9 parabolic confinement and being subject to the same ex- ternal perpendicular magnetic field have a continuous en- ergy spectrum with discrete Landau sub-bands. The Coulomb potential in Eq. (4) in the 2D wire is described by u ( r − r $^{′}$) = e 2 κ $^{√}$( x − x $^{′}$) 2 + ( y − y $^{′}$) 2 + η $_{2}$, (16) with the small convergence parameter ( η/a$_{w}$ ) = 0 . 01 to facilitate the two-dimensional numerical integration needed for the matrix elements (4). After the GME (9) has been transformed to the in- teracting many-electron basis by the unitary transfor- mation obtained by the diagonalization of H$_{S}$ (3) we truncate the RDO (8) to 32 MES. For the bias range 0 . 0 ≤ Δ µ = µ$_{L}$ − µ$_{R}$ ≤ 1 . 7 meV used here 10 SES are sufficient to obtain these lowest 32 states with good accu- racy. We will be omitting singly occupied states of high energy that should not be relavant for the parameters here. The natural strength of the Coulomb interaction will only give us MES that are occupied by one or two electrons in the energy range 0 to 6 meV covered by the 32 MES. Since in the partitioning approach [ H$_{S}$, H$_{L}$ ] = 0 we have to construct T l qn as a non-local overlap of φ$_{n}$ and ψ L,R q on the contact regions C$_{l}$ , l = L, R : 8 T l qn = ∫ C$_{l}$ d r d r ′ $^{(}$ψ ∗ $_{ql}$( r ) φ$_{n}$ ( r ) g l $_{qn}$( r , r $^{′}$) + h.c. $^{)}$. (17) g l $_{qn}$( r , r $^{′}$) = g l $_{0}$exp $^{[}$− δ l $_{1}$( x − x $^{′}$) 2 − δ l $_{2}$( y − y $^{′}$) 2 ] × exp$^{(}$ −| E$_{n}$ − ε$_{ql}$ | Δ l E ) . (18) As before ε$_{ql}$ is the energy spectrum of lead l , and E$_{n}$ is the energy of the SES numbered by n in the quan- tum wire. The quantum number q for the states in leads represents both the discrete Landau band number and a continuous quantum number that can be related to the momentum of a particular state. Here we use the pa- rameters δ$_{1}$a 2 w = δ$_{2}$a 2 w = 0 . 25, Δ LR E = 0 . 25 meV, and g LR 0 = 40 meV for B = 1 . 0 T. The domain of the over- lap integral for the leads is ± 2 a$_{w}$ into the lead or the system for x and x ′ from each end of the wire at ± L$_{x}$/ 2 and between ± 4 a$_{w}$ for y and y $^{′}$, see Ref. (8) for an exact definition. All the SES will be coupled to the leads, but the coupling strength will depend on the character of the SES, whether it is an edge- or bulk state and other finer geometrical details that is brought about by the magnetic field. The right chemical potential µ$_{R}$ is held at 1 . 4 meV and the transport properties are calculated for different values of the bias Δ µ by varying µ$_{L}$ . Figure 10 compares the total occupation of all one- electron and two-electron MES for the interacting system at two different values of the bias. At, Δ µ = 0 . 2 meV we see that almost solely FIG. 9: (a) The total current in the left lead carried by N - particle states and the total charge. for µ$_{L}$ = 1 . 5 and for µ$_{R}$ = 0 . 65. (b) The occupation number of the N -particle states. (c) The total current in the left lead carried by N - particle states for µ$_{L}$ = 1 . 2 and µ$_{R}$ = 0 . 2. (d) The occupation number of the N -particle states and the total charge.