Abstract
Unconventional quasiparticles emerging in the fractional quantum Hall regime^{1,2} present the challenge of observing their exotic properties unambiguously. Although the fractional charge of quasiparticles has been demonstrated for nearly three decades^{3,4,5}, the first convincing evidence of their anyonic quantum statistics has only recently been obtained^{6,7} and, so far, the socalled scaling dimension that determines the propagation dynamics of the quasiparticles remains elusive. In particular, although the nonlinearity of the tunnelling quasiparticle current should reveal their scaling dimension, the measurements fail to match theory, arguably because this observable is not robust to nonuniversal complications^{8,9,10,11,12}. Here we expose the scaling dimension from the thermal noise to shot noise crossover and observe an agreement with expectations. Measurements are fitted to the predicted finitetemperature expression involving both the scaling dimension of the quasiparticles and their charge^{12,13}, in contrast to previous charge investigations focusing on the highbias shotnoise regime^{14}. A systematic analysis, repeated on several constrictions and experimental conditions, consistently matches the theoretical scaling dimensions for the fractional quasiparticles emerging at filling factors ν = 1/3, 2/5 and 2/3. This establishes a central property of fractional quantum Hall anyons and demonstrates a powerful and complementary window into exotic quasiparticles.
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Main
Exotic quasiparticles could provide a path to protected manipulations of quantum information^{15}. Yet their basic features are often challenging to ascertain experimentally. The broad variety of quasiparticles emerging in the regimes of the fractional quantum Hall effect constitutes a prominent illustration. These are characterized by three unconventional properties^{1,2}: (1) their charge e* is a fraction of the elementary electron charge e; (2) their anyonic quantum statistics is different from that of bosons and fermions; and (3) the dynamical response to their injection or removal along the propagative edge channels is peculiar, ruled by a ‘scaling dimension’ Δ different from the trivial Δ = 1/2 of noninteracting electrons. In the simplest Laughlin quantum Hall states, at filling factors ν = 1/(2n + 1) (\(n\in {\mathbb{N}}\)), the fractional anyon quasiparticles have a charge e* = νe, an exchange phase θ = νπ and a scaling dimension Δ = ν/2. Despite four decades of uninterrupted investigations of the quantum Hall physics, experimental confirmations of the predicted scaling dimension remain lacking, including for Laughlin fractions.
Such a gap may seem surprising because Δ plays a role in most transport phenomena across quantum point contacts (QPCs), the basic building block of quantum Hall circuits. Indeed, the elementary tunnelling process itself consists in the removal of a quasiparticle on one side of a QPC and its reinjection on the other side, whose time correlations are set by Δ (refs. ^{1,2}). In Luttinger liquids, the scaling dimension of the quasiparticles is related to the interaction strength, also referred to as the interaction parameter K, which notably determines the nonlinear I–V characteristics across a local impurity^{16}. Consequently, the knowledge of Δ is often a prerequisite to connect a transport observable with a property of interest. Furthermore, as straightforwardly illustrated in the Hong–Ou–Mandel setup^{17,18,19}, Δ naturally rules timecontrolled manipulations of anyons, which are required in the perspective of topologically protected quantum computation based on braiding^{15}. In this work, the scaling dimension of fractional quantum Hall quasiparticles is disclosed from the thermal noise to shot noise crossover, as recently proposed^{12,13}. The observed good agreement with universal predictions establishes experimentally the theoretical understanding and completes our picture of the exotic fractional quantum Hall anyons.
Characterizing exotic quasiparticles
The first unconventional property of quantum Hall quasiparticles that has been established is their fractional charge e*. Consistent values were observed by several experimental approaches^{3,4,5,20,21,22,23,24,25,26}, with the main body of investigations based on shotnoise measurements across a QPC. In this case, the scaling dimension can be cancelled out, leaving only e*, by focusing on the ratio between shot noise and tunnelling current (the Fano factor) at high bias voltages. The nonstandard braiding statistics of fractional quasiparticles turned out to be more challenging to observe. Convincing evidences were obtained only recently, through Aharonov–Bohm interferometry^{6,27}, as well as from noise measurements in a ‘collider’ geometry^{7,28,29,30}. Note that, whereas the latter strategy is particularly versatile, the noise signal is also entangled with the scaling dimension^{18,19,31,32}, which complicates a quantitative determination of the anyon exchange phase θ (ref. ^{28}). Finally, the scaling dimension of the quasiparticles was previously investigated through measurements of the nonlinear I–V characteristics of a QPC^{33,34,35}. However, no reliable value of Δ could be obtained for the fractional quasiparticles of the quantum Hall regime (see ref. ^{36} for an observation in a circuit quantum simulator and ref. ^{37} for a good match on the I(V) of tunnelling electrons across a (ν = 1)–(ν = 1/3) interface). Indeed, the I–V characteristics is generally found at odds with the standard model of a chiral Luttinger liquid with a local impurity (see, for example, refs. ^{2,14,33,38} and references therein). Most often, a fit is impossible or only by introducing extra offsets and with unrealistic values for e* and Δ (refs. ^{29,34,35,39}).
The puzzling I–V situation motivated several theoretical investigations. A simple possible explanation for the data–theory mismatch is that the shape of the QPC potential, and therefore the quasiparticle tunnelling amplitude, is affected by external parameters, such as an electrostatic deformation induced by a change in the applied bias voltage, the temperature or the tunnelling current itself^{10}. Other possible nonuniversal complications include an energydependent tunnelling amplitude^{11}, further edge modes either localized^{8} or propagative^{40,41} and Coulomb interactions between different edges^{9}. In this context, the scaling dimension was connected to different, arguably more robust proposed observables, such as deltaT noise^{42,43}, thermal noise to shot noise crossover^{12,13} and thermal Fano factor^{44}.
A proven strategy to cancel out nonuniversal behaviours consists in considering a wellchosen ratio of transport properties, as illustrated by the Fano factor F successfully used to extract e*. Recently, it was proposed that the same F could also give access to the scaling dimension of the quasiparticles, when focusing on the lower bias voltage regime in which the crossover between thermal noise and shot noise takes place^{12,13}. As further detailed later on, the predicted evolution of F along the crossover exhibits a markedly different width and overall shape depending on the value of Δ.
