$-(1/\pi)\mbox{Im}{\bf G}(\omega+i0^+)$ at $\omega=0$. The minima at $V=\pm 0.519$ correspond to a metallic phase. $\Omega$ as well as the different contributions are symmetric functions of $V$. As ${\Sigma}(V,\omega)={\Sigma}(-V,\omega)$, however, this symmetry is trivial and does not yield an additional physical phase (see also Appendix \[sec:gofs\]). Due to the lower $\Omega$ the metallic phase is stable as compared to the insulating one.

![ $\Omega$ as a function of $\epsilon_{\rm c}$ for $V=0.519$ (metal) and $\epsilon_{\rm a}=2$. $U=4$. Inset: impurity- and bath-orbital filling, $n_{\rm c}$ and $n_{\rm a}$, as functions of $\epsilon_{\rm c}$. []{data-label="fig:e2"}](fig3.eps){width="78mm"}

![ $\Omega$ and the impurity- and bath-orbital filling, $n_{\rm c}$ and $n_{\rm a}$, as functions of $\epsilon_{\rm a}$ for $V=0.519$ and $\epsilon_{\rm c}=0$. $U=4$. []{data-label="fig:e3"}](fig4.eps){width="78mm"}

The minimum at $V=0.519$ is actually a saddle point if the entire space of variational parameters is considered. This is demonstrated by Fig. \[fig:e2\] which shows $\Omega$ as a function of $\epsilon_c$ for fixed $V=0.519$ and $\epsilon_a=2$. While $\Omega(V)$ is at a minimum for $V=0.519$, $\Omega(\epsilon_c)$ is at a [*maximum*]{} for $\epsilon_c=0$. In the $(V,\epsilon_c)$ space one therefore encounters a saddle point. As already noted in Sec. \[sec:sft\], there is no reason to expect an extremum in general. It is worth mentioning that 