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A review of the phenomenology and microscopy of cuprate superconductors is presented, with particular attention to universal conductance features, which reveal the existence of two electronic subsystems. The overall electronic system consists of $1+p$ charges, where $p$ is the doping. At low dopings exactly one hole is localized per planar copper-oxygen unit, while upon increasing doping and temperature, the hole is gradually delocalized and becomes itinerant. Remarkably, the itinerant holes exhibit identical Fermi-liquid character across the cuprate phase diagram. This universality enables a simple count of carrier density and yields comprehensive understanding of the key features in the normal and superconducting state. A possible superconducting mechanism is presented, compatible with the key experimental facts. The base of this mechanism is the interaction of fast Fermi-liquid carriers with localized holes. A change in the microscopic nature of chemical bonding in the copper-oxide planes, from ionic to covalent, is invoked to explain the phase diagram of these fascinating compounds., Comment: Dedicated to Prof. Karl Alex M\"uller on the occasion of his 95th birthday

An arbitrary derivative of a Vandermonde form in $N$ variables is given as $[n_1\cdots n_N]$, where the $i$-th variable is differentiated $N-n_i-1$ times, $1\le n_i\le N-1$. A simple decoding table is introduced to evaluate it by inspection. The special cases where $0\le n_{i+1} - n_i \le 1$ for $0, Comment: Author's final version accepted in J. Math. Phys., 15 pages, 1 figure

A simple description of Bose-Einstein condensation for non-interacting bosons is provided, based on an easily derived quantum correction to the canonical partition function. It is distinguished by the explicit appearance of the factor N !, which enhances the classical probability of finding bosons in the same state, and an elementary demonstration, that the transition should be sharper in temperature as the number of particles in the system increases. The reasoning is made explicit for the usual example of the ideal-gas dispersion.

We study the effect of antiferromagnetic (AF) correlations in the three-band Emery model, with respect to the experimental situation in weakly underdoped and optimally doped BSCCO. In the vicinity of the vH singularity of the conduction band there appears a central peak in the middle of a pseudogap, which is in an antiadiabatic regime, insensitive to the time scale of the mechanism responsible for the pseudogap. We find a quantum low-temperature regime corresponding to experiment, in which the pseudogap is created by zero-point motion of the magnons, as opposed to the usual semiclassical derivation, where it is due to a divergence of the magnon occupation number. Detailed analysis of the spectral functions along the (pi,0)-(pi,pi) line show significant agreement with experiment, both qualitative and, in the principal scales, quantitative. The observed slight approaching-then-receding of both the wide and narrow peaks with respect to the Fermi energy is also reproduced. We conclude that optimally doped BSCCO has a well-developed pseudogap of the order of 1000 K. This is only masked by the narrow antiadiabatic peak, which provides a small energy scale, unrelated to the AF scale, and primarily controlled by the position of the chemical potential., Final version as accepted in EPJ B, 13 pages, 8 figures

The parameter regime in which a Brinkmann-Rice (BR) transition appears near half-filling is investigated for a model of of one kind of electrons traversing a plane randomly tiled with CuO4 molecules, simulating the copper-oxide planes of high-T c superconductors. As the hole doping is increased, the BR transition evolves continuously into a state characterized by Kauzmann-like plateaus in the entropy vs. temperature curves. Despite clear analogies with the glass transition, these are equilibrium properties of the model. This is because the spin interactions, responsible for ordering in real space, are not included.

An effective one-particle spectrum has been derived for up-spins in the presence of static down-spins, with which they interact through a repulsive Hubbard term. This is done by finding the stable thermodynamic limit of the partition function, and expanding it around the overdoped limit. The effective model is purely fermionic, without slave fields. In the three-band case, an in-gap band appears for the up-spins, with the chemical potential moving into the charge-transfer gap already before half-filling. Due to the precise treatment of disorder, avoiding any averaging of the occupation number of the electrons at a given site, the entropy is a non-trivial function of doping, despite a k- space formulation and the in-gap fermions having a large Fermi surface.

Understanding the physical properties of unconventional superconductors as well as of other correlated materials presents a formidable challenge. Their unusual evolution with doping, frequency, and temperature has frequently led to non-Fermi-liquid (non-FL) interpretations. Optical conductivity is a major challenge in this context. Here, the optical spectra of two archetypal cuprates, underdoped HgBa$_2$CuO$_{4+\delta }$ and optimally-doped Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta }$, are interpreted based on the standard Fermi liquid (FL) paradigm. At both dopings, perfect frequency-temperature FL scaling is found to be modified by the presence of a second, gapped electronic subsystem. This non-FL component emerges as a well-defined mid-infrared spectral feature after the FL contribution -- determined independently by transport -- is subtracted. Temperature, frequency, and doping evolution of the MIR feature identify a gapped rather than dissipative response. In contrast, the dissipative response is found to be relevant for pnictides and ruthenates. Such an unbiased FL/non-FL separation is extended across the cuprate phase diagram, capturing all the key features of the normal state and providing a natural explanation why the superfluid density is attenuated on the overdoped side. Thus, we obtain a unified interpretation of optical responses and transport measurements in all analyzed physical regimes and all analyzed compounds., Comment: 19 pages, 17 figures