This investigation implements the characterization of the scaling dimension from the Fano factor crossover on four different quantum Hall quasiparticles: (1) the e* = e/3 quasiparticles, observed at ν = 1/3 (ref. ^{4}) and along the outer edge channel of conductance e^{2}/3h at ν = 2/5 (refs. ^{20,24}), of predicted Δ = 1/6 (ref. ^{1}); (2) the e* = e/5 quasiparticles observed along the inner edge channel of conductance e^{2}/15h at ν = 2/5 (refs. ^{20,24}), of predicted Δ = 3/10 (refs. ^{1,45}); (3) the e* = e/3 quasiparticles observed at ν = 2/3 (refs. ^{25,46}), of predicted Δ = 1/3 (ref. ^{45}); (4) the electrons at ν = 3 of trivial Δ = 1/2. See Methods for further details on the predictions.
Experimental implementation
The measured sample is shown in Fig. 1, together with a schematic representation of the setup. It is nanofabricated from a Ga(Al)As twodimensional electron gas (2DEG) and immersed in a strong, perpendicular magnetic field corresponding to the quantum Hall effect at filling factors ν ∈ {1/3, 2/5, 2/3, 3}. Lines with arrows show the chiral propagation of the current along the sample edges. QPCs are formed in the 2DEG by field effect, within the opening of metallic split gates (yellow). We characterize a QPC by the gatecontrolled transmission ratio τ ≡ I_{B}/I_{inj}, with I_{B} the backscattered current and I_{inj} the incident current along the edge channel under consideration. The sample includes five QPCs nominally identical except for their orientation and the presence for QPC_{W} of a surrounding gate (labelled TG in the inset of Fig. 1). This surrounding gate allows us to test the possible influence on the scaling dimension of the local 2DEG density, of an enhanced screening of the longrange Coulomb interactions^{9} and of an increased sharpness of the electrostatic edgeconfinement potential^{40}.
The noise is measured using two cryogenic amplifiers (one is shown schematically). The gain of the noiseamplification chains, and the electronic temperature within the device, were obtained from their relation to thermal noise (Methods). One amplifier (top left in Fig. 1) measures the backscattered (tunnelling) current noise \(\left\langle \delta {I}_{{\rm{B}}}^{2}\right\rangle \) for any addressed QPCs. A second amplifier (not shown) measures the forward current fluctuations δI_{F} transmitted specifically across QPC_{E}. In practice, we focus on the excess noise with respect to zero bias: S(V) ≡ ⟨δI^{2}⟩(V) − ⟨δI^{2}⟩(0).
Scalingdimension characterization
In previous characterizations of the charge e* of fractional quantum Hall quasiparticles, the shot noise is usually plotted as a function of the backscattered current I_{B} and e* is extracted by matching the highbias slope ∂S/∂I_{B} with 2e*(1 − τ), in which 1 − τ corrects for tunnelling correlations at finite τ (ref. ^{47}). Even in this representation, which puts the emphasis on the larger highbias shot noise, a visually discernible and experimentally relevant difference allows us to discriminate between predicted and trivial Δ, as illustrated in Fig. 2a–c. Continuous blue lines show the excess shot noise of quasiparticles of trivial Δ = 1/2 and of charge e/3 (Fig. 2a,c) or e/5 (Fig. 2b), which is given by the broadly used phenomenological expression^{14,47}:
Note that, for composite edges with several electrical channels (such as ν ∈ {2/5, 3}), τ ≡ I_{B}/I_{inj}, in which I_{B} and I_{inj} refer to the dc transmission ratio and currents along the specific edge channel of interest (Methods). The continuous lines of a different colour in the main panels of Fig. 2a–c show the excess noise for the predicted quasiparticle scaling dimension Δ = 1/6 (red, Fig. 2a), Δ = 3/10 (purple, Fig. 2b) and Δ = 1/3 (green, Fig. 2c) obtained from^{8,12,13}:
Here ψ is the digamma function and 1 − τ the ad hoc amplitude factor used for extracting e* from the shotnoise slope at high bias (beyond the perturbative limit τ ≪ 1 in which equation (2) is rigorously derived). Whereas equation (2) reduces to equation (1) for Δ = 1/2, for smaller Δ, the shot noise emerges at a lower voltage. Intuitively, this can be connected through the time–energy relation to the slower decay of correlations at long times (as t^{−2Δ}). For the quasiparticles {e/3, Δ = 1/6} predicted at ν = 1/3, the apparent width of the crossover is more than twice narrower than for Δ = 1/2 (Fig. 2a). The difference is smaller for the quasiparticles {e/5, Δ = 3/10} and {e/3, Δ = 1/3} because Δ is closer to 1/2 (Fig. 2b,c). Nevertheless, as can be straightforwardly inferred from the scatter of the data, it remains in all cases larger than our experimental resolution on the noise. We can already notice that the illustrative shotnoise measurements shown in Fig. 2a–c are closer to the parameterfree prediction of equation (2) with the expected Δ. Note that this agreement is accompanied by a puzzling I–V characteristics as previously mentioned (see τ(V) in insets and also in Extended Data Fig. 9).
For the present aim of characterizing Δ from the thermal noise to shot noise crossover, the Fano factor F ≡ S/2eI_{B}(1 − τ) of bounded amplitude at high bias is better suited^{12,13}. It is plotted against the relevant variable eV/k_{B}T (see equation (2)) in Fig. 2d–f, with symbols and coloured lines corresponding to the noise shown in the panel immediately above. Notably, the effect of Δ < 1/2 on F is not limited to an increased slope at low bias, which could—in principle—be attributed to a temperature lower than the separately characterized T, but results in marked changes in the overall shape of F(eV/k_{B}T). In particular, for Δ = 1/6, the Fano factor is nonmonotonous (red line in Fig. 2d). The increasing steepness while reducing Δ combined with an overall change of shape enables the extraction of this parameter from a fit using equation (2). Qualitatively, the value of F at large bias only reflects e*/e, the overall crossover shape (such as a nonmonotonous dependence at Δ < 1/4) only involves Δ and the lowbias slope is a combination of both e* and Δ. The results of such fits (minimizing the data equation (2) variance) are shown as black continuous lines in Fig. 2d–f, together with the corresponding fitting parameters e* and Δ (the temperature being fixed to the separately determined T ≃ 31 mK).
