Quantum Phase Diagram of a Moiré-Hubbard Model
QQuantum Phase Diagram of a Moir´e-Hubbard Model
Haining Pan, Fengcheng Wu, and Sankar Das Sarma
Condensed Matter Theory Center and Joint Quantum Institute,Department of Physics, University of Maryland, College Park, Maryland 20742, USA
We theoretically study a generalized Hubbard model on moir´e superlattices of twisted bilayers, andfind very rich filling-factor-dependent quantum phase diagrams tuned by interaction strength andtwist angle. Strong long-range Coulomb interaction in the moir´e-Hubbard model induces Wignercrystals at a series of fractional filling factors. The effective lattice of the Wigner crystal is controlledby the filling factor, and can be triangle, rectangle, honeycomb, kagome, etc, providing a singleplatform to realize many different spin models on various lattices by simply tuning carrier density.In addition to Wigner crystals that are topologically trivial, interaction-induced Chern insulatorsemerge in the phase diagram. This finding paves a way for engineering interaction-induced quantumanomalous Hall effect in moir´e-Hubbard systems where the corresponding single-particle moir´e bandis topologically trivial.
Introduction. — Twisted bilayers with a long-period moir´e pattern provide versatile platforms to studystrongly correlated physics, as many-body interactionsare effectively enhanced in narrow moir´e bands. It hasbeen theoretically proposed that a generalized Hubbardmodel can be simulated in twisted bilayers based ongroup-VI transition metal dichalcogenides (TMDs) [1, 2],which have fewer low-energy degrees of freedom com-pared to twisted bilayer graphene [3–5] and therefore,allow quantum simulations of model Hamiltonians. Re-cent experiments [6–11] performed using a variety of tech-niques on twisted bilayer TMDs found compelling ev-idence of correlated insulators (CIs) not only at integerfilling factors (i.e., one electron or hole per moir´e cell) butalso at a series of fractional filling factors. The CIs at theinteger filling factors are driven primarily by the on-siterepulsion in the Hubbard model, while those at factionalfilling factors are interpreted as generalized Wigner crys-tals [6, 9–11] induced by the long-range Coulomb repul-sion. The observed abundant correlated insulating statesin twisted bilayer TMDs call for thorough theoretical in-vestigations of this intriguing 2D Moir´e-Hubbard system.In this Letter, we theoretically study a generalizedHubbard model on triangular moir´e lattice realized intwisted bilayer TMDs. We show that the quantum phasediagram at a given fractional filling factor contains a richset of competing phases that can be tuned by the twistangle θ and the dielectric environment. We also find thatthe phase diagram depends nontrvially on the filling fac-tor. When interaction is much greater than the kineticenergy, Wigner crystals generally form to minimize thelong-range Coulomb interaction. The effective lattices ofWigner crystals depend sensitively on the filling factor,and can be triangle, rectangle, honeycomb, kagome, etc.After the electron spin degree of freedom is taken intoaccount, spin models on distinct lattices can be simu-lated in this system by simply tuning the carrier density,leading to a variety of charge and spin ordered phases.In competition with these states derived from Wignercrystals, interaction-induced Chern insulators also ap- ( a ) ( b )( c ) FIG. 1. (a) The effective triangular lattice formed in the moir´epattern. (b) The single-particle moir´e band ε s ( k ) of Eq. (1)at θ = 3 ◦ , where s can be ↑ or ↓ . The dashed line marks thecontour at the van Hove energy. (c) The correlated insulatinggap at representative rational filling factors ν . pear in the phase diagram, which is remarkable since thenon-interacting band structure in the model is topologi-cally trivial. Here Chern insulators arise spontaneouslyfrom effective fluxes that are spontaneously generated ei-ther by nontrivial spin texture or by interaction-inducedcomplex hopping phases. We elaborate our results bypresenting calculated rich quantum phase diagrams atrepresentative fractional filling factors, and discuss theirexperimental implications. Model. — We study a moir´e-Hubbard model defined as a r X i v : . [ c ond - m a t . s t r- e l ] A ug follows H = (cid:88) s (cid:88) i,j t s ( R i − R j ) c † i,s c j,s + 12 (cid:88) s,s (cid:48) (cid:88) i,j U ( R i − R j ) c † i,s c † j,s (cid:48) c j,s (cid:48) c i,s , (1)where R i represents the position of site i in a triangu-lar