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Le Liu, Shihao Zhang, Yanbang Chu, Cheng Shen, Yuan Huang, Yalong Yuan, Jinpeng Tian, Jian Tang, Yiru Ji, Rong Yang, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Dongxia Shi, Jianpeng Liu, Wei Yang, Guangyu Zhang

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[Isospin competitions and valley polarized correlated insulators in twisted double bilayer graphene](https://mdr.nims.go.jp/datasets/78b128c1-2239-4fa8-98ae-7f6b3d5b1720)

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Isospin competitions and valley polarized correlated insulators in twisted double bilayer grapheneARTICLEIsospin competitions and valley polarizedcorrelated insulators in twisted double bilayergrapheneLe Liu1,2, Shihao Zhang 3, Yanbang Chu1,2, Cheng Shen 1,2, Yuan Huang 4, Yalong Yuan1,2,Jinpeng Tian 1,2, Jian Tang1,2, Yiru Ji1,2, Rong Yang 1,5, Kenji Watanabe 6, Takashi Taniguchi 7,Dongxia Shi1,2,5, Jianpeng Liu 3,8, Wei Yang 1,2,5✉ & Guangyu Zhang 1,2,5✉New phase of matter usually emerges when a given symmetry breaks spontaneously, whichcan involve charge, spin, and valley degree of freedoms. Here, we report an observation ofnew correlated insulators evolved from spin-polarized states to valley-polarized states intwisted double bilayer graphene (TDBG) driven by the displacement field (D). At a high field |D | > 0.7 V/nm, we observe valley polarized correlated insulators with a big Zeeman g factorof ~10, both at v= 2 in the moiré conduction band and more surprisingly at v=−2 in themoiré valence band. Moreover, we observe a valley polarized Chern insulator with C= 2emanating at v= 2 in the electron side and a valley polarized Fermi surface around v=−2 inthe hole side. Our results demonstrate a feasible way to realize isospin control and to obtainnew phases of matter in TDBG by the displacement field, and might benefit other twisted ornon-twisted multilayer systems.https://doi.org/10.1038/s41467-022-30998-x OPEN1 Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China. 2 School of PhysicalSciences, University of Chinese Academy of Sciences, Beijing 100190, China. 3 School of Physical Sciences and Technology, ShanghaiTech University,Shanghai 200031, China. 4 Advanced Research Institute of Multidisciplinary Science, Beijing Institute of Technology, Beijing 100081, China. 5 Songshan LakeMaterials Laboratory, Dongguan 523808, China. 6 Research Center for Functional Materials, National Institute for Materials Science, 1-1 Namiki, Tsukuba305-0044, Japan. 7 International Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan.8 ShanghaiTech Laboratory for Topological Physics, ShanghaiTech University, Shanghai 200031, China. ✉email: wei.yang@iphy.ac.cn; gyzhang@iphy.ac.cnNATURE COMMUNICATIONS |         (2022) 13:3292 | https://doi.org/10.1038/s41467-022-30998-x | www.nature.com/naturecommunications 11234567890():,;http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-30998-x&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-30998-x&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-30998-x&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-30998-x&domain=pdfhttp://orcid.org/0000-0002-5787-5022http://orcid.org/0000-0002-5787-5022http://orcid.org/0000-0002-5787-5022http://orcid.org/0000-0002-5787-5022http://orcid.org/0000-0002-5787-5022http://orcid.org/0000-0001-7196-9239http://orcid.org/0000-0001-7196-9239http://orcid.org/0000-0001-7196-9239http://orcid.org/0000-0001-7196-9239http://orcid.org/0000-0001-7196-9239http://orcid.org/0000-0002-7005-1319http://orcid.org/0000-0002-7005-1319http://orcid.org/0000-0002-7005-1319http://orcid.org/0000-0002-7005-1319http://orcid.org/0000-0002-7005-1319http://orcid.org/0000-0002-8783-3878http://orcid.org/0000-0002-8783-3878http://orcid.org/0000-0002-8783-3878http://orcid.org/0000-0002