Enhancement mechanism of the electron g-factor in quantum point contacts
aa r X i v : . [ c ond - m a t . m e s - h a ll ] A p r Enhancement mechanism of the electron g-factor in quantum point contacts
Gr´egoire Vionnet
School of Physics, The University of New South Wales, Sydney, NSW 2052, Australia andInstitute of Theoretical Physics, ´Ecole Polytechnique F´ed´erale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
Oleg P. Sushkov
School of Physics, The University of New South Wales, Sydney, NSW 2052, Australia (Dated: October 9, 2018)The electron g-factor measured in a quantum point contact by source-drain bias spectroscopy issignificantly larger than its value in a two-dimensional electron gas. This enhancement, establishedexperimentally in numerous studies, is an outstanding puzzle. In the present work we explain themechanism of this enhancement in a theory accounting for the electron-electron interactions. Weshow that the effect relies crucially on the non-equilibrium nature of the spectroscopy at finite bias.
PACS numbers: 72.25.Dc, 73.23.Ad, 71.70.Gm, 73.21.Hb
A quantum point contact (QPC) is a narrow quasi-1Dconstriction linking two 2D electron gas (2DEG) reser-voirs. It is essentially the simplest mesoscopic systemwhich makes it interesting both for technological appli-cations and on a fundamental level. Experimental studiesof QPCs started with the discovery of the quantisationof the conductance in steps of G = 2 e /h [1, 2], which isa single-particle effect well understood theoretically [3].Many-body interactions/correlations in QPCs were firstundoubtedly identified in the “0.7-anomaly” of the con-ductance and the g-factor enhancement [4], and a fewyears later in the zero bias anomaly (ZBA) of the con-ductance [5]. Since their discovery, these effects havebeen the subject of numerous experimental studies, seee.g. Refs. [6–10]. In spite of 20 years of studies thereis no consensus about the mechanism of the 0.7 & ZBA.We believe that they are both due to the enhanced in-elastic electron-electron scattering on the top of the QPCpotential barrier [11–13]. However, there are alternativetheoretical models of these effects based on various as-sumptions, see e.g. Refs. [14–16]. In the present work wedo not address the 0.7 & ZBA, but consider the mecha-nism underlying the electron g-factor enhancement. Weshow that a simple saddle-point potential model com-bined with local electronic interactions is sufficient tocapture the relevant physics. There are two previoustheoretical works related to this problem, Refs. [13, 17].Ref. [17] considers the usual Landau Fermi liquid ex-change interaction mechanism of the g-factor enhance-ment in an infinitely long quantum wire. This mechanismcan hardly be relevant for a QPC since the length of thequasi-1D channel connecting the leads is much shorterthan the spin relaxation length. Ref. [13] addresses areal QPC and points out a magnetic splitting enhance-ment effect. While this effect does exist, we will showbelow that it is exactly cancelled out in a source-drainbias spectroscopy experiment and therefore does not ex-plain the observed phenomenon.We consider the conduction band electrons in a semi-conductor. Due to the spin-orbit interaction in the va-lence band, the value of the single electron g-factor can be very different from its vacuum value. For example inGaAs, g = − .
