Correlation-Induced Insulating Topological Phases at Charge Neutralityin Twisted Bilayer Graphene

Yuan Da Liao,1,2 Jian Kang ,3,* Clara N. Breiø,4 Xiao Yan Xu,5 Han-Qing Wu,6

Brian M. Andersen,4,† Rafael M. Fernandes,7,‡ and Zi Yang Meng 8,§

1Beijing National Laboratory for Condensed Matter Physics and Institute of Physics,Chinese Academy of Sciences, Beijing 100190, China

2School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China3School of Physical Science and Technology & Institute for Advanced Study, Soochow University,

Suzhou, 215006, China4Niels Bohr Institute, University of Copenhagen, Lyngbyvej 2, 2100 Copenhagen, Denmark

5Department of Physics, University of California at San Diego, La Jolla, California 92093, USA6School of Physics, Sun Yat-Sen University, Guangzhou, 510275, China

7School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA8Department of Physics and HKU-UCAS Joint Institute of Theoretical and Computational Physics,

The University of Hong Kong, Pokfulam Road, Hong Kong SAR, China

(Received 1 June 2020; accepted 24 November 2020; published 22 January 2021)

Twisted bilayer graphene (TBG) provides a unique framework to elucidate the interplay betweenstrong correlations and topological phenomena in two-dimensional systems. The existenceof multiple electronic degrees of freedom—charge, spin, and valley—gives rise to a plethora ofpossible ordered states and instabilities. Identifying which of them are realized in the regime ofstrong correlations is fundamental to shed light on the nature of the superconducting and correlatedinsulating states observed in the TBG experiments. Here, we use unbiased, sign-problem-freequantum Monte Carlo simulations to solve an effective interacting lattice model for TBG at chargeneutrality. Besides the usual cluster Hubbard-like repulsion, this model also contains an assisted-hopping interaction that emerges due to the nontrivial topological properties of TBG. Such a nonlocalinteraction fundamentally alters the phase diagram at charge neutrality, gapping the Dirac cones evenfor infinitesimally small interactions. As the interaction strength increases, a sequence of differentcorrelated insulating phases emerge, including a quantum valley Hall state with topological edgestates, an intervalley-coherent insulator, and a valence bond solid. The charge-neutrality correlatedinsulating phases discovered here provide the sought-after reference states needed for a compre-hensive understanding of the insulating states at integer fillings and the proximate superconductingstates of TBG.

DOI: 10.1103/PhysRevX.11.011014 Subject Areas: Condensed Matter Physics,Topological Insulators

I. INTRODUCTION

The recent discovery of correlated insulating and super-conducting phases in twisted bilayer graphene (TBG) [1–3]and other moire systems [4–7] sparked a flurry of activity to

elucidate and predict the electronic quantum phases realizedin their phase diagrams [8–68]. Because the low-energybands of TBG have a very small bandwidth, of about10 meV at the magic twist angle, the Coulomb interaction,which is of the order of 25 meV, is expected to play afundamental role in shaping the phase diagram [1,13,20,23].Indeed, insulating states have been reported at all commen-surate fillings of the moire superlattice [10], signaling theimportance of strong correlations. Besides correlations,topological phenomena have also been reported, includinga quantum anomalous Hall (QAH) phase [69,70].An important issue is the nature of the quantum ground

state at charge neutrality, characterized in real space by fourelectrons per moire unit cell and in momentum space byDirac points at the Fermi level. Experimentally, a large

*jkang@suda.edu.cn†bma@nbi.ku.dk‡rfernand@umn.edu§zymeng@hku.hk

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 11, 011014 (2021)

2160-3308=21=11(1)=011014(17) 011014-1 Published by the American Physical Society

charge gap characteristic of an insulating state was reportedin transport measurements in Ref. [10] and in STMmeasurements in Ref. [11], despite no obvious alignmentwith the underlying hBN layer. The fact that this gap is notobserved in all devices has been attributed to inhomoge-neity [10]. Theoretically, because the electronic states inTBG have several degrees of freedom—spin, valley, andsublattice—various possible ground states can emerge.Indeed, Hartree-Fock calculations of the continuum modelat charge neutrality found various possible phases, such asorbital-magnetization density waves, valley polarizedstates, and states that spontaneously break the threefoldrotational symmetry of the moire lattice [44,48,49,53,59,61,71,72]. To distinguish among these different pos-sibilities, and to search for novel ordered states in TBG, it isdesirable to employ a method that is not only unbiased butthat can also handle strong correlations.Large-scale quantum Monte Carlo (QMC) simulations

provide an optimal tool, limited only by the finite latticesizes. Although such a limitation makes it impossible tosimulate a model with thousands of carbon atoms per moireunit cell, it is very well suited to solve lattice models on themoire length scale. At charge neutrality, the noninteractingpart of the model has only Dirac points at the Fermi level.The crucial part of the model, however, is the interactingpart, which governs the system’s behavior in the strong-coupling regime. At first sight, based on the analogy withother strongly correlated models, it would seem sufficientto consider a cluster Hubbard-like repulsion as the maininteraction of the problem. Previously, some of us usedQMC to simulate this model, which does not suffer fromthe infamous fermionic sign problem [63,64]. The resultwas a variety of valence-bond insulating states, which,however, only onset at relatively large values of theinteraction U, of the order of several times the bandwidthW. Below these large values, the system remained in theDirac semimetal phase.However, microscopically, the full interaction of the

lattice model can be derived from projecting the screenedCoulomb repulsion on the Wannier states (WSs) of TBG.The latter turn out to be quite different than in othercorrelated materials, as they have nodes on the sites ofthe moire honeycomb superlattice and a three-peak structurethat overlaps with Wannier functions centered at other sites[20,21,23]. Recent work has shown that this leads to theemergence of an additional and sizable nonlocal interaction,of the form of an assisted-hopping term [37,54]. This newinteraction ultimately arises from the fact that, in a latticemodel, the symmetries of the continuum model cannot allbe implemented locally, a phenomenon dubbed Wannierobstruction [23]. Therefore, the assisted-hopping interactionis not a simple perturbation but a direct and unavoidablemanifestation of the nontrivial topological properties ofTBG. This important aspect of the TBG was not taken intoconsideration in the previous QMC simulations.

In this paper, we make this important step forward bystudying the impact of the assisted-hopping interaction onthe ground state of TBG at charge neutrality via sign-problem-free QMC simulations. We find that such a termqualitatively changes the phase diagram, as compared to thecase where only the cluster Hubbard interaction is included.In particular, the Dirac semimetal phase is no longer stablebut is gapped already at weak coupling. We show that thisgap is a manifestation of a quantum valley Hall (QVH)state, characterized by topological edge states. We confirmthis weak-coupling result by unrestricted Hartree-Fock(HF) calculations of the same model simulated by QMC.The HF calculations, well suited for weak interactions, alsoshow that the QVH state is a robust property of the weak-coupling regime and is directly connected to the assisted-hopping term. As the interaction strength increases, adifferent type of insulating phase arises, displaying inter-valley coherence (IVC) order. This on-site IVC orderbreaks the spin-valley SU(4) symmetry of the interactingpart of the model, resembling recently proposed ferromag-netic-like SU(4) states proposed to emerge in TBG atcharge neutrality and other integer fillings [37,48]. Uponfurther increasing the interaction, a columnar valencebond solid (cVBS) insulator state appears, favored bythe Hubbard-like interaction [63,64,73,74]. Importantly,the presence of the assisted-hopping term makes the QVHand IVC states accessible already for substantially smallervalues ofU=W, as compared to the case where there is onlyHubbard repulsion. Therefore, the experimental observa-tion of such quantum states in TBG at charge neutralitywould provide strong evidence for the importance ofnonlocal, topologically driven interactions in this system.

II. MODEL, SYMMETRY ANALYSIS,AND METHOD

Our lattice Hamiltonian H for spinful fermions on themoire superlattice consists of a noninteracting tight-bindingtermH0 and an interaction termH⬡. An important propertyof the narrow bands of TBG is their fragile topology,resulting in the phenomenon known as Wannier obstruc-tion, which prevents the construction of localized Wannierorbitals that locally implement the symmetries of thesystem of coupled Dirac fermions (i.e., the so-calledcontinuum model) [23]. There are essentially two waysto overcome the Wannier obstruction: (i) include additionalremote bands (at the expense of adding more Wannierorbitals and the associated interactions) or (ii) implementone of the symmetries of the continuum model nonlocally(at the expense of adding longer-range hopping parame-ters). In case (ii), the Wannier orbitals, denoted by theoperators cilσ , live on the sites i of the dual honeycombmoire superlattice and are labeled by spin σ ¼ ↑;↓ and 2orbital degrees of freedom, l ¼ 1, 2 (roughly correspondingto the two valleys) [20,21]. In case (i), the Wannier orbitals

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can live on the sites of the triangular moire superlattice, andadditional orbital quantum numbers are required.Since we are interested in the strong-coupling regime, it is

crucial to project the screened Coulomb repulsion onto thenonobstructed low-energy WSs. This was done in Ref. [37]for case (ii), which implemented the C2T symmetry non-locally, where C2 refers to twofold rotations with respect tothe z axis and T to time reversal. This process was found togive rise to a nonlocal assisted-hopping interaction, besidesthe more standard Hubbard-like repulsion.More specifically,the interacting Hamiltonian in this case is given by the sumof two contributions [37],

H⬡ ¼ UX⬡

ðQ⬡ þ αT⬡ − 4Þ2: ð1Þ

Here, U sets the overall strength of the Coulomb interaction.The two terms in Eq. (1), illustrated in Fig. 1(a), consistof the cluster charge Q⬡ ≡P

j∈⬡ðnj=3Þ, with nj ¼Plσ c

†jlσcjlσ , and the cluster assisted hopping T⬡ ≡P

j;σðic†jþ1;1σcj;1σ−ic†jþ1;2σcj;2σþH:c:Þ. The index j¼

1;…;6 sums over all six sites of the elemental hexagonin the honeycomb lattice.The cluster charge term Q⬡ is analogous to the Hubbard

on-site repulsion in the standard Hubbard model; it extendsover the entire hexagon because of the screening length set

by the separation between the gates in a TBG device andthe overlap betweenWSs of neighboring sites. In particular,the Wannier wave functions are not peaked at the honey-comb sites but instead are extended and peaked at thecenters of the three neighboring hexagons [20,21,23].Therefore, one single WS overlaps spatially with otherWSs on neighboring sites, leading to the cluster chargingterm Q⬡. On the other hand, the origin of the assisted-hopping term T⬡ is topological; i.e., it comes preciselyfrom the fragile topology of TBG. This can be seen fromthe derivation of the coefficient α, which controls therelative strength of the two interactions. It is the overlap oftwo neighboring WSs in a single hexagon, given as [37]

