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arXiv:cond-mat/9910012 · Physical review. B, Condensed matter

Pairing correlations in a generalized Hubbard model for the cuprates

Liliana Arrachea, Armando A. Aligia

Published Apr 1, 2000• 21 citations
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Abstract

Using numerical diagonalization of a $4\ifmmode\times\else\texttimes\fi{}4$ cluster, we calculate on-site s, extended-$s,$ and ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ pairing correlation functions (PCF's) in an effective generalized Hubbard model for the cuprates, with nearest-neighbor correlated hopping and next-nearest-neighbor hopping ${t}^{\ensuremath{'}}.$ The vertex contributions to the PCF's are significantly enhanced, relative to the ${t\ensuremath{-}t}^{\ensuremath{'}}\ensuremath{-}U$ model. The behavior of the PCF's and their vertex contributions, and signatures of anomalous flux quantization, indicate superconductivity in the d-wave channel for moderate doping and in the s-wave channel for high doping and small U.

DOI
10.1103/physrevb.61.9686

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Using numerical diagonalization of a 4\ifmmode×\else\texttimes\fi44\ifmmode\times\else\texttimes\fi{}44\ifmmode×\else\texttimes\fi4 cluster, we calculate on-site s, extended-s,s,s, and dx2\ensuremath−y2{d}_{{x}^{2}\ensuremath{-}{y}^{2}}dx2\ensuremath−y2​ pairing correlation functions (PCF's) in an effective generalized Hubbard model for the cuprates, with nearest-neighbor correlated hopping and next-nearest-neighbor hopping t\ensuremath′.{t}^{\ensuremath{'}}.t\ensuremath′. The vertex contributions to the PCF's are significantly enhanced, relative to the t\ensuremath−t\ensuremath′\ensuremath−U{t\ensuremath{-}t}^{\ensuremath{'}}\ensuremath{-}Ut\ensuremath−t\ensuremath′\ensuremath−U model. The behavior of the PCF's and their vertex contributions, and signatures of anomalous flux quantization, indicate superconductivity in the d-wave channel for moderate doping and in the s-wave channel for high doping and small U.
We calculate on-site s, extended s and ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ pairing correlation functions in a generalized Hubbard model for the cuprates, for parameters appropriate for electron-doped systems, using numerical diagonalization of a $4\ifmmode\times\else\texttimes\fi{}4$ cluster. We find indications of d-wave superconductivity for small doping $(\ensuremath{\sim}0.1$ electrons per unit cell) and s-wave superconductivity for overdoped systems $(\ensuremath{\sim}0.5$ electrons per unit cell) or small U. The magnitude of the pairing correlation functions and the vertex contributions to them are in general much smaller than in the hole-doped case. We also present results for the spin-structure factor.
We consider an extended Hubbard model with nearest-neighbor correlated hopping and next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$ obtained as an effective model for cuprate superconductors. Using a generalized Hartree-Fock BCS approximation, we find that for high enough ${t}^{\ensuremath{'}}$ and doping, antiferromagnetism is destroyed and the system exhibits d-wave superconductivity. Near optimal doping we consider the effect of antiferromagnetic spin fluctuations on the normal self-energy using a phenomenological susceptibility. The resulting superconducting critical temperature as a function of doping is in good agreement with experiment.
The Hubbard and closely related t-J models are exciting platforms for unconventional superconductivity (SC). Through state-of-the-art density matrix renormalization group calculations using the grand canonical ensemble, we address open issues regarding the ground-state phase diagram of the extended t-J model on a square lattice in the parameter regime relevant to cuprate superconductors. On large 8-leg cylinders, we demonstrate that the pure t-J model with only nearest-neighbor hoppings and superexchange interactions, for a wide range of doping ( δ = 0.1 to 0.2 ), hosts robust d-wave superconductivity possibly coexisting with weak unidirectional pair density wave. Furthermore, a small next nearest neighbor hopping t 2 suppresses pair and charge density waves, resulting in a uniform d-wave SC phase in both electron- and hole-doped cuprate model systems. Our work validates the t-J model as a proper minimum model for the emergence of superconductivity in cuprate superconductors.
