# Fileset

[s41467-024-50475-x.pdf](https://mdr.nims.go.jp/filesets/644def10-c340-4ced-b783-f4faafe45246/download)

## Creator

[Yungi Jeong](https://orcid.org/0000-0001-5709-208X), Hangyeol Park, [Taeho Kim](https://orcid.org/0009-0007-3480-3222), [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), [Jeil Jung](https://orcid.org/0000-0003-2523-0905), [Joonho Jang](https://orcid.org/0000-0003-4380-102X)

## Rights

[Creative Commons BY Attribution 4.0 International](https://creativecommons.org/licenses/by/4.0/)

## Other metadata

[Interplay of valley, layer and band topology towards interacting quantum phases in moiré bilayer graphene](https://mdr.nims.go.jp/datasets/824c6362-e4ca-4532-8f74-8e791671fefd)

## Fulltext

Interplay of valley, layer and band topology towards interacting quantum phases in moiré bilayer grapheneArticle https://doi.org/10.1038/s41467-024-50475-xInterplay of valley, layer and band topologytowards interacting quantum phases inmoiré bilayer grapheneYungi Jeong 1,2, Hangyeol Park1,2, Taeho Kim 2, Kenji Watanabe 3,Takashi Taniguchi 4, Jeil Jung 5,6 & Joonho Jang 1,2In Bernal-stacked bilayer graphene (BBG), the Landau levels give rise to anintimate connection between valley and layer degrees of freedom. Adding amoiré superlattice potential enriches the BBG physics with the formation oftopological minibands — potentially leading to tunable exotic quantumtransport. Here, we presentmagnetotransportmeasurements of a high-qualitybilayer graphene–hexagonal boron nitride (hBN) heterostructure. The zero-degree alignment generates a strong moiré superlattice potential for theelectrons in BBG and the resulting Landau fan diagramof longitudinal andHallresistance displays a Hofstadter butterfly patternwith a high level of detail. Wedemonstrate that the intricate relationship between valley and layer degrees offreedom controls the topology of moiré-induced bands, significantly influen-cing the energetics of interacting quantum phases in the BBG superlattice. Wefurther observe signatures of field-induced correlated insulators, helical edgestates and clear quantizations of interaction-driven topological quantumphases, such as symmetry broken Chern insulators.Bernal bilayer graphene (BBG) is the simplest member of therhombohedral stacked multilayer graphene family. By breakingthe inversion symmetry of BBG with a perpendicular electricdisplacement field D, BBG shows a tunable gap opening and vanHove singularities which can lead to a cascade of symmetry bro-ken states1–4. Also, electrons in BBG have a layer degree of free-dom correlated with the valley degree of freedom in a specificway, which makes it possible to investigate the proximity effect ofsubstrates surrounding BBG. When BBG is aligned with hexagonalboron nitride (hBN), the slight lattice mismatch (~1.8%) of thelayers creates a periodic moiré superlattice potential. Especiallyin this system, the zero-degree rotation between BBG and hBN isknown to be the most stable with the global configurationalenergy minimum; thus it is expected to have very low superlatticedisorder. This superlattice potential is expected to modify theband structure near the K and K’ points and induces secondaryDirac points and flat bands with non-trivial topologicalproperties5. Remarkably, in the presence of a perpendicularmagnetic field, the so-called Hofstadter’s butterfly appears due tothe interplay between superlattice potential and magnetic field6–9.When the magnetic flux per superlattice unit cell becomes arational multiple of the flux quantum, a commensurabilitybetween the cyclotron orbits and the lattice periodicity leads toregain a lattice translational symmetry to form extended elec-tronic states known as Brown–Zak (BZ) quasiparticles10–13. TheseBZ quasiparticles, satisfying the new Bloch equation, movethrough the lattice as if the magnetic field is zero. Away fromthese commensurable magnetic field values, BZ quasiparticleReceived: 5 October 2023Accepted: 12 July 2024Check for updates1Center for Correlated Electron Systems, Institute for Basic Science, Seoul 08826, Korea. 2Department of Physics and Astronomy, and Institute of AppliedPhysics, Seoul National University, Seoul 08826, Korea. 3Research Center for Electronic and Optical Materials, National Institute for Materials Science, 1-1Namiki, Tsukuba305-0044, Japan. 4ResearchCenter forMaterialsNanoarchitectonics, National Institute forMaterials Science, 1-1Namiki, Tsukuba305-0044,Japan. 5Department of Physics, University of Seoul, Seoul, Korea. 6Department of Smart Cities, University of Seoul, Seoul, Korea.e-mail: joonho.jang@snu.ac.krNature Communications |         (2024) 15:6351 11234567890():,;1234567890():,;http://orcid.org/0000-0001-5709-208Xhttp://orcid.org/0000-0001-5709-208Xhttp://orcid.org/0000-0001-5709-208Xhttp://orcid.org/0000-0001-5709-208Xhttp://orcid.org/0000-0001-5709-208Xhttp://orcid.org/0009-0007-3480-3222http://orcid.org/0009-0007-3480-3222http://orcid.org/0009-0007-3480-3222http://orcid.org/0009-0007-3480-3222http://orcid.org/0009-0007-3480-3222http://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/0000-0003-2523-0905http://orcid.org/0000-0003-2523-0905http://orcid.org/0000-0003-2523-0905http://orcid.org/0000-0003-2523-0905http://orcid.org/0000-0003-2523-0905http://orcid.org/0000-0003-4380-102Xhttp://orcid.org/0000-0003-4380-102Xhttp://orcid.org/0000-0003-4380-102Xhttp://orcid.org/0000-0003-4380-102Xhttp://orcid.org/0000-0003-4380-102Xhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-50475-x&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-50475-x&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-50475-x&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-50475-x&domain=pdfmailto:joonho.jang@snu.ac.krfeels