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Saurabh Kumar Srivastav, Ravi Kumar, Christian Spånslätt, [K. Watanabe](https://orcid.org/0000-0003-3701-8119), [T. Taniguchi](https://orcid.org/0000-0002-1467-3105), Alexander D. Mirlin, Yuval Gefen, Anindya Das

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[Determination of topological edge quantum numbers of fractional quantum Hall phases by thermal conductance measurements](https://mdr.nims.go.jp/datasets/6ba611e1-f4f4-4e55-ace6-74d6c583903c)

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Determination of topological edge quantum numbers of fractional quantum Hall phases by thermal conductance measurementsnature communicationsArticle https://doi.org/10.1038/s41467-022-32956-zDetermination of topological edge quantumnumbers of fractional quantum Hall phasesby thermal conductance measurementsSaurabh Kumar Srivastav 1,9, Ravi Kumar1,9, Christian Spånslätt 2,K. Watanabe 3, T. Taniguchi 3, Alexander D. Mirlin4,5,6,7, Yuval Gefen4,8 &Anindya Das 1To determine the topological quantum numbers of fractional quantum Hall(FQH) states hosting counter-propagating (CP) downstream (Nd) andupstream (Nu) edge modes, it is pivotal to study quantized transport both inthe presence and absence of edgemode equilibration.While reaching the non-equilibrated regime is challenging for charge transport, we target here thethermal Hall conductance GQ, which is purely governed by edge quantumnumbersNd andNu. Our experimental setup is realizedwith a hexagonal boronnitride (hBN) encapsulated graphite gated single layer graphene device. Fortemperatures up to 35mK, ourmeasuredGQ at ν = 2/3 and 3/5 (with CPmodes)match the quantized values of non-equilibrated regime (Nd +Nu)κ0T, whereκ0T is a quanta of GQ. With increasing temperature, GQ decreases and even-tually takes the value of the equilibrated regime ∣Nd −Nu∣κ0T. By contrast, atν = 1/3 and 2/5 (without CP modes), GQ remains robustly quantized at Ndκ0Tindependent of the temperature. Thus, measuring the quantized values ofGQ in two regimes, we determine the edge quantum numbers, which opens anew route for finding the topological order of exotic non-Abelian FQH states.In the quantumHall (QH) regime, transport occurs in one-dimensionalgapless edge modes, which reflect the topology of the bulk fillingfactor ν. In integer QH (IQH) states and in a certain subclass of frac-tional QH (FQH) states, only downstream edge modes (Nd of them)exist, whose chirality is dictated by the direction of the applied mag-netic field1,2. At the same time, the edge structure of a majority of FQHstates, including, in particular, the “hole-like” states (1/2 < ν < 1), ismorecomplicated. In addition to the downstreamedgemodes, the presenceof upstreammodes (Nu) leads to complex transport behavior1–6. In thissituation, the measured values of the electrical conductance (Ge)depends on the extent of the charge equilibration between thecounter-propagating downstream and upstreammodes. For example,the ν = 2/3 state hosts two counter-propagating modes: a downstreammode, ν = 1, and an upstream ν = 1/3 mode3. With full charge equili-bration, the two-terminal conductance Ge becomes7–10 2e2/3h; on theother hand, in the absence of charge equilibration, Ge is equal to8,104e2/3h. Theobservationof a crossover from4e2/3h to 2e2/3h is essentialto establish the proposed edge structure. This crossover has indeedbeenobserved in carefully engineered double-quantum-well structure,allowing control of the equilibration11. At the same time, a similardemonstration is lacking in experiments on a conventional edge (theboundary of a ν = 2/3 FQH state), whereGe is always found to be 2e2/3h.Received: 18 February 2022Accepted: 23 August 2022Check for updates1Department of Physics, Indian Institute of Science, Bangalore 560012, India. 2Department of Microtechnology and Nanoscience (MC2), Chalmers Universityof Technology, S-412 96 Göteborg, Sweden. 3National Institute of Material Science, 1-1 Namiki, Tsukuba 305-0044, Japan. 4Institute for Quantum Materialsand Technologies, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany. 5Institut für Theorie der Kondensierten Materie, Karlsruhe Institute ofTechnology, 76128 Karlsruhe, Germany. 6Petersburg Nuclear Physics Institute, 188300 St. Petersburg, Russia. 7L. D. Landau Institute for Theoretical PhysicsRAS, 119334 Moscow, Russia. 8Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel. 9These authors contributedequally: Saurabh Kumar Srivastav, Ravi Kumar. e-mail: anindya@iisc.ac.inNature Communications |         (2022) 13:5185 11234567890():,;1234567890():,;http://orcid.org/0000-0002-0498-4217http://orcid.org/0000-0002-0498-4217http://orcid.org/0000-0002-0498-4217http://orcid.org/0000-0002-0498-4217http://orcid.org/0000-0002-0498-4217http://orcid.org/0000-0001-6746-5433http://orcid.org/0000-0001-6746-5433http://orcid.org/0000-0001-6746-5433http://orcid.org/0000-0001-6746-5433http://orcid.org/0000-0001-6746-5433http://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/0000-0002-6310-1576http://orcid.org/0000-0002-6310-1576http://orcid.org/0000-0002-6310-1576http://orcid.org/0000-0002-6310-1576http://orcid.org/0000-0002-6310-1576http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-32956-z&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-32956-z&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-32956-z&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-32956-z&domain=pdfmailto:anindya@iisc.ac.inThe reason is that the small value of the charge equilibration lengthmakes it difficult to access the nonequilibrated regime. A smalldeviation from 2e2/3h indicating a beginning of the crossover towards4e2/3h was observed for the spin-unpolarized ν = 2/3 FQH state12.Measurements of the thermal conductance have recentlyemerged as a powerful tool to detect the edge structure of FQHstates13–18. Such measurements are highly useful for “counting” edgemodes and can also detect charge neutralMajoranamodes16,19. For IQHstates and FQH states with only downstream modes, the quantizedthermal conductance is given by GQ =Ndκ0T, where κ0 =π2k2B=3h, kB isthe Boltzmann constant, h is the Planck constant, and T is thetemperature14. A schematic illustration of the heat flow for such a state(ν = 1/3 in this example) is depicted in Fig. 1a. On the other hand, forhole-like FQH states, the presence of upstream modes renders thevalue of GQ strongly dependent on the extent of thermal equilibrationbetween CP modes. This leads to a crossover8 of GQ from a none-quilibrated quantized value of (Nd +Nu)κ0T to the asymptotic value offull equilibration ∣Nd −Nu∣κ0T. While the fully equilibrated and none-quilibrated limiting cases of GQ have been reported in disparate GaAs/AlGaAs based 2DEG devices15,16,20, and in graphene only the none-quilibrated values have been observed18, an in situ crossover of GQfrom the nonequilibrated to the fully equilibrated limit in a singledevice has remained unattainable.The observation of crossover in GQ has remained one of the long-standing challenges on the path to reveal the detailed edge structureof the FQH states. For example, for ν = 2/3, the by now “standard”model of the edge (based on the hierarchy construction21,22) suggestsone downstream and one upstream mode3,4,23. At the same time, a 2/3edge with two co-propagating downstream modes1 would also corre-spond to a fully legitimate FQH edge from the point of view of generaltheory23. For the first case, GQ should exhibit a crossover with tem-perature. In the nonequilibrated regime, L≪ ‘Heq, where L is the channellength and ‘Heq is the thermal equilibration length,GQ = 2κ0T, whereas inthe equilibrated regime, ‘Heq ≪ L, the GQ will exhibit asymptoticvalue ≈0κ0T. Such a crossover of GQ is schematically depicted inFig. 1b, c. On the other hand, for the second case, GQ will be inde-pendent of temperaturewith a value of 2κ0T. Similarly, the ν = 3/5 edgemodel corresponding to the hierarchy construction harbors one(a)(e)-ISIST0T0TMCGCGHot spot2 μmCASAgraphene boundary(d)Full thermal equilibrationNo thermal equilibrationS1R(b) (c)��������GQ = 2 κ0T�������GQ ≈ 0 κ0T��������GQ = 1 κ0Tdownstream mode (D)upstream mode (U)downstream mode (D)upstream mode (U)downstream mode (D)S2D1D2TM0.1 0.2 0.3 0.4VBG(V)0204060801000246810R(k)R(k) 3/52/31/32/51IS1,VS1IR,VS1IS1,VTIS1,VRB = 10 TTbath = 20 mKFig. 1 | Schematics of heat transport onQHedges,measurement setup, andQHresponse of device. a Heat transport at the edge of ν = 1/3 state along a singledownstream mode. The chirality of the downstream mode is clockwise. b Heattransport at the edge of ν = 2/3 state in nonequilibrated regime. Heat from the hotreservoir is carried awaybyboth downstreamandupstreammodes. The chirality ofupstream mode is anticlockwise. c Heat transport at the edge of ν = 2/3 state inthe equilibrated regime. The gradient of the color along the edges represents thequalitative temperature profile. In the long-length limit (L→∞), the heat carriedaway from the hot reservoir comes back to it via other edgemodes, which leads to avanishing thermal conductance. d False colored SEM micrograph of the device,shown with the measurement schematic. The graphene boundary is marked with awhite dashed line. For illustrative purposes, the device is depicted with a ν = 1 edgestructure. For thermal conductance measurements, currents IS and −IS are fedsimultaneously at contacts S1 and S2. Due to the power dissipation near the central,floating contact, the electron temperature increases to TM. The electrical andthermal conductances aremeasured respectively at low frequency (23Hz) and highfrequency (~740kHz) with an LCR resonant circuit. eQH response: The black line isthe resistance RS1 (VS1/IS1) measured at source contact ‘S1’ as a function of VBG atB = 10 T and temperature 20 mK. The blue line shows themeasured resistance (VR/IS1) at the contact ‘R’. The green line shows the measured resistance (VT/IS1) at thecontact ‘T’. The red curve shows the resistance VS1/IR measured at the contact ‘S1’,while the current is injected at the contact ‘R’ and encodes the longitudinal