resonances in a $d$-wave superconductor.[@stavropoulos; @zah; @andersen03] We therefore consider a series of models for the microscopic nature of the Zn potential which might possibly account for the experimental observations, and propose additional experimental tests to distinguish between them.

Formalism
=========

The starting point of our study is given by the $d$-wave BCS Hamiltonian $${\mathcal{H}}_0=\sum_{{\mathbf{k}},\sigma} \xi_{\mathbf{k}}
\hat{c}^\dagger_{{\mathbf{k}}\sigma} \hat{c}_{{\mathbf{k}}\sigma}
+ \sum_{{\mathbf{k}}} \left( \Delta_{\mathbf{k}}
\hat{c}^\dagger_{{\mathbf{k}}\uparrow}
\hat{c}^\dagger_{-{\mathbf{k}}\downarrow} + {\rm H.c.} \right),$$ where $\xi_{\mathbf{k}}$ denotes the quasiparticle dispersion and $\Delta_{\mathbf{k}}=\frac{\Delta_0}{2}  (\cos k_x - \cos k_y)$ is the $d$-wave pairing gap. In terms of the Nambu spinor $\hat{\psi}^\dagger_{{\mathbf{k}}}=(\hat{c}^\dagger_{{\mathbf{k}}\uparrow},\hat{c}_{-{\mathbf{k}}\downarrow})$, the corresponding Matsubara Green’s function can be expressed as $$\label{nambugreensfunctions}
{\mathcal{G}}^{0}({\mathbf{k}},i\omega_n)=\frac{i\omega_n\tau_0+\xi_{\mathbf{k}}\tau_3
+\Delta_{\mathbf{k}}\tau_1}{(i\omega_n)^2-E_{\mathbf{k}}^2},$$ where $E_{\mathbf{k}}^2=\xi_{\mathbf{k}}^2+\Delta_{\mathbf{k}}^2$, and $\tau_i$ denote the Pauli matrices. In real-space, the perturbation due to $\delta$-function potential (magnetic) impurities of strength $V_3$ ($V_0$) in the diagonal $\tau_3$ ($\tau_0$) channel is given by $${\mathcal{H}}^{\prime}_{imp}({\mathbf{r}},{\mathbf{r}}')=\hat{\psi}^\dagger_{{\mathbf{r}}}
\left[ \left( V_3\tau_3 + V_0\tau_0 \right)
\delta({\mathbf{r}})\delta({\mathbf{r}}') \right]
\hat{\psi}_{{\mathbf{r}}'}\label{potH}.$$ Likewise, local modulations in the hopping ($\delta t$) or superconducting gap ($\delta \Delta$) enter as $${\mathcal{H}}^{\prime}_{\delta}({\mathbf{r}},{\mathbf{r}}')=\hat{\psi}^\dagger_{{\mathbf{r}}}
\left[  -\delta t({\mathbf{r}},{\mathbf{r}}') \tau_3 - \delta
\Delta({\mathbf{r}},{\mathbf{r}}') \tau_1 \right]
\hat{\psi}_{{\mathbf{r}}'}.$$

![(Color online) LDOS in a $d$-wave superconductor with a Ni impurity at the center of a 5$\times 5$ patch of (a-b) suppressed ($\delta \Delta=-0.05t$), (c-d) constant ($\delta \Delta=0$) 