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Wenmin Yang, [David Perconte](https://orcid.org/0000-0001-7480-8123), [Corentin Déprez](https://orcid.org/0000-0002-6211-0307), [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Sylvain Dumont, Edouard Wagner, Frédéric Gay, [Inès Safi](https://orcid.org/0000-0002-8173-9944), [Hermann Sellier](https://orcid.org/0000-0002-1439-1044), [Benjamin Sacépé](https://orcid.org/0000-0001-5943-9999)

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[Evidence for correlated electron pairs and triplets in quantum Hall interferometers](https://mdr.nims.go.jp/datasets/c784eb83-c9fd-414a-80ef-f9874dfe40ee)

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Evidence for correlated electron pairs and triplets in quantum Hall interferometersArticle https://doi.org/10.1038/s41467-024-54211-3Evidence for correlated electron pairs andtriplets in quantum Hall interferometersWenmin Yang1,5, David Perconte 1,5, Corentin Déprez 1, Kenji Watanabe 2,Takashi Taniguchi 3, Sylvain Dumont1, Edouard Wagner1, Frédéric Gay1,Inès Safi 4, Hermann Sellier 1 & Benjamin Sacépé 1Thepairing of electrons is ubiquitous in electronic systems featuring attractiveinter–electron interactions, as exemplified in superconductors. Counter-intuitively, it can also be mediated in certain circumstances by the repulsiveCoulomb interaction alone. Quantum Hall (QH) Fabry–Pérot interferometers(FPIs) tailored in a two–dimensional electron gas under a perpendicularmagnetic field have been argued to exhibit such an unusual electron pairing,seeminglywithout attractive interactions. Here,we showevidence in grapheneQHFPIs, revealing not only a similar electron pairing at bulk filling factor νB = 2,but also an unforeseen emergence of electron tripling characterized by afractional Aharonov–Bohm flux period of h/3e (h is the Planck constant and ethe electron charge) at νB = 3. Leveraging plunger–gate spectroscopy, wedemonstrate that electron pairing (tripling) involves correlated charge trans-port on two (three) entangledQHedge channels. This spectroscopy indicates aquantum interference flux periodicity determined by the sum of the phasesacquired by the distinct QH edge channels having slightly different interferingareas. Phase jumps observed in the pajama maps can be accounted for by thefrequency beating between pairing/tripling modes and the outer interfer-ing edge.The quantum Hall effect is known to host a wide range of correlatedand symmetry–protected phases. Coulomb repulsion plays a centralrole in it, shaping the structure of QH edge channels1, inducing(pseudo) spin–polarized QH ferromagnets2, or generating fractionalquantum Hall states3 with anyonic excitations that may be useful fortopological quantum computation4.In 2015, a surprise came with the observation of the pairing ofelectrons in QH interferometers. Choi and co-workers5 found in GaAsFabry–Pérot interferometers (FPIs) defined by two quantum pointcontacts (QCPs) in series6 an anomalous Aharonov–Bohm (AB) effectwith halved flux–periodicity, h/2e. The specific configurations identi-fiedwere the presence of at least twoQH edge channels in the FPI, thatis, a bulk filling factor νB > 2, and interference from the outer channelwhile the inner one forms a closed loop. Strikingly, this electron pair-ing was confirmed by quantum shot noise that evidenced an effectivecharge e* ~ 2e (refs. 5,7), pointing conspicuously toward correlatedelectron–pair transport.The analogy with Cooper pairing in superconductors is tantaliz-ing, however, the resemblance is only apparent since there are noattractive, non–Coulombic interactions, nor evidence of a macro-scopic condensate. On the theoretical front, most efforts to date havefailed to describe this baffling phenomenon8–10. Yet, an effectivedynamical pairing via the exchange of neutralons10 has been put for-ward, but cannot capture all phenomenology5,7,11.Received: 25 July 2024Accepted: 31 October 2024Check for updates1Univ.GrenobleAlpes,CNRS,Grenoble INP, InstitutNéel,Grenoble, France. 2ResearchCenter for FunctionalMaterials, National Institute forMaterials Science,Tsukuba, Japan. 3International Center for Materials Nanoarchitectonics, National Institute for Materials Science, Tsukuba, Japan. 4Université Paris-Saclay,CNRS, Laboratoire de Physique des Solides, Orsay, France. 5These authors contributed equally: Wenmin Yang, David Perconte.e-mail: benjamin.sacepe@neel.cnrs.frNature Communications |        (2024) 15:10064 11234567890():,;1234567890():,;http://orcid.org/0000-0001-7480-8123http://orcid.org/0000-0001-7480-8123http://orcid.org/0000-0001-7480-8123http://orcid.org/0000-0001-7480-8123http://orcid.org/0000-0001-7480-8123http://orcid.org/0000-0002-6211-0307http://orcid.org/0000-0002-6211-0307http://orcid.org/0000-0002-6211-0307http://orcid.org/0000-0002-6211-0307http://orcid.org/0000-0002-6211-0307http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-8173-9944http://orcid.org/0000-0002-8173-9944http://orcid.org/0000-0002-8173-9944http://orcid.org/0000-0002-8173-9944http://orcid.org/0000-0002-8173-9944http://orcid.org/0000-0002-1439-1044http://orcid.org/0000-0002-1439-1044http://orcid.org/0000-0002-1439-1044http://orcid.org/0000-0002-1439-1044http://orcid.org/0000-0002-1439-1044http://orcid.org/0000-0001-5943-9999http://orcid.org/0000-0001-5943-9999http://orcid.org/0000-0001-5943-9999http://orcid.org/0000-0001-5943-9999http://orcid.org/0000-0001-5943-9999http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-54211-3&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-54211-3&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-54211-3&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-54211-3&domain=pdfmailto:benjamin.sacepe@neel.cnrs.frwww.nature.com/naturecommunicationsHere, we opt for a different platform–the graphene QH FPI12–15 touncover new insights into this phenomenon. By leveraging thehigh–tunability of its plunger gate12 and conducting systematicout–of–equilibrium transport measurements, we