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Jesse C. Hoke, Yifan Li, Julian May-Mann, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Barry Bradlyn, Taylor L. Hughes, Benjamin E. Feldman

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[Uncovering the spin ordering in magic-angle graphene via edge state equilibration](https://mdr.nims.go.jp/datasets/ecce1a76-2de5-42a1-bb76-7f290f494fee)

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Uncovering the spin ordering in magic-angle graphene via edge state equilibrationArticle https://doi.org/10.1038/s41467-024-48385-zUncovering the spin ordering in magic-anglegraphene via edge state equilibrationJesse C. Hoke 1,2,3, Yifan Li1,2,3, Julian May-Mann1,4, Kenji Watanabe 5,Takashi Taniguchi 6, Barry Bradlyn 4, Taylor L. Hughes4 &Benjamin E. Feldman 1,2,3The flat bands in magic-angle twisted bilayer graphene (MATBG) provide anespecially rich arena to investigate interaction-driven ground states. Whileprogress has been made in identifying the correlated insulators and theirexcitations at commensuratemoiré filling factors, the spin-valley polarizationsof the topological states that emerge at high magnetic field remain unknown.Here we introduce a technique based on twist-decoupled van der Waals layersthat enables measurement of their electronic band structure and–by studyingthe backscattering between counter-propagating edge states–the determina-tion of the relative spin polarization of their edge modes. We find that thesymmetry-broken quantum Hall states that extend from the charge neutralitypoint in MATBG are spin unpolarized at even integer filling factors. The mea-surements also indicate that the correlatedChern insulator emerging fromhalffilling of the flat valence band is spin unpolarized and suggest that its con-duction band counterpart may be spin polarized.The relative twist angle between adjacent van der Waals layers pro-vides a powerful tuning knob to control electronic properties. In thelimit of large interlayer twist, the misalignment leads to a mismatchin the momentum and/or internal quantum degrees of freedom oflow-energy states in each layer, resulting in effectively decoupledelectronic systems1–7. This decoupling can be sufficiently pro-nounced to realize independently tunable quantumHall bilayers thatsupport artificial quantum spin Hall states2 or excitoniccondensation6,7. In the opposite regime of low-twist angle, a moirésuperlattice develops, and can lead to extremely flat electronic bandswith prominent electron-electron interaction effects. The archetypallow-twist example is magic-angle twisted bilayer graphene(MATBG)8–10, which has been shown to support symmetry-brokenquantum Hall states9,11–13 as well as correlated Chern insulators (ChIs)at high magnetic fields11,14–22. However, a full understanding of thenature of these states, including their spin and valley polarization,has so far remained elusive.Combining large and small interlayer twists in a single deviceprovides an approach to probe microscopic details of correlatedground states in moiré systems23–25. Such a device would yield elec-tronically decoupled flat and dispersive bands, which can be used tointerrogate each other. In some ways, this is reminiscent of other two-dimensional heterostructures which host bands of differing character.One notable example is mirror-symmetric magic-angle twisted trilayergraphene (MATTG) and itsmultilayer generalizations26–32, whichcanbedecomposed into flat MATBG-like bands that coexist with more dis-persive bands. However, these bands hybridize at a nonzero dis-placement field, whereas a twist-decoupled architecture provides fullyindependent bands. This enables control over the relative filling oflight and heavy carriers, including in a bipolar (electron-hole) regime.Crucially, in a perpendicularmagnetic field, such a device can realize aquantum Hall bilayer with co- or counter-propagating edge modes.Because the inter-edge mode coupling depends on their respectiveinternal degrees of freedom2,33, the effects of edge backscattering onReceived: 19 February 2024Accepted: 30 April 2024Check for updates1Department of Physics, Stanford University, Stanford, CA 94305, USA. 2Geballe Laboratory for Advanced Materials, Stanford, CA 94305, USA. 3StanfordInstitute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA. 4Department of Physics and Institute forCondensedMatter Theory, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA. 5Research Center for Electronic andOptical Materials, NationalInstitute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan. 6Research Center for Materials Nanoarchitectonics, National Institute for MaterialsScience, 1-1 Namiki, Tsukuba 305-0044, Japan. e-mail: bef@stanford.eduNature Communications |         (2024) 15:4321 