# Fileset

[PhysRevX.15.011045.pdf](https://mdr.nims.go.jp/filesets/8799a97d-8058-46c7-91c1-2dddb765f7e6/download)

## Creator

Dacen Waters, Anna Okounkova, Ruiheng Su, Boran Zhou, Jiang Yao, [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, Matthew Yankowitz

## Rights

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

## Other metadata

[Chern Insulators at Integer and Fractional Filling in Moiré Pentalayer Graphene](https://mdr.nims.go.jp/datasets/d62c8c95-e7ef-4bf8-9506-7f7b22963afc)

## Fulltext

Chern Insulators at Integer and Fractional Filling in Moiré Pentalayer GrapheneChern Insulators at Integer and Fractional Filling in Moiré Pentalayer GrapheneDacen Waters ,1,* Anna Okounkova ,1,* Ruiheng Su ,2,3 Boran Zhou ,4 Jiang Yao ,1 Kenji Watanabe ,5Takashi Taniguchi ,6 Xiaodong Xu,1,7 Ya-Hui Zhang,4 Joshua Folk ,2,3 and Matthew Yankowitz 1,7,†1Department of Physics, University of Washington, Seattle, Washington 98195, USA2Quantum Matter Institute, University of British Columbia,Vancouver, British Columbia, V6T 1Z1, Canada3Department of Physics and Astronomy, University of British Columbia,Vancouver, British Columbia, V6T 1Z1, Canada4Department of Physics and Astronomy, Johns Hopkins University, Baltimore, Maryland 21205, USA5Research Center for Electronic and Optical Materials, National Institute for Materials Science,1-1 Namiki, Tsukuba 305-0044, Japan6Research Center for Materials Nanoarchitectonics,National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan7Department of Materials Science and Engineering,University of Washington, Seattle, Washington 98195, USA(Received 12 November 2024; revised 7 January 2025; accepted 5 February 2025; published 27 February 2025)The advent of moiré platforms for engineered quantum matter has led to discoveries of integer andfractional quantum anomalous Hall effects, with predictions for correlation-driven topological states basedon electron crystallization. Here, we report an array of trivial and topological insulators formed in a moirélattice of rhomobohedral pentalayer graphene (R5G). At a doping of one electron per moiré unit cell(ν ¼ 1), we see a correlated insulator with a Chern number that can be tuned between C ¼ 0 and þ1 by anelectric displacement field. This is accompanied by a series of additional Chern insulators with C ¼ þ1originating from fractional fillings of the moiré lattice—ν ¼ 1=4, 1=3, and 2=3—associated with theformation of moiré-driven topological electronic crystals. At ν ¼ 2=3 the system exhibits an integerquantum anomalous Hall effect at zero magnetic field, but further develops hints of an incipient C ¼ 2=3fractional Chern insulator in a modest field. Our results establish moiré R5G as a fertile platform forstudying the competition and potential intertwining of integer and fractional Chern insulators.DOI: 10.1103/PhysRevX.15.011045 Subject Areas: Condensed Matter Physics,Materials ScienceI. INTRODUCTIONMoiré materials with flat bands are ideal platforms forstudying the interplay between strongly correlated andtopological states of matter [1–5]. Early work in this fieldestablished the existence of a wide range of correlation-driven topological states within the magnetic subbands ofthe Hofstadter butterfly spectrum, including states thatbreak the translational symmetry of the moiré lattice (called“symmetry-broken Chern insulators”) and others featuringfractionalized quasiparticles [fractional Chern insulators(FCI)] [6,7]. Recently, remarkable progress has beenmade in achieving analogous states in the absence of anexternal magnetic field using moiré lattices with flat bandsand suitable topological properties. Prominent examplesinclude the integer and fractional quantum anomalous Hall(IQAH and FQAH) states found in twisted bilayer MoTe2[8–11] and rhombohedral multilayer graphene aligned withhexagonal boron nitride (h-BN) [12–14], as well as Cherninsulator states with discrete translational symmetry break-ing found in several twisted graphene structures [15–17].Forming Chern insulators in moiré lattices requiresspontaneous valley polarization, resulting in brokentime-reversal symmetry [4,5]. Many-body gaps can beopened at integer fillings of the moiré flat bands, resultingin the IQAH effect when the filled bands have nonzerototal Chern number. Topological gapped states can alsoform at fractional fillings of the moiré bands, but requirethe assistance of additional correlation-driven mechanisms.FQAH states are a notable example, forming anyonicquasiparticles but otherwise not needing to break any ofthe remaining symmetries of the system. Alternatively,*These authors contributed equally to this work.