α ¼ −iZ⬡drw�

1;1ðrÞw2;1ðrÞ; ð2Þ

where w1;1ðrÞ is the wave function of the WS at site 1 of thehexagon with the valley index 1, and w2;1ðrÞ is the WSat site 2 of the hexagon. Note that the integral is taken onlyinside the single hexagon. Although α is generally acomplex number, its phase can always be removed by agauge transformation. As argued in Ref. [37], the sizablevalue of α comes from the topological obstruction to thefully symmetric WSs. If the bands are topologically trivial,all the symmetries can be locally implemented forthe WSs. As a consequence, the WSs are C00

2 symmetric

l=1 l=2

U/W10 12

0.1

0.2

0.3

0.4

0.5

0.6

0 2 4 6 8

Orbital 1

Orbital 2

T

(a) (b)

Q

FIG. 1. Ground-state phase diagram at charge neutrality obtained via QMC simulations. (a) Schematics of the model: Each lattice siteon the dual moire honeycomb lattice contains two valleys l ¼ 1, 2 (red and green triangles) and spins σ ¼ ↑;↓ (not shown), with spin-valley SU(4) symmetry. The interactions act on every hexagon and consist of the cluster charge term Q⬡ (yellow dots) and the assisted-hopping interaction term T⬡ (blue arrows). (b) Ground-state phase diagram, spanned by the U=W and α axes, obtained from QMCsimulations. The y axis at U ¼ 0 (dashed line) stands for the Dirac semimetal phase. At very small U, the ground state is a QVH phasecharacterized by emergent imaginary next-nearest-neighbor hopping with complex conjugation at the valley index, as illustrated by thered and green dashed hoppings with opposite directions. The system has an insulating bulk but acquires topological edge states. Uponfurther increasing U, an IVC insulating state is found, which breaks the SU(4) symmetry at every lattice site by removing the valleysymmetry. Because it preserves the lattice translational symmetry, it is ferromagnetic-like. The cVBS insulator, which appears after theIVC phase, breaks the lattice translational symmetry and preserves the on-site SU(4) symmetry. Note that there is a reemergence of theIVC phase for the largest interactions probed. The phase transitions between QVH and IVC (blue line), between the IVC and cVBS(black line), and between the cVBS and IVC (red line) are all first order.

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and have the same parity. While two neighboring WSsoverlap in two neighboring hexagons and sum to 0 becauseof the orthogonality, the two overlaps are equal since C00

2

symmetry relates them together. Therefore, each onevanishes, leading to α ¼ 0. The sizable value of αmanifeststhe nontrivial topological properties of the narrow bands. InRef. [37], α was found to be 0.23 based on Koshino’smodel, without including the lattice relaxation [75]. Sincethe relaxation will inevitably change α, we will not fix itsvalue here but instead study the phase diagram for a widerrange of α.The nonlocal implementation of the C2T symmetry also

results in many longer-range hopping parameters in H0

[20]. Ideally, one would like to solve the model containingall of these tight-binding terms, but such a model would, ingeneral, suffer from the sign problem and cannot beefficiently simulated with QMC, which has the advantageof being unbiased and applicable even for large interactionvalues. In contrast, the interaction term H⬡ alone can besolved with QMC without the sign problem, despite thepresence of the nonlocal interaction T⬡ (see discussions inthe Appendix A). Therefore, because we are interested inthe strong-coupling regime, we opt to keep the full non-trivial interaction term and simplify the tight-bindingHamiltonian in order to circumvent the sign problem,

H0 ¼ −tXhijilσ

ðc†ilσcjlσ þ H:c:Þ: ð3Þ

This simple nearest-neighbor band dispersion displaysDirac points at charge neutrality and can be simulated withsign-problem-free QMC at charge neutrality (four electronsper hexagon once averaging over the lattice), which weassume hereafter. Moreover, we set the hopping parametert ¼ 1 and use the bare bandwidthW ¼ 6t as the energy unitin the remainder of the paper.We emphasize that, in the strong-coupling regime, we

expect that it is the nontrivial structure of the projectedinteractions that will determine the ground state and notthe bare tight-binding dispersion. Below, we provideevidence that this is indeed the case. Thus, the crucialpoint is that the topologically nontrivial properties of theTBG band structure are already incorporated in theinteracting part of our model, which inherits them fromthe projection of the screened Coulomb interaction on thenontrivial WSs.An interesting feature of H⬡ is its emergent SU(4)

symmetry describing simultaneous rotations in spin andorbital spaces. To illustrate this feature, we introduce thespinor ψ i ¼ ðci1↑; ci1↓; ci2↑; ci2↓ÞT and rewrite the inter-actions as

Q⬡ ¼ 1

3

Xi∈⬡

ψ†iψ i; ð4Þ

T⬡ ¼ iXi∈⬡

ψ†iþ1T0ψ i þ H:c:; ð5Þ

with T0 ¼ diagð1; 1;−1;−1Þ denoting a diagonal matrix.Consider the unitary transformation

ψ i∈A → Uψ i and ψ i∈B → T0UT0ψ i; ð6Þ

where U is an arbitrary 4 × 4 unitary matrix and AðBÞ arethe two sublattices of the honeycomb lattice. It is clear thatboth Q⬡ and T⬡ are invariant under this transformation.On the other hand, the kinetic term H0 is not invariantunder the transformation given by Eq. (6), thus leaving thewhole Hamiltonian only Uð1Þ × SUð2Þ × SUð2Þ symmet-ric, i.e., the valley U(1) symmetry and the two independentspin SU(2) rotations for the two valleys [23,43]. Thus,strictly speaking, the SU(4) symmetry is exact for H⬡ butonly approximate for H0.To solve the model H ¼ H0 þH⬡ nonperturbatively,

we employ large-scale projection QMC simulations[63,64]. This QMC approach, employed in several previousstudies [63,64,73,76–79], provides results about the T ¼ 0ground state, the correlation functions (which are usedto determine broken symmetries), and the electronic spectra(both single-particle and collective excitations). Asexplained above, despite the presence of the assisted-hopping interaction, the model at charge neutrality doesnot suffer from the sign problem (see Appendix A fordetails). Thus, it can be efficiently simulated by introducingan extended auxiliary bosonic field that dynamicallycouples to the electrons on a hexagon—in contrast tothe standard Hubbard model, where the auxiliary field islocal. Details about the projection QMC implementation, aswell as comparison with results from exact diagonalization,are discussed in Appendixes A and B.We also complemented the unbiased QMC simulations

with self-consistent HF calculations, which are well suitedfor the weak-coupling regime and can be employed evenwhen additional terms are included in H that introduce asign problem for QMC. The HF approach is fully unre-stricted in the sense that H0 þH⬡ is mean-field decoupledin all channels and free to acquire any value in site, spin,and valley space. Further technical details, including theresulting coupled set of (real space) self-consistencyequations, can be found in Appendix C. In the regimeof weak interactions, we find excellent agreement betweenthe results obtained from HF and QMC. Importantly, in thesame Appendix, we also extend the HF calculations toinclude longer-range hopping terms in H0 and find that theresults are similar. This result supports our aforementionedexpectation that the nontrivial structure of the projectedinteractions, arising from the fragile topology of TBG,dominates the ground-state properties of the system, at leastat charge neutrality.

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III. QUANTUM VALLEY HALL PHASE,INTERVALLEY-COHERENT INSULATOR, AND

VALENCE-BOND SOLID

The QMC-derived phase diagram for the ground states atcharge neutrality is shown in Fig. 1(b) as a function ofU=W and α. We emphasize that while U gives the overallmagnitude of the total interaction term, α is proportional tothe relative strength between the assisted-hopping andcluster-charge terms. We find that three types of correlatedinsulating phases emerge in the phase diagram: the QVHphase, the IVC phase, and the cVBS.The QVH phase is the ground state for small U values

and is characterized by a gap in the single-particlespectrum. This gap can be extracted from the imaginary-time decay of the Green’s function along a high-symmetrypath of the Brillouin zone (BZ), Gðk; τÞ ∼ e−ΔspðkÞτ.Figure 2(a) shows the enhancement of the single-particlegap at the K point of the BZ as a function of U for afixed α ¼ 0.45 (blue points). Together with Fig. 2(b), onesees that the gap opens at the entire BZ at infinitesimally

small U. In many honeycomb lattice models, the Diraccone at the K point is protected by a symmetry, and thesemimetal phase is robust against weak interactions[63,64,73,76]. In TBG, however, the relevant symmetryC2T cannot be implemented locally due to the topologicalWannier obstruction, which opens up the possibility of veryweak interactions gapping out the Dirac cone.In our QMC simulations, for any nonzero α that we

investigated, a gap appears even for the smallest values ofU probed. This result suggests a weak-coupling origin ofthis phase. To verify it, we perform HF calculations on thesame lattice model. The results, shown by the red points inFig. 2(a), are in very good agreement with the QMC results.We also use HF to investigate the stability of the gap againstchanging the phase that appears in the assisted-hoppingterm T⬡ [37]. This phase can be gauged away, at theexpense of introducing complex hopping terms in H0,which introduce a sign problem to the QMC simulations.However, they do not affect the efficiency of the HFalgorithm. As discussed in Appendix C, our analysisconfirms that the onset of the QVH phase is robust andappears regardless of the phase of T⬡.Importantly, we find that the gap completely disappears

when α ¼ 0, in agreement with Ref. [64]. Combined withthe fact that the gap opens for small interaction values whenα ≠ 0, this suggests that the origin of the gap can beunderstood from a mean-field decoupling of the cross termP

⬡ Q⬡T⬡ of the interaction in Eq. (1). This cross termcan be rewritten as

X⬡

Q⬡T⬡ ¼ iX⬡

X6i;j¼1

X2l;m¼1

ð−1Þmðc†i;lc†jþ1;mcj;mci;l−H:c:Þ;

ð7Þ

where l and m are valley indices and the spin index isomitted for simplicity. The terms with j ¼ i − 1 and j ¼ ivanish after summing over different hexagons. In the weak-coupling limit, we can do a mean-field decoupling anduse hc†i;lciþ1;mi ∝ δlm, due to the nearest-neighbor hoppingterm present in H0. The cross term then becomes

X⬡

Q⬡T⬡ ∝ −iX⬡

X6i¼1

X2l¼1

ð−1Þl

× ðc†i;lciþ2;l þ c†i−2;lci;l − H:c:Þ: ð8Þ

Thus, the cross term of the interaction naturally induces animaginary hopping between next-nearest neighbor in theweak-coupling limit. As a consequence, the mean-fieldHamiltonian becomes two copies (four, if we consider thespin degeneracy) of the Haldane model [80,81], leading to aChern number of �1 for the two different valleys. Hencewe dub this phase QVH it is illustrated inthe corresponding inset in Fig. 1(b). We verify that our

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 510-4

10-3

10-2

10-1

100

U/W = 0.25, L = 6U/W = 0.25, L = 12U/W = 2.00, L = 6U/W = 2.00, L = 12

Ged

ge

0

1

2

3

4 U/W=0.25U/W=0.50U/W=1.00U/W=1.50U/W=2.00

/W

sp

M K M

(b)

(c)U/W

0

1

2

3

4QMCHF

0.5 1 1.5 20

(K

)/W

sp

(a)

FIG. 2. QVH and gapless edge states. (a) Single-particle gapΔspðKÞ=W at the K point as a function of U=W for α ¼ 0.45,extracted from both QMC (blue points) and HF calculations (redpoints). For QMC, the spatial system size is L ¼ 12. The Diracsemimetal is gapped out at the smallest U values probed.(b) Single-particle gap extracted from QMC with L ¼ 12 alonga high-symmetry path of the Brillouin zone. (c) Topologicalnature of the QVH phase manifested by valley-polarized edgestates. Here, we compare the edge Green’s function for valleyl ¼ 1 and spin ↑ at U=W ¼ 0.25 (inside the QVH phase) andU=W ¼ 2.0 (inside the IVC phase). It is clear that gapless edgemodes only appear in the former case, highlighting the topo-logical nature of the QVH phase.

