Hartree-Fock calculations of a finite inhomogeneous quantum wire

We use the Hartree-Fock method to study an interacting one-dimensional electron system on a finite wire, partially depleted at the center by a smooth potential barrier. A uniform one-Tesla Zeeman field is applied throughout the system. We find that with the increase in the potential barrier, the low density electrons under it go from a non-magnetic state to an antiferromagnetic state, and then to a state with a well-localized spin-aligned region isolated by two antiferromagnetic regions from the high density leads. At this final stage, in response to a continuously increasing barrier potential, the system undergoes a series of abrupt density changes, corresponding to the successive expulsion of a single electron from the spin-aligned region under the barrier. Motivated by the recent momentum-resolved tunneling experiments in a parallel wire geometry, we also compute the momentum resolved tunneling matrix elements. Our calculations suggest that the eigenstates being expelled are spatially localized, consistent with the experimental observations. However, additional mechanisms are needed to account for the experimentally observed large spectral weight at near $k=0$ in the tunneling matrix elements.


I. INTRODUCTION
One-dimensional (1D) electronic systems have proved to be a very fruitful field in the studies of interacting many-body systems.The infinite homogeneous one-dimensional electron system has been extensively studied.At high density n ≫ a −1 B , where a B = ε h2 /me 2 is the Bohr radius, the low en- ergy physics of the system is well described by the Luttinger model 1 , with spatially extended electronic states as well as separate spin and charge excitations propagating at different speeds v s and v c . 2 At low density n ≪ a −1 B , a system with a long range interaction can be best described as a fluctuating Wigner crystal, with electrons being confined around their equilibrium positions by their mutual repulsion, though quantum fluctuations prevent a true long range order.The excitations in this case are the density fluctuations (plasmon) of the Wigner crystal and the spinon excitation from the Heisenberg antiferromagnetic spin chain created by the exchange of the neighboring localized electrons through a barrier formed by their mutual repulsions 3 .For a system with a short range interaction, at low densities n ≪ d −1 , where d is the range of the interaction, the charge sector can be described as a weakly interacting gas of spinless fermions, and the spin sector can again be described as a Heisenberg antiferromagnetic chain with an appropriate exchange constant J.
Theoretical analyses have also considered the case of a finite wire, with either sharp or soft confinement at the ends 4,5,6 .By contrast, the case of a spatially inhomogeneous system, with a low-density region in the center of the wire, has not been extensively explored.In recent experiments, Steinberg et al. 7,8 used a negatively charged metal gate to partially deplete the central region of a finite quasi-one-dimensional wire, and studied the low density region by means of momentumconserved tunneling from a parallel "semi-infinite" wire with higher electron density.They found a striking transition from a regime of extended electronic states to a regime of apparently localized states, as the electron density at the center of the wire is lowered by the negative gate voltage.At a critical value, the electrons at the Fermi level seem to change abruptly from an extended state with well-defined momentum into a localized state with a wide range of momentum com-ponents.In the extended state regime, the tunneling measurements show a smooth variation of the electron density in the wire as a function of the gate voltage.In contrast, in the localized regime, the tunneling only occurs at a series of discrete resonant gate voltages, corresponding to the successive expulsion of a single electron from a Coulomb blockaded region that is somehow formed under the repulsive gate.Transport measurements show that the electrical conductance along the wire is much smaller than e 2 /h when the electron-density under the center gate is low enough to be in the localized regime.Furthermore, measurements of momentum conserved tunneling from a second parallel wire show a dramatic change in behavior in the localized regime, as we shall discuss further below.
Motivated by the above experiments, we have turned to the Hartree-Fock method to investigate the physical properties of a system of interacting electrons on a finite wire with a barrier potential at its center, with a special focus on the evolution of the low density electrons.
In previous work, Matveev 3 studied the case of transport properties of 1D interacting electrons through an adiabatic barrier, and concluded that the conductance is 2e 2 /h at low temperature and e 2 /h at high temperature 9 .However, he did not explore the regime where the electron density under the barrier is nearly depleted and the two terminal conductance becomes much smaller than e 2 /h.
Mueller 6 explored the crossover from the non-magnetic state to the Wigner crystal antiferromagnetic state when reducing the electronic density in a finite wire, using a restricted Hartree-Fock method.He mostly considered a finite wire that is relatively uniform in the center region, under no external magnetic field.In Appendix B of his paper, he briefly considered a wire with an additional potential barrier in the center, and found a low density Wigner-crystal like regime under the barrier.He did not further investigate the density and spin evolution of his system as the density under the barrier is further depleted, nor did he study the momentum-dependent tunneling amplitude in the case with a low density center region.
Meir and coauthors 10,11 studied the formation of magnetic moments in a quantum point contact(QPC) in a two-dimensional geometry using spin-density-functional the- FIG.1: A schematic illustration of the geometric configurations and the potential and density profiles considered in this paper.ory(SDFT).They found that as the density inside QPC rises above pinch-off, a magnetic moment forms inside the opening channel.In longer QPC, the magnetic moments take the form of an antiferromagnetically ordered chain.The conducting channels inside the QPC can be roughly modelled as a one-dimensional system with a smooth potential barrier, and the antiferromagnetic order under the barrier they found is consistent with one of the magnetic phases we found in our study.However, unlike the QPC system, our strictly onedimensional system in a strong magnetic field further becomes ferromagnetic in low density region near depletion.Furthermore in our model we use a modified form of electron interaction to take into account of the screening in the tunneling experiments described below.
The paper is organized as following.In Sec.II we introduce our basic model, our choice of parameters and the numerical method we employed.In the Sec.III, we present the physical picture of successive magnetic phases our system goes through as we increase the potential barrier and the crossovers between phases.In Sec.IV, we make a more detailed analysis of the form of the wavefunction at Fermi energy near depletion.Motivated by the experimental measurements by Steinberg et al. 7,8 , we also compute the momentum dependence of the tunneling matrix elements for our system.In Sec.V, we compare our results to the Hartree-Fock calculation for an infinite homogeneous system and a non-interacting inhomogeneous system, and discuss the implications of the electrical conductance measurement from our calculation.In Sec.VI we summarize our results and their comparisons with experiments.

