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Patrick Knüppel, Jiacheng Zhu, Yiyu Xia, Zhengchao Xia, Zhongdong Han, Yihang Zeng, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Jie Shan, Kin Fai Mak

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[Correlated states controlled by a tunable van Hove singularity in moiré WSe2 bilayers](https://mdr.nims.go.jp/datasets/2e46e990-751f-48fc-bfe2-b7f5ae20786e)

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Correlated states controlled by a tunable van Hove singularity in moiré WSe2 bilayersArticle https://doi.org/10.1038/s41467-025-57235-5Correlated states controlled by a tunable vanHove singularity in moiré WSe2 bilayersPatrick Knüppel 1,5, Jiacheng Zhu1,5, Yiyu Xia1, Zhengchao Xia 1,Zhongdong Han1, Yihang Zeng 1, Kenji Watanabe 2, Takashi Taniguchi 2,Jie Shan 1,3,4 & Kin Fai Mak 1,3,4Twisted transition metal dichalcogenide (TMD) bilayers have enabled thediscovery of superconductivity, ferromagnetism, correlated insulators, and aseries of new topological phases of matter. However, the connection betweenthese electronic phases of matter and the underlying band structure singula-rities has remained largely unexplored. Here, combining magnetic circulardichroism and exciton sensing measurements, we investigate the influence ofa vanHove singularity (vHS) on the correlatedphases in bilayerWSe2with twistangle between 2 and 3 degrees. By tuning the vHS across the Fermi level usingelectric and magnetic fields, we observe Stoner ferromagnetism below moirélattice filling one and Chern insulators at filling one. The experimental obser-vations are supported by the continuum model band structure calculations.Our results highlight the prospect of engineering electronic phases of matterin moiré materials by tunable van Hove singularities.In two dimensions, a divergence in the density of states (DOS), alsoknown as a vanHove singularity (vHS), arises froma saddlepoint in theelectronic band structure. The presence of a vHS near the Fermi levelsignificantly enhances the electron-electron interactions which oftenlead to electronic instabilities and new phases of matter1,2. However,the Fermi level in conventional bulkmaterials cannot be easily tuned tocross the vHS. The emergence ofmoirématerials3–7, which support thehighly tunable Fermi levels and electronic band structures, has pro-vided a platform to engineer electronic phases of matter by bringingtogether the vHS and the Fermi level.Among themoirématerials, twisted homo-bilayers of TMDs, suchas MoTe2 and WSe2, have recently attracted significant attentionbecause they possess flat moiré bands with finite valley-contrastingChern numbers8–15. A suite of correlated and topological states,including the integer16 and fractional Chern insulators17–20, integer21and fractional quantum spin Hall insulators22, superconductivity23,24,ferromagnetism25 and metal-insulator transitions26–29, has been repor-ted. The electronic band structure calculations show a saddle point inthe topmost moiré valence band located at the m-point of the moiréBrillouin zone. The calculations also show that an electric fieldperpendicular to the sample plane, which controls the interlayerpotential difference8,9,30, can widely tune the electronic band structureincluding the location of the vHS26,28–32. It has been suggested thatproximity of the vHS to the Fermi level can affect the stability of thecorrelated insulators at integer moiré lattice fillings10,30,31,33. However,the general effect of the van Hove singularities on symmetry-breakingground states at generic fillings has remained elusive.Here we investigate hole-doped twisted WSe2 (tWSe2) bilayerswith twist angle between 2 and 3 degrees. We demonstrate a ferro-magnetic metal phase below filling one and a Chern insulator at fillingone by tuning the vHS across the Fermi level using the electric andmagnetic fields. These states are identified by combining reflectivemagnetic circular dichroism (MCD)34 and exciton sensingmeasurements35. Specifically, the former probes the valley polariza-tion, which is connected to the spin polarization, because of spin-momentum locking—a property inherited from the transition metaldichalcogenide (TMD) monolayers36. The latter probes the sample’selectronic incompressibility through the sensitivity of the sensorexcitons to their dielectric environment; it also determines the Chernnumber of the insulating states through their dispersion with anReceived: 10 December 2024Accepted: 14 February 2025Check for updates1Laboratory of Atomic and Solid-State Physics and School of Applied and Engineering Physics, Cornell University, Ithaca, NY, USA. 2National Institute forMaterials Science, Tsukuba, Japan. 3Kavli Institute at Cornell for Nanoscale Science, Ithaca, NY, USA. 4Max Planck Institute for the Structure and Dynamics ofMatter, Hamburg, Germany. 