resis-tance. Robust fractional plateaus at 3he2 ,5h2e2,5h3e2, and3h2e2 are clearly visible. The legenddefines the current sources and voltage probes for each curve. The subscripts of Iand V correspond to the current-fed contact and the voltage-probe contact,respectively.Article https://doi.org/10.1038/s41467-022-32956-zNature Communications |         (2022) 13:5185 2downstream and two upstreammodes7,23, and as a resultGQwill have acrossover from 3κ0T to 1κ0T. However, there exist also alternativetopologies (encoded by so-called K-matrices23) corresponding to ν = 3/5. In particular, one can imagine a ν = 3/5 edge with three co-propagating downstream modes1, and in this scenario GQ would beindependent of temperature with a value of 3κ0T. Furthermore, thevalue of GQ can reveal the possible edge reconstruction of the QHstates24,25. For example, for ν = 1/3, the edge reconstruction by a pair ofcounter-propagating modes26 would increase the number of modesfrom 1 to 3, implying a crossover of GQ from 3κ0T at low T (none-quilibrated regime) to 1κ0T at higher T. Similarly, for ν = 2/3, the edgereconstruction would increase the number of modes27 from 2 to 4,which would result in GQ = 4κ0T at low temperature (nonequilibratedregime). Thus, the observation of crossover inGQ and its precise valuescan determine the exact topological number of the FQH edges.Achieving this goal would further help to settle the topological orderof more complex non-Abelian even-denominator FQH states.In this work, we report on thermal conductancemeasurements asa function of temperature (T) of electron-like (ν = 1/3 and 2/5) and hole-like (ν = 2/3 and 3/5) FQH states, realized in a hBN encapsulatedgraphite-gated high-mobility single layer graphene device. Our keyfindings are the following: (1) At the base temperature (lowest bathtemperature Tbath, ~ 20mK), GQ for 2/3 and 3/5 is found to be 2κ0T and3κ0T, respectively, and remain constant up to ~35mK. (2) With furtherincrease of temperature, GQ for 3/5 decreases, saturating at 1κ0T forT ≳ 50mK. The similar crossover of GQ is observed for 2/3 too and GQdrops to a value ~0.5κ0T at 60mK, continuing to decrease toward zero.The observed values ofGQmatches with the theoreticalmodels for thehole-like FQH states with CP modes from the nonequilibrated limit of(Nd +Nu)κ0T to the equilibrated limit of ∣Nd −Nu∣κ0T. For ν = 2/3, theheat transport in the equilibrated regime is of diffusive character, withthe limiting value ∣Nd −Nu∣κ0T ≈0 that is approached in a power-lawway as a function of temperature. (3) For 1/3 and 2/5 FQH states, GQ isfound to be 1κ0T and 2κ0T, respectively, independent of the electrontemperature and matches with the expected GQ =Ndκ0T without CPmodes. These observations further confirm that there is no edgereconstruction in our device.ResultsDevice schematic and responseTo measure the thermal conductance, we have used a graphite-gatedgraphene device, where the graphene is encapsulated between twohBN layers. The details of the device fabrication is described inMethods as well as in Supplementary Note 1. One of the importantlength scales of the device is the separation between the graphene andthe screening graphite layer, which is ~25 nm and comparable to themagnetic length scale. It has been theoretically predicted28 that forsuch cases edge reconstruction can be avoided (see SupplementaryNote 12). We will below show that our measured GQ confirm theabsence of edge reconstruction for our device. Similar to our previouswork17,18, our device consists of a small floating metallic reservoir,which is connected to graphene channel via one-dimensional edgecontacts, as shown in Fig. 1d. To measure the electrical conductance,we used the standard lock-in technique whereas the thermal con-ductance measurement was performed with noisethermometry15–18,29,30 (see Supplementary Fig. 2). In Fig. 1e, the blackcurve represents the measured resistance RS1 (VS1/IS1) at the sourcecontact (‘S1’) as a function of the graphite gate voltage (VBG). Welldeveloped plateaus appear at ν = 13,25,35, and23. The blue curve shows themeasured resistance RR = VR/IS1 along the reflected path (at contact ‘R’)from the floating contact. Similarly, the green curve shows the mea-sured resistance RT =VT/IS1 along the transmitted path (at contact ‘T’)from the floating contact. Measured resistances along the reflectedand transmitted paths are identical, and exactly half of the resistancemeasured at the source contact, suggesting equal partitioning ofinjected current to both the transmitted and reflected side (see Sup-plementary Note 4, and Supplementary Fig. 5). In fact, the equiparti-tion of the current on both sides of the floating contact in Fig. 1e firmlyestablishes two important points: (i) it rules out the presence of anyappreciable reflection coefficient at the interface of graphene andthe floating contact (see Supplementary Fig. 6 for details), and (ii) thepositions of the plateaus at the same gate voltage suggest the sameelectronic density on both sides of the floating contact. The