II. MODEL
We consider a system of one-dimensional interacting electrons in a wire of length L = 6 µm with periodic boundary conditions.A uniform magnetic field B is applied throughout the system, which couples only to the spins in our model, and which, in most of our calculations, we set to 1T .In the experiments, 7,8 a magnetic field of 1 − 3T was typically applied.
With the above-mentioned experiments in mind, we assume the electrons interact via a Coulomb potential with both a short range and a long range cutoff.The short range cutoff comes from the finite width of the experimental wire.We model it by simply modifying a 1/z potential to 1/ √ z 2 + W 2 , where z = x − x ′ is the separation of two electrons along the wire and W is the short range cutoff, roughly on the order of half of the width of the wire.This density-independent short range cutoff is appropriate in our case of a sharp confinement transverse to the direction of the wire formed by the cleaved edge overgrowth.The long range cutoff is the result of the screening effect from the higher density wire parallel to the short wire in the tunneling experiment.We model it by putting a second wire, which is simplified to be infinitely long and perfectly conducting, parallel to the finite wire under study, at a center to center distance d, as shown in Fig. 1.The resulting form of the interaction U(z) can be easily derived, as discussed in Fiete et al.. 5 At short distance x ≪ W , U levels off smoothly as 1/ √ z 2 + W 2 , whereas at long distance x ≫ d it decays much more rapidly than Coulomb potential 1/z.Following the experimental setup, in this paper we choose W = 0.01 µm and d = 0.031 µm, and choose the strength of Coulomb interaction to correspond the value in the bulk GaAs, yielding a Bohr radius a B ≈ 0.01 µm.
In the experiments, a negatively charged 2 µm long metal gate at 0.5 µm above the finite wire is used to reduce the density at the center region of the wire, as illustrated in Fig. 1.To approximate the effect of the gate, we use a smooth bare barrier potential of the form Here L g is the length of the potential barrier, which in our calculation we choose to be the length of the experimental metal gate 2 µm.L s controls the sharpness of the edge of the potential barrier, which we choose to be on the order of the 0.5 µm, the distance from the gate to the finite wire.The quantity V g will be referred to, below, as "gate voltage", although it is actually only proportional to (minus) the applied voltage V G .
The normalization is such that V g is the bare potential at the center of the barrier region.The spatial form of this potential can be seen in Fig. 2.
In the experiments, more than one transverse mode in the quantum wires can be occupied.Correspondingly, in the wire of our model, we maintain two separate subbands of electrons, which we assume to interact only through the Hartree term to the electrons in the other subband.This is equivalent to the assumption that in the experimental wire, one can ignore any effects of scattering or exchange between electrons in different transverse modes.In our model, an energy difference of 42meV separates the bottom of the two subbands, corresponding to the energy separation of the lowest two transverse modes in a square well of width 0.02 µm.
In summary, our Hartree-Fock Hamiltonian can be written as: x (µm) The shape of the bare potential barrier V G , the Hartree potential V H and their sum, at "gate voltage" V g = 67.2meV .The density distribution at this gate voltage is shown at the bottom of Fig. 4. All potentials are symmetric around x = 0, and only the center part x < 2 µm is shown.The Fermi energy E F of the electrons is also shown.
Here m * ≈ 0.067m e is the effective electron mass in bulk GaAs, g * ≈ 0.44 is the effective g−factor in bulk GaAs, and µ B is the Bohr magneton.ψ * (x) is the complex conjugate of ψ(x).σ = (↑, ↓) is spin index.b = 0, 1 is the subband index: ∆ 0 = 0meV for ψ σ 0 in the first subband and ∆ 1 = 42meV for ψ σ 1 in the second subband.The summation over i in com- puting the Hartree potential V H and Fock kernel V F is over all the occupied states in a specific spin and subband.Notice, as discussed in the previous paragraph, in computing the Fock potential kernel V σ b F , we only sum over the occupied states with the same spin σ and in the same subband b as the eigenstate it is acting on.
The numerical method we use in our calculation is a restricted Hartree-Fock method 12 .The electron spins are required to be either parallel or anti-parallel to the applied magnetic field, so canted spin structures are not allowed.In the Appendix, however, we consider the effects of canting in an infinite homogeneous wire, and we argue that canting would have a negligible effect on results for the inhomogeneous system, for the parameters of interest to us.
Starting from solutions to the non-interacting potential barrier problem, we iteratively use the Hartree-Fock method until the convergence between iterations is achieved.Throughout the calculation, we fix the total number of electrons in the finite wire to be N = 1000, whereas the occupation numbers in each spin/subband species remain free to change.
In this paper, we focus on the depletion of the first subband under the barrier.In this regime, the second subband is fully depleted under the barrier and is only occupied in the outer x (µm) FIG. 4: Densities of spin-up and spin-down electrons in the center region, showing the emergence of antiferromagnetic order with increasing V g .All densities are symmetric around x = 0, and only the center part of the first subband electron densities x < 0.9 µm is shown.
regions where the total density is high.Consequently, in our calculations, the second subband serves mostly as a reservoir for the electrons under the barrier.