5These authors contributed equally: Patrick Knüppel, Jiacheng Zhu. e-mail: jie.shan@cornell.edu; kinfai.mak@cornell.eduNature Communications |         (2025) 16:1959 11234567890():,;1234567890():,;http://orcid.org/0000-0002-6193-3440http://orcid.org/0000-0002-6193-3440http://orcid.org/0000-0002-6193-3440http://orcid.org/0000-0002-6193-3440http://orcid.org/0000-0002-6193-3440http://orcid.org/0000-0002-3339-8575http://orcid.org/0000-0002-3339-8575http://orcid.org/0000-0002-3339-8575http://orcid.org/0000-0002-3339-8575http://orcid.org/0000-0002-3339-8575http://orcid.org/0000-0003-2941-5958http://orcid.org/0000-0003-2941-5958http://orcid.org/0000-0003-2941-5958http://orcid.org/0000-0003-2941-5958http://orcid.org/0000-0003-2941-5958http://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/0000-0003-1270-9386http://orcid.org/0000-0003-1270-9386http://orcid.org/0000-0003-1270-9386http://orcid.org/0000-0003-1270-9386http://orcid.org/0000-0003-1270-9386http://orcid.org/0000-0003-1097-475Xhttp://orcid.org/0000-0003-1097-475Xhttp://orcid.org/0000-0003-1097-475Xhttp://orcid.org/0000-0003-1097-475Xhttp://orcid.org/0000-0003-1097-475Xhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-025-57235-5&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-025-57235-5&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-025-57235-5&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-025-57235-5&domain=pdfmailto:jie.shan@cornell.edumailto:kinfai.mak@cornell.eduwww.nature.com/naturecommunicationsexternally applied magnetic field, as described by the Streda formula.Our results elucidate the impact of the vHS on symmetry-breakingground states at generic doping densities in moiré materials andprovide insight into how to design robust magnetism in correlatedmaterials in general.Results and discussionPhase diagram of tWSe2Figure 1a illustrates the dual-gateddevice structureof tWSe2 employedin this study. The top and bottom gate voltages (Vtg and Vbg) inde-pendently control the moiré lattice filling factor (ν) and the electricfield (E) perpendicular to the sample plane. (Filling factor ν = 1 isdefined as one hole per moiré unit cell, which corresponds to a half-filled moiré valence band.) A WS2 monolayer, separated from thesample by a thin hexagonal boron nitride (hBN) spacer (about 1 nmthick), is used as the exciton sensor. We first focus on a 2.7° tWSe2sample and demonstrate the effect of twist angle at the end. Unlessotherwise specified, all results are obtained at sample temperatureT = 1:6 K. See Methods for details on the device fabrication, bandstructure calculations, and optical measurements.Twisted WSe2 bilayers form a honeycomb or triangular moirélattice depending on the twist angle37. The schematic in Fig. 1a illus-trates a honeycombmoiré lattice with two sublattices centered at theMX and XM (M=W; X = Se) stacking sites. Figure 1b is the electronicband structure (left panel) and the energy-dependent DOS (rightpanel) for E =0 calculated using the continuum model described inRefs. 8,9. Only the first two moiré valence bands of the K-valley statesare illustrated. They carry Chern number C = +1. The correspondingmoiré bands of the K’-valley states are a time-reversal copy and carryChern number C = −1. The DOS shows a vHS that is aligned with thesaddle point in the first moiré band, and a second vHS, with the saddlepoint in the secondmoiré band. The Fermi level crosses the first vHS athole filling ν � 0.7. Supplementary Fig. 1 shows the band structureunder large electric fields, where the two layers are decoupled and thebands are more dispersive.Figure 1c, d are the phase diagrams of tWSe2 as a function of ν andE. We determine ν and E using the applied gate voltages and gatecapacitances, which are calibrated using the observed quantumoscillations under high magnetic fields perpendicular to the sampleplane (Methods and Supplementary Fig. 2). Figure 1c shows the spec-trally averaged reflection contrast (RC) of the intralayer exciton reso-nances of the sample. (Representative raw spectra are included inSupplementary Fig. 3).We use it to identify the layer-hybridized regionfor small electric fields and the layer-polarized region for large electricfields. Because the intralayer exciton resonance of a layer is sensitive todoping in the layer, it shows the strongest E dependence at theboundary between the two regions (dashed lines), across which one ofthe layers becomes undoped. The electric field at the boundaryincreases with ν because the electrostatics requires a larger electricfield to fully polarize a higher charge density to one of the layers. Thephase diagram is symmetric about E =0 after removing a small build-infield of E0 � 13 mV/nm (likely from the device structure asymmetry).These results are fully consistent with the reported phase diagram oftWSe2 (Refs. 23,24) and tMoTe2 (Refs. 8,9,17,18).Figure 1d shows the spectrally averaged RC of the sensor 2 sexciton. Representative raw spectra as a function of filling factor areshown in Supplementary Fig. 4 (see Methods for spectral analysis).Throughout the measurement, the WS2 sensor, which has a type-IIband alignment withWSe2 (Ref. 38), has been kept charge neutral. Anincompressible state in the sample manifests a blue shift and anenhanced spectral weight of the sensor 2 s exciton due to thereduced dielectric screening35,39. We identify several correlatedinsulators at commensurate fillings, ν = 1, 1/3, 1/4, and 1/6, in thelayer-hybridized region. These states become compressible in thelayer-polarized region, where the moiré bands are more dispersiveand the correlation effects are weaker. The phase diagram appearsFig. 