Robustness of observations
Focusing on the Fano factor cancels out some of the nonuniversal behaviours, but not all of them. Of particular concern are the disorderinduced resonances, which could result in a Coulombdominated sequential tunnelling with a strong effect on the Fano factor. This probably happens in the fractional quantum Hall regime in which QPCs often exhibit narrow peaks and dips in their transmission τ versus gate voltage (see insets in Fig. 3). Accordingly, for some gate voltages, we find that an accurate fit of the noise data is not possible with equation (2), whatever e* and Δ. In such cases, the resulting fitted values are meaningless. This was transparently addressed with a maximum variance criteria between data and best fit. If the fitdata variance is higher, the extracted e* and Δ are discarded (see Methods). This same procedure was systematically applied to all the noise measurements performed over a broad span of gate voltages controlling τ (the full dataset, including discarded fits and analysis code, is available in a Zenodo deposit).
The values of e* and Δ obtained while spanning the gate voltage of the same QPC_{E} are shown versus τ(V = 0) in Fig. 3 for each of the three investigated fractional quasiparticles (see Methods for electrons at ν = 3). We find remarkably robust scaling dimensions (and charges) close to the predictions, shown as horizontal lines. In particular, although the nature of the tunnelling quasiparticles is eventually going to change at τ → 1, we observe that Δ and e* extracted with equation (2) (which is exact only at τ ≪ 1) remain relatively stable over a broad range of τ. Such a stability matches previous e* measurements, including a particularly steady e/5 (ref. ^{48}). Figure 3a shows data points obtained in the ν = 1/3 plateau. A statistical analysis of the ensemble of these points yields ⟨Δ⟩ ≃ 0.167, with a spread of σ_{Δ} ≃ 0.023, which is to be compared with the prediction Δ = 1/6 ≃ 0.1667. The dataprediction agreement on Δ is at the level of, if not better than, that on e* (often found slightly lower than expected). Similar sweeps are shown in Fig. 3b,c for the inner channel of conductance e^{2}/15h at ν = 2/5 (Fig. 3b) and at ν = 2/3 (Fig. 3c). Note that a few data points at ν = 2/5 and at ν = 2/3 are shown as pairs of ‘×’ (e*/e) and ‘+’ (Δ) instead of open and closed symbols (Fig. 3b,c). This indicates an anomalous fitted value of the charge e*, off by about 50% or more from the wellestablished prediction \({e}_{{\rm{th}}}^{* }=e/5\) and \({e}_{{\rm{th}}}^{* }=e/3\), respectively (dasheddotted lines). Because this suggests a nonideal QPC behaviour (for example, involving localized electronic levels), we chose not to include these relatively rare points in the data ensemble analysis of Δ (they remain included in the statistical analysis of e*). For this reduced dataset composed of 15 measurements along the inner channel at ν = 2/5, we obtain ⟨Δ⟩ ≃ 0.327 (σ_{Δ} ≃ 0.078), which is to be compared with the predicted Δ = 3/10 of e/5 quasiparticles. Last, at ν = 2/3, the gatevoltage sweep shown in Fig. 3c gives ⟨Δ⟩ ≃ 0.249 (σ_{Δ} ≃ 0.029), close to the predicted Δ = 1/3 ≃ 0.33. Note, however, that in this more complex case, with counterpropagating edge modes and the emergence of a small plateau versus gate voltage at τ ≃ 0.5 (inset in Fig. 3c), the noise interpretation is not as straightforward, especially when τ is not small (see Methods for further tests and discussions).
The robustness and generic character of these Δ observations are further established by repeating the same procedure in different configurations: (1) on several QPCs, with different orientations with respect to the Ga(Al)As crystal; (2) for several temperatures T; (3) for several topgate voltages V_{tg} controlling the density in the vicinity of QPC_{W}; (4) by changing the magnetic field, both along the ν = 1/3 plateau and to ν = 2/5 on the outer edge channel. Figure 4 recapitulates all our measurements (283 in total), including conventional electrons at ν = 3. Each point represents the average value ⟨e*/e⟩ (diamonds) or ⟨Δ⟩ (triangles) and the corresponding standard deviation obtained while broadly spanning the gate voltage of the indicated QPC (individually extracted e* and Δ are provided in Methods). See also Methods for consistent conclusions from an alternative fitting procedure in which Δ is the only free parameter (e* being fixed to the wellestablished prediction and focusing on low voltages e*V ≤ 2k_{B}T).
Conclusion
Fano factor measurements previously used to investigate the charge of tunnelling quasiparticles also allow for a consistent determination of their scaling dimension, from the width and specific shape of F(eV/k_{B}T). Combined with a systematic approach, the resulting observations of Δ establish longlasting theoretical predictions for the fractional quantum Hall quasiparticles at ν = 1/3, 2/5 and 2/3. This approach could be generalized to other quasiparticles, with the potential to shed light on the nonAbelian quasiparticles predicted at evendenominator filling factors. It may also be applied to other lowdimensional conductors.