establish a new QHedge channel spectrometry that allows us to identify the exact chan-nels involved coherently in the electron pairing. This spectroscopyallows us to conclude that pairing consistently occurs when thenumber of edge channels exceeds one. The pairing’s weight sig-nificantly grows with an increase in the filling factor, ultimately leadingto visible frequency beating in the pajamamap at a filling factor higherthan 2. Furthermore, at filling factor νB = 3, we uncover evidence ofcorrelated transport involving three electrons over the three distin-guishable edge channels. Our systematic exploration of the flux andenergy bias parameter space gives key insights into a complex inter-play between edge channels and their interactions.The QH FPIs are made with hBN–encapsulated graphene depos-ited onto a graphite gate acting as a back–gate electrode. Two QPCsare electrostatically defined by a set of two palladium split–gateelectrodes12,16. The FPIs are equipped with a plunger–gate electrode totune the effective area enclosed by the QH edge channels. Several 1Dohmic contacts17 allow us to source and drain current and probe vol-tages across the FPI. Figure 1a shows an atomic force microscopytopography of the device studied in the main text, which has an FPIcavity area of 2.2 ± 0.2μm2. Importantly, the FPI is defined by thepristine, non–etched edges of the graphene flake, ensuring confine-ment of the QH edge channels to within a few magnetic lengths of thecrystal edge, without any edge reconstruction18, as well as by split andplunger gates. All experiments are performed at a magnetic field of 14T and a temperature of 0.01 K. Partial pinch–off of the inner channel asoverlaid in Fig. 1a, b yields conductance oscillations shown in Fig. 1cwith negative slope in the magnetic field, B, versus plunger gate vol-tage, Vpg, plane, which is characteristic of AB quantum interference fora flux periodicity of h/e12,19.ResultsGate–spectroscopy fingerprint of QH edge channelsThe considerable advantage of graphene FPI over conventional semi-conductors is the absence of a bandgap, which allows one to achieve avery large electrostatic tuning of the charge carrier density rangingfrom the electron states to the hole states. Figure 1g illustrates thistunability with conductance oscillations versus plunger–gate voltage,Vpg, from −5 to 0V, reflecting the quantum interference of the innerchannel at νB = 2 (same configuration as in Fig. 1a, b). The Fouriertransform of these oscillations in a small sliding window gives theplunger–gate dependence of the oscillation frequency12. The resultinggate–spectroscopy shown in Fig. 1h reveals three peaks of decreasingamplitudes (see inset) that relate to the first harmonic frequency fpgand the next two harmonics 2fpg and 3fpg. This indicates quantuminterference occurring over two and three turns of the inner channelloop, thus providing a clear signature of the interferometer’s highcoherence. Importantly, each peak diverges at the same plunger gatevoltage V cpg = � 0:28V, which corresponds to the expulsion of theinner channel from under the gate when the filling factor under thegate reaches νpg ~ 1. This divergence is channel–specific12 and providesan unambiguous indicator of the QH edge channels involved in theinterference.One new aspect of our measurement approach is the systematicacquisition of IV curves at every point of the interference patterns inFig. 1c, g, enabling us to simultaneously explore the complete para-meter space of energy, plunger gate voltage, and magnetic field (seethe plunger gate–dependent oscillations at various bias voltages inSupplementary Movie 1). Figure 1c, g are extracted from thebias–dependence data at zero bias. The bias voltage dependence ofthe oscillation frequency yields a checkerboard pattern illustrated inFig. 1e in a restricted gate voltage range, reflecting the additional phaseshift acquired by the injected electrons at finite energy. This check-erboardpattern canbe accurately simulated in Fig. 1f, asoutlined in theMethods section. In turn, this enables us to compute the Fouriertransform of the oscillations at each bias voltage (see oscillation fre-quency dispersion at various bias voltages in Supplementary Movie 2)and to extract the bias voltage dependence of each harmonics dis-played in Fig. 1i. The resulting oscillatory lobe structure of each har-monic is best fitted with a Gaussian decay for the energy relaxation(see ref. 12), and provides the Thouless energy of the interferometeredge ETh = hv/L = 135 ± 4 μV (bias–oscillation period), whereL = 3.4 ± 0.2μm is the length of the interfering channel between twoQPCs and v is the edge velocity. The harmonics then yield ETh/N (seeFig. 1i inset), where N is the respective harmonic index, providing anassessment of v = (1.1 ± 0.1) × 105 ms−1 consistent with our previouswork12.Electron pairing on two coherently coupled channelsElectron pairing emerges in our interferometer in the presence of twoedge channels by interfering with the outer channel while keeping theinner channel localized in the interferometer cavity (see inset sche-matics in Fig. 2a). Gate spectroscopy shown in Fig. 2a reveals thepairing frequency (orange dashed line) that is almost twice that of theinner channel interference unveiled previously in Fig. 1h. Since an areavariation of one flux quantum at a fixed magnetic field) is ΔA =ϕ0/B = αΔVpg = α/fpg, where α is the (non–linear) lever arm of the gate andΔVpg = 1/fpg the plunger–gate oscillation period, a frequency doublingtherefore signals an abnormal flux periodicity of h/2e similar to thatreported in GaAs5,7,11.The frequency doubling is also evidenced by the presence of aresidual peak at half the frequency highlighted with a red dashedline that coincides with the frequency of the outer channelh/e–periodic interference, the latter being independently char-acterized by its spectroscopy at a different filling factor wherepairing is sub–dominant (Supplementary Fig. 9). Inspecting