1 | Phase diagram of 2.7-degree twistedWSe2 bilayers. a Left: schematic of adual-gated twisted WSe2 bilayer device with a WS2 monolayer sensor. Few-layergraphite is used as the gate electrodes, and hexagonal boron nitride (hBN), the gatedielectrics and spacer between the sample and sensor. Top and bottom gate vol-tages (Vtg and Vbg) control the moiré lattice filling factor ν and the out-of-planeelectric field E. Right: honeycombmoiré lattice with sublattices at the MX (orange)and XM (blue) sites (M=W, black circles; X = Se, grey circles). b Left: continuummodel band structure with the first two moiré valence bands of the K-valley statealong the directions of γ � κ �m� γ in the moiré Brillouin zone (E =0). Right:density of states (DOS) showing van Hove singularities (vHS). c, d Spectrally inte-grated reflection contrast (RC) of the sample intralayer exciton (c) and the sensor2 s exciton (d) as a function of filling factor ν and electric field E at 1.6 K. Thedashedlines extracted from (c) (seemain text) separate the layer-hybridized region for lowfields from the layer-polarized region for high fields. Black arrows mark the cor-related insulators at fractional filling factors.Article https://doi.org/10.1038/s41467-025-57235-5Nature Communications |         (2025) 16:1959 2www.nature.com/naturecommunicationsasymmetric about E = 0 because the sensor is placed above the topWSe2 layer.Stoner ferromagnetismWe examine themagnetic properties of tWSe2 by performing theMCDmeasurements (see Supplementary Fig. 4 for MCD spectra around theintralayer exciton resonances and Methods for extracting the spec-trally averaged MCD, which we refer to as MCD below). Figure 2ashows MCD as a function of ν and E in the absence of magnetic fields.Weobserve a hot spot of spontaneousMCDaround ν =0:8 and E =0. Itcorresponds to a compressible region of the phase diagram (Fig. 1d).The MCD exhibits a clear magnetic hysteresis with a coercive field ofabout 10mT (Fig. 2b). As temperature increases, the spontaneousMCD and the magnetic hysteresis gradually weaken and vanish aboveabout 3 K (inset of Fig. 2b). These results support that the MCD hotspot corresponds to a ferromagnetic metal.Figure 2c is MCD under a small out-of-plane magnetic field ofB=0:5 T. The result at higher magnetic fields is included in Supple-mentary Fig. 5. Except in the ferromagnetic region, the MCD increaseslinearly with magnetic field for small fields, and is proportional to themagnetic susceptibility22. The susceptibility is substantially enhancedin the layer-hybridized region below filling one. It is also higher inregions near the correlated insulators at ν = 1/3, 1/4, and 1/6. Abovefilling one, the region with enhanced susceptibility disperses withelectricfield, exhibiting an arrowhead-like feature. The dashed lines, atwhich the MCD drops to 0.03, provide a guide to the eye of theboundary of this region.To gain insight into themagnetic properties of tWSe2, we performband structure calculations under varying electric fields. Figure 2dillustrates the electronic DOS as a function of ν and E in the experi-mentally relevant region of the phase diagram. Line cuts atrepresentative electric fields are shown in Supplementary Fig. 1. HighDOS is observed in the layer-hybridized region below filling one, wherethe moiré bands are relatively flat. The vHS is located near ν =0:7 forE =0; it continuously shifts towards higher filling factors with reducedDOS as electric field increases. The evolution of the vHS with electricfield has been verified by transport measurements23,26,28 (Supplemen-tary Fig. 6) although the precise location of the vHS in (ν, E) isdependent on the twist angle, the choice of the continuum modelparameters, and possibly also the interaction effects which are notaccounted for in the single-particle continuum model.The measured magnetic susceptibility is well correlated with thecalculated electronic DOS except the regions near the correlatedinsulators and is substantially enhanced at the vHS. This is expectedbecause the magnetic susceptibility of a Landau Fermi liquid is pro-portional to the electronic DOS40. (In the correlated insulators, theband picture breaks down, and the susceptibility arises from the localmagnetic moments, that is, the spins of the localized holes). Remark-ably, ferromagneticorder is stabilizednear the vHSaround E =0wheretheDOS is the highest. Themagnetic phase space ismuch smaller thanthat of the enhanced DOS or susceptibility. This supports the Stonermechanism: the ferromagnetic metal is driven by strong Coulombrepulsion and enhanced by high DOS. The Stoner criterion1, UDF>1,expressed in terms of the strength of Coulomb repulsion U and thesingle-particle DOS at the Fermi level DF , provides a qualitativethreshold for magnetism. This picture is further supported by theobservation of a second ferromagnetic metal phase near the secondvHS in tWSe2 around E =0 (Supplementary Fig. 7).Chern insulatorsNext, we examine the effect of an out-of-plane magnetic field on thecorrelated insulators at ν = 1. We focus on the case of E =0. Figure 3aacbdMCD0.0 0.05 0.1E (mV/nm)0.0 0.5 1.51.0B = 0 T70035-70-35E (mV/nm)0.0 0.5 1.51.0B = 0.5 T70035-70-35MCD 0.2-0.20-0.10.1B (T)0.0 0.20.1-0.1-0.25K3K1.6K0.3-0.31 3 50.20.0T (K)2 40.1E (mV/nm)0.0 0.5 1.51.0DOS (eV-1 nm-2)0 4 8 1270035-70-35Fig. 2 | Stoner ferromagnetism. a Spontaneous magnetic circular dichroism(MCD) as a function of moiré lattice filling factor ν and out-of-plane electric field Eat 1.6 K. The dashed lines (from Fig. 1c) separate the layer-hybridized and layer-polarized regions. b Magnetic-field dependence