red curvein Fig. 1e shows the resistance RS1 =VS1/IR measured at contact ‘S1’,while the current is injected from the contact ‘R’. This resistance in thisconfiguration has the same properties as a longitudinal resistance: inthe absence of bulk transport, the voltage VS1 is determined by theequilibriumpotential of the ground contact D1. The observation of thevanishing resistance plateaus further supports the formation of welldeveloped FQH states. In Supplementary Fig. 7, we show plots analo-gous to Fig. 1e but measured at elevated temperatures within ourworking temperature range—without detectable changes either inresistancevalues or in equipartition of currents. It should benoted thatthe measured resistance values in Fig. 1e at the source, reflected andtransmitted contacts suggest full charge equilibration in our device(see Supplementary Note 6 and Supplementary Table 2).Thermal conductance measurementIn contrast to our previous works17,18, to measure the thermal con-ductance, we simultaneously inject the DC currents IS and −IS at twocontacts S1 and S2, respectively. Both injected currents flow towardsthe floating reservoir. This is done in order to keep the potential of thefloating contact to be the same as that of all drain contacts. In thisconfiguration, the dissipated power at the floating reservoir due toJoule heating is given as P = I2SνG0(see Supplementary Note 3). Thispower dissipation leads to increase of the electron temperature in thefloating reservoir. The new steady state temperature TM is determinedby the heat balance relation15–18,29,31,32P = JQ = JeQðTM ,T0Þ+ Je�phQ ðTM ,T0Þ=0:5Nκ0ðT2M � T20Þ+ Je�phQ ðTM ,T0Þð1ÞHere, JeQðTM ,T0Þ is the electronic contribution of the heat current viaNchiral edge modes, and Je�phQ ðTM ,T0Þ is the heat loss via electron-phonon cooling, and T0 is the electron temperature of the coldreservoirs. The temperature TM is obtained by measuring the excessthermal noise15–18,29 along the outgoing edge channels using theNyquist-Johnson relationSI = νkBðTM � T0ÞG0 ð2ÞFor our hBN encapsulated graphite-gated device18, the electron-phonon contribution (second term in Eq. (1)) was found to benegligible for Tbath < 100mK (see Supplementary Note 9 and Supple-mentary Fig. 12). From Eq. (1), one finds N, which yields the soughtthermal conductanceGQ =Nκ0T. It shouldbe noted that Eq. (2) remainsvalid if there is a quasi-equilibrium state characterized by a hot Fermidistribution function with temperature TM, which is satisfied for ourdevice as the dwell time for the electrons in the metallic floatingcontact is longer than the electron–electron interaction time orthermalization time (see Supplementary Note 3). In Fig. 2, we show thedetailed procedure to extract the quantized GQ at the bathtemperature, Tbath ~ 20mK. Note that for each bath temperature, weexperimentally determine the electron temperature, T0 of the deviceand for our system Tbath ≈ T0 (see Supplementary Note 2, Supplemen-tary Fig. 3, and Supplementary Table 1).The measured excess thermal noise SI is plotted as a function ofcurrent IS for ν = 2/3 and 3/5 in Fig. 2a, d, respectively. The resultingheating of the floating reservoir is made manifest by the increase inexcess thermal noise with the application of the source current IS. TheArticle https://doi.org/10.1038/s41467-022-32956-zNature Communications |         (2022) 13:5185 3noise and current axes of Fig. 2a, d are converted to TM and JQ, yieldingFig. 2b for ν = 2/3 and Fig. 2e for ν = 3/5, respectively. To extractGQ, theheat current JQ is plotted as a function of T2M � T20 for ν = 1/3 (red) and2/3 (black) in Fig. 2c and for ν = 2/5 (red) and 3/5 (black) in Fig. 2f. Thesolid circles represent the experimental data, while the solid lines arethe linearfits withGQ = 1.00κ0T (red) and 2.01κ0T (black) for ν = 1/3 and2/3, respectively, in Fig. 2c and GQ = 2.02κ0T (red) and 3.02κ0T (black)for ν = 2/5 and 3/5, respectively, in Fig. 2f. To further study the tem-perature dependence of the thermal conductance, JQ is plotted as afunction of T2M � T20 at several values of the bath temperature for ν = 2/3 in Fig. 3a and for 3/5 in Fig. 3b. An analogous plot is shown for ν = 1/3(solid circles) and 2/5 (solid stars) in Fig. 3c. The slopes of the linear fitsto thedata in thesefigures allowus to extract the values ofGQ.Whereasthe data for the 2/3 and 3/5 states show an explicit dependence of GQon bath temperature, the thermal conductance remains independentof the temperature for the 1/3 and 2/5 states, Fig. 3c. Note that thethermal conductance measurement was performed at the middle ofeach QH plateau.In Fig. 4a,weplot the thermal conductanceGQ (extracted fromtheslope of the linear fits to the data in Fig. 3 as a function of the bathtemperature for ν = 1/3 (red), 2/5 (blue), 2/3 (magenta), and 3/5 (black).As can be seen in Fig. 4a for ν = 1/3 (red) and 2/5 (blue), the values GQFig. 