Observing separate spin and charge Fermi seas in a strongly correlated one-dimensional conductor
P. M. T. Vianez, Y. Jin, M. Moreno, A. S. Anirban, A. Anthore, W. K. Tan, J. P. Griffiths, I. Farrer, D. A. Ritchie, A. J. Schofield, O. Tsyplyatyev, C. J. B. Ford
OObserving separate spin and charge Fermi seas in astrongly correlated one-dimensional conductor
P. M. T. Vianez, , ∗ Y. Jin, M. Moreno, A. S. Anirban, A. Anthore, W. K. Tan, J. P. Griffiths, I. Farrer, , † D. A. Ritchie, A. J. Schofield, O. Tsyplyatyev, , ∗ C. J. B. Ford , ∗ Department of Physics, Cavendish Laboratory, University of Cambridge,Cambridge, CB3 0HE, UK Departamento de F´ısica Aplicada, Universidad de Salamanca, Plaza de la Merced s/n,37008 Salamanca, Spain Universit´e de Paris, C2N, 91120 Palaiseau, France Department of Physics, Lancaster University, Lancaster, LA1 4YB, UK Institut f¨ur Theoretische Physik, Universit¨at Frankfurt, Max-von-Laue Straße 1,60438 Frankfurt, Germany † Present address: Department of Electronic & Electrical Engineering, University of Sheffield,3 Solly Street, Sheffield, S1 4DE, UK. ∗ To whom correspondence should be addressed; E-mail: [email protected],[email protected], [email protected]
Abstract
An electron is usually considered to have only one type of kinetic energy, but could ithave more, for its spin and charge, or by exciting other electrons? In one dimension (1D),the physics of interacting electrons is captured well at low energies by the Tomonaga-Luttinger-Liquid (TLL) model, yet little has been observed experimentally beyond thislinear regime. Here, we report on measurements of many-body modes in 1D gated-wiresusing a tunnelling spectroscopy technique. We observe two separate Fermi seas at highenergies, associated with spin and charge excitations, together with the emergence of threeadditional 1D ‘replica’ modes that strengthen with decreasing wire length. The effectiveinteraction strength in the wires is varied by changing the amount of 1D inter-subbandscreening by over 45%. Our findings demonstrate the existence of spin-charge separationin the whole energy band outside the low-energy limit of validity of the TLL model, andalso set a limit on the validity of the newer nonlinear TLL theory. a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b any-body systems cannot be explained by studying their individual components, with in-teractions often giving rise to collective excitations from which an array of qualitatively newquasi-particles starts to emerge. This is particularly striking in one dimension (1D), as heregeometrical confinement alone imposes strong correlations in the presence of any interactions,leading to well-known non-Fermi-liquid phenomena such as spin-charge separation [1]. Over-all, the behaviour of 1D interacting systems in the low-energy regime is well captured by theTomonaga-Luttinger-Liquid (TLL) model [2–4], and has been extensively tested in carbon nan-otubes [5–7], semiconductor quantum wires [8–10], antiferromagnets [11] and more, recently,cold-atom chains [12]. The model, which assumes a linearised single-particle dispersion, is ex-pected to only be valid close to the Fermi points, where nonlinearities are still weak. However,pronounced consequences of band curvature have just very recently started to be explored exper-imentally [13–15]. At the same time, modelling such systems is a long-standing open problem.The phenomenologically introduced Nonlinear Luttinger Liquid [16] predicts that spin-chargeseparation no longer exists since the holons are made unstable by the nonlinearities, with a mix-ture of spinons and holons being responsible for the power-law threshold behaviour around thespectral edges [17]. Another microscopic theory [18] anticipates that extra 1D ‘replica’ modesshould emerge as a function of system length. Some signatures of these have been observedin the past [19, 20] but experimental results beyond the linear regime that have enough resolu-tion and clarity to distinguish between different predictions made by various models has beenlacking.Here, we measure the spectral function for the spin and charge excitations well beyond thelinear regime using a tunnelling spectroscopy technique that allows mapping of the system bothin energy and momentum. In the amplitude of our signal, we observe for the first time how thetwo branches of the linear TLL modes evolve away from the Fermi points in both particle andhole sectors into two fully formed Fermi seas, with different densities, that consist of purelyspin or charge quasi-particles, identified by comparison with the spectra predicted by the 1DFermi-Hubbard model [21]. This result gives a direct proof that the spin and charge collectiveexcitations both remain stable in the whole conduction band, well beyond the low-energy limitof the original TLL model where their existence was first established. We are able to tune thedegree of screening of the Coulomb interaction by changing the confinement in our wires andso the number of occupied subbands. This is accompanied by a variation of approximately 45%of the two-body interaction energy and allows us to trace the change in the difference of thetwo masses (for the nonlinear spin and charge quasi-particles). Measuring wires of differentlengths, we are also able to observe up to three separate 1D nonlinear ‘replica’ modes of thespinon type that systematically emerge as the length decreases.Our experiment consists of a tunnelling spectrometer made on a GaAs / Al . Ga . As het-erostructure with two parallel quantum wells (QWs), grown by molecular-beam epitaxy (MBE).We measure the tunnelling conductance G = d I/ d V DC between the 1D wires and the 2D layerat lattice temperatures T ∼
300 mK (see Fig. 1A), where I is the tunnelling current while V DC the DC bias applied between the layers. Tunnelling occurs when filled states in one systemhave the same energy and momentum as empty states in the other, therefore ensuring that both2 igure 1. Mapping a 1D system via magnetotunnelling spectroscopy. ( A ) Schematic representation of the 1D-2D spectrometer device. Wemeasure momentum-resolved tunnelling to and from an array of 1D wires (only one wire shown here for simplicity) and a 2D electron system,and map the elementary excitations in each system by measuring the tunnelling conductance while varying both their energy ∆ E ∝ V DC andmomentum ∆ k ∝ B . Current flows from the source into the wire and tunnels between the layers in order to reach the drain. ( B ) Scanningelectron micrographs of the various surface gates present in our device. See Supplementary Methods for details on gate operation and how toset up the tunnelling regime. Inset: Air-bridge interconnections between surface gates. ( C ) 1D wire subbands participating in the tunnellingprocess. We observe between four 1D subbands, when the wires become defined as the wire-gate voltage V WG decreases from just below thebottom of the 2D band, and one, and then pinch-off of the wires. energy and momentum are conserved. In order to map the dispersion of each system, a negative(positive) voltage V DC applied to the 1D wires provides energy for tunnelling from (to) 1D statesbelow (above) the Fermi level, while an in-plane magnetic field B perpendicular to the wiresboosts the momentum, offsetting the spectral functions of each system by ∆ k = eBd/ ¯ h , where e is the electronic charge and d the separation between the wells. The differential tunnellingconductance G displays resonant peaks corresponding to maximal overlap of the offset spectralfunctions (dispersion relations). The device therefore behaves as a spectrometer, with the well-characterised 2D system being used to probe the less-well understood spectral function of the1D system.We use a surface-gate depletion technique in order to establish separate contacts to eachwell. Our 1D system consists of an array of ∼