III. MAGNETIC PHASES
At high densities, we find that the first subband under the gate is essentially unpolarized, as shown in Fig. 3.At an electron density ρ = 40 ∼ 50 µm −1 , or ρ σ 0 = 20 ∼ 25 µm −1 per spin, an antiferromagnetic order emerges at the low density region under the barrier, as shown in Fig. 4. The antiferromag- ) x (µm) FIG. 5: The transition from antiferromagnetic order to spin aligned at the center region, in the form of a spin-aligned region which expands rapidly from the center.All densities are symmetric around x = 0, and only the center part of the first subband electron densities x < 0.9 µm is shown.netic order parameter, the staggered magnetization M, grows steadily as the density decreases with the increase of V g .There is no sharp transition between the nonmagnetic and the antiferromagnetic solutions.
At ρ = ρ * ∼ 20 µm −1 under the barrier, a spin aligned center region appears and rapidly expands, as shown in Fig. 5.As seen in the figure, the spin aligned region at the center is sandwiched by two strongly antiferromagnetic regions on each side.Within a narrow range of V g , the spin aligned region expands to a maximum length, containing N f = 10 electrons, as shown in the bottom of Fig. 5. From that point on to its full depletion, the center spin aligned region undergoes a series of transitions, each representing the expulsion of one electron from the spin aligned region.In contrast with the nonmagnetic and antiferromagnetic regimes, where the electronic density ρ(x) under the barrier changes smoothly with V g , here the ρ(x) in the spin aligned center region of the barrier varies discontinuously with an increase in V g .Figures 6-8 show the details of one of such transitions, with the number of electrons in the spin aligned region changing from N f = 8 to N f = 7 at V * g ≈ 71.75 meV .In Fig. 6, we see a crossover in the total energy of the Hartree-Fock ground state E, where the N f = 8 solution has a lower energy for V g < V * g and the N f = 7 solution becomes the ground state for V g > V * g .In terms of the occupation numbers of the states of different subbands and spins, this transition corresponds to the expulsion of one electron from the spin up fist subband to the spin up second subband.
In general, as V g increases, the transitions in our calculation always involve an expulsion of one spin-up first-subband electron to the second subband outside the center region.But spin flip transitions also happen: there are transitions showing the spin-down second subband absorbing the expelled electron from spin-up first subband.In Fig. 7, the detailed density changes in one transition are shown clearly.The spin aligned region in each of the solutions is quite well isolated by the ) x (µm) FIG. 7: Electronic density under the barrier, before and after the transition shown in Fig. 6.Only the densities of first-subband spin-up electrons are plotted since electron densities in other states do not change greatly during the transition.The spin density labeled of N f = 8 is taken at the V g = 71.74meV right before the crossover in Fig. 6, and the one labeled N f = 7 is taken at the V g = 71.76meV right after it.
antiferromagnetic regions on its sides.The first subband spindown electron density ρ ↓0 , which is not shown in Fig. 7 for the sake of clarity, drops steeply to zero for |x| ≤ 0.45 µm, where ρ ↑0 rises sharply, on both sides of the transition.The short antiferromagnetic regions on the two sides are only slightly shifted in the transition, whereas the center spin-aligned region undergoes the significant change from having eight peaks to having seven.By plotting the integrated density in the spinaligned region, Fig. 8 shows that this change in ρ ↑0 indeed amounts to the expulsion of almost a whole electron from the region.The slight deficiency from unity is due to the slightly changed length of the spin aligned region, and possibly some  small residual density from the spin aligned states extending into the antiferromagnetic sides.
It may be possible to detect the existence of the central spinaligned region by the application of a magnetic field B parallel to the quantum wire in question.For a state with N spin aligned electrons in the center region, the magnetic field will result a Zeeman energy shift E Z = Ng * mu B B T /2, where g * is the g * ≈ 0.44 is the effective g-factor in bulk GaAs, mu B is the Bohr magneton, and B T = B 2 + B 2 is the strength of the total magnetic field.Due to this energy shift, the transition voltage from the N to the N + 1 spin-aligned electron state will be shifted to a larger value, because the N + 1 electron state energy will be lowered by ∆E Z = g * mu B /2 relative to the N electron state.Such a shift may be detectable for a large change in the combined field B. For example, in the N = 8 to N = 7 transition discussed above, a change δ B T = 4T in the combined field will result a shift of ∆E Z ≈ 0.092meV.This is about one fifth of the typical spacing between transition "gate voltage" V * g , which is approximately 0.45meV.

