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Aifei Zhang, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Patrice Roche, Carles Altimiras, François D. Parmentier, Olivier Maillet

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[Ballistic-to-diffusive transition in engineered counterpropagating quantum Hall channels](https://mdr.nims.go.jp/datasets/a56ee666-e412-40ac-a0a7-6dca16c4d1b0)

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Ballistic-to-diffusive transition in engineered counterpropagating quantum Hall channelsPHYSICAL REVIEW RESEARCH 7, L042037 (2025)LetterBallistic-to-diffusive transition in engineered counterpropagating quantum Hall channelsAifei Zhang ,1 Kenji Watanabe ,2 Takashi Taniguchi ,3 Patrice Roche ,1 Carles Altimiras ,1François D. Parmentier ,1,4 and Olivier Maillet 1,*1Université Paris-Saclay, CEA, CNRS, SPEC, 91191 Gif-sur-Yvette, France2Research Center for Electronic and Optical Materials, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan3Research Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan4Laboratoire de Physique de l’Ecole normale supérieure, ENS, Université PSL, CNRS, Sorbonne Université,Université Paris Cité, F-75005 Paris, France(Received 28 May 2025; revised 23 August 2025; accepted 14 October 2025; published 19 November 2025)Exotic quantum Hall systems hosting counterpropagating edge states can show seemingly nonuniversaltransport regimes, usually depending on the size of the sample. We experimentally probe transport in a quantumHall sample engineered to host a tunable number of counterpropagating edge states. The latter are coupled byLandauer reservoirs, which force charge equilibration over a tunable effective length. We show that charge trans-port is determined by the balance of up- and downstream channels, with a ballistic regime emerging for unequalnumbers of channels. For equal numbers, we observe a transition to a critical diffusive regime, characterized bya diverging equilibration length. Our approach allows simulating the equilibration of hole-conjugate states andother exotic quantum Hall effects with fully controlled parameters using well-understood quantum Hall states.DOI: 10.1103/3d25-wpthQuantum Hall (QH) insulators are characterized by agapped, electrically insulating bulk, and conduction alongtheir edge via a quantized number of one-dimensional chiraledge channels. In most cases, the edge channels copropagate;i.e., they have the same chirality. This leads to extremelyrobust transport properties, even in the presence of stronginteractions between edge channels [1,2]: edge transport isthen ballistic and dissipationless, and the Hall conductanceis exactly given by νe2/h, where e is the electron charge,h is Planck’s constant, and ν = nh/eB is the filling factorcharacterizing the topology of the QH states (n is the carrierdensity and B is the perpendicular magnetic field). However,the edge channels can also be counterpropagating; this isnotably the case of the quantum spin Hall (QSH) effect, whichhosts two counterpropagating channels with opposite spin po-larizations [3], but it is also expected for hole-conjugate statesof the fractional quantum Hall (FQH) effect such as ν = 2/3[4]. There, Coulomb interactions and interchannel tunnelingcan lead to equilibration along the edge [5], drastically im-pacting transport properties, which become length-dependent.Theoretical [6–12] and experimental [13–18] investigationsof charge and heat transport in hole-conjugate FQH states,combined with the general observation of imperfect conduc-tance quantization in the QSH effect [3,19–24], point towarddifferent transport regimes depending on the number of equili-brating edge channels and their respective conductances. If the*Contact author: olivier.maillet@cea.frPublished by the American Physical Society under the terms of theCreative Commons Attribution 4.0 International license. Furtherdistribution of this work must maintain attribution to the author(s)and the published article’s title, journal citation, and DOI.overall upstream and downstream conductances are different,the transport can be ballistic, with a quantized conductanceand negligible dissipation at large scale. If upstream anddownstream conductances are equal, however, the transportbecomes diffusive, with inexact conductance quantization anddissipation all along the edge. Remarkably, this applies to bothcharge and heat transport, which can lead