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Le Liu, Xin Lu, Yanbang Chu, Guang Yang, Yalong Yuan, Fanfan Wu, Yiru Ji, Jinpeng Tian, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Luojun Du, Dongxia Shi, Jianpeng Liu, Jie Shen, Li Lu, Wei Yang, Guangyu Zhang

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[Observation of First-Order Quantum Phase Transitions and Ferromagnetism in Twisted Double Bilayer Graphene](https://mdr.nims.go.jp/datasets/666beb2a-389d-43e1-8266-0f86e7dcfe8d)

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Observation of First-Order Quantum Phase Transitions and Ferromagnetism in Twisted Double Bilayer GrapheneObservation of First-Order Quantum Phase Transitions and Ferromagnetismin Twisted Double Bilayer GrapheneLe Liu,1,2,‡ Xin Lu,3,‡ Yanbang Chu,1,2 Guang Yang,1,2 Yalong Yuan,1,2 Fanfan Wu,1,2 Yiru Ji,1,2Jinpeng Tian,1,2 Kenji Watanabe ,4 Takashi Taniguchi,5 Luojun Du,1,2 Dongxia Shi,1,2,6 Jianpeng Liu,3,7Jie Shen,1,2,6 Li Lu,1,2,6 Wei Yang ,1,2,6,* and Guangyu Zhang 1,2,6,†1Beijing National Laboratory for Condensed Matter Physics and Institute of Physics,Chinese Academy of Sciences, Beijing 100190, China2School of Physical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China3School of Physical Sciences and Technology, ShanghaiTech University, Shanghai 200031, China4Research Center for Functional Materials, National Institute for Materials Science,1-1 Namiki, Tsukuba 305-0044, Japan5International Center for Materials Nanoarchitectonics, National Institute for Materials Science,1-1 Namiki, Tsukuba 305-0044, Japan6Songshan Lake Materials Laboratory, Dongguan 523808, China7ShanghaiTech Laboratory for Topological Physics, ShanghaiTech University, Shanghai 200031, China(Received 13 March 2023; revised 5 June 2023; accepted 22 June 2023; published 7 August 2023)Twisted graphene multilayers are highly tunable flatband systems for developing new phases of matter.Thus far, while orbital ferromagnetism has been observed in valley-polarized phases, the long-range ordersof other correlated phases as well as the quantum phase transitions between different orders mostly remainunknown. Here, we report an observation of Coulomb-interaction-driven first-order quantum phasetransitions and ferromagnetism in twisted double bilayer graphene (TDBG). At zero magnetic field, thetransitions are revealed in a series of steplike abrupt resistance jumps with a prominent hysteresis loopwhen either the displacement field (D) or the carrier density (n) is tuned across a symmetry-breakingboundary near half filling, indicating a formation of ordered domains. It is worth noting that the goodtunability and switching of these states give rise to a memory performance with a large on/off ratio.Moreover, when both spin and valley play the roles at finite magnetic field, we observe abundant first-orderquantum phase transitions among normal metallic states from the charge-neutral point, orbital ferromag-netic states from quarter filling, and spin-polarized states from half filling. We interpret these first-orderphase transitions in the picture of phase separations and spin-domain percolations driven by multifieldtunable Coulomb interactions, in agreement with the Lifshitz transition and the Hartree-Fock calculations.The observed multifield tunable domain structure and its hysteresis resembles the characteristics ofmultiferroics, revealing intriguing magnetoelectric properties. Our result enriches the correlated phasediagram in TDBG for discovering novel exotic phases and quantum phase transitions, and it will benefitother twisted moiré systems as well.DOI: 10.1103/PhysRevX.13.031015 Subject Areas: Magnetism, Mesoscopics,Strongly Correlated MaterialsI. INTRODUCTIONPhase transitions are expected to occur at the instabilitywhen thermodynamic properties (density of states, suscep-tibility, or compressibility) or quasiparticle scattering ratesdiverge. In the strong-coupling regime, i.e., U=W ≥ ∼1where U is the Coulomb repulsion energy and W is thekinetic energy, a rich interplay of charge, spin, and orbitaldegrees of freedom contribute to versatile symmetry-breaking phase diagrams where different correlated orderscompete or coexist. Twisted graphene multilayers emergeas highly tunable flatband systems to realize these exoticphases, such as correlated insulators [1–9], superconduc-tivity [10–17], and magnetism [7,8,11,18–22]. In these*Corresponding author.wei.yang@iphy.ac.cn†Corresponding author.gyzhang@iphy.ac.cn‡These authors contributed equally to this work.Published by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution tothe author(s) and the published article’s title, journal citation,and DOI.PHYSICAL REVIEW X 13, 031015 (2023)2160-3308=23=13(3)=031015(9) 031015-1 Published by the American Physical Societyhttps://orcid.org/0000-0003-3701-8119https://orcid.org/0000-0002-3925-0352https://orcid.org/0000-0002-1242-4391https://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevX.13.031015&domain=pdf&date_stamp=2023-08-07https://doi.org/10.1103/PhysRevX.13.031015https://doi.org/10.1103/PhysRevX.13.031015https://doi.org/10.1103/PhysRevX.13.031015https://doi.org/10.1103/PhysRevX.13.031015https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/flatband systems, the presence of van Hove singularities(VHSs) [3,9,12,23] and cascade transitions [24–26]contribute significantly to instabilities that lead to thesymmetry-breaking isospin polarizations