# Fileset [s41467-024-52927-w.pdf](https://mdr.nims.go.jp/filesets/288548e1-b287-44f5-ba8f-6c9267319881/download) ## Creator Simrandeep Kaur, [Tanima Chanda](https://orcid.org/0009-0003-3516-8890), Kazi Rafsanjani Amin, Divya Sahani, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Unmesh Ghorai, [Yuval Gefen](https://orcid.org/0009-0002-4553-6039), G. J. Sreejith, [Aveek Bid](https://orcid.org/0000-0002-2378-7980) ## Rights [Creative Commons BY-NC-ND Attribution-NonCommercial-NoDerivs 4.0 International](https://creativecommons.org/licenses/by-nc-nd/4.0/) ## Other metadata [Universality of quantum phase transitions in the integer and fractional quantum Hall regimes](https://mdr.nims.go.jp/datasets/33b536cf-b759-4c57-8963-b7b2d101ac3a) ## Fulltext Universality of quantum phase transitions in the integer and fractional quantum Hall regimesArticle https://doi.org/10.1038/s41467-024-52927-wUniversality of quantum phase transitions inthe integer and fractional quantum HallregimesSimrandeep Kaur1,8, Tanima Chanda 1,8, Kazi Rafsanjani Amin2,8, Divya Sahani1,Kenji Watanabe 3, Takashi Taniguchi 4, Unmesh Ghorai5, Yuval Gefen 6,G. J. Sreejith7 & Aveek Bid 1Fractional quantum Hall (FQH) phases emerge due to strong electronicinteractions and are characterized by anyonic quasiparticles, each dis-tinguished by unique topological parameters, fractional charge, and statistics.In contrast, the integer quantumHall (IQH) effects can be understood from theband topology of non-interacting electrons. We report a surprising super-universality of the critical behavior across all FQH and IQH transitions. Con-trary to the anticipated state-dependent critical exponents, our findings revealthe same critical scaling exponent κ = 0.41 ± 0.02 and localization lengthexponent γ = 2.4 ± 0.2 for fractional and integer quantum Hall transitions.From these, we extract the value of the dynamical exponent z ≈ 1. We haveachieved this in ultra-high mobility trilayer graphene devices with a metallicscreening layer close to the conduction channels. The observation of theseglobal critical exponents across various quantum Hall phase transitions wasmasked in previous studies by significant sample-to-sample variation in themeasured values of κ in conventional semiconductor heterostructures, wherelong-range correlated disorder dominates. We show that the robust scalingexponents are valid in the limit of short-range disorder correlations.The quantumHall (QH) effect, observed in a two-dimensional electrongas subject to a perpendicular magnetic field, realizes multiple quan-tum phase transitions (QPT) between distinct insulating topologicalstates1. The magnetic field B quenches the electronic kinetic energyinto disorder-broadened discrete Landau energy levels (LL). All elec-tronic single-particle states are localized, barring those at a specificcritical energy Ec near the center of each LL, which are extended2–7.When the Fermi energy lies between the extended states of two suc-cessive LLs, the system is in a distinct topological phase characterizedby a quantized value of Hall resistance Rxy and vanishingly smalllongitudinal resistance Rxx. As the Fermi energy approaches Ec, thelocalization length ξ characterizing the single-particle states divergesas ξ ~ ∣E−Ec∣−γ while the slowest time-scale diverges as τ ~ ξz ~ ∣E−Ec∣−zγ8,9.The exponent γ governs the critical divergence of the localizationlength as the filling fraction or magnetic field approach the criticalvalues and z governs the divergence of the coherence length withdecreasing temperatures10. From the finite-size scaling theory10,11,dRxy=dνν = νc / T�1=zγ ð1ÞReceived: 6 May 2024Accepted: 24 September 2024Check for updates1Department of Physics, Indian Institute of Science, Bangalore 560012, India. 2Department of Microtechnology and Nanoscience, Chalmers University ofTechnology, 412 96Gothenburg, Sweden. 3Research Center for Electronic andOptical Materials, National Institute for Materials Science, 1-1 Namiki, Tsukuba305-0044, Japan. 