400 highly regular quantum wires formed in the3pper layer by using a set of wire gates (WGs) fabricated on a Hall bar via standard electron-beam lithography and connected by air bridges (see Fig. 1B and inset). Use of an array averagesout impurities, length resonances and charging effects as well as increasing the overall strengthof the measured signal. For the shorter devices, the air bridges are crucial for ensuring thatgood uniformity is obtained along the entire length of the wire, which would otherwise becomenarrower at one end if instead all the gates were joined by a continuous metal strip. Current isinjected into the 1D wires via a small region, ‘p’, 0.45 µ m wide. Unlike the wires, however, thisregion is 2D in nature and can be readily distinguished from the 1D signal in the measured datasince its density is different. We use the unconfined weakly interacting 2D electron gas in thebottom well as a well-understood spectrometer.A plot of d G/ d B vs B and V WG shows U-shaped curves, one per 1D subband (Fig. 1C).We start our experiment by choosing V WG so that there is just one 1D subband occupied. Fig.2C shows an example of such a measurement, with conductance through the sample beingmeasured as a function of energy ( ∝ V DC ) and momentum ( ∝ B ). The 1D Fermi wave vector k = ed ( B + − B − ) / h is determined from the crossing points ( B − and B + ) along the V DC = 0 line. The electron density in the wires is n = 2 k /π , which then gives the interactionparameter r s = 1 / (2 a B n ) , where a B is the Bohr radius of conduction electrons in GaAs. Thedensity can be controlled by tuning V WG , reducing it down to n ∼ µ m − before the wirespinch off.The curves drawn over the data in Fig. 2C are those expected from single-electron tunnellingprocesses. Unavoidable ‘parasitic’ tunnelling coming from the narrow 2D injection (‘p’) region(marked by the dashed black curves) produces a background in the form of a set of parabolicdispersions, which can be subtracted once separately mapped (with the wire gates pinched off).On the other hand, dashed blue curves reveal the elementary excitations of the 2D lower well,as probed by the 1D wires.The other strong features resemble the parabolae seen in 2D-2D tunnelling, for the 1D sys-tem. However, no single set of parabolae fits the data well (see Supplementary Notes) indicatingthe presence of spin and charge modes. In order to identify the spin and charge modes in the1D tunnelling signal, we interpret them using the dispersion of the 1D Fermi-Hubbard model inthe semiconductor limit, in which many-body spectra are described completely by the Lieb-Wuequations [21]. This system of nonlinear coupled equations is solved for two types of momen-tum states, k c j (for charge) and k s j (for spin degrees of freedom), which for the ground stateform two filled Fermi seas marked by full circles of two different colours in Fig. 2B (for moredetails see Supplementary Discussion). An excitation, say an electron tunnelling out of thewire, removes one charge and one spin simultaneously to reassemble a free electron, markedby a pair of green and magenta dashed circles in Fig. 2B. Placing the hole of one type at itscorresponding Fermi energy and moving the other one through the band describes the spectrumof the purely holon or spinon modes. While the momentum of these collective excitations as awhole is well-defined due to the translational invariance, with k = k F − ∆ P c or k = k F − ∆ P s ,the constituent degrees of freedom form non-equidistant distributions of their (quasi-)momentadepending in detail on the interaction strength and the positions of the two holes, owing to the4 igure 2. Two Fermi seas. ( A ) Dispersion for an interacting 1D system (grey: continuum of many-body excitations, green lines: spinonmodes, magenta lines: holon modes). ( B ) Solutions of Lieb-Wu equations for charge ( k c j ) and spin ( k s j ) degrees of freedom describing the pureholon (top) and spinon (bottom) excitations of an electron tunnelling out of the 1D chain. ( C ) Maps of tunnelling conductance ( G ) differentialsd G/ d V DC (top) and d G/ d B (bottom) vs DC-bias V DC and in-plane magnetic field B , for a µm -long device. Superimposed curves markall possible single-electron tunnelling processes, including the resonance process of the spin and charge modes marked by dashed green andmagenta lines, respectively. ( D ) d G/ d B zoomed in around the + k F point at negative biases (hole sector) where both spin (S) and charge(C) lines can be seen. ( E ) d G/ d V DC above + k F , showing the absence of a spinon in the particle sector (expected along green dashed line).( F ) Ratio of holon-to-spinon masses and spinon-to-holon velocities vs interaction parameter r s for devices of different lengths, extracted fromfitting a few 1D subbands. ( G ) G vs V DC at B = 0 T for pure parasitic 2D-2D tunnelling (wires pinched off) and the combination of the two1D-2D processes shown schematically. k c j and k s j for the two kinds of pure excitations produces the two dispersions drawn as the magenta andgreen solid lines in Fig. 2A.Around the ± k F points, these curves are almost linear, characterised by two different slopes v c and v s , parameters of the spinful TLL model. These two velocities are related microscopi-cally to the Hubbard interaction parameter U [22, 23], and the spectral function predicted bythe linear TLL theory displays two strong peaks on these two branches [24, 25], which havebeen measured in semiconductor quantum wires [8, 9]. Away from the Fermi points, the spec-tra of holons and spinons extend naturally to the nonlinear region, evolving into two separatecurves that are close to parabolae described by masses m c and m s , respectively. These shapesindicate formation of two separate Fermi seas by the nonlinear excitations. Their dispersionscross the Fermi energy at two different pairs of Fermi points ( ± k F and ± k F ) since the numberof holons is twice the number of spinons for the spin-unpolarised wires in our experiments,making the densities for the two kinds also different by the same factor (see SupplementaryDiscussion). The ratio of their masses depends on U , deviating further from the free-particlevalue m c / (2 m s ) = 1 with increase of the interaction strength.The dispersion of the strongest features in the experimental 1D signal (marked by dashedgreen and magenta lines in Fig. 2C) cannot be interpreted essentially by a single parabola (seedetails in Supplementary Notes), instead requiring the two-parabola picture predicted by theFermi-Hubbard model. By detecting only two modes in our data, which match the dispersionsof pure excitations of the two different kinds depicted in Fig. 2A, we thus find that two distinctFermi seas are formed by the nonlinear spinon and holon quasiparticles out of the many-bodycontinuum away from the Fermi points. The only exception is the particle part of the spinonmode in Fig. 2E, which we do not detect in our data. The spinon Fermi sea alone has alreadybeen observed by neutron scattering in antiferromagnetic spin chains realised in insulating ma-terials [26–28] as a spectral edge with a nonlinear dispersion separating the multi-spinon con-tinuum from a forbidden region [29, 30]. In the present experiment, the charges are delocalisedas well, permitting us to see both Fermi seas at the same time.By tuning the confinement in the wires (see Fig. 1C) we are also able to change the num-ber of occupied subbands and their respective densities, therefore changing r s by a significantamount. Such statistics collected from a range of samples in Fig. 2F show a systematic trend oflarger deviations of the observed m c / (2 m s ) ratio from its non-interacting value with increasing r s . The ratio of the Luttinger parameters v s /v c simultaneously extracted from the same data (seeas an example Fig. 2D) exhibits a very similar dependence on r s . Having analysed the main fea-tures of the dispersions, we inspect the line shapes for various tunnelling processes that can beclearly seen at zero magnetic field in Fig. 2G, where the 2D-2D ‘parasitic’ contribution peaksat large negative V DC . On the other hand, at positive V DC the main contribution to the 1D-2Dsignal comes from a superposition of both 1D and 2D tunnelling, marked by the magenta andblue dashed curves respectively, and as predicted by our two Fermi seas model.To analyse the 1D dispersion further, we contrast it with the simulated tunnelling conduc-tance map between a non-interacting 1D system and a 2DEG (see Fig. 3A). Note how, unlike in6ig. 2C, it is possible to fit both sectors of the map by a single 1D curve (dashed magenta). Thisis because, in the absence of interactions, the opposite spin states are degenerate, leaving roomfor only a single Fermi sea. We start by examining the region just above B + (i.e. + k F ), where aclear feature not accounted for by our non-interacting simulation can be observed, see Fig. 3B.Here, the tunnelling conductance peak broadens and splits, with one boundary following the1D holon mode while the other branches away from it. In order to isolate it from any potentialbackground contamination, we used a gate running over most of the ‘parasitic’ region in orderto move its resonance signatures away from B + . We also observe that this extra feature is notvisible once the wires are past pinch-off, and that it is independent of the ‘parasitic’ tunnellingsignal.The mode-hierarchy picture for fermions [18, 19] predicts its levels to be separated by thespectral strengths of the excitations that are proportional to powers of R /L , where R is thelength scale related to the interaction and L is the length of the system. We have observed thisfeature in samples with wire lengths ranging from 1 µ m to 18 µ m, with all devices being mappedat very similar densities and Fermi energies, making them otherwise similar in R . In all of themthe strength of the mode marked by the dotted blue line in Fig. 3B, which is a ‘replica’ of theparabola