and the resulteddelicate ground states. Thus far, orbital ferromagnetismwith magnetic domains have been observed in valley-polarized phases [7,8,11,18–22] where Coulomb interac-tion and band topology play important roles. However, thelong-range order of other correlated phases mostly remainsunknown. In particular, the spin-related phenomena arerare, and direct evidence of the magnetic order of the spin-polarized insulator has not been observed yet.Here, we focus on the instability of a phase diagramwhen spin fluctuations are critical in the vicinity of a spin-polarized correlated insulator in twisted double bilayergraphene (TDBG). With unique spin-polarized correlatedinsulators [2–5] and VHS-like phase boundary [3,23],TDBG is an excellent platform to study the spin-relatedphases and phase transitions. Importantly, by sweeping thecarrier density n or displacement field D back and forth,we observe first-order phase transitions at zero magneticfield with a series of Barkhausen-like resistance jumpsand electrical hysteresis. We further demonstrate thedominant role of the spin order by performing both in-plane and out-of-plane magnetotransport measurements,and interpret these first-order phase transitions in thepicture of correlation-driven spin-domain percolations. Inaddition, we observe abundant first-order phase transitionsamong normal metallic states from the charge-neutral point,orbital ferromagnetic states from quarter filling, and spin-polarized states from half filling.II. RESULTSA. D-field and doping-tunable first-order phasetransitions with hysteresis at B= 0 TFigure 1(a) shows a typical phase diagram at T¼30mK,i.e., a color mapping of longitudinal resistance Rxx as afunction of n and D, for the TDBG device with a twistedangle of 1.35°, and Fig. 1(b) is the corresponding Rxy at aFIG. 1. Gate-voltage-driven first-order phase transitions at the halo boundary. (a),(b) Rxx and Rxy maps as a function of ν and D. TheHall resistance is measured at B⊥ ¼ 1 T. The black dots in (a) correspond to the points of Rxy ¼ 0 in (b). (c) Top panel: thermalactivation energy gaps versus D at ν ¼ 2. Bottom panel: calculated correlated gap as a function of interlayer potential difference Ud atν ¼ 2. The corresponding noninteracting Fermi surfaces are depicted in blue lines. (d) Schematics of different phases in (a) and (b). i, ii,iii correspond to phase transitions among the three phases of NM, SPCIs, and SPM. Black arrows correspond to directions of spin. Graydashed lines correspond to Fermi levels. (e) ΔR maps limited in the white dashed box of (a). (f),(g) D- and doping-driven hysteresisloops. The black (red) line corresponds to the forward (backward) sweep direction. (h) Transitions between low- and high-resistancestates by applying a sequential pulse voltage on the back gate.LE LIU et al. PHYS. REV. X 13, 031015 (2023)031015-2small magnetic field. The mappings reveal a correlatedinsulator at half filling in the moiré conduction band(ν ¼ 2) surrounded by a halolike structure, consistent withprevious works [2–5,23,27]. Such correlated states havebeen regarded as the signature of spontaneous symmetrybreaking [3,23] in which spin degeneracy is lifted inside thehalo. The halo structure being a phase boundary separatestwo kinds of phases: the normal metal (NM) outside the haloand the correlated states inside the halo. Accompanied bysymmetry breaking, the band structure will be reconstructedby electron-electron interaction [28] leading to the spin-polarized metal (SPM, at ν ≠ 2) or the spin-polarizedcorrelated insulators (SPCIs, at ν ¼ 2) as shown schemati-cally in Fig. 1(d). When the Fermi level is tuned across thehalo boundary at ν < 2, the Fermi pocket changes fromelectron-type to hole-type with vanishing Rxy at the boun-dary, depicted by the white contour in Fig. 1(b) andcorresponding black contour in Fig. 1(a). These transportbehaviors are related to the saddle-point-type VHS, at whichthe density of states (DOS) diverges. The topology of Fermisurface changes when the Fermi level crosses VHS, knownas the Lifshitz transition. As shown in Fig. 1(c), the transportgap at ν ¼ 2 grows with jDj from 0.2 to 0.5 V=nm slowlyand then drops rapidly to zero within a range of 0.05 V=nm.The different trends on two sides suggest an abrupt phasetransition between SPCIs and normal metal at large jDj.These observations are coincident with our Hartree-Fockcalculations which show a sharp decreasing correlated gap.These results strongly suggest a first-order phase transitionoccurring between normal metal and SPCIs [iii in Fig. 1(d)]at large jDj.Smoking-gun evidence of the first-order phase transitionat B ¼ 0 T is revealed in the transfer curves when the gatevoltages are swept back and forth in the white dashedbox of Fig. 1(a). In Figs. 1(f) and 1(g), the longitudinalresistance Rxx shows a hysteresis loop as D or ν sweepsacross the phase boundary in opposite directions. This loopis independent of the sweep speed of the gate voltage(Supplemental Material Note 2 [29]), and yet is sensitive toactivating current (Supplemental Material Note 3 [29]) andtemperature (Supplemental Material Note 6 [29]), demon-strating an intrinsic first-order phase transition betweennormal metal and SPCIs. In addition, multiple jumps, mostlikely Barkhausen jumps [30], in hysteresis loops indicatethe formation of orderly domains. Note that the abrupttransitions as well as the hysteresis at zero magnetic fieldsare well reproduced in a separate TDBG device D2 with atwist angle of 1.21° (Supplemental Material Note 9 [29]).Considering the spin polarization of the insulator, weconclude that there exist multiple spin-polarized ferromag-netic insulating domains in this regime. By taking thedifferenceΔR ¼ jRxxðþÞ − Rxxð−Þjwhereþ=− representsthe sweep direction, we map out the regime where first-order phase transitions occur, shown by the blue color inFig. 