Abstract:
A model of hybridized bosons and fermions is studied beyond the mean field approximation. The divergent boson self-energy at zero temperature makes the Cooper pairing of fermions impossible.The frequency and momentum dependence of the self- energy and the condensation temperature $T_{c}$ of initially localized bosons are calculated analytically. The value of the boson condensation temperature $T_{c}$ is below $1K$ which rules out the boson-fermion model with the initially localized bosons as a phenomenological explanation of high-temperature superconductivity. The intra-cell density-density fermion-boson interaction dominates in the fermion self-energy. The model represents a normal metal with strongly damped bosonic excitations. The latter play the role of normal impurities.

Abstract:
The resonance Wigner scattering of charged bosons (small bipolarons) in a random potential leads to logarithmically divergent low-temperature resistivity as observed in several cuprates. Unusual temperature dependence of resistivity of $La_{2-x}Sr_{x}CuO_{4}$ as well as of the Hall effect is quantitatively described in a wide temperature range providing an evidence for $2e$ charged Bose-liquid in high-$T_{c}$ cuprates.

Abstract:
A single-particle spectral density is proposed for cuprates taking into account the bipolaron formation, realistic band structure, thermal fluctuations and disorder. Tunnelling and photoemission (PES) spectra are described, including the temperature independent gap observed both in the superconducting and normal states, the emission/injection asymmetry, the finite zero-bias conductance, the spectral shape in the gap region and its temperature and doping dependence, dip-hump incoherent asymmetric features at high voltage (tunnelling) and large binding energy (PES).

Abstract:
We show that objections raised by Chakraverty $et$ $al$ (Phys. Rev. Lett. 81, 433 (1998)) to the bipolaron model of superconducting cuprates are the result of an incorrect approximation for the bipolaron energy spectrum and misuse of the bipolaron theory. The consideration, which takes into account the multiband energy structure of bipolarons and the unscreened electron-phonon interaction clearly indicates that cuprates are in the Bose-Einstein condensation regime with mobile charged bosons.

Abstract:
There is a growing understanding that transport properties of complex oxides and individual molecules are dominated by polaron physics. In superconducting oxides the long-range Froehlich and short-range Jahn-Teller electron-phonon interactions bind carriers into real space pairs - small bipolarons with surprisingly low mass but sufficient binding energy, while the long-range Coulomb repulsion keeps bipolarons apart preventing their clustering. The bipolaron theory numerically explains high Tc values without any fitting parameters and describes other key features of the cuprates. The same approach provides a new insite into the theory of transport through molecular nanowires and quantum dots (MQD). Attractive polaron-polaron correlations lead to a "switching" phenomenon in the current-voltage characteristics of MQD. The degenerate MQD with strong electron-vibron coupling has two stable current states (a volatile memory), which might be useful in molecular electronics.

Abstract:
A single vortex in the charged Bose gas (CBG) has a charged core and its profile different from the vortex in neutral and BCS superfluids. Lower and upper critical fields of CBG are discussed. The unusual resistive upper critical field, Hc2(T), of many cuprates and a few other unconventional superconductors is described as the Bose-Einstein condensation field of preformed bosons-bipolarons. Its nonlinear temperature dependence follows from the scaling arguments. Exceeding the Pauli paramagnetic limit is explained. Controversy in the determination of Hc2(T) of cuprates from kinetic and thermodynamic measurements is addressed in the framework of the bipolaron theory.

Abstract:
Extending the BCS theory towards the strong electron-phonon interaction (EPI), a charged Bose liquid of small bipolarons has been predicted by us with a further prediction that the highest superconducting critical temperature is found in the crossover region of the EPI strength from the BCS-like to bipolaronic superconductivity. Later on we have shown that the unscreened (infinite-range) Froehlich EPI combined with the strong Coulomb repulsion create \emph{superlight} small bipolarons, which are several orders of magnitude lighter than small bipolarons in the Holstein-Hubbard model (HHM) with a zero-range EPI. The analytical and numerical studies of this Coulomb-Froehlich model (CFM) provide the following recipes for room-temperature superconductivity: (a) The parent compound should be an ionic insulator with light ions to form high-frequency optical phonons, (b) The structure should be quasi two-dimensional to ensure poor screening of high-frequency phonons polarized perpendicular to the conducting planes, (c) A triangular lattice is required in combination with strong, on-site Coulomb repulsion to form the small superlight bipolaron, (d) Moderate carrier densities are required to keep the system of small bipolarons close to the Bose-Einstein condensation regime. Clearly most of these conditions are already met in the cuprates.

Abstract:
The common wisdom that the phonon mechanism of electron pairing in the weak-coupling Bardeen-Cooper-Schrieffer (BCS) superconductors leads to conventional s-wave Cooper pairs is revised. An inevitable anisotropy of sound velocity in crystals makes the phonon-mediated attraction of electrons non-local in space providing unconventional Cooper pairs with a nonzero orbital momentum in a wide range of electron densities. As a result of this anisotropy quasi-two dimensional charge carriers undergo a quantum phase transition from an unconventional d-wave superconducting state to a conventional s-wave superconductor with more carriers per unit cell. In the opposite strong-coupling regime rotational symmetry breaking appears as a result of a reduced Coulomb repulsion between unconventional bipolarons dismissing thereby some constraints on unconventional pairing in the Bose-Einstein condensation (BEC) limit. The conventional phonons, and not superexchange, are shown to be responsible for the d-wave symmetry of cuprate superconductors, where the on-site Coulomb repulsion is large.

Abstract:
The nonlinear normal state diamagnetism reported by Lu Li et al. [Phys. Rev. B 81, 054510 (2010)] is shown to be incompatible with an acclaimed Cooper pairing and vortex liquid above the resistive critical temperature. Instead it is perfectly compatible with the normal state Landau diamagnetism of real-space composed bosons, which describes the nonlinear magnetization curves in less anisotropic cuprates La-Sr-Cu-O (LSCO) and Y-Ba-Cu-O (YBCO) as well as in strongly anisotropic bismuth-based cuprates in the whole range of available magnetic fields.

Abstract:
Semiconducting-like low-temperature in-plane resistivity indicates that there are no remnants of superconductivity above the resistive phase transition at T > Tc in underdoped cuprates. The model with the chemical potential pinned near the mobility edge inside the charge-transfer optical gap describes quantitatively the Nernst effect, thermopower, diamagnetism and the unusual low-temperature resistivity of underdoped cuprates as normal state phenomena above Tc.