an effective magnetic field Bef f =B� Bp=q (penetration of pflux quanta per q superlattice unit cells.) and form mini Landaufans radiating from B =Bp=q lines. Because BZ quasiparticles canbe defined for each rational p=q, a fractal-like self-repeatingstructure - called the Hofstadter butterfly14–19 − would emerge.In addition to single-particle effects, electron–electroninteractions can play a significant role in these systems due to therelatively flat moiré bands, resulting in a large density of states atcertain energies. These interactions lead to quantum correlatedphases, and their interplay with the band topology adds anotherlayer of controllability and richness to the physics of BBG/hBNsystems. By tuning the D field, which affects the interlayerpotential, one expects to control the topological properties andelectrical energy bandwidth of the system, offering a pathway toexplore a range of interacting topological quantum states andtheir phase transitions, and potentially realize devices where thetopological property is dynamically controlled. Understanding ofthe interplay between the valley and layer physics of BBG and thesuperlattice potential from hBN thus is critical. However, despitethe desirable combination of the stable configuration and elec-tric- and tunability, the electron transport properties in a BBG/hBN moiré superlattice are relatively less explored compared tothe other moiré systems, mostly due to the technical difficulty inmaking high-quality samples with both top- and bottom-gateelectrodes and robust ohmic transport contacts necessary in astrong magnetic field and superlattice potential. In this article, wereport a magneto transport study of a high mobility zero-degreealigned BBG/hBN heterostructure with dual graphite gates andfour high-transparent contacts that allow successful longitudinaland Hall measurements. The high quality of the sample and thezero-degree alignment allow us to observe a large variety ofChern insulator states and interaction-driven states even underrelatively low magnetic fields, revealing an intricate interplay ofvarious quantum degrees of freedom.ResultsDevice characterizationBBG/hBN aligned samplewas fabricated by a conventional dry-transfermethod using PDMS and PC film20 (Fig. 1a). Figure 1b shows thestacking structure of the dual-gated BBG/hBN aligned device. Sinceconventional metal contacts can open gaps in the BBG21,22, an addi-tional graphite layer was used to ensure that contact resistanceremains sufficiently low even when strongmagnetic and displacementfields are applied in cryogenic temperatures. We performed four-probe electrical transport measurements of the sample. Figure 1cshows the resistance as a function of carrier density at zero magneticand displacement field. Due to the superlattice potential imposed bythe lattice mismatch between hBN and BBG, induced energy gap orDOSminimum is expected at the energies below and above the chargeneutral point (CNP)7,8. Actually, a recent experiment supports DOSminimum rather than full gap23 (see also Supplementary Note 9). Weindeedobserve twosatellite peaks on either side of CNP atpositive andnegative densities, n= ±2:397× 1012 cm�2ð= ±4n0Þ, where thesuperlattice-induced isolated energy bands are either completely fullor empty. From the value, we estimate that the lattice constant of thesuperlattice is 13.88nmanddetermine that the twist anglebetween theBBG and the top hBN is 0 degree within our experimental accuracy.Figure 1d shows the Hall mobility as a function of carrier density, andthe mobility near the CNP is about 200,000 cm2 V s�1 comparable tohigh mobility GaAs24–26 or suspended graphene27,28, indicating that thesample is of very high quality. Unlike the free-standing BBG, we noticethat the mobility dips near the full filling the isolated superlattice μ (cm2 /Vs)-2 -1 0 1 2x0246 Rxx (kΩ)-2 -1 0 1 2n (1012/cm2)104105106a Top gateBottom gateI+I-V+V-hBNhBN (aligned)grgrgrgrAb VtVbcdFig. 1 | Device geometry and zero magnetic field transport measurements.a Optical microscope image of the aligned BBG/hBN heterostructure device. Scalebar, 15μm. The crystalline axis of bottom hBN and BBG was aligned at almost 0degrees. Inset: Schematic of the moiré pattern seen in the 0 degree aligned BBG/hBN device. b Schematic diagram of the BBG/hBN aligned device. gr denotes thegraphite flake. Top and bottom gates make it possible to tune the carrier densityandD field simultaneously. And the graphite contact layerwas additionally insertedto make good ohmic contact to the BBG. c Longitudinal resistance versus carrierdensity at zero magnetic and displacement field. Satellite peaks were observed atn = ± 2:397× 1012 cm�2 on either side of the CNP peak. d Hall mobility versus car-rier density at zero magnetic and displacement field. Mobility decreases as thecarrier density approaches the superlattice full fillings from CNP.Article https://doi.org/10.1038/s41467-024-50475-xNature Communications |         (2024) 15:6351 2bands, probably due to the small number of effective charge carriersnear the full filling of moiré band.Magneto transport measurementFigure 2 shows longitudinal and Hall resistance measurements in thepresence of a perpendicular magnetic field (see also SupplementaryFig. 