4Research Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan.5Department of Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India. 6Department of CondensedMatterPhysics, Weizmann Institute of Science, Rehovot 76100, Israel. 7Indian Institute of Science Education and Research, Pune 411008, India. 8These authorscontributed equally: Simrandeep Kaur, Tanima Chanda, Kazi Rafsanjani Amin. e-mail: aveek@iisc.ac.inNature Communications | (2024) 15:8535 11234567890():,;1234567890():,;http://orcid.org/0009-0003-3516-8890http://orcid.org/0009-0003-3516-8890http://orcid.org/0009-0003-3516-8890http://orcid.org/0009-0003-3516-8890http://orcid.org/0009-0003-3516-8890http://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/0009-0002-4553-6039http://orcid.org/0009-0002-4553-6039http://orcid.org/0009-0002-4553-6039http://orcid.org/0009-0002-4553-6039http://orcid.org/0009-0002-4553-6039http://orcid.org/0000-0002-2378-7980http://orcid.org/0000-0002-2378-7980http://orcid.org/0000-0002-2378-7980http://orcid.org/0000-0002-2378-7980http://orcid.org/0000-0002-2378-7980http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-52927-w&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-52927-w&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-52927-w&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-52927-w&domain=pdfmailto:aveek@iisc.ac.inwww.nature.com/naturecommunicationsHere, ν = nh/eB, n is the areal charge-carrier density, h is the Planckconstant, e is the electronic charge, and T is the temperature. Oneadditionally defines the scaling exponent κ = 1/zγ11–13 that governs thistemperature dependence of the slope of Rxy as well as the width of theRxx peak at the transition. The values of these three critical exponents(of which only two are independent) have been argued to be universal,with γ ≈ 2.3, κ ≈0.42, and z = 1 for all IQH transitions4,10,11,14,15.Low temperatures and high magnetic fields enhance the effectiveelectron-electron interactions, producing a richer set of the fractionalquantum Hall (FQH) phases at rational filling fractions16. The questionthen arises: Can IQH and FQH phase transitions be analyzed using a‘unified’ scaling framework17? While the IQH phases originate from thetopology of the single particle electronic Chern bands18, the FQHphasesare crucially underlain by strong electronic interactions. These aremarked by distinct electronic correlations, topological order, groundstate degeneracy, and topological entanglement. The transitionbetween FQH plateaus is driven by a proliferation of anyonic quasi-particles (characterizedbyquasiparticle statistics and fractional charge).This picture may suggest that the critical behavior at the transitionsdepends on the specifics of the topological FQH states involved and isalso different from the analogous transitions in the IQH regime.Experimental investigations of scaling in the IQH regime havereported κ varying between 0.16 ≤ κ ≤0.81 (Supplementary Informa-tion, Supplementary Note 13). This wide variation has been attributedto varying disorder correlation lengths with a universal critical beha-vior seen only in samples with short-range disorder19,20. This lack of atight constraint on κ has hindered any claims of their universality.Similar experimental investigations of scaling laws at transitionsbetween FQH phases are scarce21–23. A recent experimental study onextremely high-mobility 2D electron gas confined to GaAs quantumwells found the value of κ in the FQH regime to be non-universal, thisobservation being attributable to long-range disorder correlation23.Thus, despite over three decades of study, the fundamental questionof the values of the critical exponents across quantum Hall transitions(integer and fractional) remains unsettled14,23,24.This article reports the experimental observation of a surprisingsuper universality in the scaling exponents for transitions betweenvarious IQHand FQHphases in trilayer graphene.Wemeasureboth thescaling exponent κ and the localization length exponent γ indepen-dently over several integer-to-integer, integer-to-fractional, andfractional-to-fractional Quantum Hall