theplunger–gate evolution of those frequencies, we see that bothpairing and outer channel frequencies diverge at V cpg = � 0:96V, avalue corresponding to a filling factor underneath the plungergate νpg = 0.05, in agreement with the expulsion of the outerchannel from the plunger gate area.However, unlike the harmonics in Fig. 1h, the pairing frequency isnot exactly twice that of the outer channel: At Vpg = −4.8 V, one finds105 V−1 and 45 V−1, respectively. To understand this discrepancywe addon the gate–spectroscopy the inner channel frequency (first harmonicmeasured in Fig. 1h) as a black dashed line in Fig. 2a,which leads us to acentral finding of this study: The pairing frequency is not double theAB frequency but the sum of the distinct inner and outer channelfrequencies. This is seenwith the orange dashed line in Fig. 2a,which isconstructed from the sum of the black (inner channel) and red (outerchannel) dashed lines, and which fits remarkably well with the pairingfrequency dispersion. Here the different frequencies for the inner andouter channels stem fromtheir slightly different positionswith respectto the plunger and QPC split gates, as discussed in the following, andthus their different effective areas.This finding is particularly striking and insightful as it demon-strates that, although localized, the inner channel influences thequantum interference of the outer channel and such an unusual pair-ing frequency. The system therefore behaves as if a correlated exci-tation propagates on both inner and outer channels, therebyaccumulating the sum of the AB phases of both channelsφ= heHinnerA:dl+heHouterA:dl, where A is the vector potential.Examining the bias voltage–dependence of thisgate–spectroscopy displayed in Fig. 2b shows that the pairing modeand theouter channels exhibit nearly the samebias voltageperiodicity,that is, Thouless energy, confirming the fact that the pairing frequencyis not a harmonic of the outer channel interference. Here, it is evidentArticle https://doi.org/10.1038/s41467-024-54211-3Nature Communications |        (2024) 15:10064 2www.nature.com/naturecommunicationsthat pairing prevails over the h/e–periodic contribution from the outerchannel.We delve further into the evolution of pairing with respect tochanges in filling factor and bias voltage in Supplementary Fig. 9. At alow filling factor (Vbg = 1.2 V, νB = 1.7), pairing is present but withnotably lower prominence at all bias voltages (SupplementaryFig. 9a,d). However, pairing significance increases and becomesdominant as the filling factor rises (Supplementary Fig. 9b, c, e,f). Theinfluence of bias voltage on the relative weight of each frequencybecomes particularly noticeable when pairing and outer frequencieshave similar amplitudes, as observed at Vbg = 1.58 V (νB = 2, Supple-mentary Fig. 9e).Notably, the zero–bias pajama map depicted in Fig. 2c displays adistinct discontinuity in the tilted AB constant phase lines, deviating50150250350450Frequency     (V-1)h(mV-1)pg = -1.3 V0Experiment Calculation|FFT| (arb. units)0 1ff 1/     (mT-1) 100 100 2000.5(70, 0.36)1/Δ  pg (V-1)  pgV  qpc2VDVI  qpc1VBVDC-200 -100 0 100 200Bias voltage         (μV)1020|FFT|  (arb. units)1.21.41.61st Harmonic2nd Harmonic-4 -2 -1-3-1.5 0-1 -0.5-23rd Harmonic60 (μV)012011/3 1/2-3.5 -3 -2.5-4-4.5-50.50.1 0.3 |FFT| (arb. units)-4.5-4.7Vpg0.5-6 -5 -4 -3 -2 -1 0  1  Plunger gate filling factor     (mV)-0.100.1-2.7-2.85 -2.75-2.8 -2.7-2.85 -2.75-2.8Index 1/N1.21.6c (mT)δB 012345-3.2 -3.15 -3.1Plunger gate voltage   pg (V)0 1       |FFT| (arb. units)1.3 1.7    (e²/h)GDV δG ΔB E ThVPlunger gate voltage   pg (V)V VδG           (e²/h)DDC  (e²/h)G  DPlunger gate voltage   pg (V)VVfi1st2nd3rd   pg   ν  (e²/h)G  DV  DC0-0.200.2-0.200.2          (e²/h)Dgie fdb caFig. 1 | Inner edge channel interference at Vbg = 1.8 V (νB = 2.26). a Atomic forcemicroscope image of hBN encapsulated graphene. Ohmic contacts (rough sur-face), plunger gate and quantum point contact (QPC, smooth surface) are in yel-low. Inner and outer edges are depictedwith black and red.Dashed blue linemarksinvisible graphene.Diagonal conductanceGD = I/VD, where I andVD are current anddiagonal voltage. Scale is 1 μm b Schematic of edge configuration. c Pajama map,with QPC transmission T1 = 0.67 and T2 = 0.66, reveals AB--dominatedinterference19. Inset: Fast Fourier transform (FFT). d GD versus Vpg from −4.7 V to−4.5 V at zero bias VDC. e, f Checkerboard extracted from Supplementary Movie 1,and its calculated counterpart (Methods). g Oscilations at 0 < Vpg < −5 V, with biasdependence in Supplementary Movie 1. h Gate spectroscopy, sliding FT (with a0.175V window) on oscillations at VDC = −28 μV. Full bias dependence in Supple-mentary Movie 2. First harmonic (blue line) diverges at V cpg; second and thirdharmonics in dashed red and yellow. Inset: FFT at Vpg = −1.3 V shows three fre-quency peaks. i, Lobe structures: first (blue), second (red), and third (yellow)harmonic (Methods). Inset: Thouless energy ETh versus inverse harmonic index.Article https://doi.org/10.1038/s41467-024-54211-3Nature Communications |        (2024) 15:10064 3www.nature.com/naturecommunicationsfrom the standard pattern shown in Fig. 1b. The Fourier transformreveals two contributions associated with the pairing frequency andthe outer edge frequency, as shown in the inset.Phase jumpsTaking into account these two oscillation frequencies, a simple fre-quency beating can describe the observed regular discontinuities.Figure 3 demonstrates how to reproduce such discontinuities inpajama maps obtained from the outer edge interference at Vbg = 1.2 V(νB = 1.7), as illustrated in Fig. 3c.Wefirst simulate in Fig. 3b the oscillation of the inner channel datashown in Fig. 3a. In the data shown in Fig. 3c, a distinct negative con-stant phase line (see black arrows) correlates with the frequency of thebare outer channel, allowing for the extraction of its frequency, whichwe simulate in Fig. 3d. We then compute a signal whose frequency isthe sum of the