of MCD of the hot spot in (a) atrepresentative temperatures. Clear magnetic hysteresis is observed at lowtemperatures. Inset: temperature dependence of the spontaneous MCD. c Same as(a) under magnetic field B=0:5 T. The dashed lines are a guide to the eye of theboundary of the region with enhanced magnetic susceptibility. d Calculated elec-tronic density of states (DOS) as a function of ν and E showing the evolution of thevan Hove singularity (vHS) with E. All results are for 2.7° twisted WSe2.Article https://doi.org/10.1038/s41467-025-57235-5Nature Communications |         (2025) 16:1959 3www.nature.com/naturecommunicationsdisplays MCD as a function of ν and B (lower panel) and magneticsusceptibility as a function of ν (upper panel). Away from the ferro-magnetic metal around ν =0.8, MCD increases with magnetic field tillfull spin/valley polarization is reached. (The ferromagnetic metalshows spontaneous MCD and diverging susceptibility.) Fig. 3b illus-trates the field dependenceof normalizedMCDby the saturation valuefor several filling factors around one. We quantify the saturation fieldBs by using the value at which the normalized MCD reaches 0.85. Thesaturation field increases rapidly with filling factor, from zero nearν =0:8 to several tesla above ν = 1. At Bs , the Zeeman energy is suffi-cient to overcome the exchange energy for full spin/valley polariza-tion. The latter has been independently estimated through the Curie-Weiss analysis of the temperature dependence of the magnetic sus-ceptibility (Methods). In addition, there is a weak kink in the fielddependence of MCD before saturation for fillings around one. It ismore clearly seen in the derivative of MCD (dashed red line) as a localminimum followed by a maximum near Bs. This suggests a metamag-netic transition and the transition field is close to Bs.Figure 3cdisplays the evolutionof the correlated insulators at ν = 1with magnetic field. The lower panel shows the sensor 2 s excitonresponse as a function of ν and B; the upper panel is a line cut at thehighest applied field of B = 8.75 T; the dotted line denotes the satura-tion field. For small fields, the insulator at ν = 1 does not disperse withmagnetic field. Hence, it has Chern number C =0 based on the Stredaformula and is topologically trivial. For fields above Bs, the state turnsinto two insulators with C = 0 and 1. A potential candidate for the low-field state is a valley-coherent insulator30, whereas the emergence ofthe Chern insulator with C = 1 at high fields is compatible with thespin/valley-contrasting Chern bands in 2.7° tWSe2. The coexistence oftwo types of insulators after magnetic saturation suggests that thesecompeting states are close in their ground state energies. This is dis-tinct from tMoTe2, where only the Chern insulator is observedregardless of the field (Refs. 17,18). (The insulating states at fractionalfilling factors have C =0 and are likely generalizedWigner crystals35,41.)We perform the continuum model band structure calculationsunder varying magnetic fields to elucidate the effect of the vHS on theabcdBsMCD0.0 0.085 0.17846B (T)020 0.5 1.51.00.00.20.4RC0.2 0.4846B (T)020 0.5 1.51.00.00.20.4RCSensor 2s8.75TBsC=1C=0B (T)MCD (norm.)0.0 0.5 1.51.00.02.52.0BSDifferential susceptibility (norm.)0.80.40.60.00.21.00.97 1.00 1.03 1.060.20.40.60.81.0DOS (eV-1 nm-2)5 10846B (T)020 0.5 1.51.01Fig. 3 | Magnetic-field tuned Chern insulator. a Magnetic circular dichroism(MCD) as a functionofmoiré latticefilling factor ν andout-of-planemagneticfieldBunder zero electric field. Top:filling factor dependence ofmagnetic susceptibility χextracted from the small-field MCD. b Magnetic-field dependence of the normal-izedMCD for representative filling factors around one (solid lines, left axis) and thederivative of MCD for ν = 1 (dashed red line, right axis). MCD reaches 85% of thefully saturated value (dashed lines) at the saturation field Bs . c Spectrally integratedoptical reflection contrast (RC)of the sensor 2 s excitonasa functionof ν andB. Thedashed lines show the expecteddispersion for stateswithChern numberC =0and 1near ν = 1. Top: line cut at the highest field of B=8:75 T. The dotted line in (a, c)denotes the saturation field Bs . d Calculated electronic density of states (DOS) as afunction of ν and B showing Zeeman splitting of the van Hove singularity (vHS,dashed lines). All measurements were performed at 1.6 K. All results are for 2.7°twisted WSe2.Article https://doi.org/10.1038/s41467-025-57235-5Nature Communications |         (2025) 16:1959 4www.nature.com/naturecommunicationscorrelated insulators at ν = 1. For simplicity, we only consider theZeeman effect. Figure 3d illustrates the calculated electronic DOS as afunction of ν and B. The vHS with large DOS (dashed lines) splits intotwo under B. These features have also been identified in transportstudies1,29. The vHS that disperses to higher filling factors crosses ν = 1at about B � 4 T. The qualitative agreement between this field and thesaturation field suggests a scenario in which the vHS tuned to theFermi level (for ν = 1) by themagnetic field induces the transition fromthe spin/valley unpolarized to polarized states and the emergence ofthe Chern insulator.Twist angle dependenceFinally, we demonstrate that the observed effect of a vHS near theFermi level on the correlated states is general for tWSe2 