11234567890():,;1234567890():,;http://orcid.org/0000-0003-2531-835Xhttp://orcid.org/0000-0003-2531-835Xhttp://orcid.org/0000-0003-2531-835Xhttp://orcid.org/0000-0003-2531-835Xhttp://orcid.org/0000-0003-2531-835Xhttp://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/0000-0001-6327-1076http://orcid.org/0000-0001-6327-1076http://orcid.org/0000-0001-6327-1076http://orcid.org/0000-0001-6327-1076http://orcid.org/0000-0001-6327-1076http://orcid.org/0000-0002-4962-0548http://orcid.org/0000-0002-4962-0548http://orcid.org/0000-0002-4962-0548http://orcid.org/0000-0002-4962-0548http://orcid.org/0000-0002-4962-0548http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-48385-z&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-48385-z&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-48385-z&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-48385-z&domain=pdfmailto:bef@stanford.edutransport can be used to identify spin/valley flavor polarization of theflat moiré bands.Here we report transport measurements of a dual-gated, twistedtrilayer graphene device that realizes electrically decoupled MATBGand monolayer graphene (MLG) subsystems. By tracking features inthe resistance as a function of carrier density and displacement field,we demonstrate independently tunable flat and dispersive bands andshow that transport measurements can be used to simultaneouslydetermine the thermodynamic density of states in each subsystem.Furthermore, in the regime of counter-propagating MLG and MATBGedge modes in a magnetic field, we use longitudinal and non-localresistance measurements to infer the spin order within the MATBGsubsystem–both for symmetry-broken quantumHall states emanatingfrom the charge neutrality point (CNP), and for the primary sequenceof ChIs. Our work clarifies the microscopic ordering of correlatedstates in MATBG and demonstrates a powerful generic method toprobe internal quantum degrees of freedom in two-dimensional elec-tron systems.ResultsTwist-decoupled flat and dispersive bandsAn optical image of the device is shown in Fig. 1a, with a side view ofindividual layers schematically illustrated in Fig. 1b. Aswedemonstratebelow, the bottom two graphene layers have a twist of 1.11° and displaybehavior consistent with typical MATBG samples, while the topmostgraphene layer is electrically decoupled because of the larger inter-layer twist of ~5−6° (seeMethods). The whole device is encapsulated inhexagonal boron nitride (hBN) and has graphite top and bottomgates.This dual-gated structure allows us to independently tune the totalcarrier density ntot = (CbVb +CtVt)/e and applied displacement fieldD = (CtVt −CbVb)/(2ϵ0), where Cb(t) and Vb(t) are the capacitance andvoltage of the bottom (top) gate, e is the electron charge, and ϵ0 is thevacuum permittivity. The applied displacement field shifts the relativeenergies of states in each subsystem and, therefore, controls how thetotal carrier density is distributed between them (Fig. 1c).We first describe electronic transport through the device at zeromagnetic field. The longitudinal resistance Rxx is largest along a curveat low/moderate D, with multiple fainter, S-shaped resistive featuresextending outward, i.e. approximately transverse to it (Fig. 1d). Thisphenomenology arises from electronic transport in parallel throughthe MLG and MATBG subsystems. Specifically, the strongly resistivebehavior occurs when theMLG is at its CNP (solid black line in Fig. 1d).Relatively higher peaks in Rxx along this curve reflect insulating statesin MATBG. Analogously, when the carrier density in MATBG is fixed toan insulating state, Rxx remains elevated even as the carrier density intheMLG is adjusted. This leads to the resistive S-shaped curves (suchasthe dashed white line in Fig. 1d; see discussion below).The peaks in Rxx centered near ntot = ± 2.8 × 1012 cm−2 correspondto the single-particle superlattice gaps atmoiré filling factor (numberof electrons per unit cell) s = ±4. From these densities, we extract atwist angle of θ = 1.11° between the bottom two layers, and similarmeasurements using different contact pairs show that there is littletwist angle disorder in these two layers (Supplementary Fig. 1).Intermediate resistance peaks are also present at s = 0, 1, ±2, and 3(Fig. 1d, f). These peaks are consistent with the correlated insulatorsthat have been previously observed in MATBG8,12,13,34–39, and theypersist as the MLG is doped away from its CNP (SupplementaryFig. 2). At higher temperatures, another peak develops near s = −1(Supplementary Fig. 3), matching prior reports of a Pomeranchuk-like effect in MATBG40,41.Our characterization demonstrates the ability to independentlytune the carrier density in each subsystem, and hence shows that thesubsystems are effectively decoupled. This further allows the MLG toact as a thermodynamic sensor for the MATBG, similar to schemes inwhich a sensing graphene flake is isolated by a thin hBN spacer fromthe target sample20,28,40,42. By tracking the resistive maxima when theμMLG= 0μMLG= 0μMATBG= 0μMLGμMATBGμMLG μMATBGD ~ -| |EnergyDOS DOSMLG MATBG3 2 1 0 1 2 3ntot (1012cm2)1.00.50.00.51.0D(Vnm1 )Rxx ( )B = 0 T3 2 1 0 1 2 3ntot (1012cm2)Rxx ( )B = 2 Tad eb c(e, e)(h, h)(h, e)(e, h)top gatebottom gatehBNhBN~1.1o>5o3 2 1 0 1 2 3s2010010201.7 K4.2 K15 K02468Rxx(k)MATBG(meV)fgB = 0B = 0102103104102103Fig. 1 | Twist-decoupled monolayer graphene (MLG) and magic-angle