the paper, we will use an alternate notation which further distinguishes the two types of conduction electron orbitals, those with a partner $f$ orbital and those for which the partner has been removed.) As the notation suggests, $C^{\alpha \gamma}({\bf j}) $ is independent of ${\bf j_0}$ for translationally invariant geometries. The Fourier transform of $C^{\alpha \gamma}({\bf j})$ is the magnetic structure factor, $$\begin{aligned}
S^{\alpha \gamma}({\bf q}) = 
%% \frac{1}{N} \sum_{\bf j} c^{\alpha \gamma}({\bf j})
\sum_{\bf j} C^{\alpha \gamma}({\bf j})
\, e^{i {\bf q} \cdot {\bf j}}. 
\label{eq:Sq} \end{aligned}$$ In addition to these equal time correlation functions, we also measure appropriate unequal time quantities including the magnetic susceptibility, $$\begin{aligned}
%% \chi^{\alpha\gamma}({\bf q}) = \frac{1}{N} \sum_{\mathbf{i} \mathbf{j}} 
\chi^{\alpha\gamma}({\bf q}) =  \sum_{\mathbf{j}} 
\int_{0}^{\beta} d\tau \langle 
S^{\alpha,\, -}_{{\bf j_0}+{\bf j}}(\tau)
S^{\gamma,\, +}_{{\bf j_0}}(0)
%% S^{z}_{\mathbf{i}}(\tau) S^{z}_{\mathbf{j}}(0) 
\rangle
\, e^{i {\bf q} \cdot {\bf j}} \, ,
\label{eq:hom.chi}\end{aligned}$$ For $\chi$ we will mostly examine the uniform case, ${\bf q}=0$. Although we have defined both the equal time correlations, Eq., and susceptibility, Eq., in terms of the $xy$ ($+-$) spin components, these are, by symmetry, equivalent to those in the $z$ direction.

Metal-insulator transitions are characterized via the electronic compressibility ($\kappa$) and the dc conductivity ($\sigma_{dc}$). 