In a nearly‐filled energy band where scattering of electrons is hampered, a pairing interaction between two electrons and quanta in a solid (e.g., phonons or spin‐resonances or some other excitation of electronic origin) can still occur in the s‐channel where a field quantum mediates the coalescence of two fermions into a new composite particle. By describing the attractively‐interacting electrons as a gas mixture of individual electrons plus bosons made up of two electrons, the dependence on momentum of the energy of single electrons is investigated. When the net interaction BCS dimensionless coupling constant λ and the ratio with the maximum energy of a quantum mediating the pairing to the Fermi energy increases, the coalescence of electrons at temperatures higher than the Bose–Einstein condensation (BEC) temperature of bosons leads to the appearance of steep (waterfall‐like) changes in the high‐energy electron spectrum of the binary boson–fermion (BF) gas mixture. These waterfall‐like anomalies seen in high‐ superconductors (HTSCs) are in fact a characteristic property of a BF gas mixture and reflect the splitting of the conduction band into two subbands emerging as the number of two‐electron bosons increases.
In a nearly‐filled energy band where scattering of electrons is hampered, a pairing interaction between two electrons and quanta in a solid (e.g., phonons or spin‐resonances or some other excitation of electronic origin) can still occur in the s‐channel where a field quantum mediates the coalescence of two fermions into a new composite particle. By describing the attractively‐interacting electrons as a gas mixture of individual electrons plus bosons made up of two electrons, the dependence on momentum of the energy of single electrons is investigated. When the net interaction BCS dimensionless coupling constant λ and the ratio with the maximum energy of a quantum mediating the pairing to the Fermi energy increases, the coalescence of electrons at temperatures higher than the Bose–Einstein condensation (BEC) temperature of bosons leads to the appearance of steep (waterfall‐like) changes in the high‐energy electron spectrum of the binary boson–fermion (BF) gas mixture. These waterfall‐like anomalies seen in high‐ superconductors (HTSCs) are in fact a characteristic property of a BF gas mixture and reflect the splitting of the conduction band into two subbands emerging as the number of two‐electron bosons increases.
An extended boson–fermion (BF) model is introduced describing the evolution of attractively interacting single electrons into a binary mixture of unpaired electrons plus incoherent bosonic Cooper pairs as absolute temperature is lowered. The new model is employed to determine how the system, upon cooling, exhibits critical temperatures for: a) incoherent boson formation below as well as b) for creating a coherent boson fluid below . Critical temperatures and are related with the dynamical parameters of the attractively interacting fermion system. It is shown that the introduction of a uniform Coulomb interaction in an anisotropic BF binary gas mixture model allows finding not only a self-consistent description of a pseudogap and a dome-shaped -vs-doping behaviour, but also to predict, even in a phonon-mediated many-electron dynamics, of an energy–momentum dispersion that is either gapped or not depending on direction. This manifests itself in high- superconductors as curves in momentum space along which the superconducting gap vanishes, hence giving rise to the so-called “Fermi arcs” observed in angle-resolved-photoemission spectroscopy experiments.
A boson–fermion (BF) quantum-statistical binary gas mixture model of high- T c superconductors (HTSCs) recently introduced consists of resonant bosonic Cooper electron pairs (CPs) in chemical and thermal equilibrium with single unpaired electrons. Here, this model is refined and extended to include: i) the anisotropy of the original BF vertex interaction causing boson formation/disintegration and ii) momentum-independent Coulomb repulsions between electron charge carriers. It is shown that pair breakings due to Coulomb repulsion depend on the separation between boson and fermion spectra. Specifically, as such a separation shrinks, the pair-breaking ability of the Coulomb interaction weakens and disappears altogether at the Bose–Einstein condensation (BEC) T c , i.e., at the temperature at which a complete softening of bosons occurs due to boson self-energy renormalization. Simultaneous inclusion of both effects produces “islands” in momentum space of incoherent CPs above the Fermi sea as temperature is lowered. These islands grow upon further cooling and merge together just before T c is reached. The BF model thusly extended now predicts a pseudogap phase in 2D HTSCs with lines of points on the Fermi surface along which the pseudogap vanishes, hence explaining the origin of the temperature-dependent “Fermi arcs” observed in experiments.
A recent boson-fermion (BF) binary gas mixture model is extended to include: (i) anisotropy of the BF interaction and (ii) momentum-independent Coulomb repulsions. It is applied to account for the peculiarities of the pseudogap observed as function of absolute temperature T and concentration x of holes doped onto the CuO 2 planes and to study the further transformation of the pseudogap into the real superconducting gap, as T is lowered. Using two-time Green functions it is shown that pair breakings depend on the separation between the boson and fermion spectra of the BF mixture. As this separation shrinks, the pair-breaking ability of the Coulomb interaction weakens and disappears at the BEC T c , i.e., at the T below which a complete softening of bosons occurs. Simultaneous inclusion of both effects (i) and (ii) produces, as T is lowered, "islands" in momentum space of incoherent Cooper pairs above the Fermi sea. These islands grow upon further cooling and merge together just before T c is reached. The new extended BF model predicts a pseudogap phase in 2D high-T c superconductors with lines of points, or loci, on the Fermi surface along which the pseudogap vanishes. This explains the origin of T-dependent "Fermi arcs" observed in ARPES experiments.
