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Yiwei Chen, Yan Huang, Qingxin Li, Bingbing Tong, Guangli Kuang, Chuanying Xi, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), [Guangtong Liu](https://orcid.org/0000-0001-9436-3395), [Zheng Zhu](https://orcid.org/0000-0001-7510-9949), [Li Lu](https://orcid.org/0000-0003-2317-1382), Fu-Chun Zhang, [Ying-Hai Wu](https://orcid.org/0000-0002-0565-6178), Lei Wang

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[Tunable even- and odd-denominator fractional quantum Hall states in trilayer graphene](https://mdr.nims.go.jp/datasets/7aed48ca-9308-4d3a-8b10-5eb945c245e1)

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Tunable even- and odd-denominator fractional quantum Hall states in trilayer grapheneArticle https://doi.org/10.1038/s41467-024-50589-2Tunable even- and odd-denominatorfractional quantum Hall states in trilayergrapheneYiwei Chen1,12, Yan Huang1,12, Qingxin Li1,12, Bingbing Tong2,3, Guangli Kuang4,Chuanying Xi4, Kenji Watanabe 5, Takashi Taniguchi 6,Guangtong Liu 2,3,7 , Zheng Zhu 8, Li Lu 2,3,7, Fu-Chun Zhang8,9,10,Ying-Hai Wu 11 & Lei Wang1,10Fractional quantum Hall (FQH) states are exotic quantum many-body phaseswhose elementary charged excitations are anyons obeying fractional braidingstatistics. While most FQH states are believed to have Abelian anyons, theMoore–Read type stateswith even denominators – appearing at half filling of aLandau level (LL) – are predicted to possess non-Abelian excitations withappealing potential in topological quantum computation. These states, how-ever, depend sensitively on the orbital contents of the single-particle LLwavefunctions and the LL mixing. Here we report magnetotransport mea-surements on Bernal-stacked trilayer graphene, whose multiband structurefacilitates interlaced LL mixing, which can be controlled by external magneticand displacement fields. We observe robust FQH states including even-denominator ones at filling factors ν = − 9/2, − 3/2, 3/2 and 9/2. In addition, wefine-tune the LL mixing and crossings to drive quantum phase transitions ofthese half-filling states and neighbouring odd-denominator ones, exhibitingrelated emerging and waning behaviour.Electrons confined in a two-dimensional system under a perpendicularmagnetic field develop quantized energy levels. And fully filling suchLandau levels (LLs) one by one gives rise to the integer quantum Hallstates1. Within one LL, the strong Coulomb interaction dominates overkinetic energy in the highly degenerated LL flatband, and furtherconduces to the emergence of FQH states at certain fractional fillingsν2. In most cases, the denominators of ν are odd integers, which findsan explanation in that FQH states can be understood effectively asinteger quantum Hall states of composite fermions3. An importantexception to the odd-denominator rule is the 5/2 FQH state found inthe second LL of GaAs4. Based on extensive experimental and theo-retical investigations5–9, the most probable explanation of this state isthe Moore–Read theory10 and its extensions, which provide us thePfaffian, anti-Pfaffian, and particle-hole symmetric PfaffianReceived: 6 March 2024Accepted: 10 July 2024Check for updates1National Laboratory of Solid-StateMicrostructures, School of Physics, NanjingUniversity, Nanjing 210093,China. 2BeijingNational Laboratory for CondensedMatter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China. 3Hefei National Laboratory, Hefei 230088, China. 4AnhuiProvince Key Laboratory of Condensed Matter Physics at Extreme Conditions, High Magnetic Field Laboratory of the Chinese Academy of Science, Hefei230031, China. 5Research Center for Electronic and Optical Materials, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan.6Research Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan. 7Songshan Lake MaterialsLaboratory, Dongguan 523808, China. 8Kavli Institute of Theoretical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China. 9CASCenter for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100049, China. 10Collaborative InnovationCenter of Advanced Microstructures, Nanjing University, Nanjing 210093, China. 11School of Physics and Wuhan National High Magnetic Field Center,Huazhong University of Science and Technology, Wuhan 430074, China. 