lattice formed in the moir´e pattern [Fig. 1(a)], s isthe spin index, t and U are, respectively, the hoppingparameter and the interaction strength. As proposed inRefs. [1, 2, 12], the model in Eq. (1) can be simulated intwisted TMD heterobilayers as well as homobilayers. Fordefiniteness, we use twisted homobilayer WSe (tWSe )as the model system in this work, and Eq. (1) is then con-structed following our previous work [12] for low-energyholes in the first moir´e valence band at ± K valleys. Herewe use c † i,s to represent the hole operator, and s = ↑ and ↓ are locked to + K and − K valleys respectively. We definea filling factor ν as (1 / N ) (cid:80) i,s c † i,s c i,s , which counts thenumber of holes per moir´e cell ( N is the total number ofmoir´e sites in the system). The charge neutrality point ofthe semiconducting twisted bilayer corresponds to ν = 0.For simplicity, we assume that no external out-of-planedisplacement field is applied to WSe , and then the modelin Eq. (1) respects emergent spin SU(2) symmetry and C point group symmetry. An important advantage ofthe moir´e platform is that both the hopping parametersand the interaction strength are highly tunable. Gener-ally speaking, the moir´e bandwidth becomes narrower atsmaller twist angle (larger moir´e period) and many-bodyinteraction effects become more prominent [1, 2, 13]. Weshow the twist-angle dependence of t and U in the supple-mental material [14]. In the calculation of U , we projecta screened Coulomb interaction ( e /(cid:15) )(1 /r − / √ r + d )to the low-energy moir´e states, where (cid:15) is the backgrounddielectric constant that is tunable by the dielectric envi-ronment and d/ (cid:15) as a free pa-rameter and d , which is also experimentally controllable,to be 60 nm in calculations.We perform self-consistent mean-field (MF) Hartree-Fock studies of the moir´e Hubbard model at representa-tive filling factors with a variety of initial ansatze thatrange from Wigner crystals (which can be derived fromthe classical Coulomb model [14]) to topological states.At a given fractional filling factor, we generally find mul-tiple solutions to the Hartree-Fock equation, and theirenergetic competitions give rise to rich quantum phasediagrams. An overview of our results is illustrated inFig. 1(c) showing the interaction-induced gap E G at ra-tional ν with a denominator up to 4. In our theory, the CIat the integer filling ν = 1 is a Mott insulator, and its gapis primarily determined by the on-site repulsion [12]. CIsat fractional fillings often require the presence of off-siterepulsion and generally have smaller charge gaps. The ϵ θ - Metallic1 Tetrahedron Kagome breakingKagome breakingNormal state ( a ) A A AA A AA A AB B BB B BC CC CC CD DD D ( b ) AF:
A B C DA B C D
FM:Tetrahedron:
A BCD
Kagome breaking ( c ) Kagome breaking ( d ) FIG. 2. (a) The quantum phase diagram at ν = 1 / θ and (cid:15) . Some phases are illustrated in (b)-(d). (b)In the AF and FM phases, A and B sublattices are dominantlyoccupied, while C and D sublattices are less occupied. In theAF phase, spin polarization is antiparallel on A and B , butvanishes on C and D . In the FM phase, all sites have parallelspin polarization but different densities. In the tetrahedronphase, the four sublattices have equal density but differentspin orientations that extend a solid angle of 4 π . (c) and (d)show the kagome phases with C z and T symmetry breaking,respectively. relative trend of our calculated E G in Fig. 1 as a func-tion of ν agrees well with a recent experiment in Ref. 9,which provides confidence in the validity of our theory. ν = 1 / ν = 1 / θ and (cid:15) . When interaction is strong(small (cid:15) ), a Wigner crystal with a stripe charge den-sity wave (CDW) forms [Fig. 2(b)], and hosts a coupled-chain spin Heisenberg model. Our MF results show thatthe Heisenberg model has an antiferromagnetic (AF) ex-change coupling, as an AF phase has a lower energy com-pared to the ferromagnetic (FM) phase for small (cid:15) . Wheninteraction decreases by increasing (cid:15) , the stripe CDWgradually weakens and the FM phase becomes energeti-cally more favorable. Therefore, charge and spin order-ings are closely related. By further decreasing the inter-action strength, CDW can completely disappear but theFM ordering can remain, which leads to a FM metallicphase.In addition to these relatively simple charge- and spin-ordered phases, we also find three more exotic phasesin Fig. 2(a): one tetrahedron phase and two kagomephases. In the tetrahedron phase, there is spin order-ing but no charge ordering. The spin texture on the fourmagnetic sublattices forms a tetrahedron, which leads toa real-space Berry flux of π for electronic motion alongeach triangular plaquette. We numerically verify thatthe tetrahedron phase is a Chern insulator with a Chernnumber of |C| = 1. This phase arises because our non- ϵ θ AF - Triangle AF - Honeycomb - MetallicFM - MetallicNormal state ( a ) AF - Triangle ( b ) AF - Honeycomb - Metallic ( c ) FIG. 3. (a) The quantum phase diagram at ν = 1 /
