stationarity at $\epsilon_c = 0$ is consistent with the requirements of particle-hole symmetry. For any $\epsilon_c \ne 0$ the impurity model is asymmetric. This can be seen from the inset where the average occupations of the impurity and of the bath site are plotted as functions of $\epsilon_c$. The total particle number $\langle N' \rangle = \sum_\sigma 
(\langle c_{\sigma}^\dagger c_{\sigma} \rangle + 
\langle a^\dagger_{\sigma} a_{\sigma}\rangle) = n_{\rm c} + n_{\rm a}$ ($i$ and $k=2$ fixed) is constant: $\langle N' \rangle = 2$.

With respect to the third variational parameter $\epsilon_a$, the grand potential $\Omega$ is at a maximum for $\epsilon_a = 2 = \mu$, see Fig.  \[fig:e3\]. Again, this value is required by particle-hole symmetry. If $\epsilon_a$ exceeds a certain critical value (away from the stationary point), the ground-state of the reference system $H'$ no longer lies within $N'=2$ subspace but is found in the $N'=1$ or $N'=3$ subspace, respectively (see inset of Fig. \[fig:e3\]). While $\Omega'$ is continuous at the level crossing, the symmetry of the ground state changes. Consequently, there is a discontinuous change of the trial self-energy which implies a discontinuous change of $\Omega$.

Consider now the [*original*]{} model at slightly modified parameters, e.g. $U$, $\mu$, or 