12These authors contributed equally: Yiwei Chen, Yan Huang, Qingxin Li.e-mail: gtliu@iphy.ac.cn; yinghaiwu88@hust.edu.cn; leiwang@nju.edu.cnNature Communications |         (2024) 15:6236 11234567890():,;1234567890():,;http://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/0000-0001-9436-3395http://orcid.org/0000-0001-9436-3395http://orcid.org/0000-0001-9436-3395http://orcid.org/0000-0001-9436-3395http://orcid.org/0000-0001-9436-3395http://orcid.org/0000-0001-7510-9949http://orcid.org/0000-0001-7510-9949http://orcid.org/0000-0001-7510-9949http://orcid.org/0000-0001-7510-9949http://orcid.org/0000-0001-7510-9949http://orcid.org/0000-0003-2317-1382http://orcid.org/0000-0003-2317-1382http://orcid.org/0000-0003-2317-1382http://orcid.org/0000-0003-2317-1382http://orcid.org/0000-0003-2317-1382http://orcid.org/0000-0002-0565-6178http://orcid.org/0000-0002-0565-6178http://orcid.org/0000-0002-0565-6178http://orcid.org/0000-0002-0565-6178http://orcid.org/0000-0002-0565-6178http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-50589-2&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-50589-2&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-50589-2&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-50589-2&domain=pdfmailto:gtliu@iphy.ac.cnmailto:yinghaiwu88@hust.edu.cnmailto:leiwang@nju.edu.cnwavefunctions as candidates11–14. The elementary charged excitationsof these states obey non-Abelianbraiding statistics andmaybe utilizedto perform fault-tolerant quantum computation that are topologicallyprotected at the fundamental level15. In recent years, even-denominator FQH states have also been observed in several othersystems and some of them are believed to host non-Abeliananyons16–23.We build high quality Bernal stacked trilayer graphene (TLG)devices here and report even-denominator FQH states, to our knowl-edge, for the first time in this system, as well as a plethora of odd-denominator ones. Extensive investigations have been recently carriedout on monolayer and bilayer graphene (MLG, BLG)17–21, and experi-mental attempts have also beenmade on TLG. However, FQH states inTLG remained elusive with only tenuous traces of odd-denominatorones speculated24. Compared to MLG and BLG, TLG possesses richerand more delicate band structure tunability25. Under zero displace-ment field, the band structure of TLG can be decomposed to a com-bination of MLG and BLG (with hopping parameters that are differentfrom the actual monolayer and bilayer systems), this fact by no meansimplies that the physics of TLG is a trivial repetition ofMLG andBLG. Inthe presence of a magnetic field, the LLs originate from the MLG andBLGparts are not separated in energy but intersectwith each other. If avertical displacement field is introduced, the decomposition is nolonger valid as the MLG and BLG parts hybridize. Each single-particleeigenstate is a superposition of the solutions in the non-relativistic(NR) Landau problem and its weights in different NR levels vary withexternal fields. The separations between LLs can be tuned to generatemany different orderings. For the 5/2 state in GaAs4, the NR second LLis sandwiched between the lowest and third LLs. In contrast, one levelin TLG that is similar to the NR second LL may be surrounded fromabove and below by other levels that have various different orbitalcontents. The LL mixing between them is very sophisticated and maylead to intricate competition between strongly correlated states.ResultsThe structure of our TLG devices is depicted in Fig. 1a, where twographite gates are separated from TLG by insulating hBN layers (seeSupplementary Fig. 1 for optical images of our device). By applyingvoltages Vtg on the top gate and Vbg on the bottom gate, the carrierdensity n and the displacement field D can be tuned independently as:n = (CbVbg +CtVtg)/e and D = (CbVbg −CtVtg)/2, where Cb, Ct are averagegeometric capacitances for the bottom and top gates. The latticestructure of TLG is shown in Fig. 1b together with the Slonczewski-Weiss-McClure (SWMc) parameters in its tight-binding description26.The potential difference between the top and bottom layers caused bytheD-field is denoted as 2Δ1. An additional variableΔ2 was proposed tocharacterize the intrinsic charge imbalance between the outer andmiddle layers25. The low-energy band of TLG forD = 0mV/nm is shownin Fig. 1c, where MLG-like linear and BLG-like quadratic componentscanbediscerned.When aperpendicularmagneticfieldB is applied, theFig. 