1(e), along the halo boundary from ν ¼ ∼2 toν ¼ ∼1.7. Notably, the transition regime is intimatelyrelated to the VHS in single-particle band structure calcu-lated by a continuum model [bottom panel of Fig. 1(c)].The Fermi surface at ν ¼ 2 changes from annular withtwo pockets for medium interlayer potential difference Ud(60–80 meV) to a simple surface with one single electron-type pocket for large Ud > 80 meV. This coincidencesuggests the divergent DOS across the Lifshitz transitionplays a crucial role in the first-order phase transition, withthe satisfied Stoner criteria U � DOSðEFÞ > 1 (U standsfor the correlation strength) inducing ferromagnetic insta-bilities of Fermi surfaces [31–34].As an initial demonstration, we show that such stableand gate-tunable first-order transitions could be useful formemory. This is achieved by applying a pulse voltage onthe back gate in the phase-transition regime. As shown inFig. 1(h), a pulse voltage as small as 20 mV could induce atransition between the low-resistance state of a few hundredohms (Ω) and high-resistance state of approximately 22 kΩwith highly tunable nature. The switching of such resis-tance states suggests that TDBG is a potential candidate forthe memory devices working at cryogenic temperature.B. Ferromagnetic first-order phase transitionsIn addition to the D-driven and the doping-driven ones,we also observe magnetic field B-driven hysteresis. Let usfirst focus on the perpendicular magnetic field (B⊥)dependence for a fixed ν ¼ 1.95 in Fig. 2(a), where theupper panel shows a color mapping of RxxðD;B⊥Þ atT ¼ 100 mK, and the lower panel is a correspondinghysteresis ΔRðD;B⊥Þ. Clearly, decent nonzero ΔR existsonly at a low magnetic field smaller than 1.5 T, and itfollows the boundary that separates the metallic stateswith low resistances and the insulating states with highresistances in the phase diagram. Figure 2(d) shows onerepresentative RðBÞ curve at D ¼ −0.527 V=nm, and itgives a pronounced resistance hysteresis loop when thesweeping direction of B⊥ is changed. The hysteresisincludes multiple steplike transitions, indicating the for-mation of multiple ferromagnetic insulating domains,and the whole loop is mirror symmetric with respectto B ¼ 0 T. For a comparison, we perform similar mea-surements at parallel magnetic fields (Bk) for a fixedν ¼ 1.95 in the same range of D, as shown in Figs. 2(b)and 2(e). The resulted phase diagram and the hysteresisloop at Bk are identical to those at B⊥ < 1.5 T, suggestingthat the magnetic first-order transitions and the hysteresisare due to spin degrees of freedom, instead of orbital.The origin of spin order is further supported by thevanishing hysteresis and the reverse direction of phaseboundary at B⊥ > 1.5 T, as well as the lower R or ΔR atB⊥ compared to those at Bk, where the orbital Zeemaneffect competes with the spin polarization [21,35].The details of the isospin competition are discussed laterin the observation of other first-order phase transitions.OBSERVATION OF FIRST-ORDER QUANTUM PHASE … PHYS. REV. X 13, 031015 (2023)031015-3C. Instabilities of the first-order transitionsThe first-order transitions and the hysteresis are sup-pressed with increasing T. For instance, the B-drivenhysteresis reduces as T is increased, and it totallydisappears at T ¼ 3.5 K in Figs. 2(c) and 2(f); similarly,the D-driven hysteresis at B ¼ 0 T disappears at aroundT ¼ ∼2.5 K (Supplemental Material Note 6 [29]). Thesuppression is also observed when the applied current (I)is increased [Supplemental Material Figs. 2(a)–2(c) [30] ].Aside from the suppression of hysteresis, it needs abigger critical field to realize the first-order transitionwhen the T or I is increased. For instance, at v ¼ 1.95 andD ¼ −0.527 V=nm, the critical fields (Bc) of the B-drivenfirst-order transitions are revealed in the RðB⊥; TÞ map-ping in Fig. 2(c), where B is swept from negative topositive, and T is swept from low to high temperature.Note that there are two critical fields, a smaller Bc1(negative) for the insulator-to-metal transition and abigger one Bc2 (positive) for the metal-to-insulator tran-sition indicated as the black and the red dots in Fig. 2(c),respectively. The critical field follows a power-lawrelation, i.e., Bc − B0 ∼ Tα, where B0 is the critical fieldat the zero-temperature limit, and α represents the power-law coefficient. The critical field corresponds to theminimum spin Zeeman energy 1=2gsμBBc for nucleationof ferromagnetic insulating domains, while the temper-ature represents the minimum thermal fluctuation kBT tobreak the ferromagnetic order. Here, gs ¼ 2 is the spin gfactor, μB is the Bohr magneton, and kB is the Boltzmannconstant. We find that B0 is around zero for Bc1 and anonzero value for Bc2, and α is between 2 and 3 for bothBc1 and Bc2. It is notable that almost the same α isobserved in another device D2 (Supplemental MaterialNote 9 [29]), suggesting a universal role in this criticalphenomenon. Last but not least, the zero-field resistanceRxxðB ¼ 0 TÞ in Fig. 