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[Dacen Waters](https://orcid.org/0000-0003-3588-0039), [Ruiheng Su](https://orcid.org/0000-0003-1135-0498), Ellis Thompson, [Anna Okounkova](https://orcid.org/0009-0001-8713-214X), Esmeralda Arreguin-Martinez, [Minhao He](https://orcid.org/0000-0002-4769-0438), Katherine Hinds, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Xiaodong Xu, Ya-Hui Zhang, [Joshua Folk](https://orcid.org/0000-0003-4455-5609), [Matthew Yankowitz](https://orcid.org/0000-0002-5637-9203)

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[Topological flat bands in a family of multilayer graphene moiré lattices](https://mdr.nims.go.jp/datasets/3e2d1421-8b38-479d-91c1-0c3a47a789ea)

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Topological flat bands in a family of multilayer graphene moiré latticesArticle https://doi.org/10.1038/s41467-024-55001-7Topological flat bands in a family ofmultilayer graphene moiré latticesDacenWaters 1,2,10, Ruiheng Su 3,4,10, Ellis Thompson1,10, AnnaOkounkova 1,Esmeralda Arreguin-Martinez5, Minhao He 1,6, Katherine Hinds5,Kenji Watanabe 7, Takashi Taniguchi 8, Xiaodong Xu1,5, Ya-Hui Zhang9,Joshua Folk 3,4 & Matthew Yankowitz 1,5Moiré materials host a wealth of intertwined correlated and topological statesof matter, all arising from flat electronic bands with nontrivial quantum geo-metry. A prominent example is the family of alternating-twist magic-anglegraphene stacks, which exhibit symmetry-broken states at rational fillings ofthemoiré band and superconductivity close to halffilling.Here,we introduce asecond family of twisted graphenemultilayersmade up of twisted sheets ofM-andN-layer Bernal-stackedgrapheneflakes. Calculations indicate that applyingan electric displacement field isolates a flat and topological moiré conductionband that is primarily localized to a single graphene sheet below the moiréinterface. Phenomenologically, the result is a striking similarity in the hier-archies of symmetry-broken phases across this family of twisted graphenemultilayers. Our results show that this family of structures offers promisingnew opportunities for the discovery of exotic new correlated and topologicalphenomena, enabled by using the layer number to fine tune the flat moiréband and its screening environment.Twisting twomonolayer graphene sheets by anangle of θ≈ 1. 1° createsmagic-angle twisted bilayer graphene (MATBG), in which several newphases of matter have been realized1–8. Amuch broader range of novelphysics can be unveiled in closely related structures having three ormore sheets of graphene. Early experiments investigated the stronglycorrelated and topological physics arising in twisted monolayer-bilayer and double-bilayer graphene, noting intriguing similaritiesbetween the two systems that were, nevertheless, qualitatively distinctfrom MATBG9–20. Whereas the investigation of MATBG has expandedto alternating-twist structures up to five layers21–25, the study of struc-tures that include Bernal-stacked components has until now beenlimited to just those two, despite many others in the twisted M +Nfamily carrying predictions of closely related flat bands (tM +N, whereM and N are positive integers representing the number of Bernal-stacked graphene layers twisted atop one another)26.From a symmetry perspective, tM +N structures differ funda-mentally from the family of alternating-twist magic-angle graphenestacksby thebreakingofC2z symmetry (in-plane rotationby 180°). Thisallows a gap to be opened between the lowest valence and conductionbands at charge neutrality by an electric displacement field, D. Thecollective Berry curvature of many graphene layers contributes to anon-zerovalleyChernnumber of themoirébands, yielding topologicalelectronic states when interactions generate a spontaneous valleypolarization. Experimentally, an intriguing result is that the correlatedReceived: 25 June 2024Accepted: 22 November 2024Check for updates1Department of Physics, University of Washington, Seattle, WA, USA. 2Intelligence Community Postdoctoral Research Fellowship Program, University ofWashington, Seattle, WA, USA. 3Stewart Blusson Quantum Matter Institute, University of British Columbia, Vancouver, BC, Canada. 4Department of PhysicsandAstronomy, University of BritishColumbia, Vancouver, BC,Canada. 5Department ofMaterials Science and Engineering, University ofWashington, Seattle,WA, USA. 6Department of Physics, Princeton University, Princeton, NJ, USA. 7Research Center for Electronic and Optical Materials, National Institute forMaterials Science, Tsukuba, Japan. 8Research Center forMaterials Nanoarchitectonics, National Institute for Materials Science, Tsukuba, Japan. 9Departmentof Physics and Astronomy, Johns Hopkins University, Baltimore, MD, USA. 10These authors contributed equally: Dacen Waters, Ruiheng Su, Ellis Thompson.e-mail: jfolk@physics.ubc.ca; myank@uw.eduNature Communications |        (2024) 15:10552 11234567890():,;1234567890():,;http://orcid.org/0000-0003-3588-0039http://orcid.org/0000-0003-3588-0039http://orcid.org/0000-0003-3588-0039http://orcid.org/0000-0003-3588-0039http://orcid.org/0000-0003-3588-0039http://orcid.org/0000-0003-1135-0498http://orcid.org/0000-0003-1135-0498http://orcid.org/0000-0003-1135-0498http://orcid.org/0000-0003-1135-0498http://orcid.org/0000-0003-1135-0498http://orcid.org/0009-0001-8713-214Xhttp://orcid.org/0009-0001-8713-214Xhttp://orcid.org/0009-0001-8713-214Xhttp://orcid.org/0009-0001-8713-214Xhttp://orcid.org/0009-0001-8713-214Xhttp://orcid.org/0000-0002-4769-0438http://orcid.org/0000-0002-4769-0438http://orcid.org/0000-0002-4769-0438http://orcid.org/0000-0002-4769-0438http://orcid.org/0000-0002-4769-0438http://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/0000-0003-4455-5609http://orcid.org/0000-0003-4455-5609http://orcid.org/0000-0003-4455-5609http://orcid.org/0000-0003-4455-5609http://orcid.org/0000-0003-4455-5609http://orcid.org/0000-0002-5637-9203http://orcid.org/0000-0002-5637-9203http://orcid.org/0000-0002-5637-9203http://orcid.org/0000-0002-5637-9203http://orcid.org/0000-0002-5637-9203http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-55001-7&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-55001-7&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-55001-7&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-55001-7&domain=pdfmailto:jfolk@physics.ubc.camailto:myank@uw.eduwww.nature.com/naturecommunicationselectronic