†Contact author: myank@uw.eduPublished by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution tothe author(s) and the published article’s title, journal citation,and DOI.PHYSICAL REVIEW X 15, 011045 (2025)2160-3308=25=15(1)=011045(8) 011045-1 Published by the American Physical Societyhttps://orcid.org/0000-0003-3588-0039https://orcid.org/0009-0001-8713-214Xhttps://orcid.org/0000-0003-1135-0498https://orcid.org/0000-0001-5669-8450https://orcid.org/0000-0001-5848-8000https://orcid.org/0000-0003-3701-8119https://orcid.org/0000-0002-1467-3105https://orcid.org/0000-0003-4455-5609https://orcid.org/0000-0002-5637-9203https://ror.org/00cvxb145https://ror.org/03rmrcq20https://ror.org/03rmrcq20https://ror.org/00za53h95https://ror.org/026v1ze26https://ror.org/026v1ze26https://ror.org/00cvxb145https://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevX.15.011045&domain=pdf&date_stamp=2025-02-27https://doi.org/10.1103/PhysRevX.15.011045https://doi.org/10.1103/PhysRevX.15.011045https://doi.org/10.1103/PhysRevX.15.011045https://doi.org/10.1103/PhysRevX.15.011045https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/electrons can spontaneously break translational symmetryto form topological electronic crystal (TEC) states exhibit-ing an IQAH effect. Examples include the anomalous Hallcrystal, recently considered for moiré and nonmoiré rhom-bohedral pentalayer graphene (R5G) [18–28], and moiré-driven topological electronic crystals in other twistedgraphene lattices [15–17]. Understanding the interplaybetween integer and fractional QAH states at fractionalband filling presents a critical open challenge for the field.Here, we study an array of correlated insulating statesarising in a moiré lattice of R5G aligned to h-BN, with aperiod of 10.8 nm. Recent pioneering work on this systemrevealed the emergence of FQAH states at several Jain-sequence fractions for ν < 1, all arising in a device with aslightly larger moiré period of 11.5 nm [13]. In contrast, ourdevice instead exhibits IQAH states originating from bothinteger and fractional fillings of the electron-doped moiréconduction band. The insulating states are most robust nearcommensurate band fillings (ν ¼ 4n=ns ¼ 1=3, 2=3, and 1,where ns is the moiré superlattice density [29]), butnevertheless extend over a wide range of moiré bandfilling, 0 < ν ⪅ 1. In a magnetic field, there are signaturesof both integer and fractional Chern insulators associatedwith ν ¼ 2=3. A second device, with an even smaller(8.8 nm) moiré period, does not exhibit any such correlatedinsulators at commensurate band fillings, further hinting atthe key role of the moiré potential in generating these states(see Supplemental Material and Fig. S18 [29]).II. TRIVIAL AND TOPOLOGICAL CORRELATEDINSULATORSFigure 1(a) shows a schematic of our device, in whichR5G is nearly aligned to one h-BN dielectric but mis-aligned from the other. The top and bottom graphite gatesenable independent control over n and D, the latter ofwhich we define to be positive when available states in theconduction band are pushed away from the moiré interface.Figure 1(b) shows a map of the longitudinal resistance ρxxof the device taken over a wide range of n and D. Weextract the density needed to fully fill the lowest moirébands (ν ¼ �4) to be ns ¼ 3.98 × 1012 cm−2, correspond-ing to a moiré period of 10.8 nm and a twist angle betweenthe R5G and h-BN of θ ¼ 0.90° (assuming a 1.7% latticemismatch between graphene and h-BN). Overall, thesalient features we see are consistent with those reportedin a prior study of moiré R5G [13], including a series ofinsulating states at the charge neutrality point (ν ¼ 0) andresistive states at various integer values of ν correspondingto either correlated or single-particle band insulators [29].Figure 1(c) shows a continuum model band structurecalculation for an interlayer potential difference of δ ¼þ150 meV [29]. The calculation predicts that the lowestmoiré conduction band (colored in purple) is gapped fromthe highest moiré valence band, but overlaps the secondconduction band. Both of these features are consistentwith the experiment, in which there is an insulatingstate at ν ¼ 0 but a metallic state at ν ¼ 4 for large positiveD [see Fig. 1(d) for a representative measurement atD ¼ 0.740 V=nm, corresponding to the purple dashed linein Fig. 1(b)].Figures 1(e) and 1(f) show enlarged maps of the field-symmetrized longitudinal (ρxx) and antisymmetrized Hall(ρxy) resistances in the high-D region between ν ¼ 0 andν ≈ 1, outlined by the black dashed square in Fig. 1(b).The key features seen in these maps are summarized inFig. 