3. Two ofthe phases are illustrated in (b) and (c). interacting moir´e band at ν = 1 / ∼ / C z symmetry or time-reversal T symmetry.In the C z -breaking kagome phase, the interaction-renormalized effective hopping parameters from a siteto its nearest neighbors on opposite directions becomedifferent but remain real [Fig. 2(c)], which leads to a va-lence bond solid insulator that is topologically trivial.In the other phase with T breaking, the effective hop-ping parameters acquire complex phases with a patternshown in Fig. 2(d). This T -breaking kagome phase withspontaneously-induced fluxes of φ in the triangles and − φ in the hexagons is analogous to the Haldane modelon honeycomb lattice [16], and is a Chern insulator with |C| = 1 [14]. The topological kagome phase arising froma generalized Hubbard model on a triangular lattice hasnot been reported previously and provides a new mech-anism to realize quantum anomalous Hall effect in a re-alistic experimental system. ν = 1 / . — In the quantum phase diagram at ν = 1 / √ × √ (cid:15) , and 120 ◦ AF orderwith a 3 × × ϵ θ AF FMFM - MetallicNormal state ( a ) A AA AA A AABB B BB BB BC C CC C CC C C ( b ) AF:FM:
A B CA B C
FIG. 4. (a) The quantum phase diagram at ν = 2 /
3. (b) AFand FM phases on an effective honeycomb lattice. with dominant occupancy form an effective honeycomblattice and host collinear AF ordering. ν = 2 / . — The Wigner crystal at ν = 2 / ν = 1 /
3, and forms a honeycomb lattice [Fig. 4],where spins develop collinear AF order in the strong in-teraction limit as expected from an effective Heisenbergmodel. By decreasing interaction, there is a transitionfrom AF to FM spin orderings with the same √ × √ ν = 1 / /
3, but they are not energeti-cally favorable within our explored parameter space [14]. ν = 1 / . — At ν = 1 /
4, there are two types of Wignercrystals: (1) a 2 × × √ (cid:15) and large θ . In both phases, the effective spin exchange interactionis weak because of the large separation (small hopping)between the primarily occupied sites, and therefore, AFand FM spin orderings closely compete in energy. Wealso find a Chern insulator state at ν = 1 / ν = 1 / T symmetrybreaking, but it is energetically unfavorable [14]. ν = 3 / . —We find 7 symmetry-breaking phases in thephase diagram at ν = 3 /
4, as shown in Fig. 6(a). For (cid:15) <
5, we find two types of Wigner crystals, (1) a kagomelattice [Fig. 6(b)] for θ < . ◦ , and (2) an anti-stripe lat-tice [Fig. 6(d)] for θ > . ◦ , which are, respectively, dualto the 2 × × √ ν = 1 /
4. We find that AF spin ordering has lower en-ergy compared to FM spin ordering on both the kagomeand anti-stripe lattices for (cid:15) <
5. It is important tonote that both lattices with AF spin exchange couplingsare frustrated and can host a large number of degener-ate classical magnetic ground states, which could lead toquantum spin liquid states when quantum fluctuations inthe spin sector are taken into account.For (cid:15) >
5, we find a FM phase on the kagome lattice ,and the associated CDW gradually melts as (cid:15) increases, ϵ θ AF - Triangle AF - StripeFM - StripeFM - Triangle Normal state ( a ) AF - Triangle ( b ) FM - Triangle ( c ) AF - Stripe ( d ) FM - Stripe ( e ) FIG. 5. (a) The quantum phase diagram at ν = 1 /
4. (b)120 ◦ AF and (c) FM spin structures on the 2 × × √ and finally vanishes, leading to a FM 1 × × ◦ AF phase that has only spin density wave but noCDW, as illustrated in Fig. 6(e).Finally, we find two collinear AF phases that are de-rived from the kagome phases at ν = 1 /
2. Noting that3 / / /
2, we can construct collinear AF phaseswith effective filling factors of 1 / ↑ sector and1 / ↓ sector. Spin ↑ and ↓ states, respectively,occupy sites on kagome and triangular lattices that aredual to each other. On the kagome lattice formed by spin ↓ states, C z or T symmetry can be further broken, asin the case of ν = 1 /
2, leading to the two AF phasesillustrated in Figs. 6(f) and 6(g) that are respectivelytopologically trivial and nontrivial [14].