states appearing in t1 + 2 are rich with emergenttopology9–12, and are furthermore remarkably similar to many of thestates seen in t2 + 214–20. The intertwined correlated and topologicalstates are most similar when the direction of D is oriented from themonolayer to the bilayer of t1 + 2, which localizes the conduction bandmostly to the bilayer side.In this work, we extend the study of tM +N graphene structures toinclude configurations as thick as six total layers, focusing on t1 + 3,t2 + 3, t1 + 4, and t2 + 4. Despite the inclusion of these additional layersof graphene in the moiré structure, we find striking commonalities inboth the non-interacting and correlated physics across this entirefamily. Continuum model calculations indicate that these unexpectedsimilarities likely arise because low-energy states in the conductionband are localized to just three graphene sheets, irrespective of thetotal number of layers in the structure: the two layers at the twistedinterface and one more immediately adjacent. Ultimately, the result isthat adding graphene sheets above and below does not significantlyaffect the band structure, simply protecting the trio of active layersand strengthening the roles of topology and correlations in shapingthe electronic system.ResultsIsolated moiré–localized flat bandsWe first compare the band structures of various representative tM +Nstructures (t1 + 2, t1 + 3, t2 + 3, t2 + 4, Fig. 1a–d) as predicted by theBistritzer–MacDonald continuum model3 (see Methods). Upon incor-porating an interlayer potential arising from an external D pointingfrom the thinner to the thicker graphene constituent, the moiré con-duction band in each of these structures (purple in Fig. 1a–d) becomesrelatively flat and isolated by gaps to both the moiré valence band andhigher moiré conduction band. Other gaps between neighboringbands can also open for each particular tM +N construction, with thedetails depending sensitively on θ and both themagnitude and sign ofD owing to broken mirror symmetry (see Methods).Band structure predictions for each of these structures arecorroborated bymaps of the longitudinal resistivity, ρxx. The data arecollected as a function of top- and back-gate voltages, shown inFig. 1e–h after converting the gate voltages to moiré band filling, ν,and displacement field, D (see Methods). In qualitative agreementwith the band structure calculations, all exhibit insulating states atthe charge neutrality point (ν = 0) and at full-filling of the lowestmoiré valence and conduction bands (ν = ±4) over certain ranges ofD, marked by large values of ρxx that exceed h/e2 and diverge as thetemperature is lowered (h is Planck’s constant and e is the charge ofthe electron).It is not immediately obvious that isolated flat bands would forminmany of these tM +N structures. Bernal graphene films with three ormore layers feature multiple low-energy bands27,28, all of which musthybridizewith the bands from the other twisted constituent to yield anisolated moiré band. To help explain how different tM +N construc-tions form similar moiré flat bands, Fig. 1i–l shows calculations of thelayer-resolved local density of states (LDOS) at full-filling of the lowestmoiré conduction band (i.e., integrated over the purple bands inFig. 1a–d).Fig. 1 | Non-interacting featuresof tM +Ngraphene. aBand structure calculationsfor t1 + 2 at its nominally optimal twist angle (θ = 1.13°) and interlayer potential(δ = 50meV); defined as when the moiré conduction band (purple curve) has thenarrowest dispersion while being simultaneously gapped (grey shaded regions)from both neighboring bands. Energy is measured with respect to the bottom ofthe moiré conduction band. The calculated valley Chern number, Cv, is shown forthe moiré conduction band (see Methods for details). Similar calculations forb t1 + 3 (θ = 1. 3°, δ = 50meV); c t2 + 3 (θ = 1.45°, δ = 90meV); and d t2 + 4(θ = 1.15°, δ = 50meV). e Resistivity at zeromagnetic field as a function of ν andD ina t1 + 2 device at θ = 1.13°. Gray color denotes experimental artifacts where negativeresistance is observed, attributed to the effects of highly resistive states. Similarmeasurements for f t1 + 3 with θ = 1.29°; g t2 + 3 with θ = 1.50°; and h t2 + 4 withθ = 1.15°. Positive D is defined as pointing from the thin component to the thickcomponent for all systems. Measurements taken at T = 1.7 K, except for e, whereT =0.3 K. i–l Layer–resolved LDOS calculated for the moiré conduction band,corresponding to the layer combinations andparameters inpanels (a–d). TheLDOSis normalized to the maximum value in each panel. Red and orange shadingsdelineate theM and N layers above and below the twisted interface. The interlayerpotential used in the calculations corresponds to positive D in the experiment.Article https://doi.org/10.1038/s41467-024-55001-7Nature Communications |        (2024) 15:10552 2www.nature.com/naturecommunicationsConsidering first the t1 + 2 structure (Fig. 1i), the moiré potentiallocalizes the LDOS on a triangular lattice of ABB-stacked sites, withmost weight appearing on the graphene sheet one below the moiréinterface. The layer-resolved LDOS configuration of t1 + 2 recurs in thethicker tM +N structures, with the additional layers of graphene awayfrom the twisted interface carrying a comparatively small density ofstates. In this sense, the t1 + 2 structure can be considered as the basicbuilding block of all of the thicker tM +N constructions. Given thattheir low-energy bands are all similarly localized nearby the moiréinterface, it is natural to expect that the physics of all of these tM +Nstructures