IV. WAVEFUNCTIONS AND MOMENTUM CONSERVED TUNNELING
In the momentum-conserved tunneling experiments of experiments of Steinberg et al., electrons tunnel between the finite wire and a parallel "infinite" wire, while conserving their momentum in the wire direction.A magnetic field B perpendicular to the cleaved edges defining the two quantum wires gives a controllable momentum boost q B = eBd/h to the electrons tunneling between the wires, where d is the distance between the wires.At low temperature and small source-drain bias, the tunneling conductance G T (B,V g ) ∝ |M| 2 , where the matrix element M has the following physical interpretation: 4 where we have defined a "quasi-wavefunction" with |N being the N-electron many-body ground state in the finite wire, ψ(x) being the electron annihilation operator at position x in the finite wire, and k = q B ± k F , where k F is the Fermi wavevector in the infinite wire.(We neglect, here, electron-electron interactions in the infinite wire.) The momentum dependence of |M| can be extracted from the magnetic-field dependence of the Coulomb blockade peak. 8n the localized regime, expulsion of an electron from the region under the gate is signaled by a vertical stripe in a color plot of the tunnel conductance in the plane of gate-voltage V G and magnetic field B. (See the left panel of Fig. 9.) The momentum dependence of |M(k)| 2 is obtained by integrating the intensity across a given vertical stripe, at a fixed value of the magnetic field, and comparing the results for different values of B. As seen in the right panel of Fig. 9, except for the last peak, the momentum dependence of the |M(k)| 2 , found in the experiments, typically shows to two wide peaks, as well as a broad momentum distribution between the peaks, signaling relatively localized wavefunctions.
In our calculation, the N − 1 and N-electron states should be the complete Slater determinants of the eigenstates of the corresponding Hartree-Fock Hamiltonians.As a simplifying approximation, we may assume that after expelling one electron, the rest of the eigenstates are not affected.In this case Ψ N eff is simply the wavefunction of the electron being expelled and M(k) is its Fourier transform.|M(k)| 2 computed this way for the transition N f = 8 to N f = 7 is shown as the solid line in Fig. 10.The dashed curve is the result obtained by using the full Slater determinants of the (N − 1) and N-electron solutions to compute the matrix elements.This shows an orthogonality-catastrophe-type reduction to the overall spectral density weight.Both of the calculated matrix elements show a relatively broad momentum distribution,  consistent with the experimental result.However, the experimental result shows heavy spectral weight near k = 0, between the two largest peaks at k = ±k max , and little weight outside them, whereas our calculations show little weight between the largest peaks and considerable weight outside, see Fig. 10.This discrepancy suggests that additional mechanisms are needed to explain the finer details of observed momentum distribution |M(k)| 2 .
Insight into the calculated shape of M(k) can be gained by looking at the wavefunction in position space for the electron being expelled.Fig. 11 shows the wave function in position space, for which the squared Fourier transform is the solid curve in Fig. 10.We see that the wave function has relatively large weight in the central spin-aligned region, roughly for |x| < 0.5 µm.However, it also has significant weight outside the barrier, and in the transition region between.As will be discussed further below, the wave function is qualitatively similar to what one would expect in the WKB approximation for a state slightly above the top of a smooth potential barrier.The separation between successive zeroes of the wave function is largest near the center, where the amplitude is largest, and it decreases monotonically in the transition region, where the amplitude decreases gradually and the particle velocity increases.The Fourier transform M(k) has its largest amplitude at a value |k| = k m corresponding to the spacing between zeroes in the center region, and has additional weight at larger wavevectors, corresponding to the smaller spacing of zeroes in the transition region.By contrast, the experimental results look like what one would find for a particle confined in a soft potential well, where the zeroes of the wave function would be closest together near the center of the well, and be farther apart at the two ends.In this case, the Fourier transformed wave functions have significant weight for |k| larger than the peak value k m , but very little weight at larger |k|. 4,5In either case, one finds zeroes in M(k), and oscillations in the amplitude, arising from interference between contributions at spatial points x and −x.