2(c) shows an insulatinglike behav-ior at T < 1 K of approximately 6 kΩ, and then itsuddenly drops below 300 Ω at T ¼ 1 K, showing ametallic behavior at T > 1 K; alternatively, such aninsulator-to-metal transition could also be achieved atT < 1 K by applying a large current of 50 nA, as shown inSupplemental Material Note 2 [29].FIG. 2. Spin-polarized ferromagnetic insulators. (a) Rxx andΔR as a function ofD and B⊥ at ν ¼ 1.95. (b) Rxx andΔR as a function ofD and Bk at ν ¼ 1.95. (c) Rxx as a function of B⊥ and T at ν ¼ 1.95 and D ¼ −0.527 V=nm. Critical magnetic fields Bc are marked byblack (B⊥ < 0) and red (B⊥ > 0) points. The gray line corresponds to the power-law fitting. Inset: Rxx versus T at B⊥ ¼ 0. (d),(e) B⊥-and Bk-driven hysteresis loops at D ¼ −0.527 V=nm and T ¼ 100 mK marked by the green star in (a). (f) B-driven hysteresis loops atdifferent temperatures. The solid (dashed) line corresponds to the forward (backward) sweep direction.LE LIU et al. PHYS. REV. X 13, 031015 (2023)031015-4D. Phase separation and percolationsof spin-ordered domainsThe first-order transitions and the hysteresis driven byeither displacement, carrier density, or magnetic field areobserved at metal-insulator transition [36] and could beinterpreted within a picture of phase separation [37,38].The free energy of first-order transitions could havemultiple local minimum points in the order parameterspace, with a stable state at zero and a metastable stateat finite value (Supplemental Material Note 7 [29]). Theinstability of the phase diagram is driven by an interplay ofU=W (which can be effectively tuned by D and n) andmagnetic field, which favors the domain nucleation andgrowth, against thermal energy as well as the couplingsbetween domains and the surrounding metallic electron sea,which melts the domains.The D-field-dependent first-order phase transitions arecaptured qualitatively in the abovementioned model. First,away from the phase boundary [Fig. 3(a)], it is a metallicstate with R < 3 kΩ and no hysteresis. This agrees with areduced U=W away from the halo boundary, wherenegligible ferromagnetic insulating domains are sur-rounded by metallic electron sea, and thus the electricalconduction is dominated by metallic electrons. Then, as Dapproaches the phase boundary [Fig. 3(b)], it becomes ametastable insulating state with Rxx ∼ 10 kΩ at B ¼ 0 T,and multiple resistance jumps emerge in hysteresis as B ischanged. This is the case of large U=W, which might resultin multiple domains densely distributed in real space atB ¼ 0 T, and the magnetic field would boost the growth ofdomains [Fig. 3(c)]. At the critical fields, some percolationpaths suddenly disappear, which would contribute to theresistance jumps. Lastly, when D is tuned inside thecorrelated insulating phase [Fig. 3(d)], i.e., Rxx ∼ 75 kΩ,the hysteresis effect still exists, but the resistance smoothlychanges with magnetic field. The result suggests theabsence of globally preferred spin orientation for differentdomains at zero magnetic field. The external magnetic fieldprovides an anisotropic energy E ¼ 1=2gsμBB by theZeeman effect, and it tends to align the spin orientationfrom different domains, as shown in the bottom panel ofFig. 3(e). The presence of domains with different spinorientation might be due to the inhomogeneity of twistedangle and unexpected strain despite the most delicatesample fabrication [39–43]. A slight inhomogeneity inmoiré systems alters the ground state on a microscopicscale, inducing a phase separation near the phase boundary.E. Competing orders and abundantfirst-order transitionsNext, we discuss the competing orders and the resultedabundant first-order transitions in the phase diagramwhen the valley polarization starts to set in. Figures 4(a)and 4(b) are Rxxðn; BÞ and Rxyðn; BÞ color mappings atD ¼ −0.51 V=nm, respectively. Here, Rxx and Rxy aresymmetrically and asymmetrically processed, respectively,in order to eliminate the crosstalk (see SupplementalMaterial Note 1 [29]). There are four different regimesin the phase diagram, and evident phase boundaries arereflected in the Hall resistance measurements. Figure 4(c)shows three representative RxyðBÞ curves at differentfillings, from which Hall carrier density (νH) could beobtained by linear fitting with low field data. At ν ¼ 1.48(top panel), the linear fit by the black dashed line yieldsνH ¼ 1.43. The observation of νH ∼ ν indicates a fourfolddegenerate Fermi surface adiabatically evolving fromν ¼ 0, a single-particle picture described by the continuummodel [44,45]. At ν ¼ 2.3, the Hall carrier density followsνH ∼ ν − 2, agreeing with a twofold degenerate Fermisurface developing from ν ¼ 2. The phase boundarybetween SPCIs and SPM stays unchanged in theperpendicular magnetic fields, indicating that both statesFIG. 