1(g). Most obviously, a stripelike region cuts diago-nally across the center of the maps in Figs. 1(e) and 1(f), inwhich one or both of ρxx and ρxy are very large. The systemis a trivial insulator for ν ⪅ 1=2–2=3 (depending on theprecise value of D), previously attributed to the formationof a Wigner crystal with period larger than the originalmoiré lattice [13]. There is also a pocket centered atν ¼ 2=3 featuring large ρxy and a deep suppression ofρxx. This behavior instead indicates the formation of atopological gapped state. Finally, there is a vertical featureat ν ¼ 1 that exhibits a large anomalous Hall effect (AHE)for D larger than a critical value of Dc ≈ 0.82 V=nm,but diverging ρxx with no AHE for D < Dc [Figs. 1(h)and 1(i)].III. TOPOLOGICAL PHASE TRANSITION AT ν= 1To probe the nature of these states, we plot Landau fandiagrams for two values of displacement field [Figs. 2(a)and 2(b)]. At D ¼ 0.910 V=nm, we see a correlated Cherninsulator emerging from B ¼ 0 at ν ¼ 1, along with addi-tional associated quantum Hall states to its right at higherfield. A line cut at B ¼ 0.75 T in Fig. 2(c) confirms that thecorrelated Chern insulator state exhibits the anticipatedvalues of ρxx ≈ 0 and ρxy ≈ h=e2, where h is Planck’sconstant and e is the charge of the electron. Comparison tothe Streda formula [30,31], ð∂n=∂BÞ ¼ Cðh=eÞ, furtherindicates that the Chern number of this state is C ¼ þ1 (seethe white dashed line at the top of the Landau fans). Welabel gapped states by their Chern number and band fillingupon extrapolation to B ¼ 0 following the conventionðC ¼ a; ν ¼ bÞ≡ Cab, such that this state is Cþ11 .The Landau fan taken at D ¼ 0.740 V=nm exhibitsmarkedly different behavior. At ν ¼ 1, ρxx far exceedsh=e2 and ρxy exhibits diverging behavior with an abruptsign reversal across integer filling [see line cuts in Fig. 2(d)taken at B ¼ 0.20 T]. The insulating state does notdisperse with B, and associated quantum Hall states emergeroughly symmetrically to its left and right. Collectively,this behavior is consistent with a topologically trivialstate C01. Apparently, the critical displacement field,Dc ≈ 0.82 V=nm, where the AHE at ν ¼ 1 vanishescorresponds to a phase transition between two topologi-cally distinct states with associated Chern numbers ofC ¼ 0 and þ1. Such a phase transition typically requiresDACEN WATERS et al. PHYS. REV. X 15, 011045 (2025)011045-2a gap closure and reopening; we do not see evidence for thisin our device, potentially due to disorder.Many recent theory works [18–20,22–28,32,33] haveconsidered the nature of the Chern insulator previouslyreported at ν ¼ 1 [13] in R5G aligned to h-BN. In manyother moiré systems, a correlated gap opens at ν ¼ 1 whenthe fourfold isospin degeneracy is lifted by interactions[4,5]. But in R5G, the second moiré conduction bandoverlaps the first; as a result, this mechanism alone wouldnot be expected to open a gap. Instead, the additionalformation of a topological electronic crystalline order via alarge spatial redistribution of the charge density is believedto open the gap [18–28]. However, a definitive under-standing of the gap at ν ¼ 1 remains elusive because theputative electronic crystal is commensurate with the moirélattice, and is thus challenging to distinguish unambigu-ously from conventional moiré Chern insulators [34].Nevertheless, the basic TEC framework is consistent withthe C ¼ 1 state we observe at ν ¼ 1. The trivial C ¼ 0insulator at smallerD can be formed similarly by lifting thesame isospin degeneracies and crystallizing the electrons,but needs the filled states to instead have a total Chernnumber of zero.To better understand the possible nature of the gappedstates at ν ¼ 1, we compare our results to Hartree-Fock(HF) calculations performed in a simplified model wherethe moiré potential is artificially set to zero (more realisticcalculations with nonzero moiré potential yield similarconclusions; see Ref. [29]). Figures 2(e) and 2(f) show thecalculated band structure at ν ¼ 1 both with and without(a) (c) (e) (g)(f) (h) (i)(d)(b)h-BNh-BNGraphiteGraphiteAlignedFIG. 1. Device transport characterization and correlated states at ν ≤ 1. (a) Schematic of the device. R5G is encapsulated betweenflakes of h-BN with graphite top and bottom gates. The R5G is misaligned from the bottom h-BN, and has a small twist angle ofθ ¼ 0.90° with the top h-BN to form a moiré superlattice [29]. (b) Map of ρxx taken at B ¼ 0 over a wide range of the n −D parameterspace. The data are acquired in multiple submeasurements to mitigate measurement artifacts [29]. The gray colored data in the colorscale indicate that ρxx was measured to be negative, indicating either a highly resistive state or poor equilibration of the contacts [29].