0 0.25 0.5 0.75 1 1.25 0 20 A( BA BA+DX BA+DX+DPH 0 0.25 0.5 0.75 1 1.25 0 20 A( SCBA SCBA+DX SCBA+DX+DPH full SC (BA+DX) 11 tions including the second-order diagrams Σ DX e -vib and/or Σ GW e $_{-vib}$[ G $^{SCBA}$] in which the Green’s functions are the cor- responding SCBA Green’s functions. Finally, we perform fully self-consistent calculations, in which the Green’s functions are calculated in a self-consistent manner with the first and second-order diagrams included within each iteration of the self- consistency loop. Our rationale for choosing to do calculations in this manner is as follows. We are able to test the different lev- els of approximation and see very precisely the effects of vertex corrections and polarization are on both the bare Green’s functions G$_{0}$ and the SCBA-level Green’s func- tions. We will also show in section III D 1 that by using the SCBA Green’s functions rather than G$_{0}$ as a start- ing point, one achieve a better convergence in the cal- culations. In particular we will show later on in Section III D 1 that in some cases, for example the off-resonant transport regime when γ$_{0}$/ω$_{0}$ ≳ 0 . 6 − 0 . 7), the use of the bare G$_{0}$ Green’s functions as a starting point for a fully self-consistent calculation with second-order diagrams ac- tually gives unphysical results. Additionally, fully self- consistent calculations are extremely computationally in- tensive, and hence it is both interesting and useful to ex- plore the range of parameters for which the second-order- diagram corrections to the SCBA calculations give a suf- ficiently accurate description of the Green’s functions in comparison to the fully self-consistent calculations. FIG. 10: (Color online) Equilibrium spectral function for the off-resonant electron transport regime in the weak/intermediate electron-vibron coupling. Non self- consistent (top panel) and self-consistent (bottom panel) calculations were performed with first-order Hartree and Fock-like (BA, solid lines), plus second-order DX (dashed lines), plus DPH (squares) electron-vibron self-energy dia- grams. 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. Self-consistent calculations give better spectral functions, with vibron side-band peak closer to the main peak. The second-order DX diagram narrows the side-band peaks and bring them even closer to the main peak as expected. The second-order DPH diagram does not con- tribute much in the off-resonant regime. In this regime and for small electron-vibron coupling, second-order corrections to SCBA calculations are a good approximation to correspond- ing full self-consistent calculations. 1. Off-resonant regime at equilibrium Figure 10 shows the spectral functions of the off- resonant transport regime at equilibrium, calculated for weak/intermediate electron-vibron coupling ( γ$_{0}$/ω$_{0}$ = 0 . 50) and for different diagrams and levels of self- consistency. expansion theory beyond weak coupling $^{64,77}$. The effects of the second-order DPH diagram in the spectral function are virtually nil for the case of weak/intermediate coupling and the off-resonant regime. This might not be that surprising since the DPH diagram corresponds to a Fock-like diagram with a renormalised vibron propagator. The renormalization of the vibron is due to a single electron-hole bubble. However in the off-resonant transport regime, the spectral function is al- most empty (in the case of electron transport) or almost full (in the case of hole transport) which implies that there are not many electron-hole excitations available in this transport regime. Hence the polarization (i.e. the contribution of the electron-hole bubble) is very small, subsequently giving very small values for the DPH self- energy. As an example, we have checked our numerical values in the case shown in figure 10 and we have found, as expected, that the maximum values of Σ DPH ,r e -vib are 30 to 50 times smaller than the maximum values of Σ DX ,r $_{-vib}$. As already mentioned in sections III A and III B, the energy separation between the main peak and the first vibron side-band peak (which should be equal to the vi- bron energy, here ω$_{0}$ = 0 . 40) is not well reproduced by BA-based (Hartree-Fock based) calculations and is much larger than ω$_{0}$ . The self-consistency introduced in the calculations give a marginally smaller energy separation. The effects of the second-order DX diagram are firstly to bring the vibron side-band peak closer to the main peak, hence giving an energy separation closer to the ex- act value; and secondly, one observes a strong narrowing and a larger amplitude of the vibron side-band peak in both self-consistent and non-self-consistent calculations. Both these effects thus qualitatively modify the spectral functions towards better agreement with the exact results as shown in Section III C. It is, however, worth mentioning that including the DX diagram in the calculation does not greatly affect the po- sition of the main peak, a result which may be under- stood from the fact that the dynamical polaron shift is a quantity difficult to obtain exactly from a perturbation Furthermore, in the off-resonant regime and for weak-ish electron-vibron coupling, it seems that the fully self-consistent (full SC (BA+DX) curve in fig- ure 10) results G r $_{full SC}$( ω ) are well approximated