formed by nonlinear spinons, decreased as the B field was increased away from thecrossing point. However, once the background has been subtracted and G has been normalisedby length, one can see that qualitatively the decay away from B + was slower the shorter the 1Dsystem, as predicted, with the signal vanishing at higher momenta away from + k F .In order to further establish a length-dependent emergence of spin-type ‘replica’ modes, wehave looked at two other sectors of the tunnelling map, see Fig. 3C and D. We initially reportedthe first mode between ± k F as an inverted (spinon) shadow band symmetric to the 1D (spinon)mode [20]. According to the nonlinear theory of Luttinger liquids [16, 31], in the main | k | < k region of the hole sector, the edge of support (defined as the hole excitation with the smallestpossible energy for a given momentum) is predicted [17, 32] to coincide with the spinon massshell, whose dispersion (cid:15) s ( k ) we have already observed to be very close to a parabola in ourexperiment. Similarly, in the main region of the particle sector, the edge of support is alsopredicted to be given by the inverted spinon mass shell − (cid:15) s ( k ) in Fig. 2A. Consistent with thenonlinear theory, a symmetric inverted replica was seen in the particle sector, opposite to themain 1D subband, in all mapped devices, up to 5 µ m (see Fig. 3C). This feature can also be seenin the full DC-B maps in Fig. 2C. According to the mode-hierarchy picture, a length dependencesimilar to that of Fig. 3B is also expected to be observed since this is also a sub-leading mode.Although such dependence is not particularly clear from − µ m, the replica mode was seento not present at all for the 18 µ m wire.Similarly, another ‘replica’ mode is also predicted to exist at k F < k < k F , symmetric tothe sub-leading spinon mode shown in Fig. 3B, but in the hole sector. Only for the shortest,1 µ m device is a feature consistent with this picture starting to be observed, hinting that a fullobservation of this mode would probably only happen at sub-micron lengths. Nevertheless, asseen in Fig. 3D, both modes evolve in tandem with each other as the 1D channels are squeezedtowards pinch-off. This further establishes that these features are 1D in nature and cannot7 igure 3. A Hierarchy of Modes. ( A ) Simulated map of the differential tunnelling conductance d G/ d B vs V DC and B , between a 1Dnon-interacting system (magenta) and a 2DEG (black). In the absence of interactions the spinon and holon dispersions are degenerate witheach other. ( B ) d G/ d V DC (left) and d G/ d V DC (centre), for devices of different lengths, as labelled. Right column: G vs V DC at variousfields B > B W + for the data in the matching plots to the left; ‘x’ and ‘+’ symbols on each curve indicate the position of the fitted dispersionsin the particle sector for the holon branch and the first-order spinon ‘replica’, respectively— G stays high between the two. ( C ) d G/ d B for B < B W + , showing the zeroth-order spinon ‘replica’ mode (dotted magenta) in the particle sector for a variety of different-length devices. ( D )d G/ d V DC for a µm device at a variety of different wire-gate voltages V WG . The first-order spinon ‘replica’ mode responds to changes in V WG , completely disappearing once the wires are pinched off. Symmetric to it, in the hole sector, a kink in conductance can be observed, onlyvisible in our shortest µm devices. Conductance has been normalised by device length in B , C and D . originate from the ‘parasitic’ injection region. All three replica features discussed emerge asthe effective length of the 1D system is reduced, compatible with the mode hierarchy picturewhere a level hierarchy emerges controlled by system’s length. We attribute the different lengths8t which they occurred in this experiment, with the first ‘replica’ only visible below µ m, thesecond below µ m and the third ‘replica’ only at µ m, to different numerical prefactors thatare still unknown theoretically for spinful systems.Up to now we have confined our analysis to dispersion maps in the single-subband regime.In the current geometry, however, we are also able to vary the number of occupied subbandsup to four, by tuning the wire-gate voltage V WG until the upper layer starts to become 2D whencarriers delocalise between the wires. While the emergent hierarchy of modes becomes almostimpossible to see in the data with more than one subband occupied, the parameters of twoFermi seas can still be quite reliably extracted, see Fig. 4A and Supplementary Notes. Variationof the number of subbands provides us with an additional tool for assessing the microscopicinteraction parameter of the Hubbard parameter U in our experiment at the quantitative level.The macroscopic dimensionless parameter controlling the Hubbard model in 1D is [33] γ = 0 . λ F a Ut , (1)where a is the lattice parameter of the underlying crystal. For γ < , the weakly interactingelectrons are almost spin-degenerate, having double occupancy for each momentum state, asfor free particles. For γ > , each momentum state is occupied by only one electron due tostrong Coulomb repulsion. Such a dependence of the system’s behaviour on γ is qualitativelythe same as the dependence on r s in all dimensions, reflecting the ratio of the total interactionenergy to the kinetic energy.We have varied the number of occupied subbands all the way up to four, and fitted thebottom two using the model of two Fermi seas to extract the ratio of masses and velocitiesin the same fashion as in Fig. 2C. Fitting the same data with the dispersion produced by theHubbard model, we obtain the values of γ that correspond to these ratios for each individualsubband and for each subband occupancy in Fig. 4B, which allows the data points from multiplewires with a variety of densities to be collapsed onto the same curve. Comparing the alreadyextracted values of r s with γ for all measurements in Fig. 4C, we find that the two dimensionlessparameters are approximately proportional to each other with a proportionality coefficient of ’ . . We interpret the still-observable discrepancy as a manifestation of the screening effectthat is not captured by r s but is taken into account explicitly in the Hubbard model via the two-body interaction energy U . The latter is proportional to the integral of the screened Coulombpotential, in which only the screening radius is changed in our experiment.By means of the relation in Eq. (1) we extract the evolution of U as a function of the numberof occupied subbands, as shown in Fig. 4D, for both the 1 st and 2 nd lowest subbands, in different-length systems. Data corresponding to four occupied subbands was excluded, as its proximityto the non-interacting limit made the fitting less reliable. Similarly, fitting to the dispersionsof the third and fourth subbands was not attempted owing to the lack of sharp features andoverall increase in map complexity. Nevertheless, two clear trends emerge: first, U decreasesas more subbands are progressively filled, resulting in relative reductions of about ∼ % forthe 1 st subband and of ∼ % for the nd subband; second, the bottom subband seems to be9 igure 4. 1D-1D screening. ( A ) d G/ d B of a µm device mapped in the multi-subband regime. Dashed black line marks the location of thesubtracted 2D-2D ‘parasitic’ signal. Arrows point to the location of spinon (s) and holon (h) modes, in both the hole (-) and particle (+) sectors,for each occupied subband (1, 2). ( B ) Macroscopic dimensionless Hubbard model parameter γ (see text for discussion) vs mass and velocityratio, for devices of different lengths, as extracted by fitting the bottom two subbands. The Hubbard model can reproduce well the observedexperimental dependence (dashed black). ( C ) γ vs the interaction parameter r s , where an approximate linear dependence can be seen. Thedashed curve corresponds to fitting using only data from the single-subband-occupancy regime. Note that, even when allowing for errors, allremaining points fall systematically below this line, indicating the presence of 1D-1D inter-subband screening. ( D ) Hubbard parameter U/t vs number of occupied subbands as extracted from γ . The asymmetry in screening between the first and second subbands is expected from theirdifference in densities. 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Author contributions
P.M.T.V., Y.J., M.M., A.S.A, A.A. and W.K.T. fabricated the experimental devices, with P.M.T.V.and Y.J. performing the transport measurements shown. J.P.G. did the electron-beam lithogra-phy and I.F. and D.A.R. grew the heterostructure material. P.M.T.V, O.T and C.J.B.F. analysedthe data. O.T. and A.S. developed the theoretical framework. O.T. performed the calculations.C.J.B.F. supervised the experimental side of the project. All authors contributed to the discus-sion of the results. P.M.T.V., O.T. and C.J.B.F. wrote the manuscript.
