8 0 0.5 1 1.5 2 Bias (V) -0.15 -0.1 -0.05 0 Hartree potential -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 0 25 50 75 100 125 A( L R (a) Fock -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 0 25 50 75 100 125 A( L R (b) Hartree Fock FIG. 8: Non-equilibrium spectral function A ( ω ) for the reso- nant transport regime for different applied biases for (a) the Fock diagram only, and (b) the Fock and Hartree diagrams. The curves are shifted vertically (+10 in y -axis) for clarity, with the values of the left and right chemical potentials given by the triangular symbols, here we have a symmetric potential drop. Adding the Hartree diagram breaks the electron-hole symmetry. The other parameters are the same as for Figure 6 except for ε 0 = +0 and η V = 0 . 5. FIG. 7: (color online) Hartree potential versus applied bias. The Hartree potential is almost constant in the low bias regime, and hence it could be neglected in the calculations, because it just corresponds to a global energy reference shift. However for larger biases, it is much more influential as the non-equilibrium electron density varies a lot with the applied bias. The parameters are the same as in figure 6. ration between both side-band peaks and the main peak is ∼ 0 . 34 , > ω$_{0}$ for the parameters we used. Figure 7 shows the Hartree potential (i.e. the real part of Σ H,r e $_{-vib}$) plotted against applied bias for different values of the electronic level ε$_{0}$ . The solid line corresponds to the value of ε$_{0}$ used to calculate the spectral functions of the lower panel in Figure 6. The Hartree potential is small and almost constant in the low bias region. Hence, in the quasi-equilibrium regime, such a potential could be neglected in the calculations, because it simply corre- sponds to an energy reference shift. However when the bias window [ µ$_{L}$, µ$_{R}$ ] starts to encompass features in the spectral functions (either the main peak or vibron side- band peaks), the corresponding non-equilibrium electron density starts to vary substantially with the applied bias. Thus the Hartree potential shows a strong dependence on the value of the bias (as also shown in Ref. [60]), until the saturation regime is reached and the Hartree poten- tial is once again constant for very large biases. The asymptotic saturation value of the Hartree potential is dependent on the value of the electronic level ε$_{0}$ as one would expect. The first drop in the value of the Hartree potential happens at ˜ ε$_{0}$ , with the subsequent steps, which become progressively smaller and broader, at ˜ ε$_{0}$ + ∼ 0 . 34 and then ˜ ε$_{0}$ + nω$_{0}$ . tained from one element of the Green’s functions which are themselves dependent on the values of the Hartree potential. We now consider what happens when we apply a bias in the resonant transport regime. We apply a symmetric potential drop (i.e. the potential of the left electrode is raised by an amount ηeV while that of the right elec- trode drops by the same amount). This allows us to keep the electron-hole symmetry and see under which circum- stances the electron-hole symmetry is broken. This behaviour indicates that one cannot in principle neglect the Hartree diagram contribution in the calcu- lations, unless one is interested in calculating only the properties of the system for a bias range for which the electron density is (almost) constant $^{93}$. It is worth mentioning that it is not straightforward to relate the modification of the peak positions in the two panels in figure 6 to the Hartree potential alone. In self-consistent calculations, a highly non-linear sys- tem needs to be solved, since the Hartree potential is ob- The spectral functions for the Fock diagram are shown in figure 8(a). As we have no Hartree term, the spec- tral function is electron-hole symmetric at equilibrium. Moreover, as we have chosen a symmetric potential drop ( η = 0 . 5) the spectral functions stay symmetric for all 2. Non-equilibrium resonant transport