Abstract
Power laws in physics have until now always been associated with a scale invariance originating from the absence of a length scale. Recently, an emergent invariance even in the presence of a length scale has been predicted by the newlydeveloped nonlinearLuttingerliquid theory for a onedimensional (1D) quantum fluid at finite energy and momentum, at which the particle’s wavelength provides the length scale. We present experimental evidence for this new type of power law in the spectral function of interacting electrons in a quantum wire using a transportspectroscopy technique. The observed momentum dependence of the power law in the highenergy region matches the theoretical predictions, supporting not only the 1D theory of interacting particles beyond the linear regime but also the existence of a new type of universality that emerges at finite energy and momentum.
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Introduction
Power laws play an important role in physics and they are generally associated with a scale invariance originating from the absence of a length scale. The most notable example is in continuous phase transitions, where the diverging correlation length means that microscopic details become irrelevant and universality classes characterise the exponents^{1,2,3,4,5,6}. Response functions associated with the dynamics of manifestly scaleinvariant soft excitations in a quantum system, such as Xray absorption in a metal^{7,8,9,10} or the spectral function of a Tomonaga–Luttinger liquid (TLL)^{11,12,13,14,15}, likewise display power laws around zero momentum. Recently, the possibility of invariance emerging even in the presence of a length scale was predicted by the newly developed nonlinear Luttingerliquid theory for a onedimensional (1D) quantum fluid at finite energy and momentum^{16}. Here, the length scale is determined by the particle’s wavelength.
In 1D, the effects of electron–electron interactions are amplified strongly, structuring free electrons into collective excitations. These excitations are charge and spin density waves within the TLL theory, which approximates the electron dispersion relation with a linear energy–momentum dependence. The TLL spectral function (which gives the probability of finding an electron with a particular energy and momentum) is zero at energies below the dispersion of the collective mode, and follows a power law in energy measured from the threshold defined by that spectrum; the corresponding exponent depends on the interaction strength^{17,18}. To go beyond the linear approximation of the energy–momentum dependence and take into account the parabolicity of the dispersion of free electrons, the mobileimpurity model was developed^{19}, leading to a nonlinear hydrodynamic theory that extends the lowenergy universality of a TLL to finite energy, corresponding to excitations from far below the Fermi energy to just above it. The exponent becomes momentum dependent through a finite curvature of the spectraledge dispersion that changes with the momentum, defining the momentum dependence as a unique feature of the nonlinear hydrodynamics in 1D. For electrons (fermions with spin 1/2), this dispersion is close to parabolic^{20} and the mobileimpurity model with spin and charge degrees of freedom predicts an essential dependence of the threshold exponent on momentum away from the Fermi points^{21}.
Here, we probe the electron dispersion and threshold exponents experimentally by measuring the momentum and energyresolved tunnelling between neighbouring 1D and 2D systems, formed within the two quantum wells of a GaAs/AlGaAs doublewell heterostructure. We find experimental evidence for the new type of power law in the spectral function. As we probe the energies below the bottom of the 1D dispersion, the tunnelling current drops away more slowly than that predicted by a noninteracting model and cannot be explained by a power law with a momentumindependent exponent. This excess conductance is instead consistent with the momentumdependent exponent predicted by the new mobileimpurity model described above^{20,21}. This measurement supports not only the 1D theory of interacting particles beyond the linear regime, but also the existence of a new type of universality that emerges at finite energy and momentum. The result is a significant stepping stone towards a systematic understanding of a wider variety of manybody systems, ranging from quantum optics to highenergy and solidstate physics.