1, 2 for various D field values). In Fig. 2a, the dark regions of lowlongitudinal resistance are identified as incompressible states withedge states. The effects of the superlattice in the spectrum are readilynoticeable; at the intersections between the horizontal lines atϕ=ϕ0 =p=q and lines of the conventional integer quantum Hall states, minifans appear to form the fractal-like Hofstadter’s butterfly. In the Hallresistance measurements in Fig. 2c, we directly identify the effectivemagnetic field Bef f = B� Bp=q felt by BZ quasiparticles with signchanges of Hall resistance atϕ=ϕ0 = p=q, forming a horizontal patternof white strips. We also find the D field-induced insulating gap at CNPabruptly closes nearB=9T due to the superlattice effects (green arrowin Fig. 2a; see also Supplementary Fig. 3). In the low magnetic fieldregion near CNP, the Landau fan looks qualitatively similar to the caseof an intrinsic BBG with the spin–valley subbands of Landau levelsresolved evenbelowB = 1 Tbut, in highermagneticfields, the splittingsbecome less distinct and even disappear, due to the dominant effect ofthe moiré potential leading to significant overlaps between LL sub-bands, while ushering in the appearance of moiré-induced incom-pressible states.Each incompressible state in the Landau fan spectrum followsthe Diophantine equation n=n0 = tðϕ=ϕ0Þ+ s, where the slope t givesthe total Chern number of occupied bands proportional to the Hallconductance, and the n-intercept s gives the number of chargestrapped in the superlattice unit cell. Unlike the conventional quan-tum Hall states (s =0), nonzero s states have charge density and Hallconductance decoupled by a strong superlattice potential19,29. Toinvestigate these states thoroughly, we plotted a Wannier diagram inFig. 2b, where the light gray lines represent all possible trajectoriesallowed by the Diophantine equation. Then, we overlaid additionalcolored lines that correspond to the observed insulators in Fig. 2a.First, the dark gray lines correspond to the conventional integerquantum Hall effect (IQHE) (integers t and s =0), and the black linesidentify the Chern insulators (CI) (integer t, integer s≠0); all thesestates can appear in a non-interacting single-particle Hofstadterspectrum. On the other hand, due to particle interaction, moreincompressible features appear in the data especially whenRxy (h/e2)-6(-12,0)D = 91 mV nm-1, T = 30 mK(-8,0) (-4,0)-4 -2 0 2 400.10.20.30.40.5/002468101214B (T)-7 -6 -5 -4 -3 -2 -1 000.10.20.30.40.5/0-4 -3 -2 -1 002468101214B (T)n/n0n/n0 n/n00 10 Rxx (kΩ)-1 1ab cFig. 2 | Magneto transport in a BBG/hBN moiré superlattice. a Longitudinalresistance, as a function of normalized carrier density andmagnetic flux,measuredat T= 30mK with D =91mVnm�1. Here, n is the carrier density, n0 is the densitycorresponding to full filling of the superlattice, ϕ is the magnetic flux per super-lattice unit cell and ϕ0 =h=e is the magnetic flux quantum. The red arrows point tosome Landau levels that are strongly influenced by the moiré potential and exhibithigh resistance. The green arrow indicates the point where the bandgap at CNPcloses under the effect ofmoiré potential. The color scale is truncated at 10 kΩ. bAWannier diagram to denote observed incompressible states identified in a. Weassume thatboth spin and valleydegeneracies are lifted, andonly showup to tj j≤8.Light gray lines indicate incompressible states allowed by Diophantine equationassuming spin and valley degeneracy are lifted. Four classes of trajectories aredistinguished by color: Integer quantum Hall insulators (gray; integer t, s =0),Fractional Quantum Hall insulators (green; fractional t, s =0), moiré potentialinducedChern insulators (black; integer t, integer s, s ≠0), Symmetry-brokenCherninsulators (red; integer t, fractional s). The blue and orange horizontal linesrepresent the first and second order Brown–Zak oscillations (ϕ=ϕ0 = 1=q,2=q),respectively. c Hall resistance data in the hole-doped region at the same tem-perature and D field.Article https://doi.org/10.1038/s41467-024-50475-xNature Communications |         (2024) 15:6351 3measured at T =30mK. For example, the green lines indicate thefractional quantum Hall effect (FQHE) (fractional t, s =0), where thewell-known FQHE at ν =m� 1=3 ðm= ± 1, ± 3Þ30 are visible as low asB= 5T, reflecting the high quality of the electron system. And severalred lines denote symmetry broken Chern insulator (SBCI) (integer t,fractional s) whose trajectories are located at fractional filling ofmoiré Chern bands. We emphasize that most of the Chern numbersof the incompressible states are independently confirmed by the Hallquantization measurements.D field tunable valley-selective moiré effectThe electronic system displays dramatic changes in spectra whentuned with D field, due to BBG’s layer degrees of freedom and itspeculiar property associated with the valley degrees of freedom. Inparticular, striking bright fork-like features appear and stronglydepend on D field (red arrows in Fig. 2a, Supplementary Fig. 1). Thesefork-like features, when plotted as a function ofD and n as in Fig. 3a–c,appear as thick bright lines. Interestingly, for higher Landau levels(LLs) with N ≥ 2 (Fig. 3a–b), the bright lines move in the oppositedirection to the case for the zero-energy Landau levels (ZLLs) (Fig. 3c,Supplementary Fig.4). This phenomenon is depicted in the schematicof Fig. 3d. To explain this observation, we consider the energies of theLLs of BBG given by31,ϵ0 =12 ξU + Esσϵ1 =12 �_ωcγ1� �ξU + Esσ +Δ10ϵ±N = ± _ωcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiN N � 1ð Þp� _ωc2γ1ξU + Esσ N ≥ 2ð Þ8>>><>>>:ð1ÞHere, N is the orbital index of Landau levels, ωc is the cyclotronfrequency of carriers in BBG, ξ is the valley index (+1 for K valley, −1for K’ valley), γ1 is the interlayer hopping parameter32,33, U is theinterlayer potential given as U =Dd0 (d0: interlayer distance ofBBG), Es is the spin-dependent energy splitting, that includes theZeeman energy, to lift spin degeneracy of the bands, and Δ10∝ γ4B(γ4 is the skew tunneling term from non-dimer carbon site to dimercarbon site32,33) is an