3. Phase separation and percolation near the phase boundary. (a),(b),(d) B⊥-driven hysteresis loops at different D andT ¼ 100 mK. (c),(e) Schematics of phase separation and percolation. Red regions correspond to spin-polarized insulating domains,where the black arrow is the orientation of spin polarization. Green regions correspond to normal metal. The red arrow is the direction ofexternal magnetic field.OBSERVATION OF FIRST-ORDER QUANTUM PHASE … PHYS. REV. X 13, 031015 (2023)031015-5are spin polarized. At ν ¼ 1.8 close to the SPCIs (middlepanel), the Hall response can be divided in two parts, theordinary Hall effect νH ∼ ν in low magnetic field (<1.8 T),and the additional anomalous Hall effect in high magneticfield (>1.8 T). The observed Hall effect follows Rxy ¼B=ðneÞ þ RAxy, where the first term corresponds to theordinary Hall effect, and the second term RAxy ∼M (mag-netization) represents the anomalous Hall effect [46].Generally, the intrinsic anomalous Hall effect implies alarge Berry curvature of the energy band. In TDBG, twovalley-polarized sub-bands could carry opposite Chernnumbers that are associated with the valley-contrastingorbital magnetism [21,44,45,47], and they are separatedfrom each other with the increase of B⊥ due to the orbitalZeeman effect. Consequently, a valley-polarized (VP) statewith orbital ferromagnetism and a finite Berry curvaturebecomes a ground state.The competition of spin and valley is also revealed in thephase boundary between the SPCIs and VP states. At lowfields B⊥ < 1.35 T, the phase boundary between SPCIsand NM extends to lower doping levels with an increasingB⊥ due to the dominating spin Zeeman effect. At B⊥ >1.35 T, VP states emerge, and it gradually takes up mostof the phase space with increasing B⊥. The observationsuggests that VP states have lower energy than otherground states at high magnetic fields. In addition, whilethe phase boundary between NM and VPM keeps extend-ing nonlinearly to a lower carrier density with increasing B,that between VPM and SPCIs shifts to a higher densityalmost linearly. The different tendency indicates a strongercompetition of spin and valley polarization near ν ¼ 2,where the spin-polarized states resist the invasion of valleypolarization. In addition, the four different phases contrib-ute a series of phase transitions. The phase transitionbetween NM and VPM [Fig. 4(d)] is a ferromagneticfirst-order phase transition contributing abrupt resistancejumps and hysteresis. Those between VPM and SPCIs[Fig. 4(e)], as well as NM and SPCIs [Fig. 4(f)], are also afirst-order transition, evident from the hysteresis. Note thatall first-order transitions occur near the halo boundary,accompanied by isospin competitions. The halo boundary,also being a phase boundary with diverging DOS, givesFIG. 4. Competing phase diagram and abundant first-order transitions. (a),(b) Rxx and Rxy maps as a function of v and B⊥ atD ¼ −0.51 V=nm. In (b), we label different phases as NM, VPM, SPCIs, and SPM. (c) Rxy versus B⊥ at different filling factors. Hallcarrier densities are extracted by linear fitting. Right figures are schematics of DOS in different phases. Top panel: schematic of DOS offourfold degenerate bands. Middle panel: schematic of DOS of valley-polarized bands. Bottom panel: schematic of DOS of spin-polarized bands. (d)–(g) Doping-driven hysteresis loops of symmetrized Rxx between different phases. All linecuts correspond to yellowdashed lines in (a).LE LIU et al. PHYS. REV. X 13, 031015 (2023)031015-6birth to the large isospin fluctuation that might be respon-sible for the first-order transition; by contrast, the transitionbetween SPCIs and SPM is continuous and nonhysteretic[Fig. 4(g)] due to the same symmetry breaking induced bythe spin polarization.F. First-order phase transitions at quarter fillingThe orbital Zeeman effect in high perpendicular mag-netic field will induce fully isospin polarization near ν ¼ 1and ν ¼ 3. As shown in Fig. 5(a), two new symmetry-breaking Fermi surfaces emanate from quarter filling atB⊥ ¼ 2 T. Here, the value of ν − νH measures thedeviation of the band filling before and after the symmetrybreaking, and it indicates the degree of isospin polarization[48]. The state spreading from ν ¼ 2 to ν ¼ 1 along thehalo boundary corresponds to the VPM mentioned above.The linecut in Fig. 5(b) shows ν − νH ∼ 1 (blue region),suggesting the VPM is actually the incipience of the isospinfully polarized state at ν ¼ 1. The series of Landau levelsemanating from ν ¼ 1 in VPM also suggest the emergingsymmetry-breaking Fermi surface (Supplemental MaterialNote 10 [29]). The other state takes up the area aroundν ¼ 3 and experiences a correlated gaplike transition atν ¼ 3. The linecut also shows ν − νH ∼ 3 [violet region inFig. 5(b)]. These states at quarter filling are most likely thespin- and valley-polarized state (SVP) according to thetheoretical calculations [35,49]. Furthermore, as shown inFigs. 5(c) and 5(d), the phase transitions between VPM andSVP (quarter filling) and SPM (half filling) with gate-voltage-driven hysteresis loops turn out to be the first-orderphase transitions. The perpendicular magnetic-field-drivenhysteresis loops for these two states also suggest the keyrole of orbital magnetization at quarter filling [11,22][Figs. 5(e) and 5(f)].III. CONCLUSIONS AND OUTLOOKIn TDBG, we observe correlated first-order phasetransitions between SPCIs and normal metal at zeromagnetic field, driven by carrier density and displacementfield independently. The observations agree well with thescenario of the Lifshitz transition as well as the rapidlydecreasing energy gaps at the phase boundary fromHartree-Fock calculations. These observations are impor-tant in that they unveil the long-standing mystery aboutthe nature of the halo boundary and SPCIs [3–5,23] byproviding smoking-gun evidence for the spin-polarizedferromagnetism. We also observe identical first-ordermetal-insulator transitions when either in-plane or out-of-plane magnetic field is applied, further demonstratingthe existence of spin-polarized ferromagnetic domains.Moreover, we