Methods
Sample
The sample is nanofabricated by electronbeam lithography on a Ga(Al)As heterojunction forming a 2DEG buried at 140 nm, of density n = 1.2 × 10^{11} cm^{−2} and of mobility 1.8 × 10^{6} cm^{2} V^{−1} s^{−1}. The 2DEG mesa is first delimited by a wet etching of 105 nm, deeper than the Si δdoping located 65 nm below the Ga(Al)As surface. The large ohmic contacts (schematically shown as circles in Fig. 1) used to drive and measure the quantum Hall edge currents are then formed 100–200 μm away from the QPCs by electronbeam evaporation of a AuGeNi stack, followed by a 50s thermal annealing at 440 °C. A 15nm layer of HfO_{2} is grown by thermal atomic layer deposition at 100 °C over the entire mesa, to strongly reduce a gateinduced degradation of the 2DEG that could complicate the edge physics. This degradation is generally attributed to unequal thermal contractions on cooling^{49} or a deposition stress, which could also modulate the edge potential carrying the quantum Hall channels along the gates. In previous works, we observed a change in the behaviour of QPCs, including in the thermal noise to shot noise crossover, that was correlated with their orientation^{28,50} (see also source versus central QPCs in refs. ^{7,29}), which we suspect to result from such gateinduced complication. Here the Ti (5 nm)–Au (20 nm) gates used to form the QPCs are evaporated on top of the HfO_{2}. The five QPCs, of different orientations with respect to the Ga(Al)As crystal, have nominally identical geometries. The split gates have a nominal tiptotip distance of 600 nm and a 25° tipopening angle prolonged until a gate width of 430 nm. Largerscale electronbeam and optical images of the measured device are shown in Extended Data Fig. 1. The relatively important gate width (about three times the 2DEG depth) was chosen to reduce possible complications from Coulomb interactions between the quantum Hall edges across the gates^{9,51} and to better localize the tunnelling location when the QPC is almost open (for less negative gate voltages)^{11}. The nominal separation between the split gates controlling QPC_{W} and the surrounding metal gate is 150 nm. A highmagnification picture of QPC_{W} with its surrounding metal gate is shown in Extended Data Fig. 2. Note that all the gates were grounded during the cooldown.
Measurement
The sample is cooled in a cryofree dilution refrigerator and electrically connected through measurement lines both highly filtered and strongly anchored thermally (see ref. ^{52} for details). Final RC filters with CMS components are positioned within the same metallic enclosure screwed to the mixing chamber that holds the sample: 200 kΩ–100 nF for gate lines, 10 kΩ–100 nF for the bias line and 10 kΩ–1 nF for lowfrequency measurement lines. Note a relatively important filtering of the bias line, which prevents an artificial rounding of the thermal noise to shot noise crossover from the flux noise induced by vibrations in a magnetic field. The differential QPC transmission ∂I_{B}/∂I_{inj} = 1 − ∂I_{F}/∂I_{inj} is measured by standard lockin techniques at 13 Hz. A particularly small ac modulation is applied on V (of rms amplitude \({V}_{{\rm{ac}}}^{{\rm{rms}}}\approx {k}_{{\rm{B}}}T/3e\)) to avoid any discernible rounding of the thermal noise to shot noise crossover. The transmitted and reflected dc currents used to calculate τ and F are obtained by integrating with the applied bias voltage the corresponding lockin signal \({I}_{{\rm{B,F}}}(V)={\int }_{0}^{V}(\partial {I}_{{\rm{B,F}}}/\partial V){\rm{d}}V\).
A specific QPC is individually addressed by completely closing all the other ones. For the composite edges at ν = 2/5 and ν = 3, the characterizing current transmission ratio τ refers to the current transmission along the specific channel of interest. Explicitly, at ν = 2/5, the transmission τ along the inner edge channel is given by the ratio between measured (total) \({I}_{{\rm{B}}}^{{\rm{meas}}}\) (only the inner channel of interest is backscattered, the outer channel is fully transmitted, as attested by a broad and noiseless e^{2}/3h plateau) normalized by the current Ve^{2}/15h injected along the inner channel: \(\tau ={I}_{{\rm{B}}}^{{\rm{meas}}}/(V{e}^{2}/15h)\). For the outer channel at ν = 2/5, the fully backscattered inner edge channel current Ve^{2}/15h is removed from the measured total \({I}_{{\rm{B}}}^{{\rm{meas}}}\) and the result is normalized by the current Ve^{2}/3h injected along the outer edge channel: \(\tau =\left({I}_{{\rm{B}}}^{{\rm{meas}}}V{e}^{2}/15h\right)/\left(V{e}^{2}/3h\right)\).
Noise measurements are performed using specific cryogenics amplification chains connected to dedicated ohmic contacts, through nearly identical L–C tanks of resonant frequency 0.86 MHz (refs. ^{53,54}). The noise ohmic contacts are located upstream of the ohmic contacts used for lowfrequency transmission measurements, as shown in Fig. 1. A dc block (2.2 nF) at the input of the L–C tanks preserves the lowfrequency lockin signal. For the particular case of QPC_{E}, the forward (transmitted) current fluctuations δI_{F} are also measured, which gives us access to \(\left\langle \delta {I}_{{\rm{F}}}^{2}\right\rangle \) and to the crosscorrelations \(\left\langle \delta {I}_{{\rm{B}}}\delta {I}_{{\rm{F}}}\right\rangle \). Apart increasing the signaltonoise ratio for QPC_{E}, this allows us to confirm that \(\left\langle \delta {I}_{{\rm{B}}}^{2}\right\rangle \) matches the more robust crosscorrelation signal^{55}.
The device was immersed in a magnetic field close to the centre of the corresponding Hall resistance plateaus, except when a shift δB is specifically indicated. The data at ν = 1/3, ν = 2/5, ν = 2/3 and ν = 3 were obtained at B = 13.7 T (13.2 T for δB = −0.5 T), 11.3 T, 6.8 T and 1.5 T, respectively. See vertical arrows in Extended Data Fig. 3 for the localization of these working points within a magnetic field sweep of the device along B ∈ [4, 14] T (ν ∈ [1/3, 1]).
Thermometry
The electronic temperature inside the device is obtained by the noise measured at thermal equilibrium, with all QPCs closed. For temperatures T ≥ 30 mK (up to the maximum T ≃ 55 mK), we find at ν = 1/3 and ν = 3 that the measured thermal noise is linear with the temperature readings of our calibrated RuO_{2} thermometer. This establishes the good thermalization of electrons in the device with the mixing chamber, as well as the calibration of the RuO_{2} thermometer. Accordingly, we indifferently get T ≥ 30 mK from the equilibrium noise or the equivalent RuO_{2} readings. At the lowest investigated temperatures of approximately 15 mK, the RuO_{2} thermometers are no longer reliable and T is obtained from the thermal noise by linearly extrapolating from S (T ≥ 30 mK). Note that the S(T) slope was not recalibrated at ν = 2/3 but its change from ν = 1/3 was calculated from the separately obtained knowledge of the L–C tank circuit parameters, see the next section.