twistedbilayer graphene (MATBG). a Optical image of the device. The scale bar is 2μm.b Schematic of the device structure and interlayer angles. The twisted trilayergraphene is encapsulated in hexagonal boron nitride (hBN) and has graphite topand bottom gates. c Band diagram of the combined MLG-MATBG system. Thedisplacement field D modifies the energies of states in each subsystem and,therefore, tunes the relative chemical potential μi of each subsystem i at fixed totalcarrier density ntot. d, e Longitudinal resistance Rxx as a function of ntot and D, atzero magnetic field B and at B = 2 T, respectively. Black solid (white dashed) linesdenotewhere theMLG (MATBG) is at its charge neutrality point (CNP). Parenthesesindicate which carrier types are present in the MLG and MATBG, respectively: eindicates electrons and h indicates holes. f Rxx as a function of moiré filling factor sat B =0 and at various temperatures T where the MLG is at its CNP (solid blackcurve ind).g μMATBG as a functionof s atB =0, as extracted from (d) and analogousdata at other temperatures.Article https://doi.org/10.1038/s41467-024-48385-zNature Communications |         (2024) 15:4321 2MLG is at its CNP, and using amodel that accounts for the screening ofelectric fields by each layer (Supplementary Note 2), we extract theMATBG chemical potential μMATBG (Fig. 1g). We find a total change ofchemical potential across the flat bands of δμ ≈ 40meV, with non-monotonic dependence on filling that matches previous reports of asawtooth in inverse compressibility14,21,40,41,43. Similarly, we can deter-mine the MLG chemical potential as a function of its carrier densityμMLG(nMLG) by fitting it to the S-shaped resistive features in Fig. 1d,which occur at fixed s inMATBG (SupplementaryNote 2). Thesematchthe scaling μMLG / sgnðnMLGÞjnMLGj1=2 that is expected for the Diracdispersion of graphene. We observe similar behavior in a second tri-layer device, where MLG-like states are decoupled from a bilayer gra-phene moiré system with a 1.3° twist angle (Supplementary Fig. 4),suggesting this is a generic phenomenon that is widely applicable inmultilayer heterostructures.Electronic decoupling is also evident when we apply a perpendi-cular magnetic field B, where the energy spectrum of MLG consists ofLandau levels (LLs), and a Hofstadter butterfly spectrum develops inMATBG. Figure 1e shows Rxx as a function of ntot and D at B = 2T,revealing staircase-like patterns which reflect crossings of theMLG LLsand MATBG states (Hall resistance Rxy is plotted in SupplementaryFig. 5). Vertical features at constant ntot occur when the MLG is in aquantumHall state; their extent (inD) is proportional to the size of thegap between LLs. As the displacement field tunes the relative energiesof states in each subsystem, transitions occur when graphene LLs arepopulated or emptied. These cause each feature associated with aMATBG state to shift horizontally in density by the amount needed tofill a fourfold degenerate LL, nLL = 4eB/h, where h is Planck’s constantand the factor of four accounts for the spin and valley degrees offreedom (e.g., see dashed white line in Fig. 1e).Quantum Hall edge state equilibrationIn a magnetic field, the decoupledMLG andMATBG realize a quantumHall bilayer in which either carrier type (electron or hole) can be sta-bilized in either subsystem. This results in co- (counter-)propagatingedgemodes when the respective carrier types are the same (different).Additionally, because the device is etched into aHall bar after stacking,the edges of MLG and MATBG are perfectly aligned. Crucially, in thecounter-propagating regime, the measured resistance encodes infor-mation about the efficiency of scattering between the edge modes ineach subsystem (Supplementary Note 3), which depends on theirinternal quantum degrees of freedom. We expect that atomic scaleroughness at the etched edge of the device enables large momentumtransfer, and therefore anticipate efficient coupling irrespective of thevalley (in MLG and MATBG) and moiré valley (in MATBG). However,assuming the absence of magnetic disorder, edge states having dif-ferent spins should remain decoupled, whereas those with the samespin can backscatter and exhibit increased longitudinal resistance(Fig. 2a). Probing Rxx, therefore, allows us to deduce the relative spinpolarization of edge states in MLG and MATBG.We first focus on low carrier density and high magnetic field,where the behavior of each subsystem i is well described by quantumHall states having filling factors νi = nih/eB emanating from theirrespectiveCNPs.A sharppeak inRxx emerges at combinedfilling factorνtot = 0, flanked by several quantum Hall states at other integer νtot(Fig. 2b). These features exhibit a series of D-field tuned transitions asthe relative filling of MLG and MATBG changes. The data encompassMLG states with ∣νMLG∣≤2. Importantly, prior work has shown thatMLGedge modes at νMLG = ±1 have opposite spin and valley quantumnumbers, whereas those at νMLG = ±2 are spin unpolarized33. Combin-ing this information with the measured resistance enables us to3 2 1 0 1 2 