{D}^2}
\sqrt{{D}^2-(\omega\pm {t}_{\perp})^2}$.

The renormalized two band model {#sec:orgbfa56e3}
-------------------------------

In order to better analyze our results in the subsequent sections, it is convenient to introduce here a simple low-energy parametrization of the two-band system. We may think of this as a non-interacting renormalized two-band model (R2B). In a normal (i.e. Fermi liquid) metal the self-energy is well-behaved at low frequencies[^1]. In the present case we expand around $\omega =0$ the self-energies of the mean-field equations , and introduce a quasiparticle residue $Z$:

$$\label{eq:Z}
Z^{-1} = 1-\frac{\partial\Re \Sigma_{B/AB} (\omega)}{\partial \omega} \bigg |_{0} =
1-\frac{\partial\Re \Sigma_{11} (\omega)}{\partial \omega} \bigg |_{0}$$

and a renormalized intra-dimer hopping $t_\perp$ as:

$$\label{eq:tpeff}
\tilde{t}_\perp = t_\perp \mp \Re \Sigma_{B/AB}(\omega) |_{0} = t_\perp + \Re \Sigma_{12}(\omega) |_{0}$$

The DOS of the R2B model then reads,

$$\label{eq:low_energy_dos}
\rho^{R2B}_{B/AB}(\omega) \sim \frac{2}{\pi D^2}\,
\sqrt{D^2 - \big( \frac{\omega}{Z} \pm \tilde{t}_{\perp}\big)^2}$$

that corresponds to two heavy effective bands with dispersion $E_\epsilon^{B/AB} = \mp Z\tilde{t}_\perp + Z\epsilon$, where the effective mass renormalization is $m^*/m = 1/Z$. The overlap between the two band-edges is given by $2\eta$, where

$$\eta = ZD - Z\tilde{t}_\perp .
\label{eq:transition_orderparam}$$

Thus, for $\eta>0$ we have a metal state, and for $\eta<0$(i.e. splitting dominates) the DOS opens a gap (see 