Noiseamplification chains calibration
The gain factors \({G}_{{\rm{F,B}}}^{{\rm{eff}}}\) between raw measurements of the autocorrelations, integrated within a frequency range [f_{min}, f_{max}], and the power spectral density of current fluctuations \(\left\langle \delta {I}_{{\rm{F,B}}}^{2}\right\rangle \) are obtained from:
with R_{tk} ≃ 150 kΩ the effective parallel resistance accounting for the dissipation in the considered L–C tank and s_{F,B} the slope of the raw thermal noise versus temperature. The crosscorrelation gain factor is simply given by \({G}_{{\rm{FB}}}^{{\rm{eff}}}\simeq \sqrt{{G}_{{\rm{F}}}^{{\rm{eff}}}{G}_{{\rm{B}}}^{{\rm{eff}}}}\), up to a negligible reduction (<0.5%) owing to the small difference between the two L–C tanks. In practice, the thermal noise slopes s_{F,B} were only measured at ν = 1/3 and ν = 3. The changes in \({G}_{{\rm{F,B}}}^{{\rm{eff}}}(\nu )\) at ν ∈ {2/3, 2/5} from the gains at ν = 1/3 are obtained from:
with the tank impedance given by \({Z}_{{\rm{tk}}}^{1}(f)={R}_{{\rm{tk}}}^{1}+{\left({\rm{i}}{L}_{{\rm{tk}}}2{\rm{\pi }}f\right)}^{1}+{\rm{i}}{C}_{{\rm{tk}}}2{\rm{\pi }}f\), in which L_{tk} ≃ 250 μH and C_{tk} ≃ 135 pF (see Methods in ref. ^{50} for details about the calibration of the tank parameters). At ν ∈ {2/3, 2/5, 1/3}, we integrated the noise signal in the same frequency window f_{min} = 840 kHz and f_{max} = 880 kHz. At ν = 3, a larger window f_{min} = 800 kHz and f_{max} = 920 kHz takes advantage of the larger bandwidth of roughly νe^{2}/2πhC_{tk}.
Noise tests
Among various experimental checks, we note: (1) the effect of a dc bias voltage on the noise when each of the QPCs are either fully open or fully closed, which is found here to be below our experimental resolution. The present imperceptible ‘source’ noise could have resulted from poor ohmic contact quality, incomplete electron thermalization in the contacts or dc current heating of the resistive parts of the bias line; (2) the effect of the QPC transmission on the noise at zero dc bias voltage, which is negligible at our experimental resolution. This rules out a possibly higher electron temperature in the ohmic contact connected to the bias line with respect to one connected to a cold ground, which would translate into an increase in \(\left\langle \delta {I}_{{\rm{B}}}^{2}\right\rangle \) at τ = 1 compared with τ = 0. It also shows that the vibration noise in the bias line at frequencies well below 1 MHz does not translate into a discernible broadband excess shot noise for intermediate values of τ.
Fitting details
The extracted values of e* and Δ shown in Fig. 3 and in Extended Data Figs. 7 and 8 represent the bestfit parameters minimizing the variance between the shotnoise data and equation (2). Only the meaningful points are shown and included in the statistical analysis. These fulfil two conditions: (1) an accurate fit of the data can be achieved and (2) the charge does not deviate too much from the expected value. Condition (1) requires a quantitative assessment of the fit accuracy. For this purpose, we used the coefficient of determination R^{2} and chose to apply the same threshold to all the data taken in similar conditions. Specifically, we automatically discarded fits of R^{2} < 0.9965 at ν = 1/3 and for the outer channel at ν = 2/5, R^{2} < 0.9966 for the inner channel at ν = 2/5 and R^{2} < 0.9968 at ν = 2/3. The number of S(V) sweeps discarded by condition (1) is important, twothirds of the total number (mostly when τ is too close to 0 or 1). We checked that the overall results are only marginally affected by the specific threshold value (within reasonable variations). All the points that satisfy condition (1) are shown and included in the statistical analysis of the quasiparticle charge. Condition (2) is subsequently applied to deal with situations in which the fitting charge is found at odds with the predicted value. Specifically, we discarded S(V) sweeps for which the charge is found to be more than 44% off, that is, \({e}^{* } < 0.56{e}_{{\rm{th}}}^{* }\) or \({e}^{* } > 1.44{e}_{{\rm{th}}}^{* }\). The former happens at small τ with a small QPC gate voltage. This gate voltage might not be enough to deplete the gas under the QPC gates, which could make tunnelling happen in several places along the gates and not only located at their tip, deviating from the model of a point contact. The latter occurs in the socalled strong backscattering regime, in which the nature of the tunnelling quasiparticles is expected to change. Indeed, in the weak backscattering regime (τ ≪ 1), the tunnelling barrier between the two edges is made of the electron gas in the fractional quantum Hall regime that selects the quasiparticles. However, in the strong backscattering regime (τ → 1), the tunnelling barrier between the two edges is made of vacuum that will select electrons. The points that do not satisfy condition (2) are shown with different symbols and not included in the statistical analysis of Δ. They represent a small fraction (5%) of the data satisfying condition (1).
A complementary fitting procedure was used to further establish the robustness of our results. In Extended Data Fig. 4, we summarize the extracted Δ obtained by fitting the thermal noise to shot noise crossover of S(V) using for e* the theoretically predicted value. The fits are performed on the same set of S(V) sweeps as for the main fitting procedure for Δ (obeying the two abovementioned conditions (1) and (2)). The voltage bias extension of these fits is reduced to e*V ≤ 2k_{B}T to limit the weight of the shot noise that is only sensitive to e*.