Acknowledgements
The authors would like to thank Leonid Glazman for assistance and helpful comments.13 bserving separate spin and charge Fermi seas in astrongly correlated one-dimensional conductor
All tunnelling devices measured in this work were fabricated using double-quantum-well het-erostructures, grown via molecular-beam epitaxy (MBE), comprised of two identical 18 nmGaAs quantum wells separated by a 14 nm Al . Ga . As tunnel barrier. On each side ofthe barrier there were 40 nm Si-doped layers of Al . Ga . As (donor concentration . × m − ), with the lower and upper spacers being respectively 40 nm and 20 nm wide. Thisresulted in electron concentrations of about 3 (2.2) × m − with mobilities of around 120(165) m V − s − in the top (bottom) well, as measured at 1.4 K. A 10 nm GaAs cap layer wasused to prevent oxidation. The distance from the upper well to the surface was ∼
70 nm.The electrical (surface) structure of the device was fabricated on a 200 µ m-wide Hall bar.Contacts to both layers were established using AuGeNi Ohmic contacts. Electron-beam lithog-raphy was used to define a split gate (SG), a mid-line gate (MG), a barrier gate (BG) and acut-off gate (CG)—used in setting up the tunnelling conditions—together with an array of wiregates (WG)—used in defining the experimental 1D system (see Fig. S1A). The length of thewire gates was varied from 1–18 µ m. They were separated by 0.15–0.18 µ m gaps, and had awidth of 0.1–0.3 µ m. A‘parasitic’ injection region also ran across the entire width of the mesa,with a fixed width of 0.45 µ m. A parasitic gate (PG) was used to modulate its density. Alldimensions, particularly regarding the wire-region were carefully chosen in order to achieveminimal modulation of the lower-well carriers.The tunnelling set-up was achieved as follows: first, the SG was negatively biased in orderto pinch off both layers underneath, followed by positively biasing the MG in order to opena narrow conducting channel in the top well. At the other end of the device, the BG and theCG were biased enough to pinch off only the top layer. Under these conditions, any currentinjected through one of the Ohmic contacts had to have tunnelled between the layers in orderto be detected (Fig. S1B).Our spectroscopy technique consists of a low-noise, low-temperature measurement of thetunnelling current between the two 2DEG layers, which is given by [1] I ∝ Z d k d E [ f T ( E − E F1D − eV DC ) − f T ( E − E F2D )] × A ( k , E ) A ( k + ed ( n × B ) / ¯ h, E − eV DC ) , (S1)where e is the electron charge, f T ( E ) is the Fermi-Dirac distribution function, d is the centre-to-centre well separation, n is the unit normal to the 2D plane, B = − B ˆ y is the magnetic-fieldvector, ˆ y is the unit vector in the y -direction and A ( k , E ) and A ( k , E ) are the spectral func-tions of the 1D and 2D systems respectively, with the corresponding Fermi energies being E F1D and E F2D . The tunnelling current between the two layers in then proportional to the overlap1 a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b igure S1. Vertical tunnelling device ( A ) Scanning electron microscopy (SEM) images of the tunnelling device,showing the split (SG), mid-line (MG), barrier (BG), wire (WG) and parasitic (PG) gates. The cut-off (CG) gateis not used in this experiment and is always biased together with BG. Several samples were fabricated to vary thelength of WG from 1–18 µ m (pictured, 1 µ m). The bottom micrographs show, respectively, the lower and upperends of the wire array ( ∼
400 wires). In order to increase the uniformity of the 1D system, we developed a novelair-bridge technique to avoid having to use a connecting backbone structure (see [2]). ( B ) Gate operation andsetting of tunnelling conditions. We start by negatively biasing SG (1), followed by positively biasing MG so thatconductance is allowed only in the upper well (UW) (2). Next, we negatively bias both CG (3) and BG (4) butenough to only deplete the UW. Under this configuration, any signal measured between the ohmic contacts mustresult from direct tunnelling between each well. Inset: By varying WG and/or PG one can observe, respectively,1D-2D and 2D-2D tunnelling between the wells (5 and 6). ( C ) Side profile of the tunnelling device. It consists ofa double quantum-well heterostructure with a centre-to-centre distance of about ∼
32 nm. integral of their spectral functions. We induce an offset eV DC between the Fermi energies of thetwo systems by applying a DC bias V DC between the layers. Similarly, an offset in momentumcan also be obtained by applying a magnetic field of strength B parallel to the 2DEG layers.When the field direction is along the (in-plane) y -direction, the Lorentz force then shifts themomentum of the tunnelling electrons in the x -direction by edB . Put together one can there-fore map the dispersion of each system with respect to one another by measuring the differentialconductance G = d I/ d V in both energy and momentum space. Hubbard modelInteracting electrons in our quantum wires can be described by the 1D Fermi-Hubbard model, H = − t L/a X j =1 ,α = ↑ , ↓ (cid:16) c † jα c j +1 ,α + c † jα c j − ,α (cid:17) + U L/a X j =1 n j ↑ n j ↓ , (S2)where c jα are the second-quantisation operators in real space obeying the Fermi commutationrelations n c jα , c † j α o = δ jj δ αα , α is the spin-1/2 index ↑ or ↓ , n jα = c † jα c jα is the local densityoperator for each of the spin species, L is the length of the wire, and a is the lattice parameterof the host crystal. The two microscopic constants of the model are the hopping amplitude t ,2escribing the kinetic energy, in the first term and the two-body interaction energy U in thesecond term. The local nature of the density-density interaction in this model allows for onlya single microscopic constant U that takes into account at the same time the strength of theCoulomb interaction governed by the electronic charge e and the screening radius R , which wecan tune in our experiment.The many-body eigenstates of the Hubbard model in 1D were constructed in [3, 4]. Theydescribe the amplitude of finding all N particles at a given set of sites on the lattice j , . . . , j N = j and with a given configuration of their spins α , . . . , α N = α : Ψ = P j , α a jα c † j α · · · c † j N α N | i .These amplitudes have the form of a superposition of plane waves, a jα = X P A P Q α e i ( P k a ) · ( Q j ) , (S3)where Q is the permutation that orders all N coordinates such that Qj < · · · < Qj N , (S4)the momenta of N particles are k = k , . . . , k N , and P P is the sum over all permutations ofthe charge momenta, like in the Slater determinant for free particles. However, unlike for freeparticles, the sign under the permutation of a pair of coordinates is not − but is rather a phasefactor A P Q α that also depends on the spin configuration Q α as A P Q α = ( − P Q X R Y ≤ l
2) = 2 πJ m , (S8)with ϕ ( x ) = − (cid:18) txU (cid:19) , (S9)where N non-equal integers I j and M non-equal integers J m define the solution for the orbital k j and the spin λ m momenta of an N -electron state for a given value of the microscopic parame-ter U/t . This solution also gives the eigenenergy of the many-electron state as E = ta P Nj =1 k j and its momentum as k = P Nj =1 k j . These simultaneous quantisation conditions for both spinand charge degrees of freedom are a system of N + M connected nonlinear equations for anyfinite U .Two Fermi seasBoth the charge and the spin momenta correspond to non-equal integer numbers I j and J m since the 1D wave function in Eq. (S3) becomes zero for any pair of equal k j or λ m , whichare obtained immediately from Eqs. (S7, S8) when a pair of I j or J m are equal. In principle,4uch an emergent picture is described by two different Fermi seas, in which both kinds ofmodes are filled with spin or charge excitations up to some finite densities, just as the usualFermi sea is filled with non-interacting fermions up to a finite Fermi energy. The differencefor the problem with interactions is a more complicated relation between the two types ofthe occupation numbers, visualised by a set of I j and J m , and the actual distributions of thetwo types of momenta, k j and λ m . This relation is not just a linear quantisation conditionfor each momentum of the free particles independently but is rather given by the solution ofthe system of essentially nonlinear coupled equations (S7) and (S8) for any finite interaction U . For instance, the quantisation conditions given by the Lieb-Wu equation in Eqs. (S7, S8)alter qualitatively the equilibrium properties by changing parameters of the two Fermi seasor by changing the excitation energies of different modes, making their dispersions generallyinteraction-dependent.Our quantum wires remain unpolarised up to the maximum field of T that we use inour experiments. The number of spin-up and down electrons is equal within 10% (5% below B + ), and we therefore assume M = N/ spin particles. For the holon Fermi sea, whichhas twice as many filled states as the spinon Fermi sea, the Lieb-Wu equations give the twodistributions of their momenta at the infinite-interaction point U = ∞ , see Fig. S2A. We definethe Fermi momentum k F by the free electrons, i.e. by N/ fermions that are doubly degeneratewith respect to spin without