Signature of anyonic statistics in the integer quantum Hall regime

Anyons are exotic low-dimensional quasiparticles whose unconventional quantum statistics extend the binary particle division into fermions and bosons. The fractional quantum Hall regime provides a natural host, with the first convincing anyon signatures recently observed through interferometry and cross-correlations of colliding beams. However, the fractional regime is rife with experimental complications, such as an anomalous tunneling density of states, which impede the manipulation of anyons. Here we show experimentally that the canonical integer quantum Hall regime can provide a robust anyon platform. Exploiting the Coulomb interaction between two copropagating quantum Hall channels, an electron injected into one channel splits into two fractional charges behaving as abelian anyons. Their unconventional statistics is revealed by negative cross-correlations between dilute quasiparticle beams. Similarly to fractional quantum Hall observations, we show that the negative signal stems from a time-domain braiding process, here involving the incident fractional quasiparticles and spontaneously generated electron-hole pairs. Beyond the dilute limit, a theoretical understanding is achieved via the edge magnetoplasmon description of interacting integer quantum Hall channels. Our findings establish that, counter-intuitively, the integer quantum Hall regime provides a platform of choice for exploring and manipulating quasiparticles with fractional quantum statistics.

Integer and fractional quantum Hall effects 1 are thought of as fundamentally separate.The main features of the integer quantum Hall (IQH) states are well described within the single-particle fermionic picture [1][2][3][4] .In contrast, fractional quantum Hall (FQH) states inherently stem from strong Coulomb interactions, giving rise to anyons -composite quasiparticles which carry a fractional charge and exhibit anyonic exchange statistics 5,6 .Abelian anyons acquire a phase upon exchange, whereas non-abelian anyons undergo a deeper transformation into different states 7 .Demonstrating the exchange statistics of anyons, in particular non-abelian, is a crucial stepping stone towards realizing topological quantum computing 8 .
Anyonic exchange statistics can be revealed via interferometry, whereby anyons along the edge move around those in the bulk, and acquire a braiding (double exchange) phase 6,[9][10][11] .An alternative probe, not requiring involved heterostructures with built-in screening, is provided by a mixing process at an 'analyzer' quantum point contact (QPC) 12 .If the impinging quasiparticle beams are dilute (Poissonian), the outgoing current cross-correlations carry a signature of anyonic statistics [12][13][14] .These dilute beams are created upstream by sources typically realized by voltage-biased QPCs set in the tunneling regime.The signature of anyonic statistics becomes particularly straightforward with two symmetric sources, a configuration often referred to as a 'collider' 14 .In this simple case, the cross-correlations for free fermions vanish, whereas a negative signal is con-sidered to be a strong marker of anyonic statistics 6,14,15 .Such negative current cross-correlation signatures of anyons at filling factors ν = 1/3 and 2/5 have recently been demonstrated 13,[16][17][18] .
However, FQH states present complications which impede the analysis and further manipulation of anyons: the edge structure is often undetermined between several alternatives 19 , the tunneling density of states generally presents an anomalous voltage dependence 20 , and the decoherence along the edge appears to be very strong [9][10][11] .A promising alternative path is provided by the insight that fractional charges propagating along the edges of IQH states should also behave as anyons 7,21,22 , although they are not topologically protected, unlike fractional bulk quasiparticles.Indeed, the exchange phase of two quasiparticles of charge e * propagating along an integer quantum Hall channel 22 is π(e * /e) 2 .This exchange phase can be linked to a dynamical Aharonov-Bohm effect 24 .In practice, such IQH anyons could be obtained e.g. by driving the edge channel with a narrow voltage pulse, from the charge fractionalization across a Coulomb island, or by exploiting the intrinsic Coulomb coupling between co-propagating edge channels 7,22,25,26 .The present work proposes and implements the latter strategy, and demonstrates the anyonic character of the resulting fractional charges from the emergence of negative cross-correlations.
We focus on the filling factor ν = 2, which has two copropagating edge channels and constitutes the most simple, canonical and robust IQH state with interacting channels.The edge physics is well described by a chiral Luttinger model involving two one-dimensional channels with a linear dispersion relation and short-range Cou- lomb interactions 20,[27][28][29][30] .This theory, which has been successful in explaining many experimental findings such as multiple lobes in a Mach-Zehnder interferometer [31][32][33] , spin-charge separation 34,35 or noise measurements 36 , reformulates interacting fermionic edge states as two free edge magnetoplasmon (EMP) modes via bosonization.In the limit of weak inter-channel coupling, each EMP mode is localized in one different channel and the system can be mapped back into the free electron picture.In contrast, at strong coupling the two EMP modes are fully delocalized between the two quantum Hall channels and correspond to a charge mode, with identical charge density fluctuations on both channels, and a neutral mode, with opposite density fluctuations 21,26,37 .Experimentally, typical Al(Ga)As devices at ν = 2 often appear to be close to the strong coupling regime 34,35,38 .
Here we exploit such an inter-channel distribution of EMPs at strong coupling to split electrons into fractional charges, similarly to the theoretical proposal in [26].We start by injecting electrons into a single edge channel with a voltage-biased QPC.Then, downstream from the QPC, each injected electron progressively splits into two wave-packets.Assuming strong coupling, one is solely built upon charge EMPs propagating at velocity v c , and the other is constructed from neutral EMPs and has a slower velocity v n .If we consider separately the quantum Hall channel where the electron is injected, both wave-packets carry a fractional charge of e/2, whereas in the other channel they have opposite charges ±e/2 (see Fig. 1a) 26,39 .Such fractional wave-packets propagate non-dispersively.Considered individually, they are predicted to behave as abelian anyons with non-trivial exchange phase 6,7,21,22 .
To experimentally address the anyon character of fractional charges propagating along integer quantum Hall channels, we measure the current cross-correlations at the output of a 'collider' in the stationary regime (see Fig. 1a,c).As for the fractional quantum Hall ver-sion of the device 14 , direct anyon collisions are very rare and can be ignored in the relevant dilute beam limit 7,15 .The cross-correlation signal stems instead from a braiding in the time-domain between incident anyons and particle-hole pairs spontaneously excited at the QPC 7,12,13,15,22,[40][41][42] , as illustrated in Fig. 1b.At integer filling factors, the pairs are always formed of fermionic particles (electrons and holes), whereas in the fractional quantum Hall regime they can consist of anyons.Therefore the time-braiding considered here takes place between two different types of quasiparticles, of fractional and integer charges.The braiding (double exchange) phase 2θ acquired in such heterogeneous cases characterizes the so-called mutual quantum statistics.It is predicted to take the fractional value of 2θ = π (compared to 0 (mod 2π) for the braiding of fermions or bosons).Note that while the braiding mechanism is equally relevant for a single incident beam of fractional quasiparticles or for two symmetric beams, the latter 'collider' setup allows for a qualitative test of the unconventional anyon character from the mere emergence of non-zero crosscorrelations at the output 14,15,41,42 .
The sample, shown in Fig. 1c, is nanostructured from an Al(Ga)As heterostructure and measured at 11 mK and 5.2 T. It consists of two source QPCs (metallic split gates colored red) located at a nominal distance d = 3.1 µm from the central 'analyzer' QPC (yellow gates).If not stated otherwise, all QPCs are set to partially (fully) reflect the outer (inner) edge channel, and the analyzer QPC is tuned to an outer edge channel transmission probability τ c ≃ 0.5.A negative voltage is also applied to the non-colored gates to reflect the edge channels at all times, as schematically depicted.Lowfrequency current auto-correlations ⟨δI 2 1 ⟩ and ⟨δI 2 2 ⟩ on the left and right side, respectively, and cross-correlations ⟨δI 1 δI 2 ⟩ across the analyzer QPC are measured simultaneously.In the following, the excess noises are denoted . Spectroscopy of the electron energy distribution f (ε).The shape of f (ε) reflects the inter-channel coupling regime and informs on the conditions for a complete charge fractionalization at the central QPC.One source is voltage biased at Vs, here with τs ≈ 0.