additional energy difference between N =0and 1. Rxx (kΩ)  Rxx (kΩ) Rxx (kΩ) Rxx (kΩ)-2 -1.5 -1-0.100.1012345012345-1 0 1-0.100.10.200010203040N = 3B = 3.8T  -8-8-8-12N = 2 -8 -4 -4 4N = 0,1(ZLL) B = 5.5T B = 8.83T B=6.5T, T=30mK 00l l l l u u u ul l u ul l u uCarrier density 0D fieldD field N = 2 N = 3 N = 0,1 +: K valley: K’ valley-Energy-2 -1.5 -1-0.0500.050.10 20-2 -1.5 -1-8 -4-5(-3,-1) (-2,-1)-6-7(-4,-1)(-2,-1)(0,-2)(1,-2)(-1,-1)(2,-2)-7-6-5314313(-3,-1)-2 -1.6 -1.2n/n0 n/n0n/n0 n/n0n/n0-0.100.1D (V nm-1)D (V nm-1)adef g hb c0 0.5/ 0-40-200204060Energy (meV)0 0.5/ 0K’ valleyC = -1 C = +1 ΔC = -2K valleyC = +1C = -1 ΔC = +2--Fig. 3 | Degeneracy lifted Landau levels with moiré superlattice potential.a–c Longitudinal resistance Rxx as a function of carrier density andD field atB = (a)3.8 T, (b) 5.5 T, and (c) 8.23 T ðϕ=ϕ0 = 1=3Þ atT = 30mK.Ndenotes the orbital indexof the Landau level (LL) of BBG, and white-colored numbers indicate the Landaufilling factors ν of integer quantum Hall states. Integer quantum Hall states appeardark due to its dissipationless edge state, while metallic states that partially fill theLL have relatively high resistance and appear bright. d Schematic of fillingsequences in a–c. l (lower) and u (upper) denote the layer polarization of eachstate. e Schematic of LL energy spectrum. Each state is denoted by jNξσi (orbital,valley, and spin). The tendency of the energy with respect to the D field is derivedfrom a single-particle model, and the filling sequence of zero-energy LL (N =0,1)additionally takes into account Coulomb interactions50 (see also SupplementaryNote 7). In d–e colors indicate the type of Nξ , with solid lines for spin down anddashed lines for spin up. And, the bold lines indicate levels that are stronglyinfluencedby themoiré potential ina–c. fmoiré-induced electronic energy spectracalculated for each valley. Note the Chern number ΔC of the moiré-induced iso-lated bands is different for each valley, leading to different valley-dependentbroadening behaviors. gD field tunability of Chern insulator states in N = 2 Landaulevels. h Schematic of incompressible states in g. Each gap is color-coded andlabeled ðt,sÞ. Only t values are shown for IQHE and FQHE. Each color scale istruncated at the end value of its respective colorbar.Article https://doi.org/10.1038/s41467-024-50475-xNature Communications |         (2024) 15:6351 4Because _ωc=γ1 is smaller than 1/2 for the range of magnetic fieldsused in our experiment, ϵ0, ϵ1 for ZLLs (N =0,1) and ϵN for higher LLs(N ≥ 2) have the opposite coefficients of energy shifts upon the changeof U (and thus D) for a specific valley; i.e. for a fixed valley, the layerpolarization p= � ∂ϵ=∂D of a higher LL has the opposite sign to theone of a ZLL. Thus, the energy of the K (K’) valley should increase(decrease) with increasing D field in ZLLs, while it decreases (increase)in higher LLs. The schematic in Fig. 3e based on this valley analysis fullyexplains the behavior of the bright features in Fig. 3a–c. It thus stronglysuggests that the two out of the four symmetry-broken subbands of aLL selectively experience the stronger superlattice effect and that allthe bright bands are actually of a specific valley (denoted as K’ valleyfor both ZLLs and LLs with N ≥ 2) experiencing more scattering thatresults in a higher resistance. We point out that this finding is some-what counterintuitive and contrary to the previous beliefs7,8 in that thedegrees of freedom that determine the effect of the moiré potentialare not the layers but actually the valleys (Fig. 3d, e). We further per-formed numerical simulation (Supplementary Fig. 15) that supportsthe phenomenology of significant subband broadenings for one of thevalleys. Interestingly, the topology of the superlattice-induced isolatedbands play an important role here; the combined Chern number of theinduced isolated bands has the exact opposite value for each valley asshown in Fig. 3f (see also Supplementary Fig. 16), and we find that theobserved valley-selective moiré effect originates from the disparatemagnetic field responses of the isolated bands with opposite Chernnumbers because the band edges merge at lower magnetic fields forthe isolated band (of K’ valley) with negative Chern number, inducingmore severe modification of LL spectra5,34,35. More discussions on thepotential microscopic mechanism of this phenomenon are in Supple-mentary Note 6.The gate tunability further extends to the topological propertiesof the moiré bands. By applying D field, we control the valley-selectiveeffect of moiré potential, indirectly via the layer polarization, whosevalues are peculiarly intertwined with the valleys. In particular, variousChern insulator states are controlled due to the interplayof valleys andlayers upon varying the vertical D field. In Fig. 3g, h, we plotted datashowing the D field tunability of the N =2 LL. Inside each LL subbands,multiple Chern gaps appear and disappear strongly depending on Dfield values. We find that the Chern insulators transit under a certainrule: while IQHE gap at ν = 5, 6, 7 closes with LL subbands switchingtheir positions in filling sequence, the Chern insulators whose Chernnumbers t differ by 1 (or −1) but with the same s values appear in theadjacent LL subbands in a cascading fashion (Supplementary Fig. 5)This suggests the Chern insulators are valley- and spin-polarized, andadds to the idea that tuning valley or spin degrees of freedom of theChern insulators is a key to control topology in this BBG system.Correlated insulating statesAnother outstanding feature in the spectra is the existence of theinsulating states located at n=n0 = � 1,� 2 surprisingly persistentthroughout the values