to striking transportdecoupling effects in FQH channels where the electrical con-ductance is fractional while the thermal conductance is integer[17,18,25].Understanding and exploring the transition between thediffusive and ballistic regimes in equilibrating counterprop-agating edge channels is an experimental challenge, as itrequires controlling the number of equilibrating channels, thenature and strength of the equilibration process, and the abilityto probe dissipation along the edge of the sample. Inspiredby recent theoretical models [9,10,12], we have realized anexperiment where counterpropagating integer quantum Hall(IQH) channels equilibrate through a series of Ohmic contacts(Landauer reservoirs) that provide both charge redistributionand energy equilibration between channels. Our approachpresents multiple advantages, stemming from its simplicity:For any edge configuration among a wide range, we di-rectly measure the voltage of the Ohmic contacts, yieldingthe voltage drop and dissipation along the edge, as wellas the conductance and its length dependence. We emulatecharge equilibration in a QH edge with tunable numbers ofcounterpropagating channels. This allows us to observe andfully explore the transition between the ballistic and diffu-sive regimes, in agreement with exact scattering formalismcalculations.Our experiment consists of two Hall bars [devices A andB; see Fig. 1(a)] made of monolayer graphene encapsu-lated in hexagonal boron nitride [26], each individually gated2643-1564/2025/7(4)/L042037(6) L042037-1 Published by the American Physical Societyhttps://orcid.org/0009-0005-5622-4462https://orcid.org/0000-0003-3701-8119https://orcid.org/0000-0002-1467-3105https://orcid.org/0000-0001-6089-4083https://orcid.org/0000-0002-7029-752Xhttps://orcid.org/0000-0001-9319-565Xhttps://orcid.org/0000-0001-9842-0895https://ror.org/03n15ch10https://ror.org/0247p4w70https://ror.org/026v1ze26https://ror.org/026v1ze26https://ror.org/03a26mh11https://ror.org/013cjyk83https://ror.org/02en5vm52https://ror.org/03xjwb503https://crossmark.crossref.org/dialog/?doi=10.1103/3d25-wpth&domain=pdf&date_stamp=2025-11-19https://doi.org/10.1103/3d25-wpthhttps://creativecommons.org/licenses/by/4.0/AIFEI ZHANG et al. PHYSICAL REVIEW RESEARCH 7, L042037 (2025)ab} }(2,-1) (2,1)01,b 2,b 3,b 4,b3,t2,t1,tFIG. 1. (a) Schematics of the measured two-Hall bar device and paired contacts. (b) Effective Hall bar: equivalent configuration withcounterpropagating edge channel numbers set by individual filling factors νA and νB. In this example, one pair of contacts (labeled 0 here)is connected to a current source (injection contact), and Nt = 3 and Nb = 4 intermediate contacts on the top and bottom edges, respectively,are left floating, acting as effective Landauer reservoirs. Example sweep of device B’s gate voltage/density (in filling factor units νB) whilekeeping νA = 2, for top [panel (c)] and bottom edge contacts [panel (d)]. The colors assigned to each plateau with νB > 0 are used throughoutthe Letter.by a graphite back gate. Both are cooled down to 10 mKunder a 14 T perpendicular magnetic field. The Hall bars areconnected in a top-to-tail fashion, such that the kth contact ofdevice A, starting clockwise from the first contact next to thecold-grounded drain, is connected with the kth contact of de-vice B, starting anticlockwise from the first contact next to thedrain. Paired contacts share the same potential, and each pairmay be left floating (while its voltage with respect to groundis measured), put to ground, or used for current injection.This implements an effective single Hall bar [Fig. 1(b)] witha tunable number of counterpropagating channels determinedby the filling factor of each Hall bar νA/B. By convention, wedefine the edge flowing from (resp. into) the injection contact,when running clockwise on the effective Hall bar, as the top(resp. bottom) edge, labeled “t” (resp. “b”). We introduce Nt(resp. Nb) the number of intermediate floating Ohmic contactson the top (resp. bottom) edge. The gate-to-density correspon-dence of one device is determined by setting the other deviceat ν = 0, thus ensuring that current flows only in the former(see the Supplemental Material [27]). We then keep device Aat a fixed filling factor νA and tune the charge carrier densitynB of device B to obtain filling factors ranging from νB = −2to νB = 7. We inject a low-frequency (≈1 Hz) current throughthe source