for increasing values of $U$ and $t_\perp=0.3$ within the coexistence region. Metallic solutions are shown on left panels and insulating ones on the right. Black lines represent the local($-\Im m G_{11}$) Spectral function, which is the average of the bonding (dashed blue) and anti-bonding (dot dashed orange) bands, calculated at $T=0$ with IPT. \[fig:DOS\_coex\]](dimer_tp0.3_coex_spectra.pdf){width="\linewidth"}

We have described above how the metallic solution collapses discontinuously as one increases the interaction $U$. Here we shall consider the collapse of the insulator one as we come down from high $U$ towards $U_{c_1}$. The systematic behavior of the DOS is shown in Fig. \[fig:DOS\_ins\_met\] for two representative values of $t_\perp$. A smaller value $t_\perp=0.4$ where the system crosses the $U_{c_1}$ line and, for comparison, a larger value $t_\perp=0.6$ which is closer to the continuous transition line $U_{c_3}$.

![DOS for decreasing values of $U$ as the system crosses the $U_{c_1}$ transition line (blue line of Fig.\[fig:phasediag\]). Left panel is away from the tri-critical point and right panel data is close to it. Black lines represent the local($-\Im m G_{11}$) Spectral function, which is the average of the bonding (dashed blue) and anti-bonding (dot dashed orange) bands, calculated at $T=0$ with IPT. Note the discontinuous behavior 