1 | Band structure of TLG and its quantum Hall states. a Schematic of theTLG device with graphite top-gate and bottom-gate isolated by hBN. b The crystalstructure of Bernal stacked TLG and the SWMc hopping parameters in its tight-binding model, here Δ1 and Δ2 describes potential difference between layersinduced by the applied displacement field or non-uniform charge distributionrespectively. c The low-energy band structure of TLG without the displacementfield D in the vicinity of the K+ valley. d The color map of the longitudinalconductivity σxx plotted versus carrier density n and magnetic field B at 1.5 K andD =0mV/nm. The filling factors defined atB = 14 T are given below the bottomaxis.The diamond pattern at B ≈ 5 T is attributed to level crossings. A plethora of FQHstates are observed above 10 T. e σxx as a function of ν for different temperatures atB = 14 T andD =0mV/nm. The filling factors ν = 2, ± 6 and ± 10 aremarked by green,yellow, and blue shaded regions, respectively. In the range − 6⩽ ν⩽ 6, there are twoMLG levels and four BLG levels for each spin projection (see Fig. 3f).Article https://doi.org/10.1038/s41467-024-50589-2Nature Communications |         (2024) 15:6236 2linear and quadratic bands give rise to two sets of LLs that scale asffiffiffiBpand B, respectively. Some levels from these two setsmay intersect witheachotherwhen the B-field varies. This can be seen in Fig. 1dwhere theLandau fan diagram measured at 1.5 K is presented. A few level cross-ings are observed around B = 5 T for 2⩽ ν⩽ 6 as reported in previousworks24,27,28. When the B-field increases to 11 T, longitudinal con-ductivity σxx minima are observed at all integer filling factors− 10⩽ ν⩽ 10, which signifies a complete lifting of the spin and valleydegeneracies of the LLs. Meanwhile, due to the high quality of thedevices, a variety of well-developed FQH states emerge. The tem-perature dependence of the σxx curves at 14 T are displayed in Fig. 1efor many FQH states with denominator 3, whose thermal activationbehaviour clearly demonstrates their incompressible nature. Aninteresting feature is that some states in the range 6⩽ ν⩽ 10 havedifferent onset magnetic fields. FQH states at ν = 19/3,23/3,25/3,29/3are observed even before the neighbouring integer states at ν = 7, 9appear (see Supplementary Fig. 2 for σxx in the ν −B plane). In contrast,FQH states begin to develop at ν = 20/3, 22/3, 26/3, 28/3 near 10 T, atwhich point LL degeneracies at ν = 7, 9 have already been lifted. Thisphenomenon suggests that some FQH states are intimately connectedwith the unusual lifting of spin-valley degeneracy at ν = 7, 9.Next we investigate the FQH states with − 6⩽ ν⩽ 6 at lower tem-peratures in detail. The color map of σxx versus ν and D is plotted inFig. 2a with the left (right) panel of the top row showing the hole(electron) side. The FQH states are summarized in the bottom row forclarity (see Supplementary Fig. 3 for the complete map). For variousfilling factors of the form eν = ν � ½ν�= s=ð2s + 1Þ and 1 − s/(2s + 1) ([ν] isthe greatest integer less than or equal to ν and s = 1, 2), FQH states giverise to the observed minima of σxx. These states are illustrated as blueand green lines in the bottom sketch panel. Besides these odd-denominator states, even-denominator FQH states are unambiguouslyseen atν = − 9/2, − 3/2, 3/2, 9/2 andhighlightedby red lines. The ν = − 9/2 and 3/2 states can be realized at zeroD-field and remain stable over awide range of D, while the ν = − 3/2 and 9/2 states only appear when afiniteD-field is applied. In fact, the ν = 9/2 state can only be observed ina very narrow range of D. We plot σxx together with the Hall con-ductivity σxy around ν = 3/2 in Fig. 2b and the same quantities aroundν = 9/2 in Fig. 2c. The concomitant appearance of exponentially sup-pressed σxx and quantized σxy unambiguously demonstrate that FQHstates are realized atmany different fractions. Interestingly, a weak σxxminimum can be seen at ν = 20/13 close to the half-filling state at ν = 3/2, which could be a composite fermion state or a Pfaffian daughterstate29. As shown in Supplementary Fig. 4, similar features are alsoobserved in the vicinity of ν = − 9/2, − 3/2. In contrast, there is no sig-nature of daughter states associated with ν = 9/2, and FQH states areobserved clearly at eν = 2=5 and 3/5.For afixeddisplacementfield, thedependences ofσxxonB aroundν = − 9/2,− 3/2, 3/2 are presented in Fig. 3a–c. The existence of odd-denominator four-flux FQH states (ν = − 12/7, − 9/7, 12/7) underscoresthe high quality of our sample. The three even-denominator FQHstates are robust in a considerable range of magnetic field. The lineplot of σxx at B = 16 T with − 5⩽ ν⩽ − 4 is shown in Fig. 3d on which afew FQH states are indicated. The energy gaps of some states in Fig. 3dare deduced from their thermal activated behaviour and presented inFig. 