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Xiangyan Han, Yuting Zou, Qianling Liu, Zhiyu Wang, Ruirui Niu, Zhuangzhuang Qu, Zhuoxian Li, [Chunrui Han](https://orcid.org/0000-0002-6257-1103), [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Baojuan Dong, Zhida Song, [Jinhai Mao](https://orcid.org/0000-0002-9034-3642), [Zheng Han](https://orcid.org/0000-0001-5721-6206), [Zhi Gang Cheng](https://orcid.org/0000-0002-9449-6734), Zizhao Gan, [Jianming Lu](https://orcid.org/0000-0002-1558-4040)

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[Suppression of symmetry-breaking correlated insulators in a rhombohedral trilayer graphene superlattice](https://mdr.nims.go.jp/datasets/a821c4d9-2426-40fb-9f8a-038133ac2502)

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Suppression of symmetry-breaking correlated insulators in a rhombohedral trilayer graphene superlatticeArticle https://doi.org/10.1038/s41467-024-54200-6Suppression of symmetry-breakingcorrelated insulators in a rhombohedraltrilayer graphene superlatticeXiangyan Han1, Yuting Zou2,3, Qianling Liu1, Zhiyu Wang1, Ruirui Niu1,Zhuangzhuang Qu1, Zhuoxian Li1, Chunrui Han 3,4, Kenji Watanabe 5,Takashi Taniguchi 5, Baojuan Dong6,7,8, Zhida Song9, Jinhai Mao 10,Zheng Han 6,7,8,11 , Zhi Gang Cheng 2,3 , Zizhao Gan1 & Jianming Lu 1Counterintuitive temperature dependence of isospin flavor polarization hasrecently been found in twisted bilayer graphene, where unpolarized electronsin a Fermi liquid become a spin–valley polarized insulator upon heating. So far,the effect has been limited to v = +/−1 (one electron/hole per superlattice cell),leaving open questions such as whether it is a general property of symmetry-breaking electronic phases. Here, by studying a rhombohedral trilayer gra-phene/boron nitridemoiré superlattice, we report that at v = −3 a resistive peakemerges at elevated temperatures or in parallelmagnetic fields. Concomitantly,the Hall carrier density tends to reset at the integer filling, signaling spin–valleyflavor symmetry breaking. These phenomena can also be observed at v = −1 and−2 when the displacement field is large enough to suppress correlated insula-tors at low temperatures. Our results greatly expand the scope for observingthe counterintuitive temperature dependence of flavor polarization, i.e., theregimes proximal to symmetry-breaking phases where the flavor polarizationorder strongly fluctuates, encouraging more experimental and theoreticalexploration of isospin flavor polarization dynamics in flat-band moiré systems.Moiré superlattices in twisted van der Waals heterostructures havebeen found to host strongly correlated electrons, giving rise to exoticphenomena1–21 such as correlated insulators, superconductivity, fer-romagnetism, (fractional) Chern insulators, and the counterintuitiveisospin Pomeranchuk effect22–24. The Pomeranchuk effect is an entropydriven liquid-to-solid transition in 3He, in which the system symmetryis lowered with increasing temperature. By replacing the 3He atomswith flavor polarized electrons, one can observe its electronic analogyin twisted bilayer graphene (TBG)—isospin Pomeranchuk effect, wherespin–valley unpolarized electrons at low temperature turn to be flavorpolarized at high temperature. Correspondingly, the Fermi liquidgrows into an isospin-ordered insulator upon heating. Among varioustheoretical models proposed for TBG, a consensus has been reachedthat spin/valley symmetry breaking will result in energy gaps at com-mensurate fillings, whereas there are occasional exceptions that theground state has no gap but exhibits a Fermi liquid owing to theReceived: 2 June 2023Accepted: 31 October 2024Check for updates1State Key Laboratory for Mesoscopic Physics, School of Physics, Peking University, Beijing 100871, China. 2Beijing National Laboratory for CondensedMatterPhysics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China. 3Institue of Microelectroincs, Chinese Academy of Sciences, Beijing100029, China. 4University of Chinese Academy of Sciences, Beijing 100049, China. 5National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044,Japan. 6State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Opto-Electronics, Shanxi University, Taiyuan 030006, China.7Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China. 