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The Hubbard and related models serve as a fundamental starting point inunderstanding the novel experimental phenomena in correlated electronmaterials, such as superconductivity, Mott insulator, magnetism and stripephases. Recent numerical simulations indicate that the emergence ofsuperconductivity is connected with the next nearest-neighbor hopping$t^\prime$ in the Hubbard model. However, the impacts of complex inter-siteelectron interaction in the $t^\prime$-Hubbard model are less explored.Utilizing the state-of-art density-matrix renormalization group method, weinvestigate the $t^\prime$-Hubbard model on a two-leg ladder with inter-siteinteractions extended to the fourth neighbor sites. The accurate numericalresults show that the quasi-long-range superconducting correlation remainsstable under the repulsive nearest-neighbor and the next nearest-neighborinteractions though these interactions are against the superconductivity. Theground state properties are also undisturbed by the longer-range repulsiveinteractions. In addition, inspired by recent experiments on one-dimensionalcuprates chain $\mathrm{Ba}_{2-x}\mathrm{Sr}_x\mathrm{CuO}_{3+\delta}$, whichimplies an effective attraction between the nearest neighbors may exist in thecuprates superconductors, we also show that the attractive interaction betweenthe nearest neighbors significantly enhances the superconducting correlationwhen it is comparable to the strength of the nearest-neighbor hopping$t^\prime$. Stronger attraction drives the system into a Luther-Emery liquidphase. Nevertheless, with the attraction further increasing, the system entersan electron-hole phase separation and the superconducting correlation isdestroyed. Finally, we investigate the effects of on-site Coulomb interactionon superconductivity.

Robust superconducting correlation against inter-site interactions in the extended two-leg Hubbard ladder