may exhibit common features.Common features of the correlated phasesThe transport measurements in Fig. 1e–h exhibit insulating states atcertain integer values of ν beyond those predicted by the single-particle band structure. As previously observed in a variety of othermoiré systems, these correlated insulators arise as a consequence ofspontaneous symmetry breaking within the moiré flat bands due toCoulomb interactions1–25. The most prominent correlated insulatorsfor the devices in Fig. 1 appear forD >0, where the asymmetrywithD isdue to the lack of mirror-symmetry (see Methods). The differencesbetween theD >0 andD <0 correlated insulators has been explored indetail for the case of t1 + 29,12. Here, we turn our attention to analyzingthe properties of the correlated phases common across the family oftM +N structures, i.e., forD > 0, comparing to what is known about thesymmetry-broken states in t1 + 2 and t2 + 2.Figure 2 shows high-resolution zoom-ins of both longitudinal andHall resistances, ρxx and Rxy, for t1 + 2, t2 + 2, t1 + 3, and t2 + 3, alongwith simple schematics of the material structure. As was seen for thenon-interacting features in Fig. 1e–h, the qualitative arrangement ofcorrelated states are similar across all of these tM +N constructions.The most robust insulating state arises at ν = 2 in all, spanning thelargest range of D and exhibiting the largest value of ρxx at low tem-perature. Rxy reverses sign across the ν = 2 state, consistent with aninteraction-induced band gap with hole-like carriers at ν≲ 2 andelectron-like carriers at ν ≳ 2. Previous measurements of the evolutionof the ν = 2 states in t1 + 2 and t2 + 2 with in-plane magnetic field indi-cate that they are likely spin polarized9,14–16; analogous measurementsin our t1 + 3 and t2 + 3 samples are also consistent with spin-polarizedinsulators (see Supplementary Fig. 2). All samples exhibited a region ofν and D surrounding the insulator at ν = 2 characterized by a slightincrease in ρxx and an abrupt sign reversal in Rxy. These features havepreviously been explained for t1 + 2 and t2 + 2 as arising due to theformation of a spin-polarized half-metal state with a reduced isospindegeneracy of two.Resistive states additionally appear at ν = 1 and 3 in each of the ρxxmaps in Fig. 2. These feature additional sign reversals (or largeenhancements) in Rxy, indicating the formation of additionalsymmetry-broken states with no remaining isospin degeneracies.Previous studies of t1 + 2 and t2 + 2 suggest that the ground-stateordering of these odd ν phases is less consistent, as spin-valley polar-ized (SVP) states compete closely with intervalley coherent (IVC)states9–12,14–18. These two can be challenging to distinguish in anexperiment, as we discuss in more detail in the Methods.It is interesting to note that the t2 + 4 device from Fig. 1d, h, l, aswell as the additional t1 + 4, t2 + 4 and t2 + 5 devices we have studied,did not show correlated insulating states or pronounced regions ofenhanced resistivity with an Rxy sign reversal at a small magnetic field(Supplementary Figs. 3 and 10). This may be due to: (i) the smallerband gaps to higher moiré bands in these thicker structures, (ii)bands that are more dispersive in reality than those predicted bycalculations, (iii) devices made away from the optimal flat-band twistangle for each layer number construction, or some combination of allof these. Indications of a symmetry-broken Fermi surface at ν = 2 int2 + 4 did appear over a small range of magnetic field around B ≈ 4 T(see Supplementary Fig. 4). However, this correlated phase is quicklysuppressed by competing quantumHall states originating from ν = 0.Nevertheless, the new Fermi surface formed at ν = 2 in a modestmagnetic field indicates that this sample is also close to a stronglycorrelated regime.Fig. 2 | Symmetry-broken states in various tM +N systems. a Map of the long-itudinal resistivity around the correlated states in the t1 + 2 device at θ = 1.13°. Themap is symmetrized at B = ±0.5 T. b Similar map of the Hall resistance, anti-symmetrized at B = ±0.5 T. Analogous ρxxmaps for the c t2 + 2 devicewith θ = 1.30°;e t1 + 3 device with θ = 1.29°; and g t2 + 3 device with θ = 1.50°. d, f, h Analogous Rxymaps for the same set of devices. Schematics in the top panels indicate the layercombination for eachmeasurement. The measurements are performed at nominalsample temperatures of a, b T = 300mK, c–f T = 20mK, g, h T = 100mK.Article https://doi.org/10.1038/s41467-024-55001-7Nature Communications |        (2024) 15:10552 3www.nature.com/naturecommunicationsAbundance of correlated states in t2+ 3Among all the tM +N samples we have studied, t2 + 3 devices had thelargest extent of symmetry-broken phases as a function of ν and D(Fig. 3, see also Supplementary Fig. 5). In addition to exhibitingsymmetry-broken correlated insulators for both signs of D in thelowest moiré conduction band, the samples showed evidence forsymmetry broken phases distinct from the usual states seen in theother tM +N systems (see Fig. 1g, Supplementary Figs. 6, and 11).Figure 3a summarizes our experimental observations for a t2 + 3device with a 1.50° twist angle, incorporating observations fromFigs. 2g, h, and 3b, c, and Supplementary Fig. 7. States labeled in blackor dark gray are insulating or highly resistive (the unexpected gappedstate at ν =0.25 will be discussed in future work). Solid red curvesdenote sign changes in Rxy that likely correspond to van Hove singu-larities, whereas dashed black curves denote abrupt jumps in Rxywithout a sign change, possibly indicating the formation of a newinteraction-induced Fermi surface. Shaded regions in Fig. 3a corre-spond to metallic states with different isospin degeneracies. Experi-mentally, the degeneracy can be identified either from the spacing,Δν,between Shubnikov de Haas (SdH) oscillations in ν − D maps taken atfinite magnetic field (Figs. 2g and 3b) or from the Fourier transform ofSdH oscillations collected by sweeping the magnetic field at fixed gatevoltage (Fig. 3c and Supplementary Fig. 7). The degeneracy extractedfrom Δν is four outside the correlated region (e.g., Δν = 4 in the top ofthe map shown Fig. 3b), consistent with the fourfold spin and valleydegeneracy of graphene. The degeneracy is reduced to two inside theregion surrounding the correlated insulator at ν = 2 (light red inFig. 