4 III. MODELS AND RESULTS We have numerically implemented the GME method both for lattice and continuous models. The sample mod- els are: a short 1D chain with 5 sites, a 2D rectangular lattice with 12 × 10 = 120 sites, and a short quantum wire withe parabolic lateral confinement. In all cases the coupling functions have the form understanding of the underlying physics, and in particu- lar to show the presence of the Coulomb blocking effects at certain values of U or of the chemical potentials of the leads. The Coulomb matrix elements defined in Eq. (4) are calculated as V$_{nm,n}$ $^{′}$m′ = ∑ i = i ′ φ ∗ $_{n}$( i ) φ ∗ $_{m}$( i $^{′}$) U | i − i ′ | φ$_{n}$ ′ ( i ) φ$_{m}$ ′ ( i $^{′}$) . (15) χ$_{l}$ ( t ) = 1 − 2 e γt + 1 (13) In Fig.1 we show the equilibrium chemical potentials µ (0) N corresponding to ground states with 1 ≤ N ≤ 5 par- ticles against the interaction strength U . One observes a linear dependence of µ (0) N on U , with slope increasing with N . Obviously the total Coulomb energy increases both with U and N . with γ a constant parameter, such that at the initial mo- ment, which is t$_{0}$ = 0, we have χ$_{l}$ (0) = 0 (no coupling), and in the steady state, for t → ∞ , χ$_{l}$ = 1 (full coupling). Let us now briefly review the Coulomb blockade scenario. 24 Suppose the isolated sample contains N elec- trons and the chemical potentials of the leads are cho- sen such that µ (0) N < µ$_{R}$ < µ$_{L}$ < µ (0) N $_{+1}$. Then the bias window [ µ$_{R}$, µ$_{L}$ ] may include one or more of the excited configurations with N particles. In general some states with N electrons may have excitation energies exceed- ing µ$_{L}$ or even µ (0) N $_{+1}$. This situation corresponds to the Coulomb blockade phenomenon. Indeed, the addition of the ( N + 1)-th electron is energetically forbidden. Con- sequently the current in the steady state should vanish. However, shorter or longer transient currents are gener- ated by all many-body configurations in the vicinity of the bias window. In this model the two semi-infinite leads are attached to the ends of a 1D chain with 5 sites. The coupling between a lead state with wave function ψ$_{ql}$ and a sam- ple state with wave function φ$_{n}$ is given by the product between the wave functions at the contact site: A. A toy model: short 1D chain T l qn = V$_{l}$ψ ∗ $_{ql}$(0) φ$_{n}$ ( i$_{l}$ ) , (14) where 0 is the contact site of the lead l = L, R , the end sites of the sample being i$_{L}$ = 1 and i$_{R}$ = 5. Fig. 2(a) and 2(b) show the total currents in the left lead and the total charge residing in the sample for sev- eral values of the interaction strength. U is measured in units of t$_{S}$ , the hopping parameter in the sample, 7 and the time is expressed in units of $_{ℏ}$/t$_{S}$ while the cur- rent is in units of et$_{S}$/$_{ℏ}$ . The coupling constant in Eq. (13) is γ = 1. The system is initially empty and thus ρ (0) = | 00000 〉〈 00000 | . The chemical potentials of the leads, µ$_{L}$ = 5 . 25 and µ$_{R}$ = 4 . 75, are chosen such that in the absence of Coulomb interaction, i. e. for U = 0, µ (0) 4 is located within the bias window. In this case we obtain in the steady state the mean number of electrons about 3.6 and a non-vanishing current in the leads. This is understand- able, since µ (0) 4 = E$_{4}$ = 5, which is the 4-th level of the isolated sample. The occupation of this level in the steady state is about 0.6, the other states being either full or empty. Also in this case, the excited states have small contributions to the steady state current as the system tends to be in the ground state with N = 3 electrons. Those contributions may also depend on the coupling strength of individual states with the leads, but in gen- eral remain small. 25 FIG. 1: (Color online) The equilibrium chemical potentials µ (0) N for 1 ≤ N ≤ 5 as a function of the interaction strength U . The dotted lines mark the chemical potentials of the leads selected in the transport simulations shown in the next figure, i. e. µ$_{L}$ = 5 . 25 and µ$_{R}$ = 4 . 75. The reason to call this a toy model is that we can ob- tain the complete set of 2 5 = 32 MES, i. e. we do not need to cut the basis of the 5 SES. We also do not need to cut the MES basis, all matrix elements of the statisti- cal operator can be numerically calculated, even if not all of them might be important for the currents. In addition we will consider the strength of the Coulomb interaction as a free parameter U , whereas in a realistic systems this is fixed by the electron charge and the dielectric con- stant of the material. Our goal is to have a qualitative The situation may change for U = 0. For the inter- acting system, e. g. for U = 0 . 3, the system settles down in the Coulomb blockade regime, the total current be- ing almost suppressed in the steady state. This happens because the interaction pushes the chemical potentials upwards such that for U = 0 . 3 both ground states with