transitions. Contrary to theexpected picture of multiple plateau-to-plateau quantum phase tran-sitions, each with its own distinct critical properties, here we find thatfor all IQH and FQH plateau-to-plateau transitions (PT), κ = 0.41 ± 0.02,γ ≈ 2.4 ± 0.2, and z ≈ 1, closely aligned with the predictions of the scal-ing theory of localization13. Given the distinct origins of the two phe-nomena, this striking similarity of the critical exponents suggests aconnection between the IQH and FQH effects that transcends thecomposite fermion (CF) framework.We estimate the values of κ near criticality (ν≈ νc) using three dis-tinct approaches: (i) analyzing the critical divergence of dRxy/dν, (ii)probing the critical divergence of the inverse width of Rxx(T), and (iii) ascaling analysis of Rxy near the critical point. The localization exponent γis obtaineddeep in the tails of the localized regime fromthedependenceof Gxx on ν. A scaling analysis of Quantum Hall transitions for fractionaland integer states provides a second, independent way to extract γ.The realizations of these quantum phase transitions in graphene-based systems are associated with a highly tunable set of parameters.These include the ability to alter electron density, which is typicallyunachievable in semiconductor heterostructures25, the capability tomanage screening, and the option to induce bandmixing by applying adisplacementfieldD. Thisflexibility helps us establish thatweak Landaulevel mixing does not significantly affect these critical exponents.Graphene also provides a platform where the nature of disorderscattering can be controlled. This is because the electrical transportproperties of high-mobility graphene devices are dominated by short-range impurity scattering, while those of low-mobility graphene devi-ces are controlled by both short-ranged and long-ranged scatteringpotentials26,27. Thus, high-mobility graphene devices represent a nat-ural candidate to investigate the universality of scaling exponents. Ourcomparative study between graphene devices of varying mobilityshows that as long as long-range impurity scattering can be sup-pressed, the universality of scaling parameters persists, independentof the quantum Hall bulk phases involved.ResultsStandard dry transfer technique is used for the fabrication of dualgraphite-gated hexagonal-boron-nitride (hBN) encapsulated TLGdevices (Fig. 1a) (for details, see Supplementary Information, Supple-mentary Note 1)28. Figure 1b shows measurements of the longitudinalresistance Rxx and the transverse conductance Gxy versus the Landaulevel filling factor ν; the measurements were performed at B = 13 T,Fig. 1 | FQH in Bernal-stacked TLG. a Device schematic of TLG encapsulatedbetween two hBN and few-layer graphite flakes. b Line plots ofGxy (left-axis; solid redline) and Rxx (right-axis; solid blue line) versus νmeasured at B= 13T, T=20mK, andD=0V/nm. The dashed vertical lines mark the FQH states formed at correspondingν, and the arrows indicate corresponding plateaus inGxy. c Calculated band structureof Bernal stacked trilayer graphene for D=0V/nm. The four LLs of the NM=0 (TheMLG LLs are marked by the subscripts M, and orbital contents are given by thenumbers 0) band are indicated schematically. d Calculated Landau levels as a func-tion of energy E andB forD=0V/nm. The blue lines are themonolayer-like LLs, whilethe red lines are the bilayer-like LLs. The solid and dotted lines indicate the LLs fromKand K 0-valley, respectively. The solid-green line is the spin-degenerate NM=0−↑ andNM=0−↓ monolayer-like LLs that host the FQH states probed in this article.Article https://doi.org/10.1038/s41467-024-52927-wNature Communications | (2024) 15:8535 2www.nature.com/naturecommunicationsT = 20mK, and D = 0V/nm. We identify several major odd denomi-nator FQH states by prominent dips in Rxx and corresponding plateausin Gxy. Indications of developing ν = 3 + 1/5 and 3 + 2/7 states are alsoseen. Several of these FQH states are resolved at B = 4.5 T, attesting tothe high quality of the device in terms of excellent