Electron fractionalization
First we need to ensure that the device is in the strong coupling regime, and to determine under which conditions the tunneling electrons are fractionalized into well-separated e/2 charges at the analyzer QPC.
Our straightforward approach is to inject energy into one edge channel at a source, and to probe the energy redistribution at the analyzer 43 .Indeed, the fractionalization of the tunneling electron coincides with the emergence of charge pulses in the other, co-propagating edge channel and, consequently, with a transfer of energy.Furthermore, the full EMP delocalization between both channels, which is specific to the strong coupling limit, also translates into an equal redistribution of energy between the channels at long distances 44 .
In this measurement, only one source QPC is used together with the analyzer.We inject energy into the outer channel by applying a constant dc voltage bias Vs across the source QPC (e.g., V1 = Vs/2 and V3 = −Vs/2 for the left source, Fig. 1c).The resulting electron energy distribution immediately downstream of the injection point finj takes the shape of a double step (red inset in Fig. 1a), where finj(ε) = τsfFD(ε + eVs/2) + (1 − τs)fFD(ε − eVs/2), with τs the transmission probability of outer channel electrons across the source QPC, and fFD the Fermi-Dirac distribution.
The electron energy distribution spectroscopy at the analyzer is performed in the out-of-equilibrium outer edge channel by measuring the cross-correlations S12 vs the probe voltage Vp that controls the electrochemical potential of the equilibrium edge channel on the other side (e.g., V2 = V4 = Vp).The latter's cold Fermi distribution acts as a step filter [45][46][47] , up to a kBT ≈ 0.1 µeV rounding.The probed out-of-equilibrium electron energy distributions f displayed in Fig. 2 are computed from the measured S12 (inset) using 45,48 : The three panels in Fig. 2 show the evolution of f with the source bias voltage Vs, at τs ≃ τc ≃ 0.5.Whereas at low Vs = 23 µV (panel a) f remains close to a double-step function, we observe a marked relaxation towards an intermediate shape at Vs = 35 µV (panel b) and, at high bias Vs = 70 µV (panel c), f takes the shape of a broad single step closely matching the long distance prediction for the strong coupling limit (blue dashed lines).This last observation establishes that the present device is in the strong coupling limit.Furthermore, since the data for 70 µV agrees well with the long distance prediction, this indicates that the fractionalized e/2 wavepackets are already well-separated at the analyzer for a bias Vs of 70 µV.The observation at 35 µV of a different distribution function, still showing remnants of a double step, indicates an incomplete separation of the wave-packets up to this voltage.Therefore the separation into two e/2 wave-packets occurs for a source voltage bias within the range of 35 µV and 70 µV.In that case, the wave-packet time-width h/eVs 49 is smaller than the time delay between the arrival of fractionalized e/2 charges at the analyzer QPC δt = d/vn − d/vc.
Further evidence of the good theoretical description of the device is provided by the quality of the quantitative comparison between the data and the exact calculations of f at finite distance (purple continuous lines).These predictions were obtained by an extension of the theory involving a subsequent refermionization of the bosonized Hamiltonian, which enables a full access to the cross-correlations and outof-equilibrium electron distributions (see Supplementary Information).The only fitting parameter is the time delay δt.
Here it is fixed to δt = 64 ps, and its associated effective velocity d/δt = 5 × 10 4 m s −1 is comparable to EMP velocity measurements in similar samples 35 .In addition to the Supplementary Fig. 5 showing a comparison at additional inter-  3) with T = 11 mK, the independently measured temperature.c Measured excess shot noise S 12 /(τc(1 − τc)) as a function of source shot noise S Σ for a small source QPC transmission τs = 0.05/0.95(full/empty dots respectively).The purple lines display the strong inter-channel coupling prediction for δt = 64 ps.The green lines denotes the slope, i.e., the Fano factor (see Main text), yielding P ≃ −0.38/0.56 for τs = 0.05/0.95respectively.In all panels full/open circles and solid/dashed lines denote τs = 0.05/0.95respectively.mediate voltages, see also Supplementary Fig. 6 for measurements with a dilute quasiparticle beam, and Supplementary Fig. 7 for a (less-controlled) power injection in the inner edge channel.