44 [18]. This value can be measured infast processes, say in ESR, where g is not renormalisedby electron-electron interactions [19, 20]. On the otherhand, the static electron g-factor g ∗ measured for exam-ple via static Pauli magnetisation in an infinite system isenhanced compared to g due to the exchange electron-electron interactions [17, 21]. Considering that the timeof flight of an electron through a QPC is of the orderof the picosecond, how can the g-factor be renormalisedin such a fast process? We show below that the ob-served enhancement is specific to the source-drain biasspectroscopy method to measure the g-factor in QPCs.If we neglect the electron-electron interactions in theQPC the problem can be described by the saddle pointpotential created by the gates V ( x, y ) = V − mω x x + 12 mω y y , (1)with m the effective mass of the electron. The electriccurrent flows in the x -direction from the source to thedrain. The potential is separable and the QPC trans-mission problem is reduced to the solution of a one di-mensional Schr¨odinger equation with effective potential U ( x ) [3]. The potential is peaked at x = 0 where U n ( x ) ≈ U n − mω x x , U n = V + ¯ hω y ( n + 1 / . (2)Here n = 0 , , ... indicates the transverse channel. At aninfinitesimally small bias, the conductance is describedby the transmission coefficient at the Fermi level. Ap-plying an in-plane magnetic field B just spin splits theFermi level ǫ ± = µ ± g ∗ B/ µ the chemical po-tential and the Bohr magneton set to unity. The split-ting is determined by the g ∗ -factor which accounts forelectron-electron exchange interactions in the leads. Thequasi-1D channel has no significant impact on g ∗ sinceits length ( ∼ l s ∼ µ m). The energies ǫ ± andthe potential curves describing the QPC are sketched inFig.1a. FIG. 1: (a) Potential curves for the transverse channels n =0 (solid blue), n = 1 (dotted red) and n = 2 (dash-dottedgreen), and the magnetic field split chemical potential, ǫ ± .(b) Conductance in units of G = 2 e /h versus the saddlepoint potential height V . (c) Similar to (a), but at zeromagnetic field and different chemical potentials in the sourceand drain reservoirs: µ s − µ d = V sd . An electron wave function in a given transverse channel n is a combination of incident, reflected and transmittedwaves. Near the peak of the potential, the wave functionwith energy E k = k / m has the form (for k ≥
0) [11] ψ k,n ( x ) ≈ (cid:18) mv F ω x (cid:19) / ϕ ǫ n ( ξ ) ,ϕ ǫ n ( ξ ) = s e πǫ n / cosh( πǫ n ) D ν ( √ ξe − iπ/ ) ,ǫ n = ( E k − U n ) /ω x , ξ = x √ mω x . (3)Here v F is the Fermi velocity far from the barrier, D ν isthe parabolic cylinder function, ν = iǫ n − and ¯ h = 1.The sign of k indicates whether the electron is incidentfrom the left ( k ≥
0) or from the right ( k ≤ ρ ( ǫ n ) = | ϕ ǫ n (0) | Φ( ǫ n ) = π √ Z ǫ n −∞ ρ ( ǫ ′ ) dǫ ′ (4)plotted in Fig.2a. Due to semiclassical slowing, the FIG. 2: (a) Probability density at the top of the barrier ρ and integral of the probability Φ versus the electron energy.(b) Position of the conductance steps for non-interacting elec-trons in the QPC versus B at infinitesimal V sd . (c) Similarto (b) but versus the source-drain voltage V sd at B = 0. probability density at the top of the potential barrier, | ψ k,n (0) | ∝ ρ ( ǫ n ), is peaked at ǫ n ≈ . x = 0 ina given transverse channel.Before considering the g-factor enhancement, we il-lustrate the high susceptibility of a QPC to a mag-netic field B . Having the wave functions, it is easy tocalculate the induced magnetisation M ( x, y ) across theQPC which is directly relevant to the recent NMR ex-periment in Ref. [22]. Here we consider only the linearresponse to B . The only effect of electron-electron in-teractions is to replace g → g ∗ (and renormalise U n and ω x , see below). The magnetisation in the leads is M ( ∞ , y ) = m π g ∗ B and the magnetisation at the neck ofthe QPC is M (0 ,
0) = m π q π ω y ω x ρ ( µ − U ω x )g ∗ B , assumingthat only the n = 0 channel is open. The latter dependssignificantly on the energy through ρ and the maximumenhancement is fairly large, M (0 , /M ( ∞ , ≈ . ω y /ω x ≈
