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Dohun Kim, [Seyoung Jin](https://orcid.org/0009-0009-7043-0296), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Jurgen H. Smet](https://orcid.org/0000-0002-4719-8873), [Gil Young Cho](https://orcid.org/0000-0003-4957-3013), [Youngwook Kim](https://orcid.org/0000-0001-9544-5691)

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[Observation of 1/3 fractional quantum Hall physics in balanced large angle twisted bilayer graphene](https://mdr.nims.go.jp/datasets/0fa376e1-088d-4ca0-af1f-609ccc4aa039)

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Observation of 1/3 fractional quantum Hall physics in balanced large angle twisted bilayer grapheneArticle https://doi.org/10.1038/s41467-024-55486-2Observation of 1/3 fractional quantum Hallphysics in balanced large angle twistedbilayer grapheneDohun Kim1,9, Seyoung Jin 2,3,9, Takashi Taniguchi 4, Kenji Watanabe 5,Jurgen H. Smet 6, Gil Young Cho 3,7,8 & Youngwook Kim 1Magnetotransport of conventional semiconductor based double layer systemswith barrier suppressed interlayer tunneling has been a rewarding subject dueto the emergence of an interlayer coherent state that behaves as an excitonicsuperfluid. Large angle twisted bilayer graphene offers unprecedented stronginterlayer Coulomb interaction, since both layer thickness and layer spacingare of atomic scale and a barrier is no more needed as the twist inducedmomentum mismatch suppresses tunneling. The extra valley degree of free-dom also adds richness. Here we report the observation of fractional quantumHall physics at 1/3 total filling for balanced layer population in this system.Monte Carlo simulations support that the ground state is also an excitonicsuperfluid but the excitons are composed of fractional rather than elementarycharges. The observed phase transitions with an applied displacement field atthis and other fractional fillings are also addressed with simulations. Theyreveal ground states with different topology and symmetry properties.Large angle twisted bilayer graphene offers the unique opportunity toaddress fractional quantum Hall (FQH) physics in a regime with verystrong interlayer Coulomb interactions in view of the atomically thininterlayer spacing d of 0.33 nm. In the presence of a perpendicularmagnetic field of strength B, the intralayer Coulomb interactionstrength is set by the inverse of the magnetic length lB =ffiffiffiffiffiffiffiffiffiffiffih=eBp. Fortypical field strengths accessible in the laboratory, this length scale islarger by an order of magnitude and the ratio of the interlayer to theintralayer Coulomb interaction strength, equal to lB/d, can reach valuesas high as 20. This regime is inaccessible in conventional semi-conductor based bilayer systems in which typically only lB/d ~ 1 isachieved1–9. The finite quantum well thickness required to host a highquality two-dimensional electron gas and the need to separate bothquantum wells with a thick enough barrier of a larger band gapsemiconductor in order to suppress tunneling and observe unclut-tered interlayer Coulomb interaction physics impose unsurmountablerestrictions on the minimum interlayer spacing in conventional semi-conductor heterostructures.In large angle twisted bilayer graphene suppressed tunnelingcomes for free. The relative rotation of the two layers in real space alsocauses a displacement of the valley Dirac cones of the twomonolayersby the same angle in reciprocal space. This generates a largemomentum mismatch among the electronic states of the valleysbelonging to the adjacent layers. This effectively suppresses interlayertunneling10–14 and also prevents band hybridization across a largedensity range, leaving the valley and layer degrees of freedom essen-tially intact. This is in sharp contrast with the small twist angle bilayercounterpart, where the interlayer hybridization is significant at allReceived: 19 August 2024Accepted: 12 December 2024Check for updates1Department of Physics and Chemistry, Daegu Gyeongbuk Institute of Science and Technology (DGIST), Daegu 42988, Republic of Korea. 2Department ofPhysics, Pohang University of Science and Technology, Pohang 37673, Republic of Korea. 3Center for Artificial Low Dimensional Electronic Systems, Institutefor Basic Science, Pohang 37673, Korea. 