-8783-3878http://orcid.org/0000-0002-8783-3878http://orcid.org/0000-0003-2216-7584http://orcid.org/0000-0003-2216-7584http://orcid.org/0000-0003-2216-7584http://orcid.org/0000-0003-2216-7584http://orcid.org/0000-0003-2216-7584http://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/0000-0002-8564-0415http://orcid.org/0000-0002-8564-0415http://orcid.org/0000-0002-8564-0415http://orcid.org/0000-0002-8564-0415http://orcid.org/0000-0002-8564-0415http://orcid.org/0000-0002-3925-0352http://orcid.org/0000-0002-3925-0352http://orcid.org/0000-0002-3925-0352http://orcid.org/0000-0002-3925-0352http://orcid.org/0000-0002-3925-0352http://orcid.org/0000-0002-1242-4391http://orcid.org/0000-0002-1242-4391http://orcid.org/0000-0002-1242-4391http://orcid.org/0000-0002-1242-4391http://orcid.org/0000-0002-1242-4391mailto:wei.yang@iphy.ac.cnmailto:gyzhang@iphy.ac.cnwww.nature.com/naturecommunicationswww.nature.com/naturecommunicationsTwisted graphene-based moiré system is an ideal platform todiscover new quantum phases of matter1–14. The forma-tion of narrow moiré flat bands quenches the kineticenergy15, and induces correlated phases as a result of an enhancedCoulomb interaction with respect to the kinetic energy. Eachconduction or valence moiré band has four isospin flavors,involving spin and valley degree of freedoms. The interplaybetween charge, spin and valley (orbital) yields a rich phasediagram, in which some delicate yet fragile quantum phases areaccessible at extreme conditions. The pronounced electroninteraction in flat bands tend to induce a spontaneous symmetrybreaking at integer fillings16–18. For instance, valley polarizationhelps to unveil the nontrivial topological nature of the moirébands and quantized anomalous Hall effect at odd fillings inhexagonal boron nitride (h-BN) aligned twisted bilayer graphene(TBG)19,20 and ABC trilayer graphene9 as well as twistedmonolayer-bilayer graphene11,12, while spin polarization is cri-tical to the formation of correlated insulator at v= 2 inTDBG5–8,21,22. Intuitively, one might ask if it is possible to drivethe polarized states from one flavor to the other, and vice versa.This interesting question has been partially targeted in recentobservations of the Pomeranchuk effect in TBG which involves atransition of isospin from unpolarized to polarized states23,24.However, it remains open whether one could achieve a tunabletransition of polarized states from one isospin to the other, andmoreover whether this transition will lead to new phase of matter.Among many twisted graphene-based moiré systems, TDBG isa promising target to address this question based on the followingconcerns. First, TDBG is a strongly displacement field (D) tunablemoiré system25,26. The displacement field can in-situ tune therelative strength of Coulomb interactions, relative to its kineticenergy, thus acts an extra knob beyond the twist angle to tune theband structure and topological properties of TDBG. Second,correlated insulator at half filling in TDBG are spin polarized5–7,and it emerges over a wide range of twist angle from ~1.1° to~1.5°, with a finite D varying from ~0.2 to ~0.5 V/nm. Thesymmetry breaking instability occurs at the boundary betweenspin-polarized and unpolarized states in TDBG, and it give rise toquantum critical behaviors22. Besides, TDBG might, in principle,host valley-polarized states as the ground states since the per-pendicular magnetic field could induce a competition betweenspin and valley by spin and orbital Zeeman effects27, thusresulting to an isospin transition from one to the other.In this work, we report a study of isospin polarizations inTDBG. We fabricate ultra-clean TDBG devices with gold top gateand graphite bottom gate. We push the displacement field to itslimit, and importantly we unveil a transition from spin polar-ization to