Results
Principle of experiment
Our devices contain two closely spaced twodimensional electron gases (2DEGs). The upper 2DEG is depleted into 1D channels by Schottky gates. A small current may flow between the two layers due to quantum tunnelling (in the zdirection). In order for tunnelling to occur, some filled states of one system must have the same momentum and energy as some empty states of the other, as momentum and energy are conserved. In order to explore the entire dispersions of each of the systems, a DC voltage V_{DC} can be applied between the layers, providing extra (or less) energy for tunnelling. The dispersions can be thought of as being offset in energy one from the other. Likewise, a magnetic field B applied in the plane of the layers causes a Lorentz force which boosts the momentum of electrons as they tunnel between the layers. When this inplane B is perpendicular to a 1D wire (in the ydirection), the resultant momentum change is along the wire (in the xdirection). In a 2D system, the dispersion is a paraboloid in energy as a function of momentum. The corresponding density of states in energy and momentum is called the spectral function.
In Fig. 1a and b, spectral functions of two parallel 2D systems are shown, to illustrate the offsets in momentum \(\hbar k_x\) (caused by B in the ydirection) for which significant tunnelling occurs. Here, there is no energy offset (V_{DC} = 0), so tunnelling occurs at the Fermi energy E_{F} (where there are both empty and filled states). In Fig. 1a, B is quite small, so the paraboloids touch on the inside, probing the Fermi surface at \(k_x =  k_{\mathrm{F}}\), where k_{F} is the Fermi wave vector. In Fig. 1b, B has increased until the paraboloids touch on the outside, probing the Fermi surface at \(k_x = k_{\mathrm{F}}\). The green lines mark the states contributing most to the tunnelling. If V_{DC} is nonzero, empty states at the Fermi energy of one paraboloid overlap filled states of the other (or vice versa) away from E_{F}. In Fig. 1c, the 2D system is shown probing the more complex 1D spectral function (red) with multiple 1D subbands, for finite V_{DC}. The second 1D subband, for example, is split into two peaks in k_{y}, matching the Fourier transform of the corresponding spatial wave function.
The device therefore behaves as a spectrometer, using the 2D system to probe the spectral function of the 1D system (and vice versa). Figure 2a shows an overview of such a measurement, where conductance through the sample is measured as a function of energy (∝ V_{DC}) and momentum (∝ B). The conductance peaks form a set of intersecting parabolae, which correspond to the dispersions of each system. The parabolae corresponding to the 1D (2D) system are shown as solid (dashed) lines.
Design of the nanostructure
Our spectrometer devices are made with an MBEgrown GaAs/Al_{0.33}Ga_{0.67}As heterostructure with two parallel quantum wells 100 nm beneath the surface, both 18nm wide, separated by a 14nm tunnel barrier, giving a 32nm centretocentre distance d. Figure 3a is an illustration of the device structure. An array of identical finefeature wire gates (labelled WG) is fabricated on a Hall bar by electronbeam lithography. The gates are all joined together by air bridges, which are used to supply a negative voltage to deplete the upper 2DEG layer into 1D channels while leaving the lower 2DEG undisturbed. The use of air bridges allows the wires to be almost uniform along their whole length, rather than becoming narrower at one end, as would happen if all the gates were joined together at one end by a gate on the surface, which would give a much greater electric field at the 2DEG than does a gate separated by an air gap. Details of our airbridge technique will be published elsewhere (Y.J., J.P.G. and C.J.B.F., manuscript in preparation). A split pair of gates (labelled SG) and a middle gate (labelled MG) are positioned near the source contact of the Hall bar. The split gates are used to pinch off both the upper and lower 2DEG layers, while the midgate supplies a positive voltage to induce a channel in the upper 2DEG from where electrons can flow into the array. The splitgate/midgate combination ensures that electrons from the source Ohmic contact may only enter the 1D array via the upper layer. Current flows along a narrow strip from this injector to the entrance to each wire. This strip is labelled p in Fig. 3b as it provides an unwanted (parasitic) region in which tunnelling can occur. To control this region, it is covered by a gate (labelled PG in Fig. 3a). This allows the carrier density there to be varied, which changes the position of the parasitic tunnelling parabolae shown in magenta and cyan in Fig. 2b, but it has little effect on their magnitude. On the opposite side of the device, near the drain contact, there is a barrier gate labelled BG, which pinches off just the upper 2DEG, so that electrons may only leave via the lower 2DEG, which leads to the drain Ohmic contact. As illustrated by white arrows in Fig. 3b, electrons must tunnel between the 1D wires and the lower 2DEG in order to travel between the source and drain contacts.