the interactions at U = 0 . For strong enough interactions thefree-electron degeneracy is completely lifted, increasing the density of the charge particles bythe factor of two. At the same time, the spin particles develop the second Fermi sea of theHeisenberg-spin-chain type [7] with a density that is twice as small as that of the holons. Themomentum distribution in the spinon Fermi sea is non-equidistant: the spin momenta have apronounced higher density towards the Fermi points but are sparser at the bottom of the band[8]. A Fermi sea of this kind was observed using neutron scattering in non-itinerant magnets[9–11].Decreasing the interaction energy U away from the infinite point, we solve the Lieb-Wuequations (S7) and (S8) numerically, see Figs. S2B and C. Down to an intermediate value of U both Fermi seas remain stable, but below it the two-seas picture starts to change qualitatively.For weak interactions the holon Fermi sea becomes doubly degenerate, recovering the free-fermion picture, and the spinon Fermi sea collapses, becoming the spin part of the free-electronfunction that describes ± permutation signs for the electrons of different or the same spin.The dimensionless parameter γ , which controls this transition, emerges from the Hubbardmodel itself microscopically. It was identified in a quantitative analysis of the double occupancyof the electronic states [12] as γ = λ F a Ut − N P N/ l =1 λ l ( ∞ ) − ( U t ) λ l ( ∞ )+ ( U t ) , (S10)where λ F = 4 L/N is the Fermi wavelength of the free-electron gas, a is the lattice parameter,and λ l ( ∞ ) are the spin part of the solution of Eqs. (S7) and (S8) in the infinite-interaction limit U = ∞ . Turning the sum into an integral in the thermodynamic limit and evaluating it forthe unpolarised Heisenberg chain [6] we obtain the numerical value of the denominator in theabove expression, − P l . . . /N = 1 . . Thus, in the thermodynamic limit the parametercontrolling the interaction effects is γ = 0 . λ F a Ut . (S11)5 igure S3. (A) Excitation of the holon type, in which the spinon part is placed at the lowest possible momentumfor the spin excitations and the holon part is promoted to some finite momentum, which is larger than the lowestpossible momentum for the charge excitations by ∆ P c . (B) Excitation of the spinon type, in which the holonpart is placed at the lowest possible momentum for the charge excitations and the spin part is promoted to somefinite momentum, larger than the lowest possible momentum for the spin excitations by ∆ P s . (C) The dispersionsof the holon (green circles) and of the spinon (blue circles) modes consisting of the excitations in (A) and (B),respectively, obtained by solving the Lieb-Wu equations (S7, S8) for N = 550 unpolarised electrons with M =275 for interaction strength γ = 2 . The red lines are the parabolic fits for both modes. The two black lines at theFermi point k F mark the linear dispersions of holons v c and of spinons v s of the linear spinful Tomonaga-Luttingermodel. When the interactions are strong, γ > , both the spinon and the holon Fermi seas are fullydeveloped but, when γ < , the holon Fermi sea is still close to the double occupancy of thefree Fermi gas. The expression in Eq. (S11) is presented in Eq. (1) in the main text.Spin and charge excitationsBy adding a single electron with its charge and spin we add two excitations to the systemsimultaneously, one holon and one spinon. With two Fermi seas there are two options. Oneis adding a holon to its Fermi sea at the k F point and spinon to its Fermi sea at − k F point ontop of the ground state in Fig. S2A, obtaining the net momentum for the electronic excitation k = k F . Another possibility is adding the holon and the spinon at the same sides of their Fermiseas, producing the net momentum k = 3 k F . Our wires have inversion symmetry. Therefore,the mirror of this argument produces the k = − k F and the k = − k F as well, see Fig. S3C.Starting at a Fermi point, say k F , we can add a pair of excitations of the two kinds notjust at the Fermi points of their respective Fermi seas but also at some finite momenta, ∆ P c or ∆ P s , above them, see Fig. S3A and B. Change of the momentum of any of these excitationsdistorts both Fermi seas via solving the nonlinear Lieb-Wu equations (S7) and (S8): all states6n the spinon Fermi sea become generally asymmetric, in addition to being non-equidistant inthe ground state, and the holon Fermi sea as a whole shifts by the total spinon momentum,see Figs. S3A and B. However, the total momentum of the electronic excitation is a quantumnumber, which has a simple relation, k = 2 π (cid:16)P j I j + P m J m (cid:17) /L , to the whole set of thequantisation integer numbers, I j and J m , and is independent of U [5]. This result of the totalmomentum conservation law for a many-body system defines the momentum of the holon andthe spinon excitations as k = k F + ∆ P c and k = k F + ∆ P s , (S12)where ∆ P c = 2 πI N +1 /L and ∆ P s = 2 πJ M +1 /L can be visualised in a simple way due to thesame total momentum conservation for all particles. Therefore, the distance in the momentumvariable between the neighbouring points is always equal to the quantum of momentum π/L and the position of the second crossing points of the holon modes with the line of chemicalpotential in Fig. S3C always remains at the same momentum ± k F , irrespective of the value of U . The energies of these excitations, on the other hand, are more complicated. In order toevaluate them and the dispersions that they form, we need to solve the Lieb-Wu equations (S7)and (S8) numerically for different values of U , see Fig. S3C. At low energies, the solutionsform two linear dispersions with non-commensurate slopes, v c and v s ; the stronger U is, thelarger is the ratio v c /v s , and the two velocities become equal v c /v s = 1 in the free-particlelimit U = 0 . These two holon and spinon velocities are nothing but a pair of the Luttingerparameters in the spinful linear Tomonaga-Luttinger model [13, 14] at low energies. They aremanifested in the observable spectral function as a pair of divergences [15, 16].Extensions of the momenta of the excitations of two different kinds, ∆ P c and ∆ P s , awayfrom the ± k F points provide a natural generalisation of these linear modes to the nonlinearregime, see Fig. S3C. Unlike the linear regime, where there are only two branches aroundeach of the Fermi points, in the whole energy band there are three. This is a manifestation ofthe different densities of the holon and the spinon Fermi seas. Dispersions of all of the threecurves are not exactly parabolic for finite U but are close to parabolae due to an interactioneffect between the two Fermi seas, with the most significant deviations occurring for the spinonbranch in the vicinity of the Fermi points. Fitting parabolae to the numerically produced exactdispersions (red lines in Fig. S3C), gives the dependence of the two masses m c and m s [17]on the interaction parameter γ . Their ratio is presented in Fig. 4B of the main text. The twovelocities v c and v s are extracted as the linear coefficients in the dispersion of the holon andthe spinon modes at the Fermi energy, and the dependence of their ratio on γ is also presentedin Fig. 4B. We fit our experimental data with the exact dispersions of the Hubbard model toobtain the interaction parameter γ directly, and also with the two parabolae, see more details inthe section ‘Parabolic Model’ below. Interaction parameter r s in 1DThe Wigner-Seitz radius r s is used as a generic interaction parameter in Fermi systems indepen-dent of their dimensionality. Its common applicability stems from the fact that the Fermi energyis inversely proportional to the inter-particle distance squared and that the Coulomb energy isinversely proportional to the same inter-particle distance but to the first power. Therefore, thedimensionless ratio of the interaction energy to the kinetic energy is just given by r s .7 igure S4. 