperature
T∗ of incoherent electron pairing in the fermionic subsystem (manifest in form of a pseudogap), which steadily decreases with decreasing
ntot. Below
T∗ this electron pairing leads, in the normal phase, to electron-pair resonant states (Cooperons) with quasiparticle features which strongly depend on
ntot. For high concentrations, where ${n}_{B}\ensuremath{\simeq}0.5,$ correlation effects between the hard-core bosons lead to itinerant Cooperons having a heavy mass
mp, but are long lived. Upon reducing the concentration of charge carriers and consequently
nB, the mass as well as the lifetime of those Cooperons is considerably reduced. As a result, for high values of
nB, a superconducting state below
T∗ sets in at a
Tc, being controlled by the phase stiffness ${D}_{\ensuremath{\varphi}}={\ensuremath{\Elzxh}}^{2}{n}_{p}{/m}_{p}$ of those Cooperons, where
np denotes their density. Upon reducing
ntot, the phase stiffness steadily increases, and eventually exceeds the pairing energy
kBT∗. There, the Cooperons loose their well defined itinerant quasi-particle features and superconductivity gets controlled by amplitude fluctuations. The resulting phase diagram with doping is reminiscent of that of the phase fluctuation scenario for high-
Tc superconductivity, except that in our scenario the determinant factors are the mass and the lifetime of the Cooperons rather than their density.
The properties of a mixture of mutually interacting bound electron pairs and itinerant fermions (the boson-fermion model) on a lattice are further studied. We determine the superconducting critical temperature from a pseudogap phase by applying a self-consistent
T-matrix approach, which includes the pairing fluctuations and the boson self-energy effects. The analysis is made for a three dimensional cubic lattice with tight-binding dispersion for electrons and for both standard bosons and the case of hard-core bosons. The results describe the BCS-Bose-Einstein condensation crossover with varying position of the bosonic (local pair) level and give a further insight into the nature of resonance superfluidity in the boson-fermion model.
We review briefly the properties of a mixture of mutually interacting bosons (bound electron pairs) and itinerant fermions on a lattice (the boson–fermion model). The calculations of the superconducting phase transition temperature () and the phase diagram are the main concern. The self-consistent -matrix method is applied to determine the superconducting critical temperature from a pseudogap phase. The method takes into account the pairing fluctuations effects. The -matrix results for are given for a three-dimensional (3-D) cubic lattice with tight-binding dispersion of electrons and standard bosons, and they are also compared with those of the BCS- mean-field approximation. Our results describe the BCS-Bose-Einstein condensation crossover in the boson–fermion mixture with resonant interaction. The energy scales involved in the pseudogap formation are also analysed.