# Fileset [PhysRevX.12.041015.pdf](https://mdr.nims.go.jp/filesets/f2a109c9-b9ea-4b0e-a827-cfa1aea7563a/download) ## Creator Mingjie Zhang, Xuan Zhao, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Zheng Zhu, Fengcheng Wu, Yongqing Li, Yang Xu ## Rights [Creative Commons BY Attribution 4.0 International](https://creativecommons.org/licenses/by/4.0/) ## Other metadata [Tuning Quantum Phase Transitions at Half Filling in

3 L − {MoTe}_{2} / {WSe}_{2}

Moiré Superlattices](https://mdr.nims.go.jp/datasets/6f2c461d-e777-40e9-b2dd-c906ec62f04b) ## Fulltext Tuning Quantum Phase Transitions at Half Filling in 3L-MoTe2/WSe2 Moiré SuperlatticesTuning Quantum Phase Transitions at Half Filling in 3L-MoTe2=WSe2 Moiré SuperlatticesMingjie Zhang ,1,2 Xuan Zhao,1,2 Kenji Watanabe ,3 Takashi Taniguchi,4 Zheng Zhu ,5Fengcheng Wu ,6,7 Yongqing Li,1,2 and Yang Xu 1,2,*1Beijing National Laboratory for Condensed Matter Physics, Institute of Physics,Chinese Academy of Sciences, Beijing 100190, China2School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China3Research Center for Functional Materials, National Institute for Materials Science,1-1 Namiki, Tsukuba 305-0044, Japan4International Center for Materials Nanoarchitectonics, National Institute for Materials Science,1-1 Namiki, Tsukuba 305-0044, Japan5Kavli Institute for Theoretical Sciences and CAS Center for Excellence in Topological QuantumComputation, University of Chinese Academy of Sciences, Beijing 100190, China6School of Physics and Technology, Wuhan University, Wuhan 430072, China7Wuhan Institute of Quantum Technology, Wuhan 430206, China(Received 27 February 2022; revised 10 August 2022; accepted 14 September 2022; published 9 November 2022)Many sought-after exotic states of matter are known to emerge close to quantum phase transitions, suchas quantum spin liquids and unconventional superconductivity. It is, thus, desirable to experimentallyexplore systems that can be continuously tuned across these transitions. Here, we demonstrate suchtunability and the electronic correlation effects in triangular moiré superlattices formed between trilayerMoTe2 and monolayer WSe2 (3L-MoTe2=WSe2). Through transport measurements, we observe anelectronic analog of the Pomeranchuk effect at half filling of the first moiré subband, where increasingtemperature paradoxically enhances charge localization. At low temperatures the system exhibits thecharacteristic of a Fermi liquid with strongly renormalized effective mass, suggesting a correlated metalstate. The state is highly susceptible to out-of-plane electric and magnetic fields, which induce a Lifshitztransition and a metal-insulator transition (MIT), respectively. It enables identification of a tricritical pointin the quantum phase diagram at the base temperature. We explain the Lifshitz transition in terms ofinterlayer charge transfer under the vertical electric field, which leads to the emergence of a new Fermisurface and immediate suppression of the Pomeranchuk effect. The existence of quantum criticality in themagnetic-field-induced MIT is supported by scaling behaviors of the resistance. Our work shows the3L-MoTe2=WSe2 lies in the vicinity to the MIT point of the triangular lattice Hubbard model, rendering it aunique system to manifest the rich correlation effects at an intermediate interaction strength.DOI: 10.1103/PhysRevX.12.041015 Subject Areas: Condensed Matter PhysicsStrongly Correlated MaterialsI. INTRODUCTIONThe discovery of correlated phenomena in twistedbilayer graphene opens a new avenue for studying electroncorrelations in two-dimensional (2D) moiré superlattices,which host novel band structure reconstruction and exoticquantum phases [1–4]. Lately, semiconductor moiré sys-tems based on transition-metal dichalcogenide (TMDC)heterostructures have been shown to offer alternativeplatforms with appealing opportunities [5–15]. Moiréexcitons, trions, and polaritons [16–22], Mott insulators[23–25], generalized Wigner crystals at fractional fillings[26–28] (including those that spontaneously break rota-tional symmetry [29,30]), and topological phases [31] havebeen experimentally observed. It is generally accepted thatthe low-energy Hamiltonian of the TMDC moiré systemcan be effectively described by the triangular latticeHubbard model [5], which is one of the central paradigmsin describing correlated systems with geometrical frus-tration [32–37]. When doped with one charge per site (half-filled band), the large spin entropy owing to the frustrationand competing orders can give rise to the Pomeranchukeffect or an exotic nonmagnetic insulating state [e.g.,quantum spin liquids (QSLs)], depending on the interactionstrength U=W, where U is the on-site Coulomb repulsion*yang.xu@iphy.ac.cnPublished 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 12, 041015 (2022)2160-3308=22=12(4)=041015(10) 041015-1 Published by the American Physical Societyhttps://orcid.org/0000-0001-6837-0014https://orcid.org/0000-0003-3701-8119https://orcid.org/0000-0001-7510-9949https://orcid.org/0000-0002-1185-0073https://orcid.org/0000-0003-4223-8677https://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevX.12.041015&domain=pdf&date_stamp=2022-11-09https://doi.org/10.1103/PhysRevX.12.041015https://doi.org/10.1103/PhysRevX.12.041015https://doi.org/10.1103/PhysRevX.12.041015https://doi.org/10.1103/PhysRevX.12.041015https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/energy and W is the bandwidth [37]. Despite extensiveresearch efforts on the triangular lattice Hubbard model,many aspects of its rich phase diagram remain unexploreddue to the high complexity of the problem and the lackof suitable experimental platforms in different couplingregimes.In TMDC moiré homo- or heterobilayers (such astwisted WSe2 or MoTe2=WSe2) with small band offsetsbetween neighboring layers, the charge density and verticalelectric fields can be independently controlled by electro-static gates, enabling in situ tuning across quantum phasetransitions (QPTs) [38,39]. However, the precise nature ofthe QPT so far is not well understood [38,40–43]. Thecanonical problem of the Hubbard model under an