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arXiv:2406.08780 · Physical review. B./Physical review. B

Mean-field study of superconductivity in the t − J square lattice model with three-site hopping

Ke Yang, Qianqian Chen, Lei Qiao, Zheng Zhu

Published Aug 15, 2024• 2 citations
arXivHTMLPDF

Abstract

It remains an open question whether the two-dimensional single-band pure Hubbard model and its related pure $t\text{\ensuremath{-}}J$ model truly capture the superconducting order in cuprates. Recent numerical studies on this issue have raised a notable disparity in superconducting order between the pure Hubbard model and the pure $t\text{\ensuremath{-}}J$ model. Inspired by these works, we investigate the role of the three-site hopping term in $d$-wave superconductivity, such a term is usually neglected in the effective Hamiltonian of the Hubbard model, although its amplitude is of the same order as the superexchange coupling $J$ in the $t\text{\ensuremath{-}}J$ model. Our slave-boson mean-field solution demonstrates the suppression of $d$-wave superconducting order by incorporating the three-site hopping term, consistent with numerical observations by the density-matrix renormalization group. This suppression could be understood as a result of competition between superexchange interaction and three-site hopping, the former favors $d$-wave pairing while the latter favors $s$-wave pairing. We also discussed its role in quasiparticle dispersion and boson-condensation temperature. Our findings may offer an alternative understanding of the recent numerical contrasting findings in the strong-coupling regime: the absent or weak superconductivity in the pure Hubbard model, while the robust superconductivity in the $t\text{\ensuremath{-}}J$ model without including the three-site hopping term.

DOI
10.1103/physrevb.110.054514

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It remains an open question whether the two-dimensional single-band pure Hubbard model and its related pure t\ensuremath-Jt\text{\ensuremath{-}}Jt\ensuremath-J model truly capture the superconducting order in cuprates. Recent numerical studies on this issue have raised a notable disparity in superconducting order between the pure Hubbard model and the pure t\ensuremath-Jt\text{\ensuremath{-}}Jt\ensuremath-J model. Inspired by these works, we investigate the role of the three-site hopping term in ddd-wave superconductivity, such a term is usually neglected in the effective Hamiltonian of the Hubbard model, although its amplitude is of the same order as the superexchange coupling JJJ in the t\ensuremath-Jt\text{\ensuremath{-}}Jt\ensuremath-J model. Our slave-boson mean-field solution demonstrates the suppression of ddd-wave superconducting order by incorporating the three-site hopping term, consistent with numerical observations by the density-matrix renormalization group. This suppression could be understood as a result of competition between superexchange interaction and three-site hopping, the former favors ddd-wave pairing while the latter favors sss-wave pairing. We also discussed its role in quasiparticle dispersion and boson-condensation temperature. Our findings may offer an alternative understanding of the recent numerical contrasting findings in the strong-coupling regime: the absent or weak superconductivity in the pure Hubbard model, while the robust superconductivity in the t\ensuremath-Jt\text{\ensuremath{-}}Jt\ensuremath-J model without including the three-site hopping term.
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.
The $t\ensuremath{-}J$ model is a standard model of strongly correlated electrons, often studied in the context of high-Tc{T}_{c}Tc​ superconductivity. However, most studies of this model neglect three-site terms, which appear at the same order as the superexchange JJJ. As these terms correspond to pair hopping, they are expected to play an important role in the physics of superconductivity when doped sufficiently far from half filling. We present a density matrix renormalization-group study of the one-dimensional $t\ensuremath{-}J$ model with the pair hopping terms included. We demonstrate that these additional terms radically change the one-dimensional ground-state phase diagram, extending the superconducting region at low fillings, while at larger fillings, superconductivity is completely suppressed. We explain this effect by introducing a simplified effective model of repulsive hardcore bosons.
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.
Proximity to a Mott insulating phase is likely to be an important physical ingredient of a theory that aims to describe high-temperature superconductivity in the cuprates. Quantum cluster methods are well suited to describe the Mott phase. Hence, as a step toward a quantitative theory of the competition between antiferromagnetism and ddd-wave superconductivity in the cuprates, we use cellular dynamical mean-field theory to compute zero-temperature properties of the two-dimensional square lattice Hubbard model. The ddd-wave order parameter is found to scale like the superexchange coupling JJJ for on-site interaction UUU comparable to or larger than the bandwidth. The order parameter also assumes a dome shape as a function of doping, while, by contrast, the gap in the single-particle density of states decreases monotonically with increasing doping. In the presence of a finite second neighbor hopping ${t}^{\ensuremath{'}}$, the zero-temperature phase diagram displays the electron-hole asymmetric competition between antiferromagnetism and superconductivity that is observed experimentally in the cuprates. Adding realistic third neighbor hopping ${t}^{\ensuremath{''}}$ improves the overall agreement with the experimental phase diagram. Since band parameters can vary depending on the specific cuprate considered, the sensitivity of the theoretical phase diagram to band parameters challenges the commonly held assumption that the doping vs Tc∕Tcmax{T}_{c}∕{T}_{c}^{\mathit{max}}Tc​∕Tcmax​ phase diagram of the cuprates is universal. The calculated angle-resolved photoemission spectrum displays the observed electron-hole asymmetry. The tendency to homogeneous coexistence of the superconducting and antiferromagnetic order parameters is stronger than observed in most experiments but consistent with many theoretical results and with experiments in some layered high-temperature superconductors. Clearly, our calculations reproduce important features of ddd-wave superconductivity in the cuprates that would otherwise be considered anomalous from the point of view of the standard Bardeen--Cooper--Schrieffer approach. At strong coupling, ddd-wave superconductivity and antiferromagnetism naturally appear as two equally important competing instabilities of the normal phase of the same underlying Hamiltonian.
We consider the slave-fermion Schwinger-boson decomposition of an effective model obtained through a systematic low-energy reduction of the three-band Hubbard Hamiltonian. The model includes a three-site term t ′′ similar to that obtained in the large- U limit of the Hubbard model but of opposite sign for realistic or large O-O hopping. For parameters close to the most realistic ones for the cuprates, the mean-field solution exhibits d + s superconductivity (predominantly d x 2 − y 2 ) with a dependence on doping x very similar to the experimentally observed. We also obtained incommensurate peaks at wave vectors ∼ π(1,1 ± 2 x ) in the spin structure factor, which also agree with experiment.
Having cuprate superconductors in mind, we study a band-renormalization effect on a coexisting state of antiferromagnetism (AF) and superconductivity (SC) in a t-J model on a square lattice with a diagonal transfer t ′ and three-site hopping terms, using a variational Monte Carlo method. It is found that this t-J model with t ′/ t = –0.3 exhibits coexistence of AF and SC orders in lightly doped regime and robust SC for a wide range of doping rate δ . This result contrasts with the previous results of a large- U/t Hubbard model, in which the two orders are mutually exclusive and a coexisting state does not appear; the SC state appears only in heavily doped regime (δ≳ 0.2), after the pure AF state vanishes at δ ∼ 0.2.