as $V/t$ increases, while $C^{ff}(1)$ decreases. By the time $V/t \sim 2.4$ they are roughly equal. Meanwhile, $C^{d_{1}d_{1}}(1)$ remains small at all $V$. These results suggest $d_{0}$-sites play an important role in the magnetic correlations in the ground state. Fig.\[Fig:Corr\_NN\] also indicates that antiferromagnetic correlations are present between neighboring $d_{0}$ and $d_{1}$-sites \[$C^{d_{0}d_{1}}(0.5)$\], and $d_{1}$ and $f$-sites \[$C^{d_{1}f}(1)$\]. These slowly decrease with $V$. Finally, $C^{d_{0}f}(0.5)$ exhibits a large and almost constant value, indicating it too contributes substantially to ground state magnetism.

![(Color online) Finite-size scaling of the total ferromagnetic structure factor. A nonzero extrpolation to $N_{\rm cell}^{-1/2} \rightarrow 0$ indicates the presence of long range order for all the $V/t$ values shown. Finite size corrections are largest for small $V/t$ and for large $V/t$; the effective exchange coupling $J_{\rm eff}$ becomes small in both limits. See text. []{data-label="Fig:Scaling"}](Scaling.pdf)

According to Eq., the total ferromagnetic spin structure factor normalized by the number of sites is defined as $S(0) = \frac{1}{3} \sum_{\alpha \gamma} S_{\alpha \gamma}$, with $$\begin{aligned}
\label{Sq_ab}
S_{\alpha \gamma} = \frac{1}{N_{cells}} \sum_{\mathbf{i}
\mathbf{j}} \langle S^{z(\alpha)}_{\mathbf{i}}
S^{z(\gamma)}_{\mathbf{j}} \rangle \,.\end{aligned}$$ As earlier, $\alpha$ and $\gamma$ label the sites $d_{0}$, $d_{1}$ and $f$, and the sums over $\mathbf{i}$ and $\mathbf{j}$ are restricted to their 