Negative cross-correlation signature of anyon statistics
We now turn to the cross-correlation investigation of the fractional mutual braiding statistics between e/2 edge quasiparticles and electrons.Figure 3 displays the central measurement of cross-correlations in the configuration of two sources injecting symmetric dilute beams toward the analyzer.The source QPCs are biased at a voltage Vs equally distributed on the two inputs (V1,2 = −V3,4 = Vs/2, Fig. 1c), and set to Vs = 70 µV, previously established to correspond to the full fractionalization of the quasiparticles entering the analyzer.We set both source QPCs either to a transmission τs ≈ 0.05 corresponding to a dilute beam of electrons, or to τs ≈ 0.95 for a dilute beam of holes (Fig. 3a).
The relevant parameter to investigate the cross-correlation signature of anyonic statistics is the generalized Fano factor 14 where τc is the analyzer transmission, and SΣ is the sum of the current noises emitted from the two source QPCs.P carries information on the braiding statistics, with a high bias voltage limit that depends on the braiding phase 14,41,42 .If the (mutual) braiding statistics of free quasiparticles is trivial, such as for fermions or bosons, then P is zero, whereas it is non-zero otherwise.The mere observation of a non-zero P therefore provides a qualitative signature of an unconventional braiding statistics.However, we stress the importance of unambiguously establishing the underlying theoretical description of the system.For instance, negative cross-correlations could also be obtained with fermions, by phenomenologically introducing an ad hoc redistribution of energy (see Supplementary Information).Here we have established the suitability of the fractionalized charge picture through electron energy distribution spectroscopy, by comparing the observed bias voltage evolution of the energy distribution with the quantitative predictions of this model (see Fig. 2 and Supplementary Fig. 5).
In practice, P is extracted from the slope of S12/(τc(1−τc)) vs SΣ, as shown in Fig. 3c (green lines).Note that the measurements of SΣ and S12 are performed simultaneously, by exploiting the current conservation relation SΣ = S11 + S22 + 2S12.We first check that SΣ reflects the charge e of injected electrons (Fig. 3b).This is attested by the good quantitative agreement, without any fit parameter, between the data (symbols) and the shot noise prediction for electrons (orange lines) given by 48 : with T = 11 mK and τ L(R) the measured dc transmission of the left (right) source shown in Fig. 3a.
We then focus on the cross-correlation investigation of anyonic behavior.As shown in Fig. 3c, S12 ≈ 0 at low bias, up to SΣ ≈ 6 × 10 −29 A 2 Hz −1 corresponding to |Vs| ≈ 45 µV.This P ≈ 0 signals a trivial mutual statistics, which is expected in the low bias regime where the injected electron is not fractionalized at the analyzer.Then, at |Vs| ≳ 45 µV where the fractionalization takes place according to f (ε) spectroscopy, S12 turns negative with a slope of P ≃ −0.38/0.56(green solid/dashed line) for τs = 0.05/0.95respectively.The clear negative signal with a fixed slope constitutes a strong qualitative marker of non-trivial mutual braiding statistics, as further discussed below.The relationship between negative cross-correlations and anyonic mutual statistics in the dilute limit of small τs (or, symmetrically, small 1 − τs) is most clearly established in a perturbative analysis along the lines of Morel et al 7 .For τs ≪ 1 and at long distances from the source, we find (see Supplementary Information): with 2θ the mutual braiding (double exchange) phase.For quasiparticles of charges q and q ′ along an integer quantum Hall channel, theory predicts θ = πqq ′ /e 2 (see, e.g., Ref. 22).
In the present case of a braiding between incident fractional charges e/2 and spontaneously generated electron-hole pairs, we thus have θ = π/2 and P ≃ 4 π 2 ln τs (see also Ref. 28 for the same prediction, but without the explicit connection to the fractional mutual statistics).Note that in the present integer quantum Hall implementation, the relationship between P and θ is not complicated by additional parameters, such as the fractional quasiparticles' scaling dimension and topological spin that both come into play in the fractional quantum Hall regime 14,15,41,42 .However, achieving P ∝ ln τs requires large | ln τs| and thus exponentially small τs, which complicates a quantitative comparison of experimental data with Eq. ( 4).Accordingly, injecting τs = 0.05 (ln τs ≃ −3) into Eq.( 4) gives a slope P ≃ −1.2, substantially more negative than the observations.A better data-theory agreement can be obtained with a non-perturbative treatment of the sources.Indeed, the predicted slope in the high bias/long distance limit at τs = 0.05 is P ≃ −0.35 (see Eq. ( 52) in Supplementary Information), close to one of the observed values P ≃ −0.38.Although we expect the same cross-correlations and Fano factor for τs = 0.05 and 0.95, the measured value at 0.95 of P ≃ −0.56 (green dashed line) is somewhat higher, and also deviates more from the theory prediction (purple dashed line).We discuss possible reasons, such as increased inter-channel tunneling, in the Supplementary Information.The full finite bias/finite distance predictions (purple lines in Fig. 3c) also reproduce the overall shape of the measurements, although with a noticeable horizontal shift (see Supplementary Information for a discussion of possible theoretical limitations).Further evidence of the underlying anyonic mechanism is provided from the effect of the dilution of the quasiparticle beam.