3a), consistent with our inference of a spin-polarized ground state.The degeneracy is harder to discern in the regions surroundingν = 1 and 3 (seeMethods and Supplementary Fig. 8), although there is aregion of ν closely surrounding ν = 1, colored in purple, where Δν isunambiguouslyone. There is also a small pocket of a symmetry-brokenphase over a narrow range of D ≈0.4 V/nm between ν =0 and ν ≈0.21.Figure 3c shows the normalized Fourier transform of the SdH oscilla-tions measured at a fixed D cutting through this small pocket, asdenoted by the horizontal blackdashed line in Fig. 2b (seeMethods forfurther description of the analysis). We see that all isospin degen-eracies are lifted in this pocket, corresponding to a quarter-metalphase for ν <0.21. This is, to our knowledge, the first observation of aquarter-metal state in the tM +N family that is not directly associatedwith the ν = 1 insulator, pointing to the unusually strong interactions inthis system. Interestingly, it does not exhibit the anomalous Hall effect(AHE), suggesting that it may either carry a very small Berry curvatureor instead be in an IVC ordered state (Supplementary Fig. 9).We further see an unexpected insulating state at ν =0 near D = 0across several t2 + 3 devices with different twist angles (Fig. 3d andSupplementary Fig. 5). Our single-particle band structure calculationsdo not predict such a gap for any reasonable model parameters (seeMethods), suggesting that it may arise spontaneously. This hypothesisis supportedbymeasurements of the temperaturedependenceat ν = 0for the device with θ = 1.41° (Fig. 3e). The sample resistance at∣D∣ >0.3 steadily increases as the temperature is lowered (grey andblack curves in the inset), consistent with a band insulator. Near D =0,on the other hand, insulating behavior abruptly onsets below T ≤ 10 K(purple curve in the inset). Thermal activation measurements of thisstate yield a maximum gap size of Δν =0max = 1:54 ±0:05meV (see Sup-plementary Fig. 13). Although not definitive, the abrupt emergence ofthe gapped state with temperature at D ≈0 and its relatively small sizesuggests that it may arise owing to interactions, similarly to the cor-related insulators at charge neutrality in bilayer graphene29–32 andrhombohedral few-layer graphene33–36. Overall, it remains an openquestion as to why t2 + 3 exhibits the most robust and prevalentsymmetry-broken phases over a wide range of ν and D, as our bandstructure modeling does not predict that the bandwidth should besmall compared to other tM +N structures.Topological states in tM+N grapheneFinally, we turn our attention to the topological properties of thetM +N moiré bands. The AHE has been seen previously in both t1 + 2and t2 + 29,10,12,18. In the former, anomalous Hall resistances close to theFig. 3 | Correlated states in t2 + 3 graphene. a Summary phase diagram deter-mined frommagnetotransport measurements in the θ = 1.50° device (see Methodsfor the determination of different features). Black vertical features indicate insu-lating states (Ins). The ν = 1 and3 states are in shadeddarkgrey to indicate that it is aweakly–developed resistive state corresponding to an incipient insulator. Shadedregions indicate where the metallic states are either normal–metals (NM), half-metals (HM), or quarter–metals (QM). White regions within the bounding boxindicate situations in which the degeneracy cannot be uniquely determined. Solidred lines denote likely van Hove singularities (vHS) with Rxy =0. Dashed black linesdenote abrupt jumps inRxy.bMapof ρxx acquired atT = 20mK and symmetrized atB = ±0.9 T, corresponding to the black dashed box in (a). Spacing of the quantumoscillations (Δν) indicates the degeneracy of the metallic phases. Correspondingschematics indicate the degeneracy for representative regions of normal–metal(Δν = 4, top), half–metal (Δν = 2, bottom right), and quarter–metal (Δν = 1, bottomleft) phases. c Fourier transform analysis (FFT) of the SdH measurements takenalong thedashedblack line atD =0.41 V/nm in (b). The FFTamplitude is normalizedto the maximum value of the measurement, and the frequency of oscillations isnormalized to the density (see Methods for details). d Measurements of ρxx forthree t2 + 3 devices with different θ acquired around ν =0. All measurements areperformed at T = 2 K. e Temperature dependence of ρxx as a function ofD, acquiredin the θ = 1.41° device at ν =0. Line cuts at select values ofD are shown for each statein the inset, normalizedby their respective values atT = 2 K.The gray shaded regionindicates a region of the data influenced by artifacts from the electrical contacts.Article https://doi.org/10.1038/s41467-024-55001-7Nature Communications |        (2024) 15:10552 4www.nature.com/naturecommunicationsquantized values h/2e2 and h/e2 have been observed, consistent with aSVP state formed frombands with a θ-dependent valley Chern numberof eitherCv= 1 or 29,10. The AHE is typically not observed in t2 + 2 at ν = 1and 3, despite strong indications of a valley Chern number of 2,potentially indicating IVC order14–17. Nevertheless, in select cases it hasbeen observed for ν > 3, pointing to the emergence of a SVP state18.In our t1 + 3 sample, the correlated state at ν = 1 is too weaklydeveloped to determine its isospin ordering (even with an appliedmagnetic field, as shown in Supplementary Fig. 11). However, the stateat ν = 3 exhibits a clear AHE characterized by hysteresis in Rxy uponsweeping B back and forth, as shown in Fig. 4a at optimal doping anddisplacement field. Figure 4b shows the doping dependence of theAHE, characterized by the difference between forward and backwardsweeps in a magnetic field, ΔRxy = ðR"xy � R#xyÞ=2. The AHE state