of $t_\perp \to 0$ the two copies become independent and one recovers the conventional single site Hubbard model physics [@georges1996_dynamical_mean_field_theory_strongly; @moeller1999_rkky_interactions_mott_transition; @najera2017_resolving_vo2_controversy]. As an additional remark, here we should say that both the single site Hubbard and the Dimer Hubbard models at half filling have anti-ferromagnetic ground-states, which are favored in bipartite lattices. Nevertheless, the study of the MITs within the PM (meta-stable) states is important in its own right as it has been very useful to reveal the physical competition between different correlated states [@georges1996_dynamical_mean_field_theory_strongly].

![Schematic representation of the higher symmetry “rutile” lattice with one atom per unit cell and the low symmetry “monoclinic” lattice, which is a lattice of dimers. In shaded red, we represent the quantum impurity effective environment determined through the CDMFT equations. Notice that our model is defined on a semi-circular non-interacting density of states, that may be realized in a Bethe lattice. Nevertheless, here we depict a square lattice just for the sake of simplicity. \[fig0\]](Fig0.png){width="\linewidth"}

CDMFT equations and the bonding anti-bonding basis {#sec:org22c96f8}
--------------------------------------------------

The solution of the CDMFT equations are obtained in terms of the one-particle propagators $G_{\alpha,\beta}$ (with $\alpha, \beta=
1,2$), which are subject to the self-consistent condition

$${\bf G}(\omega)= \int 