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self-consistent HF calculation generates the same pattern ofimaginary next-nearest-neighbor hopping.One of the hallmarks of the Haldane model is the

existence of gapless edge modes, despite the bulk beinggapped. In the QVH phase, these edge states should bevalley polarized. To probe them, we perform QMC sim-ulations with open boundary conditions and extract theimaginary-time Green’s functions on the edge, GedgeðτÞ∼e−Δspτ. As shown in Fig. 2(c), in the regime of small U(U=W ¼ 0.25), the Green’s function on the edge decays toa constant in the long imaginary-time limit, demonstratingthe existence of a gapless edge mode in the QVH phase.To verify the existence of edge states, we also use HF tocapture the topological nature of the QVH phase. Inpractice, we open the boundaries in the system andcompute a self-consistent result with parameters as inTable I from Appendix C (t ¼ 1, α ¼ 0.45, U=W ¼ 0.5,T ¼ 2.5 × 10−5, and N ¼ 4 × 600). We find clear evidenceof edge states, as seen in Fig. 3. Note that a Chern numbercan be defined separately for each valley l ¼ 1 and l ¼ 2(with spin degeneracy). Because the valley Uð1Þ symmetryguarantees that these two Chern numbers must be equal,the whole system is characterized by one Chernnumber that takes integer values; i.e., it belongs to a Zclassification [82].Figure 2(c) also shows that, as U increases (U=W ¼ 2),

the gapless edge mode disappears, signaling a departurefrom the topological QVH phase. Clearly, the bulk remainsgapped, as shown in Fig. 2(a). The new insulating phaseis an IVC state, which spontaneously breaks the on-sitespin-valley SU(4) symmetry. In the QMC simulations, IVCorder is signaled by an enhancement of the correlationfunction CIðkÞ ¼ ð1=L4ÞPi;j∈AðBÞ eik·ðri−rjÞhIiIji; here,

the operator Ii ¼P

σðc†i;l;σci;l0;σ þ H:c:Þ, l ≠ l0, representsan “on-site hopping” between the two different valleys.Thus, the correlation function is a 2 × 2matrix in sublattice

space, i.e.,�CAAI

CBAI

CABI

CBBI

�, which has the relation

CAAI ¼ CBB

I ¼ −CABI ¼ −CBA

I . In the upper panels ofFigs. 4(a) and 4(b), we show the diagonal componentCAAI ðkÞ. The fact that the correlation function is peaked at

15

-15

0ry/am

0-15 15

3.5

0.5

LD

OS

(arb. units)

rx/am

FIG. 3. In-gap local density of states in the QVH phase. Weshow the real-space plot of the local density of states integratedover 1.66 < E=W < 2.00, with U=W ¼ 0.5. Here, rx;y is theposition of the lattice sites in units of the moire lattice spacing,am. The result is computed with open boundary conditions andparameters as in Table I from Appendix C.

0

0.1

0.2

0.3

0.4L=9L=12L=15

0 2 4 6 8 10 12U/W

0

0.4

0.8

1.2

0

0.1

0.2

0.3

0.4

0.5

0 U/W

0

0.4

0.8

1.2

(b) = 0.6

CI(

)C

B( K

)

(a) = 0.4

CI(

)C

B(K

)Re(D

K)

Im(D

K)

2 4 6 8 10 12

FIG. 4. IVC and cVBS insulating states. Correlation functionsCIðΓÞ and CBðKÞ, indicative of IVC and cVBS orders, respec-tively, as a function of U=W for (a) α ¼ 0.4 and (b) α ¼ 0.6.Linear system sizes are indicated in the legend. In both panels, theQVH-IVC transition, the IVC-cVBS transition, and the cVBS-IVC transition are all first order. The inset in panel (a) presentsthe histogram of the complex bond order parameter DK atU=W ∼ 5.3. The positions of the three peaks are those expectedfor a cVBS phase instead of a pVBS state.

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k ¼ Γ implies that the IVC order is ferromagnetic-like; i.e.,it does not break translational symmetry. Such an on-sitecoupling between opposite valleys [see the correspondinginset in the phase diagram in Fig. 1(b)] breaks the valleyUð1Þ symmetry and, hence, the SU(4) symmetry of themodel. The fact that the SU(4) symmetry-breaking patternis ferromagnetic-like is similar to recent analytical results[37,38], which focused, however, at integer fillings awayfrom charge neutrality. We also note that our IVC state isdifferent from that of Ref. [48] since our IVC phase doesnot have edge modes protected by a modified Kramerstime-reversal symmetry, as is the case of the IVC stateproposed in Ref. [48].For larger values of U=W, as shown in Fig. 4, the IVC

order fades away, but the system remains insulating. Thenew state that emerges is the cVBS insulator, characterizedby the appearance of strong nearest-neighbor bonds form-ing the pattern illustrated in the corresponding inset ofFig. 1(b). The onset of cVBS order is signaled by anenhancement of the bond-bond correlation function [63,64,73,74], CBðkÞ¼ð1=L4ÞPi;j e

ik·ðri−rjÞhBi;δBj;δi, with bond

operator Bi;δ ¼P

l;σðc†i;l;σciþδ;l;σ þ H:c:Þ and δ denotingone of the three nearest-neighbor bond directions of thehoneycomb lattice (e1, e2, and e3). For this particularcalculation, e1 was chosen.As shown in the lower panels of Figs. 4(a) and 4(b), we

find an enhanced CBðkÞ at momenta K and K0, demon-strating that the bond-order pattern breaks translationalsymmetry. However, a peak of CBðkÞ at these momentadoes not allow us to unambiguously identify the cVBSstate, as the plaquette valence-bond solid (pVBS)also displays peaks at the same momenta [63,64,73]. Tofurther distinguish the two types of VBS phases, weconstruct the complex order parameter DK ¼ ð1=L2Þ×P

i ðBi;e1 þ ωBi;e2 þ ω2Bi;e3ÞeiK·ri , with ω ¼ eið2π=3Þ. TheMonte Carlo histogram of DK is different for the two VBSphases [73,74]: For the pVBS state, the angular distributionof DK is peaked at argðDKÞ ¼ π=3, π, 5π=3, whereas forthe cVBS state, it is peaked at argðDKÞ ¼ 0, 2π=3, 4π=3.Our results, shown in the inset of Fig. 4(a), clearlydemonstrate that the cVBS order is realized in ourphase diagram.The phase boundaries in Fig. 1(b) are obtained by

scanning the correlation functions CIðΓÞ and CBðKÞ asa function ofU=W for fixed values of α. Two of these scansare shown in Fig. 4, for α ¼ 0.4 [panel (a)] and α ¼ 0.6[panel (b)]. It is clear that, as U=W increases, in both casesthe ground state evolves from QVH to IVC to cVBS andthen back to IVC. Furthermore, in the strong-coupling limitU=W → ∞, the IVC order CIðk ¼ 0Þ is independent of αand saturates at 0.5, consistent with our analytical calcu-lation at the charge neutrality point (see Appendix D). Thetransitions between IVC to cVBS are first order, as signaledby the fact that as the system size L increases, thesuppression of the IVC order becomes sharper [see, for

instance, the region around U=W ∼ 5 and U=W ∼ 11 inpanel (a)]. A similar sharp drop is also featured at theQVH-IVC transition [region around U=W ∼ 2.5 in panel(a)], indicating that the QVH-IVC and IVC-cVBS tran-sitions are all first order. It is interesting to note that, as αincreases, the values of U=W for which the IVC and cVBSphases emerge are strongly reduced.We note that in the limit of vanishing bandwidth, our

analyses in Appendixes A and D reveal that in this very-strong-coupling limit, the ground state of the system is inthe IVC phase for the range of α considered here.

IV. DISCUSSION

In this paper, we employed QMC simulations, which areexact and unbiased, to obtain the phase diagram of a latticemodel of TBG at charge neutrality. Our main result is thateven very small interaction values trigger a transition fromthe noninteracting Dirac semimetal phase to an insulatingstate. Upon increasing U, the nature of the insulatorchanges from a nonsymmetry-breaking topological QVHphase to an on-site SU(4) symmetry-breaking IVC state, toa translational symmetry-breaking cVBS phase, and thenfinally back to a reentrant IVC state. This rich phasediagram is a consequence of the interplay between twodifferent types of interaction terms: a cluster-charge repul-sion Q⬡ and a nonlocal assisted-hopping interaction T⬡.The former is analogous to the standard Hubbard repulsionand, as such, is expected to promote either SU(4) anti-ferromagnetic order or valence-bond order in the strong-coupling regime. The latter, on the other hand, arises fromthe topological properties of the flat bands in TBG. Whencombined with Q⬡, it gives rise not only to SU(4)ferromagnetic-like order but also to correlated insulat-ing phases with topological properties, such as theQVH phase.While the precise value of U=W in TBG is not known, a

widely used estimate is that this ratio is of order 1 [37].Referring to our phase diagram in Fig. 1(b), this means thatcertainly the QVH phase and possibly the IVC phase can berealized at charge neutrality, provided that α is not toosmall. While some experimental probes report a gap atcharge neutrality (Refs. [10,11]), additional experimentsare needed to establish its ubiquity among different devicesand the nature of the insulating state. The main manifes-tation of the QVH phase would be the appearance ofgapless edge states, whereas in the case of the IVC state, itwould be the emergence of a k ¼ 0 order with on-sitecoupling between the two different valleys.A number of recent insightful Hartree-Fock studies have

also reported several unusual ordered states at chargeneutrality [44,48,49,71,72]. In their approach, starting fromthe Bistritzer-MacDonald (BM) continuum wave functions,the Coulomb interactions are projected by use of thecontinuum model and typically include several remotebands. As usual, HF studies can depend crucially on the

