Dynamical Cluster Approximation results for the effect of long range hoppings on $T_c$ in Cuprates

The Dynamical Cluster Approximation along with the Quantum Monte Carlo (QMC) algorithm are employed to study the effect of long-range hoppings on the superconducting critical temperature of Cuprates. A two-dimensional $t-t'-t''-$U Hamiltonian describes the physics of copper oxide planes in this model. We perform calculations on $4$-site and $16$-site clusters. The results show a weak dependence of the maximum $T_c$ on the long-range hoppings. We see a suppression of $T_c$ due to $t'$ in the hole-doped systems. $t'$ increases the critical doping (the doping beyond which the superconducting phase disappears) in the hole-doped regime, but this doping value is decreased by including $t''$.