A. Hartree-Fock for a homogeneous system
In order to better understand the results of our calculations, it will be helpful to recall some of features of Hartree-Fock calculations for an infinite homogeneous one-dimensional electron system.At high electron densities, where electronelectron interactions are relatively weak, with no applied magnetic field, one finds a nearly-free electron gas, with a small gap at the Fermi energy, caused by a spin-density wave corresponding to the wavevector Q = 2k F .In position space, this means that the unit cell contains precisely two electrons on average, with a weak polarization, alternating between spin up and spin down, along a quantization axis that has arbitrary direction 13 .The total charge density will have a period half that of the spin density, i.e., there is just one electron in each charge period.The amplitude of the charge modulation will be proportional to the square of the spin-density amplitude, when these modulations are small.For pure Coulomb interactions, the amplitude of the spin-density modulation will fall off rapidly at high densities ρ, roughly as exp(−aρ), where a is a length of the order of the Bohr radius.If the electronelectron interaction is smooth at short distances, the amplitude of the spin-density modulation can fall off still faster with increasing ρ.For a system of finite length L, we would generally not expect to find any spin-density modulation if the antiferromagnetic coherence length, which is inversely proportional to spin-density amplitude of the infinite system, becomes larger than L.
As the electron density is lowered the amplitudes of the spin and charge modulations both grow until one reaches the situation of a Wigner crystal, where the there is strong modulation in the charge density, and there is nearly complete spin polarization, alternating up and down for successive electrons.At still lower densities, the Hartree-Fock approximation predicts a phase transition to a fully aligned ferromagnetic state.The ferromagnetic state is, of course, an artifact of the Hartree-Fock approximation, as it is known 14 that the exact ground state is a spin singlet, for B = 0.However, a state of full spin alignment should occur at low densities, for B = 0, and the Hartree-Fock approximation may be a reasonable description of this transition for B =1T.
The predicted antiferromagnetic order is also an artifact of the Hartree-Fock approximation, as quantum fluctuations would be expected to replace the long-range spin order with correlations that fall off as a power of the distance.In an infinite system, charge density modulations will also be destroyed by quantum fluctuations of the positions of the electrons on the Wigner crystal.However, these quantum fluctuations can be relatively weak when there is strong repulsion between the electrons, and a significant charge density modulation may exist in a finite system of moderate length.
Hartree-Fock calculations for an infinite system, with the same interaction potential used in our finite system, are presented in the Appendix.We discuss there also the effect of spin-canting in an applied magnetic field.
According to Fig. 13 of the Appendix, the Hartree-Fock transition to a fully spin-polarized state should occur at a density of approximately 16 electrons per micron for an infinite system with the parameters of the model under consideration, in a Zeeman field B =1T.This is similar to the density ρ * ≈ 20nm −1 , where we found the onset of a center region with full spin alignment, in our calculations for the system with a barrier.