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restrictions imposed in the search for ordered states, whichmay explain the rich variety of proposed spontaneouslysymmetry-broken phases identified from the continuumapproach, including a semimetal phase, quantum Hallinsulator, valley-Hall, and spin- and valley-polarizedphases. Recently, Ref. [48], allowing for coherencebetween the two valleys, argued that the resulting insu-lating IVC phase is the ground state at charge neutralityfor a broad parameter range. As discussed in theIntroduction of this paper, we have presented a comple-mentary approach; starting from the strongly interactinglimit, we have applied the topologically nontrivial pro-jected Coulomb interaction and utilized fully unrestrictedand unbiased numerical methods able to handle caseswhere the scale of interactions exceeds the kinetic band-width, to identify the ordered states at charge neutrality. Inqualitative agreement with some earlier studies, we locatean IVC phase from this strong-coupling approach butadditionally identify both the QVH and a translationallysymmetry-breaking cVBS phase. While HF calculationswith the Bloch states usually produce homogeneous phaseswithout breaking the translation symmetry, more recentdensity matrix renormalization group (DMRG) calcula-tions with hybrid WSs have identified the stripe phase as astrong candidate for a ground state in a toy BM modelwithout spin and valley degrees of freedom [54,66]. Inagreement with the DMRG calculations, our QMC studyfound that the increasing kinetic terms drive the systemfrom the IVC phase in the strong coupling limit into thecVBS phase in a more intermediate coupling regime.In a more general context beyond TBG, our work offers a

promising route to realize correlation-driven topologicalphases. As explained above, the topological QVH insulat-ing state appears due to the cross term in the interactionHamiltonian that contains both Q⬡ and T⬡. While repul-sive interactions similar to the charge-cluster term aregenerally expected to appear in any correlated electronicsystem, an interesting question is about the necessaryconditions for the emergence of an interaction similar tothe assisted hopping. In our case, it arises from theprojection of the standard Coulomb repulsion on WSs thatsuffer from topological obstruction. The latter, in turn, is amanifestation of the phenomenon of fragile topology [83].Thus, interacting systems with fragile topology may offeran appealing route to search for interaction-driven topo-logical states. While here the Wannier obstruction arisingfrom the fragile topology is circumvented by implementingthe C2T symmetry of the continuous model nonlocally,another route is to include the remote bands, separated fromthe narrow bands of TBG by a sizable gap. While we expectthe ground state to be the same regardless of how theWannier obstruction is avoided, it is an interesting openquestion to establish the strong-coupling phase diagram ofTBG starting from a model containing both the narrow andremote bands.

ACKNOWLEDGMENTS

We thank Eslam Khalaf, Ashvin Vishwanath, and YiZhang for insightful conversations on the subject, espe-cially on the nature of the IVC phase. We also thank OskarVafek for valuable suggestions and for pointing out amissing factor in the IVC correlation function. Y. D. L. andZ. Y. M. acknowledge support from the National KeyResearch and Development Program of China (GrantNo. 2016YFA0300502) and Research Grants Council ofHong Kong SAR China (Grant No. 17303019). H. Q.W. issupported by NSFC through Grant No. 11804401 and theFundamental Research Funds for the Central Universities.J. K. acknowledges support from the NSFC GrantNo. 12074276, and the Priority Academic ProgramDevelopment (PAPD) of Jiangsu Higher EducationInstitutions. R. M. F. is supported by the U.S.Department of Energy, Office of Science, Basic EnergySciences, Materials Sciences and Engineering Division,under Award No. DE-SC0020045. Y. D. L. and Z. Y.M.thank the Center for Quantum Simulation Sciences in theInstitute of Physics, Chinese Academy of Sciences, theComputational Initiative at the Faculty of Science andInformation Technology Service at the University of HongKong, the Platform for Data-Driven ComputationalMaterials Discovery at the Songshan Lake MaterialsLaboratory and the National Supercomputer Centers inTianjin and Guangzhou for their technical support andgenerous allocation of CPU time. J. K. thanks the KavliInstitute for Theoretical Sciences for hospitality during thecompletion of this work. Z. Y. M., J. K., and R. M. F.acknowledge the hospitality of the Aspen Center forPhysics, where part of this work was developed. TheAspen Center for Physics is supported by NationalScience Foundation Grant No. PHY-1607611.

APPENDIX A: PROJECTION QMC METHOD

1. Construction

Since we are interested in the ground-state propertiesof the system, the projection QMC (PQMC) is themethod of choice [76,79,84]. In PQMC, one can obtaina ground-state wave function jΨ0i from projecting a trialwave function jΨTi along the imaginary axis jΨ0i ¼limΘ→∞e−ðΘ=2ÞHjΨTi; then, the observable can be calcu-lated as

hOi ¼ hΨ0jOjΨ0ihΨ0jΨ0i

¼ limΘ→∞

hΨT je−Θ2HOe−

Θ2HjΨTi

hΨT je−ΘHjΨTi: ðA1Þ

To evaluate overlaps in the above equation, we performTrotter decomposition to discretize Θ into Lτ slices(Θ ¼ LτΔτ). Each slice Δτ is small, and the systematicerror is OðΔτ2Þ. After the Trotter decomposition, we have

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hΨT je−ΘHjΨTi ¼ hΨT jðe−ΔτHUe−ΔτH0ÞLτ jΨTi þOðΔτ2Þ;ðA2Þ

where the noninteracting and interacting parts of theHamiltonian are separated. To treat the interacting part,one usually employs a Hubbard-Stratonovich (HS) trans-formation to decouple the interacting quartic fermion termto fermion bilinears coupled to auxiliary fields.For the cluster interaction in Eq. (1) of the main text,

we make use of a fourth-order SUð2Þ symmetricdecoupling,

e−ΔτUðQ⬡þαT⬡−4Þ2 ¼ 1

4

Xfs⬡g

γðs⬡Þeληðs⬡ÞðQ⬡þαT⬡−4Þ; ðA3Þ

with λ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi−ΔτU

p, γð�1Þ¼1þ ffiffiffi

6p

=3, γð�2Þ ¼ 1 −ffiffiffi6

p=3,

ηð�1Þ ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ð3 − ffiffiffi

6p Þ

q, ηð�2Þ ¼ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ð3þ ffiffiffi

6p Þ

q, and

the sum is taken over the auxiliary fields s⬡ on eachhexagon, which can take four values �2 and �1. Aftertracing out the free fermionic degrees of freedom,we obtain the following formula with a constant factoromitted:

hΨT je−ΘHjΨTi ¼Xfs⬡;τg

��Yτ

Y⬡

γðs⬡;τÞe−4ληðs⬡;τÞ�

× det ½P†BðΘ; 0ÞP��; ðA4Þ

where P is the coefficient matrix of the trial wavefunction jΨTi. In the simulation, we make use of thereal-space ground-state wave function of the tight-bindingHamiltonian H0 as the trial wave function jΨTi. In theabove formula, the B matrix is defined as

Bðτ þ 1; τÞ ¼Xfs⬡;τg

eληðs⬡;τÞV · e−ΔτK ðA5Þ

and has properties Bðτ3; τ1Þ ¼ Bðτ3; τ2ÞBðτ2; τ1Þ; i.e., the Bmatrix is an imaginary-time propagator, where we havewritten the coefficient matrix of the interaction part as Vand K is the hopping matrix from the H0.

Every hexagon contains six sites, as shown in the figurebelow, so our V matrix is a block matrix; every blockcontributes a 6 × 6 matrix,

The configurational space fs⬡ði; τÞg with size L × L ×Θ is the space in which the physical observables in Eq. (A1)are computed with ensemble average. We choose theprojection length Θ ¼ 2L=t and discretize it with a stepΔτ ¼ 0.1=t. The spatial system sizes are L ¼ 6, 9, 12, 15.The Monte Carlo sampling of auxiliary fields is further

performed based on the weight defined in the sum ofEq. (A4). The measurements are performed near τ¼Θ=2.Single-particle observables are measured by the Green’sfunction directly, and many-body correlation functions aremeasured by the products of single-particle Green’s func-tions based on their corresponding form after Wick decom-position. The equal-time Green’s function is calculated as

Gðτ; τÞ ¼ 1 − RðτÞ½LðτÞRðτÞ�−1LðτÞ; ðA6Þ

with RðτÞ ¼ Bðτ; 0ÞP, LðτÞ ¼ P†BðΘ; τÞ.

2. Absence of sign problem

At the charge neutrality point, the model is sign-problem-free, as can be seen from the following analysis. DefineWσ;l;Si as the updated weight of one fixed auxiliary field atthe ith hexagon, where l ¼ 1, 2 is a valley or orbital indexand σ ¼ ↑;↓ is a spin index. From the symmetryof the Hamiltonian, W↑;l;Si ¼ W↓;l;Si . Since the model isparticle-hole symmetric at charge neutrality, one can performa particle-hole transformation (PHS) only for the valleyl ¼ 2. Then, one can focus on a fixed auxiliary field andfocus only on one spin flavor, such as spin up. Equation (A3)in the main text can then be abbreviated as Eqs. (A7)and (A8). Applying PHS for valley 2 and using the relationEq. (A9), we find that Eq. (A8) becomes Eq. (A10),

For l ¼ 1; exp

�iαηðSiÞ

�X6p¼1

ðic†pþ1;1;↑cp;1;↑ − ic†p;1;↑cpþ1;1;↑Þ þ1

3

X6p¼1

�c†p;1;↑cp;1;↑ −

1

2

���; ðA7Þ

For l ¼ 2; exp

�iαηðSiÞ

�X6p¼1

ð−ic†pþ1;2;↑cp;2;↑ þ ic†p;2;↑cpþ1;2;↑Þ þ1

3

X6p¼1

�c†p;2;↑cp;2;↑ −

1

2

���; ðA8Þ

−ic†pþ1;2;↑cp;2;↑ þ ic†p;2;↑cpþ1;2;↑ →PHS

ic†pþ1;2;↑cp;2;↑ − ic†p;2;↑cpþ1;2;↑c†p;2;↑cp;2;↑ −

1

2→PHS 1

2− c†p;2;↑cp;2;↑; ðA9Þ

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For l ¼ 2PHS; exp

�iαηðSiÞ

�X6p¼1

ðic†pþ1;2;↑cp;2;↑ − ic†p;2;↑cpþ1;2;↑Þ þ1

3

X6p¼1

�1

2− c†p;2;↑cp;2;↑

���; ðA10Þ

For l ¼ 1; e−αηðSiÞBþiαηðs⬡ÞA → e−αηðs⬡ÞBeiαηðs⬡ÞA;

For l ¼ 2PHS; e−αηðSiÞBþiαηðs⬡ÞA → e−αηðs⬡ÞBe−iαηðs⬡ÞA: ðA11Þ

Let us define the matrices A ¼ 13

P6p¼1 ðc†p;2;↑cp;2;↑ − 1

2Þ

and iB ¼ P6p¼1 ðic†pþ1;1;↑cp;1;↑ − ic†p;1;↑cpþ1;1;↑Þ—or,

equivalently, B ¼ P6p¼1 ðc†pþ1;1;↑cp;1;↑ − c†p;1;↑cpþ1;1;↑Þ.

