Quantum Anomalous Hall Effect from Inverted Charge Transfer Gap

Trithep Devakul and Liang FuDepartment of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

(Received 3 November 2021; revised 24 January 2022; accepted 9 March 2022; published 6 May 2022)

A general mechanism is presented by which topological physics arises in strongly correlated systemswithout flat bands. Starting from a charge transfer insulator, topology emerges when the charge transferenergy between the cation and anion is reduced to invert the lower Hubbard band and the spin-degeneratecharge transfer band. A universal low-energy theory is developed for the inversion of the charge transfergap in a quantum antiferromagnet. The inverted state is found to be a quantum anomalous Hall (QAH)insulator with noncoplanar magnetism. Interactions play two essential roles in this mechanism: producingthe insulating gap and quasiparticle bands prior to the band inversion, and causing the change of magneticorder necessary for the QAH effect after inversion. Our theory explains the electric-field-induced transitionfrom a correlated insulator to a QAH state in AB-stacked transition-metal-dichalcogenides bilayerMoTe2=WSe2.

DOI: 10.1103/PhysRevX.12.021031 Subject Areas: Condensed Matter PhysicsStrongly Correlated Materials

Electron correlation and band topology are two pivotalthemes of quantum matter theory, which are deeply rootedin the particle and wave aspect of electrons, respectively.The opposing traits of correlation and topology are clearlydisplayed in the contrast between a Mott insulator [1] and aChern insulator [2]. In a Mott insulator, electrons are boundto individual atoms, and their motion is inhibited by mutualCoulomb repulsion. In contrast, a Chern insulator featureschiral electrons on the boundary that refuse to localize.Despite their differences, electron correlation and bandtopology can cooperate to create fascinating states ofmatter, as exemplified by quantum Hall systems. Morerecently, the scope of topology has been greatly expandedby the discovery of topological band insulators [3,4] innumerous semiconductor materials [5–8]. Since then, therehas been great interest in interaction-induced topologicalstates in correlated electron systems such as transition metaloxides and f-electron materials [9–12]. However, afterconsiderable effort, it remains unclear whether there is acommon mechanism for topological physics in genericHubbard-type systems.This work is an attempt to provide a guiding principle for

the realization of strongly correlated topological states inmaterials with an odd number of electrons per unit cell.Building on and linking together the notions of Hubbardband, charge transfer gap, and topological band inversion,

we find a simple and natural mechanism leading tomagnetic topological insulators that exhibit noncollinearspin structures and a quantum anomalous Hall effect.For systems with an odd number of electrons per unit

cell, a large enough Coulomb repulsion U can suppressdouble occupancy and produce an insulating state. In one-band Hubbard models at large U, the single-particlespectral function consists of the lower and upperHubbard bands separated by the Mott gap [13]. Moreinteresting and relevant to our work are charge-transferinsulators [14–16] (such as the cuprate). These materialsare comprised of cations (Cu) and anions (O). Transferringan electron between the anion and the cation withoutcreating double occupancy costs energy less than U. Thephysics of charge transfer insulators is captured by two-band Hubbard models, where the band derived from anionsis located inside the Mott gap of the cation states. Theinsulating gap is thus controlled by the charge transferenergy Δ rather than U.The essence of our idea is that reducing the charge

transfer energyΔ can induce band inversion between cationand anion Hubbard bands and thereby drive a transitionfrom a Mott insulator to a topologically nontrivial state inwhich cations and anions are highly entangled. Theinsulating state that emerges after this transition can beviewed as having a negative charge transfer gap, in analogywith a negative band gap in inverted semiconductors [17].The band inversion paradigm is remarkably successful in

understanding and predicting topological band insulators[5,18–22]. A prime example is SnTe, in which the cation Snband and anion Te band are inverted around the L points inthe Brillouin zone [21]. The crucial difference here is that

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 12, 021031 (2022)

2160-3308=22=12(2)=021031(14) 021031-1 Published by the American Physical Society

we consider correlated insulators with a many-body gap athalf-filling. The quasiparticle bands that emerge in suchinsulators generally bear no resemblance to the band struc-ture in the noninteracting limit. Specifically, we show that theinversion of cation and anion Hubbard bands in an anti-ferromagnetic (AFM) charge transfer insulator leads to aChern insulator with canted AFM order, as shown schemati-cally in Fig. 1. In order for this mechanism to work, certainconditions regarding the Hubbard band structure and thecation-anion hybridization must be satisfied.In bulk materials, the charge transfer energy Δ is largely

determined by the chemistry of the underlying cation andanion. Recently, moire superlattices in semiconductorheterostructures [23–28] offered a physical realization ofcharge transfer insulators [29–31], where Δ is highlytunable by the displacement field. Remarkably, a displace-ment-field-induced transition from a correlated insulator toa quantum anomalous Hall (QAH) state has been discov-ered in AB-stacked MoTe2=WSe2 heterostructures [32,33].However, the origin and nature of this QAH state is notunderstood. Our theory explains the origin of this observedphenomenon and makes testable predictions.To illustrate the emergence of topology from inverted