Discussions. — Our MF results should be taken to bequalitative instead of quantitative, as Hartree-Fock the-ory generally overestimates the tendency towards order-ing. However, the advantage of MF theory is that it al-lows construction of a very large family of possible groundstate candidate phases. We envision that more sophisti-cated numerical approaches can be applied to the moir´eHubbard model, which could verify intriguing phasessuch as Chern insulators predicted by our theory and un-veil more exotic phases, for example, spin liquid states onthe effective kagome lattice at ν = 3 /
4, but such numer-ical methods are extremely computationally demandingand therefore, detailed results as functions of filling fac-tors, interaction strength, and twist angle as provided inour work are challenging. It is useful to mention here forcomparison that the MF theory applied on the standard2D minimal square-lattice on-site Hubbard model onlyfinds three phases (AF, FM, and paramagnet) as func-tions of interaction and filling [17]. Due to space limit, we ϵ θ ( a ) ( b )( c ) ( f )( g )( d )( e ) FIG. 6. (a) The quantum phase diagram at ν = 3 /
4. (b) AFand (c) FM spin structures on a kagome Wigner crystal. (d)AF spin structure on an anti-stripe Wigner crystal. The AFspin structures shown in (b) and (d) are mean-field results,and may not be the actual ground states because of fluctu-ations. (e) 120 ◦ AF state without charge density wave. (f)and (g) Sites with spin up (down) polarization form triangular(kagome) lattice. In (f), C z symmetry is spontaneously bro-ken, which leads to a valence band solid insulator. In (g), T symmetry is spontaneously broken due to interaction-inducedeffective flux, which leads to a Chern insulator. only present phase diagrams at rational ν with a denom-inator up to 4, but we do also find correlated insulatorsat other fractional filling factors.The predicted rich phase diagrams can lead to very richexperimental phenomena, because different phases can beaccessed by tuning experimentally controllable parame-ters (e.g., θ and (cid:15) ). Current experiments [6–11] were allperformed using hexagonal boron nitride as encapsulat-ing material. The corresponding dielectric constant (cid:15) isabout 5 −
10. For this range of (cid:15) , our calculations showthat ground states at the fractional filling factors areWigner crystals. The effective lattice of Wigner crystalscan spontaneously break threefold rotational symmetry,particularly in stripe phases at ν = 1 / /
4, whichcan be probed optically using linear dichroism [10]. Torealize the predicted Chern insulators at ν = 1 / / (cid:15) >
10) is desirable, which canbe engineered by changing the dielectric environment, forexamples, using an encapsulating material with a higherdielectric constant and reducing the distance from thesample to the metallic gate. Experimental observationof such interaction-induced Chern insulators in a sys-tem with topologically trivial single-particle bands wouldgreatly enhance the scope of quantum anomalous Hall ef-fect.This work is supported by the Laboratory for PhysicalSciences. [1] F. Wu, T. Lovorn, E. Tutuc, and A. H. MacDonald,Phys. Rev. Lett. , 026402 (2018).[2] F. Wu, T. Lovorn, E. Tutuc, I. Martin, and A. H. Mac-Donald, Phys. Rev. Lett. , 086402 (2019).[3] R. Bistritzer and A. H. MacDonald, PNAS , 12233(2011).[4] Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi,E. Kaxiras, and P. Jarillo-Herrero, Nature , 43(2018).[5] Y. Cao, V. Fatemi, A. Demir, S. Fang, S. L. Tomarken,J. Y. Luo, J. D. Sanchez-Yamagishi, K. Watanabe,T. Taniguchi, E. Kaxiras, R. C. Ashoori, and P. Jarillo-Herrero, Nature , 80 (2018).[6] E. C. Regan, D. Wang, C. Jin, M. I. Bakti Utama,B. Gao, X. Wei, S. Zhao, W. Zhao, Z. Zhang, K. Yu-migeta, M. Blei, J. D. Carlstr¨om, K. Watanabe,T. Taniguchi, S. Tongay, M. Crommie, A. Zettl, andF. Wang, Nature , 359 (2020).[7] Y. Tang, L. Li, T. Li, Y. Xu, S. Liu, K. Barmak,K. Watanabe, T. Taniguchi, A. H. MacDonald, J. Shan,and K. F. Mak, Nature , 353 (2020).[8] L. Wang, E.-M. Shih, A. Ghiotto, L. Xian, D. A. Rhodes,C. Tan, M. Claassen, D. M. Kennes, Y. Bai, B. Kim,K. Watanabe, T. Taniguchi, X. Zhu, J. Hone, A. Rubio,A. Pasupathy, and C. R. Dean, arXiv:1910.12147 [cond-mat] (2019).[9] Y. Xu, S. Liu, D. A. Rhodes, K. Watanabe, T. Taniguchi,J. Hone, V. Elser, K. F. Mak, and J. Shan,arXiv:2007.11128 [cond-mat] (2020).