6 tron excitations. The main peak corresponds to adding an electron in the single available level. This peak is broadened by the coupling to the leads, and its position in energy ˜ ε$_{0}$ is renormalised by the electron-vibron in- teraction, i.e. ˜ ε$_{0}$ is close to the static the polaron shift ˜ ε$_{0}$ ≈ ε$_{0}$ − γ 2 $_{0}$/ω$_{0}$ . The lesser peaks in the electron-transport spectral function are vibron side-band peaks arising from reso- nance with n = 1 , 2 , 3 , ... excitations in the vibration mode. These peaks correspond to vibration excitation (vibron emission) only. At zero vibron temperature, these are the only available mechanisms for vibrational excitations. We note, and discuss further in section III C, that these side-band peaks should occur at integer mul- tiples of ω$_{0}$ away from the main peak, but that for both our Fock-only and Hartree-Fock SCBA calculations the peak-peak separation is slightly wider than this. In this regime, the Hartree self-energy is negligible, be- cause most of the spectral weight is above the Fermi level and Σ H e -vib ∝ ∫ µ eq d ωA ( ω ) ∼ 0. This implies that the po- laron shift is mainly due to the Fock-like self-energy. For the off-resonant hole-transport regime ( Fig- ure 4(b)), all the features in the spectral function occur at ω < 0 and therefore correspond to hole excitations. The vibron side band peaks are at lower frequencies than the main peak because they correspond to the emission of vibrons by holes rather than electrons. When we include just the Fock-like self-energy, the hole spectral function is symmetric (with respect to the equilibrium Fermi level µ $^{eq}$) with the electron spectral function, as can be seen clearly in figure 5. Adding the Hartree self-energy, however, breaks this electron-hole symmetry. As most of the spectral weight is below µ $^{eq}$, the expression for Σ H e -vib given in equation Eq.(A15) re- duces to a constant 2 γ 2 $_{0}$/ω$_{0}$ (i.e. twice the polaron shift) as ∫ dω 2 $_{π}$iG$^{<}$( ω ) ∼ 1. As a result of this the whole spectral function is shifted to the left by this amount. -1.5 -1 -0.5 0 0.5 1 1.5 0 10 20 30 A( Fock Hartree Fock (a) Electron transport regime -1.5 -1 -0.5 0 0.5 1 1.5 0 10 20 30 A( Fock Hartree Fock (b) Hole transport regime FIG. 4: Equilibr ium (zero bias) spectral functions A ( ω ) for the off-resonant ( a) electron and (b) hole transport regime. Calculations wer e performed with the Fock-like electron- vibron diagram ( solid line) and with both the Hartree and Fock-like diagram s (symbols and dotted line). For the elec- tron transport reg ime the inclusion of the Hartree self-energy has no effect, but for the hole transport regime it shifts the entire spectral fun ction to lower energies. The parameters are ε 0 = +0 . 5( − 0 . 5) for electron (hole) transport γ 0 = 0 . 21 , ω 0 = 0 . 3 , t 0 L,R = 0 . 15 , η = 0 . 005 and µ L = µ R = µ eq = 0. 2. Resonant transport regime A. At equilibrium We now turn to the resonant transport regime, with the spectral function shown in Figure 5. Here our elec- tronic level ε$_{0}$ is broadened by the coupling to the leads and is partially filled with electrons. We can see that cal- culations performed with only the Fock-like self-energy preserve the electron-hole symmetry. The spectral func- tion presents peaks located both at positive and negative energies, which correspond to the emission of vibrons by electrons or holes respectively. As in the previous sec- tion, the inclusion of the Hartree self-energy (equation (A15)) breaks down the electron-hole symmetry and the features are shifted to lower energies with a correspond- ing modification of the spectral weights for each peak. We note here that the inclusion of the Hartree self- energy thus modifies so drastically the spectral function that it will also strongly affect the I - V characteristics of the junction in comparison to calculations performed For the off-re sonant electron transport regime (Fig- ure 4(a)) all th e features in the spectral function are above the Fermi level µ eq and hence correspond to elec- We first cons ider the spectral functions at equilibrium, with no applied bias. Figure 4 shows A ( ω ) for the off- resonant regime, and Figure 5 shows the resonant trans- port regime. In the equilib rium many-body language, the features in the spectral functions obtained at positive energies ( ω ≥ 0, above the Fermi level) correspond to electron excitations, wh ile the features at negative energies ( ω ≤ 0, below the Fer mi level) correspond to hole excitations. 1. Off-resonant transport regime