3tot0.30.20.10.00.10.20.3D(V nm1 )B = 8 TEnergyMagnetic Field     = 0MATBGMATBG2 4 6 8B (T)0.40.60.81.0RNL(h/e2 )0.51.01.52.0Rxx(h/e2 )= -4MATBG= 4/ = MLG MATBG -2/2/ = MLG MATBG 2/-20.2c e gdab fRxx (  )decouplededge modesMLGMATBGbackscatteringallowedMLGMATBGh/5e2= 0totVNL0.2 0.1 0.0 0.1 0.2D (V nm1)020406080RNL(k)h/2e2= 0tot4 T5 T6 T7 T8 TVxx-2/2 2/-2-1/1 0/0 1/-1MLG MATBG/ =020406080Rxx(k)103104Fig. 2 | Spin polarization of MATBG quantum Hall states near the CNP.a Schematic illustration of two possible scenarios for a single pair of counter-propagating edge modes. If the spins of each edge mode are aligned (top), back-scattering is allowed (orange circle). Backscattering is suppressed when thespins are anti-aligned (bottom), leading to quantum spin Hall-like behavior withRxx = h/2e2. b Rxx as a function of the total filling factor νtot = νMLG + νMATBG and D atB = 8 T. c, d Rxx and RNL, respectively measured in the configurations shown in thetop left insets, as a function ofDwhen νtot = 0. The filling factors of each subsystemfor each regime of D are indicated in the bottom inset of c. Insets in (d)schematically represent the inferred relative spin orientations (black arrows) ofedge modes inMLG (blue arrows) and MATBG (purple arrows), with orange circlesindicating backscattering between a given pair. e, f Rxx and RNL for νMATBG = ±2/∓2(red and blue, respectively) averaged over 0.1 < ∣D∣ <0.25 V nm−1. Error bars corre-spond to one standard deviation. The straight lines connecting data points areguides for the eye. g Schematic diagram of CNP MATBG Landau levels (LLs) andtheir spin characters. Gaps between LLs are depicted schematically and do notrepresent experimentally measured field dependence.Article https://doi.org/10.1038/s41467-024-48385-zNature Communications |         (2024) 15:4321 3determine the spin polarization of the MATBG quantum Hall stateswith ∣νMATBG∣≤4.When νtot = 0,MLG andMATBGhave equal and opposite filling, andRxx approaches different values depending on the number of counter-propagating edge states (Fig. 2c). At D=0, each subsystem is in aninsulating, ν=0 symmetry-broken state. Here, no bulk conduction oredge modes are anticipated, and we observe a large resistance. Near∣D∣≈0.05V/nm, νMLG/νMATBG =± 1/∓1, and Rxx reaches a minimum nearh/2e2 (Fig. 2c). This phenomenology can be explained by a pair ofcounter-propagating edge modes with opposite spins, analogous tohelical edge modes observed in large-angle twisted bilayer graphene2.This interpretation is further corroboratedby similar behavior in anothercontact pair (Supplementary Note 4), and measurements of non-localresistance RNL (Fig. 2d). Indeed, the pronounced non-local resistancesignal at νMLG/νMATBG =± 1/∓1 indicates that transport is dominated byedgemodes (see Supplementary Note 5 for a discussion of bulk effects).This is corroborated by the value of RNL, which is suppressed toward h/5e2, the quantized value predicted from the Landauer–Büttiker formulafor counter-propagating edge states in this contact configuration (Sup-plementary Note 3). We therefore conclude that similar toMLG,MATBGhas a filled spin down (up) electron- (hole-)like LL at νMATBG = 1(−1).Beyond ∣D∣ ≈0.08 V/nm, where νMLG/νMATBG = ± 2/∓2, we observelarger resistances Rxx > h/2e2 and RNL > h/5e2 (Fig. 2c, d). This suggeststhat backscattering occurs for both pairs of edge modes: if bothMATBG edge states had an identical spin, one counter-propagatingpair would remain decoupled and would lead to quantized resistanceRxx = h/2e2 and RNL = h/5e2 (Supplementary Note 3). A resistanceabove this value, as well as the large increase in resistance relative toνMLG/νMATBG = ±1/∓1, therefore both indicate that the edge states atνMATBG = ±2 are spin unpolarized (see Supplementary Note 4, 5 foradditionalmeasurements and discussion of alternative interpretationswhich we rule out as unlikely). There is some asymmetry in the mea-sured Rxx depending on the sign of D; it is comparatively lesspronounced inRNL. SinceRNL is inherently a probe of edge conduction,this suggests the observed asymmetry inRxxoriginates fromadditionalbulk current contributions, which may arise due to an electron-holeasymmetry in the strengths of different symmetry-broken states (seeSupplementary Note 5). Based on the above observations, we deducethe spin polarization of the edge modes of the MATBG LLs emanatingfrom its CNP, as illustrated in Fig. 2g.Addressing spin polarization of the Chern insulatorsIn addition to symmetry-broken quantum Hall states emerging fromthe CNP, ChIs extrapolating to nonzero s are evident in Landau fanmeasurements of Rxx and Rxy at fixed top gate voltages of ±3 V (Fig. 3).At these values, theMLG filling factor is νMLG = ±2, respectively, at highfields. Consequently, both the Chern number of the primary sequenceof quantum Hall states in MATBG (black lines in Fig. 3c, f) emergingfrom s =0, and the ChIs (colored lines) are offset by ±2. Afteraccounting for this shift, the ChIs that we observe are consistent withthe primary sequence ∣t + s∣ = 4 commonly reported inMATBG, where tis the Chern number of the