is mostpronounced close to ν = 3 and is quickly suppressed upon doping. Thecorresponding Landau fan diagram shows that the gapped state driftsto smaller ν upon applying B (Fig. 4c), with a slope indicating a Chernnumber of C = −2 as determined by the Středa formula (dndB =C eh). Theseobservations are all consistent with an incipient quantum anomalousHall effect (QAHE) owing to a SVP state at ν = 3. Band structure mod-eling predicts a valley Chern number of Cv = 2, consistent with ourobservation. The QAHE is not well developed, likely due to a combi-nation of a small energy gap and substantial magnetic disorder in thesample37. These topological properties are reminiscent of those seen int1 + 2 devices with similar twist angles10.The topological properties of t2 + 3 are more unusual. We againfind a valley Chern number of Cv = 2 in our band structure calculationsand see clear signatures of an AHE, in this case below ν = 1 (shown inFig. 4d for ν =0.75). The magnitude of the AHE is small, but persistsover a wide range of doping and vanishes very near ν = 1 (Fig. 4e).However, the corresponding Landau fandiagramexhibits an insulatingstate emerging from ν = 1 with zero slope up to high magnetic field(Fig. 4f), indicating that the ν = 1 symmetry-broken state is topologi-cally trivial. Although the C = 0 statemay arise due to IVC ordering, theapplication of a modest B should favor a first-order phase transition toa SVP state. Notably, however, the absence of such a transition to aC = 2 Chern insulator state is inconsistent with this scenario. An alter-native possibility is that, upon opening a gap at ν = 1, interactionsrenormalize the Chern number of the filled band to C =0. In this sce-nario, the AHE can arise due to Berry curvature hot spots in thereconstructed bands formed by spontaneous symmetry breaking,while the total Berry curvature of the symmetry-broken band inte-grates to zero. Further theoretical and experimental work is needed tobetter resolve the nature of this state.DiscussionTaken collectively, our measurements establish a family of moirégraphene structures composed of Bernal-stacked graphene thin-filmconstituents with a small interfacial twist. By extending prior studies oftwisted monolayer-bilayer and twisted double-bilayer graphene tothicker tM +N variants, we discover a number of striking commonal-ities in both the single-particle and correlated states across this entirefamily.The localization of low-energy states to the layers at and justbelow themoiré interface, whichunderlies this phenomenon,may alsogeneralize to moiré systems built from other vdW materials. Forinstance, a promising future direction would be to extend studies oftwisted homobilayer transition metal dichalcogenides to similar mul-tilayer constructions. This could, for example, help to establish newcontrol knobs over the recently discovered fractional quantumanomalous Hall states in twisted bilayer MoTe238–41. Accurately mod-eling these structures will require the development of new theoreticalanalyses beyond those considered here, including the effects of elec-trostatic screening of D and crystal fields at the twisted interface42. Inparallel, further experimental studies into the largely unexploredphysics of twisted monolayer-trilayer and bilayer-trilayer graphenesare very likely to reveal exciting new topological states. Our resultsthus establish a way to greatly expand the palette of topological flatbands available for study.MethodsDevice fabricationTo fabricate tM +N structures, wefirst optically identified an exfoliatedgrapheneflakewith a step, such thatoneportion of theflake isM layersthick whereas another portion is N layers. We next isolated regions ofthe M- and N-layer components using polymer-free anodic oxidationFig. 4 | Anomalous Hall effects in t1 + 3 and t2+ 3 devices. a Rxy measurement inthe t1 + 3 device (θ = 1.29°) acquired as B is swept back and forth at ν = 3.02 andD =0.523 V/nm. b Doping dependence of the AHE effect at the same displacementfield characterized by the difference in the forward and backward sweeps,ΔRxy = ðR"xy � R#xyÞ=2. c Landau fan diagram of Rxy versus B around ν ≈ 3. Thecorrelated Chern insulator emerging from ν = 3 exhibits a slope consistent withC = −2. d Rxy measurement in the t2 + 3 device (θ = 1.50°) acquired at ν =0.75 andD =0.532 V/nm. e Doping dependence of the AHE in the same device. f Landau fandiagram of ρxx versus B around ν ≈ 1. The correlated state at ν = 1 projects vertically,consistent with C =0. All data acquired at T = 20mK.Article https://doi.org/10.1038/s41467-024-55001-7Nature Communications |        (2024) 15:10552 5www.nature.com/naturecommunicationsnanolithography43–45. We used standard dry transfer techniques with apolycarbonate (PC)/polydimethyl siloxane (PDMS) stamp46 to stackisolated flakes with an interlayer twist by rotating the stage by an angleθ after the M-layer flake was picked up. These structures were encap-sulated with hexagonal boron nitride flakes and graphite gates, andthen transferred onto a Si/SiO2wafer.We used standard electron beamlithography and CHF3/O2 plasma etching to define vdW stacks into aHall bar geometry and standardmetal deposition techniques (Cr/Au)46to make electrical contact to the graphenemultilayers. Optical imagesof all completed devices measured in this work are shown in Supple-mentary Fig. 1, and a summary of our observations across all devices isprovided in Supplementary Table 1.We note that, in principle, there could be regions of the exfoliatedgraphene flakes with metastable stacking orders other than Bernal.However, these non-Bernal domains are known to relax to the ground-state Bernal stacking configuration during stacking unless great care istaken to isolate only non-Bernal domains and minimize strains duringtransfer47. Since we do not take any such precautions, it is over-whelmingly likely that all non-Bernal domains relax to the Bernalconfiguration in our final samples. Another possible ambiguity in oursample construction is AB vs. BA stacking configurations