3. This single-particle effect is, however, un-related to the g-factor measurement which we discuss inthe next paragraph.The potential curves in Fig.1a can be lowered andraised by varying the QPC potential height V . Whenthe top of a potential curve crosses either one of the ǫ ± horizontal lines, the conductance is changed by G / δ B in V , as illustrated in the plot of the con-ductance versus V in Fig.1b. Without accounting forthe electron-electron interactions in the QPC, the split-ting is δ B = g ∗ B . The standard way to represent Fig.1bis to plot the position of the steps versus magnetic fieldas shown in Fig.2b. The slope of the lines in Fig.2b is re-lated to the g-factor and this is the basis for the g-factormeasurement. Of course, only the absolute value can bedetermined, g → | g | . Unfortunately, in experiments V isunknown and only the gate voltage V g is directly accessi-ble. V is proportional to the gate voltage, V = αV g , anda non-equilibrium method known as source-drain biasspectroscopy is used to exclude the unknown coefficient α . The magnetic field is set to be zero, but a finite bias V sd directly controlled experimentally is applied acrossthe QPC. The difference between the source and thedrain chemical potentials is µ s − µ d = V sd , as illustratedin Fig.1c. Similarly to the magnetic splitting case, whenthe top of a potential curve crosses the µ s or µ d level, thedifferential conductance is changed by G /
2. Again, eachtransverse channel leads to two split-steps separated by δ sd in V . For non-interacting electrons δ sd = V sd . Theposition of the steps versus source-drain voltage is shownin Fig.2c. The QPC g-factor isg Q = ( dδ B /dB )( dδ sd /dV sd ) = ( ∂V g /∂B )( ∂V g /∂V sd ) , (5)where the derivatives are taken at the same gate voltage.The unknown coefficient α is cancelled out in the ratioin Eq. (5). Disregarding electron-electron interactions inthe QPC, g Q = g ∗ .Due to the many-body screening, the effective electron-electron Coulomb interaction is short-ranged and can beapproximated by a δ -function V c ( x , x ) = π λ r ω x m δ ( x − x ) . (6)Here we assume that the interaction is diagonal in trans-verse channels, ∝ δ n ,n . In principle there is also anoff-diagonal interaction, but it does not influence ourconclusions and is only relevant when several transversechannels are populated. The dimensionless coupling λ is the four-leg vertex function which generally dependson x and on the electron energy, λ → λ ( x, ǫ ). For ourpurposes we need only λ ( x = 0 , ǫ ) = λ ( ǫ ). We have per-formed a random phase approximation (RPA) calculationof λ ( ǫ ) for the n = 0 channel in GaAs with ω y /ω x = 3and ω x = 1meV, see Appendix B. It turns out that inthe present case the results are well approximated bydiscarding the energy-dependence and taking λ = 0 . λ ∼ . / √ ω x (with ω x in meV) in Ref. [11]. We comment further on the ǫ -dependence of λ below.The interaction leads to self-energy corrections to theelectron energy, ǫ k → ǫ k + Σ. We first consider theHartree approximation, for which the self-energy Σ isgiven by the diagram shown in Fig.3a. This is equiva-lent to a self-consistent potential of electrons which givescorrections to the height of the potential U n and to ω x .We focus on the former. The potential at the top of thebarrier is the sum of the potential V created by the gatesand the self-consistent potential created by the local den-sity of electrons. In a magnetic field, the conductancesteps arise when the top of a potential barrier in Fig.1atouches a horizontal dashed line (’ ǫ + ’ or ’ ǫ − ’). Hence theconditions for the conductance steps are’ ǫ + ’ : c n − V ω x = − g ∗ B ω x + 2 λ (cid:20) Φ(0) + Φ (cid:18) − g ∗ Bω x (cid:19)(cid:21) ’ ǫ − ’ : c n − V ω x = g ∗ B ω x + 2 λ (cid:20) Φ(0) + Φ (cid:18) g ∗ Bω x (cid:19)(cid:21) (7)where c n is a constant that depends on the transversechannel. The Φ(0)-terms are due to interactions betweenelectrons with same spins, whereas the Φ( ± g ∗ B/ω x )-terms are due to interactions between electrons withopposite spins. The latter terms yield an additional B -dependence compared to the non-interacting case.The position of the steps versus magnetic field whichfollow from Eqs.(7) for λ = 0 .
25 and ω y /ω x = 3 areshown in Fig.3b by black solid lines. For comparison, thedashed blue lines show the non-interacting case, identi-cal to Fig.2b. Fig.3b indicates a very significant Hartreeenhancement of the splitting: δ B > g ∗ B . In Ref. [13] thiseffect was reported as enhancement of the g-factor. How-ever, let us look at how the source-drain normalisationin Eq. (5) influences the answer. The conditions for thesource-drain conductance steps in the Hartree approxi- FIG. 3: (a) Hartree self-energy diagram. (b) Position ofthe conductance steps versus B at infinitesimal V sd withthe electron-electron interactions in the QPC accounted inthe Hartree approximation. Black solid lines correspond to λ = 0 .