homogeneity ofnumber density and suppression of long-range scattering (Supple-mentary Information, Supplementary Note 6).The band structure of TLG is formed of monolayer-like andbilayer-like Landau levels (Fig. 1c)—these are protected frommixing bythe lattice mirror-symmetry29. The calculated LL spectrum as a func-tion of B and energy E is shown in Fig. 1d, where blue (red) lines markthe monolayer-like (bilayer-like) LLs. For B > 8T, the ν = 2 and ν = 3arise from the spin-split NM=0−↑ and NM=0−↓ bands of themonolayer-like LLs. Here, (+ , −) refers to the two valleys, and (↑, ↓)refers to electronic spins. We confine our study to 8 T νc). From the plot of α versus T (inset of Fig. 2e) we obtainκ = 0.40 ± 0.03.To check the validity of our scaling analysis, we perform the fol-lowing error analysis: The residue in the least square fit between thescaling curves (like those shown in Fig. 2e) for each assumed value of κis calculated. This quantity, which we call fit error’, is presentedin Supplementary Information Supplementary Fig. 6 and Supplemen-tary Fig. 7 in a semi-log scale; we find that the fit error is indeedminimum for κ =0.41.Figure 3a compiles our findings. These results indicate a κ valueof 0.41 ± 0.03 uniformly observed across all probed transitionsbetween IQH and FQH states (compare with Supplementary Fig. 14of Supplementary Information). This consistency in scaling expo-nents spans various transition types, including (1) transitions fromone IQH state to another, (2) transitions among different FQH states,and (3) transitions between an IQH state and a neighboring FQHstate. It is important to emphasize that the observed universality of κgoes beyond marking an experimental confirmation of a uniformscaling law across FQH transitions in any material. Given the distinctphysics of IQH and FQH states, such constancy of the scaling expo-nent is remarkable and underscores the universal applicability of thescaling principle across QH transitions. This is the central result ofthis article.Locating the transitionThe physics of the FQH effect of electrons at a filling factor ν can bemapped onto that of IQH of CF at a filling factor νCF, with ν = νCF/(2νCF ± 1)33. It follows that the critical points for the transition betweensuccessive FQH phases at ν = νCF/(2νCF ± 1) and ν = (νCF + 1)/(2(νCF + 1)±1) occur at14,34:νc =ðνCF +0:5Þ2ðνCF +0:5Þ± 1: ð3ÞThe experimentally obtained values of νc, extracted either from thecrossing point of the Rxy isotherms or the maxima of Rxx, matchexceptionally well with the theoretical predictions (Fig. 3b) (Supple-mentary Information, Supplementary Table 1).Robustness of the critical exponents against LL mixingA non-zero vertical displacement fieldD gives rise to a complex phasediagram in TLG, with the Landau levels inter-crossing multiple times,resulting in significant LL mixing as either D or B is varied35–39. LL-mixing can change the effective interaction between the electrons32.However, as shown in Fig. 3c, it does not significantly affect the-2.00 -1.50 -1.0018241 10691210-4 10-3 10-20.9951.0001.0050.0 0.3-0.8-0.40.02.58 2.61 2.64 2.679.79.89.910.00.2 0.6 1 2468100.24 K1.60 K3.00 KRxy(k�)�c��= -1��= - 2�c� = -1 to -2|dRxy/d���=�c(�)��= 0.40 ± 0.01��104��103Rxy/Rxy(�c��� �c|�� = 0.40 ± 0.030.4 K 1.3 K(a) (c)(b) (d)(e)7 to 123 5ln(T)ln()Rxy(k�)�0.10 K0.40 K0.65 K1.00 K� = 13/5��= 8/3|dRxy/d���=�c(�)T (K)��= 0.42 ± 0.01� =8/3 to 13/5Fig. 2 | Scalingnearν = νc.Plot ofRxyversus ν for transitionbetween the (a) IQHstatesν=−2 and ν=−1 (the critical point νc =−1.5) and b the FQH states 2+2/3 and 2+3/5(νc =2.625). c Double logarithmic plot of ∣dRxy/dν∣ versus T for the PT ν=−2 and ν=−1at νc. Thedashed line is thefit to thedata points using Eq. (1).d Sameas in (c) for thePTbetween FQH states 2+2/3 and 2+3/5. e Scaling analysis of Rxy for the PT transitionbetween ν=2+ 1/3 and ν=2+2/5. The inset is a plot of T versus α in a double loga-rithmic scale (open circles); a linear fit to the data (dotted line) yields κ=0.40±0.03.