3e (see Supplementary Fig. 5 for the fitting). For the odd-denominator ones at eν = 2=5,3=7,4=9, the data can be understoodusing the composite fermion theory. A remarkable prediction of thistheory is that the energy gap decreases to zero linearly as the fillingfactor eν approaches 1/230. The gapvalues arefitted usingΔ = ℏeBeff/mCFin Fig. 3e, where Beff = ð1� 2eνÞB is the effective magnetic field forFig. 2 | Even- and odd-denominator FQH states in the filling factor range −6⩽ ν⩽ 6. a Top panel: the color map of the longitudinal conductivity σxx plottedversus ν and the displacement field D. The magnetic field is B = 14 T and the basetemperature is 15mK. FQH states with both even and odd denominators areobserved at various filling factors ν = p/q (p and q are integers) as signified byminima of σxx. The FQH states exhibit intricate evolution with D. Bottom panel:sketch map of the top panel in which the FQH states are marked for clarity. Greyshaded regions label the integer quantum Hall states. Red, blue, green lines labelthe FQH states with q = 2, 3, and 5, respectively. The line plots of σxx and the Hallconductivity σxy for 1⩽ ν⩽ 2 with B = 16 T and D =0mV/nm (b) and 4⩽ ν⩽ 5 withB = 14T,D = − 153mV/nm (c). A fewFQHstates are indicatedby thedashed lines andthe associated values of ν are given below. The short horizontal lines mark thequantized plateau of σxy.Article https://doi.org/10.1038/s41467-024-50589-2Nature Communications |         (2024) 15:6236 3composite fermions andmCF = 1.99me is the composite fermion mass.It is obvious that this rule is violated by the ν = − 9/2 FQH state whoseenergy gap is well above zero. The inset of Fig. 3e shows the evolutionof the gap at ν = − 9/2 with B. It gets larger when B increases as onewould expect for an interaction driven state.To understand the even-denominator states, we first inspect theLLs of TLG presented in Fig. 3f. The spin degree of freedom isneglected for simplicity. One simply assumes that all these levels arefor spin-up electrons and bear in mind that each level has a spin-downcounterpart. The parameter Δ2 is fixed at zero in most parts of ourdiscussion, but a small value would not be detrimental either. In gen-eral, the single-particle eigenstates are six-dimensional vectors consistof NR Landau orbitals. For one eigenstate of the TLG LL, the largestweight may reside in the NR lowest LL, then it would be denoted asNR0. If there is a substantial weight in the NR second LL, it would bedenoted asNR1. If there is nodisplacementfield, the LLs canbedividedtoMLG and BLG ones. For B ≳ 8 T and − 6⩽ ν⩽ 6, there are three levelsin eachof theK± valley and are further distinguishedby the symbolNM/B = 0,1. The subscript M/B traces the MLG/BLG origin of a level and thenumber 0/1 indicates that it is of the NR0/NR1 type. It is well-knownthat the NR second LL is favorable for realizing FQH state at half filling,as exemplified by the 5/2 state in GaAs4. Using this information, we canprovide a simple picture for the ν = − 9/2 and 3/2 states without the D-field. As shown in Fig. 3f, electrons fill all the LLs below K−,NB = 0 atν = − 6. An extra 3/2 filling of electrons are added to arrive at ν = − 9/2. Itis natural that the spin-up K−,NB = 0 level is fully occupied first. Theremaining 1/2 may enter the spin-down K−,NB = 0 level or the spin-upK−,NB = 1 level. Both single-particle and interaction effects should beincorporated to determine which scenario is realized. The Zeemannsplitting is EZ = 1.62meV at 14 T whereas the separation ϵ�10 betweenK−,NB = 1 andK−,NB = 0 is 4.59meV. If there is no interaction, electronswould populate the spin-downK−,NB = 0 level. However, it is has beenfound that maximal spin polarization is favoured by Coulomb inter-action inmany cases, so the electrons may instead occupy the spin-upK−,NB = 0,1 levels in a certain parameter regime. This leads to a half-filled NR1 level in which the Moore–Read type states could emerge. Asimilar picture has also been proposed to understand some even-denominator states in BLG17,18.This picture for ν = − 9/2 is corroborated using exact diag-onalization results on the torus. There are six quasi-degenerate groundstates (Supplementary Fig. 6), which is consistent with the predictionfor the Moore–Read type states as well as previous results in BLG31–34.This result alone cannot tell us if the state is of the Pfaffian, anti-Pfaffian, or particle-hole symmetric Pfaffian type. To this end, we havecomputed the chiral graviton spectral functions35,36. As illustrated inFig. 