MATBG subsystem11,14–20. Below, we focusprimarily on the (t, s) = (±2, ±2) ChIs, which exhibit near-zero Rxx andquantized Rxy in the co-propagating regime (Supplementary Fig. 6).Here, ChI edgemode chirality is determinedby the signof t: stateswitht >0(t <0) have electron- (hole-)like edge modes.Tuning into the bipolar (electron-hole) regime, allows us torealize counter-propagating edge modes from the MATBG ChIsand the MLG quantum Hall states. We apply the edge stateequilibration analysis to determine the spin polarization of theChIs in MATBG. For the (t, s) = (−1, −3) ChI, we find a sharpresistive feature that occurs only when νMLG = 1 (Fig. 4a, b), i.e.,when there is one pair of counter-propagating edge states. Theresistance grows with increasing B and reaches values sig-nificantly larger than h/2e2 (Fig. 4b). This indicates strong back-scattering between edge modes, and hence that both have the02468B(T)02468B(T)sV = 3 Vt V = -3 Vtabc fed4 3 2 1 0 1 2 3 4s02468B(T)4 3 2 1 0 1 2 3 43-2-3-2-3344-4-8-12 812-4-8Rxx (k�)–0.50.00.5015Rxy(h/e2)Fig. 3 | Landau fans demonstrating correlated Chern Insulators (ChIs). a, b Rxxand Rxy as a function of s and B at fixed top gate voltage Vt = 3 V. cWannier diagramindicating the strongest quantum Hall and ChI states determined from (a, b). TheChern numbers t of the MATBG states are labeled. At high fields, the total Chernnumbers of each state are offset by 2 because νMLG = 2. Black, red, orange, and bluelines correspond to states with zero-field intercepts s =0, s = ∣1∣, s = ∣2∣, and s = ∣3∣,respectively. For states with s =0, t ≡ νMATBG. Black dashed lines label the MATBGsymmetry-broken quantum Hall states −4 < νMATBG <4. d–f Same as a–c, but forVt = −3, where νMLG = −2 at high fields. Data were collected at T ≈ 300mK.Article https://doi.org/10.1038/s41467-024-48385-zNature Communications |         (2024) 15:4321 4same spin (inset, Fig. 4b). We conclude that the first flavor tooccupy the MATBG Hofstadter subbands (see SupplementaryNote 6) is spin down, consistent with expectations based on theZeeman effect.A resistive state also occurs when (t, s) = (−2, −2) and νMLG = 2(Fig. 4a).We observeRxx>h/2e2 that growswith increasingB (Fig. 4c andSupplementary Fig. 7), indicating efficient backscattering between bothpairs of counter-propagating edge modes. We obtain consistent resultsfrom both the non-local resistance (Fig. 4d) and Rxx measurements of asecond contact pair (Supplementary Note 4). We, therefore, concludethat the (−2, −2) ChI in MATBG is spin unpolarized (red inset, Fig. 4d).In contrast, we observe more moderate resistance for the(t, s) = (2, 2) ChI in MATBG when νMLG=−2 (Fig. 4a). In measurementsof Rxx (RNL) at fixed B, the resistance of this state saturates near h/2e2(h/5e2) at high B (Fig. 4c, d), with similar near-quantized Rxx in a Landaufan measurement (Supplementary Fig. 7). Together, these resultsdemonstrate that there is only partial coupling between edge modes.Thedata are consistentwith onepair of decoupled, counter-propagatingedgemodes, and another pair having allowedbackscattering. Thiswouldnaturally arise if the (t, s) = (2, 2) ChI in MATBG is spin polarized (blueinset, Fig. 4d). The data, therefore, suggest a spin polarized ground statemay be favored (see Supplementary Notes 5, 6 for further discussion).DiscussionThe observed spin orderings of both the quantum Hall states and theChIs clarify the microscopic interactions and relative strengths of dif-ferent symmetry breaking terms in MATBG. Near charge neutrality,spin unpolarized states are favored at νMATBG = ±2 for all measuredmagnetic fieldsB > 2 T (Fig. 2e, f). This is counter to expectations basedon both the conventional Hofstadter subband model (seeSupplementary Note 6) and Zeeman considerations. Specifically,moiré valley splitting14, which arises in the presence of My symmetrybreaking, or some other mixing between Hofstadter subbands isnecessary to produce a spin unpolarized state at νMATBG = ±2 (seeSupplementary Note 6). Moreover, even in the presence of moirévalley splitting, the Zeeman effect would favor spin polarization atνMATBG = ±2; our observations therefore indicate that exchange inter-actions dominate over Zeeman splitting throughout the measuredfield range and favor spin unpolarized states.Very recent theoretical work44 suggests there is a crossoverbetween spin polarized ChIs favored by the Zeeman effect at highmagnetic field and a partially spin unpolarized intervalley coherentstate favored at low magnetic field, with the former predictedto dominate at experimentally relevant fields. Our results for the(t, s) = (2, 2) ChI are consistent with the high field prediction, but thespin unpolarized states we observe at (t, s) = (−2, −2) are not. This dis-crepancy likely reflects electron-hole asymmetry in MATBG and/oratomic scale relaxation of the lattice, which are neglected in the theo-retical model. The calculations indicate close competition betweendifferent ground states, so including these effects will alter quantitativepredictions and could even lead to qualitatively different ground states,as observed