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Manabendra Kuiri, Christopher Coleman, Zhenxiang Gao, Aswin Vishnuradhan, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Jihang Zhu, Allan H. MacDonald, Joshua Folk

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[Spontaneous time-reversal symmetry breaking in twisted double bilayer graphene](https://mdr.nims.go.jp/datasets/1d43c86f-379d-4809-8ca5-3a9625839a6b)

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Spontaneous time-reversal symmetry breaking in twisted double bilayer grapheneArticle https://doi.org/10.1038/s41467-022-34192-xSpontaneous time-reversal symmetry break-ing in twisted double bilayer grapheneManabendra Kuiri 1 , Christopher Coleman 1, Zhenxiang Gao1,Aswin Vishnuradhan1, Kenji Watanabe 2, Takashi Taniguchi 3, Jihang Zhu4,Allan H. MacDonald 4 & Joshua Folk 1Twisted double bilayer graphene (tDBG) comprises two Bernal-stacked bilayergraphene sheets with a twist between them. Gate voltages applied to top andback gates of a tDBG device tune both the flatness and topology of the elec-tronic bands, enabling an unusual level of experimental control.Metallic stateswith broken spin and valley symmetries have been observed in tDBG deviceswith twist angles in the range 1.2–1.3°, but the topologies andorder parametersof these states have remained unclear. We report the observation of ananomalous Hall effect in the correlated metal state of tDBG, with hysteresisloops spanning hundreds of mT in out-of-plane magnetic field (B⊥) thatdemonstrate spontaneously broken time-reversal symmetry. TheB⊥hysteresispersists for in-plane fields up to several Tesla, suggesting valley (orbital) fer-romagnetism. At the same time, the resistivity is strongly affected by evenmT-scale values of in-plane magnetic field, pointing to spin-valley coupling or to adirect orbital coupling between in-plane field and the valley degree offreedom.The interplay between band topology and Coulomb interactions hasemerged as a frontier in the study of two-dimensional (2D) materialswith flat electronic bands1–5. Graphene-based van der Waals (vdW)heterostructures offer a flexible platform from which to investigatethis interplay, due to their native Dirac points close to the Fermi levelthat provide a resource for band topology. Flat bands may be engi-neered by clever heterostructure design1,2,6 and tuned using experi-mental knobs such as magnetic field or gate voltage7,8; the topology ofthe resulting bands is then readily controlled, whether by choice ofheterostructure stackingor by tuning applied gate voltages9,10. Startingfrom flat bands, Coulomb interactions often lead to spontaneoussymmetry breaking, yielding exotic electronic phases such as frac-tional Chern insulators11 or unconventional superconductivity2,12.The physical phenomenology that results from broken symmetryphases depends on the topology of the underlying electronic bands5,13.For example, the anomalousHall effect (AHE) is a striking experimentalsignature that emerges due to spontaneously broken time reversalsymmetry associatedwith spin or valley polarization. This broken timereversal symmetry creates an “anomalous” Hall resistivity whentransport is via bands with finite Berry curvature14. AHE reflectingorbital ferromagnetism has been reported in multi-layer vdW hetero-structures such as twisted bilayer graphene aligned with hexagonalboron nitride (hBN)15,16, and in naturally-occurring structures likeBernal-stacked (AB) bilayer graphene17.Twisted double bilayer graphene (tDBG)—two AB-stacked bilayergraphene sheets misaligned by twist angle θ—is a uniquely tunablesystem in which band topology, correlations and broken symmetryphases can be manipulated using top- and back-gate voltages to con-trol the electrostatic doping, n, and vertical displacement field, D(Fig. 1a)9,10,18. When θ ~ 1. 3°, the moiré-modified conduction band(hereafter referred to as the moiré band) can be tuned using D to benearlyflat, separated fromadispersive band at higher energy and fromReceived: 14 April 2022Accepted: 12 October 2022Check for updates1Department of Physics and Astronomy & Stewart Blusson Quantum Matter Institute, University of British Columbia, Vancouver, BC V6T 1Z4, Canada.2Research Center for Functional Materials, National Institute for Materials Science, Namiki 1-1, Tsukuba, Ibaraki 305-0044, Japan. 3International Center forMaterials Nanoarchitectonics, National Institute for Materials Science, Namiki 1-1, Tsukuba, Ibaraki, Japan. 