externalmagnetic field, which can polarize the spins and controlthe exchange interactions, is also unsettled and worthfurther investigation [44–47]. Utilizing the sensitivity ofthe interaction strength to the dielectric environment andinterlayer coupling, we fabricate a new type of moiréheterostructure consisting of trilayer MoTe2 and monolayerWSe2. The multilayer TMDC (3L-MoTe2 here)-basedsuperlattices can also potentially offer new opportunitiesto explore moiré physics (e.g., that arises from the Γ valleyinstead of the K valley) exhibiting different symmetries andstrengths of spin-orbit coupling [8,48–50]. Detailed infor-mation of the band structure analyses for bare 3L-MoTe2and the 3L-MoTe2=WSe2 heterostructure is presented inSupplemental Material [51].In this paper, we discover a rich and exotic quantumphase diagram tuned by both out-of-plane electrical andmagnetic fields in the 3L-MoTe2=WSe2. At half filling ofthe first moiré subband, we find a resistance peak thatcounterintuitively develops with increasing temperature.This behavior resembles the Pomeranchuk effect observedin He-3 and is attributed to the close vicinity of our systemto the MIT point in the triangular lattice Hubbard model.Because of the small valence band offset of the twomaterials, the hole-type charge carriers are first injectedinto the MoTe2 and the subband population fromWSe2 canbe continuously tuned by the vertical electric field, whichdrives a Lifshitz transition and alters the charge transportgreatly. The process is accompanied by a breakdown of thesingle-band Hubbard model description of the system. Themagnetic field, which suppresses spin fluctuations, caninduce a MIT following the quantum critical scaling. Weobserve continuous closure of the charge gap whenapproaching the quantum critical point from the insulatingside and divergence of effective quasiparticle mass from themetallic side.II. METHODSNatural trilayer MoTe2 (2H phase) and monolayer WSe2flakes are mechanically exfoliated and angle aligned toform 3L-MoTe2=WSe2 moiré superlattices (see Supple-mental Material [51] for more details). The two materialshave a lattice mismatch of about 7%, generating a maxi-mum moiré wavelength of 5 nm and a superlattice densityof approximately 5 × 1012 cm−2. The 3L-MoTe2=WSe2heterostructure is encapsulated by hexagonal boron nitridesheets (about 5–10 nm thick) as gate dielectrics andpatterned into a quasi-Hall bar structure [see the deviceschematic and an optical image in Fig. 1(a)] for electronictransport measurements. Top and bottom gate electrodesare made by few-layer graphite. Charge carriers withdensity n (hole type) and a vertical electric displacementfield D can be introduced by the combination of top andbottom gates (V tg and Vbg, respectively).Transport data under in-plane magnetic fields and highertemperatures (> 70 K) are acquired in a He-4 cryostat(base T ¼ 1.7 K, magnetic field B up to 9 T). All the otherdata are acquired in a He-3 cryostat (base T ¼ 300 mK, Bup to 9 T). Measurements are performed using a standardlow-frequency (9.373 Hz) lock-in technique under asmall bias voltage (1–2 mV). Voltage drops Vxx and Vxyand source-drain current (I) are recorded simultaneouslyto obtain the corresponding longitudinal and Hall resis-tances. Both the voltage drops are measured using voltagepreamplifiers with large input impedance (100 MΩ) tomaintain the measurement accuracy. Bottom Pt electrodes(that have a large work function) and large negative top-gate bias (V tg) help to obtain Ohmic hole contacts tothe 3L-MoTe2=WSe2.III. RESULTSFigure 1(b) shows the longitudinal 2D sheet resistanceρxx as a function of V tg and Vbg measured at an intermediatetemperature T ¼ 20 K. The two arrows denote the direc-tions for increasing D or n, respectively. Notably, thesample exhibits a resistance peak along a constant densityline [with n ≈ 5 × 1012 cm−2; see Supplemental Fig. S1(d)[51] ], corresponding to one hole per moiré unit cell or halffilling of the first moiré subband, namely, moiré fillingfactor ν ¼ 1. We do not observe any sharp feature corre-sponding to the ν ¼ 2 band insulating state. Instead, theresistance exhibits a broadened hump at Vbg < ∼2 V due tocharge population of the Γ valley in the 3L-MoTe2 (seemore discussions in Supplemental Material [51]). Uponincreasing the D field, the ν ¼ 1 peak disappears abruptly(near D ∼ 1.5 V nm−1). The behavior is reminiscent of arecently reported bandwidth-tuned Mott transition inMoTe2=WSe2 heterobilayers [38]. However, we find thatthe resistance peak here is gradually suppressed withdecreasing temperature and completely vanishes at thelowest T ¼ 0.3 K [see the resistance color mappings andconstant V tg line cuts at selective temperatures in Figs. 1(c)and 1(d), respectively], in sharp contrast to the Mottinsulating state [23,24,26]. In other words, the resistancepeak at half filling gets enhanced only at elevated temper-atures, indicating a heat-induced charge localization at theMINGJIE ZHANG et al. PHYS. REV. X 12, 041015 (2022)041015-2specific commensurate doping density (ν ¼ 1). Similareffects in moiré superlattices were first reported in magic-angle twisted bilayer graphene (TBG) [52,53] and referredto as the Pomeranchuk effect, which is an electronic analogof thehigher-temperature solidificationofHe-3 [32,37,54–56].This unusual behavior is due to the larger spin entropy in thelocalized state (solid phase in He-3 or singly occupied state in3L-MoTe2=WSe2). The system is then favorable to increasethe degree of localization upon heating. The neighboring localmoments in 3L-MoTe2=WSe2 have antiferromagnetic super-exchange interactions [5,23,37,38], whereas in TBG isospinpolarization is favored for the jνj ¼ 1 states [52,53].Phenomenologically, we note that a similar Pomeranchukeffect is observed in theMoTe2=WSe2 heterobilayers at a verynarrow electric field range near the critical electric field(Ec ∼ 0.65 V nm−1) that drives the MIT [38]. However, wepoint out that the underlying mechanisms could be verydifferent for the two systems (explained later).A representative ρxx versus T curve for the ν ¼ 1 state (atD ¼ 1.313 V nm−1) is shown in Fig. 1(e). The