experimentally. Our work provides an important bench-mark for future theoretical considerations, demonstrating the impor-tance of these terms, that distinct spin ordering can occur for electronand hole doping, and that antiferromagnetic exchange contributionscan be comparable to or larger in magnitude than Zeeman splitting.In conclusion, we have realized a twisted graphene multilayerconsisting of electrically decoupled MATBG and MLG subsystems. Eventhough the layers are in contact, we demonstrated that a twist-decoupled architecture provides a method to extract thermodynamic0 2 4 6 8B (T)0.000.250.500.751.00Rxx(h/ e2 )MLG = 1(t, s) = (-1, -3)2 1 0 1 20.50.00.5MLG = 2, (t, s) = ( 2, 2)MLG = 2, (t, s) = (2, 2)ac db2 4 6 8B (T)2 4 6 8B (T)0.000.250.500.751.00Rxx(h/e2 )RNL(h/e2 )h/5e2h/2e2Rxx (  )ntot (1012cm2)D(V nm1 )B = 8 TVNLVxx102103104MLG = 2(t, s) = (-2, -2)MLG = 1(t, s) = (-1, -3)MLG = -2(t, s) = (2, 2)Fig. 4 | Spin polarization of the ChIs inMATBG. a Rxx as a function of ntot andD atB = 8 T (see Supplementary Fig. 8 for the equivalent map in a non-local contactconfiguration). Black dashed circle: νMLG = 1, (t, s) = (−1, −3). Red dashed box:νMLG = 2, (t, s) = (−2, −2). Blue dashed box: νMLG = −2, (t, s) = (2, 2). b Rxx for theνMLG = 1, (t, s) = (−1, −3) state as a function of B. c, d Rxx and RNL, respectively,measured in the configurations shown in the top left insets, for νMLG= ±2, (t, s) = (∓2,∓2) states (red and blue, respectively) as a function of B. Data are averaged over0.325 < ∣D∣ <0.525 V nm−1. Error bars correspond to one standard deviation. Insetsin (d) schematically represent the inferred relative spin orientations (black arrows)of edge modes in MLG (blue arrows) and MATBG (purple arrows), with orangecircles indicatingbackscatteringbetween agivenpair. The straight lines connectingdata points are guides for the eye.Article https://doi.org/10.1038/s41467-024-48385-zNature Communications |         (2024) 15:4321 5properties and probe internal quantum degrees of freedom throughedge state equilibration. Looking forward, we anticipate its extension toother van derWaals materials, including to recently discovered systemsthat exhibit fractional quantum anomalous Hall states45–49. This devicegeometry also represents themost extreme limit of dielectric screeningof interactions34–36 in which a tunable screening layer is immediatelyadjacent to the system of interest. More generally, it provides a naturalarena to explore Kondo lattices50,51 with independently tunable densitiesof itinerant electrons and local moments, as well as an opportunity tostudy Coulomb drag between adjacent layers52.MethodsDevice fabricationThe MATBG-MLG stack was fabricated using standard dry transfertechniques with poly (bisphenol A carbonate)/polydimethylsiloxane(PC/PDMS) transfer slides53,54. Amonolayer graphene flakewas cut intothree pieces with a conductive AFM tip in contact mode. An exfoliatedhBN flake (26.5 nm) was used to sequentially pick up each section atthe desired twist angle before placing it on topof a prefabricated stackof few-layer graphite and hBN (27 nm). Finally, an additional few-layergraphite flake was added and served as the top gate. The stack wassubsequently patterned with standard electron beam lithographytechniques followed by etching to form a Hall bar geometry andmetallization to form edge contacts53.Transport measurementsTransport measurements were conducted at cryogenic tempera-tures (1.7 K unless otherwise stated) using standard lock-in tech-niques with a current bias of 5–20 nA at 17.777 Hz. Because edgecontacts are made to the etched sample, they simultaneouslymake electrical contact to all three graphene layers, and elec-tronic transport through the device reflects parallel transportthrough both MATBG and MLG subsystems. The longitudinal andtransverse resistance are symmetrized and anti-symmetrized in amagnetic field, respectively.Determination of twist angleThe twist angle θ between the pair of layers that form MATBG isdetermined by the superlattice carrier density ns =4=A≈8θ2=ffiffiffi3pa2when theMLG is at charge neutrality. Here, A is the superlattice unit cellarea and a=0.246nm is the MLG lattice constant. The twist angle canalso be independently confirmed by fitting Chern insulators in a Landaufan measurement using the Streda formula dn/dB=Ctot/Φ0, which haveintercepts at zero magnetic field at integers s=4n/ns. Both methodsyield a consistent value of θ= 1.11o ±0.05o, where the quoted uncertaintyreflects the width of the s=±4 resistive peaks. The capacitances Cb (Ct)between the bottom (top) gate and sample are accurately determinedbased on the slopes of features in Landau fans taken at constant bottom(top) gate voltages Vb (Vt). This also results in vertical features (Fig. 1e,for example) when data are plotted as a function of total carrier densityntot =CtVt/e+CbVb/e and displacement field D= (CtVt/e−CbVb/e)/(2ϵ0),where e is the electron charge, and ϵ0 is the vacuum permittivity.The relative angle between the MLG and MATBG subsystems isestimated based on the angle between the AFM-cut edges of the topand