observe abundant competing phases andaccompanied first-order phase transitions between differentground states where valley degrees of freedom play a moreimportant role.Our observations suggest an instability with strongisospin fluctuations near the halo boundary at high dis-placement field, where the combined van Hove singularityand reduced bandwidth leads to the strong Coulombinteraction effects. While the first-order transitions andthe hysteresis could be captured within a picture of phaseseparation and percolations, more delicate theory andexperiments are needed for a better understanding of thissystem. For instance, the first-order transitions and thehysteresis are highly tunable by electrical field and mag-netic field, resembling multiferroics. The complicatedflatbands in twisted multilayer systems, especially thereconstructed bands after Lifshitz transitions, might hostboth the electron and hole pockets, and the spatial sepa-ration of the electrons and holes at nonzero D could lead toa possible formation of electrical dipoles. In some circum-stances, it might eventually form a state where spin, charge,and the layer are locked, giving birth to multiferroics intwisted multilayers. The presence of both continuous phasetransition and first-order phase transition near the haloFIG. 5. First-order phase transitions and orbital magnetizationat quarter fillings. (a) A color mapping of ν-νH as a function of vand D at B⊥ ¼ 2 T. Here, SVP corresponds to spin- and valley-polarized states. (b) A linecut at D ¼ −0.44 V=nm from (a). Theblue region corresponds to symmetry-breaking states near ν ¼ 1,and the violet region corresponds to symmetry-breaking statesnear ν ¼ 3. (c),(d) Typical doping-driven hysteresis loop of Rxxbetween VPM and SPM (c) and those between SPM and SVP (d).The data in (c) and (d) correspond to the violet solid lines in (a).(e),(f) Magnetic-field-driven hysteresis loops of Rxx at ν ¼ 1.52and 3.05, respectively.OBSERVATION OF FIRST-ORDER QUANTUM PHASE … PHYS. REV. X 13, 031015 (2023)031015-7boundary also deserve more investigation in TDBG as wellas other twisted multilayers.ACKNOWLEDGMENTSWe thank Kun Jiang, Zhida Song, Kam Tuen Law,Fengcheng Wu, Yu Ye, Guoqiang Yu, Erjia Guo, andShiliang Li for useful discussions. We acknowledge supportfrom the National Key Research and Development Program(Grants No. 2020YFA0309600 and No. 2021YFA1202900),National Natural Science Foundation of China (GrantsNo. 61888102, No. 11834017, and No. 12074413), theStrategic Priority Research Program of CAS (GrantsNo. XDB30000000 and No. XDB33000000), and theKey-Area Research and Development Program ofGuangdong Province (Grant No. 2020B0101340001). J. S.acknowledges support from the Synergetic ExtremeCondition User Facility at IOP, CAS. K.W. and T. T.acknowledge support from the Elemental StrategyInitiative conducted by the MEXT, Japan (GrantNo. JPMXP0112101001), JSPS KAKENHI (GrantsNo. 19H05790, No. 20H00354, and No. 21H05233), andA3 Foresight by JSPS.[1] Y. Cao, V. Fatemi, A. Demir, S. Fang, S. L. Tomarken, J. Y.Luo, J. D. Sanchez-Yamagishi, K. Watanabe, T. Taniguchi,E. Kaxiras et al., Correlated Insulator Behaviour at Half-Filling in Magic-Angle Graphene Superlattices, Nature(London) 556, 80 (2018).[2] G.W. Burg, J. Zhu, T. Taniguchi, K. Watanabe, A. H.MacDonald, and E. Tutuc, Correlated Insulating Statesin Twisted Double Bilayer Graphene, Phys. Rev. Lett. 123,197702 (2019).[3] X. Liu, Z. Hao, E. Khalaf, J. Y. Lee, Y. Ronen, H. Yoo, D.Haei Najafabadi, K. Watanabe, T. Taniguchi, A. Vishwanathet al., Tunable Spin-Polarized Correlated States in TwistedDouble Bilayer Graphene, Nature (London) 583, 221 (2020).[4] Y. Cao, D. Rodan-Legrain, O. Rubies-Bigorda, J. M. Park,K. Watanabe, T. Taniguchi, and P. Jarillo-Herrero, TunableCorrelated States and Spin-Polarized Phases in TwistedBilayer–Bilayer Graphene, Nature (London) 583, 215(2020).[5] C. Shen, Y. Chu, Q. Wu, N. Li, S. Wang, Y. Zhao, J. Tang, J.Liu, J. Tian, K. Watanabe et al., Correlated States in TwistedDouble Bilayer Graphene, Nat. Phys. 16, 520 (2020).[6] G. Chen, L. Jiang, S. Wu, B. Lyu, H. Li, B. L. Chittari, K.Watanabe, T. Taniguchi, Z. Shi, J. Jung et al., Evidence of aGate-Tunable Mott Insulator in a Trilayer Graphene MoiréSuperlattice, Nat. Phys. 15, 237 (2019).[7] S. Chen, M. He, Y.-H. Zhang, V. Hsieh, Z. Fei, K.Watanabe, T. Taniguchi, D. H. Cobden, X. Xu, C. R.Dean et al., Electrically Tunable Correlated and Topologi-cal States in Twisted Monolayer-Bilayer Graphene, Nat.Phys. 17, 374 (2021).[8] H. Polshyn, J. Zhu, M. A. Kumar, Y. Zhang, F. Yang, C. L.Tschirhart, M. Serlin, K. Watanabe, T. Taniguchi, A. H.MacDonald et al., Electrical Switching of Magnetic Orderin an Orbital Chern Insulator, Nature (London) 588, 66(2020).[9] S. Xu, M.M. Al Ezzi, N. Balakrishnan, A. Garcia-Ruiz, B.Tsim, C. Mullan, J. Barrier, N. Xin, B. A. Piot, T. Taniguchiet al., Tunable van Hove Singularities and Correlated Statesin Twisted Monolayer–Bilayer Graphene, Nat. Phys. 17,619 (2021).[10] Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E.Kaxiras, and P. Jarillo-Herrero, Unconventional Supercon-ductivity in Magic-Angle Graphene Superlattices, Nature(London) 556, 43 (2018).[11] X. Lu, P. Stepanov, W. Yang, M. Xie, M. A. Aamir, I. Das,C. Urgell, K. Watanabe, T. Taniguchi, G. Zhang et al.,Superconductors, Orbital Magnets and Correlated States inMagic-Angle Bilayer Graphene, Nature (London) 574, 653(2019).