sampleexhibits an insulating behavior (dρxx=dT < 0) attemperatures higher than T� ∼ 40 K, below which a cross-over to a metallic behavior (dρxx=dT > 0) is observed withρxx decreasing by more than one order of magnitude (fromabout 40 to 0.3 K). At T < ∼10 K, the ρxxðTÞ can be welldescribed by a Fermi liquid behavior ρxxðTÞ ¼ AT2 þ ρ0(highlighted by the dashed blue curve), where A1=2 isproportional to the quasiparticle effective mass m�(Kadowaki-Woods scaling) [57,58] and ρ0 is the residualresistance. The value of A is extracted to be ð64.1�0.9Þ Ω=K2 or ð4.53� 0.06Þ μΩ cm=K2, considering athickness of approximately 0.7 nm for a single layer ofMoTe2. This value is comparable to those of many heavyfermion systems, indicating substantial quasiparticle-quasi-particle scatterings and a strongly renormalized effectivemass at low temperatures [59]. A maximum of ρxx ∼ 25 kΩis reached at T� ∼ 40 K, indicating kFl (with kF being theFermi wave vector and l being the mean free path) of thesystem to be about unity, as inferred from the Ioffe-Regelcriteria. T� is known as the coherence temperature character-izing the related energy scales (such as the renormalizedbandwidth W�) in strongly correlated materials [60,61].FIG. 1. Device structure and Pomeranchuk effect in 3L-MoTe2=WSe2 moiré superlattices. (a) Schematic (left) and optical micrograph(lower right) of the device with double graphite gates. The conduction channel of the device is outlined by the black curves in themicrograph. The scale bar is 10 μm. (b) 2D longitudinal resistance ρxx as a function of top and bottom gate voltages (V tg and Vbg,respectively) at T ¼ 20 K and zero magnetic field. An obvious resistivity peak is observed at half filling ν ¼ 1. The dashed line marksthe expected position of full filling ν ¼ 2. The two arrows denote the directions of increasingD and n, respectively. (c) Evolution of ρxxas a function of T. Each color map plane shows the ρxx versus V tg and Vbg at a fixed T. (d) The ρxx as a function of bottom gate voltage ata fixed top gate voltage V tg ¼ −5.6 V at different temperatures. Two dashed lines indicate the position of ν ¼ 1 and ν ¼ 2, respectively.(e) Temperature dependence and in-plane magnetic field dependence (inset) of ρxx at V tg ¼ −5.6 V and Vbg ¼ 6.2 V (ν ¼ 1 andD ¼ 1.313 V nm−1). At low temperatures (T < 10 K), ρxxðTÞ follows a Fermi liquid behavior AT2 þ ρ0 (blue dashed curve).TUNING QUANTUM PHASE TRANSITIONS AT HALF FILLING … PHYS. REV. X 12, 041015 (2022)041015-3The hence enhanced charge localization is concomitant withan increase of entropy (per quasiparticle) toward ln2 (atapproximately T�, arising from the spin degree of freedom),whereas the Fermi-liquid state has a lower entropy propor-tional to T=T� [55,56]. The contribution from the spinfluctuations to the entropy is shown later. At T > ∼T�, thequasiparticle mean free path l becomes comparable to orsmaller than the moiré wavelength of approximately 5 nm,and the transport enters an incoherent regime, giving rise tothe insulating behavior. In the inset in Fig. 1(e), we plot the in-plane magnetic field (Bk) dependences of ρxx at two differenttemperatures (T ¼ 1.6 and 30 K). Unlike the TBG at jνj ¼ 1[52,53], 3L-MoTe2=WSe2 is insensitive to the in-planemagnetic field.Now we discuss the effect of the vertical displacementfield (D) at ν ¼ 1. At different temperatures, a similar valueof the D field is observed where the resistance maximadisappear [Fig. 1(c)]. In Fig. 2(a), we plot the temperaturedependence (0.3–70 K) of ρxx at different D fields. The ρxxgradually changes from a nonmonotonic behavior to amonotonic and metallic behavior with a weaker T depend-ence at largerD fields. At low temperatures (T < ∼10 K), allresistance curves follow T2 dependences with ρxxðTÞ ¼AT2 þ ρ0 [Fig. 2(b)]. The extractedA coefficient is plotted inthe inset in Fig. 2(b) as a function of theD field. The A valuestays nearly a constant at small D fields, while it decreasesdrastically by more than 30 folds starting from an inflectionpoint at approximately 1.49 Vnm−1, which is denoted asthe transition field Dt [dashed vertical line in the inset inFig. 2(b)]. It indicates strong suppression of electronic inter-actions in the system and an approximately 80% reduction ofeffective quasiparticle mass. The high-temperature value ofρxx near Dt is close to 12.9 kΩ. On both sides of the tran-sition, the sample exhibits metallic ground states withFermi liquid behaviors. This “metal-metal” transitionobserved here contrasts the electric-field-inducedMIT foundin MoTe2=WSe2 [38]. The Dt value (corresponding to Et ¼Dt=εr ¼ 0.50 V nm−1,where εr is theout-of-planedielectricconstant of h-BN) is also smaller than the critical electric field(with Ec ¼ 0.65 V nm−1) for MIT in the heterobilayer [38].FIG. 2. Electric displacement field-driven Lifshitz transition. (a) Temperature dependences of ρxx with varying displacement fieldD atν ¼ 1 and B ¼ 0 T. (b) The same data as (a) (sharing the same color coding) with the x axis scaled in T2 below about 10 K. The dashedlines are the corresponding linear fits. Inset: fitting parameter A as a function ofD. The vertical dashed line indicates the inflection pointat D ¼ Dt. (c) The upper diagram shows ρxx as a function of out-of-plane magnetic field B⊥ and D at ν ¼ 1 and T ¼ 0.3 K. The lowerdiagram shows color-coded regions representing different states. The Landau fan (linear dashed lines) converges to Dt, with fillingfactors νL labeled accordingly. (d) Schematics showing the Lifshitz transition and evolution of band alignments (K valleys of MoTe2 andWSe2) with increasing electric displacement fieldD. The K0 valley is degenerate with the K valley at zero magnetic field and, hence, notdrawn on this schematic. Various effects (such as moiré band formation and interaction-induced mass renormalization) are neglected inthis simple schematic. Above Dt, charges are transferred between the MoTe2 and the WSe2.MINGJIE ZHANG et al. PHYS. REV. X 12, 041015 (2022)041015-4The magnetotransport studies (under out-of-plane mag-netic fields B⊥) suggest that the D field in our sampleintroduces a Lifshitz transition atDt, beyond which