valley polarization when D approaches a critical fieldD*. The transition is accompanied by new phases of matter, i.e. avalley polarized Chern insulator state at v= 2 in the electron sideand a valley polarized Fermi surface around v=−2 in the holeside, which never realized in previous twisted graphene-basedmoiré system. The valley Chern insulator shows a well quantizedHall conductance plateau at 2e2/h and correspondingly a van-ishing longitudinal component. The valley polarized Fermi sur-face shows a series of quantized Landau levels (LLs) with vLL= 0,±1, ±2, ±3, ±4 and others in the landau fan diagram.Results and discussionGeneral information of TDBG devices. The TDBG devices arefabricated by the cut and stack technique28. A big flake of AB-stacked bilayer graphene is firstly cut into two pieces then stackedup with a twist angle of 60° + θ, so called AB-BA TDBG, asshown in Fig. 1a, where θ is ~1.3° to ensure a strong electroncorrelation of TDBG5. Compared to the previously studied TDBGwith a twist angle of θ (AB-AB TDBG), AB-BA TDBG has similarband structure yet different topological properties according tothe theoretical calculations26,29–32. The stacked samples are ultra-clean with a bubble-free area over a length scale of ~20 μm, asshown in Fig. 1a and Supplementary Fig. 5a. These devices have adual-gate configuration with a graphite bottom gate and gold topgate, which allows independent tuning of the carrier density nand D. Here, n ¼ ðCBGVBG þ CTGVTGÞ=e and D ¼ ðCBGVBG �CTGVTGÞ=2ε0; where CBG (CTG) is the geometrical capacitance perarea for bottom (top) gate, e is the electron charge, and ε0 is thevacuum permittivity. These devices show a good quality with theangle inhomogeneity <0.01° within 2 μm size and mobility onorder of 105 cm2/(V·s) in Fig. 1b, c and Supplementary Fig. 5.We perform cryogenic magneto-transport measurements at abase temperature of T= 1.8 K. Figure 1d shows the longitudinalresistance (Rxx) as a function of filling factor (v) and D at B= 0 T.The filling factor is defined as v= 4n/ns, corresponding to thenumber of carriers per moiré unit cell. Here, ns= 4/A ≈ 8θ/(√3a2),where A is the area of a moiré unit cell, θ is twisted angle, and a isthe lattice constant of graphene. In Fig. 1d, two evident resistancepeaks at v=−4 and 4 correspond to the moiré band gap, and theresistance peak at v = 0 indicates a band gap opening between theconduction band (CB) and the valance band (VB) which resultsfrom D induced inversion symmetry breaking33. Correlatedinsulators at half filling v= 2 are observed at a medium D from0.3 to 0.6 V/nm in device D1 (Fig. 1d) and a similar D from 0.2 to0.5 V/nm in device D2 (Fig. 2a). These correlated insulators arespin polarized as evidenced from its positive in-plane magneticfield response (Fig. 2b, d) and a corresponding Zeeman spin gfactor gs of ~2.25 (Fig. 2f). Similarly, a spin-polarized insulator isdeveloped at v= 1 in Fig. 2b and d, with gs of ~2.35 (Fig. 2f). Allthese features both at zero magnetic field and in-plane magneticfield are in agreements with previous results of AB-AB TDBG5–7,which are also in line with the identical band structure betweenAB-BA and AB-AB TDBG26.Valley polarized correlated insulators at high displacementfields. We observe new correlated insulators in the hole side atv=−2 when D is sufficiently large. The new correlated insulatorsare manifested as new resistance peaks in the hole side at v = −2,as shown in Fig. 1e when |D | > 0.75 V/nm for device D1 and inFig. 