There are therefore two regions where electron tunnelling occurs from the upper well: (1) the 1D channels defined by the wiregate array (shown in blue in Fig. 3b), and (2) the parasitic regions surrounding the wiregate array, which allow electron flow into the array (shown in purple). We detect tunnelling from both regions in the experiment. Analysis is focused on the tunnelling from the wire array. The strength of confinement of the 1D channels can be controlled by the wire gates. The array, which contains 500 repeating units, provides a large total tunnelling area and hence a largeenough conductance to be measured with low noise. Tunnelling from the second region is parasitic—it cannot be eliminated due to device design. In order to remove the parasitic conductance contribution, a matching set of data is measured under identical conditions, but with the 1D wires just past pinchoff, such that the data contain only conductance due to parasitic tunnelling. After subtraction of these data from the conductance with the wires open, sharp features of the pregion (the magenta and cyan parabolae) are still noticeable, owing to a slight difference in the carrier densities, but in the regions of interest for the fitting to models (see later), the variation with density or field is slow, and so subtracting this background should be acceptable, though we always take into account the possibility that the parasitic contribution may have been scaled up or down by changing the effective area of the parasitic region with the wire gate.
The experiment was carried out at T < 100 mK in a ^{3}He/^{4}He dilution refrigerator (sample A) or at T ~ 330 mK in a ^{3}He cryostat (sample B). Device conductance was measured in a twoterminal phasesensitive setup, where a small AC voltage was applied as the source–drain bias and the current response measured by a lockin amplifier. The wiregate voltage is chosen to be negative enough that only the 1D states in the lowest subband are populated, though for smaller voltages, up to three 1D subbands can be observed clearly. The conductance across the sample was measured as the DC bias was swept and the magnetic field incremented.
Nonlinear phenomena
The Luttinger model is only applicable in a small range of excitation energies about the Fermi energy E_{F}, which we define as the energy of the highestoccupied electron state relative to the bottom of the 1D subband. This corresponds to V_{DC} = 0. As we can measure excitations at all energies, including far above and below E_{F}, the predictions of the new nonlinear TLL model can be tested. We have previously observed signs of another nonlinear theory, a hierarchy of modes, which predicts the socalled replicas of the 1D dispersion at higher momentum and inverted in energy, shown as red lines in Fig. 2c^{16,22}. We predicted the spectral strengths of the collective excitations forming the manybody continuum in the nonlinear regime to be inversely proportional to integer powers of the system length, separating the excitations into different levels of a hierarchical structure^{22}. We indeed observed signs of the k > k_{F} replica just above the magnetic field labelled k_{F} in Fig. 2b^{22,23}, where the linear TLL model predicts exactly zero for the density of states. The extra excitations cause G to remain high and then drop off quite rapidly along an inverted parabola below the 1D parabola at V_{DC} > 0, and this shows up as a broad red curve where \({\mathrm{d}}G/{\mathrm{d}}V_{{\mathrm{D}}{\mathrm{C}}} > 0\), inside the labelled region in the figure. This replica parabola is expected to become indistinct further along its length, as observed. In shorter wires, an order of magnitude shorter than in this experiment, signs of a full replica were observed^{24} in the principal region—the red line between the ±k_{F} points for E > E_{F} (above the axis) in Fig. 2c, where the spectral strength of even the strongest manybody excitations is expected to be weaker by a factor proportional to the inverse square of the length of the wire, compared with the excitations forming the principal parabola.
In this paper, we turn our attention to the region below the bottom of the 1D parabola, and make detailed fits of the data to models. As we will show in Fig. 4a and b, we find that the tunnelling conductance is significantly enhanced over that predicted by the noninteracting model and cannot be fitted by a model with a simple, momentumindependent powerlaw dependence, in agreement with the new mobileimpurity model.