The matrices A and B are real matrices. Then, due tothe fact that the matrix A is a diagonal matrix andAii ¼ Ajj, Eqs. (A7) and (A10) can be writtenas Eq. (A11).Because of the relations above, the total weight of

the model isP

Si W↑;1;Si �W↑;2;Si �W↓;1;Si �W↓;1;Si ¼PSi ðW↑;1;SiW

�↑;1;Si

Þ2, which is a real positive number.This result implies that the QMC simulations are sign-problem-free.

3. Strong-coupling limit

The PQMC simulations can also be applied at the strong-coupling limit, where the Hamiltonian only contains theinteraction part H⬡. We have performed the correspondingsimulations and found, as a function of α, that the system isalways in the IVC phase since the corresponding correla-tion function CIðΓÞ is close to the saturation value of 0.5.At the same time, the correlation function of the cVBSphase, CBðKÞ, approaches zero as the system sizeincreases. The results are shown in Fig. 5. This case is

consistent with the theoretical analysis in this limit dis-cussed in Appendix D.

APPENDIX B: BENCHMARK WITH EXACTDIAGONALIZATION

We employ Lanczos exact diagonalization (ED) to bench-mark the PQMC results, shown in Fig. 6. The systemcontains 2 × 2 unit cells of the honeycomb lattice withperiodic boundary conditions (16 electrons in total). Wemake use of symmetries, such as the valley Uð1Þ symmetryand the total Sz conservation for each valley, to reduce thecomputational cost of the ED. The ground state lies in thesubspace with N↑ ¼ 4, N↓ ¼ 4 in both valleys, whereN↑ðN↓Þ is the number of electrons with spin up (down)in each valley. The dimension of the ground-state subspace isabout 24 × 106. In the PQMC simulations, we choosethe linear system size L ¼ 2 and the projection lengthΘ ¼ 100=t with Trotter slice Δτ ¼ 0.0005=t. We comparethe ground-state expectation values of hH0i and of thedouble occupation as a function ofU=t at α ¼ 0.3, which areshown below. The results of both methods agree very well.

APPENDIX C: HARTREE-FOCK METHOD

To solve the TBG model within the Hartree-Fockapproach, we write the Hamiltonian in Eq. (1) of the maintext as

H⬡ ¼ UX⬡

ðQ0⬡ þ αT 0

⬡ÞðQ⬡ þ αT⬡Þ; ðC1Þ

where primes indicate independent index summations. Thedirect terms immediately give

HH⬡ ¼ 2U

X⬡

hQ0⬡ þ αT 0

⬡iðQ⬡ þ αT⬡Þ: ðC2Þ

0.1 0.2 0.3 0.4 0.5 0.6

0.1

0.2

0.3

0.4

0.5C

I( ) L=9

CI( ) L=12

CB(K) L=9

CB(K) L=12

FIG. 5. CIðΓÞ and CBðKÞ as a function of α at the strong-coupling limit. We perform the QMC simulation with projectionlength Θ ¼ 200L, interval of time slice Δτ ¼ 0.1, and spatialsystem sizes L ¼ 9, 12, and by setting U ¼ 1 as a dimensionlessconstant. The corresponding correlation function CIðΓÞ of IVC isclose to the saturation value of 0.5, and the correlation functionCBðKÞ of the cVBS is close to 0, which means the cVBSdisappears at the strong-coupling limit.

U/W

EDQMC

1 2Ni,l

n n i,l,

i,l,

H0

EDQMC

(a) (b)

10

-24

-22

-20

-18

-16

-14

-12

0.24

0.245

0.25

0.255

0.26

0.265

0 1 2 3 4 5 6 7 8 9U/W

100 1 2 3 4 5 6 7 8 9

FIG. 6. (a) Kinetic energy and (b) double occupancy as afunction of U=W for α ¼ 0.3. Red circles and blue squares witherror bars are obtained from ED and QMC, respectively.

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The exchange terms are given in Eq. (C3), whereP

all ¼ ðPl;l0¼1;2

Pσσ0

P6i;i0¼1

Þ. Manipulating and collecting terms inEqs. (C2) and (C3) yields the Hartree-Fock Hamiltonian Eq. (C3).

HF⬡ ¼ −U

X⬡

Xall

�1

9½hc†i0l0σ0cilσic†ilσci0l0σ0 þ H:c:�

þ αi3f½ð−Þl0þ1hc†i0þ1l0σ0cilσic†ilσci0l0σ0 þ H:c:� þ ½ð−Þl0 hc†i0l0σ0cilσic†ilσci0þ1l0σ0 þ H:c:�g

þ α2i2f½ð−Þl0þlhc†i0þ1l0σ0cilσic†iþ1lσci0l0σ0 þ H:c:� − ½ð−1Þl0þlhc†i0þ1l0σ0ciþ1lσic†ilσci0l0σ0 þ H:c:�

− ½ð−1Þl0þlhc†i0l0σ0cilσic†iþ1lσci0þ1l0σ0 þ H:c:� þ ½ð−Þl0þlhc†i0l0σ0ciþ1lσic†ilσci0þ1l0σ0 þ H:c:�g�;

HHF⬡ ¼ 2U

X⬡

n⬡ðQ⬡ þ αT⬡Þ −

Xall

�Xn;m

αnðl0Þαmðlþ 1Þhc†i0þnl0σ0ciþmlσi�c†ilσci0l0σ0

: ðC3Þ

Here, n;m ¼ f−1; 0; 1g, and we have defined

n⬡ ¼ hQ⬡ þ αT⬡i;

αðlÞ ¼

0B@

α−1

α0

α1

1CA ¼

0B@

ð−1Þliα1=3

ð−1Þlþ1iα

1CA:

We solve the full Hartree-Fock Hamiltonian (H ¼H0 þHHF

⬡ ) self-consistently using

hc†κcλi ¼XNϵ;η¼1

U†ϵκUληhγ†ϵγηi ¼

Xϵ

U†ϵκUλϵfðEϵ; μÞ; ðC4Þ

where κ; λ ¼ filσg, U is the unitary transformation diag-onalizing H, γ’s are the eigenvectors, and fðEϵ; μÞ is theFermi-Dirac distribution of the excitation energies Eϵ. Weexplicitly write the dependence on the chemical potential,μ, as we iterate this value to fulfill N−1P

ϵ fðEϵ; μÞ ¼ ν,where ν is the filling.We compute results at charge neutrality (ν ¼ 0.5) with a

total of 600 lattice sites and periodic boundary conditions.The calculations are fully unrestricted; thus, we iterateall ð4 × 600Þ2 mean fields and define the convergence bythe condition that

P jΔEϵj < N × 10−10, where ΔEϵ is thechange of the excitation energies from one iteration to thenext, and N is the total number of states (4 × 600). InTable I, we present an example of the HF calculations,displaying results for t ¼ 1, α ¼ 0.45, andU=W ¼ 0.5. Weset the temperature T ¼ 2.5 × 10−5 in all computations.The values in the table are the renormalized mean fields.It is evident that all hoppings within each hexagonare renormalized due to the interactions. The simplehopping renormalizations, however, do not open a gapin the Dirac cones. The gap is generated directly by themean fields hc†i;1;σci�2;1;σi ¼ −hc†i;2;σci�2;2;σi ¼ �0.0914i,

which explicitly display a spin-degenerate QVH phase, asillustrated in Fig. 1(b) of the main text.In Fig. 7(a), we show the single-particle gap in the QVH

phase with α ¼ 0.45 for several interaction strengths.Figure 7(b) displays the corresponding band structures.The Dirac cone at K in the bare bands is immediatelygapped out when including interactions. The renormaliza-tion initially flattens the bands with a significant gap at allhigh-symmetry points. As U increases, the valence bandgradually develops a peak at Γ while it is pushed downcorrespondingly at K. This behavior results in a gradualshift of the maximal gap value from Γ to K.

/W

U/W = 0.25

U/W = 0.50

U/W = 1.0

M MK

M MK M MK

U/W = 0.25

U/W = 0.50

U/W = 1.0

3.0

2.0

1.0

0.0

E/W

U/W = 0.0

(a)

(b)

E/W

-0.5

0.5

2.0

0.0

3.0

0.0

E/W

E/W

5.0

0.0

FIG. 7. QVH insulator and band structures. (a) Single-particlegap Δ=W along the high-symmetry path of the BZ with α ¼ 0.45and various interaction strengths. The Dirac cones are gapped atinfinitely small U, and the system enters the QVH state. Themaximal gap value gradually shifts from Γ to K. (b) Bandstructures at various interaction strengths with α ¼ 0.45. The top-left plot displays the bare kinetic bands in the absence of anyinteractions. The Dirac cone at K is evident and confirms thesemimetallic phase of the bare bands. The remaining three plotsof (b) present the renormalized band structures with increasingU.The two bands flatten for small U and gradually develop a peakat Γ. All bands are fourfold degenerate, as the QVH phase doesnot break the approximate SUð4Þ symmetry.

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Furthermore, while QMC cannot handle a longer-rangetight-binding model due to the sign problem, the same isnot true for our unrestricted Hartree-Fock calculation.We thus have computed a check with the tight-bindingmodel suggested in Refs. [21,22], including complex fifthnearest-neighbor hopping (t2=t ¼ 0.025� 0.1i), whichbreaks particle-hole symmetry and introduces a splittingalong the ΓM line. We find a complete, quantitativeagreement with the renormalized mean-field results pre-sented in Table I. Thus, in the weak-coupling regime, the

QVH phase is very robust to the addition of long-rangehoppings inH0. The resulting bands with and without long-range hopping can be seen in Fig. 8.Finally, we present results obtained by implementing

the interaction terms found in Ref. [37]. As mentioned inthe discussion section of the main text, we are able to solvethis model within the HF approach as it does not sufferfrom sign problems. The assisted hopping reads

T⬡ ¼X6i¼1

Xl;σ

ð−Þi−1ðc†ilσciþ1lσ þ H:c:Þ: ðC5Þ

The other terms, Q⬡ and H0, remain unchanged. To reachthis expression for T⬡, we have performed the followinggauge transformation,

cilσ → eiθl=2cilσ i odd;

cilσ → e−iθl=2cilσ i even: ðC6ÞThe transformation introduces phases in H0, effectivelycausing t to become complex. We set the phases accordingto Ref. [37], that is, θ1 ¼ −θ2 ¼ 0.743π. The renormalizedmean fields are presented in Table II, where we haveperformed the inverse gauge transformation for directcomparison with Table I. Input parameters are the sameas those used to generate Table I. The result is consistentwith the values presented in Table I and clearly also featuresa QVH phase.

