d\varepsilon\rho(\varepsilon) [ (\omega -
\varepsilon) \mathbb{I} - t_\perp \sigma_x -\mathbf{\Sigma}(\omega)]^{-1}
\label{eq:MFE}$$

where $\sigma_x$ is the $x$ Pauli matrix and $\Sigma_{\alpha,\beta}$ is the local self-energy. The calculation of the local self-energy requires the solution of the so-called local quantum impurity problem, which are often generalizations of the Kondo problem [@georges1996_dynamical_mean_field_theory_strongly]. To solve such a quantum impurity problem requires the numerical implementation of a many-body approach. Following previous work [@najera2017_resolving_vo2_controversy], here make extensive use of the Iterative Perturbation Theory (IPT) [@moeller1999_rkky_interactions_mott_transition] that allows fast and precise calculations in the whole phase diagram and at all $T$ and $T=0$. Notably, IPT becomes *exact* in the limits of $T=0$ and $t \to
0$ or $U \to 0$, *for all* $t_\perp$ (see Appendix B) and its numerical precision allows for reliable analytic continuation of the data to the real axis. We also benchmarked our IPT calculations with results from other numerical methods such as the continuous time quantum Monte Carlo (CT-QMC) [@werner2006_hybridization_expansion_impurity_solver; @seth2016_triqs] and the exact diagonalization (ED) [@georges1996_dynamical_mean_field_theory_strongly]. The former is a finite $T$ calculation and the latter is a $T=0$ one. These methods are both exact up to systematic errors, but are numerically very expensive and have the serious drawback of requiring, in 