Nonlinear models
The region of interest in this paper (shown shaded yellow in Fig. 2a) is below the bottom of the 1D dispersion, where a momentumdependent power law is predicted^{25}. In the region of interest, the parasitic contribution to the conductance mentioned above varies very slowly with B, so the slight variation in density is negligible. The data with the parasitic contribution removed are therefore compared with calculations based on three models that differ in the form of the 1D spectral function: (1) without the power law (or other effect) arising from interactions, (2) with a momentumdependent power law from interactions and (3) with a momentumindependent power law.
Model 1 is the full noninteracting model, and it contains a single parameter Γ_{NI}, the width of the disorderbroadened spectral function. The mobileimpurity model for electrons with Coulomb interactions (model 2) predicts the threshold exponents in terms of the curvature of the spinon mode in the nonlinear regime^{21}. The spectral function of a single 1D subband in the hole sector, \(A_1\left( {k_x,E} \right) \propto 1/\left {E  \varepsilon (k_x)} \right^{\alpha (k_x)}\), is measured in our experiment. For a parabolic dispersion^{20}, \(\varepsilon (k_x) = \hbar ^2\left( {k_x  k_{\mathrm{F}}} \right)^2/\left( {2m_{{\mathrm{1}}{\mathrm{D}}}K_{\mathrm{s}}} \right)\), the exponent is
(see details of the calculation in Supplementary Note 2) where the momentumdependent parameters are \(C\left( k \right) = \left( {k^2  k_{\mathrm{F}}^2} \right)/\left( {k^2/K_{\mathrm{s}}  k_{\mathrm{F}}^2K_{\mathrm{s}}/K_{\mathrm{c}}^2} \right)\) and \(D\left( k \right) = \left( {k  k_{\mathrm{F}}} \right)\left( {k_{\mathrm{F}}/K_{\mathrm{c}}^2 + k/K_{\mathrm{s}}^2} \right)/\left( {k^2/K_{\mathrm{s}}^2  k_{\mathrm{F}}^2/K_{\mathrm{c}}^2} \right)\), and K_{s} and K_{c} are the usual Luttinger parameters. By renormalisationgroup arguments, K_{s} = 1 in our experiment^{26} and we use K_{c} < 1 as a fitting parameter in both the linear and nonlinear regimes. In addition to these interaction effects, we also include the effect of disorderinduced broadening (of width Γ_{I}), which smears the threshold singularities to become maxima of the spectralfunction energy dependence. Model 3 uses a simple power law for comparison, where the dependence on momentum and K_{c} is replaced with a constantvalued α, so that the exponent is momentum independent.
Fitting in the nonlinear regime
The models are evaluated as functions of eV_{DC} (which is equivalent to the energy relative to E_{F}) and B. Since the calculation is too timeconsuming for automated fitting, we calculate the conductance for a range of values of each parameter, and compare the model to the experimental data visually in order to determine the parameters that best fit the experiment (Fig. 4). In Fig. 4, calculations of the 1D–2D tunnelling conductance using the interacting and noninteracting models are compared with our experimental data for two samples (A and B). Cuts through the data are shown as curves for several magnetic fields (marked with black lines in Fig. 2a). The data and calculations are normalised by their maximal values, which occur on the 1D parabola. The nonlinear regime is a relatively wide region at higher negative biases than the peak position (up to the lefthand edge of the yellow region in Figs. 4, 2a), which corresponds to the grey continuum of manybody excitations in Fig. 2c. Since the spectral functions of the disorderbroadened interacting 1D system and the 2DEG are convolved, there is also a finite conductance to the right of the peak. The calculated conductance from a model must therefore fit the measured conductance over a wide range of energies on either side of the peak, including the yellow shaded region in the figure down to a large negative energy that is significantly further from the peak than the broadening Γ (at least down to \(E  E_{\mathrm{F}} =  4.0\) meV), and the white part up to the blue shaded region \(\left( E  E_{\mathrm{F}} >  0.5\;{\mathrm{meV}} \right)\) on the right (which marks the zerobias anomaly, ZBA, to be described later). The models ignore any second subband (visible at low fields for sample A as an enhancement close to zero bias, about 2 meV away from the first subband).