4Physics Department, University of Texas at Austin,Austin, TX 78712, USA. e-mail: koolmanab@gmail.com; jfolk@physics.ubc.caNature Communications |         (2022) 13:6468 11234567890():,;1234567890():,;http://orcid.org/0000-0001-6705-5493http://orcid.org/0000-0001-6705-5493http://orcid.org/0000-0001-6705-5493http://orcid.org/0000-0001-6705-5493http://orcid.org/0000-0001-6705-5493http://orcid.org/0000-0001-9769-0738http://orcid.org/0000-0001-9769-0738http://orcid.org/0000-0001-9769-0738http://orcid.org/0000-0001-9769-0738http://orcid.org/0000-0001-9769-0738http://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/0000-0003-3561-3379http://orcid.org/0000-0003-3561-3379http://orcid.org/0000-0003-3561-3379http://orcid.org/0000-0003-3561-3379http://orcid.org/0000-0003-3561-3379http://orcid.org/0000-0003-4455-5609http://orcid.org/0000-0003-4455-5609http://orcid.org/0000-0003-4455-5609http://orcid.org/0000-0003-4455-5609http://orcid.org/0000-0003-4455-5609http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-34192-x&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-34192-x&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-34192-x&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-34192-x&domain=pdfmailto:koolmanab@gmail.commailto:jfolk@physics.ubc.cathe valence band below (Fig. 1b)19. For gate voltages that place theFermi energy within such a flat and isolated moiré band, transportmeasurements typically report regions of higher resistivity sometimesreferred to as ‘haloes’. Theoretical predictions9,10,20 and experimentaldata19,21–28 indicate that, within the halo regions, the moiré band istopologically non-trivial and interactions lead to broken spin and val-ley symmetries, similar to the symmetry breaking observed in quan-tum Hall ferromagnetism29,30.The preciseway inwhich spin and valley symmetries are broken intDBG remains an open problem10,19,21,24. Valley-polarized, spin-valley-locked, spin-polarized, and intervalley coherent states have all beenconsidered10. The consistent observation of an insulating state whenthemoiré band is half filled provides a valuable clue. The resistance ofthis state increases with in-plane field, B∥, suggesting that the metallicstates on either side may be spin polarized21,22,24 but leaving the ques-tion of possible valley order unaddressed19,24,31. In particular, an AHE ofthe kind reported in other graphene systems has not been reported intDBG with small θ.Here, we report the observation of a strong AHE in AB-AB stackedtDBG (θ = 1. 3°), with longitudinal (Rxx) andHall resistance (Rxy) that aresharply hysteretic under out-of-plane magnetic field (B⊥). The datademonstrate orbital (valley) time reversal symmetry breaking, whichmodifies the Hall resistivity when there is strong Berry curvature14,confirming predictions for tDBG near the top of the moiré band(Fig. 1c). Qualitative differences between in- and out-of-planemagneticfield dependence in the AHE signatures reinforce the orbital (valley)character of the ferromagnetism32,33. At the same time, both long-itudinal and transverse resistivity are sensitive to unusually small in-plane magnetic fields, an effect that is not yet understood.ResultsDevices were fabricated using established techniques1,34 (see Methods),and patterned into Hall bar geometries tomeasure Rxx and Rxy. Figure 1ashows a schematic of the device architecture andmeasurement scheme.Multiple voltage probe pairs were measured in each device, with similarbehaviour across most pairs (see SI). Mixing between Rxx and Rxy hasbeen minimized in the data as presented by reporting the field sym-metrized longitudinal resistance, henceforth labelled ρxx, and the anti-symmetrized transverse resistance, ρxy (see Methods). Reported valuesof B⊥ have been adjusted to reflect flux trapping in the superconductingmagnet (see SI). Similar behaviour is seen in two devices (D1 and D2),with twist angles θ≈ 1.31° and θ≈ 1.34° respectively; these were the onlytwo devices with twist angle near 1.3° that were measured.A typical resistivity map over top- and back-gate is shown inFig. 