25 and ω y /ω x = 3. Blue dashed lines correspond to thenoninteracting case, λ = 0. (c) Similar to (b) but versus thesource-drain voltage V sd at B = 0. mation are’ µ s ’ : c n − V ω x = − V sd ω x + 2 λ (cid:20) Φ(0) + Φ (cid:18) − V sd ω x (cid:19)(cid:21) ’ µ d ’ : c n − V ω x = V sd ω x + 2 λ (cid:20) Φ(0) + Φ (cid:18) V sd ω x (cid:19)(cid:21) . (8)The factor 2 in front of λ in Eqs.(7) and (8) arises fordifferent reasons. While in Eq.(7) it is due to the left-runners and the right-runners contributing to the density,in Eq. (8) it is due to the two spin polarisations. Due tothe coincidence of the prefactors, Eqs. (7) and (8) areidentical upon the substitution g ∗ B ↔ V sd . Hence, theplots of the position of the source-drain steps shown inFig.3c in black solid lines are identical to those in Fig.3b.(Again we show in blue dashed lines the non-interactingcase). Therefore in Eq. (5) the “enhancement” is can-celled out and g Q = g ∗ . There is no enhancement of theg-factor measured by source-drain bias spectroscopy dueto the Hartree term. Besides this analytical calculation,we have performed an equilibrium self-consistent Hartreenumerical calculation for a realistic QPC in a 3D geome-try in the adiabatic approximation, see Appendix A. Thisnumerical calculation supports the above conclusion.We now account for the Fock exchange term, for whichthe self-energy diagram is plotted in Fig.4a, and discardthe Hartree self-energy. Although in general this contri-bution to the self-energy leads to a nonlocal potential,in the δ -function approximation (6) it becomes a localpotential (generally spin-dependent). We can thereforeapply the same procedure as in the Hartree case. How-ever, the Fock self-energy is negative and for an electronwith a given spin depends only on the density of elec-trons with the same spin. Therefore, the conditions forthe conductance steps in a magnetic field read’ ǫ + ’ : c n − V ω x = − g ∗ B ω x − λ Φ(0)’ ǫ − ’ : c n − V ω x = g ∗ B ω x − λ Φ(0) . (9)It is very similar to the direct interaction case (7), butthe sign of the λ -terms is opposite and there is noterm related to interactions between electrons with op-posite spins. Therefore the exchange contribution is B -independent and the position of the conductance steps,shown in Fig.4b, are identical to the non-interacting case,Fig.2b. There is no exchange enhancement of the split-ting, δ B = g ∗ B . This is counterintuitive and very differ- FIG. 4: (a) Fock self-energy diagram. (b) Position of the con-ductance steps versus B at infinitesimal V sd with the electron-electron interactions in the QPC accounted in the Fock ap-proximation. Black solid lines correspond to λ = 0 .
25 and ω y /ω x = 3. Blue dashed lines correspond to the noninteract-ing case, λ = 0. (c) Similar to (b) but versus the source-drainvoltage V sd at B = 0. The red dotted line accounts for theenergy dependence of λ ( ǫ ) in the RPA. ent from what we know well about uniform systems [21].However, this does not imply that the g-factor measuredby source-drain bias spectroscopy is not changed. At zeromagnetic field and finite bias, including the exchange con-tribution, the conditions for the conductance steps are’ µ s ’ : c n − V ω x = − V sd ω x − λ (cid:20) Φ(0) + Φ (cid:18) − V sd ω x (cid:19)(cid:21) ’ µ d ’ : c n − V ω x = V sd ω x − λ (cid:20) Φ(0) + Φ (cid:18) V sd ω x (cid:19)(cid:21) . (10)Compared to the Hartree case (8), the interaction con-tributions have opposite signs and there is no factor 2since only electrons with same spin contribute. The po-sition of the steps versus V sd is shown in Fig.4c in blacksolid lines. For comparison, the red dotted line showsa curve taking into account the energy dependence of λ ( ǫ ) obtained from the RPA calculation. It is practicallyindistinguishable from the black solid line, thus justify- ing approximating λ by a constant. The exchange in-teraction reduces the splitting δ sd compared to the non-interacting case, δ sd < V sd . Hence from Eq. (5), theg-factor is enhanced. With