of B, as shown in Fig. 4. At B=0, there is aninsulating phase at n=n0 =0, but upon increasing the magnetic field,new insulators develop at n=n0 = −1, −2 (blue arrows in Fig. 4a). Weevaluated the bulk insulating gap of (0,−2) state by fitting the tem-perature dependence to the Arrhenius formula (Fig. 4b), and found thegap size is particularly larger than other nearby superlattice-inducedChern insulating states. This is not explainable by our simulation basedonly on the single-particle picture (Supplementary Fig. 14), and thusstrongly suggests that electron correlation strongly enhances theenergy gaps of the states. Such a correlated insulator can emergewhenparticle interaction leads to spontaneous spinor valley order and savesenergy by filling the superlattice with one particle per unit cell36. Weattribute this strong correlation to the narrow bandwidth of the iso-lated valence band between the CNP (at n=0) and superlattice-induced insulator at n=n0 = � 4. According to the simulation (Sup-plementary Fig. 13), the bandwidth is to be smaller than 50meV anddecreases upon increasing the strength of D. Certainly, most of thefractional states and correlated insulating states we have observedexist between n=n0 =0 and �4, suggestive of the narrow moiré-induced isolated band playing an important role in enhancing theinteraction effects. In addition, we changed the configuration of thecontact as in Supplementary Fig. 8b to measure the non-local resis-tance (RNL) of the sample and found that RNL is significantly large at(0,−1). We attribute the high non-local resistance of the state to ahelical edge channel of counter-propagating valleys (see Supplemen-tary Note 9). However, we cannot rule out the complicated involve-ment of spins in the edge channel, such as one in a spin–valleymagnetic state. A measurement with a large in-plane magnetic field tocontrol Zeeman energymay help to resolve the questionwhile it is outof the scope of the current work.Fractional incompressible statesThe effect of particle interaction due to the narrow bandwidth (Fig. 4)and tunable band topology (Fig. 3) further leads to exotic incom-pressible fractional states. Remarkably, in Fig. 5a–c andSupplementaryFig. 7, we clearly identify multiple phases whose values of s are Rxx (kΩ)-4 -3 -2 -1 000.10.20.30.40.5/002004006008000 5 10 15 20 251/T (K-1)100150200250300350Rxx (k)(t,s) = (0,-2)D = 91 mV nm-1, T = 30 mK / 0 = 0.39a bn/n0Δ/kB~3.43 KFig. 4 | Correlated insulating states. a Longitudinal resistance, as a function ofnormalized carrier density and magnetic flux, measured at T = 30mK withD= 91mVnm�1. Blue arrows indicate insulating states at n=n0 = −1, −2 respectively.The color scale is adjusted to avoid saturations at t =0 features. b Temperaturedependenceof longitudinal resistance of (0,−2) state atϕ=ϕ0 = 0.39 (indicated by awhite arrow in a). The gap was estimated by fitting the Arrhenius formulaRxx =R0expðΔ=2kBTÞ.Article https://doi.org/10.1038/s41467-024-50475-xNature Communications |         (2024) 15:6351 5fractional even below B = 7 T, which is 2–3 times lower than previousreports29,37,38 and another manifestation of the high quality of thissample. These observed interaction-driven states have gap trajectoriesdescribed by ðt,sÞ= ðtL,sLÞ+ νC ðΔt = tR � tL,Δs = sR � sLÞ (whereðtL,R,sL,RÞ is the neighboring left and right integer states respectively)with a fractional νC. We observed states with νC = 1/2 in bands withChernnumberΔt= −2 ((−5,1/2), (−3,1/2)) and stateswith νC = 1/3 and2/3in a band with Δt=3 ((−5,2/3), (−4,1/3)). These (integer t, fractional s)states are interpreted as symmetry broken Chern insulators (SBCIs)arising from the spontaneous doubling/tripling of the superlattice unitcell because they do not follow the theoretically expected filling factorνC of fractional Chern insulators (FCIs) in a Chern band with Δtj j>139,40.However, we still cannot exclude the possibility of partial filling of themoiré Chern band with exotic fractionalized excitations. The techni-ques that can detect charge quantization, such as shot noisemeasurements41, would behelpful to resolve this issue. In Fig. 5e–h, theenergy gaps of the SBCI states were extracted by fitting to theArrhenius formula. We can divide these four states into two relativelylarge (Fig. 5e, h) and two relatively small (Fig. 5f, g) gappairs. These twopairs have different Chern numbers of the fractal bands supportingeach state, which suggests that the band topology is closely related tothe nature of the fractional states (see also Supplementary Fig. 7f, g).Also, we observed that unlike the conventional FQH states in a mis-aligned BBG appearing in a relatively wide range of D field30,42,43, frac-tional states in our sample exist only in a narrow range of D field andseem to critically depend on the fractal band’s bandwidth and topol-ogy, defined by the band Chern number.DiscussionIn conclusion, we have shown the field-dependent interacting Hof-stadter spectrum in BBG/hBNmoiré system and discovered that, uponapplication of D field, the system’s valleys respond according to theirLL-dependent layer polarization and generate distinct spectral fea-tures that strongly depend on the valley-dependent band Chern0.5-1.5 -1 -0.5 00.20.30.4-1.5 -1 -0.50246810T (K)0 10 2 00.11Rxx (kΩ)Rxx (kΩ)(-5,1/2)0 10 20 301.522.5(-5,2/3)(-4,1/3)0 10 20 301/T (K-1)123Rxx (kΩ)0 10 20 301/T (K-1)0.010.1110(-3,1/2)-1.5 -1 -0.5 00.20.30.40.5-74 47 731-8-81 1 1-(-5,1/2)(-5,1/2)(-5,2/3)(-4,1/3) (-3,1/2)(-5,2/3)(-4,1/3)(-3,1/2)3333-777-111144111144 4-7-3-33 3051015D = 91 mV nm-1, T = 30 mKn/n0n/n0n/n0 n/n0/0/0-1.5 -1 -0.501/61/51/41/31/21Rxy(h/e2 )0102030Rxx  (kΩ)-2/ 0 = 0.393 (B = 9.7 T)00.20.40.60.81ad e fg hb cΔ/kB~1.6 K Δ/kB~110 