(For an error analysis, see Supplementary Information, Supplementary Note 7).Article https://doi.org/10.1038/s41467-024-52927-wNature Communications | (2024) 15:8535 3www.nature.com/naturecommunicationsuniversality of κ. This vital result suggests that as long as the anyonsare weakly interacting, the critical behavior of the localization-delocalization transition remains unaltered.Measurement of localization exponent γWenow focus on the localized regime, far away from Ec, marked by theblack rectangle in Fig. 4a. Given the presence of strong interactions, itis reasonable to assume that transport in this localized part of theenergy spectrum proceeds through Efros–Shklovskii (ES) type hop-ping mechanism40. The localization exponent γ determines the Tdependence of longitudinal conductance Gxx40,41:Gxx =G0e�ðT0=TÞ1=2 ð4ÞwithkBT0 / jδνjγ: ð5ÞThe pre-factor G0∝ 1/T and δν = (ν−νc). Figure 4b shows plots oflog(TGxx) versusT−1/2 at different values ofδν; the dotted lines are linear0.0 0.1 0.2 0.30.320.360.400.440.480.52�D (V/nm)0.320.360.400.440.480.52(c)(b)�-4to-3-2to-11.0 1.5 2.0 2.5 3.0 3.5(a)�c1 to 2 2 to 7/37/3to12/513/5to8/33 to 10/3 17/5to24/710/3to17/52.5 3.0 3.52.53.03.5 Rxx maximaRxy minima� c�ctFig. 3 | Scaling exponents for different PT. a Plot of κ as a function of νc corre-sponding to different PT evaluated from the maxima of derivative ðdRxy=dνÞmaxnear the critical point. The dotted vertical lines mark the experimentally obtainedνc. The light blue symbols are for the κ values obtained for trilayer graphene, andthe red symbols are for the single-layer graphene.b Plot of experimentally obtainedvalues of critical points, νc versus those theoretically calculated νct14. The trianglesare the values determined fromcrossing points of isotherms inRxy, while the circlesare determined from theRxxmaxima. The blackdashed line fits thedata points withslope= 1.00±0.002. c Plot of κ versus D for the FQH transition from ν = 8/3to ν = 13/5 states evaluated from the maxima of derivative ðdRxy=dνÞmax nearthe critical point. The error bars are determined from the least-square fits tothe data.0.8 1.2 1.6 2.00.11100.008 0.012 0.01648120.005 0.010 0.0151011021033.34 3.36 3.38 3.40 3.420123TGxx(K.e2 /h)1/T1/2 (K-1/2)0.2 K1 K�� =0.015�� =0.005(a)(d)T o(K)|�-�C|� = 2.21 �0.07(b)(c)Gxx/s(e2 /h.K)s1/2 (K-1/2)�� = 0.006 0.015��10-2Gxx(e2 /h)�0.2 K1.70 K103175��10-3Fig. 4 | Scaling exponent in the ES regime for ν = 3 + 2/5 to 3 + 3/7 transition.aPlots of theT-dependenceofGxx versusfilling factor ν for two FQH states betweenν = 3 and ν = 4. The black box marks the region where the ES analysis was carriedout.b Fit of ES Eq. (4) (dotted lines) to theGxxdata for the transition from ν = 3 + 2/5and ν = 3 + 3/7. Each set of data points is for a given value ofδν=∣ν–νc∣with νc = 3.416.The plots deviate from the expected ES behavior at high T (the region is markedwith an ellipse). c Plots of T0 versus δνCF. The dotted line is a linear fit to the data(see Eq. (5)). The slope yields the value of γ. The error bars are determined from theleast square fits to the data in (b).d Plot of scaled longitudinal conductanceGxx/s asa function of scaling parameter s = ∣δν∣γ/T for PT between ν = 17/5 and ν = 24/7. Thescatter points of different colors are for different values of ∣δν∣, and the solid blackline is fit to Eq. (6).Article https://doi.org/10.1038/s41467-024-52927-wNature Communications | (2024) 15:8535 4www.nature.com/naturecommunicationsfits to the data. The linearity of the data at low-T is consistent withtransport by the ES hoppingmechanism in the FQH regime (Eq. (4)). Athigh-T (in the region marked in Fig. 4b by a dotted ellipse), the valuesof Gxx become relatively large, and the plots deviate from a straightline. In passing, we note that as we move progressively closer to thecenter of the plateau in Rxy, where the value of Gxx ≈0 at low-T, thelinearity of the plots persists to higher temperatures. Fitting T0(estimated fromEqn (4)) and ∣δνCF∣ to Eq. (5), we find the estimated γ tolie in the range 2.3−2.6 (Fig. 4c) for FQH plateau-to-plateau transitions,very close to the predicted range of γ = 2.3−2.514. The