inner and outer ones in order to simulate the pairingcontribution in Fig. 3e. Adding now the pairing contribution and theouter contribution as occurring in the interferometer leads to Fig. 3f, inwhich a beating pattern naturally emerges and remarkably fits thedata Fig. 3c.Details regardingmethods and simulations for otherfilling factorscan be found in Supplementary Figs. 5-6. Additionally, differences inthe relative weight of pairing at various bias voltages result in diversebeating patterns and apparent phase jumps20, as shown in Supple-mentary Fig. 9j-l. This bias–dependent frequency superposition is alsovisualized in Supplementary Movie 4.Similarly, the apparent complexity of the checkerboard patternscan also be fully replicated by summing those of the pairingmode andthe bare outer edge, using the relative amplitude of each componentextracted at zero bias. Four checkerboard patterns, shown in Supple-mentary Fig. 10a,c,e,g, are obtained by partitioning the outer channelat different filling factors on the νB = 2 and νB = 3 plateaus (e.g. pairingOuterPairingPairingInnerOuter fofifp fo=    +fiInner    Outer-4 -3 -2 -1 050100150200250300350-200 -100 0 100 200-3.2 -3.18 -3.16 -3.14 -3.12 -3.112345010Frequency      (V -1)2001|FFT|  (arb. units)0.6 1100-6 -5 -4 -3 -2 -1 0 1Plunger gate filling factor     pg 0 1      |FFT| (arb. units) 00|FFT| (arb. units)0 10.5(e²/h)1/Δ  pg (V-1)V 1/      (mT-1)  ΔB  (mT) δB Plunger gate voltage    pg  (V)VPlunger gate voltage    pg  (V)VBias voltage          (μV)VDCfGDνfofifpca b(130, 0.74)Fig. 2 | Gate–spectroscopy of outer edge channel interference. Conductanceoscillations were measured by partitioning the outer edge channel with QPCtransmissionofT1 = 0.91 andT2 = 0.94 atVbg = 1.8 V (νB = 2.26).aGate spectroscopy,FFT amplitude of the oscillation versus Vpg and frequency f atVDC = − 55.5 μV. FFT isslided over the whole plunger gate voltage range with a 0.2V window. Supple-mentary Movie 3 shows the dependency of this gate–spectroscopy on the biasvoltage. The black and red dashed lines represent the frequency dispersion of theinner fi (determined from Fig. 1h) and outer fo edges, respectively. The orangedashed line, calculated as the sum of fi and fo, coincides with the strong signalattributed to pairing fp. Inset schematics show the edge channel configuration, withthe red dashed line indicating the partitioned outer channel and the orange wavyline representing inter–channel interaction. b The lobe structures of pairing(orange–colored) and the outer channel (red–colored), extracted frombias–voltage dependent gate–spectroscopy in Supplementary Movie 3. c Thepajamamap,measured at zero bias VDC =0V, shows frequency beating that persistsacross the entire bias–voltage range (SupplementaryMovie 4). Inset: 2DFFT revealstwo frequencies–the outer channel fo and pairing fp. fp is shown as the sum of the fo(red arrow) and fi (black arrow).Article https://doi.org/10.1038/s41467-024-54211-3Nature Communications |        (2024) 15:10064 4www.nature.com/naturecommunicationsand tripling regime). Employing the functional form described in theMethods but considering only the first harmonic, we have successfullysimulated the checkerboard patterns seen in Supplementary Fig. 10b,d, f, h,l.Electron tripling on three coherently coupled channelsThe observation of pairing naturally raises the question of whether theinclusion of an additional third channel could lead to a threefoldincrease in frequency, namely, electron tripling, although this has notbeen observed thus far in GaAs5,7,11,21.To address this question we set our QH FPI to bulk filling factor 3and, akin to the case of pairing, we partitioned the outer channel, whilehaving the middle and inner channels localized (Fig. 4a). Figure 4creveals a new frequency in the plunger gate spectroscopy, highlightedwith the violet dashed line, which is almost three times higher than thatof the outer channel: At Vpg = − 3.9 V, one finds 179 V−1 and 45 V−1,respectively. By overlaying the spectral dispersion of the inner (black),middle (blue), and outer (red) channels, each separately identified, wecan calculate the sum of the three. The violet dashed line denotes thissum and significantly aligns with the tripling frequency. As for the caseof pairing, the tripling frequency therefore results from the sum of thethree distinct AB phases of the three edge channels, each character-ized by a different effective area.The coherent mixing and contributions of the three channelsresult in a pajamapattern shown inSupplementaryMovie 7 that is evenmore complex than that for the pairing. Importantly, a non–negligiblepairing contribution shown in theorangedashed line in Fig. 4c remainspresent and comes from the sum of the outer and middle channelfrequencies. This suggests that pairing occurs only between the par-titioned channel and the nearest neighbor channel, the middle one inthis case. To ascertain this deduction, we have carried out anothergate–spectroscopy in a different configuration inwhichwepartitionedthe middle channel, fully transmitted the outer channel, and localizedthe inner channel (see Fig. 4b). The resulting spectroscopy displayed inFig. 4d clearly shows a pairing contribution coming from the sum ofthemiddle and inner channels frequencies, confirming that this pairingoccurs between partially transmitted and fully localized nearestneighbor channels.Interestingly, the amplitude of the pairing peak is weaker in thisconfiguration (also confirmed at slightly different filling factors inSupplementary Fig. 4b), while the middle and inner frequencies arevisible. Here, the difference with the previous configurations at fillingfactor 2 is that the pairing involves channels belonging to twodifferentLandau levels, that is, the zeroth and the first ones. Consequently, theedge channels are more spatially separated due to the large cyclotrongap compared to the case of pairing between the outer and middleedges which both belong to the zeroth Landau level. The fact thatpairing intensity increases with smaller separations between channelsstrongly suggests that inter–channel Coulomb interactions play acrucial role in pairing and tripling.1.1 1.3 0 0.151.1 1.3 0.01 0.080.30.10.30.1(e²/h) (mT) δB GD (e²/h)GD (e²/h)GD(e²/h)GD(e²/h)GD(e²/h)GDGD Calc. inner-3.3 -3.25 -3.20246Calc. outerExpt.  