mKΔ/kB~250 mK Δ/kB~1.9 K334333-Rxx (kΩ)42Fig. 5 | Interaction-driven fractional states. aA zoomed-in area of Fig. 2a (see alsoSupplementary Fig. 1f). The color scale is truncated at 15 kΩ.bTheWannier diagramto denote observed states in a. Each incompressible state line is labeled with thesame color as in Fig. 2b, and each colored region represents the single-particleChern bands with Chern numbers Δt obtained by evaluating the t-differencebetween the neighboring single-particle Chern gaps. The patterns of the Chernbands are similar for each of the N =0 (first and third from left) and N = 1 (secondand fourth) ZLLs. c Line cuts along the ϕ=ϕ0 = 0.393 line (dashed horizontal line ina and b). The blue graph on the left axis shows the Hall resistance in units of h=e2with dashed lines representing 1/t. The orange graph on the right axis shows thelongitudinal resistance. Each incompressible state is markedwith the same coloredarrow as inb and shows quantized plateau inHall resistance and dip in longitudinalresistance. d Temperature dependence of longitudinal resistance along theϕ=ϕ0 = 0.389 line. Each of the incompressible states exhibited thermal activationbehavior, with resistance tending to increase with increasing temperature (see alsoSupplementary Fig. 6). e–h Temperature dependent longitudinal resistance offractional states denoted ind. Gapswere estimatedbyfitting theArrhenius formulawith edge state Rxx =R0 expð�Δ=2kBTÞ.Article https://doi.org/10.1038/s41467-024-50475-xNature Communications |         (2024) 15:6351 6number. Also, we experimentally identified interaction-driven states inthe fractal bands based on their definitive fractional Hall quantization.Our work demonstrates that owing to the in-situ tunability with gateelectrodes and the high-quality assisted by its mechanical stability,BBG/hBN moiré system provides a highly-tunable platform for study-ing the interplay of band topology and electron correlation, and opensup exciting opportunities to explore spin–valley isospin polarizationsof interacting states in Hofstadter spectrum. Further theoreticalinvestigations on the energetics of SBCIs and FCIs and its quantitativerelationship to the Chern number of fractal bands would be highlydesired.MethodsDevice fabricationThe encapsulated hBN (aligned)/BBG/hBN device was fabricated usingthe van der Waals dry-transfer technique. The entire stack was pickedup by PDMS/PC stamp in the following order: hBN (for graphitepickup), top graphite gate, top hBN (34 nm), contact graphite, BBG,bottom hBN (61 nm), bottom graphite gate. Graphite flakes were usedonly after ensuring that they had at least 7–8 layers by optical contrast.The stack was dropped down on the pre-defined align marker patternand annealed in a vacuum furnace 500 degrees for 2 h to accumulatesmall bubbles in the stack. After electron beam lithography, aluminumwas deposited and used as an etch mask, and since the contact gra-phite isonly in contactwithone sideof theBBG, the sample is etched inthe shape of a horseshoe rather than a Hall bar. Gate and contactgraphite were edge contacted with Ti/Au metallic leads. Finally, justbefore measurement, the sample was annealed in a vacuum furnace at400 degrees for 1 h.MeasurementMeasurementswere taken in a cryogen-free dilution refrigeratorwith abase temperature of 20mK. Due to the slight temperature increasecaused by operating the superconducting magnet, most of the actualmeasurementswereperformed at 30mK. AnRC/RF electric filter and asapphire stripline heat sink44 were used to lower the electron tem-perature. Electrical measurements were performed using standardlock-in amplifier techniques. Tomeasure longitudinal resistance, anACvoltage bias of 1mVRMS, 13:33Hz was connected to the top contact ofthe device in series with a 1MΩ resistor to simulate an AC current biasof 1 nARMS. The lowest contactwas then connected to the current inputof the SR865A lock-in amplifier to measure the AC current (Fig.1a).Voltage was measured between the second and third contacts in themiddle of the device using a SR560 voltage preamplifier with a gain of1000. Then the longitudinal resistancewas defined asRxx =V=I. Due tothe large parameter space (carrier density, displacement field, mag-netic field, temperature), most of the measurements were performedby one-shot measurement or by taking a small number of measure-ments and averaging them to reduce themeasurement time. However,if the parameter sweep speed is not set appropriately, it can causemeasurement errors, so it is important to set the appropriate satura-tion time according to the measurement frequency and the timeconstant of the amplifier.Numerical simulationWeperformed a numerical simulation of the electronic energy spectrain a BBG/hBNwith zero-degree alignment by the direct diagonalizationof a Hamiltonian matrix following the continuum model proposed byBistritzer and Macdonald45. Due to the natural lattice mismatch of1.08%, electrons in the BBG experience a superlattice potentialimposed at the interface to the hBN even when the twisted angle iszero, which is known to be the energetically stable configuration afterconsidering the lattice relaxation.At afinite perpendicularmagneticfield, the followingHamiltonianbecomes numerically solvable when expressed in the bases of Landaugauge eigenfunctions (usually truncated at LLs above N ~ 200 for themanageable computation time, but still with a good approximation).