4Research Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan. 5Research Center for Electronic and Optical Materials, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan. 6MaxPlanck Institute for Solid State Research, 70569 Stuttgart, Germany. 7Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon34141, Republic of Korea. 8Asia-Pacific Center for Theoretical Physics, Pohang, Gyeongbuk 37673, Korea. 9These authors contributed equally: Dohun Kim,Seyoung Jin. e-mail: gilyoungcho@kaist.ac.kr; y.kim@dgist.ac.krNature Communications |          (2025) 16:179 11234567890():,;1234567890():,;http://orcid.org/0009-0009-7043-0296http://orcid.org/0009-0009-7043-0296http://orcid.org/0009-0009-7043-0296http://orcid.org/0009-0009-7043-0296http://orcid.org/0009-0009-7043-0296http://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/0000-0003-3701-8119http://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/0000-0002-4719-8873http://orcid.org/0000-0002-4719-8873http://orcid.org/0000-0002-4719-8873http://orcid.org/0000-0002-4719-8873http://orcid.org/0000-0002-4719-8873http://orcid.org/0000-0003-4957-3013http://orcid.org/0000-0003-4957-3013http://orcid.org/0000-0003-4957-3013http://orcid.org/0000-0003-4957-3013http://orcid.org/0000-0003-4957-3013http://orcid.org/0000-0001-9544-5691http://orcid.org/0000-0001-9544-5691http://orcid.org/0000-0001-9544-5691http://orcid.org/0000-0001-9544-5691http://orcid.org/0000-0001-9544-5691http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-55486-2&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-55486-2&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-55486-2&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-55486-2&domain=pdfmailto:gilyoungcho@kaist.ac.krmailto:y.kim@dgist.ac.krwww.nature.com/naturecommunicationsdensities15–19. The exceptional density tunability of 2D layers and theability to systematically vary the density imbalance among the twolayers across a wide range through the application of a displacementfieldwith the helpof a top andback gate further boost the versatility oflarge angle twisted bilayer graphene as a platform to study incom-pressible ground states induced by interlayer Coulomb interactions.Here, we report on a systematic study of the FQH-states thatemerge in this system and on transitions among FQH-states with dif-ferent topological and symmetry properties observed as the dis-placement field is tuned. Unanticipated FQH physics at total filling 1/3is observed for balanced population. Monte Carlo energy simulationsof suitable trial wave functions are invoked to identify the origin of theFQH features and the transitions among them.This studywasperformedon a total of four hBN encapsulated anddual-gated twisted bilayer graphene samples with twist angles of 9°,18°, 8° and 11°, referred to asdeviceD1 toD4. A schematic cross sectionof the layer sequence as well as optical microscope images of thedevices are shown in Supplementary Fig. 1 of the SupplementaryInformation (SI). Graphite layers serve as top and bottom gates andenable independent density control of the top and bottom graphenesheets.The total charge carrier density can be obtained from the rela-tionshipntot = eBνtot/h = (CTVT +CBVB)/ewithVT the topgate voltage,VBthe bottom gate voltage, CT the top gate capacitance, CB the bottomgate capacitance and νtot the total filling factor. Using the values of CTand CB values, we calculate the displacement field D=εint which con-siders the chemical potential correction and screening effect in gra-phene. Details on the effects of interlayer screening can be found inSupplementary Fig. 8 and Supplementary Note 3 of SI.ResultsMagnetotransport dataFigure 1a shows a color rendition of the longitudinal conductivity (σxx)for device D1 in the parameter plane spanned by νtot and D=εint for afixed B-field of 19 T and a temperature T of 30mK. Several FQH-statesappear as vertical dark blue features, each corresponding to a con-ductivity minimum at fixed total filling. They are marked by theirfractional total filling on the abscissa. Line cuts for three differentvalues of D=εint are shown in panels c–e. Also at νtot = 1 an incom-pressible ground state is observed. This is consistent with previousstudies11,12 that attributed this