3g, these quantities are designed to reveal the relative angularmomentum of electron pairs. It has been shown that the dominantchirality is negative (positive) for the Pfaffian (anti-Pfaffian)wavefunction36. If we only keep the K−,NB = 1 level in our calculation,particle-hole symmetry ensures that the two chiralities are the same.After incorporating LL mixing with the K−,NB = 0 level (and excludingall other levels), the Pfaffian state becomes the favored one (Supple-mentary Fig. 7). This is consistent with the possible existence of adaughter state at ν = − 5 + 7/13. In general, when aNR0/NR1 doublet has3/2 filling, one may expect to see an even-denominator FQH state. Forthe ν = 3/2 state, the levels that should be considered areK+,NB = 0 andK+,NB = 1. It should also be of the Pfaffian type in view of the weakminima at ν = 20/13 = 1 + 7/13. For the range of B that have been stu-died, itmaybe sufficient to keep these two levels, butwe shouldbear inmind that the MLG levels are not too far away in energy. If B increasesFig. 3 | Evolution of FQH states with magnetic field. a–c The color maps of thelongitudinal conductivity σxx plotted versus B and ν at 15mK. The displacementfieldD is zero in a and c and 217mV/nm in b. d σxx as a function of ν at B = 16 T. Thevertical lines and the numbersmark someFQHstates. eThe energy gaps of the FQHstates marked in d deduced from thermal activation measurements. The datapoints are drawn as blue rectangles with heights proportional to the uncertainty inArrhenius fitting. The gaps for three odd-denominator states are fitted linearlyusing the red dash line Δ = ℏeBeff/mCF (Beff = ð1� 2eνÞB is the effectivemagnetic fieldfor composite fermions, and mCF is composite fermion mass). Its inset shows thegap at ν = − 9/2 for different B. f The LLs computed theoretically at Δ1 =Δ2 = 0. TheK+ (K−) valley is represented using solid (dashed) lines and the spin degree offreedom is neglected for simplicity. The schemeof six levels is shownon the right ofthis panel. The MLG and BLG levels are marked by the subscripts M/B and theirorbital contents are given by the numbers 0/1. The splitting between the two BLGlevels in the same valley is denoted as ϵ±10. g Schematic of the chiral gravitonspectral functions I±(ω). Left panel: Chiral gravitons are excited when the relativeangular momentum of electron pairsm is changed by 2. I+(ω) [I−(ω)] is the spectralfunction of the operator that increases (decreases) m. Right panel: Particle-holesymmetry within one LL ensures that I+(ω) = I−(ω), but LL mixing causes asymmetrybetween them. ϵ10 denotes the gap between the two LLs under consideration.Article https://doi.org/10.1038/s41467-024-50589-2Nature Communications |         (2024) 15:6236 4to about 29 T, theK+,NB = 1 level would be equidistant fromK+,NB = 0and K+,NM=0. This results in exotic LL mixing that has not beenexplored in previous works and the fate of the 3/2 state would be veryinteresting. A highermagnetic field is required todo the same thing forthe K− valley.When the displacementfield is turnedon, the FQH states evolve indifferent manners. Rigorously speaking, the Landau levels can nolonger be labeled usingNB/M etc. However, it is convenient to track theevolution of each level with D and refer to them using the names atD = 0mV/nm. The disappearance of the ν = − 9/2 and 3/2 states can beexplained qualitatively based on Supplementary Fig. 8b. TheK−,NB = 0level goes downwhenD increases and eventually crosses with twoNR2type levels. In this regime, the ν = − 9/2 statewouldnot correspond to ahalf-filled NR1 type level, so no even-denominator FQH state isexpected. The experimental data in Fig. 2a indeed suggests that a levelcrossing occurs at D ≈ 80mV/nm. For the ν = 3/2 state, the K+,NB = 0andK+,NB = 1 levels crosswith each other at sufficiently largeD and theweight of the NR1 orbitals in K+,NB = 1 gradually decreases. It is thusplausible that the gradually weakens as ϵ+10 decreases, which is con-sistent with the experimental data in Fig. 2a. In contrast to these twofilling factors, a state emerges at ν = − 3/2 when D becomes sufficientlylarge. In the absence of the D-field, the spin-up K+,NB = 0 level is halffilled and no FQH state is expected. The K−,NB = 1 level moves up withD and crosses with the K+,NB = 0 and K+,NB = 1 levels at quite large D.This could occur at smaller D if a small positive Δ2 is invoked (seeSupplementary Fig. 8c). After the crossing, the electrons fully occupythe spin-up and spin-down K±,NB = 0 levels at ν = − 2. The − 3/2 statewould correspond to a half filled NR1 level as indicated in Supple-mentary Fig. 8d. It is likely of the Pfaffian type given the weak featureobserved at ν = − 2 + 7/13 (see Supplementary Fig. 4).Finally, we study the ν = 9/2 state that only appears in a very smallwindow ofD and B. The colormap of σxx around ν = 9/2 is presented inFig. 