the case of CT-QMC, the analytic continuation of the (noisy) imaginary axis results.

In the present case of half-filling (i.e., one particle per site), particle-hole symmetry holds. We further assume translational invariance and search for a paramagnetic solution. We have then: $G_{11}(\omega)=G_{22}(\omega)$, and $G_{12}(\omega)=G_{21}(\omega)$. The self-energies have similar properties. In order to simplify the discussion of the evolution of the electronic structure, it is convenient to consider the B/AB representation that renders the Green’s Functions and Self-energies *diagonal*.

\[eq:diagonal\_bands\] $$\begin{aligned}
\label{eq:diagonal_gf}
G_{B/AB} &= G_{11}\mp G_{12}\\
\label{eq:diagonal_sigma}
\Sigma_{B/AB} &= \Sigma_{11}\mp \Sigma_{12}\end{aligned}$$

In the B/AB basis the electronic structure of the non-interacting problem is particularly simple. The single particle energies $E_k^{B/AB}$ form two parallel bands of bandwidth $2D$ and are split by $2t_\perp$, i.e., $E_k^{B/AB} = \mp t_\perp + \epsilon_k$, where $\epsilon_k$ is the single particle energy of the bands for $t_\perp = 0$. In the present case, since we employ a semi-circular DOS (realized in a Bethe lattice in infinite dimensions) the lattice single particle energies $\epsilon_k$ drop their $k$-dependence [@georges1996_dynamical_mean_field_theory_strongly] and then are simply labeled by $\epsilon \in [-D,D]$ [@georges1996_dynamical_mean_field_theory_strongly]. Thus, at finite $t_\perp$ and $U=0$, the model DOS is composed of two semicircles split by $2t_\perp$, $\rho_{B/AB}(\omega) = \frac{2}{\pi 