(c) Single-particle calculated band structure with δ ¼ þ150 meV and a moiré period of 10.8 nm. This sign of δ corresponds to D > 0.The lowest moiré conduction band is colored in purple. (d) Line traces of ρxx and ρxy acquired atD ¼ 0.740 V=nm [corresponding to theposition of the purple dashed line in (b)] with B ¼ 0.2 T (not symmetrized). (e) Enlarged map of ρxx symmetrized at jBj ¼ 100 mT fromthe region of the black dashed box in (b). (f) Similar map of antisymmetrized ρxy. (g) Schematic indicating transport features seen in (e)and (f). Regions shaded in pink satisfy the condition ρxy > h=2e2. Regions shaded in purple satisfy the condition ρxx > h=e2. Regionssatisfying both conditions are also shaded in purple (since the large value of ρxy is an artifact corresponding to mixing with ρxx). Regionswith negative ρxx are shaded in purple, as they correspond to measurement artifacts in very insulating states [29]. (h) Map of Δρxy=2 ¼ðρ↑xy − ρ↓xyÞ=2 at ν ¼ 1, where the arrows indicate the direction B is swept. (i) A line cut of Δρxy=2 from (h) at B ¼ 0.CHERN INSULATORS AT INTEGER AND FRACTIONAL … PHYS. REV. X 15, 011045 (2025)011045-3the HF approximation for two different values of δ. In bothcases, the state is gapless at the single-particle level butgapped as a result of interactions. The isolated band has aChern number of C ¼ 0 for δ less than a critical value ofδc ≈ 140 meV, but C ¼ þ1 for δ > δc. This is consistentwith our observed phase transition between a C ¼ 0 andþ1 state as D is increased. At the noninteracting level, theintegrated Berry curvature up to the Fermi level at ν ¼ 1can be any arbitrary value since the band is not isolated.Figure 2(g) shows that this value is ð1=2πÞ RBZ d2kΩðkÞ ≈0.5 in the calculations for both δ ¼ 120 and 150 meV [29].When interactions open a gap, however, the filled statesbelow the Fermi level must have Berry curvature thatintegrates to a quantized value, equal to C. In this context,the gap opening simultaneously necessitates an interaction-driven modification of the Berry curvature in order tosatisfy the quantization condition. Small changes in thequantum geometry of the single-particle bands with δ canthus lead to an abrupt phase transition between otherwisesimilar states having C ¼ 0 (e.g., a generalized Wignercrystal [35]) and C ¼ 1 (e.g., a moiré-driven TEC [17]).IV. INTEGER AND FRACTIONAL CHERNINSULATORS AT ν = 2=3Next we turn to the topological states observed atfractional band filling in Figs. 1(e) and 1(f). Figure 3(a)shows a Landau fan acquired at D ¼ 0.863 V=nm, which(a)(c) (d)(b) (e)(f)(g)FIG. 2. Topological and trivial correlated insulators at ν ¼ 1. (a) Landau fan diagram of ρxx (top) and ρxy (bottom) taken atD ¼ 0.910 V=nm. The white dashed line shows the expected evolution of a C ¼ þ1 state originating from ν ¼ 1 based on the Stredaformula. (b) Similar Landau fan taken at D ¼ 0.740 V=nm. The white dashed line shows the expected evolution of a C ¼ 0 stateoriginating from ν ¼ 1. The black arrow denotes the trajectory of a quantum Hall state with filling factor of−3 originating from ν ¼ 1. Thespeckled features projecting vertically near ν ¼ 1 are artifacts due to a large contact resistance. (c) Line traces of ρxx and ρxy taken atB ¼ 0.75 T from theD ¼ 0.910 V=nm Landau fan, as indicated by the black dashed line in (a). (d), Similar line cuts taken at B ¼ 0.20 Tfrom the D ¼ 0.740 V=nm Landau fan, as indicated by the black dashed line in (b). (e) Single-particle (SP) calculations of the spatialdistribution of carrier density nðrÞ at a filling of ν ¼ 1. The left-hand (right-hand) panel is calculated with δ ¼ 150 meV (120 meV). Bothcorrespond to metallic states. The associated band structure calculations are shown below each plot. The calculation is performed for amoiré period of LM ¼ 11.1 nm with the moiré potential strength artificially set to zero [29]. (f) Similar calculations performed with theHartree-Fock (HF) method. Both correspond to insulating states, with the filled band having C ¼ þ1 (0) for δ ¼ 150 meV (120 meV),shown in orange (purple). Dashed lines at zero energy in (e) and (f) denote the Fermi energy at ν ¼ 1. The real-space densities in (e) and(f) share the same color scale. The dashed black triangles indicate the moiré unit cell. (g) Berry curvature integrated from the center to theedge of the moiré Brillouin zone (BZ) for both values of δ considered in (e) and (f). The dashed (solid) curves show the single-particle(Hartree-Fock) calculations. The inset schematic shows the moiré BZ in black, with high symmetry points labeled. The gray area depictsthe area of integration, which scales with rk, such that Hrk ¼ r2kHBZ, where HBZ is the area of the full moiré BZ.DACEN WATERS et al. PHYS. REV. X 15, 011045 (2025)011045-4cuts through a region with a ρxx minimum and concomitantρxy ≈ h=e2 surrounding ν ¼ 2=3 [see line cuts in Fig. 3(b)].Sweeping the perpendicular magnetic field at ν ¼ 2=3, ρxyswitches between ≈� h=e2 in a single hysteresis loop witha small coercive field of B ≈ 10 mT [Fig. 3(c)]. The slopeof this state in the Landau fan is consistent with a C ¼ þ1Chern insulator based on the Streda formula (i.e., Cþ12=3), asindicated by the black dashed line in Fig. 3(a). Thesefeatures are plainly incompatible with the C ¼ 2=3 FQAHstate at ν ¼ 2=3 reported previously [13], since ourobserved state has neither the appropriate Streda slope inthe Landau fan nor the appropriate quantization of ρxy fora C ¼ 2=3 state (that is, we find ρxy ¼ h=e2 rather than3h=2e2). Instead, the gapped Cþ12=3 state most naturallycorresponds to a moiré-driven TEC that spontaneouslyenlarges the unit cell area [29]. Notably, ρxy remains largeover a relatively wide range of doping, potentially indicat-ing that the crystalline order persists in some form evenupon doping away from ν ¼ 2=3.Figure 4(a) shows a Landau fan at D ¼ 0.820 V=nm,which differs at zero and small B in that it is dominated by atrivial insulating state. However, a series of topologicalstates abruptly emerge above B ≈ 2 T, including threeC ¼ þ1 states projecting to ν ¼ 1=4, 1=3, and 2=3 atB ¼ 0. These also correspond to moiré-driven TEC stateswith associated unit-cell enlargement, but require theassistance of the magnetic field to form [see line cuts inFigs. 4(b)–4(d)]. Additionally, there is an oval-shapedfeature near ν ≈ 0.6 centered at B ≈ 3.5 T, in which bothρxx and ρxy are abruptly suppressed and there are insteadquantum Hall states projecting to the charge neutralitypoint at ν ¼ 0. This region is separated from the surround-ing area of the Landau fan by a first-order phase transitionand has the same origin as the sharp curved feature in theLandau fan in Fig. 2(b). This phase transition potentiallyreflects a collapse of the crystalline electronic order [29].Remarkably, the Landau fan also contains an extremelynarrow feature in which the antisymmetrized ρxy clearlyexceeds h=e2 [see the line cut in Fig. 4(e)]. The purpledashed line near the top of the Landau fan denotes theposition and trajectory of this state, which projects pre-cisely to ν ¼ 2=3 at B ¼ 0 and has a Streda slope matchinga C ¼ þ2=3 state (i.e., Cþ2=32=3 ). These features are con-sistent with an incipient ν ¼ 2=3 FCI state emergingwith B. Although the state is not fully developed, withlarge residual ρxx and nonquantized ρxy, the observation ofρxy > h=e2 with a Streda slope implying C ¼ þ2=3 has nosimple explanation besides a field-induced FCI state.V. DISCUSSIONOur measurements reveal two unexpected new featuresof moiré R5G. First, we find that the correlated insulatorat ν ¼ 1 can have a Chern number of either C ¼ 0 or 1,depending on D. Although the nature of these two statesremains to be fully understood, the existence of the C ¼ 0insulator sharply constrains potential theoretical descrip-tions of the system. We have considered a possible model inwhich the many-body gap opening simultaneously drives asubstantial redistribution of Berry curvature, with smallchanges in the interlayer potential tipping the balance of thefilled band toward either C ¼ 0 or 1. These two states arepredicted to feature different real-space charge distributions[Fig. 2(f)], which may be detectable in future scanningprobe experiments.Second, we find that the correlated state at ν ¼ 2=3 canexhibit an IQAH with C ¼ 1, rather than the previouslyobserved FQAH with C ¼ 2=3 [13]. Such a state is mostnaturally explained by a spontaneous enlargement of theunit cell, corresponding to the formation of a Cherninsulator at integer filling of the folded Brillouin zone.(a)(b) (c)FIG. 3. IQAH state at ν ¼ 2=3. (a) Landau fan diagram of ρxx(top) and ρxy (bottom) taken at D ¼ 0.863 V=nm. The blackdashed line shows the expected evolution of a C ¼ þ1 stateoriginating from ν ¼ 2=3 based on the Streda formula. (b) Linetraces of ρxx and ρxy taken at B ¼ 30 mT from the D ¼0.863 V=nm Landau fan in (a). The blue shaded region corre-sponds to the contiguous range of ν for which ρxy > 0.9h=e2.