parameters corresponding tothe presented plots ( λ = 0 . ω y /ω x = 3) the g-factorenhancement is g Q / g ∗ ≈ . − λ one increases the enhance-ment, but then of course the single loop analysis becomesquestionable. When several transverse channels are pop-ulated, the screening must reduce the value of λ , thusreducing the g-factor enhancement in agreement with ex-periments [4].We stress that g Q is a multiplicative of g ∗ which is itselfsomewhat enhanced compared to g due to the exchangeinteraction in the leads. Implicit support for our analysiscomes from the experiments [23, 24]. They measure theelectron g-factor g via magnetic field and temperature de-pendence of the magnetic field induced spin polarisationin QPC injection. This method does not rely on source-drain bias spectroscopy and gives g ≈ g . The fact thatg = g Q supports our analysis which is consistent withg = g ∗ ≈ g , see also [25]. An explicit confirmation ofour theory would come from a conductance measurementof the g-factor which does not rely on the source-drainbias spectroscopy.In conclusion, the g-factor enhancement in the Hartree-Fock approximation is g Q / g ∗ = H H − F ≈ (1 + F ), where H and F are the Hartree and Fock electron-electron in-teraction contributions respectively. The numerator isdue to the magnetic splitting and the denominator is dueto the source-drain normalisation. Contrary to naive ex-pectations, the exchange Fock diagrams do not increasethe magnetic splitting. 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Lett. , 116802 (2009). Appendix A: Self-consistent Hartree simulation in a realistic 3D geometry
FIG. 5: (a) Model of a QPC as three equidistant two-dimensional layers: a 2DEG, a metallic plate with gates and an image2DEG. At z = 0, the gray metal gates have voltage V g and the remaining of this plane is grounded. The electron density in the2DEG at z = − d is n ( x, y ), whereas in the image 2DEG at z = d it is − n ( x, y ). (b) Top view of the 2DEG plane ( z = − d ). Thegray areas show the completely depleted regions where n ( x, y ) = 0. The regions denoted I (green) and II (blue) are treated inan adiabatic and Thomas-Fermi approximation respectively. We consider here a simple electrostatic model of a QPC that takes into account the three-dimensional geometryof the experimental set-up. We focus on the Hartree interaction and ignore any exchange effect. The system isapproximated by three equidistant two-dimensional layers, as illustrated in Fig.5a. The 2DEG is situated at z = − d ,and its image is at z = d . The constriction is formed by applying a potential V g on the gray metal gates in the layerat z = 0. We furthermore impose a zero potential outside of the gates in this plane. The total potential U ( x, y, z )is the sum of the electrostatic potential due to the metal gates and the Hartree potential U H ( x, y, z ) induced by theelectrons in the 2DEG and its image. In the plane z = 0, the total potential reads U | z =0 = U gates = (cid:26) − eV g in gates . (A1)Writing ~r = ( x, y ), the induced Hartree potential is U H ( ~r , z ) = e πε ε r Z d ~r ′ d z ′ n ( ~r ′ ) p | ~r − ~r ′ | + ( z − z ′ ) [ δ ( z ′ + d ) − δ ( z ′ − d )] . (A2)This problem is simplest upon Fourier transforming the in-plane coordinates: ˆ U ( ~q, z ) = R d ~r U ( ~r , z ) e i~q · ~r . Astraightforward calculation yieldsˆ U ( ~q ) (cid:12)(cid:12)(cid:12) z = − d = ˆ U gates ( ~q ) e − qd + e ε ε r ˆ n ( ~q ) q (cid:0) − e − qd (cid:1) (A3)where q = | ~q | .We first describe the simulation at zero bias and and we then explain how we treat the non-equilibrium case.