chargesare redistributed between different subbands or layers. Asshown in Fig. 2(c), the ρxx (at ν ¼ 1) is plotted as a functionof both D and B⊥ at the base temperature T ¼ 0.3 K. Atsmall values of D, the sample encounters a transition to aninsulating state (with ρxx > 107 Ω; see more data in Fig. 3)above about 6 T. Combined with the barely changed in-plane magnetoresistance [inset in Fig. 1(e)], the strongmagnetic anisotropy suggests an Ising type of spin-orbitcoupling of the charge carriers that likely arise from K=K0valleys of the MoTe2. It is also consistent with the fact thatthe finite out-of-plane displacement field lifts the layerdegeneracy and leads to a strong localization of K=K0valley wave functions on the topmost layer of the trilayerMoTe2 (more discussions can be found in SupplementalMaterial [51]) [62].At larger values of D, clear Shubnikov–de Haas (SdH)oscillations due to the formation of Landau levels areobserved at B⊥ > ∼3 T. The generated Landau fan (alsoindicated by the dashed lines in the lower diagram)converges to D ¼ 1.492 V nm−1 in the zero B⊥ field limit.The value coincides (while being more accurate) withthe transition field Dt discussed above and indicates theemergence of a well-defined Fermi surface at D > Dt.Hence, it corresponds to a Lifshitz transition. The lineardependences of the Landau levels on the D field indi-cate that the size of the new Fermi surface is directlyproportional to (D −Dt). A cyclotron effective mass(0.52� 0.01Þme (with me being the free electron mass)can be determined from analyzing the temperature depend-ence of the SdH oscillations (see more details in Supple-mental Fig. S3 [51]). It is close to the effective mass foundin monolayer WSe2 K=K0 valleys [63,64]. We, thus,conclude that the charge transfer induced by the verticalelectric field happens between the K=K0 valleys of theMoTe2 top layer and the adjacent WSe2 monolayer[schematics shown in Fig. 2(d)], giving rise to two sets ofquantum oscillations observed atD > Dt (see SupplementalFIG. 3. Magnetic-field-induced metal-insulator transition. (a) Evolution of the insulating state at half filling as a function of out-of-plane magnetic field B⊥. Each color plane shows the ρxx versus top and bottom gate voltages at a fixed B⊥. (b) Showing Vbg sweeps ofρxx at a fixed V tg ¼ −5.3 V under varying B⊥. Sharp resistance peaks occur at ν ¼ 1 above the critical magnetic field Bc ≈ 6 T. (c) Red(dashed black) and blue curves show ρxx and ρxy as functions of B⊥ at 0.3 K, respectively. The gate voltages are fixed to ensure ν ¼ 1and D ¼ 1.313 V nm−1, the same for (d) and (e). No hysteresis is observed upon sweeping down and up the magnetic fields (indicatedby the red and dashed black arrows, respectively). (d) Temperature dependences of ρxx at different B⊥, showing the magnetic-field-induced MIT. Inset: Dashed lines represent Arrhenius fits to extract activation gap magnitudes for B⊥ > 6 T. (e) Quantum criticalscaling analysis of the MIT. The normalized resistance [by ρc ¼ ρxxðB ¼ BcÞ] curves neatly collapse onto two branches that are almost“mirror symmetric,” with one showing an insulating behavior and the other showing a metallic behavior. The determination of thescaling temperature T0 is discussed in the main text. Top inset: magnetic field dependence of A, where A is the fitting parameter for thelow-temperature Fermi liquid. The dashed curve is a power-law fit A1=2 ∝ ðBc − B⊥Þ−0.98�0.11. Bottom inset: the scaling parameter kBT0versus B⊥ on both sides of the critical field (red and blue open symbols). Close to Bc (jB⊥ − Bcj ≤ ∼3 T), both sides show power-lawdependences with critical exponents 0.73� 0.02 (red and blue dashed curves).TUNING QUANTUM PHASE TRANSITIONS AT HALF FILLING … PHYS. REV. X 12, 041015 (2022)041015-5Fig. S4 [51]). Further evidence for assigning the valleyoccupations can be found in Supplemental Material [51].Such observations place a question of explaining the electric-field-induced MIT as the bandwidth-controlled Mott tran-sition within the framework of the single-band Hubbardmodel [38]. Since the charge transfer between MoTe2 andWSe2 can happen at rather low electric fields, another likelyscenario is that the charge population in the WSe2 layernaturally destroys the Mott insulating state in the hetero-bilayers. The Pomeranchuk effect observed near the criticalelectric field in the 1L-MoTe2=WSe2, hence, may have avery different origin comparing to our observations in arelatively large electric field range (1.1 Vnm−1 < D < Dt)where the single-band Hubbard model description is valid.In the following, we demonstrate the quantum criticalityinduced by the out-of-plane magnetic field B⊥ at ν ¼ 1 andD < Dt. We plot the dual-gate mappings of ρxx at differentB⊥ [Fig. 3(a)]. A resistance peak at ν ¼ 1 can be identifiedat approximately 5 T for D < Dt and quickly becomesprominent at higher fields [see line cuts in Fig. 3(b)]. Atfixed D fields [see an example in Fig. 3(c)], the ρxx weaklydepends on B⊥ at low magnetic fields, accompanied by alinear Hall resistance ρxy due to the Lorentz force. The low-field Hall slope gives a density of 5.4 × 1012 cm−2, close tothe expected moiré density. Both ρxx and ρxy diverge atapproximately 6 T, which is then identified as the criticalmagnetic field Bc. The B⊥-induced MIT at ν ¼ 1 is clearlyillustrated in Fig. 3(d), where we show the temperaturedependences of sheet resistance ρxx (0.3–70 K) at a fewrepresentative magnetic fields. Above Bc, the sampleexhibits characteristics of an insulator following a thermalactivation behavior ρxx ∝ EðΔ=2kBTÞ [inset in Fig. 3(d)],where kB is the Boltzmann constant and Δ is the acti-vation gap. At low temperatures and below the criticalfield (Bc), the resistance drops upon cooling, exhibitingthe characteristics of Fermi liquids. The extracted pre-factor A is plotted as a function of jB⊥ − Bcj in the upperinset in Fig. 3(e), showing a power-law dependenceA1=2 ∝ ðBc − B⊥Þ−0.98�0.11. Our result suggests a diver-gence of the effective mass (∝ A1=2) at the MIT [65,66].After normalizing ρxxðTÞ by the ρcðTÞ at 6 T (critical fieldBc) and scaling the temperature by a field-dependent T0,the resistance curves readily collapse onto two branches inthe Fig. 3(e) log-log plot. We notice the ρc has acomplicated dependence on the temperature. Possiblereasons are discussed in Supplemental Material [51]. Onthe insulating side (B⊥ > Bc), T0 is calculated from thefitting parameter Δ=kB, while on the metallic side(B⊥ < Bc), the same T0 is used as the insulating side withequal distance to the critical field at jB⊥ − Bcj <¼ 3 T.Because we do not have data at higher magnetic fields, theT0’s of B⊥ ¼ 0–2 T are picked independently to scale thecurves nearly symmetric to the insulating branches aboutρxx=ρc ¼ 1. The scaling parameter kBT0 continuouslyvanishes as it approaches the critical field [lower inset inFig. 