state to an interlayer coherent quantumHall state caused by Bose-Einstein condensation of excitons that formin the half filled lowest Landau levels of both layers. The yellow dottedlines in Fig. 1a running from ðνtot = 0,D=εint =42Þ to (2, 0) and from(0, 0) to (2, −42) highlight the location of the integer quantum Hallstates for the bottomgraphene sheet for filling νbot = 1 and 0. Similarly,white dashed lines in Fig. 1a mark the location of these integer QH-states for the top graphene layer. The region with the shape of a dia-mondboundedby the νtop(bot) = 0 and 1 lines contains a rich successionof maxima andminima in the conductivity. This is more clearly seen inFig. 1 | The longitudinal conductivity σxx recorded on device D1 as a functionoftotal filling νtot and displacement field D=εint for a fixed magnetic field of 19 Tand at a temperature of 30mK. aColor rendition of the longitudinal conductivityin the (νtot, D=εint)-plane. FQH-states related to two-flux composite fermions aremarked at the bottom abscissa with colored boxes containing the total filling.Dotted lines highlight the conductivity minima caused by the condensation of theelectrons in the top (white) or bottom (yellow) graphene layer into an integerquantumHall statewithfilling 1 or 0. The regionmarkedwith reddotted line suffersfrom metal-graphene contact issue. b Line traces of σxx as a function of the dis-placement electric field for fixed νtot of 1/3, 2/3, 4/3, 8/5 and 5/3 in panels i through vfrom left to right. c Line trace of σxx as a function of νtot for a fixed displacementfield of zero. d Same as (c), but for D=εint = � 23 mV/nm. e Same as (c), but forDv=εint = � 35:5 mV/nm.Article https://doi.org/10.1038/s41467-024-55486-2Nature Communications |          (2025) 16:179 2www.nature.com/naturecommunicationsFig. 1b plotting vertical line cuts through this data set for some of themarked fractional total fillings.For a balanced distribution of the charge carrier density amongthe two layers (i.e., zero displacement field) and when interlayerCoulomb interactions are ignored, the bilayer system is anticipated toexhibit a conductivityminimumand condense in an FQH-state only foreven numerator fractional total fillings equal to twice the filling ofsome prominent single layer FQH-state when both layers simulta-neously condense in such a state. Here, however, that is not the case. Aconductivity minimum also appears for balanced population whenνtot = 1/3. The appearance of an FQH-state at this filling can only beunderstood as the result of strong interlayer Coulomb interactions.The minimum is separated from other minima at non-zero displace-ment field by conductivity maxima suggesting a transition to a FQH-state of different nature. Also for other fractional total fillings,numerous maxima separating adjacent conductivity minima appear inthe vertical line cuts of Fig. 1b signaling transitions between FQH-statesof different characters. Even though for a balanceddensity distributionfractional states with even numerator can in principle appear in theabsence of interlayer Coulomb interactions, the actual ground statesare likely also governed by the interlayer Coulomb interaction in viewof its strength. In order to assess the origin of the FQH-state at zerodisplacement field as well as the transitions at finite displacement weresort to Monte Carlo energy simulations20,21 for the Hamiltoniandescribing large angle twisted bilayer graphene12,22 and trial wavefunctions that are potentially suitable ground states causing theobserved incompressible behavior at the fractional fillings in Fig. 1a.We note that due to the additional valley degree of freedom, thenumber of plausible candidate wave functions to check for a givenfractional filling can be rather large. The details of the modelHamiltonian and theMonte Carlo calculations aswell as the procedureto construct trial wave functions are deferred to Supplementary Notes4–7 of the SI. Due to the imperfection of the metal-graphene contactsin our devices, σxx approaches zero in the range of 1/3 < νtot < 2/3 and5 <D=εint < 20mV/nm, where the FQH-state is not observed and isindicated by a red dotted line.Theoretical phase diagramThe simulation results are summarized in Fig. 2 (see also Supplemen-tary Fig. 9 and Supplementary Notes 