a possible reference system $H'$ (c). See text for discussion. []{data-label="fig:hhp"}](fig1.eps){width="90mm"}

Let us discuss the case $n_{\rm s} \mapsto \infty$. For a homogeneous phase of the (translationally invariant) original system, $\Omega_{\bf t}[{\bf \Sigma}({\bf t}')]$ will be stationary at a homogeneous set of variational parameters: ${\bf t}'=\{ \epsilon^{(c)}_{i}, \epsilon^{(a)}_{i,k}, V_{i,k} \}=
\{ \epsilon^{(c)}, \epsilon^{(a)}_{k}, V_{k}\}$. Consequently, it is sufficient to consider one SIAM only. As the different equivalent SIAM’s are spatially decoupled, not only the self-energy but also its linear response is local: $\partial \Sigma_{ij}({\bf t}') / \partial {\bf t}' \propto \delta_{ij}$. To solve the Euler equation (\[eq:euler\]), it is thus sufficient to fulfill the “locally projected” equation $$\left( \frac{1}{{\bf G}_0^{-1}(i\omega) 
   - {\bf \Sigma}(i\omega)} \right)_{ii}
   = G'_{ii}(i\omega) \: .
\label{eq:dmft}$$ This is just the self-consistency equation of the DMFT: [@GKKR96] The SIAM parameters have to be found such that the on-site (“impurity”) Green’s function at a correlated site $i$ coincides with the on-site Green’s function of the Hubbard model which is calculated from ${\bf G}_0$ and the (“impurity”) self-energy of $H'$ by means of the Dyson equation. Therefore, one can state that the DMFT (as an approximation for any finite-dimensional system or as the exact theory in infinite dimensions) is recovered 