[12] J. M. Park, Y. Cao, K. Watanabe, T. Taniguchi, and P.Jarillo-Herrero, Tunable Strongly Coupled Superconductiv-ity in Magic-Angle Twisted Trilayer Graphene, Nature(London) 590, 249 (2021).[13] G.W. Burg, E. Khalaf, Y. Wang, K. Watanabe, T. Taniguchi,and E. Tutuc, Emergence of Correlations in AlternatingTwist Quadrilayer Graphene, Nat. Mater. 21, 884 (2022).[14] J. M. Park, Y. Cao, L.-Q. Xia, S. Sun, K. Watanabe, T.Taniguchi, and P. Jarillo-Herrero, Robust Superconductivityin Magic-Angle Multilayer Graphene Family, Nat. Mater.21, 877 (2022).[15] Y. Zhang, R. Polski, C. Lewandowski, A. Thomson, Y.Peng, Y. Choi, H. Kim, K. Watanabe, T. Taniguchi, J. Aliceaet al., Promotion of Superconductivity in Magic-AngleGraphene Multilayers, Science 377, 1538 (2022).[16] M. Oh, K. P. Nuckolls, D. Wong, R. L. Lee, X. Liu, K.Watanabe, T. Taniguchi, and A. Yazdani, Evidence forUnconventional Superconductivity in Twisted BilayerGraphene, Nature (London) 600, 240 (2021).[17] H. Kim, Y. Choi, C. Lewandowski, A. Thomson, Y. Zhang,R. Polski, K. Watanabe, T. Taniguchi, J. Alicea, andS. Nadj-Perge, Evidence for Unconventional Superconduc-tivity in Twisted Trilayer Graphene, Nature (London) 606,494 (2022).[18] A. L. Sharpe, E. J. Fox, A.W. Barnard, J. Finney, K.Watanabe, T. Taniguchi, M. A. Kastner, and D. Goldhaber-Gordon, Emergent Ferromagnetism near Three-QuartersFilling in Twisted Bilayer Graphene, Science 365, 605(2019).[19] M. Serlin, C. L. Tschirhart, H. Polshyn, Y. Zhang, J. Zhu,K. Watanabe, T. Taniguchi, L. Balents, and A. F. Young,Intrinsic Quantized Anomalous Hall Effect in a MoiréHeterostructure, Science 367, 900 (2020).[20] G. Chen, A. L. Sharpe, E. J. Fox, Y.-H. Zhang, S. Wang, L.Jiang, B. Lyu, H. Li, K. Watanabe, T. Taniguchi et al.,Tunable Correlated Chern Insulator and Ferromagnetismin a Moiré Superlattice, Nature (London) 579, 56 (2020).[21] L. Liu, S. Zhang, Y. Chu, C. Shen, Y. Huang, Y. Yuan, J.Tian, J. Tang, Y. Ji, R. Yang et al., Isospin Competitions andValley Polarized Correlated Insulators in Twisted DoubleBilayer Graphene, Nat. Commun. 13, 3292 (2022).[22] M. Kuiri, C. Coleman, Z. Gao, A. Vishnuradhan, K.Watanabe, T. Taniguchi, J. Zhu, A. H. MacDonald, and J.Folk, Spontaneous Time-Reversal Symmetry Breaking inLE LIU et al. PHYS. REV. X 13, 031015 (2023)031015-8https://doi.org/10.1038/nature26154https://doi.org/10.1038/nature26154https://doi.org/10.1103/PhysRevLett.123.197702https://doi.org/10.1103/PhysRevLett.123.197702https://doi.org/10.1038/s41586-020-2458-7https://doi.org/10.1038/s41586-020-2260-6https://doi.org/10.1038/s41586-020-2260-6https://doi.org/10.1038/s41567-020-0825-9https://doi.org/10.1038/s41567-018-0387-2https://doi.org/10.1038/s41567-020-01062-6https://doi.org/10.1038/s41567-020-01062-6https://doi.org/10.1038/s41586-020-2963-8https://doi.org/10.1038/s41586-020-2963-8https://doi.org/10.1038/s41567-021-01172-9https://doi.org/10.1038/s41567-021-01172-9https://doi.org/10.1038/nature26160https://doi.org/10.1038/nature26160https://doi.org/10.1038/s41586-019-1695-0https://doi.org/10.1038/s41586-019-1695-0https://doi.org/10.1038/s41586-021-03192-0https://doi.org/10.1038/s41586-021-03192-0https://doi.org/10.1038/s41563-022-01286-2https://doi.org/10.1038/s41563-022-01287-1https://doi.org/10.1038/s41563-022-01287-1https://doi.org/10.1126/science.abn8585https://doi.org/10.1038/s41586-021-04121-xhttps://doi.org/10.1038/s41586-022-04715-zhttps://doi.org/10.1038/s41586-022-04715-zhttps://doi.org/10.1126/science.aaw3780https://doi.org/10.1126/science.aaw3780https://doi.org/10.1126/science.aay5533https://doi.org/10.1038/s41586-020-2049-7https://doi.org/10.1038/s41467-022-30998-xTwisted Double Bilayer Graphene, Nat. Commun. 13, 6468(2022).[23] M. He, Y. Li, J. Cai, Y. Liu, K. Watanabe, T. Taniguchi, X.Xu, and M. Yankowitz, Symmetry Breaking in TwistedDouble Bilayer Graphene, Nat. Phys. 17, 26 (2021).[24] D. Wong, K. P. Nuckolls, M. Oh, B. Lian, Y. Xie, S. Jeon, K.Watanabe, T. Taniguchi, B. A. Bernevig, and A. Yazdani,Cascade of Electronic Transitions in Magic-Angle TwistedBilayer Graphene, Nature (London) 582, 198 (2020).[25] U. Zondiner, A. Rozen, D. Rodan-Legrain, Y. Cao, R.Queiroz, T. Taniguchi, K. Watanabe, Y. Oreg, F. von Oppen,A. Stern et al., Cascade of Phase Transitions and DiracRevivals in Magic-Angle Graphene, Nature (London) 582,203 (2020).[26] J. M. Park, Y. Cao, K. Watanabe, T. Taniguchi, and P.Jarillo-Herrero, Flavour Hund’s Coupling, Chern Gaps andCharge Diffusivity in Moiré Graphene, Nature (London)592, 43 (2021).[27] Y. Chu, L. Liu, C. Shen, J. Tian, J. Tang, Y. Zhao, J. Liu, Y.Yuan, Y. Ji, R. Yang et al., Temperature-Linear Resistivityin Twisted Double Bilayer Graphene, Phys. Rev. B 106,035107 (2022).[28] M. Xie and A. H. MacDonald, Weak-Field Hall Resistivityand Spin/Valley Flavor Symmetry Breaking in Magic-AngleTwisted Bilayer Graphene, Phys. Rev. Lett. 127, 196401(2021).[29] See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevX.13.031015 for theoreticalcalculations and additional experimental results.[30] H Barkhausen, Zwei Mit Hilfe Der Neuen VerstrkerEntdeckte Erscheinungen, Phys. Z. 20, 401 (1919).[31] F. Wu and S. Das, Sarma, Ferromagnetism and Super-conductivity in Twisted Double Bilayer Graphene, Phys.Rev. B 101, 155149 (2020).[32] Y.-T. Hsu, F. Wu, and S. Das Sarma, Topological Super-conductivity, Ferromagnetism, and Valley-PolarizedPhases in Moire Systems: Renormalization Group Analysisfor Twisted Double Bilayer Graphene, Phys. Rev. B 102,085103 (2020).