A two component model of negative U centers coupled with the Fermi sea of itinerant fermions is discussed in connection with high-temperature superconductivity of cuprates, and superfluidity of atomic fermions. We examine the phase transition and the condensed state of this boson-fermion model (BFM) beyond the ordinary mean-field approximation in two and three dimensions. No pairing of fermions and no condensation are found in two-dimensions for any symmetry of the order parameter. The expansion in the strength of the order parameter near the transition yields no linear homogeneous term in the Ginzburg-Landau-Gor'kov equation and a zero upper critical field in any-dimensional BFM, which indicates that previous mean-field discussions of the model are flawed. Normal and anomalous Green's functions are obtained diagrammatically and analytically in the condensed state of a simplest version of 3D BFM. A pairing of bosons analogous to the Cooper pairing of fermions is found. There are three coupled condensates in the model, described by the off-diagonal single-particle boson, pair-fermion and pair-boson fields. These results negate the common wisdom that the boson-fermion model is adequately described by the BCS theory at weak coupling.
In the model considered, the nonlocal interaction of the fermions in different sublattices of a bipartite lattice is introduced. It can also be regarded as local interaction of fermions with opposite ``hypercharge''. The corresponding term in the Hamiltonian is SU(4)-invariant and appears to be the most tractable version of the SO(5)-invariant model that unifies antiferromagnetic and d-wave superconducting order parameters. The model has been studied primarily in the weak interaction limit and in the mean field approximation. Near the half-filling the antiferromagnetic critical temperature has a peak. However, the superconducting transition takes place when the Fermi surface crosses the area where the density of states is of order of inverse coupling coefficient. Thus, in mean-field approximation, there exist an interval of values of the chemical potential, for which the system is a superconductor for arbitrary high temperatures. The temperature dependence of specific heat, Hall coefficient, and DC conductivity in the normal phase agrees with that experimentally observed in high-
Tc cuprates.
A cuprate superconductor model based on the analogy with atomic nuclei was shown by Iachello to have an su(3) structure. The mean-field approximation Hamiltonian can be written as a linear function of the generators of the su(3) algebra. Using algebraic methods, we derive the eigenvalues of the reduced Hamiltonian beyond the subalgebras u(1)⊗ u(2) and so(3) of the su(3) algebra. In particular, by considering the coherence between s- and d-wave pairs as a perturbation, the effects of the coherent term upon the energy spectrum are investigated.
Various BCS-like pairing theories are still invariably used to describe the unusual superconducting/superfluid states and properties of condensed matters without clarifying the fermionic or bosonic nature of Cooper pairs and the relevant mechanisms of superconductivity and superfluidity. We show that the bosonization of Cooper pairs and the novel superconductivity and superfluidity resulting from the condensation of an attractive Bose gas of Cooper pairs into a superfluid state in high-Tc cuprates and other exotic systems occur under the condition εF≲2εA (where εF is the Fermi energy, εA is the energy of the effective attraction between fermions). We argue that the simple BCS criterion (i.e. formation of an energy gap on the Fermi surface) is insufficient for the occurrence of unconventional superconductivity (superfluidity) and the superconducting/superfluid transition in attractive Bose systems is more λ-like than the usual Bose–Einstein condensation. The new and specific criteria for unconventional superconductivity and superfluidity are formulated. Mean-field theory of the three- and two-dimensional superfluid Bose condensates of Cooper pairs describes reasonably well the novel superconducting states (i.e., two distinct superconducting phases below the bulk superconducting transition temperature Tc and a vortex-like state above Tc) and properties (such as λ-like transition at Tc, first-order phase transition and gapless bosonic excitations somewhat below Tc or even far below Tc) of high-Tc cuprates in full agreement with the experimental findings. The unusual superconducting/superfluid states and properties of heavy-fermion and organic compounds, Sr2RuO4, 3He, 4He and atomic Fermi gases are explained more clearly by this theory.
Bosonization of Cooper pairs and novel Bose-liquid superconductivity and superfluidity in high-
Tc cuprates and other systems
The criteria for bosonization of Cooper pairs and novel Bose-liquid superconductivity and superfluidity in high-
Tc cuprates and other pseudogap matters are formulated by using the uncertainty principle and the boson mean field theory. We argue that the BCS-type s- or d-wave superconductivity occurring in the fermionic limit of Cooper pairs (which exist in ordinary metals and heavily overdoped cuprates with large Fermi energies) is not characteristic of underdoped to overdoped cuprates with low Fermi energies. The superconducting order parameter in high-
Tc cuprates and other pseudogap matters cannot be determined as the BCS-like (s- or d-wave) gap. We show that the unconventional superconductivity/superfluidity occurring in the bosonic limit of Cooper pairs exists in low Fermi energy systems where the bosonic Cooper pairs are formed at a pseudogap temperature T* above the uperconducting/superfluid transition temperature
Tc and then such bosons condenses into a Bose superfluid at
Tc. The pair condensation of attracting bosons occurs at
Tc and then their single particle condensation sets in at
Tc∗ lower than
Tc (in three dimensions (3D)) or at T=0 (in two dimensions (2D)). The coherent single particle and pair condensates of bosons exist as two distinct superfluid phases below
Tc. By solving the mean field equations for high-
Tc cuprates, the novel superconducting states (i.e., a vortex-like state existing below the temperature
Tv=Tc2D lower than T* but higher than
Tc=Tc3D and two superconducting phases below
Tc) and their properties characterized by the boson superfluid stiffness are determined and compared with the key experimental findings. The novel Bose-liquid superconductivity and superfluidity exists also in other exotic superconductors/superfluids, including quantum liquids, atomic Fermi gases and low-density nuclear matter.
We review briefly the properties of a mixture of mutually interacting bosons (bound electron pairs) and itinerant fermions on a lattice (the boson–fermion model). The calculations of the superconducting phase transition temperature () and the phase diagram are the main concern. The self-consistent -matrix method is applied to determine the superconducting critical temperature from a pseudogap phase. The method takes into account the pairing fluctuations effects. The -matrix results for are given for a three-dimensional (3-D) cubic lattice with tight-binding dispersion of electrons and standard bosons, and they are also compared with those of the BCS- mean-field approximation. Our results describe the BCS-Bose-Einstein condensation crossover in the boson–fermion mixture with resonant interaction. The energy scales involved in the pseudogap formation are also analysed.