inner edge interference   Expt.  outer edge interference   Calc. pairing     ( fi  )      ( f0 )      ( fi + f0 )     ( f0 ) +     ( fi + f0 )  (mT) δB Plunger gate voltage    pg (V)V Plunger gate voltage    pg (V)VPlunger gate voltage    pg (V)VPlunger gate voltage    pg (V)V Plunger gate voltage    pg (V)V Plunger gate voltage    pg (V)VGD GDGD GD-3.3 -3.25 -3.2-3.3 -3.25 -3.2-3.3 -3.25 -3.20246-3.3 -3.25 -3.2-3.3 -3.25 -3.2fa b d ec Calc. outer+pairingFig. 3 | Experimental and simulated analysis ofphase jumps in thepajamamapsmeasured atVbg = 1.2 V (νB = 1.7). a The pajamamap, obtained by partitioning theinner edge channel, suggests GD oscillation with a single–frequency from the inneredge fi. b Calculated pajama map with the frequency fi derived from the panel a.c Observed pajama map when partitioning the outer edge channel, where thedominant oscillations (black arrows) provides the frequency associated with thebare outer edge fo. d Calculated pajama map using the frequency fo. e The pajamamap determined by pairing oscillation frequency GD(fi + fo). Pairing frequency iscalculated by summing up fi and fo. f The diagonal conductance GD beating in thepajama map occurs due to the simultaneous presence of two oscillations withpairing frequency GD(fi + fo) and outer edge frequency GD(fo).Article https://doi.org/10.1038/s41467-024-54211-3Nature Communications |        (2024) 15:10064 5www.nature.com/naturecommunicationsReal-space distance between edge channelsA key question to assess the Coulomb interaction quantitatively is thereal spacedistancebetweenedgechannels. As inoptical interferometry,our gate–spectroscopy provides a very accurate measurement of theinterfering path, which can in turn lead to the edge channel–to–gatedistance δ(Vpg) by integrating the Vpg–dependence of an edge channelfrequency (see Methods). The integration of frequency dispersion inFig. 2a provides the representative edge distance between two channelsat νB = 2, as shown in Fig. 5a. Additionally, Fig. 5b displays the distancesamong the three edges at νB = 3. Strikingly, despite a relatively smooth|FFT|  (arb. units)0 1fifp1 fo =     + fmfp2 fm =    +fiftfmInner    Middle(partitioned) Outer50100150200250300-3 -2 -1 0350400-3 -2 -1 0-3 -2 -1 0 1 2 3|FFT|  (arb. units)0 1Plunger gate voltage    pg (V)V Plunger gate voltage    pg (V)VFrequency      (V -1)fPlunger gate filling factor     pg νfofifmInnerMiddle     Outer(partitioned) TriplingInner OuterPairing1  (middle+outer)  MiddleInner Middle-3 -2 -1 0 1 2 3 Pairing2 (inner+middle) Plunger gate filling factor     pg νa bc dFig. 4 | Electron tripling and pairing. a, b Schematic of edge channel configura-tions: interference with the outer channel (a) and the middle channel (b). Inner,middle, and outer edges are represented in black, blue, and red, respectively.Dashed lines denote the partitioned edges. The orange wavy line depicts themiddle–--outer edge interaction, and the violet wavy line illustrates the interac-tions among all three edges. c Gate spectroscopy, derived from the conductanceoscillations when partitioning the outer edge channel at Vbg = 2.5 V (νB = 2.93) withQPC transmission T1 = 0.59 and T2 = 0.5. Sliding FT was performed with a 0.45 Vwindow. The top axis is the filling factor beneath the plunger gate. Frequencies forinner fi, middle fm, and outer fo edges are marked with black, blue and red dashedlines. The orange dashed line (pairing1, fp1) is the sum of fm and fo, and the violetdashed line (tripling, ft) is the sumof all three channel frequencies. Datawere takenat VDC = 0 V, with a full set of their bias voltage dependence provided in Supple-mentary Movie 5. Prominent signals between 1 < νpg < 2 (above 250 V−1) are asso-ciated with electron tripling ft. Weaker signals at low frequencies (~100 V−1)correspond to fo and fp1. In this regime, the inner edge has been already expelledfrom beneath the plunger gate, while the middle and outer edges remain beneaththe plunger gate and are slightly displaced from the graphene crystal edge (sup-plementary information section VI for details). d, Gate–spectroscopy obtainedfromouter edge channel interference, with T1 = 0.78 and T2 = 0.81, atVDC = − 55μV.The orange dashed line here (pairing2, fp2) corresponds to the sumof fm and fi. Fullbias voltage–dependent gate spectroscopy is in Supplementary Movie 6. Lobestructures for all the channels are detailed in Supplementary Fig. 3.Article https://doi.org/10.1038/s41467-024-54211-3Nature Communications |        (2024) 15:10064 6www.nature.com/naturecommunicationselectrostatic potential around local top gate, the outer (red line) andmiddle (blue line) channels are very close, with a distance LMO com-parable to or even smaller than the magnetic length lB =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih=2πeBp,implying strongly interacting channels. For instance, we obtainLMO= 5.6 nm at Vpg = − 3 Vwith amagnetic length lB = 6.7 nm at B= 14 T.On theother hand, the inner channel (black line) that belongs to thefirstLandau level is located at LIM � 35nm (at Vpg = −3 V) away from themiddle channel. This distance can be accounted for by the largecyclotron gap between the zeroth and first Landau levels of graphene.On the contrary, along the graphene crystal edges of the FPI, the QHedge channels are known to be all confined on a scale of the order of lBto the crystal edge18,22. We can thus outline the spatial structure of edgechannels as well as the many–body Landau level spectra at the crystaledges and the pn junctions in our FPIwith the schematics in Fig. 5c. Thisprovides novel and accurate measurements of the inter–channel dis-tances, which are crucial for a further