8Hefei National Laboratory, Hefei 230088, PR China.9International Center for Quantum Materials, Peking University, Beijing 100871, China. 10School of Physical Sciences and CAS Center for Excellence inTopological Quantum Computation, University of Chinese Academy of Sciences, Beijing, China. 11Liaoning Academy of Materials, Shenyang 110167, China.e-mail: vitto.han@gmail.com; zgcheng@iphy.ac.cn; jmlu@pku.edu.cnNature Communications |         (2024) 15:9765 11234567890():,;1234567890():,;http://orcid.org/0000-0002-6257-1103http://orcid.org/0000-0002-6257-1103http://orcid.org/0000-0002-6257-1103http://orcid.org/0000-0002-6257-1103http://orcid.org/0000-0002-6257-1103http://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/0000-0002-9034-3642http://orcid.org/0000-0002-9034-3642http://orcid.org/0000-0002-9034-3642http://orcid.org/0000-0002-9034-3642http://orcid.org/0000-0002-9034-3642http://orcid.org/0000-0001-5721-6206http://orcid.org/0000-0001-5721-6206http://orcid.org/0000-0001-5721-6206http://orcid.org/0000-0001-5721-6206http://orcid.org/0000-0001-5721-6206http://orcid.org/0000-0002-9449-6734http://orcid.org/0000-0002-9449-6734http://orcid.org/0000-0002-9449-6734http://orcid.org/0000-0002-9449-6734http://orcid.org/0000-0002-9449-6734http://orcid.org/0000-0002-1558-4040http://orcid.org/0000-0002-1558-4040http://orcid.org/0000-0002-1558-4040http://orcid.org/0000-0002-1558-4040http://orcid.org/0000-0002-1558-4040http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-54200-6&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-54200-6&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-54200-6&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-54200-6&domain=pdfmailto:vitto.han@gmail.commailto:zgcheng@iphy.ac.cnmailto:jmlu@pku.edu.cnwww.nature.com/naturecommunicationsrestored flavor symmetry. The counterintuitive temperature depen-dence of flavor polarization, and the inferred local magnetic momentas proposed in previous theories and experiments25–28, gain renownedinterests about their role in shaping ground states of a flat bandmoirésystem. However, experiments so far have been mainly focusing onfilling factors v = ±1 in TBG, i.e., one electron/hole per moiré super-lattice cell; even in twistedWSe2/MoTe2 bilayer29,30, the effect emergesonly at v = −1. While signatures of the effect at v = −3 and 3.5 areavailable11,31, its existence at filling factors other than one remains to beconfirmed. Whether this counterintuitive temperature dependence isa general property of isospin flavor polarization is important for con-straining various theoretical models.As a tunable quantum simulator of the Hubbard model32,33,rhombohedral trilayer graphene (r-TLG) moiré superlattices (formedby crystallographic alignment with hexagonal boron nitride (hBN))exhibit various exotic properties such as Chern insulators, ferro-magnetismand signatures of superconductivity34–37. The advantages ofthis system include the strong tunability of electronic correlation bydisplacement fields, and more importantly, the clear and completecharacterization of its moiréless counterpart38,39. The high transpar-ency of such a system is expected to be suitable for examining theconsistency between theories and experiments.Here we focus on the non-topological regime of a hole-doped r-TLG moiré superlattice and find that at v = −3 (i.e., three holes permoiré superlattice cell) a Fermi-liquid ground state turns into a cor-related insulating peak upon either rising temperatures or applyingparallel magnetic fields. Such a counterintuitive phenomenon resem-bles that in TBG, but takes place at a distinct filling factor. In addition,this behavior is observed atv = −1 and−2when thedisplacementfield islarge enough to suppress the correlated insulators. At last, we discussthe potential theoreticalmodels including isospin Pomeranchuk effectand Kondo lattice. The wide exploration of the counterintuitive tem-perature dependence in thephasediagramof various heterostructuresmayhelp establishing a theoreticalmodel of isospinflavor dynamics incorrelated electronic systems.ResultsThe van der Waals heterostructure is schematically shown in the insetof Fig. 1a, where the r-TLG is aligned with the top hBN to generate amoiré superlattice. Top and bottom gates (Vt and Vb) are used toindependently control the displacement field D and carrier density n.Herewe define n = (Db +Dt)/e,D = (Db-Dt)/2, withDb(t)= ε0εrVb(t)/db(t), εras the relative dielectric constant of hBN, db(t) as the thickness of thebottom (top) hBN flake. For simplicity, a filling factor v = 4n/ns is used,wherens is the density required to fully fill themoiré band. As shown inFig. 