= 20$. Here, and in all subsequent figures, when not shown, the error bars are smaller than the symbol size.[]{data-label="Fig:local_chi"}](local_chi_V120.pdf)

An interesting step towards understanding $f$-orbital depleted systems is to consider an isolated impurity. In fact, there have been a number of recent studies of the alteration of the magnetic structure around impurities in heavy fermion materials and their possible description within the PAM[@Urbano07; @Seo14; @dioguardi16; @Benali16; @Mendes17]. Hence, we first discuss how depletion of a single $f$-orbital affects magnetic and spectral properties, and compare with the uniform case of the undepleted PAM. Inclusion of a magnetic defect breaks translational symmetry. We therefore generalize Eq. to the local magnetic susceptibility at site $\mathbf{i}$ by $$\begin{aligned}
\chi_{\mathbf{i}}^{\alpha\gamma} = \sum_{\mathbf{j}\in \gamma} 
\int_{0}^{\beta} d\tau 
\langle S^{\alpha,\,-}_{\mathbf{i}}(\tau) S^{\gamma\,,+}_{\mathbf{j}}(0) 
\rangle,
\label{eq:localchi}\end{aligned}$$ where the sum runs over all sites ${\bf j}$ with orbital $\gamma$. The total susceptibility, Eq., is the sum of these local susceptibilities. $\chi^{\alpha\gamma}_{\bf i}$ would be probed experimentally via nuclear magnetic resonance and, indeed, such site- and orbital-specific NMR has been used to explore spin and charge patterns in doped heavy fermion[@Seo14; @Benali16] and iron-pnictide superconductors[@dioguardi16].

We analysed a $10\times 10$ lattice, using periodic boundary conditions, with the depleted site defining the origin 