ð2Þ(where π � ξpx � ipy, pi � �i_∇i � eAi),where it is written in thebases of jψ>= ðjA1>, jB1>, jA2>, jB2>, jB>, jN>Þ. Here, jA1> and jB1 > arethe bases of the top graphene sublattices, jA2> and jB2> are ofthe bottom graphene, and jB�> and jN�> are the atomic Boronand Nitrogen subbases of hBN, respectively. Then, we have usedthe following parameters in the simulation: v=9:1 × 107cm=s,γ1 = 400meV, v3 = 9:0× 106cm=s, v4 = 4:5 × 106cm=s, UB = �1400meV and UN = +3300meV7,46,47. We set Δsub =0meV. The matrixT represents the interlayer moiré hopping between the bottom gra-phene and hBN and is written as,T =u u0u0 u� �+ eiξG01 �r u u0ω�ξu0ωξ u !+ eiξ G01 +G02ð Þ�r u u0ωξu0ω�ξ u !" #ð3Þwhere we used two different parameters u0 = 130meV,u=0:8× 130meV to account for the lattice relaxation effect48,49, andω= e2πi=3.G’s are the reciprocal lattice vectors of the latticemismatch-inducedmoiré potential. Note the four blocks boxed with dotted-lineshybridize twomonolayer graphene blocks and a hBNblock, so that theHamiltonian represents the whole BBG/hBN heterostructure. Bysetting the blocks labeled as T and Ty to 2-by-2 null matrices, onerecovers the intrinsic BBG spectra.Data availabilityThe sourcedata used in this study are available in thefigshare databaseunder accession code https://doi.org/10.6084/m9.figshare.24119286.Other data that support thefindings of this study are available from thecorresponding author upon request.Code availabilityThe codes related to the findings of this study are available from thecorresponding authors upon request.References1. Lee, K. et al. Chemical potential and quantum Hall ferromagnetismin bilayer graphene. Science 345, 58–61 (2014).2. de la Barrera, S. C. et al. Cascade of isospin phase transitions inBernal-stacked bilayer graphene at zero magnetic field. Nat.Phys. (2022).3. Seiler, A. M. et al. Quantum cascade of correlated phases in tri-gonally warped bilayer graphene. Nature 608, 298–302 (2022).4. Polshyn, H. et al. Electrical switching ofmagnetic order in an orbitalChern insulator. Nature 588, 66–70 (2020).5. Zhang, Y.-H., Mao, D., Cao, Y., Jarillo-Herrero, P. & Senthil, T. Nearlyflat Chern bands in moiré superlattices. Phys. Rev. B 99, 075127(2019).6. Dean, C. R. et al. Hofstadter’s butterfly and the fractal quantumHalleffect in moiré superlattices. Nature 497, 598–602 (2013).7. Moon, P. & Koshino, M. Electronic properties of graphene/hex-agonal-boron-nitride moir\’e superlattice. Phys. Rev. B 90,155406 (2014).8. Chen, X., Wallbank, J. R., Mucha-Kruczyński, M., McCann, E. &Fal’ko, V. I. Zero-energy modes and valley asymmetry in theArticle https://doi.org/10.1038/s41467-024-50475-xNature Communications |         (2024) 15:6351 7https://doi.org/10.6084/m9.figshare.24119286Hofstadter spectrum of bilayer graphene van der Waals hetero-structures with hBN. Phys. Rev. B 94, 045442 (2016).9. Andrews, B. & Soluyanov, A. Fractional quantum Hall states formoiré superstructures in the Hofstadter regime. Phys. Rev. B 101,235312 (2020).10. Krishna Kumar, R. et al. High-order fractal states in graphenesuperlattices. Proc. Natl Acad. Sci. 115, 5135–5139 (2018).11. Barrier, J. et al. Long-range ballistic transport of Brown–Zak fer-mions in graphene superlattices. Nat. Commun. 11, 5756 (2020).12. Krishna Kumar, R. et al. High-temperature quantum oscillationscausedby recurringBloch states ingraphene superlattices.Science357, 181–184 (2017).13. Huber, R. et al. Band conductivity oscillations in a gate-tunablegraphene superlattice. Nat. Commun. 13, 2856 (2022).14. Brown, E. Bloch electrons in a uniform magnetic field. Phys. Rev.133, A1038–A1044 (1964).15. Zak, J. Magnetic translation group. Phys. Rev. 134, A1602–A1606(1964).16. Hofstadter, D. R. Energy levels and wave functions of Bloch elec-trons in rational and irrational magnetic fields. Phys. Rev. B. 14,2239–2249 (1976).17. Wannier, G. H. A result not dependent on rationality for Bloch elec-trons in a magnetic field. Phys. Status Solidi B. 88, 757–765 (1978).18. Streda, P. Quantised Hall effect in a two-dimensional periodicpotential. J. Phys. C. Solid State Phys. 15, L1299 (1982).19. Thouless, D. J., Kohmoto, M., Nightingale, M. P. & Den Nijs, M.Quantized Hall conductance in a two-dimensional periodic poten-tial. Phys. Rev. Lett. 49, 405–408 (1982).20. Fan, S., Vu, Q. A., Tran, M. D., Adhikari, S. & Lee, Y. H. Transferassembly for two-dimensional van der Waals heterostructures. 2DMater. 7, 022005 (2020).21. Nouchi, R. Experimental signature of bandgap opening inbilayer graphene at metal contacts. Appl. Phys. Lett. 105,223106 (2014).22. Zheng, J. et al. Interfacial properties of bilayer and trilayer grapheneon metal substrates. Sci. Rep. 3, 2081 (2013).23. Bocarsly, M. et al. De Haas–van Alphen spectroscopy andmagneticbreakdown in moiré graphene. Science 383, 42–48 (2024).24. Schlösser, T., Ensslin, K., Kotthaus, J. P. & Holland, M. Landausubbands generated by a lateral electrostatic superlattice -chasing the Hofstadter butterfly. Semicond. Sci. Technol. 11,1582 (1996).25. Albrecht, C. et al. Evidence of Hofstadter’s fractal energy spectrumin the quantized Hall conductance. Phys. Rev. Lett. 86, 147–150(2001).26. Melinte, S. et al. Laterally modulated 2D electron system in theextreme quantum limit. Phys. Rev. Lett. 92, 036802 (2004).27. Bolotin, K. I. et al. Ultrahigh electron mobility in suspended gra-phene. Solid State Commun. 146, 351–355 (2008).28. Velasco, J. et al. Transport spectroscopy of symmetry-brokeninsulating states in bilayer graphene. Nat. Nanotechnol. 