1a, theD-vphase diagram taken at 1.5 K displays resistancepeaks atcharge neutral point (CNP) at v =0 and full filling point (FFP) of themoiré valence band at v = -4, and correlated peaks at fractional fillingsv = -1, -2 (See the full phase diagram of Sample 1 in SupplementaryInformation (SI), Supplementary Note 1; Sample 2 in SupplementaryNote 2 and Sample 3 in Supplementary Note 3).Symmetry-breaking phasesTo gain more insight into the phase diagram, the correspondingmapping of the normalized Hall carrier density, νH = 4nH/ns, is pre-sented in Fig. 1b, where nH = -[e(dRxy/dB)]−1 (with e being the electroncharge and Rxy representing the Hall resistance, see more in Supple-mentary Note 1.6). In this mapping, blue and red colors denote forelectron and hole carriers, respectively. There are three types oftransitions40,41 (insets of Fig. 1b): The ‘gap’means when the Fermi levelcrosses a gap, nH continuously crosses zero. The ‘reset’ denotes thebehavior that nH drops suddenly to zero but it does not change sign,which has been found at integer filling factors in twisted bilayer gra-phene due to the Coulomb-induced phase transitions. The ‘VHS’(abbreviated for Van Hove singularity) exhibits a divergent nH with azero crossing,which is typically observed at saddle-points on the Fermisurface. Accordingly, three types of dashed lines divide the phasediagram into several regions. Taking the white section line as anexample, we plot in Fig. 1c Hall resistance and carrier density to the leftand right axes, respectively. At v = −4and0, there are typical bandgapsisolating the narrow moiré band. During the electron filling of themoiré band, for −4<v < −2 electron density firstly increases linearly,then gradually resets to zero at v = −2. Subsequently, it increases againand resets at v = −1. In the region of −1<v <0, the carrier type changesfrom electron to hole by crossing VHS. The carrier reset behaviorwithin aflatbandhas beenobserved in twisted bilayer graphene,whichis accompanied by spin and valley symmetry breaking42 evidenced byreduced degeneracies. However, in the hole band of r-TLG moirésuperlattice quantumoscillation is found to be absent undermoderatemagnetic fields, prohibiting the confirmation of lifted degeneracy.In contrast, for the conduction band we can readily observe cor-related insulators andassociatedLandau levels (Fig. 1d, e).Obviously, thedegeneracy close to v= 1 is one and that close to v =2 is two. It’s inter-esting to see that the sequence of lifted degeneracies is exactly the sameas that of r-TLG without moiré superlattice38, although the onset carrierdensities of the resetmay be different. At B= 1 T, no correlated insulatorcan be seen (Fig. 1d), and the carrier reset (Fig. 1f, upper panel) is notbound to integer filling; Only when B= 3T (Fig. 1f, lower panel), theenergy gaps can be observed at integer filling, in agreement with cor-related insulators shown in Fig. 1d. To summarize, the degeneracysequence in the conduction band is not changed by themoiré potential.We tentatively apply this argument to the valence band: it is spin–valleypolarized for v=−1 and spin polarized for v =−2. The former postulationcan be firmly established for D>0 where a spin–valley polarized flavorwith a nontrivial Chern number has been observed35. The latter is sup-ported by strong magnetoresistance in in-plane magnetic field B//(Fig. 1g), forwhich the extractedg factor agreeswith a spinpolarizedgap(Fig. 1h). In addition, spin–valley polarization at v=−1 is in line with thefact that the resistance peak changes little in B// fields but increasessignificantly in B⊥ fields (Supplementary Fig. 2). To this end, we assignthe symmetry-breaking patterns, which is inherited from flavor polar-ization of a moiréless r-TLG38, to different regions of the phase diagram.Suppression of the symmetry-breaking insulator at v=−3At low temperatures, the correlated insulator at v = −3 is missed(Fig. 1a). The absence is understandable because the state with adegeneracy of three is also lacking in its moiréless counterpart38.However, at high temperature (e.g., 20 K in Fig. 2a), a resistive peakemerges within −0.45 <D < −0.25 V/nm. Also, for v = −1 the D range forcorrelated insulators increases rather than decreases. To quantify thedifference, we remove the smooth background (dashed curves inFig. 