middle graphene layers. An optical image of the three graphene(and top hBN) layers on the PC/PDMS stamp during the stackingprocess (before deposition) is shown in Supplementary Fig. 9. Fromthe image, we identify a twist angle between the top and middle gra-phene layers of about 5.5° ± 0.5°.Data availabilityData that support the findings in this study are available at https://doi.org/10.5281/zenodo.11044381. Additional datasets generated and/oranalyzed during the current study are available from the correspond-ing author upon request.Code availabilityThe codes that support the findings of this study are available from thecorresponding authors upon request.References1. Sanchez-Yamagishi, J. D. et al. QuantumHall effect, screening, andlayer-polarized insulating states in twisted bilayer graphene. Phys.Rev. Lett. 108, 076601 (2012).2. Sanchez-Yamagishi, J. D. et al. Helical edge states and fractionalquantum Hall effect in a graphene electron-hole bilayer. Nat.Nanotechnol. 12, 118–122 (2017).3. Rickhaus, P. et al. The electronic thickness of graphene. Sci. Adv. 6,eaay8409 (2020).4. Rickhaus, P. et al. Correlated electron-hole state in twisted double-bilayer graphene. Science 373, 1257–1260 (2021).5. Mreńca-Kolasińska, A. et al. Quantum capacitive couplingbetween large-angle twisted graphene layers. 2D Mater. 9,025013 (2022).6. Shi, Q. et al. BilayerWSe2 as a natural platform for interlayer excitoncondensates in the strong coupling limit. Nat. Nanotechnol. 17,577–582 (2022).7. Kim, D. et al. Robust interlayer-coherent quantum hall states intwisted bilayer graphene. Nano Lett. 23, 163–169 (2022).8. Cao, Y. et al. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices. Nature 556, 80–84 (2018).9. Cao, Y. et al. Unconventional superconductivity in magic-anglegraphene superlattices. Nature 556, 43–50 (2018).10. Bistritzer, R.&MacDonald, A.H.Moirébands in twisteddouble-layergraphene. Proc. Natl Acad. Sci. USA 108, 12233–12237 (2011).11. Choi, Y. et al. Correlation-driven topological phases in magic-angletwisted bilayer graphene. Nature 589, 536–541 (2021).12. Yankowitz, M. et al. Tuning superconductivity in twisted bilayergraphene. Science 363, 1059–1064 (2019).13. Lu, X. et al. Superconductors, orbitalmagnets andcorrelated statesin magic-angle bilayer graphene. Nature 574, 653–657 (2019).14. Yu, J. et al. Correlated Hofstadter spectrum and flavour phasediagram in magic-angle twisted bilayer graphene. Nat. Phys. 18,825–831 (2022).15. Das, I. et al. Symmetry-broken Chern insulators and Rashba-likeLandau-level crossings in magic-angle bilayer graphene. Nat. Phys.17, 710–714 (2021).16. Saito, Y. et al. Hofstadter subband ferromagnetism and symmetry-broken Chern insulators in twisted bilayer graphene. Nat. Phys. 17,478–481 (2021).17. Nuckolls, K. P. et al. Strongly correlated Chern insulators in magic-angle twisted bilayer graphene. Nature 588, 610–615 (2020).18. Wu, S., Zhang, Z., Watanabe, K., Taniguchi, T. & Andrei, E. Y. Cherninsulators, van Hove singularities and topological flat bands inmagic-angle twisted bilayer graphene. Nat. Mater. 20, 488–494(2021).19. Tomarken, S. L. et al. Electronic compressibility of magic-anglegraphene superlattices. Phys. Rev. Lett. 123, 046601 (2019).20. Park, J. M., Cao, Y., Watanabe, K., Taniguchi, T. & Jarillo-Herrero, P.Flavour Hund’s coupling, Chern gaps and charge diffusivity inmoiré graphene. Nature 592, 43–48 (2021).21. Pierce, A. T. et al. Unconventional sequence of correlated Cherninsulators in magic-angle twisted bilayer graphene. Nat. Phys. 17,1210–1215 (2021).22. Xie, Y. et al. Fractional Chern insulators in magic-angle twistedbilayer graphene. Nature 600, 439–443 (2021).23. Andrei, E. Y. &MacDonald, A. H. Graphene bilayers with a twist.Nat.Mater. 19, 1265–1275 (2020).24. 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).Article https://doi.org/10.1038/s41467-024-48385-zNature Communications |         (2024) 15:4321 6https://doi.org/10.5281/zenodo.11044381https://doi.org/10.5281/zenodo.1104438125. Mak, K. F. & Shan, J. Semiconductor moiré materials. Nat. Nano-technol. 17, 686–695 (2022).26. Park, J. M., Cao, Y., Watanabe, K., Taniguchi, T. & Jarillo-Herrero, P.Tunable strongly coupled superconductivity in magic-angle twis-ted trilayer graphene. Nature 590, 249–255 (2021).27. Hao, Z. et al. Electric field-tunable superconductivity in alternating-twist magic-angle trilayer graphene. Science 371, 1133–1138 (2021).28. Liu, X., Zhang, N. J., Watanabe, K., Taniguchi, T. & Li, J. I. A. Isospinorder in superconducting magic-angle twisted trilayer graphene.Nat. Phys. 18, 522–527 (2022).29. Park, J. M. et al. Robust superconductivity inmagic-anglemultilayergraphene family. Nat. Mater. 21, 877–883 (2022).30. Zhang, Y. et al. Promotion of superconductivity in magic-anglegraphene multilayers. Science 377, 1538–1543 (2022).31. Khalaf, E., Kruchkov, A. J., Tarnopolsky, G. & Vishwanath, A. Magicangle hierarchy in twisted graphene multilayers. Phys. Rev. B 100,085109 (2019).32. Shen, C. et al. Dirac spectroscopy of strongly correlated phases intwisted trilayer graphene. Nat. Mater. 