The value of K_{c} < 1 affects the skewness of the conduction peak. We find that \(K_{\mathrm{c}} = 0.70 \pm 0.03\) best matches the conductance peaks for both samples and all fields. Figure 5a shows the range of acceptable Γ_{I} values vs B, for model 2. Between 2.5 and 2.7 T, Γ_{I} is at a minimum and is roughly constant, as here we avoid the 2D parabola, as well as the localised states at the bottom of the 1D parabola (on the left, see Supplementary Note 1 for a discussion of their effects) and the charge line (on the right) probing at the same time manybody states well into the nonlinear regime far away from the Fermi energy. The nonzero Γ_{I} is largely caused by monolayer fluctuations in the barrier thickness, which give a spread of subband energies of this order. In Sample A, this \(\Gamma _{\mathrm{I}} = 0.32 \pm 0.03\) meV is about 20% larger than \(\Gamma = 0.25 \pm 0.03\) meV of the 2D–2D tunnelling signal (the magenta and cyan parabolae in Fig. 2b) measured without wire gates, probably because they introduce some additional disorder. We concentrate on this region because the larger Γ_{I} close to the bottom of the 1D parabola obscures the effect of K_{c} on the conductance line shape. We choose representative cuts through the data (shown as black vertical lines in Fig. 2a). The conductance is shown schematically in the lowerleft inset, and its differential (without background subtraction) in Fig. 2b. The position of the peak corresponds to the 1D parabola. The densities of the two layers are determined from the crossing points (labelled ±k_{F}) and are used to calculate E_{F}.
The noninteracting calculation (model 1, crosses) gives a sharp drop to the left of the peak, as there are no states below the parabolic dispersion. However, the full interacting calculation (based on model 2 and exemplified by the topmost fit presented in Fig. 4a) predicts exactly this, an enhancement of tunnelling there, as it allows multiple manybody excitations to be created. An example of such an excitation is marked by the green circle in Fig. 2c, composed of a hole deep below E_{F} (dband in Fig. 2d and a number of Luttingerliquid modes around E_{F} (rband)). This predicts a powerlaw dependence α on energy away from the dispersion relation/band, where α is, remarkably, a function of momentum. Note that, in calculating the tunnelling conductance, one convolves the 1D and 2D spectral functions over all momenta and energies, so the variation of α with k_{x} must be included and the result at any value of DC bias includes a range of α. Model 3 dispenses with this momentum dependence (see Fig. 4c). Neither a fit with the maximum correctly normalised (dots) nor one with the tail aligned to the data (+ symbols) matches the data at all well, as they deviate immediately from the data on one or other side of the peak. This shows that the momentum dependence of α is required to get a good fit. This figure also shows another attempt to fit the data using the noninteracting model (× symbols), where Γ is increased to match the tail, but this clearly gives an unacceptably slow decay to the right of the maximum.
Note that we observe a good fit to the nonlinear theory in both samples studied, with different wire lengths, and at different temperatures. Supplementary Figure 2 shows the results for Sample B corresponding to those shown here for Sample A in Figs. 2 and 5. In Supplementary Note 1, we consider, and exclude, other possible causes of the enhancement of tunnelling conductance below the 1D subband edge. Supplementary Figure 3, Supplementary Table I and ref. ^{23} show the dependence of the tunnelling conductance and layer densities on gate voltage for the two samples.
