The former is defined as $$\kappa = -\frac{1}{\rho^2}\frac{\partial \rho}{\partial \mu},
\label{eq:kappa}$$ where $\rho$ is the electronic density. Like the magnetic susceptibility, which can be measured either by the response of the magnetization to a small field or as a sum over spin correlation functions as in Eq., the compressibility can be obtained via evaluating the change in particle density induced by a shift in chemical potential via Eq. or by summing density correlation functions. We use the former approach here.

The conductivity, $\sigma_{dc}$, is evaluated as $$\sigma_{dc} = \frac{\beta^2}{\pi} \Lambda_{xx}(\mathbf{q=0}, \tau = \beta/2),
\label{eq:sigma_dc}$$ with $$\Lambda_{xx}(\mathbf{q}, \tau ) = 
\langle j_{x}(\mathbf{q}, \tau) j_{x}(-\mathbf{q}, 0)  \rangle \,\,.$$ $ j_{x}(\mathbf{q}, \tau) $ is the $\mathbf{q}$-$\tau$ dependent current in $x$ direction, the Fourier transform of $j_{x}(l) = -\mathrm{i} \sum_{l}t_{l
+ \hat{x}, l}^{\phantom{\dagger}} ( c^{\dagger}_{l+\hat{x},\sigma}
c_{l,\sigma}^{\phantom{\dagger}} - c^{\dagger}_{l,\sigma}
c_{l+\hat{x},\sigma}^{\phantom{\dagger}} )$. The assumptions involved in the use of Eq. to evaluate the conductivity are discussed in Refs.\[\] and are tested there for a variety of situations.

Single depletion {#Single depletion}
================

![(Color online) Local susceptibility of $f$-orbitals as a function of distance $|\mathbf{i}|$ from the ion defect. The dashed black line is the value of the magnetic susceptibility in the undepleted PAM at $\beta t 