contacts of our choice in both devices and mon-itor voltages on each pair of contacts, including the source,through standard lock-in measurements. A sweep of the backgate of device B for νA = 2 in the configuration of Fig. 1(b)is shown in Figs. 1(c) and 1(d). When device B is p-doped(νB < 0), the voltage on the top effective edge [“t”-labeledcontacts in Fig. 1(c)] remains constant and equal to that of theinjection, while voltages on all bottom edge contacts remainzero, up to some residual backscattering observable on contact1,b. This is expected since here, all channels copropagate,making our effective Hall bar behave like a regular one. Onthe contrary, for νB > 0, we observe a voltage decrease on thetop edge starting from the source contact, while we measurenonzero voltages on the bottom edge contacts, increasing to-ward the source. In that configuration, from the effective Hallbar’s perspective, edge channels from devices A and B haveopposite chiralities, enabling equilibration in the intermediatecontacts.These observations can be all captured with a simple ap-proach for chiral edge currents: The current emitted in oneIQH channel from contact k is e2Vk/h. We assume that idealequilibration occurs in each intermediate metallic contact andthat propagation in between contacts, beyond negligible bulkleakage, is fully ballistic, insofar as it does not suffer fromlocal (i.e., microscopic) equilibration. This last assumption iswell satisfied in practice, because of the physical separationbetween channels of opposite chirality. Current conservationat contact k, t leads to (νA + νB)Vk,t = νAVk−1,t + νBVk+1,t ,with VN+1 = 0 for the drain contact. We obtain the potentialat each contact k on the top edge when νA �= νB:Vk,t = 1 − exp [−(Nt + 1 − k) log (νA/νB)]1 − exp [−(Nt + 1) log (νA/νB)]V0, (1)with V0 the source voltage [27]. The same result is derivedfor voltages on the Nb bottom edge contacts, up to a swapbetween νA and νB. According to Eq. (1), for νB �= νA, thevoltage drop along the edge presents an exponential profile,highlighting a characteristic dimensionless equilibration dis-tance δ = 1/| log(νA/νB)| that should be compared with theeffective edge length Nt .This exponential profile is clearly observed in the data:Figure 2 shows voltages measured at the source and all float-ing contacts, for two different measurement configurations. Inthe first “asymmetric” configuration [Fig. 2(a)], all the con-tacts are located on the top effective edge (Nt = 7, Nb = 0),so that equilibration only occurs on the top edge and can beprobed over a large number of contacts. For (νA, νB) = (6, 0)L042037-2BALLISTIC-TO-DIFFUSIVE TRANSITION IN … PHYSICAL REVIEW RESEARCH 7, L042037 (2025)N=21,b1,t 2,t 3,t02,b 3,b 4,b4,t5,bSplitt bConductance0123458Asymmetric6 7555555555 6 7 1,t 2222,,t2222223,b3,b3,b3,b3,b3,b3,b3,b3, 4,b5,b5,b5,5,,FIG. 2. (a) Sketch of the asymmetric configuration and corresponding voltage drop Vk measurements for νA = 6, and νB = 0 (gray crosses)and 3 (orange triangles), for an injected current amplitude I0 ≈ 0.38 nA. (b) Sketch of the split configuration and corresponding voltage dropmeasurements or νA = 2 and νB = 5, with injected current amplitude I0 ≈ 0.96 nA. Open (resp. filled) dots correspond to bottom (resp. top)edge contacts. Solid lines in both panels are applications of Eq. (1) for corresponding edges and bulk filling factors, and the dotted line in panel(a) is a guide for the eye at constant Vk . (c) Two-point conductance configuration and measurements (square symbols) for (νA = 2, νB = 2, 4, 5),obtained when adding pairs of grounds on each side, while keeping the bottom rightmost contact grounded to keep a symmetric configuration.Solid lines are applications of Eq. (2). Dashed lines represent the limit values of zero (G�) and full (G∞) equilibration.