3(e)]. The B⊥ dependences of kBT0 follow a power-law behavior kBT0 ∝ jB⊥ − Bcj0.73�0.02 near both sides ofthe critical field (highlighted by the dashed red and bluecurves, respectively). The critical exponent 0.73� 0.02 isclose to the value found in the bandwidth-tuned MITreported by Li et al. [38], indicating the QPTs tuned bythe two different parameters share the same universalityclass. The half-filled Hubbard model under an externalmagnetic field has been extensively studied theoretically[44–47,67]. It has been puzzling for a long timewhether themagnetic-field-driven MIT should be continuous or first-order-like. Recent studies suggest the nature of the tran-sition depends on the correlation strength. Our observationin 3L-MoTe2=WSe2 near the critical field (with stronglyenhanced effective masses when approaching the transitionfrom the metallic side) is, in general, consistent with thetheoretical prediction of the magnetic-field-induced spin-polarized band insulator at the intermediate correlationstrength (U ∼W) [44,46,47]. However, we do not observeany hysteresis upon sweeping up and down the magneticfield [Fig. 3(c)], signifying the character of a continuousphase transition. We can then estimate the Zeeman energygμBB⊥ ∼W� ∼ kBT� to be 3.5 meV and the Landé g factorto be g ¼ ∼10. The role of orbital effects from the out-of-plane magnetic fields that could reduce the bandwidth isnot considered here.IV. DISCUSSION AND CONCLUSIONFigure 4 summarizes our main observations at half fillingof the 3L-MoTe2=WSe2 sample. In Fig. 4(a), the resistance(in logarithmic scale) is plotted in the three-dimensionalspace projected on the T −D (B⊥ ¼ 0), T − B⊥ (D ¼1.313 V nm−1), and D − B⊥ (T ¼ 0.3 K) planes. Thetemperature and magnetic field axes are drawn as kBTand gμBB⊥ to represent the thermal and Zeeman energies,respectively. First, the vertical electric field can inducecharge transfer between different subbands and cause abreakdown of the single-band description of the system. Itdrives the system from a strongly interacting metal to aweakly interacting metal (evidenced by the great reductionof the A value) accompanied by the Lifshitz transition.Second, the perpendicular magnetic field induces themetal-to-insulator QPT at Bc (highlighted by the dashedyellow curve in the D − B⊥ plane). The asterisk symbolsillustrate the T� (separating the high-temperature insulatingbehavior from the low-temperature Fermi liquids) as afunction of D and B⊥. A quantum tricritical point (high-lighted by the pink circle) can be identified at Dt ¼1.49 V nm−1 and Bc ¼ ∼6 T where phase transitionstuned by the two nearly independent parameters meetat the base temperature (see more discussions inSupplemental Fig. S4 [51]).In Fig. 4(b), we illustrate the generally accepted phasediagram of the half-filled triangular lattice Hubbard modelwith tuning the interaction strength U=W. The ground stateMINGJIE ZHANG et al. PHYS. REV. X 12, 041015 (2022)041015-6of our sample at D < Dt and B⊥ ¼ 0 is a correlatedmetal exhibiting a Fermi-liquid behavior at low temper-atures (with strongly renormalized effective mass) andPomeranchuk effects. The observations are consistent withan intermediate coupling strength in the vicinity to the leftside of the MIT [32,37,55,56] [marked by the yellow star inFig. 4(b)]. Spin fluctuations and geometric frustrations giverise to larger entropy in the exotic nonmagnetic insulatingstate (e.g., chiral QSLs [36,37]) compared with the metalphase. Pomeranchuk effect arises with the enhanced chargelocalization at higher temperatures, since it yields a gain infree energy. It can also be seen from the Maxwell relation∂S∂U��T¼ −∂F∂T��U;where double occupancy (F denoting the fraction of thecharges in doubly occupied lattice sites) decreases (namely,single occupancy increases) as a function of the temper-ature, since the entropy S is positively correlated with Unear the MIT [37,55,68]. The perpendicular magnetic-field-driven MIT likely originates from the Zeeman-field-induced spin splitting, realizing a spin-polarized bandinsulator at high fields (when the Zeeman energy excessesthe renormalized bandwidth W�) [44–47].The vicinity to the intermediate coupling region in3L-MoTe2=WSe2 provides a novel example of the richnessand complex phase diagrams that can be unveiled in moirésuperlattices. The observed Pomeranchuk effect is closelyrelated to the higher entropy in the singly occupied state athalf filling. Our observations could have implications onthe existence of the elusive QSL phase at larger correlationstrength, whereas compelling evidence is still missing. Themultiband nature of the system also suggests it as apromising platform to construct and simulate the physicsof a two-band Hubbard model [11] or a moiré Kondo lattice[69,70] with unprecedented controllability.ACKNOWLEDGMENTSThe authors thank Professor Kin Fai Mak and ProfessorCenke Xu for valuable discussions. This work was primarilyfunded by the National Key R&D Program of China(Grant No. 2021YFA1401300). Support was also providedby theNationalNatural Science Foundation ofChina (GrantsNo. 