4a, b (with one parameter fixed and the other varied). As the ν = 9/2 state emerges, the gap at ν = 23/5 closes and then reopens. Thisimplies that a phase transition has occurred between two differentν = 23/5 states. The ν = 32/7 state simply disappears when the ν = 9/2 state is observed. On the contrary, the states at ν = 22/5 and 31/7remain stable and no transition is found. By fitting the thermal acti-vation data in Fig. 4c, the gap at ν = 9/2 is found to be ~ 0.51 K. For ourtight-binding model with Δ2 = 0, the active levels at ν = 9/2 are of theNR0 type, which is unfavorable for realizing even-denominator states.If we changeΔ2 to −10meV, the Pfaffian state couldbe realized at ν = 9/2 (see Supplementary Fig. 9 and 10). This analysis is not very satisfac-tory because such a value ofΔ2 is not quite reasonable and it is difficultto explain why this state only appear in a narrow range of D. To thisend, wemay consider multi-component FQH states whose spin and/orvalley indices are not polarized. The Halperin 331 state is a well-knowntwo-component state at ν = 1/237. Another example is the Jain stateconstructed from the parton theory38,39. For a suitable range of D, twoNR0 type levels are almost degenerate. At ν = 9/2, two NR0 type levelswith the same valley index are half filled. The interaction betweendifferent valleys may be altered by valley anisotropic terms40 to sta-bilize the Jain state41. This mechanism was proposed to explain the 1/2 state observed in MLG19, and a more detailed analysis is needed tocheck if it also works for TLG.DiscussionIn summary, our results reveal the rich odd- and even-denominatorFQH states in TLG and underscore its extraordinary tunability due tointricate interplayof spin, valley, andorbital degrees of freedom.Whilethe odd-denominator states are most acceptably described by thecomposite fermion theory, other candidates such as the Read-Rezayistates with non-Abelian Fibonacci anyons42 are also possible anddeserve further investigations. On the other hand, it would be fruitfulto further explore the consequences brought by the evolution of LLeigenstates and their mixing and crossings. By varying external fields,we may switch between multiple Abelian and non-Abelian FQH statesas well as non-FQH states, and continuous quantum phase transitionsmay be feasible in some of these process. Since FQH states are notdescribed by the Landau paradigm based on symmetry breaking, theirtransitions are quite likely not captured by the standardLandau–Ginzburg–Wilson theory43. The low-energy effective theory ofFQH states generally involve Chern–Simons gauge fields44, so therecould be many exotic transitions described by strongly coupledquantum field theory.MethodsThe devices were fabricated using our “pick-up method"45 to achieve amulti-layer heterostructure with the TLG encapsulated by two flakes ofhexagonal boron nitride (hBN) and thin graphite flakes as the top andbottomgates. The stackswereannealedunder a high vacuumat350 °Cfor 2 h. Electron-beam lithography was used to write an etch mask todefine the Hall-bar geometry and the electrodes. Redundant regionswere etched away by CHF3/O2 plasma45. Finally the TLG and gates wereedge-contacted45 by e-beam evaporating thin metal layers consistingof Cr/Pd/Au (1 nm/15 nm/100nm).The transport measurements were performed in two systems, adilution fridgewith a base temperature of 15mK and a VTI fridge downto 1.5 K, and both are with superconducting magnets. All data weretaken using the standard four-terminal configuration with lock-inamplifier techniques by sourcing an AC current I between 10 and100nA at a frequency of 17.777Hz. The data of longitudinal con-ductivity σxx and Hall conductivity σxy are obtained from themeasuredresistances by σxx =ρxx=ðρ2xx +R2xyÞ and σxy =Rxy=ðρ2xx +R2xyÞ.Fig. 4 | Displacement and magnetic field-tuning of FQH states in the fillingfactor range 4⩽ ν⩽ 5. The color map of longitudinal conductivity σxx plottedversus D (with B = 14 T fixed in a) or B (with D = − 153mV/nm fixed in b) and ν at15mK. Phase transitions of FQH states are observed