9–10). In the left diagram (panela), each vertical line represents a fractional filling for which a con-ductivity minimum is observed in experiment for a portion or the fullrange of the displacement field (Fig. 1). The color of each vertical linechanges whenever a different trial wave function becomes energeti-cally more favorable. The nature of each of these wave functions inpanel a is elucidated in a square box demarcated by the same color inpanel b for zero and positive displacement field. The top circlesbounded by solid and dashed blue lines in each of these squaresrepresents the valleys in the top layer centered around the Kt and K 0tsymmetry points of the original Brillouin zone, respectively. Analo-gously, the bottom circles bounded by red solid and dashed lines referto the two inequivalent valleys in the bottom layer. The filling of eachcircle coincides with the average occupation of that valley. The exactfractionalfilling canbedeterminedwith thehelpof the top legend. Thetotal filled area within all four circles corresponds to the total filling.Each diagram visualizes the valley and layer symmetry breaking forthat wave function. Two-component wavefunctions are depicted witha dumbbell shape (Supplementary Note 5 in the SI). The type of two-component state is encodedby the color of the dumbbell and includedin the legend. More details about these wave functions as well as theirFig. 2 | Evolution of the FQH-states with displacement field. a Diagram sum-marizing the Monte Carlo energy simulations. Vertical lines at total fractional fill-ings change color depending on what wave function among the selected trial wavefunctions in Supplementary Table 3 of the SI takes on the lowest energy. Theproperties of these lowest energy state are visualized in (b). Yellow color is used forFQH-states satisfying the condition νtop = νbot. Red (blue) refers to partially layer-polarized states with νbot− νtop = 1(−1). Rest of the partially layer-polarized phaseswith νtop < νbot (νtop > νbot) are shown with light-red (light-blue) colors. Dark-red(dark-blue) is used for fully layer-polarized states with νtot = νbot (top). The back-ground color visualizes the degree of layer polarization of the twisted bilayer sys-tem. The four diagonal black dashed lines mark when νbot = 1, νtop = 0, νbot = 0, andνtop = 1. b Schematics visualizing the occupation of the top and bottom layer andthe two inequivalent valleys in each layer for the lowest energy trial wave functionas the displacement field is tuned from zero to positive values. K and K' valleys areshown by solid (K) and dashed (K′), blue (top layer) or red (bottom layer) lines. Thedegree of filling of each circle corresponds to the average occupation of that valley.The fractional filling can be determined with the help of the top right legend. Thetotal filling factor relevant for each schematic is shown at the bottom and the colorof the boundary surrounding a schematic is identical to the one used for thatdisplacement in (a). The colored dumbbells help to identify what two componentstate hasbeenused to construct thewave function (top left legend). For completelyor partially filled circles that are not part of a dumbbell, the wave function containsa one-component state such as the 1/3 Laughlin state for a valley of 1/3 filling or onefilled state for a completely filled valley.Article https://doi.org/10.1038/s41467-024-55486-2Nature Communications |          (2025) 16:179 3www.nature.com/naturecommunicationstopological properties can be found in Supplementary Tables 3 and 4and Supplementary Notes 8–10 of the SI. For negative displacementfields the samewave functions are relevant but the roles of the top andbottom layer are simply reversed.The blue and red color in the background of Fig. 2a visualizes thedensity imbalance, i.e., the degree of layer polarization. Whitecorresponds to a balanced density distribution whereas red and bluecolor refer to preferred occupation of the bottom and the top layer,respectively. A comparison of the vertical lines with color changes atfractional fillings in Fig. 2a with the dark blue vertical lines with inter-ruptions at the same fractional fillings in the magneto-transport dataplotted in Fig. 1a reveals reasonable agreement. With the help of Fig. 2we can