(c) Measurement of ρxy at ν ¼ 2=3 acquired as B is swept backand forth across zero. Arrows denote the sweep direction of themagnetic field.CHERN INSULATORS AT INTEGER AND FRACTIONAL … PHYS. REV. X 15, 011045 (2025)011045-5Hints of the C ¼ 2=3 FCI state additionally appear once afew-Tesla magnetic field is applied, pointing to a closecompetition between the integer and fractional Cherninsulator states. The appearance of both states in a magneticfield is possible because these two states bifurcate todifferent values of charge doping with B, as describedby the Streda formula, and can thus be stabilized separatelyby gating.Our results highlight the need for additional experimentsto map out the dependence of FQAH and electronic crystalstates on the precise value of the moiré period, as well asother device parameters such as strain and Coulombscreening [36]. As a first step in this direction, we showmeasurements from a second device with a slightly largertwist angle, and thus a smaller moiré period of 8.8 nm, inSupplemental Material [29]. Although many of the salienttransport features resemble those observed in Fig. 1(b),there are no correlated insulators at any integer or rationalfractional values of ν for any D [29]. The absence of suchstates provides evidence that a long-wavelength moirépotential may be necessary for seeding the formation ofinteger and fractional QAH states in R5G. Understandingthe sensitive dependence of these states on twist angle mayhelp to unravel the nature of the rich correlated phasediagram of moiré R5G.Looking forward, scanning probe studies will be crucialfor directly imaging the putative charge-ordered states inthis system, enabling a better understanding of the nature ofthe doping–and displacement field–dependent translationalsymmetry breaking. Such measurements will also behelpful for determining the potential effects of disorder,which may localize charge or create defects in the elec-tronic crystalline states. Understanding the relationshipbetween integer and fractional Chern insulators also standsas a critical open challenge for the field, especially as itrelates to determining the ultimate ground state ordering ofthe system.ACKNOWLEDGMENTSThe authors thank Long Ju for sharing preliminary data;Hart Goldman, Trithep Devakul, and David Cobden forhelpful discussions; and Yinong Zhang, Jordan Fonseca,and Jiaqi Cai for technical assistance with the AM-KPFMimaging of graphene stacking domains. Research at theUniversity of Washington on correlation-driven topologyin pentalayer graphene was solely supported as part ofProgrammable Quantum Materials, an Energy FrontierResearch Center funded by the U.S. Department ofEnergy (DOE), Office of Science, Basic Energy Sciences(a) (b)(c)(d)(e)FIG. 4. Integer and fractional Chern insulators at fractional ν. (a) Landau fan of ρxx (top) and ρxy (bottom), taken atD ¼ 0.820 V=nm.The black dashed lines show the expected evolution of C ¼ þ1 states originating from ν ¼ 1=4, 1=3, and 2=3. The purple dashed lineshows the same for a C ¼ þ2=3 state originating from ν ¼ 2=3. (b) Line traces of ρxx and ρxy taken along the trajectory indicated by theblack dashed line associated with the Cþ11=4 state in (a). The blue shaded region corresponds to the contiguous region of jρxyj > 0.9h=e2(excluding B < 4.5 T which is dominated by the trivial insulating phase). (c),(d) Similar line traces and shading for the Cþ11=3 and Cþ12=3states in (a),(c). (e) Similar line traces taken along the trajectory indicated by the purple dashed line associated with the Cþ2=32=3 state in (a).The purple shaded region corresponds to contiguous range of ν for which ρxy > h=e2 (excluding B < 3 T).DACEN WATERS et al. PHYS. REV. X 15, 011045 (2025)011045-6(BES), under Award No. DE-SC0019443. Experiments atthe University of British Columbia were undertaken withsupport from the Natural Sciences and EngineeringResearch Council of Canada; the Canada Foundation forInnovation; the Canadian Institute for Advanced Research;the Max Planck-UBC-UTokyo Centre for QuantumMaterials and the Canada First Research ExcellenceFund, Quantum Materials and Future TechnologiesProgram; and the European Research Council (ERC)under the European Union’s Horizon 2020 research andinnovation program, Grant Agreement No. 951541. D.W.was supported by the Intelligence Community PostdoctoralResearch Fellowship Program at University of Washingtonadministered by Oak Ridge Institute for Science andEducation through an interagency agreement betweenthe U.S. Department of Energy and the Office of theDirector of National Intelligence. M. Y., X. X., andA. O. acknowledge support from the State ofWashington-funded Clean Energy Institute. K.W. and T. T.acknowledge support from the JSPS KAKENHI (GrantsNo. 