1d, plotted with respect to n and D. Insulating stripes at n = 0reflect the separation of conduction from valence band by finite D,while the insulating stripe around n = 4 × 1012 cm−2 reflects full filling ofthe first conduction band. Given the 4-fold degeneracy of the band(spin and valley), full filling is achieved at four electrons per moiré cell(ν = 4), allowing the twist angle to be calculated and the relationbetween n and ν determined (see Methods). Near D ~ ± 0.4 V/nm,numerical calculations indicate a moiré band that is nearly flat andisolated both from lower and upper bands. Figure 1b shows single-particle and self-consistent Hartree calculations of the moiré bandstructure for the parameters matching experimental conditions (seecaption), illustrating the flatness of the band especially in the self-consistent calculation that highlights the role of interactions19,35.SettingD near ± 0.4 V/nm in the device, a strong insulating state atν = 2 is clearly visible, consistent with previous reports21,22,24, withFig. 1 | Twisteddoublebilayergraphene. aSchematic of a tDBGdevice consistingoftwo Bernal (AB) bilayer graphene layers stacked with a twist angle θ≈ 1. 3°. The stack isencapsulated with top and bottom hexagonal boron nitride (hBN) layers, with a gra-phitebottom-gate (Vbg) andametal top-gate (Vtg) to independently tune thedensity (n)and displacement field (D). b Calculated moiré band dispersion for tDBG with twistangle θ= 1. 3° and D=0.44V/nm. The solid (dashed) line corresponds to Hartree-Fock(single particle) calculations, with thicker lines indicating the first conduction band. HFcalculations are filling-dependent, and are made for ν=3.7. c Berry curvature (Ω)calculated for the conduction band. d Four terminal resistance, Rxx, as a function of nand D at T=20mK, B=0 for D1. The top axis shows the band filling, ν.Article https://doi.org/10.1038/s41467-022-34192-xNature Communications |         (2022) 13:6468 2surrounding 'halo' regions of higher resistivity sharply separated froma lower-resistance background. The relatively narrow range of ∣D∣ overwhich the halos appear is explained by the importance of an isolatednarrow band for strong interaction effects: at too-small values of ∣D∣there is almost nogapbetween conduction andvalenceband,while fortoo-large D the first moiré band overlaps with the second. The highquality of these devices is demonstrated by the the strong insulatingstate at ν = 3 even atB =0; this signature is known to emerge only overa narrow range of twist angle θ ~ 1. 3° 24 where the correlations aremaximum.Figure 2 a offers a higher resolution map of Rxx across the haloregion at negative D, with insulating states at ν = 2 and ν = 3 clearlyvisible. More insight into the broken-symmetry metallic states withinthe halo may be obtained from the Hall resistance, shown for B⊥ = ± 1Tin Fig. 2b. ρxy changes sign across ν = 2 within the entire halo, and alsoat ν = 3 across a narrow region at the low-D edge of the halo. Thisbehaviour implies that the 4-fold band degeneracy is brokenthroughout the halo region, and fully lifted for a narrow range of D.The most telling data comes from the magnetoresistance of thecorrelated metallic states. Figure 2c, d show ρxx and ρxy as B⊥ is sweptfrom 350mT to −350mT and back, comparing data for three {ν,D} pairs,indicated by markers in Fig. 2a. These datasets are chosen to illustratethree characteristic behaviours observed within the {ν,D} map. Outsidethe halo region (•), longitudinal magnetoresistance is weak and the dataare independent of sweep direction, consistent with the behaviour ofnon-interacting metals. Within the halo but far from ν=4 (□), ρxx showsan additional strong but narrow positive magnetoresistance, but ρxy isagain nearly linear and neither ρxx(B⊥) nor ρxy(B⊥) depend on sweepdirection. Within the halo and near ν=4 ( × ), both ρxy and ρxx arestrongly hysteretic, a clear indication of spontaneously broken timereversal symmetry giving rise to ferromagnetism with coercive fieldsnear 0.3 T (bottom panels in Fig. 2c,d). AHE was also observed for thehalo at positive D, though the hysteresis was less pronounced (see SI).The range of ν over which ferromagnetic AHE signals appear isillustrated in Fig. 2e, showing data along the D = − 0.43 V/nm line inFig. 