2c when |D | > 0.6 V/nm for device D2, in the presence of afinite perpendicular magnetic field. It is unexpected as the valanceband becomes more and more dispersive and entangles withremote bands with the increase of |D | , and it demands animportant role played by electron correlation. In principle, valleydegeneracy can be lifted in a perpendicular magnetic field34,35, asshown in Fig. 1f. Considering a Bloch electron is subjected to aweak magnetic field, the energy can be expressed as εN;σ;τ k;Bð Þ ¼εN;τ kð Þ þ σSgsμBBþmN;τ kð ÞB; where the second (third) term isthe spin (orbital) Zeeman energy. Here, N is the band index,σ = ±1 is the spin index, S= 1/2 is spin quantum number, andτ = ±1 is the valley index. In the third term, mn;τ kð Þ correspondsto the orbital magnetization, and it is opposite in different valleyK and K’ according to the time reversal symmetry. Therefore, thevalley degeneracy can be lifted with the orbital Zeeman effect. Thelast two terms can be expressed as gμBB, where g is the effective gfactor (g= S*gs= 1 for pure spin Zeeman effect).We perform Zeeman effect measurements for the newcorrelated insulator at v=−2. The resulting Zeeman thermalgap as a function of B⊥ is plotted in Fig. 1g, and from which weobtain effective Zeeman g ≈ 13.6. This insulator is also not relatedto spin degree of freedom, as it only response to perpendicularmagnetic field and it is featureless even at B||= 9 T (Fig. 2b).ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-022-30998-x2 NATURE COMMUNICATIONS |         (2022) 13:3292 | https://doi.org/10.1038/s41467-022-30998-x | www.nature.com/naturecommunicationswww.nature.com/naturecommunicationsThus, we assign the correlated insulator at v=−2 as a valley-polarized insulator.Moreover, we also observe a new correlated resistance peak atelectron side v= 2 when |D | > 0.9 V/nm, aside from theconventional spin correlated insulator at v= 2 when D is atmediate range from 0.3 to 0.6 V/nm, as shown in Fig. 1e. At ahigher B⊥, the insulator at the higher D is much morepronounced (Fig. 1e), while the spin-polarized insulator at thelower D becomes weaker and weaker and eventually disappear atB⊥ > 5 T (Supplementary Fig. 13). We plot the gap for the newinsulator at higher D as a function of B⊥ in Fig.1g, and once againwe get g ≈ 8.5, indicating a dominating role played by valleyinstead of spin. Thus, the correlated insulator at v= 2 when |D | > 0.9 V/nm is also valley polarized, similar to that at v=−2.Competition of isospin polarization between spin and valley.The disappear of spin-polarized insulator and simultaneously therise of the valley-polarized insulator at v= 2 with increase B⊥suggest a competition between spin and valley polarizations35,36.Such a competing scenario is even more pronounced in thevalence band at v=−2. The evolution of Rxx at v = −2 with B|| isstudied at a fixed B⊥= 6 T in D2, as shown in Fig. 2e. Note thatthe orbital Zeeman splitting energy remains almost unchangedand only the spin Zeeman effect need to be considered in thissituation. The peak resistance at v=−2 decreases as B|| increases,and it shows an unconventional insulating behavior where R(T)doesn’t present a well thermal activation behavior. Instead, itcould be divided into two parts, i.e. a strong B|| dependent and Tsensitive insulating behavior at T < 10 K and an almost B|| inde-pendent insulating state at T > 16 K. According to the thermalactivation behavior Rxx ¼ R0exp 4=ð2kBTÞ� �, where kB is theBoltzmann constant, we estimated the thermal energy gaps (4)from R (T) at T < 10 K in Fig. 2g. We can see that 4 decreases asB|| increases, and the spin g factor gs ≈−1.26. The negative sign ofgs indicates that spin Zeeman effect tend to close the gap of thevalley polarized insulator at v = −2, a strong evidence of com-peting instead of cooperating between spin and valley polariza-tion. Similar competing behaviors and negative gs are alsoobserved at v= 2 in device D1(Supplementary Fig. 7).Valley Chern insulator with C= 2 emanating from v= 2. Thetopological nature of the valley polarized moiré flat band can berevealed in the Landau fan diagram18,37–42. Figure 3a shows thelongitudinal