Extensive numerical studies have demonstrated that the two-dimensional single-band Hubbard model contains much of the key physics in cuprate high-temperature superconductors. However, there is no definitive proof that the Hubbard model truly possesses a superconducting ground state or, if it does, of how it depends on model parameters. To answer these longstanding questions, we study an extension of the Hubbard model including an infinite-range ddd-wave pair field term, which precipitates a superconducting state in the ddd-wave channel. Using exact diagonalization on 16-site square clusters, we study the evolution of the ground state as a function of the strength of the pairing term. This is achieved by monitoring the fidelity metric of the ground state, as well as determining the ratio between the two largest eigenvalues of the ddd-wave pair/spin/charge-density matrices. The calculations show a ddd-wave superconducting ground state in doped clusters bracketed by a strong antiferromagnetic state at half filling controlled by the Coulomb repulsion UUU and a weak short-range checkerboard charge ordered state at larger hole doping controlled by the next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$. We also demonstrate that negative ${t}^{\ensuremath{'}}$ plays an important role in facilitating ddd-wave superconductivity.
To clarify the effects of next-nearest-neighbor electron hopping ${t}^{\ensuremath{'}}$ on the spin, charge, and pairing correlations, we studied the two-dimensional $t\ensuremath{-}{t}^{\ensuremath{'}}\ensuremath{-}U$ Hubbard model using the constrained-path Monte Carlo technique. We found that a negative ${t}^{\ensuremath{'}}$ suppresses the spin fluctuations near the Brillouin zone point $M=(\ensuremath{\pi},\ensuremath{\pi})$ and the charge fluctuations near the point $X=(\ensuremath{\pi},0),$ while a positive ${t}^{\ensuremath{'}}$ has the opposite effect. For a negative ${t}^{\ensuremath{'}},$ we also found that the long-range vertex contributions to the pairing correlations are heavily suppressed, but the local contributions are enhanced. Over the entire hole doping range studied, the d-wave long-range vertex contributions to pairing dominated those of the extended s-wave vertex contributions. In general, we did not find any tendency for these correlations to grow with increasing system size. We also did a limited number of simulations with an attractive nearest neighbor density-density interaction added to the $t\ensuremath{-}{t}^{\ensuremath{'}}\ensuremath{-}U$ model. It enhanced the long-range vertex contributions to the pairing correlations only for weak values of U.
We investigate the influence of nonlocal interactions on superconducting correlations within the extended Hubbard model. In addition to the on-site Coulomb interaction and nearest-neighbor hopping, we include next-nearest-neighbor hopping together with several physically relevant nonlocal terms, namely nearest-neighbor Coulomb interaction, correlated hopping, exchange interaction, and pair hopping. Using Lanczos exact diagonalization on a 4×44\times44×4 cluster, supported by projector quantum Monte Carlo simulations for selected parameter regimes, we analyze pairing correlations in both the s- and d-wave channels. We demonstrate that nonlocal interactions exert a highly nontrivial and symmetry-dependent influence on superconducting correlations. While the on-site repulsion in cooperation with next-nearest-neighbor hopping enhances d-wave pairing tendencies, correlated hopping and nearest-neighbor Coulomb interaction strongly promote s-wave correlations, whereas exchange and pair-hopping interactions can efficiently suppress superconductivity beyond relatively small critical strengths. When all nonlocal interactions are considered simultaneously, the resulting phase diagrams reveal a complex interplay and competition between different pairing symmetries. Our results highlight the crucial role of extended interactions in shaping the pairing landscape of strongly correlated systems and demonstrate that a comprehensive treatment beyond the minimal Hubbard model is essential for a realistic description of unconventional superconductivity.