Mean Field Study of Superconductivity in the Square Lattice t-J Model with Three-Site Hopping

It remains an open question whether the two-dimensional single-band pure Hubbard model and its related pure t-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-J model. Inspired by these, 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, though its amplitude is of the same order as the superexchange coupling J in the t-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-J model without including the three-site hopping term.