Fig. \[fig:R2B\]). We will use this quantity to describe the metal to insulator transition, in the next section.

![Schematic representation of the two bands in the R2B model. This model is a simple renormalization of the non-interacting case where $Z=1$ and the intra-dimer hopping is $\tilde{t}_\perp=t_\perp$. $ZD$ denotes the renormalization of bandwidth and $2\eta$ the overlap of the two bands. \[fig:R2B\]](figR2B.pdf){width="\linewidth"}

The renormalized two-band (R2B) model may describe both metallic and insulating states, so long the $\Sigma$ remains well-behaved (i.e. linear) according to the parametrization. As it turns out the description will be valid at low frequencies throughout the metallic phase, which is a Fermi liquid. In the insulator, we shall see that it is a good approximation only within the Peierls limit, where the interaction and thus the $\Sigma$ are small. In the Mott regime this parametrization is not appropriate.

Phase diagram and insulator-metal transitions {#sec:orgcd05f74}
=============================================

In Fig. \[fig:phasediag\] we show the $U-t_\perp$ phase diagram at $T=0$, which is well-known from previous studies [@moeller1999_rkky_interactions_mott_transition; @hafermann2009_metal_insulator_transition_by_suppression_spin_fluctuations; @fuhrmann2006_from_mott_insulator_to_band_insulator; @najera2017_resolving_vo2_controversy]. We recall now its main features. There is a metallic phase for $t_\perp <
1$, and an insulator phase at high enough $U$. The IMT changes character depending on the value 