[33] H. Zhou, T. Xie, A. Ghazaryan, T. Holder, J. R. Ehrets,E. M. Spanton, T. Taniguchi, K. Watanabe, E. Berg, M.Serbyn et al., Half- and Quarter-Metals in RhombohedralTrilayer Graphene, Nature (London) 598, 429 (2021).[34] H. Zhou, L. Holleis, Y. Saito, L. Cohen, W. Huynh, C. L.Patterson, F. Yang, T. Taniguchi, K. Watanabe, and A. F.Young, Isospin Magnetism and Spin-Polarized Supercon-ductivity in Bernal Bilayer Graphene, Science 375, 774(2022).[35] S. Zhang, X. Dai, and J. Liu, Spin-Polarized NematicOrder, Quantum Valley Hall States, and Field-TunableTopological Transitions in Twisted Multilayer GrapheneSystems, Phys. Rev. Lett. 128, 026403 (2022).[36] M. Imada, A. Fujimori, and Y. Tokura, Metal-InsulatorTransitions, Rev. Mod. Phys. 70, 1039 (1998).[37] M. Fäth, S. Freisem, A. A. Menovsky, Y. Tomioka, J. Aarts,and J. A. Mydosh, Spatially Inhomogeneous Metal-Insulator Transition in Doped Manganites, Science 285,1540 (1999).[38] M. Uehara, S. Mori, C. H. Chen, and S.-W. Cheong,Percolative Phase Separation Underlies Colossal Magneto-resistance in Mixed-Valent Manganites, Nature (London)399, 560 (1999).[39] A. Uri, S. Grover, Y. Cao, J. A. Crosse, K. Bagani, D.Rodan-Legrain, Y. Myasoedov, K. Watanabe, T. Taniguchi,P. Moon et al., Mapping the Twist-Angle Disorder andLandau Levels in Magic-Angle Graphene, Nature (London)581, 47 (2020).[40] T. Benschop, T. A. de Jong, P. Stepanov, X. Lu, V. Stalman,S. J. van der Molen, D. K. Efetov, and M. P. Allan, Meas-uring Local Moiré Lattice Heterogeneity of Twisted BilayerGraphene, Phys. Rev. Res. 3, 013153 (2021).[41] C. L. Tschirhart, M. Serlin, H. Polshyn, A. Shragai, Z. Xia,J. Zhu, Y. Zhang, K. Watanabe, T. Taniguchi, M. E. Huberet al., Imaging Orbital Ferromagnetism in a Moiré ChernInsulator, Science 372, 1323 (2021).[42] S. Grover, M. Bocarsly, A. Uri, P. Stepanov, G. Di Battista,I. Roy, J. Xiao, A. Y. Meltzer, Y. Myasoedov, K. Pareeket al., Chern Mosaic and Berry-Curvature Magnetism inMagic-Angle Graphene, Nat. Phys. 18, 885 (2022).[43] C. L. Tschirhart, E. Redekop, L. Li, T. Li, S. Jiang, T. Arp,O. Sheekey, T. Taniguchi, K. Watanabe, M. E. Huber et al.,Intrinsic Spin Hall Torque in a Moiré Chern Magnet, Nat.Phys. 19, 807 (2023).[44] M. Koshino, Band Structure and Topological Properties ofTwisted Double Bilayer Graphene, Phys. Rev. B 99, 235406(2019).[45] N. R. Chebrolu, B. L. Chittari, and J. Jung, Flat Bands inTwisted Double Bilayer Graphene, Phys. Rev. B 99, 235417(2019).[46] N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, andN. P. Ong, Anomalous Hall Effect, Rev. Mod. Phys. 82,1539 (2010).[47] J. Liu, Z. Ma, J. Gao, and X. Dai, Quantum Valley HallEffect, Orbital Magnetism, and Anomalous Hall Effect inTwisted Multilayer Graphene Systems, Phys. Rev. X 9,031021 (2019).[48] Y. Saito, F. Yang, J. Ge, X. Liu, T. Taniguchi, K. Watanabe,J. I. A. Li, E. Berg, and A. F. Young, Isospin PomeranchukEffect in Twisted Bilayer Graphene, Nature (London) 592,220 (2021).[49] J. Y. Lee, E. Khalaf, S. Liu, X. Liu, Z. Hao, P. Kim, and A.Vishwanath, Theory of Correlated Insulating Behaviourand Spin-Triplet Superconductivity in Twisted DoubleBilayer Graphene, Nat. Commun. 10, 5333 (2019).OBSERVATION OF FIRST-ORDER QUANTUM PHASE … PHYS. REV. X 13, 031015 (2023)031015-9https://doi.org/10.1038/s41467-022-34192-xhttps://doi.org/10.1038/s41467-022-34192-xhttps://doi.org/10.1038/s41567-020-1030-6https://doi.org/10.1038/s41586-020-2339-0https://doi.org/10.1038/s41586-020-2373-yhttps://doi.org/10.1038/s41586-020-2373-yhttps://doi.org/10.1038/s41586-021-03366-whttps://doi.org/10.1038/s41586-021-03366-whttps://doi.org/10.1103/PhysRevB.106.035107https://doi.org/10.1103/PhysRevB.106.035107https://doi.org/10.1103/PhysRevLett.127.196401https://doi.org/10.1103/PhysRevLett.127.196401http://link.aps.org/supplemental/10.1103/PhysRevX.13.031015http://link.aps.org/supplemental/10.1103/PhysRevX.13.031015http://link.aps.org/supplemental/10.1103/PhysRevX.13.031015http://link.aps.org/supplemental/10.1103/PhysRevX.13.031015http://link.aps.org/supplemental/10.1103/PhysRevX.13.031015http://link.aps.org/supplemental/10.1103/PhysRevX.13.031015http://link.aps.org/supplemental/10.1103/PhysRevX.13.031015https://doi.org/10.1103/PhysRevB.101.155149https://doi.org/10.1103/PhysRevB.101.155149https://doi.org/10.1103/PhysRevB.102.085103https://doi.org/10.1103/PhysRevB.102.085103https://doi.org/10.1038/s41586-021-03938-whttps://doi.org/10.1126/science.abm8386https://doi.org/10.1126/science.abm8386https://doi.org/10.1103/PhysRevLett.128.026403https://doi.org/10.1103/RevModPhys.70.1039https://doi.org/10.1126/science.285.5433.1540https://doi.org/10.1126/science.285.5433.1540https://doi.org/10.1038/21142https://doi.org/10.1038/21142https://doi.org/10.1038/s41586-020-2255-3https://doi.org/10.1038/s41586-020-2255-3https://doi.org/10.1103/PhysRevResearch.3.013153https://doi.org/10.1126/science.abd3190https://doi.org/10.1038/s41567-022-01635-7https://doi.org/10.1038/s41567-023-01979-8https://doi.org/10.1038/s41567-023-01979-8https://doi.org/10.1103/PhysRevB.99.235406https://doi.org/10.1103/PhysRevB.99.235406https://doi.org/10.1103/PhysRevB.99.235417https://doi.org/10.1103/PhysRevB.99.235417https://doi.org/10.1103/RevModPhys.82.1539https://doi.org/10.1103/RevModPhys.82.1539https://doi.org/10.1103/PhysRevX.9.031021https://doi.org/10.1103/PhysRevX.9.031021https://doi.org/10.1038/s41586-021-03409-2https://doi.org/10.1038/s41586-021-03409-2https://doi.org/10.1038/s41467-019-12981-1