the B and AB bands start to overlap and realize a transition into a metal.

Before proceeding, we should avoid any confusion here by noting that the model is fully defined only after its lattice is specified. For instance, if dimers are arranged as a one dimensional system, the model is a “ladder” (the dimer rungs are perpendicular to the direction of the lattice). In two dimensions, as in the schematic Fig. \[fig0\], one would get a “bi-layer” model, where the dimer rungs connect at every site two parallel 2D layers. Those systems have qualitatively different behaviors from the one that concerns us here, namely, the physics of the three-dimensional systems such as VO$_2$. In 3D systems that have strong local interactions, one may expect the Dynamical Mean Field Theory (DMFT) to be a reasonable approximation [@georges1996_dynamical_mean_field_theory_strongly]. In fact, DFT+DMFT methods are implicitly based on such an assumption. The DMFT approach to the DHM with a dimer unit cell is, strictly speaking, a cluster-DMFT calculation, possibly the simplest instance of CDMFT. In DMFT and CDMFT it is mathematically convenient and thus customary to adopt a non-interacting semi-circular density of states (DOS), which is realized in a Bethe lattice [@georges1996_dynamical_mean_field_theory_strongly]. In 