now discuss the most likely ground state for each fractionalfilling and how the ground state changes as the displacement fieldis varied.As seen in Fig. 1b-(i) a clear minimum in the conductivity isobserved even at zero displacement field for νtot = 1/3. The MonteCarlo energy simulations suggest that the underlying ground state isthe interlayer coherent two component (333)-state (see Supplemen-tary Fig. 10 and Supplementary Note 9, SI). This state can be con-sidered the fractional analog of the (111)-state occurring at total fillingone due to the formation of excitons that share opposite chargesbetween the two layers. Bose-Einstein condensation of the excitonsinto a superfluid gives rise to incompressible behavior. The samephysical picture apparently applies here except that the excitons areformed out of quasi-particles with fractional charge 1/3 and −1/3. Thetopological properties, such as the ground state degeneracy on atorus, as well as the topological fractional excitations of this state areidentical to those of the 1/3 Laughlin state, but the wave function isspread equally and coherently across the two layers (see Supplemen-tary Table 4). When the displacement field is increased, the interlayercoherent (333)-state disappears and the fully layer-polarized, singlecomponent 1/3-Laughlin state takes over (see Fig. 2a and top square inFig. 2b-(i)). This is consistent with the displacement field driven tran-sition in the experimental data shown in Fig. 1a, b-(i).Magnetic field dependence of the νtot= 1/3 FQH stateThe νtot = 1/3 fractional quantumHall (FQH) state at zero displacementfield is observed not only in this sample but also in devices D2, D3, andD4 (Supplementary Figs. 2 and 4 and Supplementary Note 1, SI).Notably, transport data in these samples show enhanced performanceunder hole doping. Figure 3a highlights displacement field-dependentdata recorded on device D2 at a fixed νtot = −1/3 for varying magneticfields. Panel (b) shows the Hall conductivity σxy for a magnetic field of19 T, calculated using device geometry and the tensor relation:σxy =RxyðRxxðw=lÞÞ2 +R2xy×he2, ð1Þwhere w and l represent the sample width and the distance betweenvoltage probes. A quantized plateau is observed, while theminimum inσxx appears from 15 T onwards, becoming more pronounced withincreasing magnetic field. The (333) state has not previously beenreported for twisted bilayer graphene. Although a νtot = 1/3 state wasobserved in double-layer graphene separated by a thin hBN layer9, thatstudy lacked theoretical support and used Corbino geometry, limitingquantized Hall measurement.In contrast, the νtot = 2/3 FQH state in double-layer systems iscommonly observed without a displacement field, as it does not relyon interlayer Coulomb interactions. Instead, it can result from twodecoupled 1/3 Laughlin states in each layer when the interlayer dis-tance d greatly exceeds the magnetic length lB, as extensivelydocumented23–26. However, in regimes with strong interlayer interac-tions, other energetically favorable ground states may emerge. Trialwave functions considered include pseudospin singlet states26,27,interlayer/intralayer Pfaffian states28, the particle-hole conjugate of the1/3 Laughlin state as a composite fermion (CF) state21,27, Z4 Read-Rezayi states29,30, and composites of two 1/3 Laughlin states21,26. MonteCarlo simulations for νtot = 2/3 revealed that the lowest-energy groundstate at zero displacement field is an interlayer coherent pseudospinsinglet state (bottom square in Fig. 2(b)-(ii)) with electrons occupyingFig. 3 |Magneticfield dependence of σxxðD=εintÞ for νtot = −1/3 in device D2. a σxxas a function of D=εint for different magnetic field values. D=εint =0 is highlightedwith an orange line. Here, the (333) state develops. Curves are vertically offset forclarity. bHall conductivity σxy as a function of νtot. Red lines emphasize the plateaufor νtot = −1/3. The rightmost part of the curve is shown as a dotted line. The plateauis an artifact from the saturation of the lock-in amplifier as charge neutrality isapproached and the Hall resistance rises rapidly. All data were acquired atapproximately 30mK.Fig. 4 | Longitudinal conductivity in device D2 as a function of the total fillingand for different fixed values of the magnetic