model parameter space should also shed light for the classification of a variety of monoclinic transition metal oxide systems with the MO$_2$ formula [@hiroi2015_structural_instability_rutile_compounds] and the non-magnetic insulator states of other vanadates.

This paper is organized as follows: In Section II we introduce the DHM and the DMFT equations. We also describe a simple parametrization of a renormalized two band model that will be useful for the discussion of our results. In Section III we present the phase diagram and discuss the various insulator-metal transitions of the model. In particular, we describe the destruction of the correlated metal and the insulator in the Mott regime. In Section IV we present the detailed study of the Mott to Peierls crossover. We characterize the several distinct physical regimes, including one with mixed coherent and incoherent features in the Hubbard bands possibly relevant for VO$_2$. Section V is dedicated to the conclusions of our work.

The dimer Hubbard model {#sec:org34ebb63}
=======================

We focus on the dimer Hubbard Model, which is a basic and natural extension of the single band Hubbard Model.

The DHM reads,

$$\label{eq:dimer-hubbard-model}
\begin{aligned}
H=& \left[-t \sum_{\langle i, j\rangle \alpha\sigma}
c^\dagger_{i\alpha\sigma} c_{j\alpha \sigma} +\ t_{\perp}
\sum_{i\sigma} c^\dagger_{i1\sigma} c_{i2\sigma} + H.c. \right] 