APPENDIX D: STRONG-COUPLING LIMIT ATTHE CHARGE NEUTRALITY POINT

For the system at the charge neutrality point, each unitcell contains four fermions on average. Following the

E/W

-0.5

0.5U/W = 0.0

E/W

2.0

0.0

M MKM MK

U/W = 0.25

3.0

0.0

E/W

U/W = 0.50

E/W

5.0

0.0

U/W = 1.0

M MK M MK

U/W = 0.25

U/W = 0.50

U/W = 1.0

E/W

U/W = 0.0

(b)

E/W

-0.5

0.5

2.0

0.0

3.0

0.0

E/W

E/W

5.0

0.0

(a)

FIG. 8. QVH insulator and band structures with longer hop-pings. Band structures at various interaction strengths with α ¼0.45 and fifth nearest-neighbor hopping (a) t2=t ¼ 0 and(b) t2=t ¼ 0.025� 0.1i. The top-left plots in (a,b) display thebare kinetic bands in the absence of any interactions. The Diraccones at K are evident and confirm the semimetallic phase of thebare bands in both cases. The remaining six plots present therenormalized band structures with increasing U. It is evident thatthe only effect of including the fifth nearest-neighbor hopping isan emergent splitting along Γ −M. This splitting decreases withincreasing U=W.

TABLE I. Mean-field renormalization with U=W ¼ 0.50 and α ¼ 0.45. The input on ½c†ilσ ; cjl0σ � represents the renormalized mean-field parameter hc†ilσcjl0σi. The result is homogeneous and spin degenerate; hence, the listed values contain information about all sitesand flavors. Note that interactions have generated neither spin nor valley mixing. We have subtracted the bare band contributionsevaluated at ν ¼ 0.5 and ignored all mean fields with ðjMFjmax=jMFjÞ > 100.

U=W ¼ 0.50

ci;1;σ ci;2;σ ci�1;1;σ ci�1;2;σ ci�2;1;σ ci�2;2;σ ciþ3;1;σ ciþ3;2;σ

c†i;1;σ � � � � � � −0.0209 � � � �0.0914i � � � 0.0024 � � �c†i;2;σ � � � � � � � � � −0.0209 � � � ∓ 0.0914i � � � 0.0024

TABLE II. Mean-field renormalization with U=W ¼ 0.50 and α ¼ 0.45 using the model from Ref. [37]. The input on ½c†ilσ ; cjl0σ �represents the renormalized mean-field parameter hc†ilσcjl0σi. The result is homogeneous and spin degenerate; hence, the listed valuescontain information about all sites and flavors. We have subtracted the bare band contributions evaluated at ν ¼ 0.5 and ignored all meanfields with ðjMFjmax=jMFjÞ > 100.

U=W ¼ 0.50

ci;1;σ ci;2;σ ci�1;1;σ ci�1;2;σ ci�2;1;σ ci�2;2;σ ciþ3;1;σ ciþ3;2;σ

c†i;1;σ � � � � � � −0.1015 − ð−Þi0.0555i � � � ∓ 0.0945i � � � −0.0806þ ð−Þi0.0300i � � �c†i;2;σ � � � � � � � � � −0.1015þ ð−Þi0.0555i � � � �0.0945i � � � −0.0806 − ð−Þi0.0300i

YUAN DA LIAO et al. PHYS. REV. X 11, 011014 (2021)

011014-12

method applied in Ref. [37], the ground state jΨgri of theinteraction H⬡ should be annihilated by the assisted-hopping operator T⬡ for any hexagon. The most generalform of the wave function is

jΨgri ¼Yi

Xαβ

U1αU2βψ0i;α

†ψ 0i;β

†j∅i; ðD1Þ

where ψ 0i¼ðci;1;↑;ci;1;↓;ð−ÞsðiÞci;2;↑;ð−ÞsðiÞci;2;↓ÞT , andUaγ

is an arbitrary 4 × 4 matrix.

1. H0 = 0

In the case of zero kinetic energy, the manifold of theground states is described by Eq. (D1) and thus can becompared with the numerical results produced by QMC.For this purpose, we consider the correlation function:

Ii;↑ ¼ hc†i;1;↑ci;2;↑ þ H:c:i¼ ð−ÞsðiÞðU�

11U13 þ U�21U23 þ c:cÞ; ðD2Þ

Ii;↓ ¼ hc†i;1;↓ci;2;↓ þ H:c:i¼ ð−ÞsðiÞðU�

12U14 þ U�22U24 þ c:cÞ; ðD3Þ

which leads to

CAAI ¼ 1

L4

Xi;j∈A

⟪ðIi;↑ þ Ii;↓ÞðIj;↑ þ Ij;↓Þ⟫

¼ 1

L4

Xi;j∈A

ð⟪Ii;↑Ij;↑⟫þ ⟪Ii;↓Ij;↓⟫Þ

¼ 2 × ðjU11j2jU13j2 þ jU21j2jU23j2 þ jU12j2jU14j2

þ jU22j2jU24j2Þ ¼1

2:

Note here that ⟪ � � �⟫ is the average over all possible4 × 4 unitary matrices, and therefore, U�

ijUkl ¼ 14δikδlj.

Similarly, we can obtain CBBI ¼ −CAB

I ¼ −CBAI ¼ 1

2.

2. Strong-coupling limit

In this subsection, we assume that H0 is finite but smallcompared with H⬡. Since the kinetic terms break SUð4Þsymmetry, the ground-state manifold shrinks and becomesa subset of the manifold described in Eq. (D1). Our purposehere is to identify the new manifold of the ground states andshow that it is independent of the exact form of kineticterms as along as it breaks the SUð4Þ symmetry describedin the main text.For the convenience of calculation, we write Eq. (D1) in

the following form,

jΨGSi ¼Yi

ðα1c†i;1;n þ ð−ÞsðiÞα2c†i;2;mÞ

× ðγðα�2c†i;1;n − ð−ÞsðiÞα�1c†i;2;mÞ þ β1c†i;1;−n

þ ð−ÞsðiÞβ2c†i;2;−mÞj∅i; ðD4Þ

where sðiÞ ¼ 0 and 1 if the site i is on sublattice A and B,respectively. Here, n and m are two arbitrary spin quan-tization directions, and α1, α2, β1, and β2 are four complexvariables that satisfy jα1j2þjα2j2¼jγj2þjβ1j2þjβ2j2¼1.Furthermore, consider the hopping between two sites.

We show next that the energy is minimized whenjα1j ¼ jα2j ¼ 1=

ffiffiffi2

p, γ ¼ 0, and jβ1j ¼ jβ2j ¼ 1=

ffiffiffi2

p.

Applying the second-order perturbation theory, the correc-tion to the energy of the ground state is found to be

δE ¼Xn

jhnjH0jΨGSij2E0 − En

;

where n sums over all the excited states. Since it is almostimpossible to obtain the exact spectrum of the excitedstates, we will maximize the term

Pn jhnjH0jΨGSij2

instead of δE. Furthermore, note that hGSjH0jΨGSi ¼ 0,and write H0 ¼

Pij Kði; jÞ, where i and j refer to the

honeycomb lattice site; we obtain

Xn

jhnjH0jΨGSij2 ¼ kH0jΨGSik2 ¼Xij

kKði; jÞjΨGSik2:

For notational convenience, it is worth introducingf†i;a ¼

Pα ψ

0i;α

†Uαa, where U is a unitary 4 × 4 matrix,and the ground state is given by

jΨGSi ¼Yi

Y2a¼1

f†i;1f†i;2j∅i:

From Eq. (D4), it is obvious that

U1α ¼ ðα1; 0; α2; 0Þ U2α ¼ ðγα�2; β1;−γα�1; β2Þ:

We introduce the diagonal matrix

T ¼ diagðt; t; ð−ÞsðiÞþsðjÞt�; ð−ÞsðiÞþsðjÞt�Þ;

where t is the hopping constant from site j to site i forvalley 1. The kinetic terms can be written as

Kði; jÞ ¼ ψ 0i;α

†Tαβψ j;β ¼ f†i;aðU†TUÞabfj;b:

Now, it is easy to derive

CORRELATION-INDUCED INSULATING TOPOLOGICAL PHASES … PHYS. REV. X 11, 011014 (2021)

011014-13

kKði; jÞjΨGSik2

¼Xi;j

X4a¼3

X2b¼1

TiiT�jjU

�iaUibUjaU�

jb

¼Xij

TiiT�jj

�δij −

X2a¼1

U�iaUja

�UibU�

jb

¼Xi

jTiij2X2b¼1

jUibj2 −Xij

X2ab¼1

TiiT�jjU

�iaUibUjaU�

jb

¼ 8jtj2 −Xij

X2ab¼1

TiiT�jjU

�iaUibUjaU�

jb: ðD5Þ

Since the first term is independent of the form of the groundstate, we need to minimize the last term. Notice the Tmatrix can be written as

T ¼ t0I4×4 þ t1diagð1; 1;−1;−1Þ;

and t0;1 ¼ 12ðt� ð−ÞsðiÞþsðjÞt�Þ, which leads to t0t�1, is a

pure imaginary number. Thus, we find

kKði; jÞjΨGSik2 ¼ 8jtj2 − 8jt0j2 − jt1j2Xab

jMabj2; ðD6Þ

where M is a 2 × 2 matrix given by

M ¼� jα1j2 − jα2j2 2γα�1α

�2

2γ�α1α2 −jγj2ðjα1j2 − jα2j2Þ þ jβ1j2 − jβ2j

�:

Clearly, the energy due to the second-order perturbationis minimized when jα1j ¼ jα2j ¼ 1=

ffiffiffi2

p, γ ¼ 0, and

jβ1j ¼ jβ2j ¼ 1=ffiffiffi2

p.

