We study the effect of long-range hoppings on Tc for the two-dimensional (2D) Hubbard model with and without Holstein phonons using parameters evaluated from band-structure calculations for cuprates. Employing the dynamical cluster approximation (DCA) with a quantum Monte Carlo (QMC) cluster solver for a 4-site cluster, we observe that without phonons, the long-range hoppings, t′ and t′′, generally suppress Tc. We argue that this trend remains valid for larger clusters. In the presence of the Holstein phonons, a finite t′ enhances Tc in the under-doped region for the hole-doped system, consistent with local-density approximation (LDA) calculations and experiment. This is interpreted through the suppression of antiferromagnetic (AF) correlations and the interplay between polaronic effects and the antiferromagnetism.
Ehsan Khatami, A. Macridin, and M. Jarrell. "Effect of long-range hopping on Tc in a two-dimensional Hubbard-Holstein model of the cuprates" Physical Review B (2008). doi:10.1103/PhysRevB.78.060502