2a. Close to the single-particle insulator (ν > 3.75), multiplejumps are seen in up-and-down magnetic field sweeps, indicatingmulti-domain switching behaviour; this behaviour continues deepinto the insulating state. Throughout the range 3.5 ≲ ν ≲ 3.7, thehysteresis loop was wide and clean with nearly constant coercivefield, then below ν ~ 3.4 both the width and height of the hysteresisloop collapsed. The reduced width of the loops indicates lowercoercive fields for ν ≲ 3.4. The reduced height may result fromreduced valley polarization or smaller Berry curvature: numericalcalculations indicate that the strongest Berry curvature is nearkx = ky = 0, corresponding to the highest energy (last-filled) statesin the moiré band(Fig. 1c). For ν < 3.3, no Rxy hysteresis loops of theform shown in Fig. 2d, f were seen, though evidence of first-orderdomain flips was visible in the narrow region of gate voltageimmediately adjacent to the ν = 3 insulating state (see SI). Takentogether, the data are consistent with valley polarization origi-nating from the ν = 3 state but only developing into AHE withstrong Berry curvature above ν = 3.4.Interestingly, AHE hysteresis loops were nearly independent of Dwithin the halo, although outside the halo they were absent. The datain Fig. 2f represent the evolution of the AHE at fixed ν = 3.6, varying thedisplacement field from D = −0.4V/nm to D = −0.46 V/nm (Fig. 2f).This uniformity contrasts with the very narrow range of D over whichthe ν = 3 insulating state is observed.Having established a ferromagnetic AHE close to full filling of themoiré band, we turn to the question of whether spin or valley sym-metry breaking is responsible for the observed effect. Previous reportsof spin polarization in the ν = 2 insulator and presumably in the sur-rounding metallic states might suggest spin ferromagnetism as apossible source for the AHE, but the weak spin-orbit interaction36 ofgraphene makes this explanation less likely. Experimental input intoFig. 2 | Out-of-plane magnetoresistance. a Rxx as function of ν and D showing thecorrelated insulating states at ν = 2 and ν = 3, appearing in red by the colour scalechosen here, surrounded by a halo region that is metallic but higher resistance(light blue) than the surrounding areas (dark blue). b Anti-symmetrized Hallresistance, ρxy, for ∣B⊥∣ = 1T. The Hall resistance changes sign at integer fillings andalso at the boundary of the halo region. c, d Magnetic field dependence of thesymmetrized longitudinal resistance,ρxx(B⊥), and anti-symmetrizedHall resistance,ρxy(B⊥), for three valuesof {ν,D}marked ina.B⊥ is sweptback and forth, shownwithsolid (positive to negative) and dashed (negative to positive) lines. e, f ρxy(B⊥) as B⊥is swept back and forth (solid/dashed) for (e) varying fillings ν = 3.4→ 3.75 at fixedD = −0.43V/nm, and (f) varying D = −0.4→ = −0.46V/nm at fixed ν = 3.6. D1 atT = 20mK for all.Article https://doi.org/10.1038/s41467-022-34192-xNature Communications |         (2022) 13:6468 3this question is obtained from a comparison of in-plane and out-of-plane magnetoresistance: whereas the valley degree of freedom cou-ples primarily to out-of-plane field, spins are expected to couple tototal magnetic field with g = ~2.Figure 3 explores the effect ofB∥ inmoredetail. A clear illustrationof the anisotropic nature of the AHE comes from Fig. 3a, where B⊥hysteresis loops (±450mT) are shown for increasing fixed B∥. A parti-cularly robust AHE in D2 exhibits a small hysteresis loop in B⊥, withcoercive field around ~50mT, even when B∥ is held at 5T. (Equivalentdata for D1 show B⊥ hysteresis persisting above B∥ ~ 1 T, see SI.) Theweak dependence of the AHE on the in-plane component of themagnetic field, up to B∥ of several Tesla, is reminiscent of the angle-dependent hysteretic AHE observed in twisted bilayer graphenealigned to hBN33. In thatwork, the AHE signature depended exclusivelyon the out-of-plane magnetic field component, leading to the con-clusion that the AHE signal must be attributed to valley polarization.That conclusion was based on the understanding that valley order isonly weakly sensitive to B∥.DiscussionIn our measurements of tDBG, as in ref. 33, there is a qualitative dis-tinction between phenomena driven by B⊥ and B∥. But the influence ofB∥ is nevertheless very strong—much stronger (especially for small B∥)than in ref. 33. Figure 3b, c provide illustrations of this extreme B∥magnetoresistance, which was observed in both devices. Throughoutthe AHE region an anomalously sharp negative B∥-magnetoresistancewas observed (Fig. 3b), with a rounding around zero field on theB∥ ~ 1mT scale that is orders of magnitude smaller than what might beexpected when comparing gμBB∥ to kBT (kBT/gμB ~ 100mT at 100mKfor g = 2). Examining a sequence of B∥ sweeps for increasing ν, along aline of fixedD = −0.43 in the centre of the halo (Fig. 3c, c.f. Fig. 2e), thepeak appears immediately upon entering the AHE region (ν ≳ 