fact that theexponent controlling the divergence of the localization length atcriticality is almost identical for both FQH and IQH states points to aneffective model of localization that is universal across the differentstatistics of the quasiparticles in these QH phases. Furthermore, fromκ = 1/zγ ≈0.41 ± 0.03 and γ ≈ 2.3, we get z ≈ 1, as expected for a stronglyinteracting system7,9,42,43.An independent estimate of γ is obtained by casting Eq. (4) into asingle-parameter scaling form44:Gxx sð Þ= σ*se�ðT*sÞ1=2 , ð6Þwith the the scaling parameter s = ∣δνCF∣γ/T. Figure 4d shows the scalingplots of Gxx/s versus s1/2 for the PT in ES regime from ν = 3 + 2/5 toν = 3 + 3/7. We find a near-perfect data collapse for all values of δνCF inthe localized regime with γ ≈ 2.3, providing an independent validationof the universality of γ.DiscussionWe are now in a position to compare the universality of κ seen in theFQH PT in our high-mobility TLG with non-universality of the samemeasured in the high-mobility 2D semiconductors23. The largespread in the observed values of κ seen in the data in GaAs quantumwells was attributed to two main reasons23. The first is the formationof numerous emerging FQH phases between ν = 1/3 and 2/5, whichlimits the temperature range over which one observes the decreaseof the width of Rxx with T. Note that in Fig. 1b, there are two incipientFQH phases, ν = 3 + 1/5 and 3 + 2/7, between the more robust phasesν = 3 and ν = 3 + 1/3. The incipient phases are weak enough notto affect the scaling of the transition region in Rxy even at the lowesttemperature employed here. As a result, we find κ = 0.42 ±0.01 (Fig. 3a).The second reason is related to the type of disorder in thesample23. Universality in κ is observed only when the effective disorderpotential is short-ranged20, as in our graphite-gated high-mobilitygraphene devices. This is not the case in GaAs/AlGaAs systems, wherelong-range scattering potential from the impurities cannot beignored23. We fabricated graphene devices without the graphite gateelectrodes to probe the effect of long-range interactions on κ. Thegraphene channel was no longer screened from long-range Coulombfluctuations arising from the SiO2 substrate; this was reflected inreduced mobility ~2−5m2/Vs. While in these devices, we do not findFQH states, the value of κ for IQH transitions varied widely between0.45−0.64 (Supplementary Information, Supplementary Note 4), sup-porting the conclusions of ref. 23.To summarize, our principal finding is that scaling properties fortransitions involving Abelian FQH states and/or IQH phases are uni-versal. Specifically, we have demonstrated the scaling of the long-itudinal conductance (with a scaling exponent κ =0.41 ± 0.02 andlocalization exponent γ ≈ 2.3) in the IQH and FQH states in Bernal-stacked ABA trilayer graphene. This conclusion holds for plateau-to-plateau transitions between two consecutive IQH states, two FQHstates, and even between IQH and the adjoining FQH state, underliningthe universal character of the scaling. This universality of κ persistseven when an external displacement field hybridizes the Landau levelsof Bernal-stacked TLG. In fact, we find deviations from universality inthe value of κ only in devices where long-range scattering dominates.To our knowledge, ours is the first definite scaling analysis of the QPTover a series of fractional QH states.FQH phases are underlined by strongly correlated and interactingelectrons. Our results demonstrate a surprising correspondencebetween the FQH phase transitions and those of non-interactingelectrons. The results indicate a super-universality in thelocalization–delocalization transitions across distinct anyonic speciesthat represent the characteristic quasiparticles of the FQH phases.While much is known about the localization of electrons, the observedsuper universality motivates the study of localization in anyonic qua-siparticles and the mechanism that drives their conduction in thepresence of disorder and quasiparticle interactions. Our study raisesthe natural question of