[10] C. Jin, Z. Tao, T. Li, Y. Xu, Y. Tang, J. Zhu, S. Liu,K. Watanabe, T. Taniguchi, J. C. Hone, L. Fu, J. Shan,and K. F. Mak, arXiv:2007.12068 [cond-mat] (2020).[11] X. Huang, T. Wang, S. Miao, C. Wang, Z. Li, Z. Lian,T. Taniguchi, K. Watanabe, S. Okamoto, D. Xiao, S.-F.Shi, and Y.-T. Cui, arXiv:2007.11155 [cond-mat] (2020).[12] H. Pan, F. Wu, and S. Das Sarma, Phys. Rev. Research , 033087 (2020).[13] M. H. Naik and M. Jain, Phys. Rev. Lett. , 266401(2018).[14] see supplemental material for calculation details..[15] I. Martin and C. D. Batista, Phys. Rev. Lett. , 156402(2008).[16] F. D. M. Haldane, Phys. Rev. Lett. , 2015 (1988).[17] J. E. Hirsch, Phys. Rev. B , 4403 (1985).[18] T. Fukui, Y. Hatsugai, and H. Suzuki, J. Phys. Soc. Jpn. , 1674 (2005).[19] R. Yu, X. L. Qi, A. Bernevig, Z. Fang, and X. Dai, Phys.Rev. B , 075119 (2011).[20] S. Raghu, X.-L. Qi, C. Honerkamp, and S.-C. Zhang,Phys. Rev. Lett. , 156401 (2008). Supplemental Materials for “Quantum Phase Diagram of a Moir´e-Hubbard Model”
MOIR´E HAMILTONIAN
The methodology to calculate moir´e band structure forvalence band states in twisted bilayer WSe (tWSe ) isgiven in Refs. 2 and 12. Here, we briefly provide thenuermical details underlying our calculations. The moir´eHamiltonian for valence states in tWSe at + K valley is H ↑ = (cid:32) − (cid:126) ( k − κ + ) m ∗ + ∆ + ( r ) ∆ T ( r )∆ † T ( r ) − (cid:126) ( k − κ − ) m ∗ + ∆ − ( r ) (cid:33) , (S1)where m ∗ = 0 . m is the valence band effective mass( m is the rest mass of electron). The layer-dependentmomentum offset κ ± = [4 π/ (3 a M )]( −√ / , ∓ /
2) cap-ture the rotation in the momentum space, where a M = a /θ is the moir´e lattice constant and a = 3 . . Here ∆ ± ( r ) is thelayer dependent moir´e potential∆ ± ( r ) = 2 V (cid:88) j =1 , , cos( b j · r ± ψ ) , (S2)where b = [4 π/ ( √ a M )](1 ,
0) and b j with j =2 to 6 arerelated to b by ( j − π/ V and ψ char-acterize the amplitude and spatial pattern of the moir´epotential. The interlayer tunneling ∆ T ( r ) is∆ T ( r ) = w (1 + e − i b · r + e − i b · r ) , (S3)where w quantifies the interlayer tunneling strength. Inthis calculation, we choose a set of phenomenological pa-rameters at which the topmost moir´e valence band istopologically trivial: ( V, ψ, w ) =(4.4 meV, 5.9, 20 meV).We diagonalize the moir´e Hamiltonian (S1) using theplane-wave expansion based on Bloch’s theorem, andthen construct a generalized Hubbard model for the topo-logically trivial topmost valence band, which resides onan effective triangular lattice. To calculate the hoppingenergy t and Coulomb interaction U in the generalizedHubbard model, we first construct the Wannier functionand choose the gauge which ensure the bottom-layer com-ponent of the Bloch wave function at each momentum tobe real and positive at the origin in the real space. Weshift the band structures to the vicinity of zero energy bydropping the onsite energy t , and flip the sign of the hop-ping parameters t compared to those reported in Ref. [12]since here our generalized Hubbard model is constructedusing the hole operator while the moir´e Hamiltonian (S1)describes electron. The band structure and the densityof states at θ = 3 ◦ are shown in Fig. S1, where the vanHove singularity is roughly at ν = .Figure S2 shows hopping t and Coulomb interaction U up to the first three neighbors, where the hopping t increases exponentially as twist angle θ increases (moir´e ( a ) ( b ) FIG. S1. (a) The band structure and (b) density of states ofthe topmost valence moir´e band at θ = 3 ◦ . lattice constant a M decreases) and U increases approxi-mately linearly. In the numerical calculation, the threenearest neighbors are considered in the hopping term,while remote Coulomb interactions U up to 100 hexag-onal shells are considered to guarantee the convergence.To calculate U , we project the screened Coulomb inter-action V ( r ) = ( e /(cid:15) )(1 /r − / √ r + d ) onto Wannierstates. As shown in Fig. S2, interaction U ( R ) can beapproximated by V ( R ) for large R as expected. COULOMB MODEL