1. Zero bias: V sd = 0 We split the in-plane space ( x, y ) in regions I and II as shown in Fig.5b and use different approximations for thedensity n ( x, y ) in each region. The region I is defined to be where the approximation n I ( x, y ) used in this region isvalid. In region II , we use the Thomas-Fermi approximation n II ( x, y ) = m π X σ = ± (cid:16) µ + σ g ∗ B − U ( x, y ) (cid:17) θ (cid:16) µ + σ g ∗ B − U ( x, y ) (cid:17) (A4) FIG. 6: (a) Potential heights E ( j )0 of the transverse channels j = 0 , , V g . The solid lines areguides to the eye. (b) Magnetic splitting for the conduction channels j = 0 , , ∗ B . Thesolid lines are linear fits based on the data for g ∗ B > . where U ( x, y ) = U ( x, y, − d ) is the potential in the 2DEG plane. In the region I , we use the adiabatic approximationand approximate the wavefunctions as Ψ ( j ) k ( x, y ) = ψ ( j ) k ( x ) χ ( j ) x ( y ) with j labelling the transverse channels. The ψ ( j ) k ( x )are 1D scattering wavefunctions with the incident part having momentum k asymptotically far from the constriction.Hence the sign of k indicates the direction of the incident wave. We first solve numerically the one-dimensionalSchr¨odinger equation at fixed x : − m ∂ χ x ( y ) ∂y + U ( x, y ) χ x ( y ) = E x χ x ( y ) . (A5)This yields for each x a set of eigenfunctions χ ( j ) x ( y ) and eigenvalues E ( j ) x with j = 0 , , . . . . The wavefunctions ψ ( j ) k ( x ) are then found by solving − m ∂ ψ ( j ) k ( x ) ∂x + E ( j ) x ψ ( j ) k ( x ) = k m ψ ( j ) k ( x ) (A6)where the x -dependent transverse energies E ( j ) x play the role of an effective potential. Close to the constriction, wecan approximate E ( j ) x ≈ E ( j )0 − m (cid:16) ω ( j ) x (cid:17) x for which we know the exact solution as a function of the paraboliccylinder functions. The validity of this harmonic approximation defines the region I . The density in that region is n I ( x, y ) = X σ = ± X j X −∞ 10 g ∗ B . 2. Finite bias: V sd = 0 We restrict the discussion here to zero magnetic field. In the region I , we have for V sd = 0 n I ( x, y ) = 2 X j | χ ( j ) x ( y ) | n X k> | ψ ( j ) k ( x ) | θ ( µ s − k m ) + X k< | ψ ( j ) k ( x ) | θ ( µ d − k m ) o (A8) FIG. 7: Comparison between the out-of-equilibrium 1D density n x and equilibrium approximation n eq x , using as unit lengththe dimensionless ξ = √ mω x x . The chemical potentials are: (a) µ s − E = 0 . ω x and µ d − E = 0; (b) µ s − E = 0 and µ d − E = − . ω x . where the factor 2 comes from the spin degeneracy. However, we cannot write such an out-of-equilibrium expressionfor the density in the region II since the Thomas-Fermi approximation assumes local equilibrium. Therefore ourmethod cannot treat a non-equilibrium situation and we need a simplifying approximation. We are mostly interestedin what happens close to the top of the potential, i.e. for small | x | . We therefore average symmetrically around x = 0, n ( x, y ) → 12 ( n ( x, y ) + n ( − x, y )) . (A9)To justify this substitution, let’s consider the one-dimensional density for a single transverse channel n x ( x ) = X k> | ψ k ( x ) | θ ( µ s − k m ) + X k< | ψ k ( x ) | θ ( µ d − k m ) (A10)and its equilibrium approximation n eq x ( x ) = ( n x ( x ) + n x ( − x )). Considering a parabolic potential barrier, these 1Ddensities can be expressed in dimensionless densities with ξ = √ mω x x . They are compared in Fig.7 for the cases µ s − E = 0 . ω x , µ d − E = 0 and µ s − E = 0 , µ d − E = − . ω x . The equilibrium approximation moves someelectrons from the source ( x < 0) to the drain ( x > x = 0.Since ψ ( j ) k ( − x ) = ψ ( j ) − k ( x ), the substitution (A9) yields n I ( x, y ) → X σ = ± X j X −∞ FIG. 8: Diagrammatic representation of the Dyson self-consistent equation for the screened interaction in the RPA. We detail here a random phase approximation (RPA) calculation of the screened Coulomb interaction for electronsin the lowest transverse channel ( n = 0) and use it to justify the δ -function model presented in the main text.The 2D wavefunction for the lowest transverse channel is Ψ k, ( x, y ) = ψ k, ( x ) χ ( y ) with χ ( y ) = (cid:0) mω y π (cid:1) e − mωyy .The 1D bare Coulomb interaction is obtained by averaging over the transverse direction: V c ( x, x ′ ) = e πε ε r Z d y Z d y ′ | χ ( y ) | | χ ( y ′ ) | p ( x − x ′ ) + ( y − y ′ ) . (B1)In dimensionless lengths ξ = x √ mω x , V c ( ξ, ξ ′ ) = ω x Λ f ( ξ − ξ ′ , ω y ω x ) (B2)with Λ = e πε ε r q mω x ≈ . 