Unification of the Bardeen, Cooper and Schrieffer (BCS) and the Bose‐Einstein condensation (BEC) theories is surveyed in terms of a generalized BEC (GBEC) finite‐temperature statistical formalism. A vital distinction is that Cooper pairs (CPs) are true bosons that may suffer a BEC since they obey BE statistics, in contrast with BCS pairs that are “hard‐core bosons” at best. A second crucial ingredient is the explicit presence of hole‐pairs (2h) alongside the usual electron‐pairs (2e). A third critical element (particularly in 2D where ordinary BEC does not occur) is the linear dispersion relation of CPs in leading order in the center‐of‐mass momentum (CMM) power‐series expansion of the CP energy. The GBEC theory reduces in limiting cases to all five continuum (as opposed to “spin”) statistical theories of superconductivity, from BCS on one extreme to the BEC theory on the other, as well as to the BCS‐Bose “crossover” picture and the 1989 Friedberg‐Lee BEC theory. It accounts for 2e‐ and 2h‐CPs in arbitrary proportions while BCS theory can be deduced from the GBEC theory but allows only equal (50%‐50%) BE condensed‐mixtures of both kinds of CPs. As it yields the precise BCS gap equation for all temperatures as well as the precise BCS zero‐temperature condensation energy for all couplings, it suggests that the BCS condensate is a BE condensate of a ternary mixture of kinematically independent unpaired electrons coexisting with equally proportioned weakly‐bound zero‐CMM 2e‐ and 2h‐CPs. Without abandoning the electron‐phonon mechanism in moderately weak coupling, and fortuituously insensitive to the BF interactions, the GBEC theory suffices to reproduce the unusually high values of Tc (in units of the Fermi temperature TF) of 0.01–0.05 empirically found in the so‐called “exotic” superconductors of the Uemura plot, including cuprates, in contrast to the low values of Tc/TF⩽10−3 roughly reproduced by BCS theory for conventional (mostly elemental) superconductors.
Starting from the Friedberg-TD Lee Hamiltonian describing a coexisting and dynamically interacting many-particle binary boson-fermion gas mixture with a coupling $(\ensuremath{\lambda})$-dependent gap $2\ensuremath{\Delta}(\ensuremath{\lambda})$ in the boson dispersion relation for the s-wave Cooper or BCS model interaction, we deduce several observed characteristic features of high-temperature superconductors at the simplest level. Analytic expressions for both the unpaired-fermion and boson number densities, as well for the fermion chemical potential $\ensuremath{\mu}(\ensuremath{\lambda},T)$, all of which vary with the degree of bosonization and with temperature T, are derived in detail using two-time, finite-temperature Green function techniques. Simple implicit formulas are then obtained for both two and three dimensions for the pseudogap T∗ and Bose-Einstein condensation Tc temperatures in terms of $\ensuremath{\mu}(\ensuremath{\lambda},T)$ and $2\ensuremath{\Delta}(\ensuremath{\lambda})$. In particular, even at the s-wave level we find a self-consistent description of the generic phase diagram observed in cuprates, including the appearance of a pseudogap and a dome-shaped Tc vs doping behavior both of which hinge on the gapped boson spectrum.
By recognizing the vital importance of two‐hole Cooper pairs (CPs) in addition to the usual two‐electron ones in a strongly‐interacting many‐electron system, the concept of CPs was re‐examined with striking conclusions: namely, they are gapped and linearly‐dispersive resonances with a finite lifetime—but provided the ideal‐gas Fermi sea is replaced by a BCS‐correlated unperturbed ground‐state “sea.” Based on this, Bose‐Einstein condensation (BEC) theory has been generalized to include not boson‐boson interactions (also neglected in BCS theory) but rather boson‐fermion (BF) interaction vertices reminiscent of the Fröhlich electron‐phonon interaction in metals. Instead of phonons, the bosons in the generalized BEC (GBEC) theory are now both particle and hole CPs. Unlike BCS theory, the GBEC model is not a mean‐field theory restricted to weak‐coupling as it can be diagonalized exactly. In weak coupling it reproduces the BCS condensation energy, and the next‐order‐in‐coupling term increases its magnitude with respect to BCS. Each kind of CP is responsible for only half the condensation energy. The GBEC theory reduces to all the old known statistical theories as special cases—including the so‐called “BCS‐Bose crossover” picture which in turn generalizes BCS theory by not assuming that the interelectronic chemical potential equals the Fermi energy. Indeed, a BCS condensate is precisely the weak‐coupling limit of a GBE condensate with equal numbers of both types of CPs. With feasible Cooper/BCS model interelectonic interaction parameter values, and even without BF interactions, the GBEC theory yields transition temperatures [including room‐temperature superconductivity (RTSC)] substantially higher than the BCS ceiling of around 45K, without relying on non‐phonon dynamics involving excitons, plasmons, magnons or otherwise purely‐electronic mechanisms. The results are expected to shed light on the experimental search for RTSC.