Predictions
The Δ predictions indicated in the manuscript for fractional quasiparticles at ν = 1/3 and 2/5 follow the Luttinger liquid expression Δ = (e*/e)^{2}/(2Gh/e^{2}) for e* quasiparticles along a chiral 1D channel of conductance G (refs. ^{1,13,45}). Note that, in these fully chiral states (in which all channels along one edge propagate in the same direction), the quasiparticles exchange phase θ is predicted to be simply related to Δ by the relation θ = 2πΔ (see, for example, Appendix A in ref. ^{13}).
The above Luttinger expression for Δ does not apply at ν = 2/3 for e/3 quasiparticles delocalized between a 2e^{2}/3h edge channel and a neutral counterpropagating channel that further increases Δ; see ref. ^{56}. Note that, in general, the predicted link between Δ and θ for fully chiral quantum Hall edges does not hold in the presence of counterpropagating (charged and/or neutral) modes^{13}.
Filling factor 2/3
In this more complex holeconjugate state^{1,2,56}: (1) the edges are not fully chiral and found to carry a backward heat current (going upstream to the flow of electricity) and (2) the QPCs can exhibit a plateau at half transmission (see, for example, refs. ^{46,57}). The former may introduce unwanted heatinduced contributions to the noise, whereas the latter alludes to a composite edge structure. Both have possible consequences on the interpretation of the noise signal^{58,59,60}.

(1)
Nonchiral heating. As in previous works (see, for example, ref. ^{57}), we observe in our device an upstream heating (only) at ν = 2/3, through three noise signatures (see Extended Data Fig. 5). Signature 1: the strongest noise signature, seen at all temperatures (see Extended Data Fig. 5d for T ≃ 40 mK), is obtained in the configuration schematically depicted Extended Data Fig. 5a. Here the noise is measured on a contact located electrically upstream a hotspot in an adjacent voltagebiased contact (about 30 μm away, for example, the contact usually used for measuring ⟨I_{B}⟩ in Fig. 1). As shown in Extended Data Fig. 5a, the noise increase is attributed to a local heating of the noisemeasurement contact (near the location at which electrical current is emitted from this contact) by the upstream neutral heat current originating from the downstream hotspot. See Fig. 3 of ref. ^{57} for a similar observation in the same configuration. Note that, in configurations used for investigating Δ, the heat generated at the downstream grounded contacts cannot propagate to the noisemeasurement contacts, because a floating contact located in between (measuring ⟨I_{B,F}⟩; see Fig. 1) absorbs the upstream heat flow (see Appendix A and Fig. 10 in ref. ^{58} for a specific discussion). Signature 2: a weaker noise signature from a different heating mechanism is observed, only at the lowest temperature (T ≃ 17 mK in Extended Data Fig. 5e), in the configuration schematically depicted Extended Data Fig. 5b (the same configuration is labelled N → C in Fig. 4 of ref. ^{57}). Here a hotspot is created at a downstream contact biased at V. The counterpropagating neutral mode carries an upstream heat current to the QPC, which converts the increased temperature into electrical noise from a thermally induced mechanism. The signal is weaker, as would be expected from a smaller heat current through the longer distance of about 150 μm between the hotspot and the QPC (the heat propagation is diffusive owing to thermal equilibration between counterpropagating channels). In practice, it is discernible only at the lowest temperature T ≃ 17 mK and for the highest QPC sensitivity (τ ≈ 0.5). The lower effect (imperceptible here) at higher temperatures is expected from the generally more efficient relaxation to thermal equilibrium. Note that, in the configurations used to examine Δ, the contacts immediately downstream of the QPC are floating (used to measure the noise or ⟨I_{F}⟩) and, consequently, absorb the upstream heat current originating from the grounded contacts further downstream. Signature 3: a possibly more consequential signature of upstream heating is observed in the same configuration as that used to examine Δ, through an increase in the noise sum S_{Σ} ≡ S_{F} + S_{B} + 2S_{FB}. From charge conservation and the chirality of electrical current, S_{Σ} corresponds to the thermal noise emitted from the source contacts electrically upstream of the QPC (independently of any noise generated along the path, such as the partition noise at the QPC, as long as there is no charge accumulation in the investigated MHz range). In fully chiral states such as ν ∈ {1/3, 2/5, 3}, the temperature of the source contacts is independent of the applied bias V (at the emission point) and so is S_{Σ}. At ν = 2/3 and T ≃ 16 mK, this is not the case, as shown in Extended Data Fig. 5f. This increase in S_{Σ} is interpreted as the signature of a local hotspot in the approximately 150 μm upstream source contacts, by heatedup counterpropagating neutral modes generated at the voltagebiased QPC, as schematically shown in Extended Data Fig. 5c (for a previous observation of the same mechanism, see configuration labelled C → N in Fig. 4 of ref. ^{57}). In practice, we observe a fast increase followed by a near saturation at S_{Σ} ≲ 7 10^{−30} A^{2} Hz^{−1}, which is not negligible with respect to the partition noise of interest (see Fig. 2c). To limit the impact of this effect at T ≃ 16 mK and ν = 2/3, we only considered the crosscorrelation signal (S ≡ −S_{FB} = −⟨δI_{F}δI_{B}⟩) measured on QPC_{E}. Indeed, a symmetric heating of the two source contacts electrically upstream of QPC_{E} (biased at V and grounded) would not result in any change of the crosscorrelations (but instead in a thermally induced increase of the autocorrelations). See ref. ^{55} for a discussion on the stronger robustness to artefacts of crosscorrelations with respect to autocorrelations. At the higher temperatures investigated, there was no discernible change in S_{Σ} and we performed our data analysis using all the noise measurements available.

