Abstract:
The relaxation function theory for a doped two-dimensional Heisenberg antiferromagnetic system in the paramagnetic state for all wave vectors through the Brillouin zone is presented in view of low frequency response of high-$T_c$ copper oxide superconductors. We deduced the regions of long lifetime [$T \lesssim 400(1-4x)$ K] and "overdamped" [$T \gtrsim 700(1-4x)$ K] paramagnonlike excitations in the temperature ($T$)-doping index ($x$) phase diagram from plane oxygen nuclear spin-lattice relaxation rate $^{17}(1/T_1)$ data in up to optimally doped La$_{2-x}$Sr$_{x}$CuO$_{4}$ thus providing the regimes for the spin wave concept and the ''overdamped'' mode.

Abstract:
First principle band calculations based on local versions of density functional theory (DFT), together with results from nearly free-electron models, can describe many typical but unusual properties of the high-$T_C$ copper oxides. The methods and a few of the most important results are reviewed. Some additional calculations are presented and the problems with the commonly used approximate versions of DFT for oxides are discussed with a few ideas for corrections. It is concluded that rather modest corrections to the approximate DFT, without particular assumptions about strong correlation, can push the ground state towards anti-ferro magnetic (AFM) order. Spin fluctuations interacting with phonons are crucial for the mechanism of superconductivity in this scenario.

Abstract:
Band calculations for the hole doped La$_2$CuO$_4$ system show that artificial periodicities of Ba dopants can give the material different properties than from a uniform distribution of dopants. A periodicity within the planes make static pseudogaps which could be tuned to raise the density-of-states (DOS) at $E_F$ and the superconducting $T_C$. A periodic doping dependence perpendicular to the CuO planes can increase the matrix element for spin fluctuations.

Abstract:
We used high-resolution scanning tunneling spectroscopy to study the hole-doped iron pnictide superconductor Ba$_{0.6}$K$_{0.4}$Fe$_{2}$As$_{2}$ ($T_c=38$ K). Features of a bosonic excitation (mode) are observed in the measured quasiparticle density of states. The bosonic features are intimately associated with the superconducting order parameter and have a mode energy of $\sim$14 meV, similar to the spin resonance measured by inelastic neutron scattering. These results indicate a strong electron-spin excitation coupling in iron pictnide superconductors, similar to that in high-$T_c$ copper oxide superconductors.

Abstract:
Electronic Raman scattering measurements have been performed on hole doped copper oxide superconductors as a function of temperature and doping level. In the superconducting state coherent Bogoliubov quasiparticles develop preferentially over the nodal region in the underdoped regime. We can then define the fraction of coherent Fermi surface, $f_c$ around the nodes for which quasiparticles are well defined and superconductivity sets in. We find that $f_c$ is doping dependent and leads to the emergence of two energy scales. We then establish in a one single gap shem, that the critical temperature $T_{c} \propto f_{c}\Delta_{max}$ where $\Delta_{max}$ is the maximum amplitude of the d-wave superconducting gap. In the normal state, the loss of antinodal quasiparticles spectral weight detected in the superconducting state persists and the spectral weight is only restored above the pseudogap temperature $T*$. Such a dichotomy in the quasiparticles dynamics is then responsible for the emergence of the two energy scales in the superconducting state and the appearance of the pseudogap in the normal state. We propose a 3D phase diagram where both the temperature and the energy phase diagrams have been plotted together. We anticipate that the development of coherent excitations on a restricted part of the Fermi surface only is a general feature in high $T_c$ cuprate superconductors as the Mott insulating is approaching.

Abstract:
We present a unified description of the resonance peak and low-energy incommensurate response observed in high-$T_c$ cuprate superconductors. We argue that both features have a purely magnetic origin and they represent universal features of an incommensurate spin state both below and above the superconducting transition temperature. In this description the resonance peak is the reflection of commensurate antiferromagnetism. Our theoretical scenario gives an account of the main features observed in various families of superconductors and predicts those not yet observed, like a resonance peak in La$_2$NiO$_{4+x}$.

Abstract:
Self-assembling organic polymers and copper-oxide compounds are two classes of "strange" superconductors, whose challenging behavior does not comply with the traditional picture of Bardeen, Cooper, and Schrieffer (BCS) superconductivity in regular crystals. In this paper, we propose a theoretical model that accounts for the strange superconducting properties of either class of the materials. These properties are considered as interconnected manifestations of the same phenomenon: We argue that superconductivity occurs in the both cases because the charge carriers (i.e., electrons or holes) exchange {\it fracton excitations}, quantum oscillations of fractal lattices that mimic the complex microscopic organization of the strange superconductors. For the copper oxides, the superconducting transition temperature $T_c$ as predicted by the fracton mechanism is of the order of $\sim 150$ K. We suggest that the marginal ingredient of the high-temperature superconducting phase is provided by fracton coupled holes that condensate in the conducting copper-oxygen planes owing to the intrinsic field-effect-transistor configuration of the cuprate compounds. For the gate-induced superconducting phase in the electron-doped polymers, we simultaneously find a rather modest transition temperature of $\sim (2-3)$ K owing to the limitations imposed by the electron tunneling processes on a fractal geometry. We speculate that hole-type superconductivity observes larger onset temperatures when compared to its electron-type counterpart. This promises an intriguing possibility of the high-temperature superconducting states in hole-doped complex materials. A specific prediction of the present study is universality of ac conduction for $T\gtrsim T_c$.

Abstract:
An inelastic neutron scattering experiment has been performed in the high-temperature superconductor $\rm YBa_2Cu_3O_{6.89}$ to search for an oxygen-isotope shift of the well-known magnetic resonance mode at 41 meV. Contrary to a recent prediction (I. Eremin, {\it et al.}, Phys. Rev. B {\bf 69}, 094517 (2004)), a negligible shift (at best $\leq$ +0.2 meV) of the resonance energy is observed upon oxygen isotope substitution ($^{16}$O$\to^{18}$O). This suggests a negligible spin-phonon interaction in the high-$T_c$ cuprates at optimal doping.

Abstract:
A two-band Hubbard model is used to describe the band structure and phase separation (PS) in multiband superconductors, especially in cuprates. We predict a large peak in the density of states at the Fermi level in the case of optimum doping, corresponding to the minimum energy difference between the centers of two hole bands. For strong interband hybridization, a metal-insulator transition occurs near this optimum doping level. We suggest a mechanism of PS related to the redistribution of holes between two Hubbard bands rather than to the usual antiferromagnetic correlations. We show that the critical superconducting temperature $T_c$ can be about its maximum value within a wide range of doping levels due to PS.

Abstract:
We study by means of renormalization group techniques the effect that on the two-dimensional electron liquid may have the van Hove singularities observed experimentally in the copper-oxide superconductors. We find significant deviations from Fermi liquid behavior, that lead to the appearance of an unstable fixed point in the renormalization group flow of the effective coupling constant. Besides the attenuation of electron quasiparticles already known on phenomenological grounds, our approach is able to explain the reduction in the dispersion of the band as well as the pinning of the Fermi level near the singularity, as observed in the photoemission experiments.