In order to study the anisotropic superconductivity in two dimensional lattices, it has been recently proposed a generalized Hubbard model based on first- and second-neighbour correlated-hopping interactions. After considering this Hamiltonian within the BCS formalism, we obtain a system of two coupled integral equations, whose solution gives the superconducting gap and the chemical potential for each temperature and electronic density. This system of equations is usually solved in a numerical way, but the involved integrals over the first Brillouin zone (1BZ) consume a large amount of computing time since the integrand functions are extremely sharp around the Fermi surface (FS) especially for small pairing interactions. In this work, we report a new efficient way to carry out these integrals by dividing the 1BZ in regions delimited by curves close to the FS.
In order to study the anisotropic superconductivity in two dimensional lattices, it has been recently proposed a generalized Hubbard model based on first- and second-neighbour correlated-hopping interactions. After considering this Hamiltonian within the BCS formalism, we obtain a system of two coupled integral equations, whose solution gives the superconducting gap and the chemical potential for each temperature and electronic density. This system of equations is usually solved in a numerical way, but the involved integrals over the first Brillouin zone (1BZ) consume a large amount of computing time since the integrand functions are extremely sharp around the Fermi surface (FS) especially for small pairing interactions. In this work, we report a new efficient way to carry out these integrals by dividing the 1BZ in regions delimited by curves close to the FS.
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.
Many properties of the two-dimensional Hubbard model have been explored for the model parameters appropriate for strongly correlated electronic systems, especially cuprate superconductors.Most of the calculations are done using a well-established dynamical cluster quantum Monte Carlo method.Using this method, we investigate the effect of long-range hoppings on superconductivity with and without the presence of phonons on small clusters.The superconducting transition temperature, T c , is found to generally decrease with a negative next nearest neighbor hopping, t ′ .In the presence of the Holstein phonons, a finite t ′ enhances T c in the under-doped region for the hole-doped system, consistent with band structure calculations and experiment.The validity of the spin-susceptibility-mediated pairing in this model is studied and found to yield symmetries other than d-wave when a finite t ′ is considered.A new numerical algorithm for solving the embedded cluster problems is introduced, and used to calculate the thermodynamic properties of the model on larger clusters, especially those associated with the quantum critical behavior at finite doping.Our results suggest that the quantum critical point (QCP) which separates the Fermi liquid and pseudogap regions, is the second order terminus of the line of first order phase separation transition in the limit when a positive t ′ goes to zero.For small t ′ , the superconducting dome is centered at the QCP, suggesting that charge fluctuations might have a role in the pairing mechanism in this model.I am very thankful to University of Cincinnati for providing me with the opportunity to pursue my education and the support I received during my program.Part of this thesis is a result of my work as a researcher at Louisiana State University.I am also thankful to them for their support.I would like to thank my adviser, Mark Jarrell, who taught me how to be an effective researcher, provided me with the opportunities to interact with other professionals and fully supported me financially for most of the duration of the
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.
Recent refinements of analytical and numerical methods have improved our understanding of the ground-state phase diagram of the two-dimensional (2D) Hubbard model. Here, we focus on variational approaches, but comparisons with both quantum cluster and Gaussian Monte Carlo methods are also made. Our own ansatz leads to an antiferromagnetic ground state at half filling with a slightly reduced staggered order parameter (as compared to simple mean-field theory). Away from half filling, we find d-wave superconductivity, but confined to densities where the Fermi surface passes through the antiferromagnetic zone boundary (if hopping between both nearest-neighbour and next-nearest-neighbour sites is considered). Our results agree surprisingly well with recent numerical studies using the quantum cluster method. An interesting trend is found by comparing gap parameters Δ (antiferromagnetic or superconducting) obtained with different variational wave functions. Δ varies by an order of magnitude and thus cannot be taken as a characteristic energy scale. In contrast, the order parameter is much less sensitive to the degree of sophistication of the variational schemes, at least at and near half filling.
In the present work the Roth s two-pole approximation (Phys. Rev. 184, 451 (1969)) has been used to investigate the role of d− p hybridization in the normal and superconducting states of an extended d− p Hubbard model. Superconductivity with singlet dx2−y2 -wave pairing is treated by following Beenen and Edwards formalism (Phys. Rev. B 52, 13636 (1995)). In the first part of this work, the effects of the hybridization on the superconductivity, in the hole-doped regime, have been studied treating Roth s band shift within two different approximations. In the first one, the band shift has been calculated in the limit U →¥ (U is the Coulomb interaction), with zero temperature and without consider the superconducting effects. These regards, are restrict to the band shift. In the other parts of the problem, U, the temperature and the superconducting effects have been considered finites. In the second approximation, the Coulomb interaction, the temperature and the superconductivity have been considered in the calculation of some relevant correlation functions present in the Roth s band shift. The obtained results show that the hybridization acts in the sense of to suppresses the superconductivity. Also, it has been verified that the first approximation overstimates the effects of the hybridization on the superconductivity. In the second part of these work, hoppings to second-nearest-neighbors have been included in the model with the purpose of reproduces adequately the asymetries (mainly those related with the Fermi surface, band structure and phase diagram) between the hole- and electron-doped systems. Particularly, it is shown that the crossover from hole-like to electronlike Fermi surface is deeply affected by the d − p hybridization in the hole doping case. It has been verified that the effect of the hybridization is most pronounced around the saddle-points, where the superconducting gap is maximum in the particular case of dx2−y2 -wave symmetry. As a consequence, the critical temperature Tc is directly affected by the hibridization. Moreover, the obtained results suggest that in the hole doped regime, the hybridization may act on the transport properties of the system due to the sign changes of the Hall coefficient when the crossover of the Fermi surface occurs. In the electron doped case, the crossover in the Fermi surface is not verified. Nevertheless, as the hybridization suppresses the density of states near the Fermi level, the superconductivity is affected. The topology of the Fermi surface in the hole and electron doping regime agree with recent experimental ARPES results for La2−xSrxCuO4 (hole doping) and Nd2−xCexCuO4 (electron doping).