Suppression of $d$-wave superconductivity in the weakly inhomogeneous checkerboard Hubbard Model

Using a dynamical cluster quantum Monte Carlo approximation we investigate the $d$-wave superconducting transition temperature T$_{c}$ of the doped 2D Hubbard model with a weak inhomogeneity in the form of checkerboard pattern in the hoppings. The hopping within a 2 $\times $ 2 cluster (plaquette) is $t$ and the hopping between the plaquettes is $t'$ ( 0.8$t \le $ \textit{t' $\le $ t }). We find T$_{c}$ decreases monotonically with decreasing $t' $ for both fixed $U/t$ or $U/W$ ($U$ the on site Hubbard interaction and $W$ the bandwidth). The characteristic spin excitation energy scale and the strength of $d$-wave pairing interaction decrease with decreasing T$_{c}$ suggesting a strong correlation between these two quantities.