22, 316–321 (2022).33. Amet, F., Williams, J. R., Watanabe, K., Taniguchi, T. & Goldhaber-Gordon, D. Selective equilibration of spin-polarized quantum halledge states in graphene. Phys. Rev. Lett. 112, 196601 (2014).34. Saito, Y., Ge, J., Watanabe, K., Taniguchi, T. & Young, A. F. Inde-pendent superconductors and correlated insulators in twistedbilayer graphene. Nat. Phys. 16, 926–930 (2020).35. Stepanov, P. et al. Untying the insulating and superconductingorders in magic-angle graphene. Nature 583, 375–378 (2020).36. Liu, X. et al. Tuning electron correlation inmagic-angle twisted bilayergraphene using Coulomb screening. Science 371, 1261–1265 (2021).37. Yu, J. et al. Spin skyrmion gaps as signatures of strong-couplinginsulators in magic-angle twisted bilayer graphene. Nat. Commun.14, 6679 (2023).38. Morissette, E. et al. Dirac revivals drive a resonance response intwisted bilayer graphene. Nat. Phys. 19, 1156–1162 (2023).39. Nuckolls, K. P. et al. Quantum textures of the many-body wave-functions in magic-angle graphene. Nature 620, 525–532 (2023).40. Saito, Y. et al. Isospin Pomeranchuk effect in twisted bilayer gra-phene. Nature 592, 220–224 (2021).41. Rozen, A. et al. Entropic evidence for a Pomeranchuk effect inmagic-angle graphene. Nature 592, 214–219 (2021).42. Kim, S. et al. Direct measurement of the Fermi energy in grapheneusing a double-layer heterostructure. Phys. Rev. Lett. 108,116404 (2012).43. Zondiner, U. et al. Cascade of phase transitions and Dirac revivals inmagic-angle graphene. Nature 582, 203–208 (2020).44. Wang, X. & Vafek, O. Theory of correlated chern insulators in twis-ted bilayer graphene. Preprint at arXiv 2310.15982 (2023).45. Cai, J. et al. Signatures of fractional quantum anomalous hall statesin twisted mote2. Nature 622, 63–68 (2023).46. Zeng, Y. et al. Thermodynamic evidence of fractional chern insu-lator in moiré mote2. Nature 622, 69–73 (2023).47. Park, H. et al. Observation of fractionally quantized anomalous halleffect. Nature 622, 74–79 (2023).48. Xu, F. et al. Observation of integer and fractional quantum anomaloushall effects in twisted bilayer mote2. Phys. Rev. X 13, 031037 (2023).49. Lu, Z. et al. Fractional quantum anomalous Hall effect in multilayergraphene. Nature 626, 759–764 (2024).50. Kumar, A., Hu, N. C., MacDonald, A. H. & Potter, A. C. Gate-tunableheavy fermion quantum criticality in a moir\’e Kondo lattice. Phys.Rev. B 106, L041116 (2022).51. Doniach, S. The kondo lattice andweak antiferromagnetism.Phys. B+C. 91, 231–234 (1977).52. Narozhny, B. & Levchenko, A. Coulomb drag. Rev. Mod. Phys. 88,025003 (2016).53. Wang, L. et al. One-dimensional electrical contact to a two-dimensional material. Science 342, 614–617 (2013).54. Zomer, P. J., Guimarães, M. H. D., Brant, J. C., Tombros, N. & vanWees, B. J. Fast pick up technique for high quality heterostructuresof bilayer graphene and hexagonal boron nitride. Appl. Phys. Lett.105, 013101 (2014).AcknowledgementsWe thank Pablo Jarillo-Herrero, Steve Kivelson, Yves Kwan, Sid Para-meswaran, B. Andrei Bernevig, Oskar Vafek, Xiaoyu Wang, David Gold-haber-Gordon, and Aaron Sharpe for helpful discussions. This work wassupported by theQSQM, an Energy Frontier Research Center funded bythe U.S. Department of Energy (DOE), Office of Science, Basic EnergySciences (BES), under Award # DE-SC0021238. K.W. and T.T. acknowl-edge support from the JSPS KAKENHI (Grant Numbers 20H00354 and23H02052) and World Premier International Research Center Initiative(WPI), MEXT, Japan. J.C.H. acknowledges support from the StanfordQ-FARMQuantumScience and Engineering Fellowship. Part of this workwas performed at the StanfordNano Shared Facilities (SNSF), supportedby the National Science Foundation under award ECCS-2026822.Author contributionsJ.C.H. fabricated the devices. J.C.H. and Y.L. conducted transportmeasurements. B.E.F supervised the project. J.M.M., B.B., and T.L.H.provided theoretical support. K.W. and T.T provided hBN crystals. Allauthors contributed to analysis and writing of the manuscript.Competing interestsThe authors declare no competing interest.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-024-48385-z.Correspondence and requests for materials should be addressed toBenjamin E. Feldman.Peer review information Nature Communications thanks the anon-ymous reviewers for their contribution to the peer review of this work. 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To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2024Article https://doi.org/10.1038/s41467-024-48385-zNature Communications |         (2024) 15:4321 7https://doi.org/10.1038/s41467-024-48385-zhttp://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/ Uncovering the spin ordering in magic-angle graphene via edge state equilibration Results Twist-decoupled flat and dispersive�bands Quantum Hall edge state equilibration Addressing spin polarization of the Chern insulators Discussion Methods Device fabrication Transport measurements Determination of twist�angle Data availability Code availability References Acknowledgements Author contributions Competing interests Additional information