2 | Thermal conductance of fractionalQH states. a Excess thermal noise SI asa function of source current IS at ν = 2/3. The DC currents IS and −IS were injectedsimultaneously at contacts S1 and S2, respectively, as shown in Fig. 1d. b Thetemperature TM of the floating contact as a function of the dissipated power JQ atν = 2/3. c JQ (solid circles) is plotted as a function of T2M � T20 at ν = 2/3 (black) and 1/3 (red). Solid black and red lines are linear fits with GQ = 2.01κ0T and 1.00κ0T forν = 2/3 and 1/3, respectively. d Excess thermal noise SI as a function of sourcecurrent IS at ν = 3/5.eThe temperatureTMof thefloating contact as a function of thedissipated power JQ at ν = 3/5. f JQ (solid circles) is plotted as a function of T2M � T20for ν = 3/5 (black) and 2/5 (red). Solid black and red lines are linear fits with GQ =3.02κ0T and 2.02κ0T for ν = 3/5 and 2/5, respectively . The black and dashed redarrows depict the downstream and upstream modes, respectively, for each edgestructure.Fig. 3 | Temperature dependence of thermal conductances. a, b JQ (solid circles)is plotted as a function of T2M � T20 at ν = 2/3 (a) and ν = 3/5 (b) at several values ofthe bath temperature. Solid circles show the experimental data,while solid lines arelinear fits to these experimental data points. Different colors correspond to dif-ferent bath temperatures as shown in the legend. c JQ (solid circles) is plotted as afunction of T2M � T20 for ν = 1/3 (filled circles) and ν = 2/5 (filled stars) at severalvalues of the bath temperature. Different colors of the symbols correspond todifferent bath temperatures, (see legend). For all panels, the thermal conductanceGQ at each temperature is extracted from the slope of the linear fit.Article https://doi.org/10.1038/s41467-022-32956-zNature Communications |         (2022) 13:5185 4(1κ0T and 2κ0T, respectively) remain independent of the bath tem-perature. The samebehavior is found for integerQH states:GQ remainsconstant with temperature (see Supplementary Note 11 and Supple-mentary Fig. 13). On the other hand, for the hole-like 3/5 state, at thelowest bath temperature (Tbath ~ 20mK), we observe GQ ~ 3κ0T, whichremain constant up to Tbath ~ 35mK and with further increase of thetemperature, theGQdecreases and saturates to ~1κ0T forTbath ≳ 50mK.A similar crossover is observed also for the 2/3 state. For this state, atlow temperatures, GQ ~ 2κ0T is observed. When the temperatureincreases beyond 35mK, GQ starts decreasing and drops down to avalue of ~0.5κ0T at our largest temperature, Tbath ~ 60mK.To understand these results, we show the expected edge struc-tures and their corresponding thermal conductance values for thestudied FQH states in Fig. 4b. For the electron-like 1/3 and 2/5 states,there are only downstreammodes with Nd = 1 and 2, respectively, andthus, the expected GQ should be 1κ0T and 2κ0T, respectively, and willremain independent of the temperature. This is indeed seen for ourexperiment in Fig. 4a. This behavior is analogous to that for integer QHstates (See Supplementary Note 11 and Supplementary Fig. 13), whereall edgemodes also propagate downstream.On the other hand, for thehole-like 3/5 state, the temperature dependence crossover of GQ fromone quantum value to another one rules out any possibility of havingonly downstreammodes. Furtheromore, the measured values of 3κ0Tand 1κ0T, respectively, perfectly match with the nonequilibrated((Nd +Nu)κ0T) and equilibrated (∣Nd −Nu∣κ0T) regimes of GQ with Nd = 1and Nu = 2. Similarly, for 2/3, our observation rules out the theoreticalmodel with only downstreammodes, and support the crossover fromthe nonequilibrated regime of GQ to the equilibrated regime withNd =Nu = 1. The equilibrated transport in this situation is diffusive innature, so that GQ is expected to tend to zero relatively slowly (as ~1/L)in the long-length limit. Since our device channel length L is limited to~5μm, we observe a finite value of ~0.5κ0T at Tbath ~ 60mK.Approaching substantially closer the asymptotic value of 0κ0T for 2/3would be very interesting but it is not a simple task. For a given lengthL, this would require a further increase of temperature. However, wefind that then the electron-phonon cooling starts to contribute sig-nificantly, spoiling the analysis (See Supplementary Note 9 and Sup-plementary Fig. 12).Thus, measuring the quantized values of GQ at the two regimeshelps to experimentally determine the topological edge quantumnumbers. We note that, in our previous work30, the noise measure-ment confirmed the presence of CPmodes for hole-like FQH states ingraphene. At the same time, the approach of ref. 30 was unable todetect exact topological edge quantum numbers. Furthermore, inthe present study, the low-temperature values of GQ rules out anyedge reconstruction in our device for all of the QH states studied. Forexample, for a 1/3 edge, one would observe a crossover from 3κ0T to1κ0T in the presence of edge reconstruction. We find, however, 1κ0Tdown to lowest T, demonstrating that the edge reconstruction is notoperative. Similarly, for a 2/3 edge, the edge reconstruction wouldincrease the total number of edge modes from 2 to 4, and conse-quently would give rise to 4κ0T value in