Suppression of $d$-wave superconductivity in the checkerboard Hubbard model

Using a dynamical cluster quantum Monte Carlo approximation, we investigate the $d$-wave superconducting transition temperature $T_c$ in the doped two-dimensional repulsive Hubbard model with a weak inhomogeneity. The inhomogeneity is introduced in the hoppings $t^{\prime }$ and $t$ in the form of a checkerboard pattern where $t$ is the hopping within a $2\times 2$ plaquette, and $t^{\prime }$ is the hopping between the plaquettes. We find inhomogeneity suppresses $T_c$. The characteristic spin excitation energy and the strength of $d$-wave pairing interaction decrease with decreasing $T_c$, suggesting a strong correlation between these quantities.

Figure 1

(Color online) $2\times 2$ plaquette model. Here $t$ is intracluster hopping and $t^{\prime }$ is intercluster hopping.

Figure 2

(Color online) (a) $T_c\left(t^{\prime }\right)/W$ for $U/W=1$ and at 10% doping. $T_c$ is suppressed for $t^{\prime }\ne t$. (b) $T_c\left(t^{\prime }\right)/t$ for $U=6t$ and at 8% doping. $T_c$ monotonically decreases with decreasing $t^{\prime }$. (c) $T_c/t$ for $t^{\prime }=t$ and $t^{\prime }=0.88t$ as a function of doping $\delta$ when $U=8t$. (d) Low-energy density of states $N\left(\omega \right)t$ for $t^{\prime }=t$, and $t^{\prime }=0.88t$ at $T=0.11t$ for 8% doping and $U=8t$. Note the increase in low-energy spectral weight for $t^{\prime }=0.88t$.

Figure 3

(Color online) (a) Normalized values of $T_c$, $V_d$, and $P_{d0}$ as a function of $t^{\prime }$ at 8% doping, $U=6t$, and $N_c=4$. $V_d$ and $P_{d0}$ are described in the text. (b) Néel temperature $T_N/t$ for different values of $t^{\prime }$ at 2% doping for $U=8t$.

Figure 4

(Color online) (a) Magnetic structure factor $S\left(Q,\omega \right)$ for $U=6t$, $\delta =0.08$, and temperature $T=0.087t$ at $Q=\left(0,\pi \right)$ for different values of $t^{\prime }$. The location of the peak is a measure of the characteristic spin excitation energy, here defined as $J_{\text{eff}}$. The inset of (a) is blown up to show the area around peak positions. $T_c\left(t^{\prime }\right)$ and $J_{\text{eff}}\left(t^{\prime },T=0.087t\right)$ are normalized to their values at $t^{\prime }=t$, as a function of $t^{\prime }$ for (b) $U=6t$ and $\delta =0.08$, (c) $U=8t$ and $\delta =0.08$, and (d) $U=8t$ and $\delta =0.15$. Note that both normalized $T_c\left(t^{\prime }\right)$ and $J_{\text{eff}}\left(t^{\prime }\right)$ monotonically increase with $t^{\prime }$.