content_image/417395.jpg content_image/417389.jpg content_image/417386.jpg content_image/417387.jpg content_image/417388.jpg content_image/417391.jpg content_image/417392.jpg content_image/417393.jpg content_image/417394.jpg content_image/417390.jpg
diagrams, shown in this paper for the spectral functions only, are also important in the full non-equilibrium trans- port properties of the nanojunctions. The study of these effects on the non-linear conductance are currently un- dertaken and will be considered in a forthcoming paper. 

## Appendix A: Self-energies for the Born approximation and the second-order diagrams 

In this appendix, we show how to derive the expression for the self-energies corresponding to one- and two-vibron process diagrams in terms of electron-vibron coupling. 

Starting from the SSSM Hamiltonian, the general def- inition of the one-particle GF of the central region is ob- tained from the time-loop contour-ordered product of the creation and annihilation operators in the central region 

The time-loop contour C$_{K}$ contains two branches, the up- per (+) and the lower ( − ) branch. On the upper branch, time starts in the infinitely remote past and evolves for- wards, then at the turning point, which can be placed at any arbitrary time, one passes onto the lower branch where the system evolves backwards in time back to the initially non-interacting starting point at t = −∞ . 

Then any expectation value of products of opera- tor reduces to 〈 φ$_{0}$ | T$_{C}$ K ( ˆ A ( τ ) ˆ B ( τ $^{′}$) . . . S$_{C}$ K ) | φ$_{0}$ 〉 where 〈 φ$_{0}$ | . . . | φ$_{0}$ 〉 is the average over the non-interacting ground state. The operators are then given in the interaction picture, and S$_{C}$ K is the generalization of the time evolution operator on the Keldysh contour S$_{C}$ K = T$_{C}$ K ( exp {− i ∫ C K d τ ˆ V ( τ ) } ) where T$_{C}$ K is the time-ordering operator on the contour C$_{K}$ and ∫ C K d t implies integration over C$_{K}$ . And ˆ V is the “perturba- tion” to the reference Hamiltonian, which in our case would be the interaction electron-vibron as well as the coupling of the central region to the leads). 

Expanding S$_{C}$ K as a series in terms of the electron- vibron coupling Hamiltonian H$_{e}$$_{-vib}$ , one can derive the electron-vibron self-energies to any order of the electron- vibron coupling by calculating any time ordered products in the series using the usual rules of many-body perturba- tion theory, such as Feynmann diagrammatic expansion or Wick’s theorem $^{85–87}$. 

The one-vibron process self-energies, corresponding to the Hartree and Fock-like diagrams (see Figure 2) are given by 

<img src='content_image/417395.jpg'>

<img src='content_image/417389.jpg'>

## 1. Lowest order self-energies 

and 

<img src='content_image/417386.jpg'>

where τ$_{i}$ are times on the time-loop contour. 

The projections onto the real (physical) times are given by, for example, 

<img src='content_image/417387.jpg'>

for the Fock self-energy . 

The index ζ$_{i}$ = ± labels the branch of the time-loop contour corresponding to the forward ( ζ$_{i}$ = +)/ back- ward ( ζ$_{i}$ = − ) time evolution respectively. 

There are several useful relationships between the different projections (or Keldysh components) of the Green’s functions X ≡ G, D : $^{66–70}$. They are 

<img src='content_image/417388.jpg'>

or equivalently in terms of time-ordered ( t = ++), anti time-ordered ( ˜ t = −− ), greater ( > = − +) and lesser ( < = + − ) Green’s functions: 

and 

<img src='content_image/417391.jpg'>

(or equivalently X a = X t − X > = X < − X ˜ $^{t}$), and hence X r − X a = X > − X $^{<}$. 

Similar relationships exist for the self-energies Σ x ↔ X x ( x = r, a, <, >, t, ˜ t ). 

Using these relationships and taking the steady state limit, i.e. X ( t, t $^{′}$) = X ( t − t $^{′}$), and after Fourier trans- formation into an energy representation X ( ω ), we ob- tain the usual expressions 20,24,26,27,30–34,38,44–46 for the Hartree Σ H,x e $_{-vib}$( ω ) and Fock Σ F,x e $_{-vib}$( ω ) electron-vibron self- energies ( x = r, a, <, > ). 

For example, the Hartree and Fock self-energies are 

<img src='content_image/417392.jpg'>

<img src='content_image/417393.jpg'>

where y represents one of the tree Keldysh compoentns y = t, <, > . Using the relationship Σ r = Σ t − Σ $^{<}$, we find 

<img src='content_image/417394.jpg'>

<img src='content_image/417390.jpg'>