2c) that originates from thermal broadening of adjacent resistivepeaks and denote R* as the indicator that evaluates the influence of theemerging correlated gap (See Supplementary Note 1.4 for more dis-cussion). Then the contours can be compared in Fig. 2d, where sig-nificant differences could be found at v = −3, and v = −1 at a largedisplacement field (D < −0.55V/nm). Intriguingly, the recovery ofresistive peaks is also found by applying strong in-plane magneticfields (Fig. 2b). The duality between temperature and magnetic fieldsuggests that the counterintuitive behavior may stem from magneticinteraction. In the following, we firstly examine the case of v = −3.Figure 3a shows the temperature dependence of resistance atD = −0.3 V/nm, where the resistance peak at v = −3 gradually decreasesand vanishes at 1.5 K. The transfer curves at various temperatures arecompared in Fig. 3b. Corresponding nH(v) is presented in Fig. 3c (Seemore details of the analysis in Supplementary Note 1.6). Below 15 K,the absolute value of nH increases monotonically, accompanied by anemergent dip at v = −3. The dip, ascribed to the incomplete carrierArticle https://doi.org/10.1038/s41467-024-54200-6Nature Communications |         (2024) 15:9765 2www.nature.com/naturecommunicationsreset at v = −3, grows stronger at 7.5 K and then gradually diminishes atlower temperatures. This carrier reset should not be mixed with theone for v = −2. The latter starts to develop right after v = −3, and growsstronger towards lower temperature.The carrier reset at v = −3 also develops with growing (Fig. 3d).Compared to the weak temperature-driven dip, the field driven resetclose to v = −3 is much stronger. With increasing B// fields, the resetshifts towards v = −3, consistent with the varying position of resistivepeaks (Fig. 3e). The duality between temperature (Fig. 3a) and B// field(Fig. 3e) can be further confirmed in Fig. 3f, in which at elevatedtemperature 12.5 K the B// field continues to strengthen the correlatedpeak (i.e., promoting the symmetry-breaking insulator).Before discussing its mechanism, we need to firmly excludeanother mechanism, i.e., a semimetal due to the small correlation gap.For a correlated semimetal, during cooling process the resistance peakfirst increases because of the growing gap size owing to increasingcorrelation, then decreases due to suppressed electron-phonon scat-tering. While the temperature dependence of resistance seems to besimilar, its carrier density is expected to decrease monotonically dur-ing cooling down, in contrast to the growing Fermi surface in Fig. 3b.(1,2) (2,2)(3,2)(4,2)(6,2)(8,2)(10,2)(1,1)(2,1)(3,1)(4,1)(-1,1)(-2,1)aT = 1.45 KSample 1-0.2-0.3-0.5-0.1-0.4D/ε0 (V/nm)0-0.6-0.7-4 -2 0νDhBN (aligned)hBN (misaligned)0639B┴ (T)10 2ν3bd-2ν-1 0D/ε0 = -0.45 V/nm010614284120 106 14B// (T)2 84 12432∆(meV)D/ε0  (V/nm)EFB// ≠ 0B = 0ν = -2K±,↑K±,↓gμBB-0.2-0.3-0.5-0.1-0.4D/ε0 (V/nm)0-0.6-0.7 -4 -2 0D/ε0 = 0.5 V/nm-2 2νHB┴ = 1TSample 110 2 3n H (×1012 cm-2)0-1-2-2ce2-2-5 0ν-3 1-4 -1nH (×1012 cm-2)RXY(kΩ)50-5-1010-0.5100.5-1gfB┴ = 1 TB// (T)hB┴ = 3 TVHSGap Reset-0.5 (g=2.6±0.2)-0.45 (g=2.1±0.1)0-11n H (×1012 cm-2)10 2ν30639B┴ (T)D/ε0 = -0.375 V/nmν νRABC (kΩ) 0 180RXX (kΩ)102 103 104 1050 50RXX (kΩ)Fig. 1 | Flavor polarization in a r-TLG superlattice. a The D-v phase diagram at1.5 K. Inset: Device schematic showing the direction of displacement fields frommoiréless to moiré interfaces. b Normalized Hall carrier density νH = 4nH/ns atB = ± 1 T, versus v and D. Inset: definitions of three colored dashed lines for bandgap (blue), VHS (yellow) and reset (red). See details in the main text. c Profiles ofcarrier density (right axis) and Hall resistance (left axis) along the white dot line in(b) where phase boundaries are denoted by colored bar. d–f Correspondencebetween symmetry breaking and phase boundaries is evidenced in the moiréconduction band (D =0.5 V/nm). Here, symmetry breaking among spin and valleyflavors is identified by the lifted degeneracy of Landau