of small twistangles. Figure 4a–d display the evolution of the correlated insulatorswith magnetic field (upper panel) and the filling dependence ofspontaneous MCD at representative temperatures (lower panel) intWSe2 with twist angle 1.8 (a), 2.1 (b), 2.3 (c), and 2.5 degrees (d). Theblack dashed lines denote the expected dispersion for states withChern number 0 and ±1 near ν = 1. The results show that Stoner mag-netism is stabilized in all but the lowest twist angle sample (1.8°). Cherninsulators are absent at zero magnetic field; they emerge above thesaturation field for all twist angles. In general, decreasing the twistangle lowers the saturation field (denoted by arrows for ν = 1) obtainedfrom the MCD measurements (Supplementary Fig. 8). In addition, in1.8° tWSe2, a weak C = � 1 Chern insulator coexists with the C = + 1Chern insulator, whereas only the C = + 1 Chern insulator is stable forsamples with larger twist angles.In summary, combining observations from the exciton sensingand magnetization measurements with the continuum model bandstructure calculations, we develop a general understanding of theelectronic phase diagram for tWSe2 and connection to the vHS in theband structure. Our work bridges the gap37,42 between the previouslyexplored regimeof small twist angles (with strong correlation effects)16and regime of large twist angles (with weak correlation effects)26,28,29. Italso provides the basis for vHS engineering of the correlated states inacbd846B (T)020.0 1.00.5 1.52.5°0.10.00.05MCD1.6K3.0K2.0KBs846B (T)020.0 1.00.5 1.52.3°0.10.00.05MCD1.6K2.9K2.1KBs846B (T)020.0 1.00.5 1.52.1°MCD1.6K2.9K2.1K0.020.00.01Bs846B (T)020.0 1.00.5 1.51.8°MCD1.6K0.020.00.01Bs0.1 0.2RC Sensor 2sFig. 4 | Twist angle dependence. a–d Top: Spectrally integrated optical reflectioncontrast (RC) of the sensor 2 s exciton as a function ofmoiré lattice filling factor ν andout-of-plane magnetic field B under zero electric field at 1.6K. The dashed lines showthe expected dispersion for states with Chern number C=−1, 0 and 1 near ν = 1. Thearrows denote the saturation field and emergence of the Chern insulator(s). Bottom:filling dependence of the spontaneous magnetic circular dichroism (MCD) at repre-sentative temperatures. The twist angle inWSe2 bilayers is 1.8 (a), 2.1 (b), 2.3 (c) and 2.5degrees (d). The saturation field Bs (denoted by white arrows) decreases with twistangle. In 1.8° twisted WSe2, spontaneous MCD is not observed down to 1.6K and aweak C=−1 Chern insulator coexists with a C= 1 Chern insulator.Article https://doi.org/10.1038/s41467-025-57235-5Nature Communications |         (2025) 16:1959 5www.nature.com/naturecommunicationsmoiré material, such as superconductivity and exciton condensationinvolving flat Chern bands43.MethodsDevice fabricationTwisted WSe2 moiré devices were assembled using the layer-by-layerdry transfer method44. Thin flakes of hBN and graphite, monolayerWSe2, and monolayer WS2 were exfoliated onto Si/SiO2 substrates.Optical RC was used to identify the appropriate flake shape andthickness. A thin film of polycarbonate on polydimethylsiloxane wasemployed as a stamp to pick up the flakes following the sequenceshown in Fig. 1a. The complete stack was released at 180 °C onto aSi/SiO2 substrate with prepatterned platinum (Pt) gate electrodes. Tocreate the moiré superlattice, a flake of monolayer WSe2 was cut intotwo parts using an atomic force microscope tip and stacked with asmall relative twist angle of θ.Optical characterizations (RC and MCD)Optical measurements were performed in a closed-cycle cryostat(Attocube, Attodry 2100) with magnetic fields up to 9T and tempera-tures down to 1.6 K. Either a halogen lamp (for 2 s sensing andMCD) ora light emitting diode (LED, forMCD) was used as the light source. Theinput lightwas spatiallyfilteredby a single-modefiber and sent into thecryostat as a collimated beam. A low-temperature microscope objec-tive (Attocube, numerical aperture 0.8) was used to focus the lightonto the sample. The light intensity on the sample was kept below50 nW/μm2 to minimize its effects on the electronic states; negligiblechanges in the magnetization were observed by further reducing theincident power by an order of magnitude. The reflected light wascollected by the same objective and analyzed by a spectrometerequippedwith a liquid-nitrogen-cooled charge coupled device array toobtain spectrum R. The RC spectrum is defined as (ðR� R0Þ=R0, wherethe reference spectrum R0 was taken for sample at a high dopingdensity with quenched excitonic resonances.The reflective MCD was used to study the magnetic properties ofthe samples. A combination of a linear polarizer and a quarter-waveplate was used to generate a right and left circularly polarized light(σ� and σ + ) on the sample. The MCD spectrum is definedas ðR� � R + Þ=ðR� +R+ Þ, where R� and R+ are the reflection spectra forthe σ� and σ + incident light, respectively.The RC spectrum of tWSe2 depends sensitively on the dopingdensity, displacement field, and magnetic field. To account for thesechanges, we chose to average the reflectance (for combined σ� and σ +channels) over a range of wavelength (725–745 nm, or equivalently,1.66–1.71 eV) that focuses on the 1 s exciton resonance of tWSe2. Theaveraged RC is displayed in Fig. 1c. The same wavelength range wasused to compute the average of the absolute value of