fields. The FQH-states, νtot = −1/3,−4/3, and −8/5, are highlighted. Curves are vertically offset for clarity. The flatminima around νtot = 0 shown as faint dotted lines are artifacts caused by thesaturation of the lock-in amplifier.Article https://doi.org/10.1038/s41467-024-55486-2Nature Communications |          (2025) 16:179 4www.nature.com/naturecommunicationsdifferent valleys in the topand bottom layers.Whendisplacementfieldincreases, electrons shift entirely to the bottom layer, yieldingνtot = νbot = 2/3 and νtop = 0, forming a fully layer-polarized pseudospinsinglet state (top square in Fig. 2(b)-(ii)). This transition alters sym-metry but preserves topological properties (see Supplementary Fig. 13and Supplementary Notes 8–10, SI).Similarly, at νtot = 4/3 and 8/5, interlayer coherent pseudospinsinglet states dominate without displacement fields, while large dis-placement fields induce fully layer-polarized states in one layer.Intermediate fields result in mixed states involving integer (ν = 1) QHstates in one layer and fractional (ν = 1/3 or 3/5) QH states in the other(Fig. 2(b)-(iii), (iv) and Supplementary Figs. 14 and 16). Magnetic field-dependent data for νtot = −8/5, −4/3, and −1/3 on device D2 are shownin Fig. 4 (see also Supplementary Fig. 4). FQH states developdifferentlyacross cool-down cycles (see Section I, SI).For νtot = 5/3 (Fig. 1), the absence of FQH behavior at zero dis-placement field stems from challenges in constructing balanced-layertrial wave functions, such as two independent FQH states atνtop + νbot = 5/6 + 5/6. Instead, partial layer polarization near zero dis-placement field combines an integerQH state inone layerwith a 2/3CFstate in the other. Intermediate displacement fields produce a mix ofinterlayer coherent and integer QH states, transitioning to fully layer-polarized states at higher fields (Fig. 2(b)-(v), see also SupplementaryFig. 17 and Supplementary Note 10, SI).DiscussionWe conclude that large angle twisted bilayer graphene with stronglysuppressed interlayer tunneling and a layer separation on the atomicscale is a powerful test bed for the exploration of correlation physicsinduced by interlayer Coulomb interactions of unprecedentedstrength. For balanced layer population interlayer coherent statesappear such as (333)-state at νtot = 1/3, the fractional analog of thesuperfluid exciton condensate observed in the integer quantum Hallregime. This state has been challenging to detect in double-layer sys-tems based on conventional semiconductor heterostructures. How-ever, it has previously been reported in a graphene/h-BN/graphenestructure using Corbino geometry9, despite the absence of supportingtheoretical calculations. In our study, we utilized Monte Carlo energysimulations with carefully chosen trial wave functions to confirm theground state of not only the 1/3 state but also other FQH-states andtheir transitions across different filling factors.MethodsDevice fabricationWe used the dry pick-up technique with an Elvacite stamp to assemblea graphite/h-BN/twisted bilayer graphene/h-BN/graphite/h-BN (frombottom to top) heterostructure, where the thickness of the h-BN wasbetween 50 and 70 nm and the thickness of graphite was approxi-mately 5 nm.31,32 Both graphiteswere used as the top and bottomgates.The heterostructure was then annealed at 600 °C in a forming gas (Ar/H2) environment for 30min. The Hall bar geometry and edge contactwere defined using e-beam lithography and reactive ion etching with amixture of CF4 and O2 and a power of 40W. Finally, a metal electrodewas deposited with Cr/Au (5/100 nm) by e-beam evaporator with abase pressure of 5 × 10−7 torr12,13.Transport measurementsTransport measurements were performed with the help of low fre-quency lock-in techniques using an ac current of 100nA and a fre-quency of 17.777Hz in a dilution refrigerator with a base temperatureof 30mK. The samples were fabricated on top of a heavily dopedsilicon substrate with a 285 nm thick thermal oxide at the top. Apartfrom the top and back gate voltages applied to the graphite layers totune the density and the displacement field, also a voltage of 70V or−70 V (for electron or hole doping of the twisted graphene bilayer) wasapplied to the Si substrate in order to reduce the contact resistance