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grams and levels of self-consistency. We have considered the case for which real excitations of vibrons are possible V = 0 . 45 > ω$_{0}$ . 

The interpretation of the results is not as straightfor- ward as in the equilibrium case, because non equilib- rium effects are sometimes counter intuitive. However, the overall shapes of spectral functions are quite similar to that obtained within Hartree-Fock based calculations (see Section III B 1) in the sense that they present a cen- tral peak with vibron side-band peaks on both sides. 

Similarly to the resonant regime at equilibrium, the main effects of the second-order DX diagram is to narrow the width of all peaks, and to shift slightly the side-band peaks towards the main peak. 

As mentioned in the previous section, the main effect of the second-order DPH diagram is to broaden the peaks, in opposition to the effects of the DX diagram. How- ever here, the broadening in the non-equilibrium condi- tion appears less important than in the case of the res- onant regime at equilibrium. Hence full self-consistent calculations performed with both DX and DPH second- order diagrams give a narrowing of the peaks with a cor- responding increase of their amplitude in comparison to SCBA calculations. 

Finally, out of equilibrium, it can be seen that the results given by a full self-consistent calculation (curve SC(BA+DX) in Figure 13) are strongly different from a second-order correction to an Hartree-Fock calculation (SCBA+DX curve in 13) which gives an excessively nar- rowed central peak. 

This means that in the weak/intermediate electron- vibron coupling, second-order corrections to a SCBA cal- culations are only good enough at equilibrium, however at non-equilibrium full self-consistency needs to be per- formed with all diagrams of the same order. 

As we have already shown in the first-order electron- vibron diagrammatic calculations, the spectral func- tions for the off-resonant and resonant regime at non- equilibrium are qualitatively similar, in the sense that they present a central peak with vibron side-band peaks on both sides. One can compare for example the spec- tral functions obtained for Hartree-Fock-like calculations at non-equilibrium shown in Figure 6 and Figure 8. 

Hence, and again on a qualitative level, the effects of the second-order DX and DPH diagrams on the resonant case at finite bias are similar to what has been obtained for the off-resonant non-equilibrium case described in the previous section. The effects of these higher-order dia- grams on the full non-equilibrium transport properties for the different transport regimes will be presented in a forthcoming paper. 

However before turning the discussion to the linear- response properties of the system at and near equilib- rium, we would like to comment on a specific aspect of 

## 4. Resonant regime at finite bias 

<img src='content_image/396213.jpg'>

the effects of higher-order diagrams. 

The narrowing of the vibron side-band peaks and of the main central peak due to higher-order (DX) diagrams was also obtained by other authors (see for example Ref. [84]). In this paper, a different non-equilibrium approach was used. It consists of starting with an electron dressed by a vibron (a polaron) in the isolated central region, then using perturbation expansion theory (with partial resum- mation) in terms of the coupling of the central region to the non-equilibrium left and right leads. However, the results for the spectral functions in Ref. [84] were only given for the resonant transport regime at equilibrium and all calculations were performed without the Hartree- like diagram. Though we have already shown that such a diagram plays a crucial role in the spectral properties of the system in the resonant regime at and out of equi- librium. 