(2)
Noisy τ = 1/2 plateau. A small but discernible ‘plateau’ is present at τ ≃ 1/2 in the transmission versus split gate voltage of both QPC_{E} (see inset in Fig. 3c) and QPC_{SE} (see Extended Data Fig. 6). These plateaus, which are robust to the application of a bias voltage V and to temperature changes, suggest the presence of two e^{2}/3h edge channels sequentially transmitted across the QPC. In that case, there would be no partition noise at the QPC, as observed at ν = 3 and ν = 2/5. By contrast, the small τ ≃ 1/2 ‘plateaus’ at ν = 2/3 exhibit a substantial noise signal (see also, for example, ref. ^{46}). It was proposed that such noise on a τ = 1/2 plateau was resulting not from the emergence of shot/partition noise but from a heating mechanism involving the thermal equilibration between downstream charge modes and upstream neutral modes^{58,59,60}. In this picture of the QPC at τ ≈ 0.5, the fit parameters e* and Δ should not be interpreted as the charge and scaling dimension of fractional quasiparticles. Which picture more adequately describes the QPC at τ ≃ 1/2 and ν = 2/3 is not straightforward. On the one hand, whereas the smallness of the τ ≃ 1/2 ‘plateaus’ does not rule out a simple accidental explanation within the tunnelling picture (from the specific way the barrier deforms with gate voltage, possibly with nearby defects), their mere observation casts doubts on the tunnelling picture and, consequently, on the interpretation of the fit parameters e* and Δ near τ ≃ 0.5 as characterizing quantum numbers of fractional quasiparticles. On the other hand, the observation of similar values as for small transmissions, at which the noise signal originates from the tunnelling of fractional quantum Hall quasiparticles across the QPC, suggests that the same tunnelling mechanism is at work at τ ≃ 1/2. In particular, a markedly larger (overPoissonian) noise would be expected from the heating interpretation in the socalled thermally equilibrated regimes (compared with e*/e ≈ 0.3 observed here over a broad transmission range, including τ ≃ 1/2, and theoretically expected for the fractional quantum Hall quasiparticles at ν = 2/3)^{58,60}. Overall, more caution is advised on the interpretation of the extracted e* and Δ at τ ≈ 0.5 for ν = 2/3, compared with lower τ or different ν ∈ {1/3, 2/5}.
Extended data
Individual values of Δ and e*/e extracted along gate voltage spans are shown in Extended Data Figs. 7 and 8, in complement to Fig. 3.
The dc voltage dependence of the transmission τ(V) at all gatevoltage tunings in the three configurations shown in Fig. 3 are plotted in Extended Data Fig. 9. Among these are three τ(V) also shown in the insets of Fig. 2a–c.
Measured versus tunnelling noise
The measured backscattered current I_{B} can always be written as the sum I_{B} = I_{T} + I_{gnd} of the incident current I_{gnd} emitted from the ohmic contact that would solely contribute to I_{B} in the absence of tunnelling and of the tunnelling current I_{T} across the constriction. With this decomposition, the backscattered current noise reads:
with \(\langle \delta {I}_{{\rm{gnd}}}^{2}\rangle =2{k}_{{\rm{B}}}T\nu {e}^{2}/h\) the thermal noise emitted from the grounded reservoir. Note that because \(\left\langle \delta {I}_{{\rm{gnd}}}^{2}\right\rangle \) is independent of the applied bias voltage V, it cancels in the excess noise S_{B}. In the tunnelling limit (τ_{B} ≪ 1), theory predicts from detailed balance between upstream and downstream tunnelling events that the first term in the right side of equation (5) is independent of the scaling dimension Δ and given by^{61}:
The dependence on Δ of the measured noise thus solely results from the second term on the right side of equation (5), namely, 2⟨δI_{T}δI_{gnd}⟩. According to the socalled nonequilibrium fluctuation–dissipation relations for chiral systems (assuming a Vindependent Hamiltonian, as discussed below), this Δdependent contribution to the noise is simply given by^{62}:
Experimentally, ∂⟨I_{B}⟩/∂V is directly measured. Hence, in this framework, we could calculate the excess noise \({S}_{{\rm{B}}}^{{\rm{FDT}}}\) by plugging the separately measured tunnelling current and its derivative into these equations. This gives (as well as the usual ad hoc correction for large τ)
However, as illustrated in Extended Data Fig. 10, we find that equation (8) using the measured ⟨I_{B}⟩(V) does not reproduce the simultaneously measured thermal noise to shot noise crossover. This should not come as a surprise as the current and its derivative do not follow Luttinger liquid predictions. One could explain this mismatch by invoking the same possible explanation as for the data–theory discrepancy on the I–V characteristics, namely, that the shape of the QPC potential, and therefore the quasiparticle tunnelling amplitude, is affected by external parameters, such as an electrostatic deformation induced by a change in the applied bias voltage, the temperature or the tunnelling current itself^{10}. Indeed, as pointed out in ref. ^{62}, equation (7) holds if the voltage bias V only manifests through the chemical potential of the incident edge channel and not if applying V affects the tunnel Hamiltonian.
Note added in proof: Coincident to this investigation, two other works are experimentally addressing the scaling dimension of the e/3 fractional quantum Hall quasiparticles at ν = 1/3. An experiment by the team of M. Heiblum with a theoretical analysis led by K. Snizhko^{63} exploits the same thermal noise to shot noise crossover as in this work, with a focus on low voltages and assuming the predicted fractional charge (see Fig. 4 for such a singleparameter data analysis at low bias), and finds Δ ≃ 1/2. The team of G. Feve (M. Ruelle et al., submitted) relies on a different, dynamical response signature and finds Δ ≃ 1/3. In these two coincident works, the extracted scaling dimension is different from the pristine prediction Δ = 1/6 observed in this work. As pointed out in the manuscript, the emergence of nonuniversal behaviours could be related to differences in the geometry of the QPCs.
Data availability
Further information related to this work is available from the corresponding authors on reasonable request. Plotted data, raw data and dataanalysis code are available from Zenodo at https://doi.org/10.5281/zenodo.10599318 (ref. ^{64}).
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Acknowledgements
This work was supported by the European Research Council (ERC2020SyG951451) and the French RENATECH network. We thank K. Snizhko for discussions and E. Boulat for providing the τ(V, T) prediction at ν = 1/3 in Fig. 2.