inregions not covered by the graphite gate layers11.Monte Carlo simulationWe theoretically model the system of large angle twisted bilayer gra-phene by the Hamiltonian consisting of the Coulomb interaction term,the potential energy term due to the displacement field, the capacitiveenergy term due to the finite interlayer distance, and the phenomen-ological short-range interaction term (see Supplementary Note 4 of SIfor details)12,22. As briefly discussed in the main text, various states ofdifferent topology and symmetry are considered as candidate wavefunctions of the ground state. To determine the ground state, weemploy Monte Carlo method 20,21 for calculation of the energy for thecandidate wave functions. For each state, the energy is calculated from106 samples of particle configurations collected via Metropolis-Hastings algorithm. The algorithm is ahead optimized by acceptancerates and integrated autocorrelation times computed from the pre-rundata. The acceptance ratio of the algorithm is the ratio of probabilityamplitudes. The detail is in Supplementary Note 6 of SI.Data availabilitySource data for allmain text figures are included in the SupplementaryInformation. Any other data supporting the findings of this study areavailable from the corresponding author upon request. Source dataare provided with this paper.Code availabilityKey algorithms of theoretical calculations are fully clarified in theSupplementary Information.References1. Wiersma, R. D. et al. Activated transport in the separate layers thatform the νT = 1 exciton condensate. Phys. Rev. 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Half of this research was supported by the Nano andMaterial Technology Development Program through the NationalResearch Foundation of Korea (NRF) funded by Ministry of Science andICT (No. RS-2024-00444725). S.J. and G.Y.C. are supported by theSamsung Science and Technology Foundation under Project NumberSSTF-BA2002-05 and SSTF-BA2401-03, the NRF of Korea (Grant No.RS-2023-00208291, No. 2023M3K5A1094810,No. 2023M3K5A1094813,No. RS-2024-00410027) funded by the Korean Government (MSIT), theAir Force Office of Scientific Research under Award No. FA2386-22-1-4061, and the Institute of Basic Science under project code IBS-R014-D1.The work from DGIST was supported by the Basic Science ResearchProgram NRF-2020R1C1C1006914, NRF-2022M3H3A1098408 throughthe National Research Foundation of Korea (NRF) and the BrainLinkprogram funded by theMinistry of Science and ICT through the NationalResearch Foundation of Korea (2022H1D3A3A01077468). We alsoacknowledge the partner group program of the Max Planck Society.J.H.S. is grateful forfinancial support from theSPP 2244of theDFG. K.W.and T.T. acknowledge support from the JSPS KAKENHI (Grant Numbers21H05233 and 23H02052) and World Premier International ResearchCenter Initiative (WPI), MEXT, Japan.Author contributionsD.K., Y.K. and G.Y.C. conceived the project. D.K. carried out the devicefabrication and performed the low-temperature measurement withJ.H.S. and Y.K. The theory was performed by S.J. and G.Y.C. The h-BNcrystals were synthesized by T.T. and K.W. All authors contributed to themanuscript writing.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-024-55486-2.Correspondence and requests for materials should be addressed toGil Young Cho or Youngwook Kim.Peer review information Nature Communications thanks the anon-ymous, reviewers for their contribution to the peer review of this work. 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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/s41467-024-55486-2Nature Communications |          (2025) 16:179 6https://doi.org/10.1002/advs.202300574https://doi.org/10.1002/advs.202300574http://arxiv.org/abs/1007.2022https://doi.org/10.1038/s41467-024-55486-2http://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 Observation of 1/3 fractional quantum Hall physics in balanced large angle twisted bilayer graphene Results Magnetotransport data Theoretical phase diagram Magnetic field dependence of the νtot = 1/3 FQH state Discussion Methods Device fabrication Transport measurements Monte Carlo simulation Data availability Code availability References Acknowledgements Author contributions Competing interests Additional information