$T$. Clearly, the stationary point of $\Omega$ will be expected then at slightly different values of the variational parameters $V$, $\epsilon_c$, $\epsilon_a$. This implies that all physical quantities which derive from the thermodynamical potential $\Omega$ will be continuous functions of the (original) model parameters in general – irrespective of the fact that the reference system includes a finite number of degrees of freedom only: It is a typical feature of any mean-field approach that results are directly provided for the thermodynamical limit. A discontinuous jump of $\Omega$ due to a symmetry change of the ground state of the reference system (see Fig. \[fig:e3\]) usually occurs away from stationarity and is thus irrelevant physically. It is conceivable, however, that the stationary point moves to a point of discontinuity as a function a parameter of the original model. In this case the approach would generate an artifact which is a reminiscence of the finiteness of $H'$.

![ $U$ dependence of the quasi-particle weight $z = 1 / (1-d \Sigma(\omega = 0)/d\omega)$ within different approximations. BR: Brinkman-Rice (Gutzwiller) approach. [@BR70] 2-site DMFT: a non-variational two-site approach. [@Pot01] DIA, $n_{\rm s}=2$: self-energy-functional approach with a reference system $H'$ consisting of decoupled two-site impurity models. 