between thetwisted components. For example, t2 + 2 can host symmetry-brokenstates when twisted slightly away from 0° (tAB +AB) or away from 60°(tAB +BA)48. While we fabricated all of our devices by twisting slightlyaway from 0°, it is known that AB–BA stacking faults naturally exist inbilayer graphene flakes49, and these cannot be observed optically. It istherefore possible that tM +N devices with M ≥ 2 and N ≥ 2 have unin-tentional domains of AB–BA at the twisted interface. These could, inprinciple, have different band structures and valley Chern numbers.Further work is needed to carefully distinguish these potentialscenarios.Transport measurementsTransport measurements were carried out in various cryogen-freesystems using a lock-in amplifier with frequencies between 13.33 and17.77 Hz and a.c. bias between 1 and 10 nA. All data presented in themain textwere acquired inBluefors dilution refrigerators with nominalbase temperatures between 10 and 30mK unless otherwise noted.Data reported at and above 1.5 K (e.g., Fig. 3d, e) were acquired in aCryomagnetics variable temperature insert.The top and bottom gate voltages were used to independentlycontrol the carrier density, n, and perpendicular displacement field, Daccording to the following relations: n = (VtCt +VbCb)/e andD = (VtCt −VbCb)/2ϵ0, where Ct and Cb are the capacitance per unit areaof the top and bottom dielectrics, respectively, Vt and Vt are the topand bottom gate voltages, respectively, and ϵ0 is the vacuum permit-tivity. When specified, we perform field symmetrization (anti-symmetrization) of ρxx and Rxy following ρxx = [ρxx(B) + ρxx(−B)]/2 andRxy = [Rxy(B)−Rxy(−B)]/2.We estimate the twist angle, θ, between M- and N-layer flakes byfitting the sequences of quantum oscillations emerging from ν =0 andν = ±4 in Landau fan diagrammeasurements of ρxx andRxy as a functionof n and magnetic field, B. From the Landau fan fit, we determine thesuperlattice density, ns, and extract the twist angle using the relationns =42θ2ffiffi3pa2, where a =0.246 nm is the graphene lattice constant.Extraction of Fermi surface degeneracy from quantumoscillationsThe frequency of quantum oscillations, fν, is extracted from low-fieldLandau fan measurements taken at constant D (Supplementary Fig. 7)The frequency of oscillations, fB, is first extracted from the Fouriertransform (FFT) of each field sweep with respect to 1/B. The frequencyis then normalized by the total carrier density, fν = fB/(Φ0n). For thecase of a singly connected Fermi surface at the Fermi level, the inversequantity f �1ν represents the degeneracy of charge carriers47.Supplementary Fig. 7b, c shows the analysis outside of the correlatedregion of the t2 + 3 device, where there is a sharp peak in the FFT signalat f�1ν =4, consistent with the fourfold degeneracy of graphene. Var-ious line cuts are shown in the remainder of Supplementary Fig. 7, andindicate quarter metal (f �1ν = 1) and half metal (f �1ν = 2) states formedover certain ranges of n and D.Multiband transport in t2+ 3 grapheneRegions of parameter space in the t2 + 3 device shown in Fig. 3acolored in white correspond to situations in which we are unable tounambiguously determine the degeneracy of the Fermi surface.Especially around ν = 3, this ambiguity likely arises due to multipleFermi surface pockets coexisting within the moiré Brillouin zone.Supplementary Fig. 8 shows evidence for this in the form of curvedtrajectories of quantum Hall states seen in Landau fan diagrams. Suchcurved trajectories violate the Středa formula, which always predictslinear trajectories of topological gapped states, and generally arise dueto the need to fill charge carriers simultaneously into two separatebands that each have their own sequence of Landau levels50.Determination of isospin polarization at integer band fillingsWe employ a combination of out-of-plane and in-plane magnetic fieldmeasurements of ρxx and Rxy in order to infer the isospin polarizationof the correlated states seen at ν = 1, 2, and 3. In-plane magnetic fieldcouples primarily to the spin degree of freedom in graphene owing toits very weak spin-orbit coupling strength (although there can beorbital contributions in multilayer graphene samples51). Previousmeasurements of the correlated insulator at ν = 2 in t1 + 2 andt2 + 2 showed that the energy gap, as extracted from thermal activa-tion measurements, grows with in-plane field9,14–16. This behavior isconsistent with spin-polarization, as the in-plane field adds a Zeemancontribution to the energy gap. We have performed similar measure-ments in our t1 + 3 (θ = 1.29°) and t2 + 3 (θ = 1.50°) devices. Supple-mentary Fig. 2a, b shows measurements of ρxx as a function of ν in themoiré conduction band at various values of the in-planemagneticfield,B∣∣. In both devices, ρxx is highly sensitive to B∣∣ very near ν = 2, exhi-biting enhanced resistance for larger in-plane fields. These observa-tions are consistent with spin-polarized correlated insulators at ν = 2 int1 + 3 and t2 + 3. For the t1 + 3 device, we further performedtemperature-dependent measurements at several values of B∣∣. In theArrhenius plot shown in Supplementary Fig. 2c, we can extract the gapsize of the ν = 2 insulator in the thermally activated regime followingρν = 2xx / eΔν = 2=2kBT , where Δν=2 is the gap size and kB is the Boltzmannconstant. The results are shown in Supplementary Fig. 2d. The mea-sured gap size growsmonotonically as a function ofB∣∣. By fittingwith aline, assuming Δν=2(B) = gμBB +Δν=2(0), we find g ≈ 2, consistent with aspin-polarized insulating state.At ν = 1 and 3, calculations typically find the most competitiveground states to be either IVC states or SVP states51. When Cv is non-zero, these states are distinguished by the Chern number of thesymmetry-broken state, which is 0 for the IVC and non-zero for theSVP. A well-developed IVC state with non-zero Cv would be a trivialinsulator at integer band filling, whereas an SVP would exhibit theQAHE. Although these are in principle straightforward to