Other evidence for interactions
Our samples also show the following effects that were reported in previous works: (1) powerlaw suppression of the tunnelling conductance around zero DC bias (zerobias anomaly, ZBA), caused by vanishing of the tunnelling density of states of the linear TTL at the Fermi energy^{27,28}. This has been seen in carbon nanotubes^{13}, in tunnelling between two 1D wires^{14} and between a 1D wire and a 2DEG^{15}. From fitting to the powerlaw formula used in ref. ^{15} in the lowenergy regime, with exponent β, we deduce the finite range of energies around EF (blue region in Fig. 5b), where the ZBA affects G significantly (see Supplementary Note 1), so that we can exclude this region from the nonlinear or noninteracting fitting described above. From the ZBA fits, we obtain \(\beta = 0.48 \pm 0.24\). This linear TLL exponent β gives, using the relation between β and K_{c} in the endtunnelling regime from ref. ^{15}, \(K_{\mathrm{c}} = 0.59 \pm 0.13\) (sample A). (2) Separation of spinon and holon excitations in the linear regime^{14,15} close to the Fermi momentum \(\hbar k_{\mathrm{F}}\) (the red lines labelled by C and S in Fig. 2a, b and e).
The intrinsic interaction parameter K_{c} for each sample can be extracted from the slopes of the spinon and holon modes, see Fig. 2e, as \(K_{\mathrm{c}} = v_{\mathrm{s}}/v_{\mathrm{c}} = 0.76 \pm 0.07\) (sample A) and \(K_{\mathrm{c}} = 0.61 \pm 0.06\) (sample B). These two other ways of extracting K_{c} are consistent with the value of \(K_{\mathrm{c}} = 0.72 \pm 0.03\) (sample A) and \(K_{\mathrm{c}} = 0.69 \pm 0.03\) (sample B) extracted in the nonlinear regime in Fig. 4a and b, giving additional confidence that we are observing an effect of the same Coulomb interaction within a 1D system in the nonlinear regime.
Another indication of electron–electron interactions is the effective mass m_{1D} that we observe for the 1D parabola. For the calculated peaks to line up well in energy with the data in Fig. 4a, we find that \(m_{{\mathrm{1D}}} = 0.92( \pm 0.05)m^ \ast\) (sample A) and \(m_{{\mathrm{1D}}} = 0.81( \pm 0.05)m^ \ast\) (sample B), where m* is the 2D effective mass. Note that, while there is an uncertainty in the exact tunnelling distance, and hence in the conversion factor between B and momentum, we use the 2D parabolae (measured by the 1D system or the parasitic region) to calibrate the distance. The observed difference in masses is due to nonequal contributions of the interactions in different dimensions to renormalisation of the free particle mass.
Discussion
We have studied experimentally the decay of the tunnelling current below the bottom of the 1D subband. The conductance G decays more slowly than that predicted by the noninteracting theory, or by interacting theory that includes only a fixed power law α. A good fit, however, is obtained using a power law that depends on momentum, as predicted by the recent theory. This appears to be the first example of an interactiondriven variable power law.
Methods
Tunnelling current
The tunnelling current between the two 2DEG layers is given by^{29}
where e is the electron charge, \(f_T(E)\) the Fermi–Dirac distribution function, d is the spatial separation between the two layers of 2DEGs, n is the unit normal to the surface, \({\mathbf{B}} =  B{\hat{\mathbf{y}}}\) is the magneticfield vector (magnitude B), \({\hat{\mathbf{y}}}\) is the unit vector in the ydirection, A_{1} and A_{2} are the spectral functions of the 1D and 2D systems, respectively and their corresponding Fermi energies are E_{F1D} and E_{F2D}. According to Eq. (2), the tunnelling current between the two layers is proportional to the overlap integral of the spectral functions of the two layers. We can induce an offset eV_{DC} in the Fermi energies between the two layers by applying a DC bias V_{DC}. A momentum offset can be induced by a magnetic field of strength B parallel to the 2DEG layers (as shown in Fig. 3b). Assuming that the field direction is along the yaxis, the vector potential is equal to \({\mathbf{A}} = (zB,0,0)\) in the Landau gauge, and the Lorentz force shifts the momentum of the tunnelling electrons in the xdirection by \({\mathbf{p}} = \hbar {\mathbf{k}} = (  edB,0,0)\). At low temperatures, the Fermi–Dirac distributions can be approximated by a Heaviside step function θ(E).