1D wire subbands. ( A ) d G/ d V WG differential of the tunnelling conductance G with respect tothe wire-gate voltage V WG , as a function of V WG and magnetic field B perpendicular to the wires, as obtainedunder equilibrium conditions V DC = 0 . Three fully developed 1D subbands can be seen below the 2D band at V WG ≈ − . V and above the cutoff point at V WG ≈ − . V, below which the wires cannot conduct. ( B )Equilibrium 1D electron densities in both upper (circles) and lower (squares) wells for each occupied subband.Note that the electrons are not laterally confined in the bottom well and as such the lower well 1D density has nophysical meaning and is only shown for comparison. ( C ) 2D electron densities in both upper (blue) and lower(green) wells’ ‘parasitic’ regions. The overall independence of the 2D densities on the wire-gate voltage and theproximity of the (now physical) 2D electron density of the bottom well below the wire region (red) to that of theinjection region shows that the bottom 2DEG remains largely unaffected by V WG , and as such can be taken to bea well-understood 2D probe. ( D ) Subband energies relative to the chemical potential µ = 0 as a function of V WG . We can extract the value of the Fermi wavelength λ F in our experiment from the density ofthe 1D system, which can be obtained from either the 1D wire subbands dispersion ( e.g. seeFig. S4) or alternatively by using the zero-field intersect points in the tunnelling conductionmaps (see section on ‘Parabolic Model’). This gives us two independent estimates for theelectron density. Its relation to r s in 1D is as follows. In one dimension the electron density isexpressed in terms of r s as n = 12 a B r s , (S13)where a B = 4 πεε ¯ h me (S14)is the Bohr radius of electrons in GaAs, with ε ≈ and m = 0 . m e . On the other hand, the8ensity of free electrons in a 1D Fermi sea is also given by the integral over the Fermi function, n = 2 Z dk π n k = 4 λ F , (S15)where the occupation numbers are a step function, n k = θ ( k F − | k | ) , and k F = 2 π/λ F . Com-bining Eqs. (S13) and (S15) produces the following relation r s = λ F a B . (S16)This generic dimensionless interaction parameter r s plays the same role as the microscopicdimensionless interaction parameter of the 1D Fermi-Hubbard γ in Eq. (S10). However, r s does not account for screening effects, see the main text.Parabolic ModelWhen fitting the 1D tunnelling resonances observed in the dispersion maps, one can approxi-mate the exact numerical solutions arising from the Hubbard model by parabolae, see Supple-mentary Text ‘Spin and charge excitations’ and Fig. S3C for both spin- or charge-type excita-tions. On the other hand, 2D systems at low densities (see Fig. S4C), are known to be Fermiliquids with effective mass renormalised by interactions. Together these account for the dis-persion of the elementary excitations in both the upper and lower wells (UW and LW), whichcan behave as either 2D or 1D systems depending on whether the signal arises from unconfined(‘p’-, LW) or confined (wire-) regions of the device. We will now show how separation be-tween spin- and charge-type modes in 1D, each associated with different effective masses anddegeneracies, emerges naturally from the data after considering simpler models. We start byintroducing the capacitive-coupling correction, which was also applied when fitting using theHubbard model. • Modelling the capacitive effects
Tunnelling between the upper and lower layers of a device, across its dielectric barrier, isaffected by capacitance effects, which result in the observed dispersions being slightly asym-metric. Owing to the finite capacitance C between the two layers, a small increase or reductionof the electron density ± δn occurs at each side of the barrier. We have eδn = V DC C/A for the 2D system (of area A ) and eδn = V DC C/L for the corresponding 1D system (oftotal length L ), with V DC the DC-bias applied between the wells. For two layers with Fermiwavevectors k F,1 and k F,2 , where k F,1 < k
F,2 , the zero-bias crossing points B + and B − can becombined to give k F,1 = ed h ( B + − B − ) , k F,2 = ed h ( B + + B − ) , (S17)where d is the separation between the wells. Our 1D wires, once defined by the gate voltage,do indeed have lower density than the 2D system beneath, so in the wire region we take k F,1 asthe Fermi wavevector of the 1D system in the absence of interactions, and k F,2 as that of the 2Dlayer.The electron densities for 1D and 2D systems are given by n = d N d L = 2 k F,1D π (S18)9nd n = d N d A = k F,2D π . (S19)From here, we get δn = n ( V DC ) − n (0) = η i CV DC eL = 2 π ( k F,1D ( V DC ) − k F,1D (0)) (S20)and δn = n ( V DC ) − n (0) = η i CV DC eA = 12 π ( k F,2D ( V DC ) − k F,2D (0)) , (S21)which in turn gives the modified Fermi wavevectors as the inter-layer voltage V DC causes thedensities to change, k F,1D ( V DC ) = k F,1D (0) + πη i CV DC eL (S22)and k F,2D ( V DC ) = r k F,2D (0) + 2 πη i CV DC eA . (S23)Here η i = ± , with i = 1 , labelling the upper/lower layer respectively, is a sign factor basedon the experimental setup. In our experiment, η = − and η = 1 , since for V DC > the upperwell was more positive than the lower well.The bottom well is always 2D in nature, which means that k F,2 = k F,2D ( V DC ) with η = 1 .The top well, on the other hand, can behave as being either 2D, in the injection region, or 1D,in the wires. For the latter we have k F,1 = k F,1D ( V DC ) with η = − . For the former, the upper-well density is higher than in the lower well. Note also that equation (S23) reduces to equation(S22) when expanded in the low-capacitance limit, k F,2D = k F,2D (cid:18) πη i CV DC eAk F,2D (cid:19) / (S24) ≈ k F,2D + πη i CV DC eAk F,2D . (S25)Therefore, comparing Eqs. S22 and S25 we conclude that the capacitance correction for thewire region, which is 1D in nature, can nevertheless be treated as for a 2D system by trans-forming the capacitance as C/L = C/A × /k F,1D . One can also assume charge neutrality ofthe pair of layers to estimate the width of the wires w by solving C/L = w × C/A . • Estimating the capacitive coupling
We now estimate the capacitive coupling between the two wells in our setup, when applying avoltage to the upper layer and keeping the potentials of the gates and lower layer fixed. Thisis required to correct for the mapped dispersions as discussed in the previous section. In aclassical Coulomb system, the capacitance C ≡ d Q/ d V DC of a conductor is a purely geometricquantity (for charge Q on the conductor). For example, in a parallel-plate capacitor, it takes thewell-known value C = εε A/d , A being the surface area of the plates, d their separation, and ε the relative permittivity of the dielectric material in between. In systems with a low densityof states however, such as a 2DEG, this result does not strictly apply, since unlike for a perfectmetal plate, the density of states here is no longer infinite. Therefore, in contrast to a metal, a2DEG cannot, in general, perfectly screen the electrical field generated by the surface gates.10n order to account for this effect one has to consider band-filling/band-emptying in chang-ing the density of states in the 2DEG as the gate voltage is varied. This correction was initiallyproposed in [18] and can be modelled by considering two capacitors connected in series, C = 1 C G + 1 C Q , (S26)where C G is the usual geometric capacitance, C Q = e dn /dE F A is the new quantum capaci-tance, and E F is the Fermi energy as measured relative to the bottom of the band and thus variesas the occupation changes. Depending on dimensionality, the ratio of the two capacitances isthen given by C G C Q = ¯ h π εε w m ∗ e D n for gate-wire system , ¯ h πεε m ∗ e D for gate-2DEG system , (S27)where D is the distance from the wells to the surface, w is the width of the wire, and m ∗ is theelectron effective mass in GaAs. Here, we assumed n = n · w , with n and n as definedin equations (S18) and (S19), respectively. Taking w = 50 nm, D = 85 nm, m ∗ = 0 . m e , ε = 12 and n = 33 µ m − , we get C G /C Q = 0 . and C G /C Q = 0 . for a 1D and 2Dsystem, respectively. In both cases C Q (cid:29) C G and so the geometric contribution is expected todominate in any capacitance measurement.Our system consists of a GaAs/AlGaAs double-quantum-well heterostructure with transla-tional invariance along the x - and y -directions, the wells being at roughly and nm belowthe surface. A set of surface gates is used to define the quantum wires in the upper well. Thereis capacitive coupling between the wells and also between each well and the surface gates,which provide screening to the 2D electron gas.We used the COMSOL Multiphysics 5.5 software package [19] to simulate our device elec-trostatically (not self-consistently) and computed the electrical field and the potential distribu-tion in the dielectrics given the known charge distributions in each well, one of which had wiresdefined by the surface gates (see Fig. S4B and S4C). Specifically, when solving Poisson equa-tion we took U ( r ) as the potential induced by the gates and solved for it by taking U ( r ) = V SG at the gates and Neumann boundary conditions otherwise. In our simulation we accounted forthe finite width of the 2D electron systems, though this made little difference to the results.Poisson’s equation was solved using a finite-element grid, the number of nodes chosen so thatthe computation was free from finite-element effects. Finally, in our model, we ignored theeffect of the ionised donor layers on both sides of the wells since they form a static layer ofcharge which is not affected by changes to gate voltage. The capacitance values (per unit area)obtained were c = 0 . Fm − for the 2D injection region and c = 0 . Fm − for the 1Dwire region in the single-subband regime. • In order to subtract the influence of the ‘parasitic’ injection region from the overall signal wemap it beforehand by setting V WG negative enough that the wires do not conduct (see Fig.S5, below the bottom subband). The parameters of interest when fitting in this regime are,respectively, the capacitances (per unit area) of both the upper and lower wells, c and c , theeffective masses of electrons in top and bottom wells, m UW and m LW , the separation betweenthe wells, d , and the zero-bias field intercepts, B p − and B p + . Given that this region is relativelywide ( ∼ . µ m), it can be safely treated as a 2D system in both wells.11 d G/ d B ( μ S/T) -0.5-505 -100.51Magnetic fi eld B (T) V D C ( m V ) -0.20-505 -0.400.20.4Magnetic fi eld B (T) V D C ( m V ) d G/ d V DC ( μ S/mV)
Figure S5. 2D-2D Background.