resistance (Rxx) as a function of v and B⊥ atD= 0.8 V/nm and T= 1.8 K. An obvious wedge-shaped Rxxminimum emanating from v= 2 develops along the line with aslope of dn/dB= Ce/h, where C is equal to 2. The vanishing Rxxcomes together with a plateau of Hall resistance (Rxy), with anonset magnetic field of B⊥= 4.2 T. The perfect quantization ofRxy as well as corresponding zero Rxx, i.e. σxy = 2e2/h with σxx ~0, is demonstrated in Fig. 3b at a fixed B⊥= 7.3 T and Figure 3cat a fixed carrier density of v= 2.46. Here, the conductanceσxx and σxy are given by: σxx ¼ ar � Rxx=ððar � RxxÞ2 þ Rxy2Þ,Fig. 1 Valley polarized correlated insulating states in AB-BA TDBG. a Optical microscope images of the fabrication process of device D1(θ = 1. 38°). Thebilayer graphene, ABBA-TDBG and the dual-gate device are presented in turn from left to right. The bottom gate is few layers graphite (FLG) and the topgate (TG) is Ti/Au. The scale bar is shown in figure. b Four-terminal longitudinal resistance versus carrier density n at D=−0.46 V/nm between every twoadjacent bars from 1 to 16. c Hall mobility and Hall resistance versus carrier density at D = 0 V/nm. d, e Longitudinal resistance Rxx as a function of fillingfactor v and displacement field D in device D1. Left and right figures correspond to the transport data measured at B⊥= 0 T and B⊥= 3 T, respectively.f Schematic of the valley polarization. The blue and pink circles represent the orbital magnetization of K and K’ valley, respectively. Different direction ofarrows indicates the orbital magnetizations of two valleys are opposite. The blue and pink curves correspond to the valley polarized energy band induced bythe orbital Zeeman effect under the perpendicular magnetic field. g Thermal activation gaps versus perpendicular magnetic fields. The top figure shows theenergy gap at v=−2 and D=−1.24 V/nm, and the bottom figure shows the energy gap at v= 2 and D=−0.94 V/nm. Error bars are estimated accordingto the uncertainty at the thermal activation region.NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-022-30998-x ARTICLENATURE COMMUNICATIONS |         (2022) 13:3292 | https://doi.org/10.1038/s41467-022-30998-x | www.nature.com/naturecommunications 3www.nature.com/naturecommunicationswww.nature.com/naturecommunicationsFig. 2 Competition between spin and valley polarization. a–c Longitudinal resistance Rxx as a function of filling factor v and displacement field D of deviceD2 (θ= 1.21°) at B= 0 T, B|| = 9 T and B⊥= 2 T, respectively. d Top, longitudinal resistance Rxx as a function of filling factor v and in-plane magnetic field B||at D=−0.38 V/nm. Bottom, line cuts of Rxx (v, B||) from B||= 0 to B||= 9 T. e Top, longitudinal resistance Rxx as a function of filling factor v and in-planemagnetic field B|| at D= 0.73 V/nm and B? = 6 T. Bottom, line cuts of Rxx(v, B||) from B||= 0 to B||= 6.4 T. f Thermal activation gaps versus B|| at v = 1 andv= 2 corresponding to the insulating states in d. The spin g factor can be extracted from the linear fitting with the spin Zeeman effect, 4 � 2 ´ SgsμBB andS= 1/2. g Thermal activation gaps versus total magnetic field at v=−2 corresponding to the insulating state in e. All Error bars are estimated according tothe uncertainty at the thermal activation region. Inset, temperature dependence of Rxx under the tilted magnetic field. The perpendicular magnetic field isfixed at 6 T and Btotal increases from 6 T to 9 T.Fig. 3 Valley polarized Chern insulator in CB. a Longitudinal resistance Rxx as a function of filling factor v and perpendicular magnetic field B⊥ atD= 0.8 V/nm in device D2. White dash lines correspond to the LL with vLL=+6 emanating from v= 0 and the C= 2 Chern insulator emanating fromv= 2, respectively. b Line cuts of σ (v, B⊥) at B⊥= 7.3 T. The plateau within the orange color area indicates a well-quantized Chern insulator with σxx = 0and σxy = 2e2/h. c Line cuts of σ (v, B⊥) at v= 2.46. Inset, thermal activation gaps of the Chern insulator versus perpendicular magnetic field. Error bars areestimated according to the uncertainty at the thermal activation region.ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-022-30998-x4 NATURE COMMUNICATIONS |         (2022) 13:3292 | https://doi.org/10.1038/s41467-022-30998-x | www.nature.com/naturecommunicationswww.nature.com/naturecommunicationsσxy ¼ Rxy=ððar � RxxÞ2 þ Rxy2Þ, where ar is the aspect ratio of thedevice. Similar quantization with C= 2 is also observed in otherthree devices with both AB-BA and AB-AB stacking, as shownin the Supplementary Fig. 9–11. The observed quantizationwith C= 2 is in good agreement with our theoretic calculationof the valley polarized moire Chern band in TDBG (see Sup-plementary Note 1 for details), where Cv= 1 and the totalChern number C= 2*Cv by including two-fold spindegeneracy.The magnetic field dependent 4 for the C= 2 valley Cherninsulator is illustrated in the inset of Fig. 3c. The gap 4 increaseswith B⊥ and tends to saturate at high B⊥, being consistent withthe mechanism that the valley polarization becomes stronger withthe increased B⊥. We compare the gap of valley Chern insulatorwith that of the first LL (vLL=+6) emanating from v= 0 atB⊥= 6 T (Supplementary Fig. 8), and it turns out that the energygap of the former (~4.3 meV) is much larger than that of thelatter (~1.6 meV). The valley Chern insulator is also clearlydistinguished from the faint features of LLs (with vLL=+1, +3,+4) emanating from v= 2 in the fan diagram of Fig. 3a, wherethe Rxx dip with C= 2 reaches zero while the rest not. Thedistinguishment is further illustrated in the perfect quantizationof Hall conductance with C= 2 in Fig. 3b, c.Fermi surface reconstruction and Landau quantization aroundv=−2. In contrast to the topological valley subband in CB, it isa trivial valley polarized subband with C= 0 in VB according toour calculations with Hartree-Fock approximation at a large D,and the VB is generally more dispersive than CB as D isincreased (see Supplementary Note 1 for details). It goesthrough a series of transformations due to enhanced valleypolarization as B⊥ is increased. Figure 4 illustrates typical Fermisurface reconstructions driven by valley polarization at differentmagnetic field when a large D=−0.73 V/nm is applied.Essentially, the reconstruction is captured in the change of Hallfilling factors vH= 4nH/n to moiré band filling factor v, whereHall carrier density nH= B⊥/eRxy.At B⊥= 0.8 T, the magnetic field is too small to reconstruct theenergy band, and the vH in Fig. 4c varies linearly, indicated by twoblue dashed lines of vH= v and vH= v+ 4 near the chargeneutrality point (CNP) and moiré band edge, respectively. vanHove singularity (VHS) is indicated by the diverging Hall carrierdensity near v=−2, which suggests a Lifshitz transition of theFermi surface39,43. Note that VHS locates at around v=−2,corresponding to the flat band of the highest VB, based on ourband structure calculations from the continuum model (Supple-mentary information). The position of VHS changes with theincreased electric field, and it follows the white contour line inFig. 