Cross-correlations vs beam dilution
Here we explore the effect of dilution by sweeping the transmission across the two symmetric sources over the full range τs ∈ [0, 1], and we also extend our investigation to the inner edge channel where the electron fractionalization results in two pulses carrying opposite charges ±e/2 (see Fig. 1a).
Let us first consider the previous/standard configuration, with source QPCs and analyzer QPC set to partially transmit the same, outer, channel.Away from the dilute lim-its, P and S12 show a change of sign (see Fig. 4a).This results from an increasing importance of the positive contribution from the noise generated at the sources 50 with respect to the noise generated at the analyzer involving the emergence of mechanisms other than time braiding (such as collisions) 7,41 .In addition we see that the data are symmetric around τs = 0.5, due to the unchanging electron nature of tunneling particles into the IQH edges (with small deviations possibly from inter-channel tunneling, as previously mentioned).This is in contrast with the fractional quantum Hall regime where the nature of the tunneling quasiparticles changes 51 between τs ≪ 1 and 1 − τs ≪ 1.The agreement with theory observed for P (τs), and more specifically for the τs dependence in the dilute limits τs ≪ 1 and 1 − τs ≪ 1, further establishes the experimental cross-correlation signature of fractional mutual statistics.Note that the less precise agreement with the full S12 signal is reminiscent of the horizontal shift of the negative slope in Fig. 3.
We then consider in Fig. 4b the alternative configuration, where the analyzer is set to τc ≃ 0.5 for the inner edge channel (the outer edge channel, where electrons are injected at the sources, being fully transmitted, see schematics).In that case, the cross-correlation signal is always negative due to charge conservation.The positive contribution τc(1 − τc)SΣ from the partition of the current noise generated at the sources is absent since the electron tunneling at the sources does not take place in the probed inner edge channel.In the strong coupling limit where fractional charges of identical amplitude e/2 propagate on both inner and outer channel, the same unconventional braiding is expected to have the same cross-correlation consequences.The absence of source noise then simply results in an offset: , where the label inner/outer indicates the channel probed at the analyzer.For a direct comparison, Pouter − 1 is also shown (filled stars).The agreement between the two data sets provides an additional, direct confirmation that the device is in the strong coupling regime.It also experimentally establishes the robust contribution from the source noise to S12, allowing to distinguish it from the effect of time-braiding.

Discussion
Edge excitations are not characterized by topologically protected quantum numbers, in contrast to the quantum Hall bulk quasiparticles.Along the integer and fractional quantum Hall edges, the charge and the quantum statistics of such excitations can be varied continuously.In the present work, we exploit this property to form dilute beams of fractional charges which behave as anyons.This is achieved by using QPCs as electron sources in combination with the intrinsic Coulomb interaction between co-propagating integer quantum Hall channels.We establish their fractional quantum statistics by the emergence of negative current crosscorrelations between the two outputs of a downstream analyzer QPC, similarly to previous observations in the fractional quantum Hall regime.By contrast, when applying sufficiently low source bias voltages such that the tunneling electrons do not fractionalize, the absence of a cross-correlation signal coincides with their fermionic character.
We believe that the demonstrated integer quantum Hall platform opens a promising practical path to explore the emerging field of anyon quantum optics 52 .Advanced and time-resolved quantum manipulations of anyons are made possible by the large quantum coherence along the integer quantum Hall edge and the robustness of the incompressible bulk.By tailoring single-quasiparticle wave-packets, for example with driven ohmic contacts, a vast range of fractional anyons of arbitrary exchange phase becomes available along the integer quantum Hall edges, well beyond the odd fractions of π of Laughlin quasiparticles encountered in the fractional quantum Hall regime.

Sample fabrication
The device, shown in Fig. 1c of the Main text, is patterned in an AlGaAs/GaAs heterostructure forming a two-dimensional electron gas (2DEG) buried 95 nm below the surface.The 2DEG has a mobility of 2.5 × 10 6 cm 2 V −1 s −1 and a density of 2.5 × 10 11 cm −2 .It was nanofabricated following five standard e-beam lithography steps: 1. Ti-Au alignment marks are first deposited through a PMMA mask.
2. The mesa is defined by using a ma-N 2403 protection mask and by wet-etching the unprotected parts in a solution of H3PO4/H2O2/H2O over a depth of ∼100 nm.
4. The split gates controlling the QPCs consist in 40 nm of aluminium deposited through a PMMA mask.
5. Finally, we deposit thick Cr-Au bonding ports and large-scale interconnects through a PMMA mask.
The nominal tip-to-tip distance of the Al split gates used to define the QPCs is 150 nm.

Measurement setup
The sample is installed in a cryofree dilution refrigerator with important filtering and thermalization of the electrical lines, and immersed in a perpendicular magnetic field B = 5.2 T, which corresponds to the middle of the ν = 2 plateau.Cold RC filters are mounted near the device: 200 kΩ -100 nF on the lines controlling the split gates, 10 kΩ -100 nF on the injection lines and 10 kΩ -1 nF on the low frequency measurement lines.
Lock-in measurements are made at frequencies below 25 Hz, using an ac modulation of rms amplitude below kBT /e.We calculate the dc currents and QPC transmissions by integrating the corresponding lock-in signal vs the source bias voltage (see the following and Ref. 53 for details).
The auto-and cross-correlations of the currents I1 and I2 (Fig. 1c) are measured with home-made cryogenic amplifiers 54 around 0.86 MHz, the resonant frequency of the two identical tank circuits along the two amplification chains.The measurements are performed by integrating the signal over the bandwidth of [0.78, 0.92] MHz.The measurement setup is detailed in the supplemental material of Ref. 55.

Thermometry
The electron temperature in the sample is measured using the robust linear dependence of the thermal noise S(T ) ∝ T .At T > 40 mK, we use the (equilibrium) thermal noise plotted versus the temperature readout by the calibrated RuO2 thermometer.The linearity is a confirmation of the electron thermalization and of the thermometer calibration.The quantitative value of the slope provides us with the gain of the full noise amplification chain, as detailed in the next section.To determine the temperature in the T < 40 mK range, we measure the thermal noise and determine the corresponding temperature by linearly extrapolating from the S(T > 40 mK) data.The values of T obtained using the two amplification chains are found to be consistent.We also check that T corresponds to the temperature obtained from standard shot noise measurements performed individually on each QPC ahead of and during each measurement.A 1 mK higher shot noise temperature is specifically associated with the topright ohmic contact feeding the right source, and attributed to noise from the corresponding connecting line.