(no counterpropagating channel), the voltage stays constantover the whole top edge up to the drain. For νB > 0, e.g.,the (6, 3) case represented in Fig. 2(a), the voltage remainsconstant and equal to that of the source contact, until a dropoccurs within the last few contacts before the drain. Thisdrop corresponds to energy dissipation over a portion of theedge, highlighted as a “hot spot” in red in Figs. 2(a) and 2(b).The measured voltage profile is exactly matched by Eq. (1),particularly the exponential dependence parametrized by thedimensionless length δ.In the second, “split” configuration, Nt = 3 contacts arelocated on the top edge and Nb = 4 on the bottom edge,allowing probing the two edges independently. Figure 2(b)shows both voltage profiles for the (2, 5) case (where νA <νB), in excellent agreement with Eq. (1). On the top edge(where more channels come from the drain at zero potentialthan from the injection contact), the voltage rapidly drops tozero over the expected length δ. Contrarily, on the bottomedge, the voltage remains close to V0, only dropping closeto the drain over the same length δ. Therefore, the voltageprofile on a given edge (saturation value and drop region) isdictated by its dominant chirality. Furthermore, it signals aballistic behavior outside of the voltage drop region, when theeffective edge length (the number of contacts) exceeds a fewδ, as shown in Fig. 2(a). This is qualitatively reminiscent ofdissipation in a standard QH system, where the voltage alonga given chiral edge channel is constant, and only drops at thedownstream contact, with a fully localized hot spot. Thus,we use hereafter the term “downstream” to indicate the endcontact (source or drain) closer to which the voltage dropoccurs for a given edge. We also denote �Vend by the voltagedrop between the last floating contact and the downstreamcontact.In addition to the voltage profile measurements, we probethe length dependence of the two-point conductance of theeffective Hall bar. In this configuration [see Fig. 2(c)], the bot-tom contact closest to the drain is grounded to obtain a fullysymmetric device. We change the sample’s effective lengthby successively grounding pairs of contacts facing each other,starting from the drain side, thus leaving N pairs of floatingcontacts between the source and the grounded contacts. N = 0corresponds to all contacts grounded apart from the injectionone (and thus no equilibration), while N = 3 correspondsto none grounded except the bottom one closest to drain.The conductance in this configuration can be calculated fromEq. (1), for νA �= νB:G2w = G∞ coth[N + 12|log(νA/νB)|]. (2)This expression also involves the characteristic length δ =1/| log(νA/νB)| and shows an exponential convergence tothe equilibrated value G∞ = |νA − νB|e2/h for large samples(i.e., N � δ).The measured conductances are displayed in Fig. 2(c)for νA = 2 and νB = 2, 4, 5, showing an excellent agreementwith Eq. (2). We observe a clear decrease of conductancewith the number N of intermediate contacts, from a valueclose to G� = (νA + νB)e2/h, which corresponds to decou-pled channels at N = 0. Conductances are close to G∞ alreadyat N = 3. For the singular case νA = νB, the conductanceslowly decreases and remains substantially above G∞ = 0 atN = 3, and is well matched by the corresponding formulaG2w = G�/(N + 1) (see the Supplemental Material [27]).The divergence of δ translates as a voltage profile thatapproaches the linear trend as νB → νA and becomes lin-ear for the singular case νB = νA: Vk,γ=t,b = V0(Nγ + 1 − k)/L042037-3AIFEI ZHANG et al. PHYSICAL REVIEW RESEARCH 7, L042037 (2025)FIG. 3. Normalized voltage profiles Vk/V0 for all νB values atνA = 2 [panels (a)–(c)] and 6 [panels (d)–(f)]. Solid lines are derivedfrom the scattering approach. Linear profiles are singled out forνA = νB (red dotted lines).(Nγ + 1), as shown in Figs. 3(a)–(f). This corresponds to atransition to a diffusive, Ohmic regime, with a hot spot that isdelocalized over the whole edge, since all voltage drops be-tween successive contacts have the same value −V0/(Nγ + 1)independent of their position. This also appears in the conduc-tance: for νA = νB, we have G2w = 2νAe2/(N + 1)h, which isconfirmed experimentally through the slow, algebraic conver-gence of the (2, 2) conductance to zero [Fig. 2(c)].Our results highlight universal behaviors for both voltageprofiles and conductance. For νA �= νB, the potential differ-ence between counterpropagating channels is given by thevoltage difference between successive contacts, which canbe written, following Eq. (1), as �Vk,γ ≡ Vk,γ − Vk−1,γ =e−λ/δ�Vend,γ (λ = k or Nγ + 1 − k depending on the domi-nant chirality on the considered edge). Therefore, equilibra-tion is ensured for distances to downstream contact λ � δ,where �Vk ≈ 0. When λ is rescaled to the equilibration dis-tance δ, all voltage drops should collapse on a single curve andfall exponentially to zero. This is shown clearly in Fig. 4(a) forall the data presented in Fig. 3 (colors correspond to the fillingfactor combinations