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A.V., C.P., P.G., Y.S. and F.P. performed the experiments, with input from A. Aassime and A. Anthore; A.V. and F.P. analysed the data, with input from A. Anthore, C.P., P.G. and Y.S.; A.C. and U.G. grew the 2DEG; A.V., A. Aassime and F.P. fabricated the sample; Y.J. fabricated the highelectronmobility transistor used in the cryogenic noise amplifiers; A.V. and F.P. wrote the manuscript, with contributions from all authors; A. Anthore and F.P. led the project.
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Extended data figures and tables
Extended Data Fig. 1 Largescale pictures of the measured device.
a, Electronbeam micrograph. Areas with a 2DEG underneath (the mesa) appear darker. Lighter parts with bright edges are thick, gold layers used to climb down the mesa edges and as bonding pads. b, Optical image. The thick top layer made of gold appears as the brightest yellow. Ohmic contacts have staircase edges and show as a darker shade of yellow. The surface over which the HfO_{2} was deposited (dark grey) completely encapsulates the active part of the device, including ohmic contacts. The dashed rectangle indicates the boundary of the electronbeam image in panel a.
Extended Data Fig. 2 QPC with surrounding metal gate.
Electronbeam micrographs of QPC_{W}.
Extended Data Fig. 3 Magneticfield sweep.
Forward (\({\mathop{\nu }\limits^{ \sim }}_{{\rm{F}}}\), red) and backscattered (\({\mathop{\nu }\limits^{ \sim }}_{{\rm{B}}}\), black) ac voltages across a fully open QPC_{E}, in response to a fixed ac bias current Ĩ, are plotted as a function of magnetic field B. The other QPCs are fully closed. Arrows indicate at which B the different measurements were performed (except B ≃ 1.5 T at ν = 3, not shown here). At these points, the backscattered signal is zero, whereas the forward signal is well within a plateau, despite the increased mixingchamber temperature of 160 mK during this B sweep. Note that \({\mathop{\nu }\limits^{ \sim }}_{{\rm{F}}}\) does not precisely scale as h/νe^{2} along plateaus, owing to the parallel capacitance shown schematically (100 nF) and the finite ac frequency (13 Hz).
Extended Data Fig. 4 Summary of singleparameter analysis.
Symbols recapitulate the extracted scaling dimension in all explored configurations, similarly to Fig. 4 but with Δ obtained by a different procedure. The quasiparticle charge e* is here assumed to take its predicted value and the fit is performed only on the thermal noise to shot noise crossover at low bias and using Δ as the only free parameter (see text).
Extended Data Fig. 5 Signatures of upstream neutral heat flow at ν = 2/3.
The presence of a neutral heat current flowing in the opposite direction of the electrical current is assessed by noise measurements in three different configurations. a–c, Schematic illustrations of the three processes (described in the text) by which heat is created, transported upstream by the neutral modes (dashed red arrows) and detected. d–f, Noise signature of upstream heating measured in the configuration shown in the panel immediately above.
Extended Data Fig. 6 Transmission ‘plateau’ across QPC_{SE} at ν = 2/3.
The measured QPC ‘backscattering’ transmission τ across QPC_{SE} at ν = 2/3 is plotted as a continuous line versus gate voltage V_{gate}. The horizontal dashed line indicates τ = 1/2. See inset in Fig. 3c for the corresponding gatevoltage sweep of QPC_{E} at ν = 2/3.
Extended Data Fig. 7 Scaling dimension versus QPC tuning of predicted {e/3, Δ = 1/6} quasiparticles.
Individual values of extracted scaling dimension (triangles) and charge (diamonds) are plotted versus τ(V = 0) for each configuration addressing the predicted {e/3, Δ = 1/6} fractional quantum Hall quasiparticles. A few points associated with anomalously low or high charge are shown as different symbols (+, ×). Each panel corresponds to the configuration indicated within it. The average and spread of Δ indicated in the panels are calculated only on points shown as triangles and correspond to the individual symbols with error bars in Fig. 4. All measurements are at ν = 1/3 except the bottomright panel addressing the outer edge channel at ν = 2/5. The configuration corresponding to {ν = 1/3, QPC_{E}, 31 mK, δB ≃ −0.5 T} is shown in Fig. 3a.
Extended Data Fig. 8 Scaling dimension versus QPC tuning at ν = 2/3 and ν = 3.
Individual values of extracted scaling dimension Δ (triangles) and charge e*/e (diamonds) are plotted versus τ(V = 0). A few points associated with anomalously low or high charge are shown as different symbols (+, ×). Each panel corresponds to the configuration indicated within it. The average and spread of Δ indicated in the panels are calculated only on points shown as triangles and correspond to the individual symbols with error bars in Fig. 4. The upper six panels correspond to ν = 2/3, whereas the bottom panel corresponds to ν = 3. The configuration corresponding to {ν = 2/3, QPC_{E}, 31 mK} is shown in Fig. 3c.
Extended Data Fig. 9 Transmission versus dc voltage bias at different gate voltages.
The measured QPC ‘backscattering’ transmission τ is plotted versus V for the different gate voltagetunings and three QPC_{E} configurations shown in Fig. 3. Each individual tuning in each panel is shown as a line of different colour. Panels a–c correspond to the ν = 1/3, 2/5 inner channel and 2/3 configurations shown in Fig. 3a–c, respectively.
Extended Data Fig. 10 Measured S_{B} versus calculated \({{\boldsymbol{S}}}_{{\bf{B}}}^{{\bf{FDT}}}\).
Illustrative comparison at ν = 1/3 between the measured excess noise S_{B} and the value \({S}_{{\rm{B}}}^{{\rm{FDT}}}\) calculated from equation (8) (derived with the nonequilibrium fluctuation–dissipation relation in equation (7)) using the simultaneously measured ⟨I_{B}⟩(V). The noise shown here is the same as in Fig. 2a. The mismatch could be explained by the same mechanism invoked for the I–V characteristics (see Methods).
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Veillon, A., Piquard, C., Glidic, P. et al. Observation of the scaling dimension of fractional quantum Hall anyons. Nature (2024). https://doi.org/10.1038/s4158602407727z
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DOI: https://doi.org/10.1038/s4158602407727z
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