12174439 and No. 12074375) and the Strategic PriorityResearch Program of Chinese Academy of Sciences (GrantsNo. XDB28000000 and No. XDB33000000). The growth ofh-BN crystals was supported by the Elemental StrategyInitiative ofMEXT, Japan, andCREST (JPMJCR15F3), JST.FIG. 4. Phase diagram of 3L-MoTe2=WSe2 at half filling. (a) Longitudinal resistivity ρxx (in log scale) as a function of gμBB⊥,D, andkBT. The diagram shows three slices of the function. Different states can be identified by the color contrast. The low-field light blue areadenotes the Fermi liquid phase; the dark red area denotes the magnetic-field-induced insulating phase. Lifshitz transition occurs near1.5 Vnm−1 and generates a dark blue area at large D. In the dark blue area at the base temperature, Landau fan can be observed. Thepink circle, the asterisk symbols, and the dashed yellow curves highlight the tricritical point in the gμBB⊥ −D plane, the coherent-incoherent transport crossover temperature T�, and critical field Bc, respectively. (b) The expected phase diagram of the triangular latticeHubbard model. Our 3L-MoTe2=WSe2 lies on the left boundary of the MIT (denoted by the yellow arrow and star). Lower-panelschematics show the real-space charge distributions and spin orientations (denoted by the arrows) for corresponding phases. The metaland nonmagnetic insulator phases have random spin orientations, while the system favors 120° Néel order in the large U=W limit.TUNING QUANTUM PHASE TRANSITIONS AT HALF FILLING … PHYS. REV. X 12, 041015 (2022)041015-7[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] 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).[3] L. Balents, C. R. Dean, D. K. Efetov, and A. F. Young,Superconductivity and Strong Correlations in Moiré FlatBands, Nat. Phys. 16, 725 (2020).[4] E. Y. Andrei, D. K. Efetov, P. Jarillo-Herrero, A. H.MacDonald, K. F. Mak, T. Senthil, E. Tutuc, A. Yazdani,and A. F. Young, The Marvels of Moiré Materials, Nat. Rev.Mater. 6, 201 (2021).[5] F. Wu, T. Lovorn, E. Tutuc, and A. H. MacDonald, HubbardModel Physics in Transition Metal Dichalcogenide MoiréBands, Phys. Rev. Lett. 121, 026402 (2018).[6] F. Wu, T. Lovorn, E. Tutuc, I. Martin, and A. H.MacDonald, Topological Insulators in Twisted TransitionMetal Dichalcogenide Homobilayers, Phys. Rev. Lett. 122,086402 (2019).[7] K. Slagle and L. Fu, Charge Transfer Excitations, PairDensity Waves, and Superconductivity in Moiré Materials,Phys. Rev. B 102, 235423 (2020).[8] Y. Zhang, T. Liu, and L. Fu, Electronic Structures, ChargeTransfer, and Charge Order in Twisted Transition MetalDichalcogenide Bilayers, Phys. Rev. B 103, 155142 (2021).[9] T. Devakul, V. Crépel, Y. Zhang, and L. Fu, Magic inTwisted Transition Metal Dichalcogenide Bilayers, Nat.Commun. 12, 6730 (2021).[10] Y.-T. Hsu, F. Wu, and S. Das Sarma, Spin-Valley LockedInstabilities in Moiré Transition Metal Dichalcogenideswith Conventional and Higher-Order van Hove Singular-ities, Phys. Rev. B 104, 195134 (2021).[11] Y.-H. Zhang, D. N. Sheng, and A. Vishwanath, SU(4)Chiral Spin Liquid, Exciton Supersolid, and Electric De-tection in Moiré Bilayers, Phys. Rev. Lett. 127, 247701(2021).[12] H. Pan and S. Das Sarma, Interaction Range and Temper-ature Dependence of Symmetry Breaking in StronglyCorrelated Two-Dimensional Moiré Transition MetalDichalcogenide Bilayers, Phys. Rev. B 105, 041109(2022).[13] M. Biderang, M.-H. Zare, and J. Sirker, Proximity-DrivenFerromagnetism and Superconductivity in the TriangularRashba-Hubbard Model, Phys. Rev. B 105, 064504 (2022).[14] Z. Zhu, D. N. Sheng, and A. Vishwanath, Doped MottInsulators in the Triangular-Lattice Hubbard Model, Phys.Rev. B 105, 205110 (2022).[15] K. F. Mak and J. Shan, Semiconductor Moiré Materials,Nat. Nanotechnol. 17, 686 (2022).[16] E. M. Alexeev, D. A. Ruiz-Tijerina, M. Danovich, M. J.Hamer, D. J. Terry, P. K. Nayak, S. Ahn, S. Pak, J. Lee, J. I.Sohn et al., Resonantly Hybridized Excitons in MoiréSuperlattices in van der Waals Heterostructures, Nature(London) 567, 81 (2019).[17] C. Jin, E. C. Regan, A. Yan, M. Iqbal Bakti Utama, D.Wang, S. Zhao, Y. Qin, S. Yang, Z. Zheng, S. Shi et al.,Observation of Moiré Excitons in WSe2=WS2 Heterostruc-ture Superlattices, Nature (London) 567, 76 (2019).[18] K. L. Seyler, P. Rivera, H. Yu, N. P. Wilson, E. L. Ray, D. G.Mandrus, J. Yan, W. Yao, and X. Xu, Signatures of Moiré-Trapped Valley Excitons in MoSe2=WSe2 Heterobilayers,Nature (London) 567, 66 (2019).[19] K. Tran, G. Moody, F. Wu, X. Lu, J. Choi, K. Kim, A. Rai,D. A. Sanchez, J. Quan, A. Singh et al., Evidence for MoiréExcitons in van der Waals Heterostructures, Nature (Lon-don) 567, 71 (2019).[20] E. Liu, E. Barré, J. van Baren, M. Wilson, T. Taniguchi, K.Watanabe, Y.-T. Cui, N. M. Gabor, T. F. Heinz, Y.-C. Changet al., Signatures of Moiré Trions in WSe2=MoSe2 Hetero-bilayers, Nature (London) 594, 46 (2021).[21] X. Wang, J. Zhu, K. L. Seyler, P. Rivera, H. Zheng, Y. Wang,M. He, T. Taniguchi, K. Watanabe, J. Yan et al., MoiréTrions in MoSe2=WSe2 Heterobilayers, Nat. Nanotechnol.16, 1208 (2021).[22] L. Zhang, F. Wu, S. Hou, Z. Zhang, Y.-H. Chou, K.Watanabe, T. Taniguchi, S. R. Forrest, and H. Deng, vander Waals Heterostructure Polaritons with Moiré-InducedNonlinearity, Nature (London) 591, 61 (2021).[23] Y. Tang, L. Li, T. Li, Y. Xu, S. Liu, K. Barmak, K.Watanabe, T. Taniguchi, A. H. MacDonald, J. Shan et al.,Simulation of Hubbard Model Physics inWSe2=WS2 MoiréSuperlattices, Nature (London) 579, 353 (2020).[24] L. Wang, E.-M. Shih, A. Ghiotto, L. Xian, D. A. Rhodes, C.Tan, M. Claassen, D. M. Kennes, Y. Bai, B. Kim et al.,Correlated Electronic Phases in Twisted Bilayer TransitionMetal Dichalcogenides, Nat. Mater. 19, 861 (2020).[25] Y. Shimazaki, I. Schwartz, K. Watanabe, T. Taniguchi, M.Kroner, and A. Imamoğlu, Strongly Correlated Electronsand Hybrid Excitons in a Moiré Heterostructure, Nature(London) 580, 472 (2020).[26] E. C. Regan, D. Wang, C. Jin, M. I. Bakti Utama, B. Gao, X.Wei, S. Zhao, W. Zhao, Z. Zhang, K. Yumigeta et al., Mottand Generalized Wigner Crystal States in WSe2=WS2Moiré Superlattices, Nature (London) 579, 359 (2020).[27] Y. Xu, S. Liu, D. A. Rhodes, K. Watanabe, T. Taniguchi, J.Hone, V. Elser, K. F. Mak, and J. Shan, Correlated Insulat-ing States at Fractional Fillings of Moiré Superlattices,Nature (London) 587, 214 (2020).[28] X. Huang, T. Wang, S. Miao, C. Wang, Z. Li, Z. Lian, T.Taniguchi, K. Watanabe, S. Okamoto, D. Xiao et al.,Correlated Insulating States at Fractional Fillings of theWS2=WSe2 Moiré Lattice, Nat. Phys. 17, 715 (2021).