the low-T limit instead of theobserved 2κ0T.According to theoretical predictions, the crossover of GQbetween the asymptotic limits of no thermal equilibration (L≪ ‘Heq)and perfect thermal equilibration (L≫ ‘Heq) is described by a functionof the dimensionless ratio L=‘Heq, with the thermal equilibrationlength scaling as a power of temperature, ‘Heq / T�p. Explicit forms ofthe crossover functions for ν = 2/3 and ν = 3/5 states are given belowinMethods. Our experimental data are well described by these formsas shown by the solid lines in Fig. 4a. At the same time, the values ofthe exponent p that are obtained from the fits turn out to be unex-pectedly large: p = 6.3 for ν = 2/3 and p = 9.3 for ν = 3/5, well abovep = 2 expected in the vicinity of the strong-disorder fixed points6–8.This implies that the crossover GQ(T) is surprisingly sharp as afunction of temperature. While this observation remains puzzling atthis stage, several plausible equilibration mechanisms that mightyield a large p are discussed in the next section. It is worth noting thatthe asymptotic limit of GQ = 0κ0T in the equilibrated regime for 2/3 state is expected to be achieved (within our measurement accu-racy) around Tbath ~ 140mK [obtained by extending the fittedmagenta curve in Fig. 4a], which is virtually impossible to experi-mentally measure due to strongly enhanced electron-phonon cool-ing as mentioned above.DiscussionIn this section, we discuss a few additional points related to theexpected theoretical regimes of equilibration, the accuracy of ourmeasurement, the large temperature exponents of the thermal equi-libration lengths, and future implications of our observations.(a) (b)Filling Factor��� ��Edge StructureThermal conductance GQNo equilibration Full equilibration GQ = (Nd + Nu)��
 GQ = |Nd - Nu|����� � ���� ���� � ���� � �‘0’GQ (0T)20 30 40 50 60Tbath (mK)00.511.522.533.5 ����������������������������Fig. 4 | Crossover from nonequilibrated to equilibrated heat transport.a Thermal conductance GQ, as extracted from the slope of the linear fit in Fig. 3,plotted as a function of the bath temperature for ν = 1/3 (red), 2/5 (blue), 3/5 (black),and 2/3 (magenta). The horizontal dashed lines correspond to quantized values ofGQ. The solid curves (black andmagenta) are theoreticalfits of thedata that serve toextract out temperature scaling exponents (seeMethods). Error bars correspond tothe standard deviation associated with the slope of the linear fit shown in Fig. 3.b Edge structures of the studied FQH states. Solid black and dashed red arrowsrepresent downstream and upstream modes, respectively. The two right-mostcolumns show expected values of the thermal conductance GQ (in units of κ0T) inthe two limiting regimes of the heat transport.Article https://doi.org/10.1038/s41467-022-32956-zNature Communications |         (2022) 13:5185 5(1) The quantized value GQ = (Nd +Nu)κ0T of the thermal con-ductance in the nonequilibrated regime, L≪ ‘Heq, where ‘Heq is thethermal equilibration length, strictly holds if there is no back-scattering of heat at interfaces with contacts. This is fulfilled underan additional condition L≪ LT where LT ~ T−1 is the thermal length. Inthe intermediate regime LT ≪ L≪ ‘Heq, a correction to this value isexpected to emerge8,20,33. Thus, the nonequilibrated regime may, infact, be expected to be split into two plateaus, which is, however, notobserved in our experiment.(2) The experimental determination of the thermal conductancefollows the approach of several preceding works that use two implicitassumptions: (i) current fluctuations propagating from the centralcontact satisfy the thermal equilibrium distribution, implying theJohnson-Nyquist relation between the contact temperature and thenoise; (ii) all power dissipated close to the central contact heats it.When all modes propagate downstream, both these assumptionsstrictly hold. However, for edges with CP modes, the situation may besomewhatmore delicate and somedeviations from the assumptions (i)and (ii) may emerge. This issue was discussed in ref. 20, where cor-rections to the procedure of extraction of GQ were obtained thatslightly reduce the experimental value of GQ. We do not include thesecorrections in the present work. First, they would not affect the iden-tification of the asymptotic regimes. Second, the values of GQ that wefind without including these corrections agree remarkably with thequantized values, both for the nonequilibrated regime (as was alsofound for bilayer graphene in ref. 18) and in the equilibrated limit. Itremains to see which features of our device favor this remarkableagreement.Wewould like to note that the precise determination ofGQdepends on the accuracy of electron temperature and gain of theamplification chain, which are shown in details in SupplementaryNote 2 and Supplementary Fig. 3.