4b. At B⊥= 4 T, the Fermi surface of VB is greatlyreconstructed (Fig. 4e–g). The divergent carrier densities arereset to zero at v=−2 and extend to both sides along the line ofvH= v+ 2. In this case, a correlation driven energy gap replacesthe VHS and separates the VB into two valley-polarized subbandswith new VHSs near v=−1 and v=−3.The reconstruction of VB is reminiscent of the stoner criterionin ferromagnetic metal44. The paramagnetic phase becomes ofinstability if U*g(EF) > 1, where U and g(EF) are the Coulombrepulsion and density of states (DOS) at the Fermi surface,respectively. Hence, similar to the stoner mechanism, theoccupancy of VB is redistributed between two valley flavorsfacilitated by the divergent DOS at VHS, and then the orbitalZeeman effect separate the two valley-polarized subbands withthe increased B⊥.The complete Landau fan diagram is present in Fig. 4i. LLsemanating from v=−2 are observed with vLL= ±1, ±2, ±3, ±4, −5,DOSDOSvan Hove singularityvalley-polarized insulatorvv-2-2a b c de f g hi j kFig. 4 Fermi surface reconstruction and Landau fan diagram of VB. a, b, e, f Longitudinal resistance Rxx and Hall coefficient RH as a function of filling factorv and displacement field D at B⊥= 0.8 T and 4 T, respectively (device D2). c, g Line cuts of mapping at D=−0.73 V/nm show Hall filling factor vH as afunction of filling factor v. d, h Schematics of density of states at B⊥= 0.8 T and 4 T, respectively. The red dashed line corresponds to v=−2. i Longitudinalresistance Rxx as a function of filling factor v and perpendicular magnetic field B⊥ at D=−0.73 V/nm. j Schematic of LLs shown in i. Red lines correspond tothe LLs emanating from v=−2, and black lines correspond to the LLs emanating from v= 0. k Line cuts show the well-quantized σxy = ±2e2/h, −4e2/hwith almost zero Rxx (orange color bars) and incipiently quantized σxy = ±e2/h, −3e2/h, −5e2/h with finite Rxx at B⊥= 7.6 T (blue color bars).NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-022-30998-x ARTICLENATURE COMMUNICATIONS |         (2022) 13:3292 | https://doi.org/10.1038/s41467-022-30998-x | www.nature.com/naturecommunications 5www.nature.com/naturecommunicationswww.nature.com/naturecommunications−6, indicating the spin and valley degeneracy are completely lifted.The even LLs are derived from the valley-polarized subbands, whilethe odd LLs correspond to the quantum hall ferromagnets45,46which polarize the spin flavors. The difference is reflected in theenergy scale between them. As shown in Fig. 4k, even LLs withvLL= ±2, −4 reveal well quantized Hall plateaus and zero Rxx.However, odd LLs with vLL= ±1, −3, −5 show finite Rxx andincipient Hall plateau which indicates incomplete quantization. Theperpendicular magnetic field plays a key role on the reconstructionof VB. It not only interacts with opposite orbital magnetic momentsat different valleys, promoting the formation of valley-polarizedstates, but also makes the carriers do the cyclotron motion in valley-polarized subbands.We have observed new correlated insulators evolved from spin-polarized states to valley-polarized states in TDBG. The transitionof the isospin polarization is a result of the competition betweenspin and valley, driven by the displacement field and magneticfield. Moreover, we have unveiled the unique topology of theTDBG, including a quantized valley Chern insulator with C= 2emanating at v= 2 in the electron side and a valley polarized yettopologically trivial Fermi surface with C= 0 around v=−2 inthe hole side. It is worth mentioning that our devices are of highquality and state-of-the-art clean, with twist angle inhomogeneity<0.01°. Our studies shed light on the importance of thedisplacement field to tune the band topology, and our resultscould enrich the current understanding of the TDBG system andprovide references for other twisted or non-twisted multilayersystems as well.MethodsEstimation of the twisted angle. The top and bottom gate capacitance can beextracted from the Landau fan diagram and dual-gate voltage mapping. We cal-culate the carrier density n and the electric displacement field D according to thecapacitances and transform the Rxx (Vb, Vt) mapping to the Rxx (n, D) mapping.According to the formula, ns= 4/A ≈ 8θ/(√3a2), the twisted angle θ can beextracted from the location of superlattice resistance peak in the Rxx (n, D) map-ping. The obtained