The Wigner crystal as the ansatz for the Hubbardmodel is derived from a zero-temperature Coulombmodel with only the potential term in the Hubbardmodel, H Coulomb = 12 (cid:88) s,s (cid:48) (cid:88) i,j U ( R i − R j ) n i,s n j,s (cid:48) , (S4)where n i,s is the binary occupancy number of site i . Wechoose proper supercells manually and minimize the totalCoulomb energy per site by exploring various arrange-ment of occupied sites.Table I lists all the possible Wigner crystals we findto be existing in the quantum phase diagram at differ-ent filling factors and also shows the minimal short-rangeinteractions required to open a finite gap at such fillingfactors as well as the value of finite gap and energy persite correspondingly. While Table I presents analyticalresults for minimal interactions required for Wigner crys-tals, our numerical calculations include interaction U upto 100 hexagonal shells. ( a ) ( b ) ( c ) ( d ) FIG. S2. (a) and (b) | t | n and (cid:15)U n as a function of twistangle θ . (cid:15) is the effective dielectric constant. (c) U ( R ) can beapproximated to V ( R ) with a good accuracy. Here θ = 3 ◦ . (d) (cid:15)U ( q ) along Γ to K in one BZ at θ = 3 ◦ HUBBARD MODEL AND MEAN-FIELDTHEORY
The Hubbard model of Eq. (1) in the main text iscomposed of two terms: kinetic energy H and interac-tion energy H . We perform the Fourier transformationof the Hamiltonian H in the real space to the momentumspace. Therefore, the kinetic term H becomes H = (cid:88) s (cid:88) k ε s ( k ) c † k ,s c k ,s , (S5)where k is summed over the first Brillouin zone ( BZ )of the moir´e lattice, and ε s ( k ) is the non-interacting band energy dispersion calculated from the tight-binding TABLE I. Analytical results for Wigner crystals ν Wigner crystal Least U n Gap Energy1 / U , U min( U , U ) U / U , U min( U , U ) 02 / U , U min( U , U ) U / U , U min( U , U ) 01 / U , U min( U , U ) 03 / U , U min( U , U ) U / U , U min( U − U , U ) U U U model for spin s . The interaction term H in the mo-mentum space is H = 12 N (cid:88) s,s (cid:48) (cid:88) k U ( k α − k δ ) δ k α , k β , k γ , k δ c † k α ,s c † k β ,s (cid:48) c k γ ,s (cid:48) c k δ ,s , (S6)where N is the number of total sites in the lattice,and k α , k β , k γ , k δ are summed over the first BZ . Here,the interaction in the momentum space (as shown inFig. S2(d)) is U ( q ) = (cid:88) R U ( R ) e i q · R , (S7)and δ k α , k β , k γ , k δ = (cid:88) G δ ( k α + k β − k γ − k δ , G ) , (S8)where G is any moir´e reciprocal lattice vector, and δ ( . . . )is the Kronecker delta function.Using the Hartree-Fock truncation, we obtain themean-field Hamiltonian for the interaction term H int = 1 N (cid:88) s,s (cid:48) (cid:88) k U ( k α − k δ ) δ k α , k β , k γ , k δ (cid:104)(cid:68) c † k α ,s c k δ ,s (cid:69) c † k β ,s (cid:48) c k γ ,s (cid:48) − (cid:68) c † k α ,s c k γ ,s (cid:48) (cid:69) c † k β ,s (cid:48) c k δ ,s (cid:105) (S9)The Hartree-Fock state can spontaneously break thediscrete translational symmetry, and resulting unit cellcan be multiple times of the moir´e unit cell, which causesthe Brillouin zone ( (cid:122) ) to be smaller than the moir´e Bril-louin zone ( BZ ). Therefore, BZ of moir´e lattice can betessellated by multiple (cid:122) s with appropriate shift vectors Q . (See Fig. S3 for example). Therefore, we can dis- assemble the summation over the whole BZ into aggre-gates of several smaller (cid:122) s with the shifting vectors, i.e.,rewrite k = q + p , where q ∈ { Q } and p is a goodquantum number lying in the smaller (cid:122) . Thus, Eq. (S5)becomes H = (cid:88) s (cid:88) p , q ε s ( p + q ) c † p + q ,s c p + q ,s , (S10) FIG. S3. An example of (cid:122) (blue) for the tetrahedron state at ν = 1 / BZ of moir´e lattice (brown). Thelarger BZ is tessellated by the smaller (cid:122) with four shift vec-tors: Q = (0 , , Q = b M , Q = b M , Q = b M + b M where { Q } is the set of all shifting vectors, the numberof Q equals to the number of sites contained in one unitcell of the symmetry-breaking states.Therefore, the mean-field Hamiltonian becomes H HF = H + H Hartree + H Fock , (S11)where the Hartree term is H Hartree = 1 N (cid:88) s,s (cid:48) (cid:88) p , q U ( q α − q δ ) δ q α , q β , q γ , q δ (cid:68) c † p α + q α ,s c p α + q δ ,s (cid:69) c † p β + q β ,s (cid:48) c p β + q γ ,s (cid:48) (S12)and the Fork term is H Fock = − N (cid:88) s,s (cid:48) (cid:88) p , q U ( p α − p β + q α − q δ ) δ q α , q β , q γ , q δ (cid:68) c † p α + q α ,s c p α + q γ ,s (cid:48) (cid:69) c † p β + q β ,s (cid:48) c p β + q δ ,s . (S13)Here the expected value (cid:104) . . . (cid:105) is taken over all occupiedstates. We choose an initial ansatz for the Hartree-Fockstate and substitute it into the H HF . After diagonal- izating the H HF , we find the energies and wavefunctions,which are fed into the mean-field Hamiltonian H HF againto find a self-consistent state iteratively. The convergencecriterion is the total energy per site, which is defined as (cid:104) H (cid:105)N = 1 N (cid:88) s (cid:88) p , q ε s ( p + q ) (cid:68) c † p + q ,s c p + q ,s (cid:69) (S14)+ 12 N (cid:88) s,s (cid:48) (cid:88) p , q U ( q α − q δ ) δ q α , q β , q γ , q δ (cid:68) c † p α + q α ,s c p α + q δ ,s (cid:69) (cid:68) c † p β + q β ,s (cid:48) c p β + q γ ,s (cid:48) (cid:69) (S15) − N (cid:88) s,s (cid:48) (cid:88) p , q U ( p α − p β + q α − q δ ) δ q α , q β , q γ , q δ (cid:68) c † p α + q α ,s c p α + q γ ,s (cid:48) (cid:69) (cid:68) c † p β + q β ,s (cid:48) c p β + q δ ,s (cid:69) . (S16) ORDER PARAMETER OF THE WIGNERCRYSTAL
