of $t_\perp$. At values higher than $t_\perp \approx
0.7$ the transition is continuous (2nd order) along a line $U_{c_3}$ indicated in green in Fig. \[fig:phasediag\]. At smaller values of $t_\perp$ there are two lines $U_{c_1}$ and $U_{c_2}$ respectively indicated in blue and red in the figure. These are two spinodal lines of the mean field theory self-consistent solution. The metal state is destabilized for $U>U_{c_2}$ whereas the insulator state is destabilized along $U<U_{c_1}$. Thus, in between the two spinodal lines there are two different solutions of the CDMFT equations, one metallic and one insulating. At finite temperature this coexistence region shrinks, until it disappears at a critical temperature. At higher $T$ there is a crossover behavior and bad metal states [@najera2017_resolving_vo2_controversy]. This phase diagram is obtained with IPT but we have validated all its main features by extensive CT-QMC calculations [@najera2017_resolving_vo2_controversy].

The description of how these transitions take place in this basic model of strongly correlated systems has (rather surprisingly) not been investigated in detail. As we discussed before, this type of impurity model is at the core of calculations of realistic material with a dimer in the unit cell. We shall therefore describe the transitions in detail in the 