21H05233 and No. 23H02052) and World PremierInternational Research Center Initiative (WPI), MEXT,Japan. This work made use of shared fabrication facilitiesat UW provided by NSF MRSEC 2308979 with graphenedevice development supported by National ScienceFoundation (NSF) CAREER Award No. DMR-2041972.Theoretical calculations were supported by the NationalScience Foundation under Grant No. DMR-2237031.This work acknowledges usage of the millikelvin opto-electronic quantum material laboratory supported by theM. J. Murdock Charitable Trust.A. O. and D. W. developed the sample fabricationcapabilities; A. O. fabricated the sample; D. W. andA. O. measured the sample in the Yankowitz lab atUW; R. S. performed follow-up measurements in theFolk lab at UBC, which appear in many of the maintext figures, under the supervision of J. F. and in dis-cussion with D. W., A. O., and M. Y.; D. W. and A. O.analyzed the data with the assistance of R. S.; J. Y.developed the atomic force microscopy-based imagingtechnique to detect rhombohedral graphene under thesupervision of X. X.; M. Y. supervised the project; D. W.,A. O, and M. Y. wrote the manuscript with B. Z. andY.-H. Z. providing theory support; K. W. and T. T. pro-vided the h-BN crystals.The authors declare no competing interests.DATA AVAILABILITYAll data that support the findings of this study areavailable from the contact author upon request.[1] L. Balents, C. R. Dean, D. K. Efetov, and A. F. Young,Superconductivity and strong correlations in moiré flatbands, Nat. Phys. 16, 725 (2020).[2] E. Y. Andrei and A. H. MacDonald, Graphene bilayers witha twist, Nat. Mater. 19, 1265 (2020).[3] K. F. Mak and J. Shan, Semiconductor moiré materials,Nat. Nanotechnol. 17, 686 (2022).[4] K. P. Nuckolls and A. Yazdani, A microscopic perspectiveon moiré materials, Nat. Rev. Mater. 9, 460 (2024).[5] P. C. Adak, S. Sinha, A. Agarwal, and M.M. Deshmukh,Tunable moiré materials for probing Berry physics andtopology, Nat. Rev. Mater. 9, 481 (2024).[6] C. R. Dean et al., Hofstadter’s butterfly and the fractalquantum Hall effect in moiré superlattices, Nature (London)497, 598 (2013).[7] E. M. Spanton et al., Observation of fractional Cherninsulators in a van der Waals heterostructure, Science360, 62 (2018).[8] J. Cai et al., Signatures of fractional quantum anomalous Hallstates in twisted MoTe2, Nature (London) 622, 63 (2023).[9] Y. Zeng et al., Thermodynamic evidence of fractional Cherninsulator in moiréMoTe2, Nature (London) 622, 69 (2023).[10] H. Park et al., Observation of fractionally quantizedanomalous Hall effect, Nature (London) 622, 74 (2023).[11] F. Xu et al., Observation of integer and fractional quantumanomalous Hall effects in twisted bilayerMoTe2, Phys. Rev.X 13, 031037 (2023).[12] G. Chen et al., Tunable correlated Chern insulator andferromagnetism in a moiré superlattice. Nature (London)579, 56 (2020).[13] Z. Lu et al., Fractional quantum anomalous Hall effect inmultilayer graphene, Nature (London) 626, 759 (2024).[14] J. Xie et al., Even- and odd-denominator fractional quan-tum anomalous Hall effect in graphene moiré superlattices,arXiv:2405.16944.[15] Y. Xie et al., Fractional Chern insulators in magic-angletwisted bilayer graphene, Nature (London) 600, 439 (2021).[16] H. Polshyn et al., Topological charge density waves at half-integer filling of a moiré superlattice, Nat. Phys. 18, 42(2022).[17] R. Su, D. Waters, B. Zhou, K. Watanabe, T. Taniguchi, Y.-H.Zhang, M. Yankowitz, and J. Folk, Moiré-driven topologi-cal electronic crystals in twisted graphene, Nature (London)637, 1084 (2025).[18] Z. Dong, A. S. Patri, and T. Senthil, Theory of fractionalquantum anomalous Hall phases in pentalayer rhombohe-dral graphene moiré structures, Phys. Rev. Lett. 133,206502 (2024).[19] B. Zhou, H. Yang, and Y. Zhang, Fractional quantumanomalous Hall effects in rhombohedral multilayer gra-phene in the moiréless limit, Phys. Rev. Lett. 133, 206504(2024) (2023).[20] J. Dong et al., Anomalous Hall crystals in rhombohedralmultilayer graphene I: Interaction-driven Chern bandsand fractional quantum Hall states at zero magnetic field,Phys. Rev. Lett. 133, 206503 (2024).[21] Y. H. Kwan, J. Yu, J. Herzog-Arbeitman, D. K. Efetov,N. Regnault, and B. A. Bernevig, Moiré fractional ChernCHERN INSULATORS AT INTEGER AND FRACTIONAL … PHYS. REV. X 15, 011045 (2025)011045-7https://doi.org/10.1038/s41567-020-0906-9https://doi.org/10.1038/s41563-020-00840-0https://doi.org/10.1038/s41565-022-01165-6https://doi.org/10.1038/s41578-024-00682-1https://doi.org/10.1038/s41578-024-00671-4https://doi.org/10.1038/nature12186https://doi.org/10.1038/nature12186https://doi.org/10.1126/science.aan8458https://doi.org/10.1126/science.aan8458https://doi.org/10.1038/s41586-023-06289-whttps://doi.org/10.1038/s41586-023-06452-3https://doi.org/10.1038/s41586-023-06536-0https://doi.org/10.1103/PhysRevX.13.031037https://doi.org/10.1103/PhysRevX.13.031037https://doi.org/10.1038/s41586-020-2049-7https://doi.org/10.1038/s41586-020-2049-7https://doi.org/10.1038/s41586-023-07010-7https://arXiv.org/abs/2405.16944https://doi.org/10.1038/s41586-021-04002-3https://doi.org/10.1038/s41567-021-01418-6https://doi.org/10.1038/s41567-021-01418-6https://doi.org/10.1038/s41586-024-08239-6https://doi.org/10.1038/s41586-024-08239-6https://doi.org/10.1103/PhysRevLett.133.206502https://doi.org/10.1103/PhysRevLett.133.206502https://doi.org/10.1103/PhysRevLett.133.206504https://doi.org/10.1103/PhysRevLett.133.206504https://doi.org/10.1103/PhysRevLett.133.206503insulators