(3) It was pointed out above that the temperature-driven cross-over from nonequilibrated to equilibrated regime is remarkably sharpin our experiment, i.e., the parameter p controlling the scaling of theequilibration length (‘Heq / T�p) is unusually large. Theoretically, thevalue of p is controlled by irrelevant operators within therenormalization-group framework. Various mechanisms correspond-ing to such operators are known that may lead to large values of p incorrelated 1D systems. In particular, this may happen if the energyrelaxation is dominated by complex (multiparticle) interchannelprocesses7,23,34 or by nonlinearities of the quasiparticle and plasmonspectrum at the edge35–41. We leave a detailed investigation of this issuein the present context to future research.(4) Observing a crossover of the thermal conductance betweentwo asymptotic limits of the thermal equilibration is an important steptowardpinpointing the topological order ofmorecomplex FQH states.Of particular interest is the ν = 5/2 state, whose topological order is asubject of active debate. Specifically, the anti-Pfaffian state shoulddemonstrate a crossover from 4.5 to 1.5κ0T, whereas the PH-Pfaffianstate should demonstrate a crossover from 3.5 to 2.5κ0T. For thePfaffian state, GQ = 3.5κ0T independent of temperature.The findings of this work are a notable manifestation of an inter-play of equilibration (or absence thereof) and topology in FQH trans-port. While the charge transport is in the equilibrated regime, the heattransport crosses over from the nonequilibrated to equilibratedregime, with both asymptotic limits characterized by topologicallyquantized heat conductances determined by edge quantum numbers.We expect that this physics should be relevant also to other FQH statesand materials. In particular, interpretation of the experimentallymeasured thermal conductance 52 κ0T at the non-Abelian ν = 5/2 staterequires assumptions about the presence, absence, or partial characterof thermal equilibration42–46. Measurement of the full crossover fromthe nonequilibrated to equilibrated regime would permit to unam-biguously resolve this problem.MethodsDevice fabrication and measurement schemeIn our experiment, an encapsulated device (heterostructure of hBN/single layer graphene(SLG)/hBN/graphite) was made using the stan-dard dry transfer pickup technique47. Fabrication of this hetero-structure involvedmechanical exfoliation of hBN and graphite crystalson oxidized silicon wafer using the widely used scotch tape technique.First, a hBN of thickness of ~25 nmwas picked up at 90 °C using a Poly-Bisphenol-A-Carbonate (PC) coated Polydimethylsiloxane (PDMS)stamp placed on a glass slide, attached to tip of a home built micro-manipulator. This hBN flake was aligned on top of previously exfo-liated SLG. SLG was picked up at 90 °C. The next step involved thepickup of bottomhBN (~25 nm). This bottomhBNwas picked up usingthe previously picked-up hBN/SLG following the previous process.This hBN/SLG/hBN heterostructure was used to pick up the graphiteflake following the previous step. Finally, this resulting hetros-tructure (hBN/SLG/hBN/graphite) was dropped down on top of anoxidized silicon wafer of thickness 285 nm at temperature 180 °C. Toremove the residues of PC, this final stack was cleaned in chloroform(CHCl3) overnight followed by cleaning in acetone and iso-propylalcohol (IPA). After this, Poly-methyl-methacrylate (PMMA) photo-resist was coated on this heterostructure to define the contactregions in the Hall probe geometry using electron beam lithography(EBL). Apart from the conventional Hall probe geometry, we defineda region of ~5.5 μm2 area in the middle of SLG flake, which acts asfloating metallic reservoir upon edge contact metallization. AfterEBL, reactive ion etching (mixture of CHF3 and O2 gas with flow rateof 40 sccm and 4 sccm, respectively at 25 °C with RF power of 60W)was used to define the edge contact. The etching time was optimizedsuch that the bottom hBN did not etch completely to isolate thecontacts from bottom graphite flake, which was used as the backgate. Finally, thermal deposition of Cr/Pd/Au (3/12/60 nm) was donein an evaporator chamber having base pressure of ~1–2 × 10−7 mbar.After deposition, a lift-off procedure was performed in hot acetoneand IPA. This results in a Hall bar device along with the floatingmetallic reservoir connected to the both sides of SLG by the edgecontacts. The schematics of the device and measurement setup areshown in Fig. 1d. The distance from the floating contact to the groundcontacts was ~5 μm (see Supplementary Fig. 1 for optical images). Allmeasurements were done in a cryo-free dilution refrigerator having abase temperature of ~20mK. The electrical conductance was mea-sured using the standard lock-in technique, whereas the thermalconductancewasmeasuredwith noise thermometry basedon an LCRresonant circuit at resonance frequency ~740 kHz. The signal wasamplified by a home-made preamplifier at 4 K followed by a roomtemperature amplifier, and finally measured by a spectrum analyzer.Details of the measurement technique are discussed in the Supple-mentary Fig. 2.Description of the crossover from the nonequilibrated to equi-librated regimeWhen edge modes are not thermally