MCD. Theaveraged MCD is displayed in Fig. 2a. Details of the analysis are illu-strated in Supplementary Fig. 4. The magnetic susceptibility wasevaluated from the slope of MCD at small magnetic fields ( Bj j≤0:5 T).The magnetic saturation field Bs was defined as the field, at which theMCD reaches 85% of the saturation value. To evaluate the differentialsusceptibility dMCDdB (Fig. 3b), we applied a Savitzky–Golay filter to theexperimental data before taking the numerical derivative with respectto the magnetic field.Determination of the phase boundariesThe boundary between the layer-polarized and layer-hybridizedregions in the ðν, EÞ phase space was identified by the strongestdependence of the optical refection of the moiré exciton on theelectric field25. For 0:4≤ ν ≤ 1, the boundary is sharp, and the electric-field derivative of the reflection displays a clear peak. An example isshown in Supplementary Fig. 3 for ν = 1. The electric-field derivativeof the moiré exciton reflection shows a pronounced peak atEc =60 meV/nm. The dashed green line in Fig. 1c is a cubic splineextrapolation of the boundary electric field to the origin. The dashedblack line in Fig. 2c provides a guide to the eye of the boundary of theregion with enhanced magnetic susceptibility. We define theboundary at which MCD (at B=0:5 T) drops to 0.03.Exciton sensingMonolayer WS2 was used as the sensor. Changes in the samplecompressibility modulate the dielectric environment for thesensor39, which was probed by its 2 s exciton response as demon-strated in Refs. 35,45–47. The alignment of the sensor and samplevalence bands is such that the sensor remains charge neutral for theentire range of ðν, EÞ throughout this study. An example is illustratedin Supplementary Fig. 4c for E = 0. The RC spectrum of the sensoraround the 2 s resonance is displayed as a function of filling factor ofthe sample (the reference spectrum was acquired when the sensor iselectron-doped). Both the 2 s resonance energy and intensity varystrongly with filling factor of the sample. We used the 2 s resonanceintensity to represent the sample incompressibility (Fig. 1d). Toextract the 2 s resonance intensity, we first removed a broad third-order polynomial background for the entire spectral window whereboth 2 s and higher lying excitonic resonances of the sensor arepresent. We then integrated around the 2 s exciton peak (black dots,Supplementary Fig. 4d) over a 2-nm wavelength window for eachfilling factor.Twist angle calibrationWe calibrated the moiré density nM and the twist angle θ of WSe2bilayers using the quantum oscillations observed optically under anout-of-plane magnetic field of 8.8 T. The details for the sample exam-ined in the main text are shown in Supplementary Fig. 2. Supplemen-tary Fig. 2a is the MCD spectrum of the sample near the moiré excitonresonance as a function of gate voltage (or equivalently, hole density).The MCD signal oscillates due to the formation of the Landau levels(LLs). The LL period is determined to be 0.3 V (Supplementary Fig. 2b),fromwhichwededuce a hole density changeof 7 × 1011 cm-2 per volt. Inaddition, we identified insulating states through the sensor responseas a function of gate voltage and assign the first four most prominentones to be ν = 1/4, 1/3, 1, and 2. Supplementary Fig. 2c shows theseinsulating states in filling factor and gate voltage. From the data belowfilling 1, we determined nM = ð2:4±0:1Þ× 1012 cm−2 and θ= 2:7 ±0:1ð Þdegrees. The results also allow us to determine the hBN thicknessfor the top and bottom gates: dtg � 18 nm and dbg � 15 nm. Usingthese values we determined the out-of-plane electric field,E =V tg=2dtg � Vbg=2dbg � E0, where V tg and Vbg denote the top andbottom gate voltages, respectively, and E0 � 13 mV/nm is a built-infield likely from the asymmetry of the device structure in the presenceof the sensor layer. We also note that the LL period in gate voltagedecreases slightly with increasing filling factor. The nonlinear gatingeffect likely arises from the non-ohmic contact. We accounted for thiseffect by using a two-piece linear interpolation of the experimentaldata points above and below filling factor 1 (Supplementary Fig. 2c).Energy scale for magnetic saturation at ν = 1Supplementary Fig. 9 shows the temperature (T) dependence of theinversemagnetic susceptibility (1=χ) measured by the small-field MCDfor ν = 1. The dependence is well described by the Curie-Weiss law(1χ � T � TCW ) above 10 K with a Curie-Weiss temperature of TCW ��ð4:0±0:6Þ K. The negative sign indicates an antiferromagneticinteraction for the localized magnetic moments of the Mott insulator.The strength of the exchange interaction is estimated by J ��kBTCW � 0:34 meV, where kB is the Boltzmann constant. The Zee-man energy (gμBB) from the externally applied magnetic field mustovercome J to achieve magnetic saturation. Here g � 10 is the holeg-factor for WSe2 and μB is the Bohr magneton. The magnetic satura-tion is thus expectedwhen gμBB≳ J orB≳ JgμB� 0:6 T. Thefield value isArticle https://doi.org/10.1038/s41467-025-57235-5Nature Communications |         (2025) 16:1959 6www.nature.com/naturecommunicationsconsistent with the observed saturation field of 1.5 T at ν = 1 from theMCD measurement (Fig. 3b).Band structure calculationsWe used the continuum model for twisted TMD homobilayers fol-lowing Refs. 8,9 to compute the