at ν = 9/2, 32/7, 23/5. c Thetemperature dependence of σxx versus ν for D = − 153mV/nm and B = 14 T. Its insetshows the Arrhenius fitting of the thermal activation gap at ν = 9/2.Article https://doi.org/10.1038/s41467-024-50589-2Nature Communications |         (2024) 15:6236 5Data availabilityThe data that support the findings of this study are available from thecorresponding authors upon request.Code availabilityThe code that support the findings of this study is available from thecorresponding authors upon request.References1. Klitzing, K. V., Dorda, G. & Pepper, M. New method for high-accuracy determination of the fine-structure constant basedon quantized Hall resistance. Phys. Rev. Lett. 45, 494–497(1980).2. Tsui, D. C., Stormer, H. L. & Gossard, A. C. Two-dimensional mag-netotransport in the extreme quantum limit. Phys. Rev. Lett. 48,1559–1562 (1982).3. Jain, J. K. Composite-fermion approach for the fractional quantumHall effect. Phys. Rev. Lett. 63, 199–202 (1989).4. Willett, R. et al. Observation of an even-denominator quantumnumber in the fractional quantum Hall effect. Phys. Rev. Lett. 59,1776–1779 (1987).5. Halperin, B. I. & Jain, J. K. (eds) Fractional QuantumHall Effects: NewDevelopments (World Scientific Publishing, 2020).6. Radu, I. P. et al. Quasi-particle properties from tunneling in the ν = 5/2 fractional quantum Hall state. Science 320, 899–902 (2008).7. Venkatachalam, V., Yacoby, A., Pfeiffer, L.&West, K. Local chargeofthe ν = 5/2 fractional quantum Hall state. Nature 469,185–188 (2011).8. Banerjee, M. et al. Observation of half-integer thermal Hall con-ductance. Nature 559, 205–210 (2018).9. Dutta, B. et al. Distinguishing between non-Abelian topologicalorders in a quantum Hall system. Science 375, 193–197 (2022).10. Moore, G. & Read, N. Nonabelions in the fractional quantum Halleffect. Nucl. Phys. B 360, 362–396 (1991).11. Lee, S.-S., Ryu, S., Nayak, C. & Fisher, M. P. A. Particle-hole sym-metry and the ν = 5/2 quantum Hall state. Phys. Rev. Lett. 99,236807 (2007).12. Levin, M., Halperin, B. I. & Rosenow, B. Particle-hole symmetry andthe Pfaffian state. Phys. Rev. Lett. 99, 236806 (2007).13. Son, D. T. Is the composite fermion a Dirac particle? Phys. Rev. X 5,031027 (2015).14. Zucker, P. T. & Feldman, D. E. Stabilization of the particle-holePfaffian order by Landau-level mixing and impurities that breakparticle-hole symmetry. Phys. Rev. Lett. 117, 096802 (2016).15. Nayak, C., Simon, S. H., Stern, A., Freedman, M. & Sarma, S. D. Non-Abelian anyons and topological quantum computation. Rev. Mod.Phys. 80, 1083–1159 (2008).16. Falson, J. et al. Even-denominator fractional quantum Hall physicsin ZnO. Nat. Phys. 41, 347–351 (2015).17. Zibrov, A. A. et al. Tunable interacting composite fermion phases ina half-filled bilayer-graphene Landau level. Nature 549, 360 (2017).18. Li, J. I. A. et al. Even denominator fractional quantum Hall state inbilayer graphene. Science 358, 648 (2017).19. Zibrov, A. A. et al. Even-denominator fractional quantumHall statesat an isospin transition in monolayer graphene. Nat. Phys. 14,930–935 (2018).20. Kim, Y. et al. Even denominator fractional quantum Hall states inhigher Landau levels of graphene. Nat. Phys. 15, 154–158 (2018).21. Huang, K. et al. Valley isospin controlled fractional quantum Hallstates in bilayer graphene. Phys. Rev. X 12, 031019 (2022).22. Shi, Q. et al. Odd- and even-denominator fractional quantum Hallstates in monolayer WSe2. Nat. Nanotechnol. 15, 569–573 (2020).23. Hossain, M. S. et al. Valley-tunable even-denominator fractionalquantum Hall state in the lowest Landau level of an anisotropicsystem. Phys. Rev. Lett. 130, 126301 (2023).24. Stepanov, P. et al. Tunable symmetries of integer and fractionalquantumHall phases in heterostructureswithmultiple Dirac bands.Phys. Rev. Lett. 117, 076807 (2016).25. Serbyn, M. & Abanin, D. A. New Dirac points and multiple Landaulevel crossings in biased trilayer graphene. Phys. Rev. B 87,115422 (2013).26. Koshino, M. & McCann, E. Landau level spectra and the quantumHall effect of multilayer graphene. Phys. Rev. B 83, 165443 (2011).27. Taychatanapat, T., Watanabe, K., Taniguchi, T. & Jarillo-Herrero, P.QuantumHall effect and Landau-level crossing of Dirac fermions intrilayer graphene. Nat. Phys. 7, 621–625 (2011).28. Datta, B. et al. Strong electronic interaction and multiple quantumHall ferromagnetic phases in trilayer graphene. Nat. Commun. 