Tunnelling conduction differentials d G/ d V DC and d G/ d B vs magnetic fieldB ( ∝ momentum) and voltage V DC ( ∝ energy eV DC ) for a 1.7 µ m long device, mapped at V WG = − . V. Atthis voltage the wires are pinched off and do not conduct. Therefore, the spectrum observed arises solely from2D-2D tunnelling taking place between the ‘parasitic’ injection region and the bottom 2DEG, with the dashedand dash-dotted black curves corresponding to the capacitance corrected and uncorrected resonances. This is latersubtracted from the tunnelling data obtained when the wires are conducting, allowing us to separate the ‘parasitic’from the 1D-2D tunnelling signal. Note that the quality of the fit with d = const. allows us to rule out a potential d ≡ d ( B ) dependence due to the Lorentz repulsion between the wells, at least up to 7 T, which is the highest fieldused in our experiment. We found that good results were obtained when setting both m UW and m LW to be equal . m b , where m b = 0 . m e is the electron band mass in GaAs. We note that m *2D =0 . m b = 0 . m e is in very good agreement with independent work carried out in 2D systemsvery similar to ours [20–24]. From the MBE growth data we also know that d ≈ nm and,even though it is reasonable to expect deviations from this value due to monolayer fluctuations,these shouldn’t exceed a few nanometres. Finally, the zero-bias crossing points can be eas-ily determined by visual inspection of the data while the capacitive coupling was estimated bysimulating it under an electrostatic framework following the approach described in the previoussection. We found that the best results were obtained with d = 31 nm, c = 0 . Fm − and c = 0 . Fm − , in very good agreement with the expected values. Owing to extracoupling to the surface gates, the calculated and observed capacitances of the upper well areapproximately ∼ /
70 = 1 . that of the lower well.It is worth noting also that, particularly at high fields where the sign of k changes whiletunnelling, one could expect the Lorentz force to act in such a way as to force carriers in eachwell further apart. This, in practice, would translate in having d ≡ d ( B ) . However, as can beseen from Fig. S5, no such correction is needed up to at least 7 T and so we can safely rule itout for the rest of our analysis. • In the previous section we discussed how to constrain c , c , m ∗ , and d from the back-ground data, i.e. using the conductance maps with the 1D wires past pinch-off. Since theobserved modulation by the wire gate to the injection region is very small however, as can beseen by noting that the density under the parasitic and the wire regions closely match here (seeFig. S4C), it is reasonable to expect the same values to apply when fitting to the 2D signal even12s the 1D channels are now conducting.We map the 1D subbands as shown in Fig. S4A. Under the current geometry, most devicesusually display between three to four subbands below the 2D band. By setting appropriatevalues for V WG , we are able to map the dispersion of the system at different subband occupancyvalues (see an example in Fig. 4A in the main text). For now, we will restrict ourselves to thesingle-subband regime. From Fig. S4, both the Fermi energies as well as the 1D (2D) densitiesin the wire (‘parasitic’) region can be determined by using equations (S17), (S18) and (S19).The procedure followed when fitting the tunnelling signal arising from the wire region isanalogous to what was done for the ‘parasitic’ injection region. Since c , c and d havealready been obtained, the only parameters left to be determined are B w − and B w + (the zero-bias field intersects for the 1D signal, obtained similarly by visual inspection of the tunnellingmaps), m UW = m *1D (the effective mass of the 1D electrons in the upper well), and c and c LW (the capacitances per unit area in both wells). For the latter the values found, c = c LW = 0 . Fm − , were significantly lower that those predicted in COMSOL. Using thelarger values for capacitance completely distorts the dispersions and cannot be accounted forby errors in other parameters as the constraints are strong. We interpret the observed deviationsas a significantly stronger effect of interactions in 1D that push the band down, reducing therate of filling the wire, dn /dE F , or in other words, increasing the energy required to add anextra electron. This results in a decrease in the quantum capacitance C Q , therefore making thetotal capacitance C no longer solely determined by the geometric contributions as was done inour simulation, see (S26).The effective mass of electrons in the lower well in the 1D-2D regime was found to beslightly higher than in the 2D-2D scenario ( ’ . m e vs ’ . m e ) that we attribute tothe different amount of interactions, which electrons in the Fermi liquid of the bottom layerexperience when the wire-gate depletes significantly the upper layer. In other words, reductionin screening by the upper layer changes interactions and hence m ∗ . Model 1: Single Fermi Sea and Single Plasmon
Let us first assume that there is a single,uniquely determined 1D effective mass, which matches the 2D non-interacting effective mass, m *1D = m *2D . From Fig. S6A it can be seen then that while the 1D mode is nicely fitted in thehole sector ( V DC < ), the same does not happen in the particle sector ( V DC > ), particularlyat very high biases, even when correcting for capacitance. We therefore conclude that the 1Dexcitation cannot be fully captured by a single 1D effective mass. Model 2: Single Fermi Sea and Two Plasmons
We now assume that there are both spin-and charge-type excitations in our 1D system. We note that we have previously observed spin-charge separation in similar devices [25]. Under our configuration, we also know that the 1Dexcitations in the hole sector correspond to a spinon mode, while that in the particle sector to aholon mode [26]. Since each mode corresponds to a different kind of excitations, it is reason-able to expect different effective masses for each, which we label as m s and m c respectively.As before, it can be shown (see Fig. S6B) that while m s = m *2D provides a good match to thedata, even when taking m c < m *2D one still fails to fully capture the observed behaviour for thismode. This is particularly noticeable when taking the d G/ d B differential. We also note thatany fit to the holon branch in the particle sector must also catch the low-energy charge mode,that is, the holon branch in the hole sector, when extended back to it, which is clearly not thecase. Model 3: Two Fermi Seas and Two Plasmons