distinguish,there can be various complicating factors in experiments. One exam-ple is twist-angle inhomogeneity in the sample, which can greatlyobscure theQAHE. Another is that the correlated statemaynot be fullygapped at zero magnetic field, preventing a straightforward determi-nation of the topology of the state. There are also exotic forms of IVCordering that break time-reversal symmetry, leading to a metallicAHE12. Althoughband structure calculation ofCv can provide guidance,the complexity of the calculation for thick tM +N structuresmay resultin incorrect predictions. Furthermore, interactions can potentiallyrenormalize the Chern number of the symmetry-broken states atpartial band filling. In general, we are not able to unambiguouslyArticle https://doi.org/10.1038/s41467-024-55001-7Nature Communications |        (2024) 15:10552 6www.nature.com/naturecommunicationsdetermine the ground state ordering at ν = 1 and 3 in our devices,except for the select cases shown in Fig. 4 in the main text.Symmetries of tM+N structuresAll tM + N structures studied here break C2z symmetry, thusenabling a gap to be opened at the CNP by D. Structures withM = N have mirror symmetry about the twisted 2D plane, and thusexhibit (approximately) identical transport for D > 0 and D < 0. Incontrast, all structures with M ≠ N break this mirror symmetry,and thus have different transport properties depending on thesign of D. This was explored in detail for the case of t1 + 2 inrefs. 9,12, and is also evident for the new, thicker structures stu-died here (Fig. 1). In cases where correlated states are seen forboth signs of D, their properties are distinct owing to substantialdifferences in the layer-polarization of the LDOS and corre-sponding differences in the non-interacting bands.Band structure calculationsWe utilize a generalized Bistritzer–MacDonald Hamiltonian for thesingle-particle band structure calculations. The effective Hamiltoniancan be written asH =HM HyintHint HN !, ð1Þwhere HM and HN are Hamiltonians for the M- and N-layer graphene,respectively, and Hint captures the interlayer coupling of the twistedmoiré interface. The multilayer graphene Hamiltonians are given byHM =H1 � Δ1 Γ ~Γ 0Γy H1 � Δ2 Γy ~Γ0~Γ Γ H1 � Δ3 Γ0 ~Γ0Γy H1 � Δ4. ..0BBBBBBBB@1CCCCCCCCA,withH1 =0ffiffi3p2 γ0ðkx � ikyÞffiffi3p2 γ0ðkx + ikyÞ 0 !,Γ=�ffiffi3p2 γ4ðkx � ikyÞ �ffiffi3p2 γ3ðkx + ikyÞγ1 �ffiffi3p2 γ4ðkx � ikyÞ !,~Γ=12 γ2 00 12 γ5 !, ~Γ0=12 γ5 00 12 γ2 !,andΔi =δi 00 δi� �ð2ÞThe Hamiltonian is appropriately truncated according to how manylayers there are. We use hopping parameters (γ0, γ1, γ2, γ3, γ4, γ5) =(2610, 361, −20, −283, −140, 20)meV. The parameter δi captures theeffect of a potential difference across the layers. For simplicity, weassume that the potential drops uniformly across the structure with atotal magnitude given by δ =PM +Ni= 1 jδij. For example, the potentials forthe t1 + 2 system would be δ1 = δ/2, δ2 =0, and δ3 =−δ/2. By this defini-tion, δ>0 corresponds to an experimentally applied D >0 that pointsfrom the top to the bottom, or thin to thick layer as shown in Fig. 1. Inthe continuum approximation, the M- and N-layer systems are coupledwhen the Bloch wave vectors differ by q!j , whereq!0 = ð0, 0Þ, q!1 = 1=LM ð� 2πffiffi3p , � 2πÞ, q!2 = 1=LM ð2πffiffi3p , � 2πÞ, and LM=a/θis the moiré wavelength. The interlayer Hamiltonian is then given byHint =X2j =0tMα exp �i 2πj3� �exp i 2πj3� �α0B@1CA, ð3Þwhere tM = 110meV and α = 0.5.Example band structures calculated for the t1 + 3 system areshown in Supplementary Fig. 15a–c. Acrossmany tM +N constructions,we generally find overlap between the moiré conduction and valencebands with δ = 0. Further, the moiré valence band tends to be moredispersive than the moiré conduction band, consistent with ourexperimental observations that correlated states primarily occur onthe electron-doped side. We performed a series of calculations for arange of θ andΔ for all tM +N layer combinations up to t3 + 6, resultingin over 5000 individual band structures. To inform our experimentalsearch, we quantify how isolated and flat themoiré conduction band isfor each set of parameters by definingϕ= ξjΔ+4jjΔ0jδE, ð4Þwhere Δ+4 (Δ0) is the energy difference between the top (bottom) ofthe moiré conduction band and the remote conduction band (moirévalence band), and δE is the bandwidth of the moiré conduction band(Supplementary Fig. 15c). We define ξ = +1 when Δ+4 and Δ0 are bothpositive, and ξ = −1 otherwise.With this definition, ϕ becomes more positive as the moiré con-duction band becomes more flat and isolated. When ϕ is negative, themoiré conduction band overlaps with the moiré valence band and/orthe remote conduction band. Supplementary Fig. 15d–g shows δE,Δ+4,Δ0, and ϕ as function of θ and δ for the t1 + 3 system, in which we findthe optimal angle condition (defined as when ϕ achieves its largestpositive value, shown in purple) to be at θ ≈ 1.30° and δ ≈ +75meV.Detailed results for each layer combination are shown in Supplemen-tary Figs. 16–29.We summarize the results for all layer combinations inSupplementary Fig. 14, where each data point is color-coded accordingto the maximum value of ϕ that is obtained for each system. Systemsup to t2 + 3 have appreciably flat and isolatedmoiré conduction bands,but for t1 + 4, t2 + 4, and thicker, ϕ is very small or negative. For eachlayer combination in which we find that a flat and isolated moiréconduction band is predicted (i.e., ϕmax >0), we further calculate thevalley Chern number of the moiré conduction band at the optimalparameter condition (i.e., for the values of θ and δwhereϕ=ϕmax). Wefind that all systems have non-zero Cv, reaching as large as Cv = 3 forthicker layer combinations.We note that we use the continuum model predictions of thenominally optimal twist angle only as a rough guide for our experi-ments. Theoretically, it is not clear whether band flatness, band isola-tion, or some specific combination of the two is the most importantparameter to optimize for achieving strongly correlated