equilibrated, i.e. for edge lengthsL satisfying L≪ ‘Heq, the thermal conductance becomes quantized asGQ = ðNd +NuÞκ0T , ð3Þwhichmeans that every edgemodegives a contribution 1κ0T toGQ. Forfilling factors ν = 1/3, ν = 2/5, ν = 2/3, and ν = 3/5, the correspondingvalues of the thermal conductance are GQ/κ0T = 1, 2, 2, and 3, respec-tively. In fact, the validity of Eq. (3) requires that L also satisfies L≪ LT,where LT ~T−1 is the thermal length. In the intermediate regimeLT ≪ L≪ ‘Heq, a correction to this value emerges due to back-scatteringof heat at interfaces with contacts8,20,33. For the sake of simplicity, wediscard this correction in our analysis in the present work.Article https://doi.org/10.1038/s41467-022-32956-zNature Communications |         (2022) 13:5185 6In the regime of full thermal equilibration, L≫ ‘Heq, the thermalconductance becomes topologically quantized asGQ = ∣Nd � Nu∣κ0T : ð4ÞFor ν = 1/3 and 2/5we haveNu = 0, so that Eqs. (3) and (4) coincide.For such FQH edges, with only downstream modes, the thermal con-ductance is thus predicted to be GQ =Ndκ0T, independent of tem-perature. This is exactly what is observed in our experiment. On theother hand, for FQH edges with CP modes, i.e., with Nu >0, the equi-librated value (4) is smaller than the nonequilibrated value (3), so thatthere is a nontrivial crossover of GQ between the two limits. This is thecase for ν = 2/3 and ν = 3/5.For ν = 3/5, we have Nd = 1 and Nu = 2, so thatGQ/κ0T = 1. It is worthnoting that in this case, Nd −Nu = − 1, implying that the heat flowsupstream on the equilibrated edge, i.e., against the charge flowdirection. However, the present experimental setup onlymeasures theabsolute value of GQ and does not reveal the heat flow direction onindividual edge segments. The crossover function between the none-quilibrated and equilibrated regime is found to be8,9,18GQκ0T=2 + e�L=‘Heq2� e�L=‘Heq=2 + e�kTp2� e�kTp , ð5Þwhere L=‘Heq = kTp. Fitting our experimental data to Eq. (5) with fitparameters k and p, we obtain p ≈ 9.34 (in Fig. 4a).For the ν = 2/3 state, we have Nd =Nu = 1, so that the equilibratedlimiting value of GQ, Eq. (4), is zero. In this case, the crossover takesplace between ballistic heat transport in the nonequilibrated regimeand heat diffusion in the equilibrated regime8,9,18:GQκ0T=2‘HeqL+ ‘Heq=21 + kTp : ð6ÞFitting the experimental data to this form, we get the exponentp ≈ 6.34 (in Fig. 4a).Data availabilityAdditional information related to this work is available from the cor-responding author upon reasonable request. Source data are providedwith this paper.References1. Beenakker, C. Edge channels for the fractional quantum hall effect.Phys. Rev. Lett. 64, 216 (1990).2. Wen, X.-G. Chiral luttinger liquid and the edge excitations in thefractional quantum hall states. Phys. Rev. B 41, 12838 (1990).3. MacDonald, A. H. Edge states in the fractional-quantum-hall-effectregime. Phys. Rev. Lett. 64, 220–223 (1990).4. Johnson, M. & MacDonald, A. Composite edges in the ν = 2/3fractional quantum hall effect. Phys. Rev. Lett. 67, 2060 (1991).5. Wen, X.-G. Theory of the edge states in fractional quantum halleffects. 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C.S.acknowledges funding from the Excellence Initiative Nano at the Chal-mers University of Technology and the 2D TECH VINNOVA competenceCenter (Ref. 2019-00068). This project has received funding from theEuropean Union’s Horizon 2020 research and innovation programmeunder grant agreement No 101031655 (TEAPOT). K.W. and T.T.acknowledge support from the Elemental Strategy Initiative conductedby the MEXT, Japan and the CREST (JPMJCR15F3), JST.Author contributionsS.K.S. and R.K. contributed to device fabrication, data acquisition, andanalysis. A.D. contributed in conceiving the idea and designing theexperiment, data interpretation, and analysis. K.W. and T.T. synthesizedthe hBN single crystals. C.S., A.D.M., and Y.G. contributed in develop-ment of theory, data interpretation, and all the authors contributed inwriting the manuscript.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-022-32956-z.Correspondence and requests for materials should be addressed toAnindya Das.Peer review information Nature Communications thanks the anon-ymous reviewers for their contribution to the peer review of thiswork. 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To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2022Article https://doi.org/10.1038/s41467-022-32956-zNature Communications |         (2022) 13:5185 8https://doi.org/10.1038/s41467-022-32956-zhttp://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/ Determination of topological edge quantum numbers of fractional quantum Hall phases by thermal conductance measurements Results Device schematic and response Thermal conductance measurement Discussion Methods Device fabrication and measurement scheme Description of the crossover from the nonequilibrated to equilibrated regime Data availability References Acknowledgements Author contributions Competing interests Additional information