of Fig. 3).This rescaling suggests a correspondence between ourexperiment and a sample with counterpropagating channelsexperiencing charge equilibration over a continuous edge[14–16]. Indeed, the number of Landauer reservoirs may bemapped to a continuous edge’s length in units of is bareequilibration length, N ≡ L/�eq. This analogy is strengthenedby the similarity between Eq. (2) and the conductance of acontinuous edge sample as a function of its length L in thelimit of L � �eq [6,17]. As shown in Fig. 4(b), all reducedconductance data (see the Supplemental Material [27] for thefull conductance datasets for νA = (2, 6) and νB = 1 ↔ 7)(G2w − G∞)/G∞ collapse on a single curve coth(x/2) − 1,with x = (N + 1)/δ. For an effectively large sample, i.e.,N � δ, G2w converges exponentially toward its equilibratedvalue G∞, i.e., (G2w − G∞)/G∞ ∼ e−N/δ ≡ e−L/�̃eq . Here weVk-1 VkΔVk}FIG. 4. (a) Voltage drops between successive contacts, normal-ized to the maximum voltage drop �Vend, as a function of λ/δ, thedistance to the downstream contact normalized to the equilibrationdistance (see text). Solid line: e−λ/δ . (b) Normalized two-point con-ductance of the effective Hall bar in the “conductance” configuration,as a function of the scaled effective bar’s length. Solid line: applica-tion of Eq. (2). The dashed line corresponds to the asymptotic limitof a ballistic (L � �̃eq) regime, with N identified to be L/�eq.have operated a discrete-to-continuum correspondence, with�̃eq = �eq/| log(νA/νB)| the effective equilibration length thatabsorbs the contribution from filling factors measured in ourexperiment.We now discuss the relation with equilibration models usedfor hole-conjugate states [5,6,9,11,14], particularly ν = 2/3,in which charge transport exemplifies the νA �= νB case. Inthese models, the potential difference between counterprop-agating edges drives interchannel charge tunneling, leading toequilibration over an effective length �eq/|ν−1A − ν−1B | [10,14],with �eq the bare equilibration length that is usually obtainedin the Lüttinger liquid framework [6,17]. Our definition of�̃eq differs from this one, because the fundamentally differentequilibration mechanisms are at play (interedge tunneling vsreservoir redistribution). However, in the critical regime νA →νB, both definitions lead to the same divergence δ ∼ 1/|νA −νB|. In the diffusive case νA = νB, the two-point conductancein the macroscopic limit L � �eq behaves according to a uni-versal Ohmic scaling law, decaying algebraically: G2w ∝ 1/L[see Fig. 2(c)], irrespective of the microscopic mechanism ofequilibration [5,11,12,27]. This is reminiscent of the Ohmicdeviations to quantized transport observed in QSH samples[21], where the resistance scales roughly linearly with thesample’s length for large samples, and analogous to the trans-port of heat for the disordered ν = 2/3 edge, where thermalconductances are equal for charged and neutral modes, lead-ing to diffusive signatures in noise while charge transportremains ballistic [7,9,10,17,18,28,29]. In that respect, ν =2/3 is formally equivalent to the cases (2, 6) and (6, 2) thatL042037-4BALLISTIC-TO-DIFFUSIVE TRANSITION IN … PHYSICAL REVIEW RESEARCH 7, L042037 (2025)we study for charge, and would be equivalent to (2, 2) or(6, 6) for heat transport.In conclusion, our experiment explores the transitionbetween an effective ballistic edge transport (despite thepresence of counterpropagating edge states) and a criti-cal, scale-invariant diffusive regime. It conceptual simplicitymakes it applicable to a large variety of systems with counter-propagating edge channels: Beyond hole-conjugate fractionaland QSH states, it also encompasses the recent observation ofrobust quantization to the equilibrated value for the valley-polarized ν = −2 state in PbSnSe Dirac systems [30]. Apossible extension of this work is the investigation of the heatflow in such systems via noise measurements, e.g., to witnesscharge-heat separation with several fractional states, or whenadding interactions, as in the case of heat Coulomb blockade[31–33].We thank D. C. Glattli, P. Joyez, and C. Mora for helpfuldiscussions. This work was funded by the ERC (ERC-2018-STG QUAHQ), by the “Investissements d’Avenir”LabEx PALM (ANR-10-LABX-0039-PALM), and by theRegion Ile de France through the DIM QUANTIP. O.M.acknowledges funding from the ANR (ANR-23-CE47-0002CRAQUANT). 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