theoretical assessment of Cou-lomb interactions between the channels. It is also consistent with ourobservation and interpretation of weaker pairing between the middleand inner channels at νB = 3 discussed above.DiscussionInter–channel interactions typically result in fractions of the electroncharge being redistributed in edge magnetoplasmons23–27. For twoco–propagating channels with strong mutual interactions (or withequal edge velocities), electrons decompose into fast (charged) andslow (neutral) plasmonicmodes, each evolving in both channels. Theirdynamics, solved through the matrix scattering approach23 for edgemagnetoplasmons, does not give rise, according to Ref. 8, to anydominant multiple electron tunneling whose contribution to dctransport vanishes at νB = 2 and νB = 3. Amore recent theoretical work10has taken into account charge discreteness and has considered thelimit where all edges are totally pinched off at the QPCs. It has pre-dicted that an electron entering the FPI generates neutral plasmons,so–called neutralons, which are absorbed by a second tunnelingXPlunger gate voltage (V)-2 -1-440Edge channel-to-gate distance    (nm)002060-3InnerOuterbδV δ3 V 2 V 1 VXN=1E  FN=0N=-1IML MO ~ lBL EN=0N=1EFE~lBXMO ~ lBIML L Plunger gate voltage (V)-2 -1-340Edge channel-to-gate distance   (nm)08020601000Outer Middle Innerδpg<   2   V 2 <    pg <   1V V V 2 <    pg <   1   V V V V 3 <    pg <   2   V V V pg <   3   V V pg>   1   V V pg >   1   V 1 V 2 V δa cFig. 5 | Interferometric determination of edge channel–to–plunger gate dis-tance. a δ represents the distance between edge channel-to-plunger gate. Blackand red lines show δ as a function of plunger gate voltage atVbg = 1.8 V, obtainedbyintegrating each plunger gate frequency dispersion in Fig. 2a. The top schematicsillustrate edge channel configurations in three plunger gate regimes. Calculationdetails are in the Methods section. b δ for three edge channels at Vbg = 2.33 V(νB = 2.8), derived from Fig. 4c. The distance LMO between the outer and middlechannels is less than a magnetic length lB, whereas the inner is at a distance LIM �35nm from themiddle due to the large cyclotron gap between the zeroth and firstLandau levels. c The center schematic illustrates an edge configuration with apartitioned outer edge channel (dashed red line), while the inner andmiddle edgesare shown in the black and blue lines, respectively. The FPI is defined by gatessurrounding the QPC and plunger gate (color-coded in yellow), with the crystaledge delineating the remainder. The top and bottom schematics illustrate theenergy dispersion of the Landau levels along the crystal-defined and gate-definededges, highlighting noticeable differences in edge positions. Previous studies(Refs. 18,22) have assessed the crystal edge dispersion, revealing edge channelsconfined to a fewmagnetic lengths from the crystal edge. At the pn junction, gapsof broken symmetry states open in the zeroth Landau level when the local fillingfactor νB(x) reaches every quarter filling, i.e., νB(x) = −1, 0, and 1 (refs. 36,37).Article https://doi.org/10.1038/s41467-024-54211-3Nature Communications |        (2024) 15:10064 7www.nature.com/naturecommunicationselectron; this exchange of neutralons induces a dynamical attractiveinteraction, thus enhancing the average Fano factor, which couldpotentially reach 2. In addition to the assumption of strong pinchedQPCs, this enhancement requires non–universal interaction para-meters, thus couldn’t yet explain the robustness of pairing observed invarious experimental contexts. In addition, correlated tunneling ofthree electrons or more is neglected as they are less probable, whichseems plausible at higher filling factors as well. Note that at νB = 3 oneexpects strong enough inter–edge interactions to induce two neutralmodes that could enhance neutralon exchange thus pairing dynamicalattraction. The present observation of tripling at νB = 3 doesn’t favorsuch a mechanism, and can therefore, more generally, provide aselective criterion for theoretical explanations.A possible scenario would consist of two or three electronsentanglement mediated by inter–edge interactions. It was shown inRef. 28 that two electrons entangled through adouble dot tend tobunchin the singlet state, leading to a doubled Fano factor, but to antibunch inthe triplet state. In our setup, distinguishable edge states and valleycomponents could offer alternative degrees of freedom to the spin.Besides, it is worth mentioning a recent report on pairwise elec-tron tunneling into large quantum dots29, reminiscent of the FPI con-figuration. In this context, some theoretical models predict a possibleattraction–pairing–of localized electrons resulting from the mini-mization of the screened Coulomb interaction30, and even three elec-trons bunching in very specific configurations31.A different explanation of the frequency doubling and triplingbased on charging effects32,33, which has long obscured AB–dominatedinterferometry19, certainly deserves careful consideration. For strongcapacitive coupling between the interfering edge and the bulk,designated by theCoulomb–dominated (CD) regime, and similarly to asingle–electron transistor, the conductance should not depend on themagnetic field and rather oscillates with the gate voltage, whichchanges the occupancy of the device, leading to vertical lines in theB, VG plane. The interfering phase varies continuously according to therelation θ = −2πνϕ/ϕ0, thus is doubled or tripled at νB = 2 or νB = 3respectively. In our FPI at νB = 2, we have observed, for the innerchannel, a noteworthy crossover from theCD regime to the AB regime,with an increase in the filling factor on the h/2e2 quantumHall plateau,as illustrated in Supplementary Fig. 8. Nonetheless, the pairing phe-nomenon gets simultaneously weaker (stronger) when the innerchannel is CD (AB) (see Supplementary Fig. 8a,b,c,d). Moreover, alarger QH FPI with a size of 15 μm2 exhibits pairing exclusively at νB = 2,without