3.4) andgrows with larger values of ν. Beyond the consistently-observed peakin Rxx(B∥), however, the details of B∥ magnetoresistance in the AHEregion were less repeatable from scan to scan than the hysteresis in B⊥that is the primary subject of this work. The B∥ scans in Fig. 3c areneither hysteretic nor exactly reproducible, in marked contrast to theAHE hysteresis loops reported elsewhere in this work, which retracescan after scan.The striking effects of small B∥ seen in Fig. 3 have not, to ourknowledge, been reported before; they appear in both devices, andonly inside the AHE regions. These effects may be due to the coupling,albeit weak, of B∥ to valley order10. Even though this coupling is small,proportional to the thickness of the double bilayer, it competes with avalley anisotropy that is poorly understood atpresent and likely also tobeweak. Alternately, B∥ could influence the AHE via spin-orbit coupingas reported recently in twisted (single) bilayer graphene with spin-orbit interaction enhancedby proximity toWSe237. Although spin-orbitcouplingwas not intentionally enhanced in our devices, this possibilitycannot be excluded. A more complete study of the B∥-induced effectsin tDBG will be the subject of future work.The AHE signatures seen in Fig. 2 persist up to 1.8 K in D2 (similarin D1), illustrating the relatively large energy scales associated withvalley symmetry breaking in this system. Figure 4a, b show thetemperature-induced collapse of the hysteresis loop via shrinkingcoercive field for ρxy xx and ρxx xy respectively. Early measurements oftDBG noted a step-like rise in Rxx with temperature within the haloregion, which has been attributed to temperature-induced collapse ofbroken symmetry states24. An analogous rise in Rxx around 7 K isobserved in our measurements away from the AHE region (ν < 3.2,Fig. 4c). The onset of AHE with increasing ν is correlated with a sharpdecrease in the transition temperature, consistent with the Kelvin-scale collapse of the symmetry-broken state that gives rise to thehysteresis in Fig. 4a,b. The gradual shift of the transition temperaturewith increasing ν in Fig. 4, instead of the appearance of a secondsymmetry breaking transition in the AHE region, supports the notionof a correlated low-T ground state with coupled spin and valleyorder38,39.In conclusion, we have observed an AHE, signifying orbital mag-netic order, in AB-AB stacked tDBG. The ferromagnetic state occursonly within the strongly interacting ‘halo’ regions of the (ν,D) plane.Strong magnetic anisotropy suggests valley ferromagnetism, while in-plane field signatures indicate a complex interplay of broken spin/valley symmetries in the correlated metallic state of tDBG.MethodsDevice fabricationHigh quality AB-AB stacked tDBG devices were fabricated using the'modified' tear and stack technique1,40. First, a large bilayer graphene(BLG) flake ~ 70μm was exfoliated on a Si/SiO2 substrate with 285nmoxide. The BLG flake was mechanically precut into two pieces using asharp ~ 1μm diameter tungsten dissecting needle which was attachedto the micromanipulator in our transfer setup. Using a stampmade ofpolybisphenol carbonate (PC) on polydimethylsiloxane (PDMS), thetop hBN layer was picked up at T = 100 °C. Next, the BLG flake waspicked up at T = 30 °C. The stage was rotated to 1.36°, followed bypickingup the secondBLG atT = 30 °C. Then thebottomhBN layerwaspicked up at T = 100 °C, followed by graphite at T = 110 °C. Thesequence of stacking from top to bottom is hBN, tDBG, hBN and gra-phite. Finally, the entire stack was deposited on a Si/SiO2 substrate atT = 175 °C. Top gate was defined using electron beam lithography(EBL), followed by deposition of Cr/Au (5 nm/50nm). Edge contactswere formed by etching the stack in CHF3:O2 plasma, followed bydeposition of Cr/Au (5 nm/80nm) at a base pressure of ~ 1 × 10−7mbar.Finally, another step of EBL and etchingwas done to pattern the deviceFig. 