It is worth emphasizing that this result is independent ofthe exact form of the kinetic terms. As long as the hoppingsbreak the SUð4Þ symmetry, the second-order perturbationalways leads to the same manifold of the ground states.As a consequence, the ground state is an equal mixture of

two valleys. It is easy to obtain

hc†i;1;nci;2;mi ¼ ð−ÞsðiÞα�1α2;hc†i;1;nci;2;−mi ¼ hc†i;1;−nci;2;mi ¼ 0;

hc†i;1;−nci;2;−mi ¼ ð−ÞsðiÞβ�1β2: ðD7Þ

Suppose that n ¼ ðsin θ cosϕ; sin θ sinϕ; cos θÞ andm ¼ ðsin θ0 cosϕ0; sin θ0 sinϕ0; cos θ0Þ. We obtain

Ii;↑ ¼ hc†i;1;↑ci;2;↑ þ H:c:i ¼ ð−ÞsðiÞ�cos

θ

2cos

θ0

2α�1α2 þ sin

θ

2sin

θ0

2eiðϕ−ϕ0Þβ�1β2 þ c:c

�; ðD8Þ

Ii;↓ ¼hc†i;1;↓ci;2;↓ þ H:c:i ¼ ð−ÞsðiÞ�cos

θ

2cos

θ0

2β�1β2 − sin

θ

2sin

θ0

2eiðϕ−ϕ0Þα�1α2 þ c:c

�: ðD9Þ

As a consequence, when averaging over all the possible configurations of the ground states, we obtain

CAAI ¼ 1

L4

Xi;j∈A

⟪ðIi;↑ þ Ii;↓ÞðIj;↑ þ Ij;↓Þ⟫ ¼ 1L4

Xi;j∈A

ð⟪Ii;↑Ij;↑⟫þ ⟪Ii;↓Ij;↓⟫Þ

¼ 2 ×

�⟪cos2

θ

2cos2

θ0

2⟫ðjα1j2jα2j2 þ jβ1j2jβ2j2Þ þ ⟪sin2

θ

2sin2

θ0

2⟫ðjα1j2jα2j2 þ jβ1j2jβ2j2Þ

�; ðD10Þ

where ⟪ � � �⟫ refers to the average over the directions n andm, as well as the phases of α1, α2, β1, and β2. Averagingover n and m on the sphere, we obtain ⟪cos2 θ

2⟫ ¼⟪cos2 θ0

2 ⟫ ¼ ⟪sin2 θ2⟫ ¼ ⟪sin2 θ0

2 ⟫ ¼ 12. Thus, CAA

I ¼ 12.

Similarly, we can obtain CBBI ¼ −CAB

I ¼ −CBAI ¼ 1

2, which

is the same as the case of zero kinetic energy and consistentwith the QMC result in the limitU=W → ∞ in Appendix A.

[1] R. Bistritzer and A. H. MacDonald,Moire Bands in TwistedDouble-Layer Graphene, Proc. Natl. Acad. Sci. U.S.A. 108,12233 (2011).

[2] 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, CorrelatedInsulator Behaviour at Half-Filling in Magic-Angle Gra-phene Superlattices, Nature (London) 556, 80 (2018).

[3] 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).

[4] C. Shen, Y. Chu, Q. Wu, N. Li, S. Wang, Y. Zhao, J. Tang, J.Liu, J. Tian, K. Watanabe, T. Taniguchi, R. Yang, Z. Y.Meng, D. Shi, O. V. Yazyev, and G. Zhang, CorrelatedStates in Twisted Double Bilayer Graphene, Nat. Phys. 16,520 (2020).

YUAN DA LIAO et al. PHYS. REV. X 11, 011014 (2021)

011014-14

[5] X. Liu, Z. Hao, E. Khalaf, J. Y. Lee, K. Watanabe, T.Taniguchi, A. Vishwanath, and P. Kim, Tunable Spin-Polarized Correlated States in Twisted Double BilayerGraphene, Nature (London) 583, 221 (2020).

[6] 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, Nature (London) 583, 215 (2020).

[7] 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 Moire Superlattice, Nature (London) 579, 56 (2020).

[8] A. Kerelsky, L. J. McGilly, D. M. Kennes, L. Xian, M.Yankowitz, S. Chen, K. Watanabe, T. Taniguchi, J. Hone, C.Dean et al., Maximized Electron Interactions at the MagicAngle in Twisted Bilayer Graphene, Nature (London) 572,95 (2019).

[9] S. L. Tomarken, Y. Cao, A. Demir, K. Watanabe, T.Taniguchi, P. Jarillo-Herrero, and R. C. Ashoori, ElectronicCompressibility of Magic-Angle Graphene Superlattices,Phys. Rev. Lett. 123, 046601 (2019).

[10] 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).

[11] Y. Xie, B. Lian, B. Jäck, X. Liu, C.-L. Chiu, K. Watanabe, T.Taniguchi, B. A. Bernevig, and A. Yazdani, SpectroscopicSignatures of Many-Body Correlations in Magic-AngleTwisted Bilayer Graphene, Nature (London) 572, 101(2019).

[12] Y. Jiang, X. Lai, K. Watanabe, T. Taniguchi, K. Haule, J.Mao, and E. Y. Andrei, Charge Order and Broken Rota-tional Symmetry in Magic-Angle Twisted Bilayer Graphene,Nature (London) 573, 91 (2019).

[13] 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).

[14] 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).

[15] Y. Saito, J. Ge, K. Watanabe, T. Taniguchi, and A. F. Young,Independent Superconductors and Correlated Insulators inTwisted Bilayer Graphene, Nat. Phys. 16, 926 (2020).

[16] P. Stepanov, I. Das, X. Lu, A. Fahimniya, K. Watanabe, T.Taniguchi, F. H. Koppens, J. Lischner, L. Levitov, and D. K.Efetov, Untying the Insulating and Superconducting Ordersin Magic-Angle Graphene, Nature (London) 583, 375(2020).

[17] 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 MoireSuperlattice, Nat. Phys. 15, 237 (2019).

[18] G. Chen, A. L. Sharpe, P. Gallagher, I. T. Rosen, E. J. Fox,L. Jiang, B. Lyu, H. Li, K. Watanabe, T. Taniguchi et al.,Signatures of Tunable Superconductivity in a Trilayer

Graphene Moire Superlattice, Nature (London) 572, 215(2019).

[19] C. Xu and L. Balents, Topological Superconductivity inTwisted Multilayer Graphene, Phys. Rev. Lett. 121, 087001(2018).

[20] J. Kang and O. Vafek, Symmetry, Maximally LocalizedWannier States, and a Low-Energy Model for TwistedBilayer Graphene Narrow Bands, Phys. Rev. X 8,031088 (2018).

[21] M. Koshino, N. F. Q. Yuan, T. Koretsune, M. Ochi, K.Kuroki, and L. Fu, Maximally Localized Wannier Orbitalsand the Extended Hubbard Model for Twisted BilayerGraphene, Phys. Rev. X 8, 031087 (2018).

[22] N. F. Q. Yuan and L. Fu, Model for the Metal-InsulatorTransition in Graphene Superlattices and Beyond, Phys.Rev. B 98, 045103 (2018).

[23] H. C. Po, L. Zou, A. Vishwanath, and T. Senthil, Origin ofMott Insulating Behavior and Superconductivity in TwistedBilayer Graphene, Phys. Rev. X 8, 031089 (2018).

[24] C.-C. Liu, L.-D. Zhang, W.-Q. Chen, and F. Yang, ChiralSpin Density Wave and dþ id Superconductivity in theMagic-Angle-Twisted Bilayer Graphene, Phys. Rev. Lett.121, 217001 (2018).

[25] L. A. Gonzalez-Arraga, J. L. Lado, F. Guinea, and P.San-Jose, Electrically Controllable Magnetism in TwistedBilayer Graphene, Phys. Rev. Lett. 119, 107201 (2017).

[26] M. Ochi, M. Koshino, and K. Kuroki, Possible CorrelatedInsulating States in Magic-Angle Twisted Bilayer Grapheneunder Strongly Competing Interactions, Phys. Rev. B 98,081102(R) (2018).

[27] J. F. Dodaro, S. A. Kivelson, Y. Schattner, X. Q. Sun, and C.Wang, Phases of a Phenomenological Model of TwistedBilayer Graphene, Phys. Rev. B 98, 075154 (2018).

[28] H. Guo, X. Zhu, S. Feng, and R. T. Scalettar, PairingSymmetry of Interacting Fermions on a Twisted BilayerGraphene Superlattice, Phys. Rev. B 97, 235453 (2018).

[29] H. Isobe, N. F. Q. Yuan, and L. Fu, Unconventional Super-conductivity and Density Waves in Twisted Bilayer Gra-phene, Phys. Rev. X 8, 041041 (2018).

[30] J. W. F. Venderbos and R. M. Fernandes, Correlations andElectronic Order in a Two-Orbital Honeycomb LatticeModel for Twisted Bilayer Graphene, Phys. Rev. B 98,245103 (2018).

[31] F. Guinea and N. R. Walet, Electrostatic Effects, BandDistortions, and Superconductivity in Twisted GrapheneBilayers, Proc. Natl. Acad. Sci. U.S.A. 115, 13174(2018).

[32] J. Liu, J. Liu, and X. Dai, Pseudo Landau Level Repre-sentation of Twisted Bilayer Graphene: Band Topology andImplications on the Correlated Insulating Phase, Phys. Rev.B 99, 155415 (2019).

[33] 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).

[34] T. Cea, N. R.Walet, and F. Guinea, Electronic Band Structureand Pinning of Fermi Energy to van Hove Singularities inTwisted Bilayer Graphene: A Self-Consistent Approach,Phys. Rev. B 100, 205113 (2019).

CORRELATION-INDUCED INSULATING TOPOLOGICAL PHASES … PHYS. REV. X 11, 011014 (2021)

011014-15

[35] Q.-K. Tang, L. Yang, D. Wang, F.-C. Zhang, and Q.-H.Wang, Spin-Triplet f-Wave Pairing in Twisted BilayerGraphene near 1

4-Filling, Phys. Rev. B 99, 094521 (2019).

[36] J. González and T. Stauber, Kohn-Luttinger Superconduc-tivity in Twisted Bilayer Graphene, Phys. Rev. Lett. 122,026801 (2019).

[37] J. Kang and O. Vafek, Strong Coupling Phases of PartiallyFilled Twisted Bilayer Graphene Narrow Bands, Phys. Rev.Lett. 122, 246401 (2019).

[38] K. Seo, V. N. Kotov, and B. Uchoa, Ferromagnetic MottState in Twisted Graphene Bilayers at the Magic Angle,Phys. Rev. Lett. 122, 246402 (2019).

[39] Y.-H. Zhang, D. Mao, Y. Cao, P. Jarillo-Herrero, and T.Senthil, Nearly Flat Chern Bands in Moire Superlattices,Phys. Rev. B 99, 075127 (2019).

[40] 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, 1 (2019).