We define the site-resolved density as: (cid:104) n i (cid:105) = (cid:68) c † i, ↑ c i, ↑ + c † i, ↓ c i, ↓ (cid:69) , (S17) where n is the average number density at site i in oneunit cell. The order parameter of Wigner crystal is thus FIG. S4. A line cut of order parameters of Wigner crystal at θ = 3 ◦ and ν = 1 / defined as η = min i n i max i n i . (S18) η → η = 1means no charge ordering in the moir´e lattice. We presenta line cut of order parameters η as a function of the back-ground dielectric constant (cid:15) at θ = 3 ◦ and ν = 1 / η is small. At larger (cid:15) , η = 1 indicates the Wignercrystal disappears in the FM-metallic phase— each siteis evenly occupied by half holes. The FM-metallic phaseis thus a spin-polarized metal. THE ENERGY DIFFERENCE BETWEEN AFAND FM AT ν = 1 / Figure S5 shows the energy of AF-Triangle (blue)/ AF-Stripe (yellow)/ FM-Stripe (orange) relative to that ofFM-Triangle at ν = 1 /
4. In the main text, we find thecompetition of AF and FM is different in the phase dia-gram of ν = compared to other fractional ν , which weattribute to the larger site-to-site distance of the Wignercrystal and thus smaller exchange energy. Indeed, theenergy of AF and FM in the phase diagram of ν = 1 / EFFECTIVE KAGOME LATTICE AT ν = 1 / , / ,AND / : C z BREAKING VS T BREAKING
Figure S6(a) and (b) show the interaction-renormalized band structure in the kagome phasedue to the Coulomb repulsion at ν = 1 /
2, where the FM - StripeAF - StripeAF - Triangle
FIG. S5. The energy per site of three phases— AF-Triangle,AF-Stripe, and FM-Stripe— relative to that of FM-triangleat ν = and θ = 4 ◦ . Dirac cone is opened at the corner of (cid:122) due to thebreaking of C z symmetry and T symmetry respectively.The bottom two bands are occupied and fully polarizedwith spin ↑ . The Fermi energy is labeled by E F .To obtain these kagome phases, we introduce ancillaryHamiltonians. When C z symmetry breaks, we can con-struct spinless effective tight-binding model for valencebond solid insulator on the kagome lattice including onlythe nearest-neighbor hoppings, H C z = (cid:88) (cid:104) i,j (cid:105)∈{ = } tc † i c j + (cid:88) (cid:104) i,j (cid:105)∈{−} pc † i c j , (S19)where the nearest-neighbor pairs (cid:104) i, j (cid:105) are summed oversingle bonds {−} with hopping p and double bonds { = } with hopping t as shown in Fig. 2(c) in the main text,The band structure can be obtained by transforming theancillary Hamiltonian into the momentum space, i.e., H C z ( k ) = te i k ·−−→ AB + pe − i k ·−−→ AB te i k ·−→ AC + pe − i k ·−→ AC te − i k ·−−→ AB + pe i k ·−−→ AB te i k ·−−→ BC + pe − i k ·−−→ BC te − i k ·−→ AC + pe i k ·−→ AC te − i k ·−−→ BC + pe i k ·−−→ BC , (S20)0 FIG. S6. The interaction-renormalized band structure of theeffective kagome lattice with (a) C z -breaking and (b) T -breaking for θ = 3 ◦ and (cid:15) = 35 at ν = 1 / ↑ and ↓ label thespin polarization of each band, and E F indicates the Fermienergy. ( a ) π π W C ( b ) π π W C FIG. S7. Wannier center flow along b m for the topologi-cally trivial C -breaking (a) and topologically nontrivial T -breaking (b) for θ = 3 ◦ and (cid:15) = 35 at ν = 1 / where site A, B, C are defined in Fig. 2(a), and −−→