B. Inhomogeneous system without interactions
It is also useful to review what one would expect for an inhomogeneous system analogous to the wire under consideration, but without electron-electron interactions.In particular, we may consider what should happen as one varies the height of a smooth center barrier, similar to the bare potential barrier in Fig. 2 or to the self-consistent potential in that figure, including the Hartree potential but not the non-local exchange potential.By comparing this qualitative picture with the results of our Hartree-Fock calculations, we can better see whether there are features of the latter which reflect in an essential way the exchange and correlation features of a strongly interacting many-electron system.
It is important to note that the total length of our system is finite, so there will be a discrete set of energy levels for the system as a whole.As the total length is six microns, and the flat potential area under the gate is of order one to two microns, the majority of the length is outside the barrier region.Because the density of states in a non-interacting onedimensional system is inversely proportional to the electron density ρ, however, the local density of states for electrons in the barrier region can be much higher than the density of states outside, if the electron density is sufficiently low in the barrier region.Thus, if the chemical potential is fixed, and the height of the barrier is lowered below the Fermi energy, we may expect to see a closely spaced sequence of electrons entering into states whose amplitudes are highly concentrated in the center region.Indeed, if the overall system length is not too large, we may expect a large fraction of the probability density for each added electron will be located in the barrier region.
It should be noted that the upper wire in the experiments of Ref. [8] has an overall length that is not very different from the wire used in our calculations.The experimental wire is not truly isolated, but is tunnel-coupled to leads in its outer regions; so its energy levels should actually be life-time broadened.If the escape rate from the wire is smaller than the spacing between energy levels, however, the features of a finite system should be maintained.
For non-interacting electrons, if the total number of particles is fixed, rather than the chemical potential, then the electrons entering the center must be transferred from electron states outside the barrier region, (e.g., states belonging to a second subband), and the Fermi level will itself decrease each time an electron is added to the center region.If the level spacing of the outside bands is larger than the level spacing of the center region, then the spacing between gate potentials where successive electrons enter the center region will be determined by the larger energy spacing between these reservoir states.
We may also consider what would happen if one had an infinite system of non-interacting electrons, with a flat barrier of finite length in the center.Suppose the potential V (x) is zero outside the barrier region and equal to V g at the center of the barrier.If the barrier is smooth enough so that one can use the WKB approximation, then the wavefunction ψ(x) for a state with energy E slightly above V g will have an amplitude which is larger inside the central region than outside, by a factor [(E − V (x))/E] −1/4 ≈ [ρ 0 /ρ(x)] 1/2 , where ρ(x) is the cumulative (Thomas-Fermi) electron density at point x from all states with energy less than E, and ρ 0 is the elec- tron density far from the barrier.For a smooth barrier, we see that ρ(x), and hence the amplitude of the wavefunction ψ, should have a maximum in the center of the barrier, and fall off monotonically with increasing |x|.The spacing between successive zeroes of ψ should be given by 1/ρ(x), which should decrease monotonically with increasing |x|.We note that the Hartree-Fock wavefunction plotted in Fig. 11 is qualitatively consistent with these features.
If the barrier height is varied continuously at fixed Fermi energy, for an infinite system, when the WKB approximation is valid, the wavefunctions will vary continuously, and the number of particles above the barrier will likewise vary in a continuous fashion The WKB approximation will break down, however, if the energy E gets too close to the top of the barrier.For a smooth potential, one expects the WKB approximation to be valid for all but the last one or two states above the barrier.By contrast, for a flat-topped potential that falls off relatively abruptly at the ends of the barrier, deviations may more pronounced.In this case we may find a number of resonant states above the barrier, which have very small electron density outside the barrier region and which exist only in narrow energy bands.Then, as the gate voltage is varied, the number of electrons in the barrier region will increase by one FIG. 12: A schematic plot of an effective barrier potential V eff that could produce a form of the wavefunction more consistent with the results of the tunneling experiments.The dashed line is the bare potential barrier V G and the dotted line is the Fermi level E F . in a narrow region of gate voltage, as each resonance passes through the Fermi energy.In the limit of a potential V (r) that drops sharply to V = −∞ at the edge of the barrier, a wavefunction ψ(x) with weight inside the barrier region will vanish outside this region; the wavefunctions and energy levels will be discrete and will be the same as if there was an infinitely high potential at the end of the barrier.
We may also imagine a situation where the self-consistent potential V eff (x) is smooth but non-monotonic, having somehow developed a pair of maxima near the edges of the original barrier region, as illustrated in Fig 12.In this case, there will be an energy range such that E is smaller than the maximum value of V eff but larger than the value at the center of the barrier region.If this energy range is large enough, there may be a discrete series of states which are well localized in the classically allowed region, decay to a small value in the classically forbidden regions, and have only a small amplitude outside the barrier.The distance between zeroes of the wavefunction will then increase with increasing |x| in the region where the wavefunction is large.The form of |M(k)| 2 that one would obtain by taking the Fourier transform of this wavefunction will have a maximum intensity at a wave vector corresponding to the local Fermi wave vector k F at x = 0, and will have significant weight for |k| < k F , but very little weight at |k| > k F .This result is qualitatively similar to the observations of Steinberg et al., illustrated in Fig. 9.However, it is quite different from what we have obtained from our Hartree Fock calculations, illustrated in Fig. 10, where there is considerable weight at large k.
The spin-density structures obtained in our Hartree-Fock calculations suggest that a non-monotonic self-consistent potential, similar to that in Fig. 12, might in fact have been a qualitatively reasonable representation of the Hamiltonian seen by the electrons with the majority spin orientation.Because there is a fully polarized spin-down electron on either side of the central region of spin-up electrons , and because there is a strong repulsion between electrons of opposite spin, one might expect that the spin up electrons would see an effective potential maximum at the positions of the spin-down electrons.The strong antiferromagnetic order just outside this region, suggests that there might be, locally, an antiferromagnetic energy gap at the Fermi energy, so that the wavefunction of the highest energy filled state would decay further as it passes through this region.Our numerical results suggest however, that these effects, if they are present, are not strong enough to produce the type of localization one might have hoped for.
The qualitative resemblance between the calculated Hartree-Fock wavefunction and the WKB form for states above a smooth potential barrier, discussed above, appears to persist down to densities where there are only one or two electrons left at the top of the barrier.