7,156–160 (2012).29. Spanton, E. M. et al. Observation of fractional Chern insulators in avan der Waals heterostructure. Science 360, 62–66 (2018).30. Maher, P. et al. Tunable fractional quantum Hall phases in bilayergraphene. Science 345, 61–64 (2014).31. McCann, E. & Fal’ko, V. I. Landau level degeneracy and quantumHall effect in a graphite bilayer. Phys. Rev. Lett. 96, 086805 (2006).32. Slonczewski, J. C. & Weiss, P. R. Band structure of graphite. Phys.Rev. 109, 272–279 (1958).33. McClure, J. W. Band structure of graphite and de Haas–van AlphenEffect. Phys. Rev. 108, 612–618 (1957).34. Chittari, B. L., Chen, G., Zhang, Y., Wang, F. & Jung, J. Gate-tunabletopological flat bands in trilayer graphene boron-nitride Moirésuperlattices. Phys. Rev. Lett. 122, 016401 (2019).35. Lu, X. et al. Multiple flat bands and topological Hofstadter butterflyin twisted bilayer graphene close to the second magic angle. Proc.Natl Acad. Sci. 118, e2100006118 (2021).36. Cao, Y. et al. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices. Nature 556, 80–84 (2018).37. Wang, L. et al. Evidence for a fractional fractal quantum Hall effectin graphene superlattices. Science 350, 1231–1234 (2015).38. Cheng, B. et al. Fractional and symmetry-brokenChern insulators intunable Moiré superlattices. Nano Lett. 19, 4321–4326 (2019).39. Liu, Z., Bergholtz, E. J., Fan, H. & Läuchli, A. M. Fractional Cherninsulators in topologicalflat bandswithhigherChernnumber.Phys.Rev. Lett. 109, 186805 (2012).40. Möller, G. & Cooper, N. R. Fractional Chern insulators inHarper–Hofstadter bands with higher Chern number. Phys. Rev.Lett. 115, 126401 (2015).41. de-Picciotto, R. et al. Direct observation of a fractional charge.Nature 389, 162–164 (1997).42. Li, J. I. A. et al. Even-denominator fractional quantum Hall states inbilayer graphene. Science 358, 648–652 (2017).43. Huang, K. et al. Valley isospin controlled fractional quantum Hallstates in bilayer graphene. Phys. Rev. X. 12, 031019 (2022).44. Kuemmeth, F. & Marcus, C. M. Reducing noise and temperatureduring measurements in cryostats. (2015). US PatentUS20150060190A1 (2015).45. Bistritzer, R. & MacDonald, A. H. Moiré butterflies in twisted bilayergraphene. Phys. Rev. B. 84, 035440 (2011).46. Jung, J. &MacDonald, A. H. Accurate tight-bindingmodels for the πbands of bilayer graphene. Phys. Rev. B. 89, 035405 (2014).47. Zhang, L. M., Fogler, M. M. & Arovas, D. P. Magnetoelectric cou-pling, Berry phase, and Landau level dispersion in a biased bilayergraphene. Phys. Rev. B 84, 075451 (2011).48. Koshino, M. et al. Maximally localized Wannier orbitals and theextendedHubbardmodel for twisted bilayer graphene. Phys. Rev. X8, 031087 (2018).49. Zhang, Y.-H., Po, H. C. & Senthil, T. Landau level degeneracy intwisted bilayer graphene: role of symmetry breaking. Phys. Rev. B100, 125104 (2019).50. Hunt, B. M. et al. Direct measurement of discrete valley and orbitalquantum numbers in bilayer graphene. Nat. Commun. 8, 948 (2017).AcknowledgementsWegratefully acknowledge helpful discussionswith B. J. Yang. Theworkat SNU was supported by the National Research Foundation of Koreagrants funded by the Ministry of Science and ICT (Grant Nos.2019R1C1C1006520, 2020R1A5A1016518, RS-2023-00258359), theInstitute for Basic Science of Korea (Grant No. IBS-R009-D1), SNU CoreCenter for Physical Property Measurements at Extreme Physical Condi-tions (Grant No. 2021R1A6C101B418), Creative-Pioneering ResearcherProgram through Seoul National University and Samsung DS BasicResearch Program (Project No. 0409-20230298). J. J. acknowledgessupport from Samsung Science and Technology Foundation Grant No.SSTF-BA1802-06. K. W. and T. T. acknowledge support from the JSPSKAKENHI (Grant Numbers 21H05233 and 23H02052) andWorld PremierInternational Research Center Initiative (WPI), MEXT, Japan.Author contributionsY.J. and J.Jang conceived the project, Y.J. fabricated the device andperformed measurements with the help from H.P. and T.K. Y.J. andJ.Jang analyzed data and performed numerical calculations with thehelp from J.Jung. K.W. and T.T. grew the single crystal hBN. Y.J., J.Jungand J.Jang wrote the manuscript with inputs from all authors. J.Jangsupervised the overall project.Competing interestsThe authors declare no competing interests.Article https://doi.org/10.1038/s41467-024-50475-xNature Communications |         (2024) 15:6351 8Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-024-50475-x.Correspondence and requests for materials should be addressed toJoonho Jang.Peer review information Nature Communications thanks Alina Mreńca-Kolasińska, Eric Spanton and the other, anonymous, reviewer(s) for theircontribution to the peer review of this work. A peer review file is avail-able.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 4.0 International License, which permits use, sharing,adaptation, distribution and reproduction in any medium or format, aslong as you give appropriate credit to the original author(s) and thesource, provide a link to the Creative Commons licence, and indicate ifchanges were made. The images or other third party material in thisarticle are included in the article’s Creative Commons licence, unlessindicated otherwise in a credit line to the material. If material is notincluded in the article’s Creative Commons licence and your intendeduse is not permitted by statutory regulation or exceeds the permitteduse, you will need to obtain permission directly from the copyrightholder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2024Article https://doi.org/10.1038/s41467-024-50475-xNature Communications |         (2024) 15:6351 9https://doi.org/10.1038/s41467-024-50475-xhttp://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/ Interplay of valley, layer and band topology towards interacting quantum phases in moiré bilayer graphene Results Device characterization Magneto transport measurement D field tunable valley-selective moiré effect Correlated insulating states Fractional incompressible states Discussion Methods Device fabrication Measurement Numerical simulation Data availability Code availability References Acknowledgements Author contributions Competing interests Additional information