Calibration of the noise amplification chain
For each noise amplification chain i ∈ {1, 2}, the gain factor G eff i between current noise spectral density and raw measurements needs to be calibrated.From the slopes s1 and s2 of S(T > 40 mK) measured, respectively, for the amplification chain 1 and 2 (see Thermometry), and the robust fluctuationdissipation relation S(T ) = 4kBT Re[Z] with Z the frequency dependent impedance of the tank circuit in parallel with the sample, we get: with RH = h/2e 2 the Hall resistance of the sample, and R = 150 kΩ (153 kΩ) the separately obtained effective parallel resistance due to the dissipation in the tank circuit connected to the same port.With G eff 1 and G eff 2 given by the above relation, the gain for the cross-correlation signal reads G eff 12 = G eff 1 G eff 2 thanks to the good match of the two resonators.For more details see Ref. 53.

Differential (ac) and integral (dc) transmission Source transmissions:
The transmissions of the left and right source QPC τL,R are defined as the ratio between the dc current transmitted across the QPC and the dc current e 2 V3,4/h incident on one side of the QPC for the considered outer edge channel (i.e., half of the total injected current), yielding: In the specific case where the sources are set to partial transmission of the inner channel (case of Fig. 7 in the Supplementary Information), source transmissions of the inner channel can similarly be defined as Analyzer transmission: Whereas above we use dc transmissions for the sources, in all measurements the relevant transmission of the analyzer is the ac transmission, obtained as In particular, there is no dc bias of the analyzer in the symmetric source configuration probing the quantum statistics.
measurements; I.P. and P.G. wrote the manuscript with contributions from all authors; A.An. and F.P. led the project.

A. Introduction
In this Section we outline the theoretical description of the electron collider using a model which can be solved via refermionization techniques.We also present the results for the asymptotics of the noise in the small tunneling limit.
The notation in the 'Non-perturbative theory' Section is different from the rest of the paper in order to keep the correspondence with the previous work Ref. 1 and with an upcoming theoretical publication providing further details (D.Kovrizhin, in preparation).
The schematics of the model is shown in Supplementary Fig. 1 for the noise measurement between channels 1 ↓ and 2 ′ ↓ .In the Main text this is outlined as the default configuration, i.e., injection and measurement in the outer channel.Other configurations can be calculated in a similar manner, and the theory will be detailed in the above mentioned upcoming article.
Let us consider a system with four edge states 1, 1 ′ , 2, 2 ′ , each of which is carrying two co-propagating edge channels (↑, ↓), where the arrows denote the spin.The biasing scheme of biasing source quantum point contacts with 0 and eV in Supplementary Fig. 1 is equivalent to biasing with −eV /2 and +eV /2 (used in the experiment) due to invariance of the observables under a global potential shift.
We model the non-interacting edge channels by the free-fermion Hamiltonian with linear dispersion (assuming the same Fermi-velocity vF in every channel).This is a standard description of edge states at integer filling factors.In addition, we assume that electrons on each edge interact via short-range interactions with strength g which has the dimension of a velocity.This model of interactions provides a good description of previous experiments at filling factor ν = 2, where interactions are usually sufficiently strong to overcome any asymmetry between channels 1-3 .We note that in the experimental setup the long-range Coulomb interactions are expected to be screened by the metallic gates over a typical length scale of a few hundred nanometers of the order of the distance between 2DEG and nearest gate.
The Hamiltonian of the system shown in Supplementary Fig. 1 reads: Here d is the distance between the source QPC1,2 and the analyzer QPCS.Ψηs(x) are fermion annihilation operators and ρηs(x) are fermion density operators on the corresponding edge channels (ηs).v1,2,S are the tunneling amplitudes connected to the transmission probabilities T1,2,S across QPC1,2,S (T1,2,S = sin 2 (|v1,2,S|/ℏvF ) in the non-interacting case).See Ref. 4 for details of the refermionization approach with the same notations as in this Section.
The operators obey the following commutation relations: and the density operator is written in terms of the bosonic fields as In this bosonic representation, the Hamiltonian in Eq. ( 10) can be written as:

C. Refermionization
Here we show how to refermionize the Hamiltonian in Eq. ( 18), which allows one to obtain the exact expressions for the noise in the presence of interactions.To express the current correlators we will use refermionization, which permits us to map our model onto a system of non-interacting fermions for each QPC separately.This does not mean that there is no dependence of the noise on the interactions because the transformations between the new fields and the original fields will generate an interaction-dependent contribution to the noise.
We start with the refermionization of the four channels (1 For this purpose we first introduce new bosonic operators χS+(x), χA−(x), χA+(x), χS−(x), which are related to the original bosonic operators ϕηs via the transformation χT = U ϕ T where U is the following 4 × 4 matrix: Note that the dispersions of the bosons corresponding to χA,S+ and χA,S− are given by the velocities v+ = vF + g and v− = vF − g, respectively.
We would like to evaluate current correlation functions at non-equal times for the channels 1 ↓ and 2 ′ ↓ at some position after the QPCS.For that we need to have expressions for the currents in these channels in terms of refermionized operators.The currents in terms of the original fermions can be obtained from the Heisenberg equations of motion for the density operators (in the interaction representation, where the tunneling is treated as an interaction term in the Hamiltonian): The corresponding currents are expressed in terms of the original fermions in the interaction representation as Î1↓ (x, t) = −e(vF ρ1↓ (x, t) with ρηs given by Eq. ( 17).In the following we will omit the number operators Nηs appearing in Eq. ( 17) in order to simplify the notations.Since they transform in the same way as the fields under linear transformation with the matrix U , we will be able to restore them at the end.Using the inverse transformation U −1 (U −1 = U ) we can write the currents at position d in terms of the transformed density operators ρS,A,±, which are related to the χ fields via an equation analogous to Eq. ( 17), as We can rewrite these expressions in the more convenient form where we have defined We can now proceed with the calculations of the noise, obtained by integrating in time the following current correlator We now proceed with the above correlator.Whereas the last two terms cancel out as Î0(d, t1,2) and Î1(d, t2,1) commute, it is not the case of the first two terms.After refermionization, one can write the Hamiltonian for the four channels coupled by QPCS in terms of free fermions with the standard tunneling term ., with ṽS being the renormalized tunneling strength 4 directly related to the transmission probability TS measured in the experiment.Note that H ref T S does not affect the fields S±.The fermion operators ΨA ± are transformed by QPCS as denoting the transmission and reflection amplitudes as tS and rS (|tS| 2 ≡ TS, |rS| 2 ≡ RS ≡ 1 − TS), and where d + and d − are the positions just after and just before QPCS, respectively.Let us start with ⟨ Î0(d, t1) Î0(d, t2)⟩.Because the operator Î0 does not transform under the action of H ref T S , we can write the current Î0 in terms of the original bosonic fields φ: We note that the operators on the right hand side are given in the Heisenberg representation, which includes the interactions and the tunneling at both QPC1 and QPC2.We now need to refermionize the subsystems connected by QPC1 and QPC2 (e.g., we refermionize separately channels with primed indices, and channels with unprimed indices).In order to do that we introduce operators χ(x) related to transformations of the bottom four channels and operators χ′ (x) related to top four channels ( χ′ Using these transformations we can write the current operator Î0(d, t) as We note that there is no coherence between primed and unprimed terms as well as between S+, S− terms because they are not connected by refermionized QPC1,2 (as with QPCS, only A+, A− and A ′ + , A ′ − channels are connected by QPC1,2 correspondingly).
The operators in the current (taken after QPC1,2) should be related to the operators before QPC1,2.For this purpose, we use and the similar transformation of A+ operators at 0 + .These transformations allow us to write the current correlators (noting that the A+ and A− operators on the right hand side are incoherent because they are taken before QPC1, at position 0 − ).Denoting as δτ ≡ t1 − t2 − d/v eff , where v eff = (1/v− − 1/v+) −1 is an effective velocity, we obtain (for voltage-dependent terms) where double brackets denote normal-ordering of the current operators, and are free-fermion Green functions.In order to evaluate this correlator we need to know the chemical potentials of the channels in the refermionized representation.These chemical potentials follow the refermionization prescription (and can be obtained using matrix U ), which gives for this voltage setup (the distribution function setting would correspond to different values, see Ref. 1) This gives: where we note that the result is independent of the position d.
Similarly, after some algebra, we find for the correlator of I1 operators the following expression (shown here again for voltage-dependent terms) (37)   where we have introduced the interacting Green functions defined as The first two terms in Eq. ( 37) involve solely the non-interacting Green functions (GA ± ,G A ′ ± ) and can be expressed using the values of the chemical potentials given Eq. ( 35) as 2(RS − TS) 2 (R1T1 + R2T2) cos(eV δτ /ℏ)/δτ 2 , in a similar form as Eq. ( 36).This gives:

Zero temperature noise
Here we only present the result at zero temperature, and the finite-temperature result can be obtained in a similar way.The finite temperature version used to calculate numerically the cross-correlations shown in Figs. 3 and 4 in the Main text and Supplementary Figure 10 is provided in the subsection 'Finite temperature expressions'.
Integrating over δτ , the T = 0 correlators given by Eqs. ( 36) and ( 39) yield the cross-correlation noise at zero frequency as a function of the interacting Green functions G 1↓ and G 2 ′ ↓ : where ṽ = √ v+v−.Note that the noise S 1↓2 ′ ↓ from Eq. ( 40) can be written as a sum of the non-interacting contribution given in Eq. ( 5) and an interacting contribution where The interacting Green functions can be obtained from refermionization, for example we have where the function K1↓ (d, t) is defined as Here, the averages are taken with respect to the filled Fermi seas at the corresponding chemical potentials, and the scattering matrix S1 corresponds to the rotation of fermions ΨA + , ΨA − by QPC1 in the standard way (see also Eq. ( 27) for QPCS).We have defined the particle number operators 1,4 NA ± (d, t) for the refermionized channels with velocities v± as where the fermion operators are given in the interaction representation with tunneling at QPC1 treated as interaction.These particle number operators count the number of particles passing position x = 0 in a time window (−d/v±, −d/v± + t).We note that functions K have a form similar to the full counting statistics (FCS).At large distances the two exponents are uncorrelated, and these functions can be analysed by the methods developed for the FCS.At intermediate distances we have to rely on calculations using fermionic determinants 4 .In order to numerically calculate their values, we first obtained analytically the matrix elements of the FCS exponents with respect to the filled Fermi seas in the corresponding channels with one extra particle/hole.Then, using these matrix elements in the expressions given in terms of fermionic determinants, we calculated the functions K(d, t) (details will be provided in D. Kovrizhin in preparation, see also Appendix A in Ref. 4).
Similarly, the Green function for channel 2 ′ ↓ is obtained as where The results at T = 0 could be further simplified in the limit of large distances d → ∞, where the exponents decouple (so they are independent of the distance d), and we consider the symmetric case T1 = T2 where S (0) 1↓2 ′ ↓ = 0. We have in this limit where the averages are taken with respect to the non-interacting Fermi seas transformed by the scattering matrix S1,2.Note first that in the case of T1 = T2 = 1/2 one can observe numerically that the averages are well-described by the analytical expression (see also the Supplementary Material of Ref. 1): Using this expression (which seems to be exact, but we do not have a proof), one can write the noise in the following form We note that the cross-correlations are positive, compared to zero cross-correlations in the case of T1 = T2 in the non-interacting limit.