levels associated with cor-related insulators (d, e). The degeneracy is one for v = 1 and two for v = 2. The phaseboundaries (bandgap, VHS and reset) can also be found at low (1 T in the upperpanel of (f) and high (3T in the lower panel) magnetic fields. g At D = −0.45 V/nm,the resistance peak of v = −2 is found to increase with parallel B fields. h The energygap at v = −2 shows a g factor of the order of 2 in B// fields. Error bars are estimatedfrom the uncertainty in the range of the simply activated regime. Inset: Schematicof spin polarization induced gap at half filling in parallel fields.Article https://doi.org/10.1038/s41467-024-54200-6Nature Communications |         (2024) 15:9765 3www.nature.com/naturecommunicationsExtension to more filling factorsThe counterintuitive temperature dependent isospin flavor polarizationis not limited to v=−3. As shown in Fig. 4a, the phenomenon also existsat v=−1 at D=−0.7V/nm (see more in Supplementary Note 1.5). The B//dependence is also depicted in Fig. 4c. As expected, the resistive peak atv=−1 becomes more significant at higher temperature and in-planemagnetic field. Corresponding carrier reset shows similar dependenceon temperature (Fig. 4b) and B// field (Fig. 4d). Note that the carrier resetand resistive peak still survive at 1.5K, although severely suppressed. Fora complete suppression, amuch lower temperature and/or a higherD areneeded (Fig. 4e, and Supplementary Fig. 18 and Supplementary Note 2).For v = −2, the temperature dependence of resistive peaks andcarrier reset are similar to that of v = −1. One should be cautious aboutthe field dependence, since the intrinsic spin polarized gap at v = −2may also give rise to similar behavior. Nevertheless, the overall simi-larity between these three filling factors probably suggests the sameworking mechanism.DiscussionProximity to boundaries of symmetry-breaking phasesThe requirement for observing the counterintuitive effect can beprobed by examining the similarities between r-TLGmoiré superlatticeand other systems. The most significant phenomenon takes place atv = −3 in r-TLG, whereas it happens at v = ±1 in TBG. Superficially, theyare different filling factors. However, in r-TLG the carrier at v = −3 is ofelectroncharacteristics inferred fromHall effects (Fig. 1b),which isdueto the proximity of VHS to CNP under a large displacement field.Consequently, the state at v = −3 stands for one electron per super-lattice cell. This is actually similar to the case of TBG in which thePomeranchuk effect occurs for one electron/hole per superlattice cell.More importantly, both of them stay between a phase with a fulldegeneracy (i.e., four around v = 0 and +/−4) and a phase with lifteddegeneracy (i.e., two around v = +/−2). The reduction in degeneracyindicates spin–valley symmetry breaking at v = −2, resulting in a fer-romagnetic order (with ordered magnetic moment), e.g., spin polar-ization in r-TLG moiré superlattice. This rule is also valid for the effectat v = −1 and −2 in r-TLG and v = −1 in MoTe2/WSe2, where a large Ddrives the metal-insulator transition and the counterintuitive phe-nomenon is observed on themetallic side. The insulating phase itself isof spin/valley symmetry breaking: although a long-range magneticorder may be not available29, abundant spatially localized magneticmoments are expected to exist. Note that, for the small D where theinsulating phase also disappears, at present we cannot firmly excludeanothermechanism, i.e., the weak correlation at a smallDwill lead to anormal metal. This is because the behavior of carrier reset cannot bereliably characterized by Hall effects due to the proximity to VHS(Fig. 1b). Overall, fluctuation of ordered phases resulted by tuningcarrier density or displacement field may be the prerequisite forobserving this phenomenon. Otherwise, when the interaction of localmoment is strong, a ferromagnetic ordermaybepreferred, resulting ina symmetry-breaking insulator.Possible theoretical modelsWe now discuss possible mechanisms for the observed phenomena.One is the isospin Pomeranchuk effect22,23, in which local moment offlavor polarized electrons is developed from unpolarized Fermi liquidupon increasing temperature. In the r-TLG moiré superlattice, thegrowing feature of the carrier reset at v = −3 upon increasingT = 20 K