whether the universality observed in this con-text applies to transitions between other topological phases withfractional excitations, such as fractional Chern insulators45.MethodsDevice fabricationDevices of dual graphite gated ABA trilayer graphene (TLG) hetero-structures were fabricated using a dry transfer technique (for details,see Supplementary Information Supplementary Note 1). Raman spec-troscopy and optical contrast were used to determine the number oflayers and the stacking sequence. The devices were patterned usingelectronbeam lithography followedby reactive ion etching and thermaldeposition of Cr/Pd/Au contacts. Dual electrostatic gates were used tosimultaneously tune the areal number density n = [(CtgVtg +CbgVbg)/e +no] and the displacement field D = [(CbgVbg−CtgVtg)/2ϵ0 +D0] acrossthe device. Here Cbg(Ctg) is the capacitance of the back gate (top gate),and Vbg(Vtg) is the voltage of the back gate (top gate). The values of Ctgand Cbg are determined from quantum Hall measurements. no and Doare the residual number density and electric field due to unavoidableimpurities in the channel.Transport measurementsThe electrical transport measurements were performed in a dilutionrefrigerator (with a base temperature of 20mK) at low frequency(11.4Hz) using standard low-frequency measurement techniques, witha bias current of 10 nA.Data availabilityThe authors declare that the data supporting the findings of this studyare available within the main text and its Supplementary Informationand at https://doi.org/10.6084/m9.figshare.26809147.v1. Other rele-vant data are available from the corresponding author upon request.Code availabilityThe codes that support the findings of this study are available from thecorresponding author upon request.References1. Klitzing, K. V., Dorda, G. & Pepper, M. New method for high-accuracy determination of the fine-structure constant based onquantized hall resistance. Phys. 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A.B. acknowledges funding from U.S. ArmyDEVCOM Indo-Pacific (Project number: FA5209 22P0166) and Depart-ment of Science and Technology, Govt of India (DST/SJF/PSA-01/2016-17). K.W. and T.T. acknowledge support from the JSPS KAKENHI (GrantNumbers 21H05233 and 23H02052) and World Premier InternationalResearch Center Initiative (WPI), MEXT, Japan. G.J.S. thanks CondensedMatter Theory Center and Joint Quantum Institute, University of Mary-land College Park, for their hospitality during the preparation of thismanuscript. Y.G. acknowledges the support by the Deutsche For-schungsgemeinschaft (DFG) through grant No. MI 658/10-2 and RO2247/11-1, the US-Israel Binational Science Foundation 2022391, and theMinerva Foundation. Y.G. is the incumbent of the InfoSys chair at IISc.Author contributionsS.K., T.C., K.R.A., D.S., and A.B. conceived the idea of the study, con-ducted the measurements, and analyzed the results. T.T. and K.W. pro-vided the hBN crystals. U.G., G.J.S., and Y.G. developed the theoreticalmodel. All the authors contributed to preparing the manuscript.Competing interestsThe authors declare no competing interests.Article https://doi.org/10.1038/s41467-024-52927-wNature Communications | (2024) 15:8535 6www.nature.com/naturecommunicationsAdditional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-024-52927-w.Correspondence and requests for materials should be addressed toAveek Bid.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/s41467-024-52927-wNature Communications | (2024) 15:8535 7https://doi.org/10.1038/s41467-024-52927-whttp://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 Universality of quantum phase transitions in the integer and fractional quantum Hall regimes Results Critical exponents near FQH plateau-to-plateau transitions Locating the transition Robustness of the critical exponents against LL mixing Measurement of localization exponent γ Discussion Methods Device fabrication Transport measurements Data availability Code availability References Acknowledgements Author contributions Competing interests Additional information