symmetries[@li16], owing to the minus-sign problem[@loh90; @troyer05]. For this reason, our focus is on half-filling, $\mu = \epsilon^{f} = \epsilon^{d} = 0$, where the sign problem is absent. Fortunately, this density is of considerable interest, both because of the strong magnetic order favored by commensurate filling, and by the materials for which half-filling is appropriate (e.g. the undoped parent compounds of the cuprate superconductors). Depleting $f$-orbitals, i.e. removing them from the lattice, preserves particle-hole symmetry (PHS). This is true regardless of the number or pattern of the removed sites, in much the same way that PHS is present for arbitrary (including position-dependent) choices of the energy scales $t, V$, and $U_f$, as long as the hopping only connects sites on opposite sublattices of a bipartite lattice.

We concentrate on the following observables: The magnetic features of the Hamiltonian will be characterized by the real space spin-spin correlation function, $$\begin{aligned}
C^{\alpha \gamma}({\bf j}) = 
\langle 
S^{\alpha,\, -}_{{\bf j_0}+{\bf j}} 
S^{\gamma,\, +}_{{\bf j_0}} 
\rangle 
=
\langle 
c^{\alpha \,\,\dagger}_{{\bf j_0}+{\bf j} \, \downarrow} 
c^{\alpha \phantom{\dagger}}_{{\bf j_0}+{\bf j} \, \uparrow} 
c^{\gamma \,\,\dagger}_{{\bf j_0} \, \uparrow} 
c^{\gamma \phantom{\dagger}}_{{\bf j_0} \, \downarrow} 
\rangle \,\,,
\label{eq:Cspin}\end{aligned}$$ where the orbital indices are $\alpha, \gamma = d,f$. (Later in 