Modelling details
The tunnelling rate is proportional to each spectral function, which gives the probability density to find an electronic state at a given point of energy–momentum space. The spectral function can be obtained via a Fourier transform of the realspace Green function^{30}. The latter is expressed in terms of electron wave functions. In a free 2D space, the electron wave function is a plane wave, so the spectral function is a delta function: \(A_2({\mathbf{k}},E) = \delta \left( {E  \varepsilon ({\mathbf{k}})} \right)\), where ε(k) is the dispersion relation \(\hbar ^2(k_x^2 + k_y^2)/2m\). To account for disorder broadening, the spectral function is convolved with a Lorentzian function with spread Γ:
Experimentally, the gateinduced 1D channels have finite transverse confinement potentials (instead of being infinitely narrow and hence having infinite subband spacing). For this reason, the 1D spectral function depends on the transverse dimension, ky. The confinement potential can be treated as a parabolic quantum well, whose electron wave function is given by a quantum harmonicoscillator solution^{31}. The confinement results in energy levels known as 1D subbands. The spectral function of the 1D system is given by the Fourier transform of the wave functions summed over all subbands, which contribute to conduction (i.e. below E_{F}), and convolved with a Lorentzian function to account for broadening. Without considering the effects of interactions, the 1D spectral function is
where n is the 1D subband index, \(a = m_{{\mathrm{1}}{\mathrm{D}}}\omega /\hbar\) is a finite width of the wire in the ydirection, \(E_n\left( {k_x} \right) = \hbar^2 k_x^2/(2m_{{\mathrm{1}}{\mathrm{D}}}) + \hbar \omega \left( {n + 1/2} \right)\) is the parabolic dispersion of each subband, \(\hbar \omega\) is the energy spacing of the subbands and H_{n}(x) are the Hermite polynomials. As was shown in the main text, the 1D spectral function derived from the mobileimpurity model is \(A_1(k_x,E) \propto 1/E  \varepsilon (k_x)^{\alpha (k_x)}\) (see details in Supplementary Note 2) and the momentum dependence of the exponent is given by Eq. (1). The 1D spectral function that includes the effects of interactions is therefore
where a finite number of 1D subbands n is taken into account.
The integral in Eq. (2) was evaluated numerically using Mathematica, which gave the tunnelling current across the sample and was used to calculate the conductance after taking the derivative with respect to eV. The calculation and the experimental results were normalised to their own conduction peak values and compared in Fig. 4.
Data availability
Data associated with this work are available at the University of Cambridge data repository (https://doi.org/10.17863/CAM.39711).
Code availability
Code associated with this work is available as part of the data available at the above link.
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Acknowledgements
This work was supported by the UK EPSRC [Grant Nos. EP/J01690X/1 and EP/J016888/1]. O.T. acknowledges support from the German DFG through the SFB/TRR 49 programme. L.G. acknowledges support from NSF DMR Grant No. 1603243.
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Project planning: C.J.B.F., O.T., L.I.G. and A.J.S.; MBE growth: I.F. and D.A.R.; ebeam lithography: J.P.G.; sample fabrication: Y.J. and M.M.; transport measurements: Y.J., M.M., A.A., W.K.T. and C.J.B.F.; analysis of results and theoretical interpretation: Y.J., C.J.B.F. and O.T.
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Jin, Y., Tsyplyatyev, O., Moreno, M. et al. Momentumdependent power law measured in an interacting quantum wire beyond the Luttinger limit. Nat Commun 10, 2821 (2019). https://doi.org/10.1038/s41467019106132
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DOI: https://doi.org/10.1038/s41467019106132
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