Dilute beam asymptotics
In the case of dilute beams with equal transmission T1 = T2 = T → {0, 1} and at d → ∞ we can use the asymptotics developed in the theory of FCS.At short times the product of the exponents in the Eq. ( 48) has a quadratic behaviour 1 − αt 2 , where constant α depends on the tunneling, so the integral converges at short times.This contribution is trivial, and below we focus on the long-time asymptotics.
The long-time asymptotics can be calculated using the Fisher-Hartwig approach for the Toeplitz matrices developed in Ref. 5. Note that in the Supplemental Material of Ref. 1 regarding the equilibration of edge states, we showed a comparison of these asymptotics with the numerical results, and it was pointed out that the asymptotics break down at tunneling T=1/2 (see above).
Let us first reproduce the equations obtained in Ref. 5. We consider a double-step Fermi distribution at zero temperature where n0(ε) = θ(−ε) is a step function, µ2 > µ1 are the chemical potentials, R = 1 − T is the reflection probability of the QPC, and V = µ2 − µ1 is bias voltage.We further introduce the following constants, taking δ the fractionalization parameter in ⟨e −iδ N (t) ⟩ to be δ = π and assuming as well as the dephasing time With these definitions, the asymptotics of e −iπ N (t) at long times (normalised to the equilibrium value), are given by the following expression obtained in Ref. 5 Now let us use these expressions to calculate the noise.We need the absolute value of ∆(t), which reads This function has an exponential decay times a power-law, and for small reflections R ≪ 1 we get In order to obtain the tunneling-dependent contribution to the noise we need to integrate these asymptotics where we have introduced a cutoff τ .Note that if we do not assume the smallness of the reflection R, then we have to calculate the following integral: One can express this integral in terms of a special function (where we introduce a dimensionless cutoff Θ = eV τ /ℏ), but we only need its asymptotic development at small R, which gives which has the most singular terms containing logarithms independent of the cutoff, as well as the constant term and the term proportional to R, which do depend on the cutoff.Using the structure of this expansion with respect to R, we can obtain the numerical values of the cutoff-dependent terms by fitting the numerical results obtained from the theory.This gives the following terms in the noise asymptotics at small R Finally, we can rewrite this in the notation used in the Main text in terms of the generalized Fano factor P and using the symmetry τ → 1 − τ : For τs = 0.05 we obtain P ≃ −0.35, close to the experimental value in Fig. 3 of the Main text.

Finite temperature expressions
Here, we provide the finite temperature expression for the cross-correlation noise at zero frequency in the standard collider configuration shown Supplementary Fig. 1.The following expression, obtained along the same lines described above for T = 0, was computed numerically for the data-theory comparison: We also provide the finite temperature expression for the cross-correlation noise at zero frequency in the alternative configuration where the analyzer QPCS is tuned to couple the two inner edge channels (2 ′ ↑ and 1 ↑ , which are not directly excited by the source QPC1,2) with the transmission probability TS: Therefore, this alternative description of the system with interacting fermions can also lead to non-zero cross-correlations with the same change of sign between the dilute regime and the τs ∼ 0.5 regime, due to the relative importance of the positive source noise redistribution and negative cross-correlation contribution.Moreover, as seen in Supplementary Fig. 4, the shape of the prediction for both models is similar, although the quantitative values are different.We therefore conclude that the mere presence of negative cross-correlations cannot be directly attributed to the anyonic exchange phase, since a simple fermion model also qualitatively predicts it.In order to be able to attribute the negative signal to the fractional statistics of the involved charges, we need to validate the non-perturbative approach by complementary distribution measurements, as we have done in the Main text.Note that we cannot directly rule out the phenomenological fermionic theory because this would require one to compare the observed distributions with specific predictions.However this phenomenological theory does not allow one to make specific predictions regarding the evolution of the distribution as this would require to introduce a choice for the rate of inelastic collisions as a function of the exchanged energy.

V. FITTING PROCEDURE TO DETERMINE δt
We outline the procedure used to obtain the only fitting parameter of the theory, the time delay δt between the arrival of fractionalized e/2 charges at the analyzer QPC, namely δt = d/vn − d/vc, with d the distance between source and analyzer quantum point contacts and vc,n the velocities of charged and neutral mode.
In the source-analyzer configuration we measure the cross-correlations S12(Vp) which yield the distributions f (ε = eVp) (Main text Eq. ( 1)).Cross-correlations S12(Vp) contain a big contribution from the equilibrium noise : After subtraction we obtain dS12(Vp) = S12(Vp) − S 0 12 (Vp), which is very sensitive to the value of δt.We find the best value by a least-squares method in the data subset corresponding to the [59 µV,82 µV] range of bias voltage.We chose this range because it corresponds to the regime of fully fractionalized charge.As given in the Main text, we have determined δt = 64 ps.With d = 3.1 µm measured in the SEM photo (Fig. 1c, Main text), we obtain d/δt = 5 × 10 4 m s −1 .We find slightly different values of δt for left and right side, namely δtL = 68 ± 2 ps and δtR = 60 ± 2 ps.If we assume that the velocity difference between the fast and slow mode is the same on both sides, this yields the left and right distance of 3.3 ± 0.1µm and 2.9 ± 0.1µm.This is plausible if we consider the SEM photo which reveals a slightly longer distance between the source and analyzer QPC on the right-hand side (Fig. 1c, Main text).The difference can also originate from the screening details in the edge.However, since the theory is developed for equal lengths on the left and right, we adopt the mean value of δt = 64 ps which we use throughout.