III: Hartree-Fock phase diagram, magic angleregime for Chern insulator states, the role of the moirépotential and Goldstone gaps in rhombohedral graphenesuperlattices, arXiv:2312.11617.[22] T. Tan and T. Devakul, Parent Berry curvature and theideal anomalous Hall crystal, Phys. Rev. X 14, 041040(2024).[23] T. Soejima et al., Anomalous Hall crystals in rhombohedralmultilayer graphene II: General mechanism and a minimalmodel, Phys. Rev. B 110, 205124 (2024).[24] Z. Dong, A. S. Patri, and T. Senthil, Stability of anomalousHall crystals in multilayer rhombohedral graphene, Phys.Rev. B 110, 205130 (2024).[25] X. Shen et al., Stabilizing fractional Chern insulators viaexchange interaction in moiré systems, arXiv:2405.12294.[26] Y. Zeng, D. Guerci, V. Crepel, A. J. Millis, and J. Cano,Sublattice structure and topology in spontaneously crystal-lized electronic states, Phys. Rev. Lett. 132, 236601 (2024).[27] K. Kudo, R. Nakai, and K. Nomura, Quantum anomalous,quantum spin, and quantum valley Hall effects in pentalayerrhombohedral graphene, Phys. Rev. B 110, 245135 (2024).[28] V. Crépel and J. Cano, Efficient prediction of superlatticeand anomalous miniband topology from quantum geometry,Phys. Rev. X 15, 011004 (2025).[29] See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevX.15.011045, whichincludes methods, Supplemental Text, and SupplementalFigs. 1–18.[30] P. Streda, Quantised Hall effect in a two-dimensionalperiodic potential, J. Phys. C 15, L1299 (1982).[31] P. Streda, Theory of quantised Hall conductivity in twodimensions, J. Phys. C 15, L717 (1982).[32] J. Yu, J. Herzog-Arbeitman, Y. H. Kwan, N. Regnault, andB. A. Bernevig, Moiré fractional Chern insulators IV:Fluctuation-driven collapse of FCIs in multi-band exactdiagonalization calculations on rhombohedral graphene,arXiv:2407.13770.[33] K. Huang, S. Das Sarma, and X. Li, Fractional quantumanomalous Hall effect in rhombohedral multilayer gra-phene with a strong displacement field, arXiv:2408.05139[Phys. Rev. B (to be published)].[34] S. A. Parameswaran, Anomalous Hall crystals or moiréChern insulators? Spontaneous versus explicit translationalsymmetry breaking in graphene pentalayers, J. ClubCondens. Matter Phys. (2024).10.36471/JCCM_January_2024_02[35] E. C. Regan et al., Mott and generalized Wigner crystalstates in WSe2=WS2 moiré superlattices, Nature (London)579, 359 (2020).[36] X. Liu et al., Tuning electron correlation in magic-angletwisted bilayer graphene using Coulomb screening, Science371, 1261 (2021).DACEN WATERS et al. PHYS. REV. X 15, 011045 (2025)011045-8https://arXiv.org/abs/2312.11617https://doi.org/10.1103/PhysRevX.14.041040https://doi.org/10.1103/PhysRevX.14.041040https://doi.org/10.1103/PhysRevB.110.205124https://doi.org/10.1103/PhysRevB.110.205130https://doi.org/10.1103/PhysRevB.110.205130https://arXiv.org/abs/2405.12294https://doi.org/10.1103/PhysRevLett.132.236601https://doi.org/10.1103/PhysRevB.110.245135https://doi.org/10.1103/PhysRevX.15.011004http://link.aps.org/supplemental/10.1103/PhysRevX.15.011045http://link.aps.org/supplemental/10.1103/PhysRevX.15.011045http://link.aps.org/supplemental/10.1103/PhysRevX.15.011045http://link.aps.org/supplemental/10.1103/PhysRevX.15.011045http://link.aps.org/supplemental/10.1103/PhysRevX.15.011045http://link.aps.org/supplemental/10.1103/PhysRevX.15.011045http://link.aps.org/supplemental/10.1103/PhysRevX.15.011045https://doi.org/10.1088/0022-3719/15/36/006https://doi.org/10.1088/0022-3719/15/22/005https://arXiv.org/abs/2407.13770https://arXiv.org/abs/2408.05139https://doi.org/10.36471/JCCM_January_2024_02https://doi.org/10.36471/JCCM_January_2024_02https://doi.org/10.1038/s41586-020-2092-4https://doi.org/10.1038/s41586-020-2092-4https://doi.org/10.1126/science.abb8754https://doi.org/10.1126/science.abb8754 Chern Insulators at Integer and Fractional Filling in Moiré Pentalayer Graphene I. INTRODUCTION II. TRIVIAL AND TOPOLOGICAL CORRELATED INSULATORS III. TOPOLOGICAL PHASE TRANSITION AT &nu;=1 IV. INTEGER AND FRACTIONAL CHERN INSULATORS AT &nu;=2/3 V. DISCUSSION ACKNOWLEDGMENTS DATA AVAILABILITY References