35 for typical experimental parameters in GaAs ( ǫ r = 13, m = 0 . m e , ω x ≈ f ( ξ, α ) = απ Z d η Z d η ′ e − α ( η + η ′ ) p ξ + ( η − η ′ ) → | ξ | | ξ | → ∞ q απ | ln | ξ || | ξ | → . (B3)Here α = ω y /ω x . Furthermore, because the QPC is not isolated, there will be an additional screening from the metalgates that will be important for large | ξ | . We model this qualitatively by using the bare interaction V c ( ξ, ξ ′ ) = ω x Λ f ( ξ − ξ ′ , ω y ω x ) e − ( ξ − ξ ′ )2 τ = ω x ˜ V c ( ξ, ξ ′ ) (B4)with τ = 10. The somewhat arbitrary choice of τ has, however, very little influence on the results presented below.The screened interaction V c ( ξ, ξ ′ ) = ω x ˜ V c ( ξ, ξ ′ ) is found by solving numerically the Dyson self-consistent equation˜ V c ( ξ, ξ ′ ) = ˜ V c ( ξ, ξ ′ ) + Z d ξ d ξ ˜ V c ( ξ, ξ )Π( ξ , ξ ) ˜ V c ( ξ , ξ ′ ) . (B5)The RPA approximation of this relation is depicted diagrammatically in Fig.8. In this approximation, the staticpolarisation reads,Π( ξ , ξ ) = X σ = ± X δ ,δ = ± Z ˜ µ + σ g ∗ B ωx −∞ d ǫ π Z ∞ ˜ µ + σ g ∗ B ωx d ǫ π φ ǫ ,δ ( ξ ) φ ∗ ǫ ,δ ( ξ ) φ ǫ ,δ ( ξ ) φ ∗ ǫ ,δ ( ξ ) ǫ − ǫ (B6)where ˜ µ = µ − U ω x and the δ j = ± indicate the sign of k , i.e. the direction of the electron.Because the system is non-uniform, the polarisation is not uniform and V c ( ξ, ξ ′ ) = V c ( ξ − ξ ′ ). The dimensionlessinteraction ˜ V c (0 , η ) between an electron at the centre of the QPC and an electron at position η is compared to thebare interaction in Fig.9a for ˜ µ = 0, B = 0 and ω y = 3 ω x = 3meV. The dimensionless interaction ˜ V c (1 , η ) betweenan electron at ξ = 1 and an electron at position η for the same parameters is shown in Fig.9b.The short-ranged screened interaction justifies the δ -function approximation V c ( ξ, ξ ′ ) → ω x π λ ( ξ, ˜ µ, B ) δ ( ξ − ξ ′ ) (B7)with the coupling constant λ ( ξ, ˜ µ, B ) = 1 π Z ∞−∞ ˜ V c ( ξ, η )d η . (B8)0 FIG. 9: (a) Screened and bare dimensionless interactions between an electron at the centre of the QPC and an electron atposition η for ˜ µ = 0, B = 0 and ω y = 3 ω x = 3meV. (b) Same as (a) but for an electron at ξ = 1 and an electron at position η . The coupling constant at ξ = 0 and ξ = 1 is plotted as a function of ˜ µ in Fig.10a for B = 0. As ˜ µ becomes negativeand the constriction depopulates, the screening becomes less effective. For ˜ µ < ∼ − . 5, the contact approximationbecomes questionable as the screened interaction gets longer-ranged and eventually tends to the bare interaction.This is qualitatively different from the RPA result for a uniform quantum wire which predicts an unphysical vanishingcoupling constant as the density vanishes (as a consequence of the divergence of the density of state). This issue doesnot arise in the present case of a QPC.In the magnetic splitting, the step-positions (for the lowest transverse channel) are obtained when ǫ + = U (lowerline going down in e.g. Fig.3b of the main text) or ǫ − = U (lower line going up). The coupling constant λ as afunction of magnetic field B with either one of these conditions satisfied is shown in Fig.10b. This result applies also tothe source-drain splitting ( ǫ ± → µ s/d , g ∗ B → V sd ) by applying an analogous equilibrium approximation as in sectionA 2. Remarkably, even though the coupling λ can in principle vary significantly, as in Fig.10a, it only varies by < ∼ B (or V sd ), thus explaining whythere is almost no perceptible difference in Fig.4c of the main text between the result with energy-dependent coupling(red dots) and with constant coupling λ = 0 . 25 (black lines). FIG. 10: (a) Coupling constant λ ( ξ, ˜ µ, B ) at ξ = 0 (centre of the QPC) and ξ = 1 as a function of ˜ µ for ω y = 3 ω x = 3meVand B = 0. (b) Coupling constant λ ( ξ, ˜ µ, B ) at the centre of the QPC ( ξ = 0) as a function of the magnetic field B at thepositions of conductance steps. The condition ǫ − = U yields the lower line going up (in e.g. Fig.3b of the main text) whereasthe condition ǫ + = U00