An exact-diagonalization technique on small clusters is used to study low-lying excitations and superconductivity in the two-dimensional negative-U Hubbard model. We first calculate the Bogoliubov-quasiparticle spectrum, condensation amplitude, and coherence length as functions of the coupling strength U/t, thereby working out how the picture of Bogoliubov quasiparticles in the BCS superconductors is affected by increasing the attraction. We then define the Cooper-pair operator as a spatially extended composite boson and make a variational evaluation of its internal structure. We thereby calculate the single-particle spectral function of the Cooper pair and obtain the dispersion relation for its translational motion. The dynamical density correlation function of the pairs is also calculated. We thus demonstrate the applicability of our numerical method for gaining insight into low-lying excitations of models for the intermediate coupling superconductivity relevant to cuprate materials.
We consider a lattice version of the ``negative-
U′′ Hubbard model in three dimensions, to study the crossover from a superconducting phase transition (dominated by phase fluctuation) to a regime where the amplitude of the superconducting order parameter controls the critical temperature. Starting from a fermionic Hamiltonian, we give a microscopic derivation of the effective action in terms of the relevant physical fields: the modulus and phase of the superconducting order parameter. Furthermore, by employing to the resulting coarse--grained quantum
XY model the phase fluctuation algebra between number and phase operators (given by the Euclidean group
E2), we map the ``phase-only'' action onto a solvable quantum spherical model. We establish the self--consistent theory (involving both fermionic and bosonic degrees of freedom), and calculate the superconducting phase coherence transition temperature
Tc as a function of the coupling strength
∣U∣, exhibiting a maximum where the behavior crosses over from BCS to Bose-Einstein (BE) condensation. We examine a pseudogap in the normal state which emerges naturally as a precursor of superconductivity at the second characteristic temperature
Tg due to a state with bound pairs but without long--range phase coherence. Furthermore, we demonstrate that the reduced gaps of the model $2{\ensuremath{\Delta}}_{0}{/k}_{B}{T}_{g}\ensuremath{\sim}4$ is almost independent on the pairing strength
∣U∣/t, whereas $2{\ensuremath{\Delta}}_{0}{/k}_{B}{T}_{c}$ increases with increasing
∣U∣---in striking analogy with the phenomenology of the high--temperature cuprate superconductors when interpreted in terms of the BCS-BC crossover scenario.
The electronic $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}U\text{\ensuremath{-}}J$ model for the
CuO2 plane in high-
Tc superconducting cuprates is elaborated. It incorporates both charge transfer
U and exchange correlation
J with direct
t and diagonal ${t}^{\ensuremath{'}}$ hopping. Making use of the Yang concept of the off diagonal long range order (ODLRO) in the reduced density matrix of the electronic system we study the loss of quantum wave coherence (and thereby superconductivity) due to the Coulomb interaction. To this aim the second-quantized Hamiltonian of the model is translated to the phase representation with the help of the topologically constrained path integral formalism. As a result the electrons emerge as composite particles consisting of of spin-carrying neutral fermions and charged topological bosons in a form of ``flux tubes'' with the quantum phase variable dual to the local electron density. The charge carrying bose field forms a collective variable representing the correlated ``sliding'' motion of the entire electronic system. We found that the spin gap $\ensuremath{\Delta}$ which appears in one-particle spectrum is the result of valence bond pairing of neutral fermions without ODLRO and deduce the solution for $\ensuremath{\Delta}$ with ``
d-wave'' symmetry. In the bosonic sector we derive the effective quantum rotor model along with microscopic phase stiffnesses setting the energy scale for superconductivity and point out the special role played by the diagonal hopping ${t}^{\ensuremath{'}}$. We explore the relationship of the quantum rotor phase model and the Bose-Hubbard system emphasizing the ring topology of quantum phase variable. Furthermore, we obtain the ground state phase diagrams which facilitate the Mott insulator-superconductor transition scenario. We elucidate the role of the chemical potential (doping) and the huge degeneracy of the effective bosonic system in achieving ODLRO with long-range phase coherence in the presence of large Coulomb repulsion. Finally, we discuss a number of relevant issues pertaining to the the present work which include: theoretical aspects of the hierarchy of the energy scales in cuprates, ``spin-charge separation'' concepts, ``local-pair'' and ``phase-fluctuation'' scenarios as well as experimental findings related to the role of the ``
d-wave'' symmetry of the spin gap and diagonal hopping ${t}^{\ensuremath{'}}$ in cuprates.
We calculate the spectral weights of the optical conductivity of a three-dimensional lattice version of the ``negative-
U′′ Hubbard model of a strongly correlated superconductor in the intermediate to large-
U limit. Starting from a tight-binding Fermionic Hamiltonian, we pass to an effective action in terms of the modulus and phase of the superconducting pairing field as the relevant Bosonic variables. Furthermore, we calculate the temperature and U-interaction dependence of the oscillator strength and find that below the pair formation temperature ${T}^{\ensuremath{\star}}$ the optical integral is not conserved, implying the lowering of the kinetic energy at
Tc upon entering the phase coherent superfluid state.
By means of the functional-integral formulation we discuss the occurrence of superconductivity resulting from Bose-Einstein condensation within a model describing preformed pairs which coexist in thermal equilibrium with unpaired fermions. At saddle-point level we study the thermodynamic properties of the model as functions of the boson excitation energy $\ensuremath{\nu},$ discussing the results in connection with the phenomenology of the pseudogap state of the superconducting cuprates. For temperatures above the Bose-Einstein condensation temperature
Tc we also analyze the effects of Gaussian fluctuations on the critical temperature and on the boson spectrum. Finally, we compare the bosonic effective action which controls thermal fluctuations of the mixture to the time-dependent Ginzburg-Landau functional of an interacting fermionic system at intermediate coupling.
The phase boundary between normal and superconducting phases of the negative-U Hubbard model is obtained within a self-consistent weak-coupling approach known as the ``pseudopotential parquet'' approximation. The shortcomings of simpler self-consistent approaches, including the Hartree-Fock and fluctuation-exchange approximations, are outlined. For intermediate values of U, pseudopotential parquet predicts (i) a maximum in the superconducting transition temperature
Tc at an electron filling 〈n〉 of approximately 0.92; and (ii) a sharp decrease in
Tc for values of 〈n〉 closer to 1. The scale of transition temperatures is consistent with results from quantum simulations.