C. Electrical conductance
A striking feature of the experimental results is the occurrence of a sharp drop in the conductance through the finite wire in a regime where there were still several electrons in the region below the center gate.(Herewe refer to the two terminal conductance G measured through contacts at two ends of the finite wire, and not the tunnelling conductance G T for current flowing between the finite wire and the semi-infinite wire).For a one-dimensional system of non-interacting electrons, in a potential V (x) that vanishes outside a central region, one can relate the electrical conductance, using the Landauer-Buttiker formula, to the transmission probability for an incident electron at the Fermi energy.For a potential barrier which is symmetric under reflection, the transmission probability, in turn, can be expressed in terms of the phase shifts for states of even and odd parity.In the Hartree-Fock approximation, however, this analysis is complicated by several factors.Although electron-electron interactions in the leads are relatively weak because of the high electron density there, they would still give rise to antiferromagnetic order, in a lead of infinite length, as discussed above.Thus, in principle there should always be an energy gap at the Fermi energy, and phase shifts cannot be defined.In practice, this should not be a serious problem for our system, because the calculated energy gap is extremely small at high densities, and one could estimate the conductance from phase shifts at energies outside of the energy gap, but still close to the Fermi energy.
A more significant problem arises from the fact that our computations use a system of finite length, and we have only a discrete set of energy levels.Analyzing these wave functions, we may obtain even-parity and odd-parity phase shifts at a discrete set of energies, but we do not obtain both even and odd phase shifts at a single energy.We can obtain some estimate of the phase shifts for an infinite system, however, by looking at the alternation between energy levels for even and odd numbered wave functions of a given spin and band index, in the finite system.
For an energy high above the barrier, we expect that WKB is a good approximation, which means that an incident particle is transmitted with essentially no reflection, corresponding to a conductance of e 2 /h per spin.This means that there is no difference in the phase shifts for even and odd parity.In our model calculations, we assume periodic boundary conditions at the ends of the wire.Then, for a large but finite system, when there is negligible reflection at the barrier, we expect energy levels to occur in pairs, with even and odd parity states that are nearly degenerate.
For an energy well below the barrier, where there is nearly total reflection, the even and odd parity phase shifts should differ by approximately π/2.Then, with periodic boundary conditions, we expect energy levels to alternate between even and odd parity states, with nearly equal spacings between them.
The energy spacings we find in our Hartree-Fock calculations are in good agreement with these expectations provided the Fermi level is not too close to the barrier top.Thus we have near perfect transmission when the Fermi level is well above the barrier, and near perfect reflection when it is well below.However, we have not been able to analyze the conductivity in the most interesting region, when there are only a few electrons in the spin-polarized region at the top of the barrier, essentially because our system size is too small, and we do not have enough energy levels in that region.
Finally, we note that the Landauer-Buttiker conductance discussed above applies to a wire that is connected to its leads by adiabatic, non-reflecting contacts.In the experiments by Steinberg et al. 7,8 , the contacts from the finite wire to the two dimensional electron gas(2DEG) are not perfectly adiabatic, and will add contact resistance to any resistance discussed above.

