content_image/443864.jpg content_image/443865.jpg
applied biases. Constraining the symmetry in this way allows us to concentrate on the effects of increasing the bias, and we can see that the most important modifi- cations of the spectral function are in the width of the central and satellite peaks. The vibron side-band peaks show only a very small change with increasing applied bias. 

Figure 8(b) shows the spectral function when we add in the Hartree term while keeping all other variables un- changed. We note immediately that the electron-hole symmetry is broken, even at equilibrium. In addition the main peak is shifted towards negative energies with re- spect to the Fock-only calculation, and the right-hand vi- bron side-band peak is completely suppressed—the whole spectral function takes on the qualitative appearance of a spectral function in the hole-dominated regime (com- pare with Figure 4). On increasing the bias, the width and shape of the left-hand vibron side-band peaks is var- ied, and for higher biases new vibron side-band peaks appear above the main peak, until at high biases the spectral function is very similar in shape to that for the off-resonant spectral function (Figure 6). 

## C. A comparison with inelastic scattering techniques 

In this section, we will check the validity of the self-consistent Born approximation (SCBA), i.e. self- consistent calculations using only the Hartree and Fock diagrams, versus another method which gives more ex- act results as far as electron-vibron coupling is concerned. Since NEGF is a many-body perturbation expansion the- ory, by definition it does not contain all diagrams even though self-consistency allows us to achieve a partial re- summation of a subclass of diagrams. 

As an alternative to NEGF, one can calculate the transport properties using an extension of conventional scattering theory to include the interaction of incoming single-particle states with some bosonic degrees of free- dom inside the central region of interest $^{16,64,74}$. 

This technique, termed the multi-channel inelastic scattering technique (MCIST) $^{16,64}$, has the advantage of being an exactly solvable problem, even in the presence of many non-interacting electronic states coupled to many vibration modes in the central region $^{16}$. MCIST is based on many-body perturbation theory for polaron and it is exactly solvable in the sense that MCIST is treating the electron-vibron coupling to all orders. In the language of polaron theory, MCIST contains all the diagrams corre- sponding to the electron-vibron interaction (in the corre- sponding transport regime). To be more precise, MCIST contains all orders of crossing and non-crossing diagrams in terms of the bare vibron propagator $^{64,75–77}$. In con- ventional polaron theory for one electron, there are no diagrams with electron-hole loops in them $^{77}$. 

However, MCIST is a single-particle scattering tech- nique and treats the statistics of the Fermi seas of the 

left and right leads only in an approximate manner. In the language of NEGF, this means that the results given by MCIST are only valid for a specific transport regime (as we will see below). 

In the case of the SSSM model, the retarded Green’s function of the central region has the usual form 

<img src='content_image/443864.jpg'>

Within MCIST, the retarded electron-vibron self-energy containing all orders of the electron-vibron coupling is expressed as a continued-fraction as shown analytically in Refs. [64] and [76]: 

<img src='content_image/443865.jpg'>

where we recall that G r 0 is the retarded GF of the central region connected to the leads, i.e. G r $_{0}$( ω ) = [ g r $_{0}$( ω ) − 1 − Σ r $_{leads}$( ω )] − $^{1}$, without electron-vibron coupling. 

As show in Ref. [64], the lowest Born approximation, fully consistent with SCBA (Hartree-Fock calculations as shown above), is given in MCIST by an electron-vibron self-energy equivalent to Eq. (29) but where the integer n factors at each level of the continued fraction are all replaced by the integer n = 1. 

Obviously, this approximate substitution is good enough in the (very) weak electron-vibron coupling for which only the first level of the continued fraction con- tributes the most $^{64}$. 

Below we compare the spectral functions obtained by exact MCIST calculations, by MCIST approximated to the SCBA (Hartree-Fock) level, and to NEGF-SCBA cal- culations. 

The spectral functions obtained from MCIST are shown in figure 9 for the weak/intermediate electron- vibron coupling regime ( γ$_{0}$/ω$_{0}$ = 0 . 5) and for the strong electron-vibron coupling regime ( γ$_{0}$/ω$_{0}$ = 0 . 8). The overall lineshapes correspond to a main peak with vi- bron side-band peaks located only above the main peak. These are typical results fully consistent with an off- resonant transport regime situation. Furthermore, the spectral functions obtained with BA-based approxima- tion (MCIST-SCBA and NEGF-SCBA) are virtually identical, especially in the intermediate ( γ$_{0}$/ω$_{0}$ = 0 . 5) to very weak (not shown here) electron-vibron coupling regime. For strong electron-vibron coupling, one obtains the same peak positions, however the amplitude of the peaks (especially the main peak) is slightly different. 

Since MCIST calculations will always give similar line- shapes independent of the value of ε$_{0}$ , one can con- clude that MCIST calculations are only valid for the off-resonant transport regime at and near equilibrium. MCIST is not able to reproduce the spectral features 