single-particle band structure and theDOSof tWSe2. Specifically, themoiréHamiltonian for the valence bandstates at the K-valley readsHK =�_k22m* +Δb rð Þ+ Vz2 ΔT ðrÞΔyT ðrÞ �_k22m* +Δt rð Þ � Vz20@1A: ð1ÞThe moiré Hamiltonian for the K’-valley states is a time-reversalcopy ofHK . Herek is themomentum,m* = 0.43m0 is the effective holemass of WSe2 (m0 denoting the free electron mass), Δb, tðrÞ is thebottom (top) layer energy and ΔT ðrÞ is the interlayer tunnelingamplitude as a function of the spatial position r in the moiré unit cell.An interlayer potential differenceVz , which canbe tuned by the out-of-plane electric field, is also introduced. In the long moiré period limit,Δb, t rð Þ andΔT ðrÞ are smooth functions of r and canbe approximated asΔb, t rð Þ=2VPj = 1, 3, 5 cosðGjr±ψÞ and ΔT rð Þ=w 1 + eiG2r + eiG3r� �whichsatisfy all the symmetry constraints of the moiré superlattice. Here Gjis the reciprocal lattice vectors (lattice constant a = 3.317 Å) and(V,ψ, w) = (9meV, 128°, 18meV) fromRef. 9. describe themoiré depth,shape and interlayer tunneling strength, respectively. TheHamiltonianwas cut off at the 5th shell inmomentumspace. The computedDOSwassmoothed using a Gaussian filter with a full width at half maximum of1meV. To compare to experiments, we converted the interlayerpotential difference to the electric field using a dipolemoment of 0.26e⋅nm48. To include the effect of an out-of-plane magnetic field, weadded a Zeeman energy shift between the K- and K’-valleys, using ahole g-factor of 10. Specifically, the band structure was first calculatedwithout accounting for the magnetic field; the bands for the K- and K’-valley states were then displaced in energy by 0.58meV per tesla andcombined to obtain an approximation to the band structure at finitemagnetic fields.Data availabilityThe Source Data underlying the figures of this study are available withthepaper. All rawdata generatedduring the current study are availablefrom the corresponding authors upon request. Source data are pro-vided with this paper.References1. Stoner, E. C. Collective electron ferromagnetism. Proc. R. Soc.Lond. Ser. A. Math. Phys. Sci. 165, 372–414 (1997).2. Bardeen, J., Cooper, L. N. & Schrieffer, J. R. Theory of super-conductivity. Phys. Rev. 108, 1175–1204 (1957).3. Mak, K. F. & Shan, J. Semiconductor moiré materials. Nat. Nano-technol. 17, 686–695 (2022).4. Kennes, D. M. et al. Moiré heterostructures as a condensed-matterquantum simulator. Nat. Phys. 17, 155–163 (2021).5. 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).6. Andrei, E. Y. &MacDonald, A. H. Graphene bilayers with a twist.Nat.Mater. 19, 1265–1275 (2020).7. Andrei, E. Y. et al. Themarvels ofmoirématerials.Nat. Rev. Mater.6,201–206 (2021).8. Wu, F., Lovorn, T., Tutuc, E., Martin, I. & MacDonald, A. H. Topolo-gical Insulators in twisted transition metal dichalcogenide homo-bilayers. Phys. Rev. Lett. 122, 086402 (2019).9. Devakul, T., Crépel, V., Zhang, Y. & Fu, L. Magic in twisted transitionmetal dichalcogenide bilayers. Nat. Commun. 12, 6730 (2021).10. Pan, H., Wu, F. & Das Sarma, S. Band topology, Hubbard model,Heisenbergmodel, and Dzyaloshinskii-Moriya interaction in twistedbilayer WSe2. Phys. Rev. Res. 2, 033087 (2020).11. Li, H., Kumar, U., Sun, K. & Lin, S.-Z. Spontaneous fractional Cherninsulators in transition metal dichalcogenide moiré superlattices.Phys. Rev. Res. 3, L032070 (2021).12. Yu, H., Chen, M. & Yao, W. Giantmagnetic field frommoiré inducedBerry phase in homobilayer semiconductors. Natl Sci. Rev. 7,12–20 (2020).13. Zhang, X.-W. et al. Polarization-driven band topology evolution intwisted MoTe2 and WSe2. Nat. Commun. 15, 4223 (2024).14. Jia, Y. et al. Moiré fractional Chern insulators. I. First-principlescalculations and continuummodels of twisted bilayerMoTe2. Phys.Rev. B 109, 205121 (2024).15. Xu, C.,Mao, N., Zeng, T. & Zhang, Y.MultipleChern bands in twistedMoTe2 and possible non-Abelian states. Phys. Rev. Lett. 134,066601 (2025).16. Foutty, B. A. et al. Mapping twist-tuned multiband topology inbilayer WSe2. Science 384, 343–347 (2024).17. Zeng, Y. et al. Thermodynamic evidence of fractional Chern insu-lator in moiré MoTe2. Nature 622, 69–73 (2023).18. Cai, J. et al. Signatures of fractional quantum anomalous Hall statesin twisted MoTe2. Nature 622, 63–68 (2023).19. Park, H. et al. Observation of fractionally quantized anomalous Halleffect. Nature 622, 74–79 (2023).20. Xu, F. et al. Observation of integer and fractional quantum anom-alous hall effects in twisted bilayer MoTe2. Phys. Rev. X 13,031037 (2023).21. Kang, K. et al. Double quantum spin hall phase inmoiréWSe2.NanoLett. 24, 14901–14907 (2024).22. Kang, K. et al. Evidence of the fractional quantum spin Hall effect inmoiré MoTe2. Nature 628, 522–526 (2024).23. Guo, Y. et al. Superconductivity in 5.0° twisted bilayerWSe2.Nature637, 839–845 (2025).24. Xia, Y. et al. Superconductivity in twisted bilayer WSe2. Nature 637,833–838 (2025).25. Anderson, E. et al. Programming correlated magnetic states withgate-controlled moiré geometry. Science 381, 325–330 (2023).26. Wang, L. et al. Correlated electronic phases in twisted bilayertransition metal dichalcogenides. Nat. Mater. 19, 861–866 (2020).27. Xu, Y. et al. A tunable bilayer Hubbardmodel in twistedWSe2. Nat.Nanotechnol. 