[41] F. Wu and S. D. Sarma, Collective Excitations of QuantumAnomalous Hall Ferromagnets in Twisted Bilayer Gra-phene, Phys. Rev. Lett. 124, 046403 (2020).

[42] X.-C. Wu, A. Keselman, C.-M. Jian, K. A. Pawlak, and C.Xu, Ferromagnetism and Spin-Valley Liquid States in MoireCorrelated Insulators, Phys. Rev. B 100, 024421 (2019).

[43] N. Bultinck, S. Chatterjee, and M. P. Zaletel, Mechanismfor Anomalous Hall Ferromagnetism in Twisted BilayerGraphene, Phys. Rev. Lett. 124, 166601 (2020).

[44] S. Liu, E. Khalaf, J. Y. Lee, and A. Vishwanath, NematicTopological Semimetal and Insulator in Magic AngleBilayer Graphene at Charge Neutrality, arXiv:1905.07409.

[45] Y. Alavirad and J. D. Sau, Ferromagnetism and Its Stabilityfrom the One-Magnon Spectrum in Twisted Bilayer Gra-phene, arXiv:1907.13633.

[46] S. Chatterjee, N. Bultinck, and M. P. Zaletel, SymmetryBreaking and Skyrmionic Transport in Twisted BilayerGraphene, Phys. Rev. B 101, 165141 (2020).

[47] D. V. Chichinadze, L. Classen, and A. V. Chubukov, Nem-atic Superconductivity in Twisted Bilayer Graphene, Phys.Rev. B 101, 224513 (2020).

[48] N. Bultinck, E. Khalaf, S. Liu, S. Chatterjee, A. Vishwanath,and M. P. Zaletel, Ground State and Hidden Symmetry ofMagic-Angle Graphene at Even Integer Filling, Phys. Rev.X 10, 031034 (2020).

[49] J. Liu and X. Dai, Theories for the Correlated InsulatingStates and Quantum Anomalous Hall Phenomena inTwisted Bilayer Graphene, arXiv:1911.03760.

[50] R. M. Fernandes and J. W. Venderbos, Nematicity with aTwist: Rotational Symmetry Breaking in a Moire Super-lattice, Sci. Adv. 6, eaba8834 (2020).

[51] Y. Zhang, K. Jiang, Z. Wang, and F. Zhang, CorrelatedInsulating Phases of Twisted Bilayer Graphene at Com-mensurate Filling Fractions: A Hartree-Fock Study, Phys.Rev. B 102, 035136 (2020).

[52] C. Repellin, Z. Dong, Y.-H. Zhang, and T. Senthil, Ferro-magnetism in Narrow Bands of Moire Superlattices, Phys.Rev. Lett. 124, 187601 (2020).

[53] J. Liu and X. Dai, Anomalous Hall Effect, Magneto-OpticalProperties, and Nonlinear Optical Properties of TwistedGraphene Systems, npj Comput. Mater. 6, 57 (2020).

[54] J. Kang and O. Vafek, Non-Abelian Dirac Node Braidingand Near-Degeneracy of Correlated Phases at Odd IntegerFilling in Magic-Angle Twisted Bilayer Graphene, Phys.Rev. B 102, 035161 (2020).

[55] S.-M. Huang, Y.-P. Huang, and T.-K. Lee, Slave-RotorTheory on Magic-Angle Twisted Bilayer Graphene, Phys.Rev. B 101, 235140 (2020).

[56] C. Lu, Y. Zhang, Y. Zhang, M. Zhang, C.-C. Liu, Z.-C. Gu,W.-Q. Chen, and F. Yang, Chiral SO(4) Spin-ChargeDensity Wave and Degenerate Topological Superconduc-tivity in Magic-Angle-Twisted Bilayer-Graphene, arXiv:2003.09513.

[57] S.-Y. Li, Y. Zhang, Y.-N. Ren, J. Liu, X. Dai, and L. He,Experimental Evidence for Orbital Magnetic MomentsGenerated by Moire-Scale Current Loops in TwistedBilayer Graphene, Phys. Rev. B 102, 121406 (2020).

[58] Y. Wang, J. Kang, and R. M. Fernandes, Topological andNematic Superconductivity Mediated by Ferro-SU(4) Fluc-tuations in Twisted Bilayer Graphene, arXiv:2009.01237.

[59] T. Wang, N. Bultinck, and M. P. Zaletel, Flat Band Top-ology of Magic Angle Graphene on a Transition MetalDichalcogenide, arXiv:2008.06528.

[60] M. Christos, S. Sachdev, and M. Scheurer, Superconduc-tivity, Correlated Insulators, and Wess-Zumino-WittenTerms in Twisted Bilayer Graphene, arXiv:2007.00007.

[61] V. Kozii, M. P. Zaletel, and N. Bultinck, Superconductivityin a Doped Valley Coherent Insulator in Magic AngleGraphene: Goldstone-Mediated Pairing and Kohn-Luttinger Mechanism, arXiv:2005.12961.

[62] W.-Y. He, D. Goldhaber-Gordon, and K. T. Law, GiantOrbital Magnetoelectric Effect and Current-Induced Mag-netization Switching in Twisted Bilayer Graphene, Nat.Commun. 11, 1650 (2020).

[63] X. Y. Xu, K. T. Law, and P. A. Lee, Kekule Valence BondOrder in an Extended Hubbard Model on the HoneycombLattice with Possible Applications to Twisted BilayerGraphene, Phys. Rev. B 98, 121406(R) (2018).

[64] Y. Da Liao, Z. Y. Meng, and X. Y. Xu, Valence Bond Ordersat Charge Neutrality in a Possible Two-Orbital ExtendedHubbard Model for Twisted Bilayer Graphene, Phys. Rev.Lett. 123, 157601 (2019).

[65] Y. Da Liao, X. Y. Xu, Z. Y. Meng, and J. Kang, CorrelatedInsulating Phases in the Twisted Bilayer Graphene, arXiv:2009.10076.

[66] T. Soejima, D. E. Parker, N. Bultinck, J. Hauschild, and M.P. Zaletel, Efficient Simulation of Moire Materials Using theDensity Matrix Renormalization Group, Phys. Rev. B 102,205111 (2020).

[67] F. Xie, A. Cowsik, Z.-D. Song, B. Lian, B. A. Bernevig, andN. Regnault, TBG VI: An Exact Diagonalization Studyof Twisted Bilayer Graphene at Non-Zero Integer Fillings,arXiv:2010.00588.

[68] O. Vafek and J. Kang, Towards the Hidden Symmetry inCoulomb Interacting Twisted Bilayer Graphene: Renorm-alization Group Approach, arXiv:2009.09413.

[69] A. L. Sharpe, E. J. Fox, A. W. Barnard, J. Finney, K.Watanabe, T. Taniguchi, M. Kastner, and D. Goldhaber-Gordon, Emergent Ferromagnetism near Three-QuartersFilling in Twisted Bilayer Graphene, Science 365, 605(2019).

YUAN DA LIAO et al. PHYS. REV. X 11, 011014 (2021)

011014-16

[70] M. Serlin, C. Tschirhart, H. Polshyn, Y. Zhang, J. Zhu, K.Watanabe, T. Taniguchi, L. Balents, and A. Young, IntrinsicQuantized Anomalous Hall Effect in a Moire Heterostruc-ture, Science 367, 900 (2020).

[71] M. Xie and A. H. MacDonald, Nature of the CorrelatedInsulator States in Twisted Bilayer Graphene, Phys. Rev.Lett. 124, 097601 (2020).

[72] T. Cea and F. Guinea, Band Structure and Insulating StatesDriven by Coulomb Interaction in Twisted Bilayer Gra-phene, Phys. Rev. B 102, 045107 (2020).

[73] T. C. Lang, Z. Y. Meng, A. Muramatsu, S. Wessel, and F. F.Assaad, Dimerized Solids and Resonating Plaquette Orderin SUðNÞ-Dirac Fermions, Phys. Rev. Lett. 111, 066401(2013).

[74] Z. Zhou, D. Wang, Z. Y. Meng, Y. Wang, and C. Wu, MottInsulating States and Quantum Phase Transitions of Cor-related SUð2NÞ Dirac Fermions, Phys. Rev. B 93, 245157(2016).

[75] P. Moon and M. Koshino, Energy Spectrum and QuantumHall Effect in Twisted Bilayer Graphene, Phys. Rev. B 85,195458 (2012).

[76] Z. Y. Meng, T. C. Lang, S. Wessel, F. F. Assaad, and A.Muramatsu, Quantum Spin Liquid Emerging in Two-Dimensional Correlated Dirac Fermions, Nature (London)464, 847 (2010).

[77] X. Y. Xu, K. S. D. Beach, K. Sun, F. F. Assaad, and Z. Y.Meng, Topological Phase Transitions with SO(4) Symmetry

in ð2þ 1ÞD Interacting Dirac Fermions, Phys. Rev. B 95,085110 (2017).

[78] Y.-Y. He, X. Y. Xu, K. Sun, F. F. Assaad, Z. Y. Meng, andZ.-Y. Lu, Dynamical generation of topological masses indirac fermions, Phys. Rev. B 97, 081110(R) (2018).

[79] Y. Liu, W. Wang, K. Sun, and Z. Y. Meng, DesignerMonte Carlo Simulation for the Gross-Neveu-YukawaTransition, Phys. Rev. B 101, 064308 (2020).

[80] F. D. M. Haldane, Model for a Quantum Hall Effect withoutLandau Levels: Condensed-Matter Realization of the“Parity Anomaly”, Phys. Rev. Lett. 61, 2015 (1988).

[81] M. Hohenadler, Z. Y. Meng, T. C. Lang, S. Wessel, A.Muramatsu, and F. F. Assaad, Quantum Phase Transitionsin the Kane-Mele-Hubbard Model, Phys. Rev. B 85, 115132(2012).

[82] Y.-Y. He, H.-Q. Wu, Y.-Z. You, C. Xu, Z. Y. Meng, andZ.-Y. Lu, Bona Fide Interaction-Driven Topological PhaseTransition in Correlated Symmetry-Protected TopologicalStates, Phys. Rev. B 93, 115150 (2016).

[83] H. C. Po, H. Watanabe, and A. Vishwanath, Fragile Top-ology and Wannier Obstructions, Phys. Rev. Lett. 121,126402 (2018).

[84] F. Assaad and H. Evertz, World-Line and DeterminantalQuantum Monte Carlo Methods for Spins, Phonons andElectrons, in Computational Many-Particle Physics, editedby H. Fehske, R. Schneider, and A. Weiße (Springer, Berlin,Heidelberg, 2008), pp. 277–356.

CORRELATION-INDUCED INSULATING TOPOLOGICAL PHASES … PHYS. REV. X 11, 011014 (2021)

011014-17