VI. DISTRIBUTIONS A. Distributions at intermediate bias voltages
In Supplementary Fig. 5 we expand on the data shown in Fig. 2 of the Main text and show the full evolution of the distribution function for bias voltages between 12 µV and 82 µV.The injection and measurement take place on the outer channel.We see the relaxation from the double-step at lower bias into a single broader-step distribution in the range 47-70 µV.
At Vs = 82 µV we have some inter-channel tunneling starting to take place, see Section XI below.

B. Distributions at low transparency
In order to verify that the charge fractionalization in the dilute limit does not deviate from the expected behavior, we measured the distributions at source transmission τs = 0.05 and τs = 0.95, shown in Supplementary Fig. 6.In the inset we show S12 used to obtain f (ε) by derivation, see Eq. (1) in the Main text.
We see that the finite-and infinite-time predictions (purple and blue curve) are very close, and explain the data very well.With the same reasoning outlined in the Main text, from this we conclude that the full fractionalization has taken place for dilute beams.We shall use this result for the cross-correlation 'collider' measurement.

IX. CROSS-CORRELATIONS IN THE FULL BIAS AND TEMPERATURE RANGE
In Supplementary Fig. 10 we show the cross-correlations as function of source transparency in the full bias range and for the additional temperature of 21 mK.As in Fig. 4 in the Main text the left/right column corresponds to the analyzer set to the outer/inner channel, respectively.

Figure 1 .
Figure 1.Experimental setup.a, In the presence of two strongly coupled quantum Hall channels at ν = 2, tunneling electrons e (individual red wave-packets) progressively split into two pairs (circled).The fast 'charge' pair (blue background) consists of two copropagating e/2 wave-packets, one in each channel, whereas the slow 'neutral' pair (green background) consists of opposite ±e/2 charges.The fractionalized e/2 charges propagate toward a central QPC (yellow split gates) of transmission τc, used to investigate their quantum statistics from the outgoing current cross-correlations.The strong coupling regime and the degree of fractionalization at the level of the central QPC are established separately through the evolution of the electron energy distribution function f (ε) from a non-equilibrium double step (red inset) to a smoother function (magenta inset).b, Illustration of the time-braiding mechanism, whereby an impinging fractionalized e/2 charge (red) braids with an electron-hole pair (black) spontaneously excited at the central QPC.c, E-beam micrograph of the sample.The two copropagating edge channels are drawn as black lines with arrows indicating the chirality.The aluminum gates used to form the QPCs by field effect are highlighted in false colors (sources in red, central analyzer in yellow).A negative voltage is applied to the non-colored gates to reflect the edge channels at all times.Tunneling at the sources is controlled by the applied dc voltages V 1,2,3,4 and through their gate-controlled transmission probability τs.

Figure 3 .
Figure 3. Cross-correlation signature of fractional statistics with symmetric dilute beams.a Measured left/right source QPC dc transmission as a function of bias voltage, shown in light/dark blue, respectively.b Sum of sources' shot noise S Σ vs source bias voltage Vs.The orange lines display Eq. (3) with T = 11 mK, the independently measured temperature.c Measured excess shot noise S 12 /(τc(1 − τc)) as a function of source shot noise S Σ for a small source QPC transmission τs = 0.05/0.95(full/empty dots respectively).The purple lines display the strong inter-channel coupling prediction for δt = 64 ps.The green lines denotes the slope, i.e., the Fano factor (see Main text), yielding P ≃ −0.38/0.56 for τs = 0.05/0.95respectively.In all panels full/open circles and solid/dashed lines denote τs = 0.05/0.95respectively.

Figure 4 .
Figure 4. Cross-correlations vs dilution of symmetric beams.Main panels and insets show, respectively, the generalized Fano factor P and the renormalized cross-correlations S 12 (Vs =70 µV)/(τc(1 − τc)) vs the outer edge channel transmission τs of the symmetric source QPCs.Symbols are data points.Blue lines are high bias/long δt predictions.Purple lines are S 12 (Vs =70 µV)/(τc(1 − τc)) predictions at δt = 64 ps.a, The cross-correlation signal and corresponding Pouter (open circles) are measured by partially transmitting at the central QPC (τc ≈ 0.5) the same outer edge channel (black) where electrons are tunneling at the sources (see schematics).This is the standard 'collider' configuration.b, The cross-correlation signal and corresponding P inner are obtained by setting the central QPC to partially transmit (τc ≈ 0.5) the inner edge channel (grey), whereas electrons are tunneling into the outer edge channel at the sources (see schematic).In this particular configuration, the source shot noise does not directly contribute to the cross-correlation signal.Filled symbols in the main panel display Pouter − 1, with Pouter the data in (a) and −1 corresponding to the subtraction of the source shot noise.
Cross-correlations vs dilution at intermediate voltages and higher temperature.a, c, Cross-correlations S 12 /(τc(1 − τc)) as function of source QPC transmission τs at 11 mK (a,b) and 21 mK (c, d).Extension of Fig. 4 from the Main text.Injection takes place in the external channel at all times, whereas the measurement is done on the external channel (a, c, e) or the internal channel (b, d, f ).The solid lines in a-d are the prediction for δt = 64 ps.Data at each bias voltage and their prediction have the same color (cf.legend).e, f, Fano factors extracted at T = 21 mK with the central QPC partially transmitting the outer (e) and inner (f ) channel.Blue lines correspond to the high bias/large δt prediction.Full stars in panel f display Pouter − 1.
5, and the same probe voltage Vp is applied across the other one (see schematic in a).Circles and triangles show data points with the voltage biased source QPC on the left and right side, respectively.Purple continuous lines and blue dashed lines represent exact theoretical predictions in the strong coupling regime for a time delay between charge and neutral pairs of δt = 64 ps and ∞, respectively (see Supplementary Information).Insets: Cross-correlations S 12 versus probe voltage Vp.Main panels: f (ε) obtained by differentiation of S 12 , see Eq. (1) with τc ≃ 0.5.a,b,c: Data and theory at T ≃11 mK for a source voltage Vs = 23 µV, 35 µV, and 70 µV, respectively.