VI. SUMMARY
In summary, from our Hartree-Fock calculations, we have developed a picture of successive magnetic phases in the low density region of an inhomogeneous one-dimensional electron systems in a uniform magnetic field.The depleted electrons under the barrier first enter an antiferromagnetic phase, then, near depletion, part of the lowest density electrons become spin aligned and get isolated from the high density region outside the barrier by two antiferromagnetic regions sandwiching it.The final stage of depletion takes the form of successive expulsion of a single electron from the spin aligned region, resulting successive periods of relative insensitivity of the spin aligned electron density to V g , followed by the sudden rearrangement due to the expulsion of one electron.
The most serious discrepancy between our calculations and the experimental results of Steinberg et al is the form factor for momentum-conserved tunneling in the regime where there are of the order of four to ten electrons under the central gate.Our matrix elements have too much weight at large momenta, It is not clear what is the source of this discrepancy, However, it may be that potential fluctuations due to residual disorder are important.A small random potential due to charged impurities set back from the wire may have little effect on the mean free path for relatively high electron densities, but could lead to strong back scattering and localization at very low densi- ties, where the kinetic energy can be small, and where the small value of 2k F permits backscattering from potential fluctuations of relatively long wave length.As mentioned earlier, calculations near the depletion of the upper band at a smaller V g , give results similar to those obtained near the depletion of the lowest band.We find, again, a spin-aligned central region, sandwiched by antiferromagnetic regions on each sides.A similar phenomenon of sudden expulsions of a single localized electron from the spin aligned parts are also observed.This is consistent with the experimental observation of similar localization behavior upon the depletion of the second subband 8 .Small potential fluctuations due to impurities may again be important for explaining the experimental results for momentum-dependent tunneling in this regime.ual spins will cant towards the direction of the applied field.Hence, it may be asked whether allowing canting of the spins in a Hartree-Fock calculation would significantly change our results.To clarify this issue we have carried out both restricted and unrestricted Hartree-Fock calculations on a uniform infi-nite electron system in a magnetic field.The interaction between electrons is the same as we used before, and the magnetic field is also B = 1T .As shown in Fig. 13 and Fig. 14, the effects of canting in terms of both the energy gains and the changes in spin density are small.Canting is only relevant to our zero-temperature calculation when the antiferromagnetic solution is the ground states.Even near the transition density ρ * = 16 µm −1 from the antiferromagnetic ground state to the fully spin aligned ground state, when canting is the greatest, the energy gained by allowing canting is only 2.64 µeV per electron.
Extrapolating these results to the system with a barrier, we see that even if we allowed ten antiferromagnetic electrons on each side of the solution shown in Fig 7 to cant, the total energy gain would be minuscule compared with the level spacing at Fermi energy there.At ρ * , in the homogeneous system, the magnetization parallel to the uniform magnetic field, in the canted state, is less than one tenth of the antiferromagnetic magnetization perpendicular to it.Thus, our calculations for the homogeneous system suggest that allowing canting in the Hartree-Fock calculation of the inhomogeneous one dimensional wire would not give qualitatively different results from our restricted calculations above.
We remark that for a homogeneous system in zeromagnetic field, with the same electron-electron interaction as above, the onset of ferromagnetism in the Hartree-Fock approximation would occur at a density ρ * ≈ 14 µm

1 FIG. 3 :
FIG.3:The density distributions ρ σ b (x) for the different spin states and subbands in the ground state, in a high density regime where the center region is non-magnetic.Identifications of the indices σ , b for the curves are shown in the inset.For the lower subband (b = 0), the densities of the two spin states are indistinguishable, and are shown by a single curve.All densities are symmetric around x = 0, and only the center part x < 2 µm is shown.

FIG. 8 :
FIG.8: I ρ is the integrated number of spin up electrons in the lower band within the center region defined by −x 0 ≤ x ≤ x 0 , where x 0 = 0.498 µm.As a function of the gate voltage V g , we follow the lowest energy solution across the crossover shown in Fig.6, i.e. the I ρ on the left of the crossover is computed from the solution N f = 8, and the I ρ on the right from N f = 7.The drop in I ρ represents the expulsion of approximately 0.88 electron from within the center region.

FIG. 9 :
FIG. 9: (Color Online) High resolution measurement of the localized features from Steinberg et al. 8 .On the left is the raw tunneling data dG/dV G .On the right are the corresponding tunneling rates Γ(B), extracted from the fitting of the localized features to Coulomb blockade lineshape at each magnetic field.Γ(B) is proportional to |M(k)| 2 , where M is defined in Eq. 5 and B controls the momentum of the tunneling electron.

FIG. 10 :
FIG. 10: |M(k)| 2 for the transition shown in Fig. 6, from N f = 8 to N f = 7.The solid line is the squared Fourier transform of the wavefunction being expelled.The dotted line is computed from the overlap between full Slater determinants.

FIG. 11 :
FIG.11:This is the highest occupied eigenstate of the solution at V g = 71.74meV , right before the crossover the Fig.6.The wavefunction is normalized and only the center region is plotted.

FIG. 13 :
FIG.13:The energy difference per electron ∆ between the antiferromagnetic and the fully spin aligned solutions in an infinite uniform electron system, as a function of electron density.Both systems are in a uniform magnetic field B = 1T .The dotted curve is the canted solution and the solid curve is the solution with spin colinear to B.

FIG. 14 :
FIG.14:The spin polarization in x and z direction in the canted solution at density ρ = 16 µm −1 , right at the transition from the antiferromagnetic to the spin-aligned ground state.Due to the magnetic field in x direction, a small paramagnetic component is developed in addition to the dominant antiferromagnetic magnetization in z direction.Only a single unit cell, consisting of two electrons, are shown.The total density is also shown as the dotted line.
This plot shows the crossover, as a function of the barrier height V g , of the ground state energy E of two Hartree-Fock solutions, labeled by the number of electrons N f in spin aligned region at the center.For the sake of clarity, a quadratic function of V g is subtracted from each of the ground state.