[29] C. Jin, Z. Tao, T. Li, Y. Xu, Y. Tang, J. Zhu, S. Liu, K.Watanabe, T. Taniguchi, J. C. Hone et al., Stripe Phases inWSe2=WS2 Moiré Superlattices, Nat. Mater. 20, 940(2021).[30] H. Li, S. Li, E. C. Regan, D. Wang, W. Zhao, S. Kahn, K.Yumigeta, M. Blei, T. Taniguchi, K. Watanabe et al.,Imaging Two-Dimensional Generalized Wigner Crystals,Nature (London) 597, 650 (2021).[31] T. Li, S. Jiang, B. Shen, Y. Zhang, L. Li, Z. Tao, T. Devakul,K. Watanabe, T. Taniguchi, L. Fu, J. Shan, and K. F. Mak,Quantum Anomalous Hall Effect from Intertwined MoiréBands, Nature (London) 600, 641 (2021).[32] K. Aryanpour, W. E. Pickett, and R. T. Scalettar, DynamicalMean-Field Study of the Mott Transition in the Half-FilledMINGJIE ZHANG et al. PHYS. REV. X 12, 041015 (2022)041015-8https://doi.org/10.1038/nature26154https://doi.org/10.1038/nature26154https://doi.org/10.1038/nature26160https://doi.org/10.1038/nature26160https://doi.org/10.1038/s41567-020-0906-9https://doi.org/10.1038/s41578-021-00284-1https://doi.org/10.1038/s41578-021-00284-1https://doi.org/10.1103/PhysRevLett.121.026402https://doi.org/10.1103/PhysRevLett.122.086402https://doi.org/10.1103/PhysRevLett.122.086402https://doi.org/10.1103/PhysRevB.102.235423https://doi.org/10.1103/PhysRevB.103.155142https://doi.org/10.1038/s41467-021-27042-9https://doi.org/10.1038/s41467-021-27042-9https://doi.org/10.1103/PhysRevB.104.195134https://doi.org/10.1103/PhysRevLett.127.247701https://doi.org/10.1103/PhysRevLett.127.247701https://doi.org/10.1103/PhysRevB.105.041109https://doi.org/10.1103/PhysRevB.105.041109https://doi.org/10.1103/PhysRevB.105.064504https://doi.org/10.1103/PhysRevB.105.205110https://doi.org/10.1103/PhysRevB.105.205110https://doi.org/10.1038/s41565-022-01165-6https://doi.org/10.1038/s41586-019-0986-9https://doi.org/10.1038/s41586-019-0986-9https://doi.org/10.1038/s41586-019-0976-yhttps://doi.org/10.1038/s41586-019-0957-1https://doi.org/10.1038/s41586-019-0975-zhttps://doi.org/10.1038/s41586-019-0975-zhttps://doi.org/10.1038/s41586-021-03541-zhttps://doi.org/10.1038/s41565-021-00969-2https://doi.org/10.1038/s41565-021-00969-2https://doi.org/10.1038/s41586-021-03228-5https://doi.org/10.1038/s41586-020-2085-3https://doi.org/10.1038/s41563-020-0708-6https://doi.org/10.1038/s41586-020-2191-2https://doi.org/10.1038/s41586-020-2191-2https://doi.org/10.1038/s41586-020-2092-4https://doi.org/10.1038/s41586-020-2868-6https://doi.org/10.1038/s41567-021-01171-whttps://doi.org/10.1038/s41563-021-00959-8https://doi.org/10.1038/s41563-021-00959-8https://doi.org/10.1038/s41586-021-03874-9https://doi.org/10.1038/s41586-021-04171-1Hubbard Model on a Triangular Lattice, Phys. Rev. B 74,085117 (2006).[33] T. Yoshioka, A. Koga, and N. Kawakami, Quantum PhaseTransitions in the Hubbard Model on a Triangular Lattice,Phys. Rev. Lett. 103, 036401 (2009).[34] H.-Y. Yang, A. M. Läuchli, F. Mila, and K. P. Schmidt,Effective Spin Model for the Spin-Liquid Phase of theHubbard Model on the Triangular Lattice, Phys. Rev. Lett.105, 267204 (2010).[35] T. Shirakawa, T. Tohyama, J. Kokalj, S. Sota, and S. Yunoki,Ground-State Phase Diagram of the Triangular LatticeHubbard Model by the Density-Matrix RenormalizationGroup Method, Phys. Rev. B 96, 205130 (2017).[36] A. Szasz, J. Motruk, M. P. Zaletel, and J. E. Moore, ChiralSpin Liquid Phase of the Triangular Lattice HubbardModel: A Density Matrix Renormalization Group Study,Phys. Rev. X 10, 021042 (2020).[37] A. Wietek, R. Rossi, F. Šimkovic IV, M. Klett, P. Hansmann,M. Ferrero, E. M. Stoudenmire, T. Schäfer, and A. Georges,Mott Insulating States with Competing Orders in theTriangular Lattice Hubbard Model, Phys. Rev. X 11,041013 (2021).[38] T. Li, S. Jiang, L. Li, Y. Zhang, K. Kang, J. Zhu, K.Watanabe, T. Taniguchi, D. Chowdhury, L. Fu et al.,Continuous Mott Transition in Semiconductor Moiré Super-lattices, Nature (London) 597, 350 (2021).[39] A. Ghiotto, E.-M. Shih, G. S. Pereira, D. A. Rhodes, B.Kim, J. Zang, A. J. Millis, K. Watanabe, T. Taniguchi, J. C.Hone et al., Quantum Criticality in Twisted TransitionMetal Dichalcogenides, Nature (London) 597, 345 (2021).[40] S. Ahn and S. Das Sarma, Disorder-Induced Two-Dimensional Metal-Insulator Transition in Moiré Transi-tion Metal Dichalcogenide Multilayers, Phys. Rev. B 105,115114 (2022).[41] J. Zang, J. Wang, J. Cano, A. Georges, and A. J. Millis,Dynamical Mean-Field Theory of Moiré Bilayer TransitionMetal Dichalcogenides: Phase Diagram, Resistivity, andQuantum Criticality, Phys. Rev. X 12, 021064 (2022).[42] Y. Xu, X.-C. Wu, M. Ye, Z.-X. Luo, C.-M. Jian, and C. Xu,Interaction-Driven Metal-Insulator Transition with ChargeFractionalization, Phys. Rev. X 12, 021067 (2022).[43] D. C. Sunghoon Kim and T. Senthil, Continuous MottTransition in Moiré Semiconductors: Role of Long-Wave-length Inhomogeneities, arXiv:2204.10865v1.[44] L. Laloux, A. Georges, and W. Krauth, Effect of a MagneticField on Mott-Hubbard Systems, Phys. Rev. B 50, 3092(1994).[45] J. Bauer and A. C. Hewson, Field-Dependent Quasipar-ticles in the Infinite-Dimensional Hubbard Model, Phys.Rev. B 76, 035118 (2007).[46] M.M. Wysokiński and J. Spałek, Properties of an AlmostLocalized Fermi Liquid in an Applied Magnetic FieldRevisited: A Statistically Consistent Gutzwiller Approach,J. Phys. Condens. Matter 26, 055601 (2014).[47] W. Zhu, D. N. Sheng, and J.-X. Zhu, Magnetic FieldDependent Dynamics and Field-Driven Metal-to-InsulatorTransition of the Half-Filled Hubbard Model: A DMFTþDMRG Study, Phys. Rev. B 96, 085118 (2017).[48] Z. Zhan, Y. Zhang, P. Lv, H. Zhong, G. Yu, F. Guinea, J. A.Silva-Guillén, and S. Yuan, Tunability of Multiple UltraflatBands and Effect of Spin-Orbit Coupling in Twisted BilayerTransition Metal Dichalcogenides, Phys. Rev. B 102,241106(R) (2020).[49] M. Angeli and A. H. MacDonald, Γ Valley Transition MetalDichalcogenide Moiré Bands, Proc. Natl. Acad. Sci. U.S.A.118, e2021826118 (2021).[50] L. Xian, M. Claassen, D. Kiese, M.M. Scherer, S. Trebst,D. M. Kennes, and A. Rubio, Realization of Nearly Dis-persionless Bands with Strong Orbital Anisotropy fromDestructive Interference in Twisted Bilayer MoS2, Nat.Commun. 12, 5644 (2021).