8,14518 (2017).29. Levin, M. & Halperin, B. I. Collective states of non-Abelian quasi-particles in a magnetic field. Phys. Rev. B 79, 205301 (2009).30. Du, R. R., Stormer, H. L., Tsui, D. C., Pfeiffer, L. N. & West, K. W.Experimental evidence for new particles in the fractional quantumHall effect. Phys. Rev. Lett. 70, 2944–2947 (1993).31. Apalkov, V. M. & Chakraborty, T. Stable Pfaffian state in bilayergraphene. Phys. Rev. Lett. 107, 186803 (2011).32. Papić, Z., Thomale, R. & Abanin, D. A. Tunable electron interactionsand fractional quantumHall states in graphene. Phys. Rev. Lett. 107,176602 (2011).33. Snizhko, K., Cheianov, V. & Simon, S. H. Importance of interbandtransitions for the fractional quantum Hall effect in bilayer gra-phene. Phys. Rev. B 85, 201415 (2012).34. Zhu, Z., Sheng, D. N. & Sodemann, I.Widely tunable quantumphasetransition from Moore-Read to composite fermi liquid in bilayergraphene. Phys. Rev. Lett. 124, 097604 (2020).35. Liou, S.-F., Haldane, F. D. M., Yang, K. & Rezayi, E. H. Chiral gravitonsin fractional quantumHall liquids. Phys. Rev. Lett. 123, 146801 (2019).36. Haldane, F. D. M., Rezayi, E. H. & Yang, K. Graviton chirality andtopological order in the half-filled Landau level. Phys. Rev. B 104,L121106 (2021).37. Halperin, B. I. Theory of the quantizedHall conductance.Helv. Phys.Acta 56, 75 (1983).38. Jain, J. K. Incompressible quantum Hall states. Phys. Rev. B 40,8079–8082 (1989).39. Moran, N., Sterdyniak, A., Vidanović, I., Regnault, N. & Milovanović,M. V. Topologicald-wave pairing structures in Jain states. Phys. Rev.B 85, 245307 (2012).40. Kharitonov, M. Phase diagram for the ν=0 quantum Hall state inmonolayer graphene. Phys. Rev. B 85, 155439 (2012).41. Wu, Y. Two-component parton fractional quantum Hall state ingraphene. Phys. Rev. B 106, 155132 (2022).42. Read, N. & Rezayi, E. Beyond paired quantum Hall states: Paraf-ermions and incompressible states in the first excited Landau level.Phys. Rev. B 59, 8084–8092 (1999).43. Sachdev, S. Quantum Phase Transitions (Cambridge UniversityPress, Cambridge, 2011).44. Wen, X.-G. Topological orders and edge excitations in fractionalquantum Hall states. Adv. Phys. 44, 405–473 (1995).45. Wang, L. et al. One-dimensional electrical contact to a two-dimensional material. Science 342, 614–617 (2013).AcknowledgementsL.W. acknowledges support from the National Key Projects for Researchand Development of China (Grant Nos. 2021YFA1400400,2022YFA1204700), National Natural Science Foundation of China (GrantNo. 12074173) and Natural Science Foundation of Jiangsu Province(Grant No. BK20220066). Y.H.W. acknowledges support from theNational Natural Science Foundation of China (Grant No. 12174130). Z.Z.acknowledges the National Natural Science Foundation of China (GrantNo.12074375), the Strategic Priority Research Program of CAS (GrantArticle https://doi.org/10.1038/s41467-024-50589-2Nature Communications |         (2024) 15:6236 6No.XDB33000000). K.W. and T.T. acknowledge support from the JSPSKAKENHI (Grant Numbers 21H05233 and 23H02052) andWorld PremierInternational Research Center Initiative (WPI), MEXT, Japan. F.C.Z.acknowledges partial support fromChinaMinistry of Sci and Tech (grant2022YFA1403902), Priority Program of CAS grant No XDB28000000,NSFC grant No 11674278, and Chinese Academy of Sciences undercontract No. JZHKYPT-2021-08. G.L. acknowledges the support by theNSFC of China (grant No. 9206520), the National Basic Research Pro-gram of China from the MOST (grant No. 2022YFA1602803), and theStrategic Priority Research Program of the Chinese Academy of Sci-ences (grant No. XDB33010300). Thiswork is supported by the SynergicExtreme Condition User Facility and by the Innovation Program forQuantum Science and Technology (Grant No. 2021ZD03026001).Author contributionsL.W. conceived and designed the experiment. Y.C., Y.H., and Q.L. fab-ricated the samples. Y.C., Y.H., Q.L., G.K., C.X., B.T., G.L., and L.L. per-formed the transport measurements. Y.C., L.W., Y.H.W., F.C.Z., and Z.Zanalyzed the data. Y.H.W. conducted theoretical analysis. K.W. and T.T.supplied the hBN crystals. Y.C., Y.H.W. and L.W. wrote the manuscriptwith input from all co-authors.Competing interestsThe authors declare no competing interest.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-024-50589-2.Correspondence and requests for materials should be addressed toGuangtong Liu, Ying-Hai Wu or Lei Wang.Peer review information Nature Communications thanks Thiti Taycha-tanapat, and the other, anonymous, reviewer(s) for their contribution tothe peer review of this work. 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