twist angle is further corrected by the Brown-Zak oscillations inthe Landau fan diagram. As shown in Supplementary Fig. 10, Rxx shows dips atΦ/Φ0= 1/q, where Φ= AB⊥ is the magnetic flux per moiré unit cell, Φ0= h/e is themagnetic flux quantum, and q is a positive integer.Estimation of the Zeeman g factor. The Zeeman splitting energy, including thecontributions from both spin and orbital magnetization, can be expressed as4 ~ 2gμBB, where g corresponds the effective g factor. The value of g can beextracted from the linear fitting of thermal activation energy gap versus magneticfield. Under the in-plane magnetic field, only the spin Zeeman effect needs to beconsidered, hence 4 ~ 2gμBB= 2SgsμBB, where S= 1/2 is the spin quantumnumber and gs is spin g factor. To clearly show the role of spin, we present ourresults using gs instead of g in this situation.Calibration of the magnetic field direction. The change of the magnetic fielddirection is achieved by rotating the sample with the attocube stage. Then thedirection is further calibrated by measuring Hall resistance at different rotatingangle. We measured the Hall resistance at Vtg= Vbg= 0 V from −3 T to 3 T inperpendicular and parallel direction, as shown in Supplementary Fig. 15. Theresponse of Hall resistance totally come from the cyclotron motion of the electronunder the perpendicular magnetic field. Hence, the value of the Hall resistanceunder the parallel magnetic field can be used to calibrate the direction of themagnetic field. The Rxy is about −30 ohm at B||= 3 T, which is equivalent to theRxy at B⊥= 0.03 T. 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K.W. and T.T. acknowledge support from the ElementalStrategy Initiative conducted by the MEXT, Japan (Grant No. JPMXP0112101001), JSPSKAKENHI (Grant Nos. 19H05790, 20H00354 and 21H05233) and A3 Foresight by JSPS.Author contributionsW.Y. and G.Z. supervised the project; L.L., W.Y., G.Z. designed the experiments; L.L.,Y.C. fabricated the devices with assistance from Y.Y., J.T., Y.J., J.T., R.Y., D.S.; L.L., Y.C.,C.S., Y.H. performed the magneto-transport measurement; S.Z. and J.L. performed thecalculations; K.W. and T.T. provided hexagonal boron nitride crystals; L.L., W.Y., G.Z.,analyzed the data; W.Y., L.L., G.Z. wrote the paper with the input from all the authors.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version contains supplementary materialavailable at https://doi.org/10.1038/s41467-022-30998-x.Correspondence and requests for materials should be addressed to Wei Yang orGuangyu Zhang.Peer review information Nature Communications thanks Jeong Min Park and theanonymous reviewers for their contribution to the peer review of this work. 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To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2022NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-022-30998-x ARTICLENATURE COMMUNICATIONS |         (2022) 13:3292 | https://doi.org/10.1038/s41467-022-30998-x | www.nature.com/naturecommunications 7https://doi.org/10.1038/s41467-022-30998-xhttp://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/www.nature.com/naturecommunicationswww.nature.com/naturecommunications Isospin competitions and valley polarized correlated insulators in twisted double bilayer graphene Results and discussion General information of TDBG devices Valley polarized correlated insulators at high displacement fields Competition of isospin polarization between spin and valley Valley Chern insulator with C = 2 emanating from v = 2 Fermi surface reconstruction and Landau quantization around v = −2 Methods Estimation of the twisted angle Estimation of the Zeeman g factor Calibration of the magnetic field direction Data availability References References Acknowledgements Author contributions Competing interests Additional information