the presence of the inner or outer channel frequencies (seeSupplementary Fig. 2). In that case, no signature of the CD regime isobserved,most likely due to a smaller charging energy. In addition, thetheoretical analysis in Ref. 32 is valid in the presence of edge recon-struction, so that highly dense localized states in compressible stripesare close enough to the edges to feel strong edge–bulk interactions.Nonetheless, we have shown the absence of edge reconstruction at thecrystal edges of our devices, thus pointing towards an incompressiblebulk18. Although the question remains open, these concordant obser-vations suggest that the CD regime is not related to the pairing andtripling phenomena, in agreement with the conclusions of Ref. 5.Beyond the edge–bulk interactions, interactions between the chiraledges need tobemorecarefully considered and could lead us to revisitthe interpretation of such a transition9.A concurrent work20 with a similar observation of frequencydoubling proposes an explanation based on phase shifts induced bythe discrete addition of charges in the inner channel strongly coupledto the outer one. The key idea is that the charge on the outer edge,which controls the FPI phase θ, is itself dependent, through inter–edgecoupling, on the inner edge charge. In this model, the ground–stateenergy variations are expandedwith respect to thoseof the charges onboth channels. Thus, contrary to8,9, the plasmonic modes are ignored;which could be justified for a short enough inner edge whereelectron–hole pairs creation costs high energy. This also differs frommodeling interactions by Gaussian fluctuating phase in order to fit theobserved voltage dependence of the AB oscillations in Fig. 1a. Inprinciple, one could generalize the same argument to νB = 3. None-theless, we don’t attribute the observed discontinuities in the pajamato phase jumps renormalized by inter–edge interactions, but rather tobeating between AB oscillations of the outer channel and a pairingmode. The way these discontinuities vary with the bias voltage (seeSupplementary Movie 4) would also question their emergence from acharge addition mechanism.In addition, it is not yet established that a pure electrostaticapproach could account for the electron transfer 2 by 2 observed inlarge quantum dots29 and for shot noise measurements in inter-ferometers basedonGaAs devices5,7. Indeed, for symmetric ac voltagesat two QPCs, the Fano factor is given universally by the charge of thedominant tunneling process, independently on the form, strength andrange of interactions, but canbe increased by non–equilibriumexcitedstates34 (as in Ref. 9). The argument in Ref. 20 is also based on awell–defined charge number for the outer edge, which is questionablefor almost open QPCs for which pairing has been observed as well7.Therefore, correlated phenomena beyond this electrostatic approx-imation might enter into play in this electron pairing and triplingobserved in QH interferometers.MethodsSample fabricationThehBN–encapsulated graphene heterostructurewas assembled fromexfoliated flakes using the van der Waals pick–up technique17 anddeposited onto a graphite flake serving as the back–gate electrode.The substrates are highly doped Si wafers with a 285nm thick SiO2layer. The flake thicknesses are 27 nm for the graphite, 45.5 nm for thebottom hBN, and 27.5 nm for the top hBN. Contacts and electrostaticgates were patterned using e–beam lithography, and Cr/Au wasdeposited for the contacts after etching the heterostructure with aCHF3/O2 plasma. Pd was deposited for the electrostatic gates, pre-ceded by a slight O2 plasma etching to remove resist residues on hBNand ensure a homogeneous electrostatic potential beneath the gate.MeasurementsAll measurements were performed in a dilution fridge with a basetemperature of 0.01 K at 14 T. Themeasurement setup and filtering aredescribed in Refs. 12 and 22. Systematic current–voltage character-istics were measured in a four–terminal configuration as illustrated inFig. 1a with an acquisition card (NI–6346 from National Instruments).IV curve takes about 10 s, with oversampling enabling us to averageabout 1000 samples per data point. The diagonal voltage drop acrossthe interferometer was measured with a differential FET amplifier(DLPVA–100–F–D from Femto GmbH). A homemade multichannel20–bit digital–analog converter (DAC) was used to adjust the variousgate voltages, with noise levels below 7.5 nV/ffiffiffiffiffiffiHzp, and a long timeresolution of 1ppm. The DAC electronic includes an ultra–stable vol-tage reference LTZ1000 from Linear Technology. Differential resis-tance data were obtained by numerically differentiating thecurrent–voltage characteristics.Checkerboard patternIn the presence of a single harmonic numbered n, the oscillationdependence with bias voltage can be described by the functionalform12:Goscn =An β cos n× 2πφϕ0� 2L_veVβ� �� ��+β cos n× 2πφϕ0+2L_veVβ� �� ��exp �ðeV Þ2σ2n !,ð1ÞArticle https://doi.org/10.1038/s41467-024-54211-3Nature Communications |        (2024) 15:10064 8www.nature.com/naturecommunicationswhere β and β are asymmetry parameters describing how symmetric isthe voltage drop on the two sides of the interferometer. An is the nthharmonic oscillation amplitude. eV is the voltage applied between thesource anddrain. L is the lengthof the interfering channel between twoQPCs, v is the edge channel velocity, and φ is the AB flux picked up bythe electrons. The phenomenological Gaussian energy decay describ-ing phase fluctuations of the interfering edge channel due to Coulombinteractions or the electric noise in the non–interfering edgechannels35 fits best our data. The checkerboardpattern in Fig. 1e is verywell reproduced by the sumof the three first harmonicsGosc =PnGoscn ,with β =0.4, β =0:6, A1 = 0.25e2/h, A2 = 0.052e2/h, A3 = 0.0125e2/h,σ1 = 100 μeV, σ2 = 80 μeV, and σ3 = 65 μeV, as shown in Fig. 1f. In thepresence of several edge channel contributions to the oscillations,the oscillation dependence