3 | Anisotropic magnetoresistance. a Hysteresis in ρxy(B⊥) for fixed in-planemagnetic fields (B∥) up to 5 T. The curves are offset by 600Ω for clarity (D2,ν = 3.57, D = −0.43 V/nm, T = 100mK). b A sharp peak in Rxx(B∥) is observed atB∥ = 0 throughout the AHE region. The slight offset in Rxx, comparing up anddown traces away from B = 0, is caused by additional heating of the sample whensweeping the magnet away from zero field. Inset: finer scan just over the peakillustrates rounding on the B∥ = 1mT field scale (D1, ν = 3.43, D = −0.42 V/nm,T = 100mK). c Evolution of the Rxx(B∥) magnetoresistance with ν. The peakappears immediately upon entering the AHE region (ν ≳ 3.4). Difference betweenup and down traces illustrates irreproducibility of B∥ sweeps at low field (D2,D = −0.43 V/nm, T = 300mK, B⊥ = 0).Article https://doi.org/10.1038/s41467-022-34192-xNature Communications |         (2022) 13:6468 4in a Hall bar geometry. The top and bottom hBN thickness wasdetermined by atomic force microscope.Transport measurementsThe transport measurements were performed in four terminal con-figuration, using standard lock in (SRS830) techniques at f ~ 13.3Hzwith a current bias I = 1 nA. The experiment was carried out in a Blue-fors cryogen-free dilution refrigerators equipped with a 3-axis magnetenabling simultaneous control of in- and out-of-plane magnetic fields.Except where noted, measurements were carried out at the nominalbase temperature of the refrigerator, with the mixing chamber at10–15mK, but heating due to magnetic field sweeps raised the sampletemperature up to 50–200mK depending on sweep conditions.Twist angle, θ, was calculated from the density corresponding tofull filling of the moiré miniband n(ν = ± 4) = 8θ2=ffiffiffi3pa2, wherea =0.246 nm is the lattice constant of graphene. In D1, ν = ± 4 corre-sponds n = ± 4.05 × 1012cm−12, indicating a twist angle θ ≈ 1.31°. In D2,ν = ± 4 corresponds n = ± 4.16 × 1012cm−12 giving θ ≈ 1.34°.Top gate (Vtg) and bottom gate (Vbg) voltages were used to controlthe charge carrier density (n) and the displacement field (D) indepen-dently. The density is given by n= (CbgVbg+CtgVtg)/e and the displace-ment field is given by D= ∣CbgVbg−CtgVtg∣/2ϵ0, where Cbg (Ctg) are thebottom(top) gate capacitances per unit area, e is the electronic chargeand ϵ0 is the vacuum permittivity. We label the field symmetrized (anti-symmetrized) data for Rxx (Rxy) as ρxx (ρxy), whereρxxðB, "Þ= Rxx ðB,"Þ+Rxx ð�B,#Þ2WL , ρxxðB, #Þ= Rxx ðB,#Þ+Rxx ð�B,"Þ2WL and the anti-symmetrized Hall resistance ρxyðB, "Þ=RxyðB,"Þ�Rxyð�B,#Þ2 ,ρxyðB, #Þ=RxyðB,#Þ�Rxyð�B,"Þ2 , using ↑,↓ to indicate the magnetic fieldsweep direction.Data availabilityAdditional information related to this work is available from the cor-responding author upon reasonable request. Source data are providedwith this paper.Code availabilityThe code used for the purposes of analyzing data is also available fromthe corresponding author upon request.References1. Cao, Y. et al. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices. Nature 556, 80 (2018).2. 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A.H.M. was supported by was supported by the U.S. Depart-ment of Energy, Office of Science, Basic Energy Sciences, under Awards# DE-SC0019481 and DE-SC0022106. K.W. and T.T. acknowledge sup-port from JSPS KAKENHI (Grant Numbers 19H05790, 20H00354 and21H05233).Author contributionsM.K. fabricated the devices, with help from A.V.; M.K., C.C., and Z.G.performed transport measurements and analysed the data; M.K., J.F.,J.Z. and A.H.M. interpreted the data; M.K. and J.F. wrote the manuscript.K.W. and T.T. provided the hBN crystals.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-022-34192-x.Correspondence and requests for materials should be addressed toManabendra Kuiri or Joshua Folk.Reprints and permissions information is available athttp://www.nature.com/reprintsPublisher’s note Springer Nature remains neutral with regard to jur-isdictional claims in published maps and institutional affiliations.Open Access This article is licensed under a Creative CommonsAttribution 4.0 International License, which permits use, sharing,adaptation, distribution and reproduction in any medium or format, aslong as you give appropriate credit to the original author(s) and thesource, provide a link to the Creative Commons license, and indicate ifchanges were made. 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To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2022Article https://doi.org/10.1038/s41467-022-34192-xNature Communications |         (2022) 13:6468 6https://doi.org/10.1038/s41467-022-34192-xhttp://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/ Spontaneous time-reversal symmetry breaking in twisted double bilayer graphene Results Discussion Methods Device fabrication Transport measurements Data availability Code availability References Acknowledgements Author contributions Competing interests Additional information