We analyze solutions to a superconducting gap equation based on the two-dimensional Hubbard model with nearest and next-to-nearest neighbor hopping. The Cooper pair potential can be calculated exactly and expressed in terms of elliptic functions. The Fermi surfaces at finite temperature and chemical potential are calculated based on the exact two-body S-matrix of the Hubbard model using the formalism we recently developed, which allows variation of hole doping. The resulting solutions to the gap equation are strongly anisotropic, namely largest in the anti-nodal direction, and zero in the nodal directions of the Brillouin zone. For U/t = 13 and t' /t =-0.3, appropriate to BSCO, and a physically natural choice for the cut-off, our self-contained analytic calculations yield the gap in the anti-nodal direction Delta/t = 0.06 and a maximum Tc/t = 0.04 at hole doping h=0.15. For phenomenological fits to the Fermi surfaces for cuprates, we obtain the comparable value Tc/t = 0.03 at optimal doping, both in good agreement with experiments. The superconducting gap is non-zero for all hole-doping h < 0.35, and increases all the way down to zero doping, suggesting that it evolves into the pseudogap.
Quasiparticle bands of the two-dimensional Hubbard model are calculated using the Roth two-pole approximation to the one-particle Green's function. Excellent agreement is obtained with recent Monte Carlo calculations, including an anomalous volume of the Fermi surface near half-filling, which can possibly be explained in terms of a breakdown of Fermi liquid theory. The calculated bands are very flat around the (\ensuremath{\pi},0) points of the Brillouin zone in agreement with photoemission measurements of cuprate superconductors. With doping there is a shift in spectral weight from the upper band to the lower band. The Roth method is extended to deal with superconductivity within a four-pole approximation allowing electron-hole mixing. It is shown that triplet p-wave pairing never occurs. A self-consistent solution with singlet ${\mathit{d}}_{{\mathit{x}}^{2}\mathrm{\ensuremath{-}}{\mathit{y}}^{2}}$-wave pairing is found and optimal doping occurs when the van Hove singularity, corresponding to the flat band part, lies at the Fermi level. Nearest-neighbor antiferromagnetic correlations play an important role in flattening the bands near the Fermi level and in favoring superconductivity. However, the mechanism for superconductivity is a local one, in contrast to spin-fluctuation exchange models. For reasonable values of the hopping parameter the transition temperature
Tc is in the range 10--100 K. The optimum doping ${\mathrm{\ensuremath{\delta}}}_{\mathit{c}}$ lies between 0.14 and 0.25, depending on the ratio U/t. The gap equation has a BCS-like form and 2${\mathrm{\ensuremath{\Delta}}}_{\mathrm{max}}$/
kTc\ensuremath{\simeq}4.
The two-dimensional Hubbard model is studied for small values of the interaction strength (
U of the order of the hopping amplitude
t), using a variational ansatz well suited for this regime. The wave function, a refined Gutzwiller ansatz, has a BCS mean-field state with
d-wave symmetry as its reference state. Superconducting order is found for densities
n<1 (but not for
n=1). This resolves a discrepancy between results obtained with the functional renormalization group, which do predict superconducting order for small values of
U, and numerical simulations, which did not find superconductivity for $U\ensuremath{\lesssim}4t$. Both the gap parameter and the order parameter have a domelike shape as a function of
n with a maximum for $n\ensuremath{\approx}0.8$. Expectation values for the energy, the particle number, and the superconducting order parameter are calculated using a linked-cluster expansion up to second order in
U. In this way large systems (millions of sites) can be readily treated and well converged results are obtained. A big size is indeed required to see that the gap becomes very small at half filling and probably tends to zero in the thermodynamic limit, whereas away from half filling a finite asymptotic limit is reached. For a lattice of a given size the order parameter becomes finite only above a minimal coupling strength
Uc. This threshold value decreases steadily with increasing system size, which indicates that superconductivity exists for arbitrarily small
U for an infinite system. For moderately large systems the size dependence is quite irregular, due to variations in level spacings at the Fermi energy. The fluctuations die out if the gap parameter spans several level spacings.
By introducing multiple-site correlation functions, we propose a hierarchical Green’s function approach, and apply it to study the characteristic properties of a two-dimensional (2D) square lattice Hubbard model by solving the equation of motions of a one-particle Green’s function and related multiple-site correlation functions. Under a cut-off approximation and taking the Fourier representation of multiple-site correlation functions, we obtain an analytical expression of one-particle Green’s function with static correlation functions. Then we calculate the spectral density function of electrons, and obtain that besides two main peaks corresponding to the lower and upper Hubbard bands in the spectral density function, there emerge some novel states between these two main peaks, and the total spectral weight of these emerged states is proportional to the hole doping concentration [Formula: see text]. Meanwhile, there also emerge some collective modes related to possible charge/spin density wave and/or electronic pairing density wave ordering states. This calculation is completely consistent with the spectroscopy observations of the cuprate superconductors in normal states. On the other hand, the appearance of the static correlation functions in the one-particle Green’s function can be used to describe the intertwined orders observed in the normal state of the cuprate superconductors.
We compute the two-particle quantities relevant for superconducting correlations in the two-dimensional Hubbard model within the dynamical cluster approximation. In the normal state we identify the parameter regime in density, interaction, and second-nearest-neighbor hopping strength that maximizes the d_{x^{2}-y^{2}} superconducting transition temperature. We find in all cases that the optimal transition temperature occurs at intermediate coupling strength, and is suppressed at strong and weak interaction strengths. Similarly, superconducting fluctuations are strongest at intermediate doping and suppressed towards large doping and half filling. We find a change in sign of the vertex contributions to d_{xy} superconductivity from repulsive near half filling to attractive at large doping. p-wave superconductivity is not found at the parameters we study, and s-wave contributions are always repulsive. For negative second-nearest-neighbor hopping the optimal transition temperature shifts towards the electron-doped side in opposition to the van Hove singularity, which moves towards hole doping. We surmise that an increase of the local interaction of the electron-doped compounds would increase T_{c}.
A generic Hamiltonian, which incorporates the effect of the orbital contraction on the hopping amplitude between nearest sites, is studied both analytically at the weak coupling limit and numerically at the intermediate and strong coupling regimes for a finite atomic cluster. The effect of the orbital contraction due to hole localization at atomic sites is specified with two coupling parameters V and W (multiplicative and additive contraction terms). The singularity of the vertex part of the two-particle Green’s function determines the critical temperature Tc and the relaxation rate Γ(T) of the order parameter at temperature above Tc. Unlike the case in conventional BCS superconductors, Γ has a non-zero imaginary part which may influence the fluctuation conductivity of the superconductor above Tc. We compute the ground state energy as a function of the particle number and magnetic flux through the cluster, and show the existence of the parity gap Δ appearing at the range of system parameters consistent with the appearance of the Cooper instability. Numeric calculation of the Hubbard model (with U>0) at arbitrary occupation does not show any sign of superconductivity in a small cluster.