nearest pairs of sites along the $x$ (or $y$) direction, as a function of the hybridization $V/t$. The sign of the correlations $C^{\alpha\gamma}(\mathbf{r})$ is always positive for {$\alpha \gamma
$}={$d_{0}d_{0}$}, {$d_{1}d_{1}$}, {$ff$} and {$d_{0}f$}, and negative for {$d_{0}d_{1}$} and {$d_{1}f$} even at larger $r$ (not shown). This is consistent with Shen’s theorem[@Shen96] for the KLM, which asserts that, on bipartite lattices, $C^{\alpha \gamma}(\mathbf{r})$ is always positive for sites on the same sublattice, and always negative for sites on different sublattices.

![(Color online) Homogeneous spin structure factor $S({\bf q}=0)$ versus temperature for $V/t=1.6$ and $U_{f}/t=4.0$.[]{data-label="Fig:SF"}](SF_total_V16.pdf)

In the undepleted case, short range spin correlations decline in magnitude upon crossing the AF-singlet quantum critical point at $V_c
\sim t$; e.g., see Ref.. Figure \[Fig:Corr\_NN\], which resolves the spin correlations by orbital type, is useful in isolating the origin of the long range order which we will show to exist, later in this Section. In particular, as seen in the figure, some of the short range correlations grow as $V/t$ increases, contrary to the behavior in the regular PAM. For small hybridization, $C^{ff}(1)$ dominates the other correlations; it is almost two orders of magnitude larger than $C^{d_{0}d_{0}}(1)$, for example. However, $C^{d_{0}d_{0}}(1)$ is strongly enhanced 