Based on the previous analyses we willnow consider the spectroscopic predictions of the 1D Fermi-Hubbard model (see details in theSupplementary Text ‘Spin and charge excitations’ above and Fig. S3C), where we now assumenot only different effective masses m s and m c but also different densities, and therefore different13 igure S6. Single-subband regime: One Fermi Sea. Tunnelling-conductance differentials d G/ d V DC andd G/ d B as a function of the DC-bias V DC and the in-plane magnetic field B , for a 1.7 µ m device, mapped at V WG = − . V. ( A ) Model 1, with a single plasmon type, where m ∗ = m ∗ = 0 . m b . The dashed blackcurves indicate the location of the subtracted 2D-2D ‘parasitic’ tunnelling signal mapped in Fig. S5. The dashedand dash-dotted magenta curves mark the capacitance corrected and uncorrected resonances arising from the tun-nelling between the LW ground states and the UW wire region. They reveal the dispersion of the elementaryexcitations in the 1D UW wire region. Similarly, the dashed and dash-dotted blue curves mark the location of theresonances resulting from the reverse tunnelling process, between the UW ground states and the LW, revealingthe dispersion of the 2D LW. ( B ) Same as in ( A ) but now fitted using Model 2, where two types of excitations,spin and charge, are allowed. The dashed black and blue lines represent the same dispersions as before. The greencurves correspond to a spinon mode with m s = m ∗ . The magenta lines, on the other hand, mark a holon modewith m c < m ∗ . Inset: Spin-charge separation (dashed black ‘S’ and ‘C’ lines) near the + k F point at low bias. f , is parabolic, before correcting for capacitance. That gives us f ( k ) = ¯ h k m s − E sF = ¯ h m s ( k − k F ) (S28)and f ( k ) = ¯ h ( k + k F ) m c − E c F = ¯ h m c (cid:2) ( k + k F ) − k F (cid:3) (S29)where E sF , m s and E cF , m c are the respective Fermi energies and the renormalised masses, with k F ≡ k F , UW for simplicity. The charge mode, forming between − k F and + k F , has a densityof states half that of the spinon excitation, between ± k F . An analogous expression can also beobtained for the holon branch between − k F and +3 k F . Since v = d ω/ d k = ¯ h − d E/ d k , we get v s = 1¯ h d f d k (cid:12)(cid:12)(cid:12)(cid:12) k = k F = ¯ hkm s (cid:12)(cid:12)(cid:12)(cid:12) k = k F = ¯ hk F m s (S30)and v c = 1¯ h d f d k (cid:12)(cid:12)(cid:12)(cid:12) k = k F = ¯ h ( k + k F ) m c (cid:12)(cid:12)(cid:12)(cid:12) k = k F = 2¯ hk F m c , (S31)therefore arriving at v c v s = K s K c = 2 m s m c . (S32)Equation (S32) allows us to relate the charge-to-spin velocity ratio with the phenomenolog-ical Luttinger parameters K c,s , which account for the renormalisation of the effective masses m s and m c due to the 1D confinement. We have m s = m b K s and m c = m b K c , with K c = K s .The ratio K c /K s is a good estimate of the interaction strength. Note also that the extra factorof two in the third term of (S32) arises from the assumption of different densities of states ofthe quasiparticles in each Fermi sea, or in other words because k F is twice as large for holonsas it is for spinons. For repulsive interactions, since K s > and K c < , we have m c < m s .In Fig. S7A we fit the data using the model with two Fermi seas. The dashed curves wereobtained using the parabolic model described in this section while the open-circle lines cor-respond to solutions of the Fermi-Hubbard model. Both models are in very good agreementacross all momentum and energy range experimentally probed. Note how, unlike before, theholon mode emanating from + k F is now fully captured both in the hole and particle sectors.The line shapes at zero magnetic field for various tunnelling processes are also shown in Fig.S7B. The tunnelling resonance peak at large negative bias (black dashed line in Fig. S7A)corresponds to 2D-2D ‘parasitic’ tunnelling, while at positive bias the main contribution toconductance comes from both 1D and 2D processes. Unlike the 2D-2D peak however, somesignificant amount of broadening can be observed. We interpret this as the superposition ofboth the 2D (dashed blue) and 1D (dashed magenta) dispersions, as predicted by our model.This feature is completely missed by previous models, see Fig. S6. • Multiple-subband Occupancy
Every time a new subband starts being occupied, another four parameters ( B w − , B w + , m s and m c )need to be added. The capacitances c and c also need to be updated, with their values nowlying somewhere in between the single-subband 1D-2D and the 2D-2D regimes. This leads toa total of 6+4+4+4=18 parameters when we are in the three subband regime.15 igure S7. Single-subband regime: Two Fermi Seas and Fermi-Hubbard Model. ( A ) Tunnelling-conductancedifferentials d G/ d V DC and d G/ d B as a function of the DC-bias V DC and the in-plane magnetic field B , for a1.7 µ m device, mapped at V WG = − . V and fitted using the two Fermi seas model, see text. Dashed blackand blue curves have the same meanings as before, see Fig. S6. The dashed green and magenta lines mark the1D dispersions for spinons and holons, respectively. The open-circle curves mark the corresponding solutions ofthe Fermi-Hubbard model for N = 54 . Inset: Spin-charge separation (dashed black ‘S’ and ‘C’ lines) near the+ k F point at low-bias. ( B ) Conductance G line cut at B = 0 T, showing both the 2D-2D and 1D-2D tunnellingresonance peaks. The dashed curves mark the same resonances as described in A . igure S8. Multiple-subband Regime Tunnelling conductance differentials d G/ d V DC and d G/ d B plotted asa function of the DC-bias V DC and the in-plane magnetic field B for a 1.7 µ m device, mapped with ( A ) two( V WG = − . V) and ( B ) three ( V WG = − . V) occupied subbands, respectively. The dashed colouredcurves have the same meaning as before (see Fig. S6 and S7). Insets show spin-charge separation near the + k F point for the bottom-most occupied subband.
17e have restricted our analysis, where applicable, to the bottom two subbands even whenmore are occupied, as it becomes progressively more difficult to accurately and reliably analysethe data, see Fig. S8 and Fig. 4 in the main text. One of the difficulties is that, as the number ofsubbands is increased, it becomes visually harder to extract B w − for most subbands. We knowhowever that a high degree of symmetry exists between all 1D subbands (see Fig. S4A). Sincethe 1D channels have approximately parabolic confinement potentials, the subband dispersionsshould approximately follow that of a harmonic oscillator. Therefore, we have B + , + B − , B + , + B − , , (S33)where B ± ,n denote the crossing points between the nth-subband and the B axis at V DC = 0 V.Similarly, one can look at the crossing point V , n between the n th-subband and the V DC axis at B = 0 T, in which case we obtain eV ,n = ± ¯ h m ? ( k F,UW, n − k F,LW ) = ⇒ V ,n = ± ed m ? B + ,n B − ,n . (S34)Here, k F,UW, n refers to the Fermi wavevector of the spinon mode of the n th subband while ± labels the lower and upper layers, respectively. The second equality is obtained by converting k F s into B ± ,n using (S17). Combining both (S33) and (S34) we arrive at ( B + , − B − , ) = ( B + , + B − , ) − B + , B − , = ⇒ B + , − B − , s(cid:18) B + , + B − , (cid:19) ± V , med . (S35)Equation (S35) can be used to determine the position of the second subband by varying V , and plotting the dispersions with the resultant B ± , until the best match is obtained. Since B + , is usually well-observed in the data, this effectively imposes a very strong constraint on B − , .Another difficulty of the multiple-occupancy regime has to do with extracting the holonand spinon masses, m c and m s . The first is usually obtained by looking at the holon branchemerging from + k F at high bias. However, as more subbands come into play these becomeharder to resolve separately. Similarly to what was done in the single-subband regime, we canobtain a second constraint on m c by looking at the peak broadening at B = 0 T. On the otherhand, m s is extracted by matching to the spinon-mode dispersion in the hole sector. This isusually clearly observed for at least the bottom two subbands. Nevertheless, since the spin andcharge velocities can only be obtained for the bottom-most occupied subband, this ratio cannotbe compared to the independently obtained mass ratios, as was done when only one subbandwas occupied. Overall, this translates into a slight increase of the error with higher subbandoccupancy, as seen from Fig. 2F and 4 in the main text. References [1] G. D. Mahan.
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