Abstract:
We compute the single-particle inverse lifetime, along with the conductivity-derived scattering rate, for a metallic system in an s-wave superconducting state. When both electron-phonon and electron-impurity scattering are included, we find that while these scattering rates are in qualitative agreement, in general quantitative agreement is lacking. We also derive results for the quasiparticle lifetime within the BCS framework with impurity scattering, which makes it clear that impurity scattering is suppressed for electrons near the Fermi surface in the superconducting state.

Abstract:
In a d-wave superconductor with elastic impurity scattering, not all the available optical spectral weight goes into the condensate at zero temperature, and this leads to residual absorption. We find that for a range of impurity parameters in the intermediate coupling regime between Born (weak) and unitary (strong) limit, significant oscillator strength remains which exhibits a cusp like behavior of the real part of the optical conductivity with upward curvature as a function of frequency, as well as a quasilinear temperature dependence of the superfluid density. The calculations offer an explanation of recent data on ortho-II YBa$_2$Cu$_3$O$_{6.5}$ which has been considered anomalous.

Abstract:
Using a maximum entropy technique within a finite band Eliashberg formalism we extract from recent high accuracy nodal direction angular resolved photo-emission spectroscopy (ARPES) data in optimally doped Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ (Bi2212) a quasiparticle electron-boson spectral density. Both normal and superconducting state with d-wave gap symmetry are treated. Finite and infinite band results are considered and contrasted. We compare with results obtained for the related transport spectral density which follows from a similar inversion of optical data. We discuss the implication of our results for quasiparticle renormalizations in the antinodal direction.

Abstract:
We compute the single particle inverse lifetime, evaluated in the superconducting state. Within the BCS framework, the calculation can be done non-perturbatively, i.e. poles can be found well away from the real axis. We find that perturbative calculations are in good agreement with these results, even for non-zero impurity scattering. With electron-phonon scattering added to the problem, we use the Eliashberg equations, with impurity scattering, to calculate the quasiparticle inverse lifetime perturbatively. In all cases we find that the inverse lifetime is significantly suppressed in the superconducting state, particularly in the presence of impurity scattering. We also compute the far-infrared and microwave conductivity, and describe procedures for extracting scattering rates from conductivity measurements. While these procedures lead to scattering rates in qualitative agreement with the inverse lifetime, we find that quantitative agreement is lacking, in general.

Abstract:
Information on the nature of the dominant inelastic processes operative in correlated metallic systems can be obtained from an analysis of their AC optical response. An electron-boson spectral density can usefully be extracted. This density is closely related to the optical scattering rate. However, in the underdoped region of the high Tc cuprate phase diagram a new energy scale (the pseudogap) emerges, which alters the optical scattering and needs to be taken into account in any fit to data. This can influence the shape and strength of the recovered boson spectral function. Including a pseudogap in an extended maximum entropy inversion for optimally doped Bi-2212 is more consistent with existing data than when it is left out as done previously.

Abstract:
We use a maximum entropy technique to obtain the electron-boson spectral density from optical scattering rate data across the underdoped region of the Bi_2Sr_{2-x}La_xCuO_6 (Bi-2201) phase diagram. Our method involves a generalization of previous work which explicitly include finite temperature and the opening of a pseudogap which modifies the electronic structure. We find that the mass enhancement factor \lambda associated with the electron-boson spectral density increases monotonically with reduced doping and closer proximity to the Mott antiferromagnetic insulating state. This observation is consistent with increased coupling to the spin fluctuations. At the same time the system has reduced metallicity because of increased pseudogap effects which we model with a reduced effective density of states around the Fermi energy with the range of the modifications in energy set by the pseudogap scale.

Abstract:
We calculate the $c$-axis kinetic energy difference between normal and superconducting state for coherent and for incoherent interlayer coupling between CuO$_{2}$ planes. For coherent coupling the ratio of the missing conductivity spectral weight to the superfluid density is equal to one and there is no violation of the conventional sum rule, but for the incoherent case we find it is always greater than one whatever the nature of the impurity potential may be. To model more explicitly YBa$_{2}$Cu$_{3}$O$_{7-x}$ around optimum doping, which is found to obey the sum rule, we consider a plane-chain model and show that the sum rule still applies. A violation of the sum rule of either sign is found even for coherent coupling when the in-plane density of electronic states depends on energy on a scale of the order of the gap.

Abstract:
In recent years a unified phenomenological picture for the hole doped high-T_c cuprates has emerged for a spin and charge spectroscopy. Spectral anomalies have been interpreted as evidence of charge carrier coupling to a collective spin excitation present in the optical conductivity, in ARPES (angular resolved photoemission), and in tunneling data. These anomalies can be used to derive an approximate picture of a charge carrier-exchange boson interaction spectral density I^2 chi(omega) which is then be used within an extended Eliashberg formalism to analyze normal and superconducting properties of optimally doped and overdoped cuprates. This paper reviews recent developments and demonstrates the sometimes astonishing agreement between experiment and theoretical prediction.

Abstract:
Based on an attractive $U$ Hubbard model on a lattice with up to second neighbor hopping we derive an effective Hamiltonian for phase fluctuations. The superconducting gap is assumed to have s-wave symmetry. The effective Hamiltonian we finally arrive at is of the extended XY type. While it correctly reduces to a simple XY in the continuum limit, in the general case, it contains higher neighbor interaction in spin space. An important feature of our Hamiltonian is that it gives a much larger fluctuation region between the Berezinskii-Kosterlitz-Thouless transition temperature identified with $T_{c}$ for superconducting and the mean field transition temperature identified with the pseudogap temperature.

Abstract:
The concepts, which have traditionally been useful in understanding the effects of the electron--phonon interaction in optical spectroscopy, are based on insights obtained within the infinite electronic band approximation and no longer apply in finite band metals. Impurity and phonon contributions to electron scattering are not additive and the apparent strength of the coupling to the phonon degrees of freedom is substantially reduced with increased elastic scattering. The optical mass renormalization changes sign with increasing frequency and the optical scattering rate never reaches its high frequency quasiparticle value which itself is also reduced below its infinite band value.