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arXiv:1008.5116

Superconductivity in the two-dimensional Hubbard model based on the exact pair potential

André LeClair

Published Aug 30, 2010• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.1008.5116

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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{\mathit{T}}_{\mathit{c}}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{\mathit{kT}}_{\mathit{c}}kTc​\ensuremath{\simeq}4.
A systematic diagrammatic expansion for Gutzwiller-wave functions (DE-GWF) is formulated and used for the description of superconducting (SC) ground state in the two-dimensional Hubbard model with electron-transfer amplitudes ttt (and ${t}^{\ensuremath{'}}$) between nearest (and next-nearest) neighbors. The method is numerically very efficient and allows for a detailed analysis of the phase diagram as a function of all relevant parameters (UUU, $\ensuremath{\delta}$, ${t}^{\ensuremath{'}}$) and a determination of the kinetic-energy driven pairing region. SC states appear only for substantial interactions, $U/t\ensuremath{\gtrsim}3$, and for not too large hole doping, $\ensuremath{\delta}\ensuremath{\lesssim}0.32$ for ${t}^{\ensuremath{'}}=0.25t$; this upper critical doping value agrees well with experiment for the cuprate high-temperature superconductors. We also obtain other important features of the SC state: (i) the SC gap at the Fermi surface resembles ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave only around the optimal doping and the corrections to this state are shown to arise from the longer range of the pairing; (ii) the nodal Fermi velocity is almost constant as a function of doping and agrees quantitatively with the experimental results; (iii) the SC transition is driven by the kinetic-energy lowering for low doping and strong interactions.
A variational Monte Carlo calculation is used for studying the ground state of the two-dimensional Hubbard model, including hopping between both nearest and next-nearest neighbor sites. Superconductivity with d-wave symmetry is found to be restricted to densities where the Fermi surface crosses the magnetic zone boundary. The condensate energy is much larger for hole doping than for electron doping. Superconductivity is kinetic energy driven for hole doping, but potential energy driven for electron doping. Our findings agree surprisingly well with experimental data for layered cuprates, both for electron- and hole-doped materials.
An elaborate variational wave function is used for studying superconductivity in the (repulsive) two-dimensional Hubbard model, including both nearest- and next-nearest-neighbor hoppings. A marked asymmetry is found between the ``localized'' hole-doped region and the more itinerant electron-doped region. Superconductivity with ddd-wave symmetry turns out to be restricted to densities where the Fermi surface crosses the magnetic zone boundary. A concomitant peak in the magnetic structure factor at $(\ensuremath{\pi},\ensuremath{\pi})$ clearly points to a magnetic mechanism.
Abstract The spectral properties of the doped t – t ’ Hubbard model, using parameters typical for high-temperature cuprate superconductors, and the mechanism of d-wave pairing remain among the longstanding problems of many-body fermionic materials. We used a strong-coupling Green’s function expansion around a correlated reference system, namely a particle-hole-symmetric undoped Hubbard lattice with t′=0{t}^{{\prime} }=0t′=0 t ′ = 0 , which can be treated numerically exactly using sign-problem-free lattice Quantum Monte Carlo calculations. This reference system exhibits a large antiferromagnetic Mott-Hubbard-Slater gap in the electronic spectrum. We investigate how the Mott-like spectrum is reconstructed under finite doping and nonzero t′{t}^{{\prime} }t′ t ′ using a dual-fermion-inspired perturbation expansion. For a large next-nearest-neighbor hopping t′=−0.3t{t}^{{\prime} }=-0.3tt′=−0.3t t ′ = − 0.3 t , characteristic of cuprate families with T c around 100 K, the electronic spectral function reveals a strongly renormalized flat-band feature with a pseudogap near the antinodal point. The superconducting response of this system to a small dx2−y2{d}_{{x}^{2}-{y}^{2}}dx2−y2​ d x 2 − y 2 -like external field shows a pseudogap at the antinodal point in the normal part of the Nambu Green’s function, associated with “bad-fermion” behavior in the normal phase. At the same time, the anomalous Green’s function exhibits a d-wave-like structure with zero response at the nodal point of the Brillouin zone.
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.