disconnected from the rest of the system, by (ii) switching off the hopping $t_{i,j}$ between the correlated sites and (iii) switching on a hopping $V_{i,k}$ to the bath sites. After step (i) the Hamiltonian Fig. 1b (in the figure $n_{\rm s}=5$) has an enlarged Hilbert space but the same self-energy. It is important to note that steps (i) - (iii) leave the interaction part unchanged and thus preserve the functional dependence $F[{\bf \Sigma}]$. Actually, the system $H'$ is a set of $L$ decoupled single-impurity Anderson models (SIAM) [@And61] with $n_{\rm s}$ sites each. Compared to $H$, the problem posed by $H'$ is strongly simplified. This is achieved at the cost of restricting the set of trial self-energies. In particular, as the correlated sites are decoupled in $H'$, the trial self-energies are local: $\Sigma_{ij}(i\omega,{\bf t}') \propto \delta_{ij}$. One has to consider $H'$ for arbitrary one-particle parameters, namely the on-site energies of the correlated (“c”) and of the bath sites (“a”), $\epsilon^{(c)}_{i}$ and $\epsilon^{(a)}_{i,k}$, respectively, and the hopping (“hybridization”) $V_{i,k}$ between them and take these as variational parameters in the principle $\delta \Omega_{\bf t}[{\bf \Sigma}({\bf t}')] = 0$.

![ Schematic representation of the Hubbard model $H$ (a), an equivalent model (b), and 