12 0 0.25 0.5 0.75 1 1.25 1.5 0 10 20 30 A( ) BA BA+DX SCBA SCBA+DX 0 0.25 0.5 0.75 1 1.25 1.5 0 5 10 15 20 25 A($^{ω}$ ) SCBA full SC(BA+DX) from G$_{SCBA}$ full SC(BA+DX+DPH) from G$_{SCBA}$ full SC(BA+DPH) from G$_{SCBA}$ by the results given by a second-order correction to a self-consistent Hartree-Fock (SCBA) calculations (SCBA+DX curve in figure 10) G r $_{partial SC}$( ω ) = [ G r $_{SCBA}$( ω ) − 1 − Σ DX $^{,r}$[ G$_{SCBA}$ ] ] − $^{1}$. This is an interest- ing result as it implies that the physical properties of the system, at least in these conditions, could be well de- scribed by a second-order correction of the lowest-order SC calculations, without the need to perform a fully self- consistent calculation up to the second-order. Now we turn to the analysis of results obtained for stronger electron-vibron coupling. Examples of such cal- culations are given in figure 11 for γ$_{0}$/ω$_{0}$ = 0 . 80. In the strong electron-vibron coupling regime, one obtains qual- itatively the same contributions of the second-order dia- grams as explained above for the weak coupling regime. The DPH diagram does not have a large role in the off- resonant transport regime, although slightly affecting the width of the main peak. And the DX diagram shifts the vibron side-band peak towards the main peak (and hence towards the exact results) as well as narrowing the peak width and increasing the peak amplitude. These effects are amplified in Figure 11 because the electron-vibron coupling constant γ$_{0}$ is bigger than in Figure 10. Fur- thermore, for strong electron-vibron coupling, the DX diagram modifies the energy position of the main peak and seems to give a slightly better polaron shift. There is however another interesting effect observed from such a set of calculations, which can be seen in the upper panel of Figure 11. The first iteration of a fully self-consistent calculation (including the Hartree, Fock, and DX diagrams) starting with the non-interacting GFs G$_{0}$ gives non physical results, i.e. negative values of the spectral function (see curve BA+DX in figure 11), and such an unphysical behaviour does not self-correct in the following iterations. This a well known problem, which has already been encountered in the past by sev- eral authors in the context of electron-vibron interac- tion (for example in Ref [78]) and also in the context of electron-electron interaction when considering topo- logically equivalent diagrams $^{79,8094}$. In Ref. [78], a some- what different approach than ours was used. It is based on a linked cluster expansion for the non-equilibrium steady-state regime, and negative densities of states were obtained when including second-order diagrams, and in some cases even higher-order cluster approximations did not seem to give a convergent solution at intermediate electron-vibron strength and in the presence of the Fermi seas. FIG. 11: Equilibrium spectral function for the off-resonant electron transport regime and strong electron-vibron cou- pling. Calculations were performed with different dia- grams: first-order Hartree and Fock-like (BA) and second- order (DX, DPH) electron-vibron self-energy diagrams. (Top panel): Calculations were done with no self-consistency (BA,BA+DX), partial (SCBA+DX) and full self-consistency (SCBA). (Bottom panel): full self-consistent SC(BA+...) cal- culations. The other parameters are ε 0 = +0 . 5, γ 0 = 0 . 32 , ω 0 = 0 . 4 , t 0 L,R = 0 . 15 , η = 0 . 025. The calculations show that no self-consistency gives poor results in comparison to ex- act calculations given in Figure 9. A closer spectral function to the exact result is obtained from SC(BA+DX) calculation. The second-order DPH diagram do not contribute much in the off-resonant regime. See main text for a detailed analysis of the spectral functions. nation of the reason why starting from a SCBA calcula- tions is better to achieve full self-consistency with higher- order diagram than Hartree-Fock, apart from the simple fact that an Hartree-Fock calculation is probably closer to the true interacting solution than the non-interacting solution. However, our calculations reveal that it is possible to solve such a problem by starting the fully self-consistent calculations (up to second order) from a different starting point, namely by starting from the G$_{SCBA}$ Green’s func- tions (i.e. the GFs obtained from a fully SC calculation including only the lowest-order diagrams). This is shown by the curve (SCBA+DX) in the upper panel of figure 11 and by the lower panel in which all the fully converged self-consistent results are shown. For the moment, we do not have a full physical expla- To conclude this section, we can say that in the off- resonant regime at equilibrium, the second-order DX di- agram dominates over the second order DPH diagram.