states. Evenupon choosing ametric for thenominally optimal twist angle in theory,the particular value differs for each tM +N construction and dependson detailed parameters in the continuum model which are not neces-sarily well known. Experimentally, determining the optimal twist anglefor even a single tM +N construction requires studying dozens ofdevices, which is also beyond the scope of this work. We thus leave acareful determinationof the evolutionof theoptimal twist angleon thelayer number construction to future efforts.Finally, we consider the effect of changing the Hamiltonianparameters to potentially explain the unexpected insulating state atν =0 andD ≈0 in t2 + 3 system (Fig. 3d–e).Within reasonable values ofthe tight binding parameters (γi), we find that there is always bandoverlap for twist angles between 1.33° and θ = 1.72°. The only para-meter that has a significant effect at ν =0 and D =0 is the strength ofArticle https://doi.org/10.1038/s41467-024-55001-7Nature Communications |        (2024) 15:10552 7www.nature.com/naturecommunicationsthe moiré coupling, tM. In Supplementary Fig. 30, we show bandstructure calculations varying tM around the nominal value of 110meV.Increasing tM has the effect of flattening the bands further, but doesnot create a gap at charge neutrality. Reducing tM can create a smallgap at charge neutrality, but in this case the moiré bands are muchmore dispersive andoverlapwith the remotebands. Both scenarios areinconsistent with the observations in our experiment. Taken togetherwith the temperature dependence measurements shown in Fig. 3e, weconclude that the insulating state at ν =0 and D ≈0 is most likely to bea correlated state.Berry curvature and Chern number calculationWeuse themethod from ref. 52 to calculate Berry curvature. ThemoiréBrillouin zone (MBZ) is discretized into a 50× 50 grid. For each k in theMBZ, we calculate the U(1) link:UμðkÞ=huðk+ δkμk̂μÞjuðkÞiNμðkÞ, ð5Þwhere NμðkÞ= jhuðk+ δkμk̂μÞjuðkÞij and μ = 1, 2. The Berry curvature isthen given by:ΩðkÞ= � i lnW ðkÞδk1δk2, ð6ÞwhereW(k) is U1ðkÞU2ðk+ δk1k̂1ÞU�11 ðk+ δk2k̂2ÞU�12 ðkÞ. We ensure that�π<� i lnW ðkÞ≤π. The valley Chern number is calculated as:Cv =12πZMBZd2kΩðkÞ ð7Þwhere MBZ denotes an integration over the moiré Brillouin zone.Data availabilitySource data for all the main text figures are available for this paper inthe Supplementary Information. All other data that support the find-ings of this study are available from the corresponding author uponrequest. Source data are provided with this paper.Code availabilitySource code used to perform the calculations in this paper is availablefrom the corresponding author upon request.References1. Balents, L., Dean, C. R., Efetov, D. K. & Young, A. F. Super-conductivity and strong correlations in moiré flat bands. Nat. Phys.16, 725–733 (2020).2. Andrei, E. Y. &MacDonald, A. H. Graphene bilayers with a twist.Nat.Mater. 19, 1265–1275 (2020).3. Bistritzer, R.&MacDonald, A.H.Moirébands in twisteddouble-layergraphene. Proc. Natl Acad. Sci. 108, 12233–12237 (2011).4. Suárez Morell, E., Correa, J. D., Vargas, P., Pacheco, M. & Barticevic,Z. 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The development of twisted graphenesamples is partially supported by the Department of Energy, BasicEnergy Science Programs under award DE–SC0023062 (M.Y.). XX. andM.Y. acknowledge support from the State of Washington–funded CleanEnergy Institute. D.W. was supported by an appointment to the Intelli-gence Community Postdoctoral Research Fellowship Program at Uni-versity of Washington administered by Oak Ridge Institute for Scienceand Education through an interagency agreement between the USDepartment of Energy and the Office of the Director of National Intelli-gence. E.T. and E.A.-M. were supported by grant no. NSF GRFPDGE–2140004. K.W. and T.T. acknowledge support from the JSPSKAKENHI (Grant Numbers 21H05233 and 23H02052) andWorld PremierInternational Research Center Initiative (WPI), MEXT, Japan. Y.-H.Z. wassupported by the National Science Foundation under Grant No.DMR–2237031. This work made use of shared fabrication facilities pro-vided by NSF MRSEC 2308979. This research acknowledges usage ofthemillikelvin optoelectronicquantummaterial laboratory supportedbythe M.J. Murdock Charitable Trust.Author contributionsD.W., E.T., A.O., E.A.-M.,M.H., and K.H. fabricated the devices. D.W., R.S.,E.T., and M.H. performed the measurements and analyzed data. Y.Z.wrote the code to calculate the continuummodel band structures. D.W.performed the continuummodel calculations. K.W. and T.T. grew theBNcrystals. X.X., J.F., and M.Y. supervised device fabrication, measure-ment, and data analysis. D.W., R.S., E.T., J.F., and M.Y. wrote the paperwith input from all authors.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-024-55001-7.Correspondence and requests for materials should be addressed toJoshua Folk or Matthew Yankowitz.Peer review information Nature Communications thanks Anna Seiler,Tobias Stauber who co-reviewed with Miguel Sánchez Sánchezand theother, anonymous, reviewer(s) for their contribution to thepeer reviewofthis 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/s41467-024-55001-7Nature Communications |        (2024) 15:10552 9https://doi.org/10.1038/s41467-024-55001-7http://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 Topological flat bands in a family of multilayer graphene moiré lattices Results Isolated moiré–localized flat bands Common features of the correlated phases Abundance of correlated states in t2 + 3 Topological states in tM + N graphene Discussion Methods Device fabrication Transport measurements Extraction of Fermi surface degeneracy from quantum oscillations Multiband transport in t2 + 3 graphene Determination of isospin polarization at integer band fillings Symmetries of tM + N structures Band structure calculations Berry curvature and Chern number calculation Data availability Code availability References Acknowledgements Author contributions Competing interests Additional information