[51] See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevX.12.041015 for details ofsample fabrication, band structure analyses, and moresupplemental data.[52] A. Rozen, J. M. Park, U. Zondiner, Y. Cao, D. Rodan-Legrain, T. Taniguchi, K. Watanabe, Y. Oreg, A. Stern,E. Berg et al., Entropic Evidence for a Pomeranchuk Effectin Magic-Angle Graphene, Nature (London) 592, 214(2021).[53] Y. Saito, F. Yang, J. Ge, X. Liu, T. Taniguchi, K. Watanabe,J. Li, E. Berg, and A. F. Young, Isospin Pomeranchuk Effectin Twisted Bilayer Graphene, Nature (London) 592, 220(2021).[54] R. C. Richardson, The Pomeranchuk Effect, Rev. Mod.Phys. 69, 683 (1997).[55] A. Georges and W. Krauth, Numerical Solution of thed ¼ ∞ Hubbard Model: Evidence for a Mott Transition,Phys. Rev. Lett. 69, 1240 (1992).[56] A. Georges and W. Krauth, Physical Properties of the Half-Filled Hubbard Model in Infinite Dimensions, Phys. Rev. B48, 7167 (1993).[57] M. Rice, Electron-Electron Scattering in Transition Metals,Phys. Rev. Lett. 20, 1439 (1968).[58] K. Kadowaki and S. Woods, Universal Relationshipof the Resistivity and Specific Heat in Heavy-FermionCompounds, Solid State Commun. 58, 507 (1986).[59] A. Jacko, J. Fjærestad, and B. Powell, A Unified Explan-ation of the Kadowaki-Woods Ratio in Strongly CorrelatedMetals, Nat. Phys. 5, 422 (2009).[60] M.M. Radonjić, D. Tanasković, V. Dobrosavljević, K.Haule, and G. Kotliar, Wigner-Mott Scaling of Transportnear the Two-Dimensional Metal-Insulator Transition,Phys. Rev. B 85, 085133 (2012).[61] E. Pavarini, P. Coleman, and E. Koch, Many-Body Physics:From Kondo to Hubbard. Heavy Fermions and the KondoLattice: A 21st Century Perspective (ForschungszentrumJulich, Julich, 2015).[62] H. C. P. Movva, T. Lovorn, B. Fallahazad, S. Larentis, K.Kim, T. Taniguchi, K. Watanabe, S. K. Banerjee, A. H.MacDonald, and E. Tutuc, Tunable Γ-K Valley Populationsin Hole-Doped TrilayerWSe2, Phys. Rev. Lett. 120, 107703(2018).[63] H. C. P. Movva, B. Fallahazad, K. Kim, S. Larentis, T.Taniguchi, K. Watanabe, S. K. Banerjee, and E. Tutuc,Density-Dependent Quantum Hall States and ZeemanSplitting in Monolayer and Bilayer WSe2, Phys. Rev. Lett.118, 247701 (2017).[64] M. V. Gustafsson, M. Yankowitz, C. Forsythe, D. Rhodes,K. Watanabe, T. Taniguchi, J. Hone, X. Zhu, and C. R.TUNING QUANTUM PHASE TRANSITIONS AT HALF FILLING … PHYS. REV. X 12, 041015 (2022)041015-9https://doi.org/10.1103/PhysRevB.74.085117https://doi.org/10.1103/PhysRevB.74.085117https://doi.org/10.1103/PhysRevLett.103.036401https://doi.org/10.1103/PhysRevLett.105.267204https://doi.org/10.1103/PhysRevLett.105.267204https://doi.org/10.1103/PhysRevB.96.205130https://doi.org/10.1103/PhysRevX.10.021042https://doi.org/10.1103/PhysRevX.11.041013https://doi.org/10.1103/PhysRevX.11.041013https://doi.org/10.1038/s41586-021-03853-0https://doi.org/10.1038/s41586-021-03815-6https://doi.org/10.1103/PhysRevB.105.115114https://doi.org/10.1103/PhysRevB.105.115114https://doi.org/10.1103/PhysRevX.12.021064https://doi.org/10.1103/PhysRevX.12.021067https://arXiv.org/abs/2204.10865v1https://doi.org/10.1103/PhysRevB.50.3092https://doi.org/10.1103/PhysRevB.50.3092https://doi.org/10.1103/PhysRevB.76.035118https://doi.org/10.1103/PhysRevB.76.035118https://doi.org/10.1088/0953-8984/26/5/055601https://doi.org/10.1103/PhysRevB.96.085118https://doi.org/10.1103/PhysRevB.102.241106https://doi.org/10.1103/PhysRevB.102.241106https://doi.org/10.1073/pnas.2021826118https://doi.org/10.1073/pnas.2021826118https://doi.org/10.1038/s41467-021-25922-8https://doi.org/10.1038/s41467-021-25922-8http://link.aps.org/supplemental/10.1103/PhysRevX.12.041015http://link.aps.org/supplemental/10.1103/PhysRevX.12.041015http://link.aps.org/supplemental/10.1103/PhysRevX.12.041015http://link.aps.org/supplemental/10.1103/PhysRevX.12.041015http://link.aps.org/supplemental/10.1103/PhysRevX.12.041015http://link.aps.org/supplemental/10.1103/PhysRevX.12.041015http://link.aps.org/supplemental/10.1103/PhysRevX.12.041015https://doi.org/10.1038/s41586-021-03319-3https://doi.org/10.1038/s41586-021-03319-3https://doi.org/10.1038/s41586-021-03409-2https://doi.org/10.1038/s41586-021-03409-2https://doi.org/10.1103/RevModPhys.69.683https://doi.org/10.1103/RevModPhys.69.683https://doi.org/10.1103/PhysRevLett.69.1240https://doi.org/10.1103/PhysRevB.48.7167https://doi.org/10.1103/PhysRevB.48.7167https://doi.org/10.1103/PhysRevLett.20.1439https://doi.org/10.1016/0038-1098(86)90785-4https://doi.org/10.1038/nphys1249https://doi.org/10.1103/PhysRevB.85.085133https://doi.org/10.1103/PhysRevLett.120.107703https://doi.org/10.1103/PhysRevLett.120.107703https://doi.org/10.1103/PhysRevLett.118.247701https://doi.org/10.1103/PhysRevLett.118.247701Dean, Ambipolar Landau Levels and Strong Band-SelectiveCarrier Interactions in Monolayer WSe2, Nat. Mater. 17,411 (2018).[65] V. Dobrosavljevic, N. Trivedi, and J. M. Valles, Jr.,Conductor-Insulator Quantum Phase Transitions (OxfordUniversity, New York, 2012).[66] T. Senthil, Theory of a Continuous Mott Transition in TwoDimensions, Phys. Rev. B 78, 045109 (2008).[67] S. Acheche, L.-F. Arsenault, and A. M. S. Tremblay,OrbitalEffect of the Magnetic Field in Dynamical Mean-FieldTheory, Phys. Rev. B 96, 235135 (2017).[68] F. Werner, O. Parcollet, A. Georges, and S. R. Hassan,Interaction-Induced Adiabatic Cooling and Antiferromag-netism of Cold Fermions in Optical Lattices, Phys. Rev.Lett. 95, 056401 (2005).[69] A. Kumar, N. C. Hu, A. H. MacDonald, and A. C. Potter,Gate-Tunable Heavy Fermion Quantum Criticality in aMoiré Kondo Lattice, Phys. Rev. B 106, L041116 (2022).[70] A. Dalal and J. Ruhman, Orbitally Selective Mott Phase inElectron-Doped Twisted Transition Metal-Dichalcoge-nides: A Possible Realization of the Kondo Lattice Model,Phys. Rev. Res. 3, 043173 (2021).MINGJIE ZHANG et al. PHYS. REV. X 12, 041015 (2022)041015-10https://doi.org/10.1038/s41563-018-0036-2https://doi.org/10.1038/s41563-018-0036-2https://doi.org/10.1103/PhysRevB.78.045109https://doi.org/10.1103/PhysRevB.96.235135https://doi.org/10.1103/PhysRevLett.95.056401https://doi.org/10.1103/PhysRevLett.95.056401https://doi.org/10.1103/PhysRevB.106.L041116https://doi.org/10.1103/PhysRevResearch.3.043173