By means of the continuous unitary transformation similar to a general scheme of the Renormalization Group (RG) procedure we study the issue of symmetry breaking and pairing instability in the system of interacting fermions. Constructing a generalized version of the Bogoliubov transformation we show that formation of the fermion pairs and their superconductivity/superfluidity can appear at different temperatures. It is shown that strong quantum fluctuations can destroy the long-range order without breaking the fermion pairs which may still exist as incoherent and/or damped entities. Such unusual phase is characterized by a partial suppression of the density of states near the Fermi energy and by residual collective features like the sound-wave mode in the fermion pair spectrum.
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.
We apply the flow equation method to the study of the fermion systems with pairing interactions which lead to the BCS instability signalled by the appearance of the off-diagonal order parameter.For this purpose we rederive the continuous Bogoliubov transformation in a fashion of renormalization group procedure where the low and high energy sectors are treated subsequently.We further generalize this procedure to the case of fermions interacting with the discrete boson mode.Andreev-type interactions are responsible for developing a gap in the excitation spectrum.However, the long-range coherence is destroyed due to strong quantum fluctuations.
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).
Single-particle spectra are calculated in the superconducting state for a fermionic system with an attractive interaction, as functions of temperature and coupling strength from weak to strong. The fermionic system is described by a single-particle self-energy that includes pairing-fluctuation effects in the superconducting state. The theory reduces to the ordinary BCS approximation in weak coupling and to the Bogoliubov approximation for the composite bosons in strong coupling. Several features of the single-particle spectral function are shown to compare favorably with experimental data for cuprate superconductors.
We propose a scenario for superconductivity at strong electron-electron attractive interaction, in the case when the increase of the interaction strength promotes the nucleation of the local Cooper pairs and forms a state with a spatially phase incoherent Cooper pair order parameter. We show that this state can be characterized by a pseudogap which is determined by the electron scattering by phase fluctuations. At low temperatures, however, long-range correlations between the regions with different phases become important and establish global phase coherence and hence superconductivity in the system. We develop a mean-field theory to describe a phase transition between the preformed Cooper pair and superconducting states. This scenario of superconductivity applies not only to conductors with parabolic bands but also to the flat-band systems in which flat and dispersive bands coexist and are responsible for the formation of Cooper pairs as well as their phase synchronization.