a-0.2-0.3-0.5-0.1-0.4D/ε0 (V/nm)0-0.6-0.7-4 -2 0B// = 14Tνb-0.2-0.3-0.5-0.1-0.4D/ε0 (V/nm)0-0.6-0.7-4 -2 0cν-4 -2 0ν-0.2-0.3-0.5-0.1-0.4D/ε0 (V/nm)0-0.6-0.7-4 -2 0ν-3 -1RXX (kΩ)15105020d201.5T (K)R*0 25RXX (kΩ) 0 25RXX (kΩ)Fig. 2 | Counterintuitive temperature dependence of resistive peaks. Phasediagrams at high temperatures (a) and strong in-plane magnetic fields (b). c Toevaluate the magnitude and the range of correlated peaks, a smooth background(dashed lines) are removed for transfer curves at low temperature (red) and hightemperature (black). d. Temperature induced difference of correlated resistancepeaks (R* in c) at commensurate fillings v = −1, −2 and −3. The peaks seem todisappear at low temperature for v = −3 and −1. Overlaid are phase boundariesdefined in Fig. 1b.Article https://doi.org/10.1038/s41467-024-54200-6Nature Communications |         (2024) 15:9765 4www.nature.com/naturecommunicationstemperature or B// fields is consistent with this scenario, but a directevidence of entropy is required to substantiate this proposal. Anotheris the Kondo lattice model based on the topological heavy fermionmodel43,44, where the resistance grows significantly when the heavyFermion liquid turns into a state with incoherent scatterings betweenitinerant electrons (c-electron) and localized electron (f-electron). Itwas developed specifically for twisted bilayer graphene45–50 that alsoexhibits the counterintuitive temperature dependence, and signaturesof Kondo effect – zero-bias conduction peak in the tunneling spectra –has been found10,28,51,52. Nevertheless, for the r-TLG moiré superlattice,theoretically a concretemodel is required to checkwhether the Kondolatticemodel is appropriate in this crystallographically distinct system.On the experimental side, we need to further examine the signature ofheavy Fermions (see Supplementary Note 1.3 for r-TLG), and search forthe direct spectroscopic evidence in the future.To conclude, we report transport evidences of counterintuitivetemperature dependence of isospin flavor polarization in a rhombo-hedral trilayer graphenemoiré superlattice. The observation at variousfilling factors and displacement fields enriches the exploration of thiscounterintuitive phenomenon in a quantum simulator of Hubbardmodel, shedding light on the controlled interplay between Coulomband spin interactions of correlated electronic states in moiré super-lattice systems.MethodsSample fabricationRhombohedral trilayer graphene flakes are mechanically exfoliatedfrom natural graphite crystals, whose stacking order is identified byRaman spectroscopy. Standard dry transfer using polycarbonate filmis used to sequentially pick up hexagonal boron nitride and grapheneflakes on demand. After finishing the multilayer heterostructure,Raman mapping is conducted to confirm the rhombohedral stackingorder. At last, e-beam lithography/evaporation and reactive ion etch-ing are used to define a metallic top gate (Cr/Au 5/30 nm) and one-dimensional edge contact.Electrical measurementMost transport measurements (above 1 K) were carried out in a 4Hecryostat with base temperature of 1.5 K and a superconductingmagnetup to 14 T. Unless specified otherwise, the sample temperature was atbase temperature. A standard four-probe method of constant currentwas performed. The AC current (10 nA) was supplied by StanfordResearch Systems SR830 lock-in amplifiers with a ballast resistor atfrequency of 17.777Hz. The DC gate voltages were output by twoKeithley 2400 Source Meters.The ultralow temperaturemeasurements (below 1 K) were carriedout in a dilution refrigerator (Oxford Instrument Triton500) with abase temperature of 8.7mK and a highest magnetic field of 12 T.Composite low-pass filters (LPF) are installed for every lead, eachincluding a 3-meter long thermocoax between the 3 K stage and themixing chamber stage, followed by a RC filter on the mixing chamberstage. The cutoff frequency is 400MHz with an attenuation of −100dB, minimizing the electron temperature. Lock-in amplifiers (NF5640)were used to measure the resistance, with an excitation ac current of2 nA at frequency of 17.77Hz. Bias current was supplied by aDC sourcemeter (Keithley 2612B) for the differential resistance measurements.cνn H (×1012 cm-2) -0.40-0.2-0.6dn H (×1012 cm-2)-0.40-0.2-0.6-0.8D/ε0 = -0.3 V/nm T (K)1.452.53.85.27.512.515B┴ =1 TBtotal (T)136912νRXX (kΩ)100.11001νb1.657101530T (K)νT (K)25155201030eaν0106B// (T)28412f-3 -2 -1-4-3 -2 -1 0-4 -3.5 -3 -2.5ν0106B// (T)2841214T = 12.5 KT = 1.5 K0 5RXX (kΩ)RXX (kΩ) 0 15-3 -2 -1-4 -3.5 -3 -2.5-3 -2 -1 0-4RXX (kΩ) 0 2D/ε0 = -0.3 V/nm Fig. 3 | Temperature and B field dependences at v = −3. a In the T-v diagram withD = −0.3 V/nm, the resistance peak at v = −3 gradually decreases and almost van-ishes at 1.5 K. At v = −1, a similar decrease in resistance is observed but the peakremains significant at low temperature. In contrast, the peak at v = −2 growsstronger.b Line cut from a at various temperatures. Hall carrier density is plotted asa function of temperature (c) and in-plane magnetic field (d) at v = −3 andD = −0.3 V/nm. Carrier reset manifested as a dip is maximized at T = 7.5 K (c) orB// ~ 11.96 T (Btotal = 12 T). With increasing in-plane magnetic fields, the correlatedpeak at v = -3 emerges at low temperature (T = 1.5 K in e) or is significantly enhancedat higher temperature (T = 12.5 K in f). The peak position (black dot) is shiftedgradually towards v = -3 in B// fields (e); whenever it was shifted to v = −3 by risingtemperature, the position would be fixed, regardless of B// fields (f).Article https://doi.org/10.1038/s41467-024-54200-6Nature Communications |         (2024) 15:9765 5www.nature.com/naturecommunicationsThe extraction and physical meaning of R*The parameter R*, obtained by subtracting a smooth background fromthe transfer curve R(v), is used to identify the potential regions withcounterintuitive temperature dependence of isospin flavor polariza-tion. In some circumstances, the feature of a resistance peak R(v) has anon-monotonic evolution as a function of temperature, whereas theabsolute value of the peak resistance decreases upon cooling thesample. The inconsistency obviously stems from the incomplete flavorpolarization and the coexistence of carriers with distinct character-istics, which complicate the macroscopic electronic transport. Toderive the key information on a qualitative level, the smooth back-ground is removed to highlight the behavior of R*. In essence, R*represents the variation of resistance resulted from the emergingcarrier localization, which simultaneously affects the carrier densityand mobility. Consequently, R* remains a convolution of changes incarrier density andmobility. Nevertheless, it can be taken as a sensitiveindicator to detect whether the state has an emerging gap at the Fermisurface. 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We alsoacknowledge the support from Peking Nanofab.Author contributionsJ.L., Z.C. and Z.H. supervised the project. Devices are fabricated by X.H.assistedbyQ.L., Z.W., R.N., Z.Q., Z.L. andC.H.; X.H. performed transportmeasurements with assistance from Y.Z. and B.D.; Crystallographiccharacterization was performed by X.H., Z.L. and J.M.; Theoretical dis-cussion was done by J.L., Z.S. and Z.G.; K.W. and T.T. synthesized boronnitride crystals; All authors contribute to the data analysis. X.H., Q.L.,Z.W., C.H. and J.L. wrote the paper with input from all authors.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-024-54200-6.Correspondence and requests for materials should be addressed toZheng Han, Zhi Gang Cheng or Jianming Lu.Peer review information Nature Communications thanks Louk Rade-maker, and the other, anonymous, reviewer(s) for their contribution tothe peer review of this work. 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To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.© The Author(s) 2024Article https://doi.org/10.1038/s41467-024-54200-6Nature Communications |         (2024) 15:9765 8http://creativecommons.org/licenses/by-nc-nd/4.0/http://creativecommons.org/licenses/by-nc-nd/4.0/www.nature.com/naturecommunications Suppression of symmetry-breaking correlated insulators in a rhombohedral trilayer graphene superlattice Results Symmetry-breaking phases Suppression of the symmetry-breaking insulator at v = −3 Extension to more filling factors Discussion Proximity to boundaries of symmetry-breaking phases Possible theoretical models Methods Sample fabrication Electrical measurement The extraction and physical meaning of R* Data availability References Acknowledgements Author contributions Competing interests Additional information