14 -0.2 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 50 A( SCBA SCBA+DX SC(BA+DX) SC(BA+DPH) SC(BA+DX+DPH) 0.3 0.35 0.4 0.45 0.5 0 20 40 -0.25 0 0.25 0 1 2 3 0.75 1 0 1 2 3 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. FIG. 13: (color online). Non-equilibrium spectral function (bias V = 0 . 45 > ω 0 ) for the off-resonant transport regime for the weak/intermediate electron-vibron coupling. Calcula- tions were performed self-consistently with first-order Hartree and Fock-like diagrams (SCBA curve), and with second-order DX and/or DPH diagrams (SC(BA+DX,+DPH,+DX+DPH) curves). Second-order DX correction to SCBA is also shown (SCBA+DX curve). The top-left inset shows a zoom on the central peak around ω ∼ 0 . 4, and the top-right in- sets show a zoom on the vibron side-band peaks around ω ∼ 0 . 0 and ω ∼ 0 . 8. The other parameters are ε 0 = +0 . 5 , γ 0 = 0 . 2 , ω 0 = 0 . 4 , t 0 L,R = 0 . 15 , η = 0 . 025 , η V = 1. The second-order DX diagram narrows the peaks with a slight shift of the side-band peaks towards the central peak. The narrowing is too strong in the case of partially self-consistent calculations (SCBA+DX). The second-order DPH diagram broadens the peak, but not as much as in the resonant case. Full self-consistent calculations including both second-order diagrams result in an intermediate behaviour for the modifi- cations of the spectral functions. 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. 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. 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 the effects of higher-order diagrams.