17, 934–939 (2022).28. Ghiotto, A. et al. Quantum criticality in twisted transition metaldichalcogenides. Nature 597, 345–349 (2021).29. Wei, L. et al. Linear resistivity at van Hove singularities in twistedbilayer WSe2. Proc. Natl. Acad. Sci. USA 121, e2321665121 (2024).30. Bi, Z. & Fu, L. Excitonic density wave and spin-valley superfluid inbilayer transition metal dichalcogenide. Nat. Commun. 12, 642(2021).31. Zang, J., Wang, J., Cano, J. & Millis, A. J. Hartree-Fock study of themoiré Hubbard model for twisted bilayer transition metal dichal-cogenides. Phys. Rev. B 104, 075150 (2021).32. Zang, J., Wang, J., Cano, J., Georges, A. & Millis, A. J. Dynamicalmean-field theory of moiré bilayer transition metal dichalcogen-ides: phasediagram, resistivity, andquantumcriticality.Phys. Rev. X12, 021064 (2022).33. Hsu, Y.-T., Wu, F. & Das Sarma, S. Spin-valley locked instabilities inmoiré transition metal dichalcogenides with conventional andhigher-order Van Hove singularities. Phys. Rev. B 104, 195134(2021).34. Shree, S., Paradisanos, I., Marie, X., Robert, C. & Urbaszek, B. Guideto optical spectroscopy of layered semiconductors. Nat. Rev. Phys.3, 39–54 (2021).35. Xu, Y. et al. Correlated insulating states at fractional fillings ofmoirésuperlattices. Nature 587, 214–218 (2020).Article https://doi.org/10.1038/s41467-025-57235-5Nature Communications |         (2025) 16:1959 7www.nature.com/naturecommunications36. Xiao, D., Liu, G.-B., Feng, W., Xu, X. & Yao, W. Coupled spin andvalley physics in monolayers of MoS2 and other group-VI dichal-cogenides. Phys. Rev. Lett. 108, 196802 (2012).37. Crépel, V. & Millis, A. Bridging the small and large in twisted tran-sition metal dichalcogenide homobilayers: a tight binding modelcapturing orbital interference and topology across a wide range oftwist angles. Phys. Rev. Res. 6, 033127 (2024).38. Jin, C. et al. Ultrafast dynamics in van der Waals heterostructures.Nat. Nanotech 13, 994–1003 (2018).39. Raja, A. et al. Coulomb engineering of the bandgap and excitons intwo-dimensional materials. Nat. Commun. 8, 15251 (2017).40. Landau, L. D. The theory of a fermi liquid. Zh. Eksp. Teor. Fiz. 30,1058 (1956).41. Regan, E. C. et al. Mott and generalized Wigner crystal states inWSe2/WS2 moiré superlattices. Nature 579, 359–363 (2020).42. Reddy, A. P., Alsallom, F., Zhang, Y., Devakul, T. & Fu, L. Fractionalquantum anomalous Hall states in twisted bilayer MoTe2 andWSe2.Phys. Rev. B 108, 085117 (2023).43. Xie, M., Hafezi, M. & Das Sarma, S. Long-lived topological flatbandexcitons in semiconductormoiré heterostructures: a bosonic Kane-Mele model platform. Phys. Rev. Lett. 133, 136403 (2024).44. Wang, L. et al. One-dimensional electrical contact to a two-dimensional material. Science 342, 614–617 (2013).45. Popert, A. et al. Optical sensing of fractional quantum hall effect ingraphene. Nano Lett. 22, 7363–7369 (2022).46. Gu, J. et al. Remote imprinting of moiré lattices. Nat. Mater. 23,219–223 (2024).47. Zhang, Z. et al. Correlated interlayer exciton insulator in hetero-structures of monolayer WSe2 andmoiréWS2/WSe2. Nat. Phys. 18,1214–1220 (2022).48. Li, T. et al. Continuous Mott transition in semiconductor moirésuperlattices. Nature 597, 350–354 (2021).AcknowledgementsWe thank Kaifei Kang, LiguoMa and Liang Fu for fruitful discussions. Thisworkwas supported by the USNational Science Foundation under DMR-2114535 (sensor design) and DMR-1807810 (magneto optical measure-ments). It was also funded partially by the US Air Force Office of Sci-entific Research under award no. FA9550-19-1-0390 (samplefabrication) and FA9550-20-1-0219 (modeling). The growth of hBNcrystals was supported by the Elemental Strategy Initiative of MEXT,Japan, and CREST (JPMJCR15F3), JST. This work used the CornellNanoScale Facility supported by NSF grant NNCI-2025233. We alsoacknowledge support from the David and Lucille Packard Fellowship(K.F.M.) and the Swiss Science Foundation Postdoc Fellowship (P.K.).Author contributionsP.K. and J.Z. fabricated the devices, performed the measurements, andanalyzed the data. Z.X. and Y.Z. provided data from additional devices.Y.X. and Z.H. provided electronic transportmeasurements. K.W. and T.T.grew the bulk hBN crystals. K.F.M. and J.S. oversaw the project. Allauthors discussed the results and commented on the manuscript.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-025-57235-5.Correspondence and requests for materials should be addressed toJie Shan or Kin Fai Mak.Peer review information Nature Communications thanks the anon-ymous reviewer(s) for their contribution to thepeer reviewof thiswork. 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To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2025Article https://doi.org/10.1038/s41467-025-57235-5Nature Communications |         (2025) 16:1959 8https://doi.org/10.1038/s41467-025-57235-5http://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/www.nature.com/naturecommunications Correlated states controlled by a tunable van Hove singularity in moiré WSe2 bilayers Results and discussion Phase diagram of tWSe2 Stoner ferromagnetism Chern insulators Twist angle dependence Methods Device fabrication Optical characterizations (RC and MCD) Determination of the phase boundaries Exciton sensing Twist angle calibration Energy scale for magnetic saturation at    1ν=1 Band structure calculations Data availability References Acknowledgements Author contributions Competing interests Additional information