of edge channel i with bias voltage can bedescribed by:Gosci =Ai βi cos 2πφiϕ0� 2L_vieVβi� ��+βi cos 2πφiϕ0+2L_vieVβi� ��exp �ðeV Þ2σ2i !,ð2Þwith the same notation as above. Checkerboard patterns in Supple-mentary Fig. 10 are very well reproduced by the sum of the outer andpairing conductance Gosc =Gosco +Goscp without harmonics.Gate–to–edge channel distance from gate–spectroscopyThe gate–spectroscopy is a directmeasure of the capacitancecouplingbetween the gate and the interfering edge channels12. Assuming adistance δbetween the gate and the interfering channel asdrawn in theleft inset in Fig. 5a, the lever arm of the gate is given byαðVpgÞ= LpgdδðVpgÞdVpg, where Lpg = 1.5 μm is the gate edge length12. As aresult, one cancompute thedisplacement distanceδby integrating thegate–voltage dependence of the frequency of the considered channel:δðVpgÞ= ϕ0BLpgR VpgV cpgf pgðV ÞdV , using ΔA =ϕ0/B = αΔVpg = α/fpg. Figure 5bdisplays the resulting distances for the three channels at νB = 3 com-puted from Fig. 4c.Data availabilityThe movies related to the transport data in this manuscript aredescribed in the supplementary information and are available at theonline Zenodo repository: https://zenodo.org/records/10420556.References1. Chklovskii, D. B., Shklovskii, B. I. & Glazman, L. I. Electrostatics ofedge channels. Phys. Rev. B 46, 4026–4034 (1992).2. Ezawa, Z. F. & Tsitsishvili, G. QuantumHall ferromagnets.Rep. Prog.Phys. 72, 086502 (2009).3. Halperin, B. I. & Jain, J. K. Fractional Quantum Hall Effects: NewDevelopments (World Scientific, 2020).4. Nayak, C., Simon, S. H., Stern, A., Freedman, M. & Sarma, S. D. Non-Abelian anyons and topological quantum computation. Rev. Mod.Phys. 80, 1083 (2008).5. Choi, H. et al. Robust electron pairing in the integer quantum Halleffect regime. Nat. Commun. 6, 7435 (2015).6. Van Wees, B. et al. 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B 102, 041113 (2020).35. Roulleau, P. et al. Finite bias visibility of the electronic Mach-Zehnder interferometer. Phys. Rev. B 76, 161309 (2007).Article https://doi.org/10.1038/s41467-024-54211-3Nature Communications |        (2024) 15:10064 9https://zenodo.org/records/10420556www.nature.com/naturecommunications36. Liu, X. et al. Visualizing broken symmetry and topological defects ina quantum Hall ferromagnet. Science 375, 321–326 (2022).37. Coissard, A. et al. Imaging tunable quantum Hall broken-symmetryorders in graphene. Nature 605, 51–56 (2022).AcknowledgementsWe thank A. Assouline, D. Basko, P. Degiovanni, B. Douçot, M. Heiblum,B. Rosenow, K. Snizhko, and E. Sukhorukov for valuable discussions.We thank F. Blondelle for technical support on the experimentalapparatus. Samples were prepared at the Nanofab facility of the NéelInstitute. This work has received funding from the European Union’sHorizon 2020 research and innovation program under the ERC grantSUPERGRAPH No. 866365. B.S., H.S., and W.Y. acknowledge supportfrom the QuantERA II Program, which has received funding from theEuropean Union’s Horizon 2020 research and innovation programunder Grant Agreement No 101017733. K.W. and T.T. acknowledgesupport from the JSPS KAKENHI (Grant Numbers 20H00354,21H05233 and 23H02052) and World Premier International ResearchCenter Initiative (WPI), MEXT, Japan.Author contributionsW.Y. fabricated the sample.W.Y. andD.P. performed themeasurements.W.Y. and D.P. analyzed the data with the inputs of H.S., I.S., C.D., andB.S.; I.S. provided theoretical support. K.W. and T.T. supplied the hBNcrystals. S.D., E.W., and F.G. provided technical support for the experi-ment. B.S. supervised the project. B.S., W.Y., D.P., and I.S. wrote thepaper with inputs from all the authors.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-024-54211-3.Correspondence and requests for materials should be addressed toBenjamin Sacépé.Peer review information Nature Communications thanks the anon-ymous, reviewer(s) for their contribution to the peer review of this work.A peer review file is available.Reprints and permissions information is available athttp://www.nature.com/reprintsPublisher’s note Springer Nature remains neutral with regard to jur-isdictional claims in published maps and institutional affiliations.Open Access This article is licensed under a Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License,which permits any non-commercial use, sharing, distribution andreproduction in any medium or format, as long as you give appropriatecredit to the original author(s) and the source, provide a link to theCreative Commons licence, and indicate if you modified the licensedmaterial. Youdonot havepermissionunder this licence toshare adaptedmaterial derived from this article or parts of it. The images or other thirdparty material in this article are included in the article’s CreativeCommons licence, unless indicated otherwise in a credit line to thematerial. If material is not included in the article’s Creative Commonslicence and your intended use is not permitted by statutory regulation orexceeds the permitted use, you will need to obtain permission directlyfrom the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.© The Author(s) 2024Article https://doi.org/10.1038/s41467-024-54211-3Nature Communications |        (2024) 15:10064 10https://doi.org/10.1038/s41467-024-54211-3http://www.nature.com/reprintshttp://creativecommons.org/licenses/by-nc-nd/4.0/http://creativecommons.org/licenses/by-nc-nd/4.0/www.nature.com/naturecommunications Evidence for correlated electron pairs and triplets in quantum Hall interferometers Results Gate–spectroscopy fingerprint of QH edge channels Electron pairing on two coherently coupled channels Phase jumps Electron tripling on three coherently coupled channels Real-space distance between edge channels Discussion Methods Sample fabrication Measurements Checkerboard pattern Gate–to–edge channel distance from gate–spectroscopy Data availability References Acknowledgements Author contributions Competing interests Additional information