AB, −→ AC, −−→ BC are all defined on the double bond plaquetteas shown in Fig. 2(c) in the main text. We diagonalizethe ancillary Hamiltonian (S20) to obtain the wavefunc-tion, which will be used as the initial ansatz before thefirst iteration.We calculate the Chern number of all the occupiedbands [18] in Fig. S6(a) and find |C| = 0. We alsoshow Wannier center (WC) flow [19] along one recip-rocal vector in Fig. S7, which also has zero winding.The Wannier center is defined here as the phases ofeigenvalues of a Wilson loop along a closed path L ,i.e., arg(exp (cid:0) i (cid:72) L A ( k ) dk (cid:1) ) , where A ( k ) is the non-Abelianberry connection. Here we choose the closed path L along b m and plot the Wannier center flow along the directionof b m .When T symmetry breaks, we can construct a spin-less model for the Chern insulator by imposing complexhoppings on the nearest neighbors, H T = (cid:88) (cid:104) i,j (cid:105) te iφ/ v i,j c † i c j , (S21)where v i,j = 1 (-1) if the hopping from j to i is counter-clockwise (clockwise) in the triangular plaquette in thekagome lattice, and φ is the nonzero effective flux. Tofind the band structure in the momentum space, we per-form the Fourier transformation and obtain H T ( k ) = te iφ/ cos (cid:16) k · −−→ AB (cid:17) te − iφ/ cos (cid:16) k · −→ AC (cid:17) te − iφ/ cos (cid:16) k · −−→ AB (cid:17) te iφ/ cos (cid:16) k · −−→ BC (cid:17) te iφ/ cos (cid:16) k · −→ AC (cid:17) te − iφ/ cos (cid:16) k · −−→ BC (cid:17) , (S22)where site A, B, C are defined in Fig. 2(a) in the maintext, and −−→
AB, −→ AC, −−→ BC are defined on the triangles point-ing to the right in Fig. 2(d) in the main text. The Diraccones at (cid:122) corners are gapped out as long as φ (cid:54) = nπ ,where n ∈ Z . Therefore, we choose φ = π/ b m as shown in Fig. S7(b).Similarly, at ν = 1 /
4, there are also two kinds ofkagome lattice with C z symmetry breaking and T sym- metry breaking. Figure S8(a) shows the interaction-renormalized band structures for the T symmetry break-ing and Fig. S9(a) shows its topologically nontrivial Wan-nier center flow at ν = 1 /
4. These kagome phases at ν = 1 / ν = 3 / ↓ states occupy a kagomelattice, while spin ↑ states occupy a triangular lattice.The corresponding band structure is shown in Fig. S8(b),where band structures derived from kagome (spin ↓ ) andtriangular (spin ↑ ) lattices can be identified. The Wan-nier center flow shown in Fig. 9(b) confirms that thestate has a Chern number of |C| = 1.1 FIG. S8. The interaction-renormalized band structure of theeffective kagome lattice with T breaking at (a) ν = 1 / ν = 3 / θ = 3 ◦ and (cid:15) = 35. ( a ) π π W C ( b ) π π W C FIG. S9. Wannier center flow along b m for the T symmetrybreaking case at (a) ν = 1 / ν = 3 / θ = 3 ◦ and (cid:15) = 35. EFFECTIVE HONEYCOMB LATTICE AT ν = 1 / AND / : HALDANE MODEL At ν = 1 / ν = 2 /
3, we can also construct topo-logically nontrivial states, although we find them to beenergetically unfavorable. We derive the initial ansatzfrom the Haldane model [16, 20] by introducing a nonzerophase on the next-nearest-neighbors of the honeycomb asshown in Fig. S10. For example at ν = 1 /
3, the hoppingsbetween the neighboring sites A ( B ) following the blue(red) arrows are t e iφ ( t is real); the hoppings betweenthe nearest A and B sites are the real t . The corre-sponding ancillary Hamiltonian in the momentum spaceis H HC ( k ) = t (cid:80) b cos( k · b − φ ) t (cid:80) a e i k · a t (cid:80) a e − i k · a t (cid:80) b cos( k · b + φ ) , (S23)where three a connect the three pairs of the nearest-neighbors −−→ AB , and three b connecting the next-nearestneighbors are defined as b = a − a , b = a − a , and b = a − a .We diagonalize Eq. (S23) and use wavefunction as theinitial ansatz of the Hubbard model. Figures S11(a) A AA AA A AABB B BB BB BC C CC C CC C C
A B CA B C
FIG. S10. Two Topological states at ν = 1 / ν = 2 / T breaking at (a) ν = 1 / ν = 2 / θ = 3 ◦ and (cid:15) = 35. and S12(a) show the interaction-renormalized bandstruc-tures and the corresponding nontrivial Wannier centerflow at ν = 1 / ν = 2 /
3, we can also construct a topological stateinspired by 2 / / /
3, where sites
A, B host spin ↑ with half occupancy and site C hosts spin ↓ with unity oc-cupancy. Therefore, sites A, B form a honeycomb latticeof Haldane model and sites C form a triangular lattice.Figure S11(b) shows the interaction-renormalized band-structure where the two occupied bands are polarizedwith the opposite spins: the dispersive band with spin ↑ is the lower band of the effective honeycomb lattice andthe nearly flat band with spin ↓ is from the triangularlattice. We show the corresponding Wannier center flowin Fig. S12(b), where the constant phase is associatedwith the occupied spin ↓ band on triangular lattice and2 ( a ) π π W C ( b ) π π W C FIG. S12. Wannier center flow along b m for topologicalstates at (a) ν = 1 / ν = 2 / θ = 3 ◦ and (cid:15) = 35. the other is associated with the occupied spin ↑ band onthe honeycomb lattice winding one time across b m1