Hubbard bands, we introduce a two-band Hubbard modelof spin-1

2electrons on the honeycomb lattice, with the A and

B sublattice representing the cation and anion, respectively:

H ¼ HA þHB þHAB þ UA

Xi∈A

ni↑ni↓ þ UB

Xi∈B

ni↑ni↓:

ð1Þ

Unless stated otherwise, we take UA ¼ UB ¼ U. Here, HAand HB are the tight-binding Hamiltonians within eachsublattice, while HAB is the hybridization term:

Hα ¼ −Xhi;jiα;σ

ðtαeisσνijϕαc†iσcjσ þ H:c:Þ −Xi∈α

ταΔ2

ni;

HAB ¼ −tABXhi;ji;σ

c†iσcjσ; ð2Þ

where α ¼ A, B denotes the two sublattices; σ ¼↑;↓electron spin Sz; s↑ ¼ −s↓ ≡ 1; τA ¼ −τB ≡ 1. The firstsum is over next-nearest-neighbor sites hi; jiα, whichbelong to the same sublattice α. Here, the hoppingamplitude can be complex and spin-sz dependent, describ-ing an Ising spin-orbit coupling that is allowed by sym-metry, with νij ¼ −νji ¼ �1 depending on the pathconnecting site j to i [34]. If the path turns right,νij ¼ 1; otherwise, νij ¼ −1. The special case ϕA ¼ ϕB ¼ðπ=2Þ corresponds to the original Kane-Mele model. Thesecond term is a sublattice potential, with Δ the chargetransfer energy. The last term is a hybridization term thatconnects nearest-neighbor sites hi; ji.Our model is motivated by and captures the essential

physics of �K-valley spin-polarized moire bands in tran-sition-metal-dichalcogenides (TMD) bilayers. Consider,for example, MoTe2=WSe2. Its low-energy electron statesreside primarily in the MM region of the MoTe2 layer andthe XX region of the WSe2 layer, respectively, whichtogether form a honeycomb lattice [Fig. 5(b)] [33]. Thecorresponding moire bands are well described by our tight-binding Hamiltonian H ¼ HA þHB þHAB, with the sub-lattice and spin σ corresponding to the layer and the �Kvalley, respectively. The charge transfer energy Δcorresponds to the layer bias potential, which is tunedby an applied displacement field. As we show inAppendix B, the relevant model parameters are ϕA ≈ 0and ϕB ≈ −½ð2πÞ=3�. We focus on the case of ϕA ¼ 0 in ourfollowing analysis.At Δ → ∞, our model effectively reduces to the standard

one-band Hubbard model on the triangular lattice of A sites.As is well known, at the filling n ¼ 1 considered through-out this work, the ground state at large U is a 120° AFMordered Mott insulator. For large (but finite) Δ, integratingout the B sublattice gives an effective A sublattice Hubbardmodel with complex sz-dependent hopping parameters.Consequently, the effective spin model derived from ourHubbard model at large U is an XXZ Heisenberg modelwith a Dzyaloshinskii-Moriya interaction. The reducedspin Uð1Þ symmetry results in the 120° AFM within the

FIG. 1. Illustration of our mechanism for topological Hubbardband inversion. In the top panels, we start with a charge transferinsulator with 120° xy AFM order on the A (red) sublattice. The Aquasiparticle band is split into a filled lower Hubbard band and anupper Hubbard band separated by energy U. The spin-degenerateB band lies in the Hubbard gap, resulting in a charge transferinsulator with charge transfer energy Δ. In the bottom panels,when Δ is reduced, a topological band inversion occurs. TheHubbard interaction on the B (blue) sublattice results in spinsplitting of the B bands and noncoplanar spin order, as illustrated.The filled band has nontrivial Chern number and exhibits QAH.

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xy plane, as shown in Ref. [35]. This AFM insulator servesas the starting point of our analysis below.In order to obtain the complete phase diagram of our

model, we perform a Hartree-Fock (HF) treatment of theHubbard interaction. We expect the HF treatment to givequalitatively reasonable results for insulating phases, andmore importantly, it will allow for an analytical under-standing of the key physics through the quasiparticle bandstructure. Our main findings from the HF study are fullysupported by density matrix renormalization group(DMRG) calculations, to be presented below.The HF approximation amounts to the replacement of

the Hubbard term by

HHFHub¼

U2

Xi

�nihnii− si · hsii−

1

2hnii2þ

1

2jhsiij2

�; ð3Þ

where ni ¼P

σ niσ , ski ¼ hc†iσπk

σσ0ciσ0 i, and πk are k ¼ x, y,z Pauli matrices. The HF Hamiltonian must be solved self-consistently at the filling n ¼ 1 to obtain the HF groundstate (in cases where multiple self-consistent solutions arefound, the lowest energy solution among them is chosen).In order to characterize the magnetic order, we define the

following order parameter, a matrix in spin space:

Sαζσ0σ ≡1

N

Xk

hc†ασ0ðkþζKÞcασki; ð4Þ

where ζ ¼ 0 describes ferromagnetic (FM) states andζ ¼ �1 describes AFM states with wave vector �K ¼�½ð4πÞ=3a�ð1; 0Þ. In the presence of a spin-orbit interaction,xy AFM states with ζ ¼ þ and − are distinct statesdisplaying spin configurations of opposite chirality, andthey are not degenerate [36]. For ϕA ¼ 0 and ϕB≈−½ð2πÞ=3�, the AFM Mott insulator at large Δ has ζ ¼ −1.The order parameters are shown in Fig. 2(a) as a function

of Δ near the band inversion point for ϕB ¼ −½ð2πÞ=3�,tA ¼ tB ¼ 1

2tAB ¼ t, and U ¼ 50t. We have defined the

combinations Zα ≡ 12ðSα0↑↑ − Sα0↓↓Þ and XYα ≡ jSαðζ¼−1Þ

↑↓ j,which capture the observed nonzero z FM and xy AFMorders, respectively, on the α sublattice. Figure 2(b) alsoshows the charge gap and Chern number of the HF groundstate. We observe two distinct insulating phases: an xyAFM (with ζ ¼ −) on the A sublattice at large Δ transitionsinto a canted xy AFM as Δ is decreased. This canted phase,in particular, has nontrivial Chern number jCj ¼ 1 and istherefore a QAH phase. This QAH phase with noncoplanarmagnetism appearing at reduced charge transfer energy Δis the highlight of this work.To gain insight into the origin of the QAH phase, we exa-

mine the evolution of the quasiparticle band structure as afunction of Δ. As a first step, it is useful to first derive thenoninteracting band structure at U ¼ 0. By Fourier transfor-mation, the single-particle HamiltonianH0¼HAþHBþHAB

takes the form H0 ¼P

kσ c†σkHσðkÞcσk, where c†σk ¼

ðc†Aσk; c†BσkÞT in k space, and the Bloch Hamiltonian is

HσðkÞ ¼�EAσðkÞ TσðkÞT†σðkÞ EBσðkÞ

�; ð5Þ

where

EασðkÞ ¼ −2tαXn

cosðk · an þ sσταϕαÞ −1

2ταΔ; ð6Þ

TσðkÞ ¼ −tABðe−ik·b1 þ e−ik·b2 þ e−ik·b3Þ; ð7Þ

where an ¼ afcos½ð2πnÞ=3�;sin½ð2πnÞ=3�g and bn ¼ða= ffiffiffi

3p Þfsin½ð2πnÞ=3�;− cos½ð2πnÞ=3�g.At large U, the quasiparticle band structure of the xy or

canted AFM insulator is completely different from thenoninteracting case. While the AFM order results in a

ffiffiffi3

p×ffiffiffi

3p

enlarged unit cell, this state is invariant under acombination of the unit translation and spin rotation aroundthe z axis. Thanks to this symmetry property, the descriptionof quasiparticle band structures can be simplified byperform-ing a spin-dependent momentum boost with a unitarytransformation Uζ∶c

†↑k → c†↑ðkþζKÞ; c

†↓k → c†↓ðk−ζKÞ. This

transformation preserves the z FM order and maps the xyAFM order into xy FM order, which is translationallyinvariant. After this transformation, the HF Hamiltonian,which includes the effect of magnetic order, is a 4 × 4matrix(involving the sublattice and spin) given by

(a)

(b) (d)

(c)

FIG. 2. (a) Order parameters and (b) charge gap obtained fromthe self-consistent HF Hamiltonian as a function of Δ, forparameters tA ¼ tB ¼ 1

2tAB ¼ t, ϕA ¼ 0, ϕB ¼ −½ð2πÞ=3�, and

U ¼ 50t. There is a transition from the xy AFM phase to thenoncoplanar QAH phase with Chern number jCj ¼ 1 as Δ isreduced. The gap is also shown for U ¼ 30t. (c) Quasiparticleband structure, with shift ζ ¼ −, obtained from the HF Hamil-tonian atΔ ¼ 7t. (d) Berry curvature of the filled topological band.

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HHFζ ðkÞ ¼

�H↑ðk − ζKÞ þ U2S0↓↓ − 1

2USζ↑↓

− 12UðSζ↑↓Þ� H↓ðkþ ζKÞ þ U

2S0↑↑

�;

ð8Þ

where Sζσ0σ ¼ diagðSAζσ0σ; SBζσ0σÞ.In the limit Δ → ∞, the two sublattices are decoupled

and only the A sublattice is occupied at the filling of n ¼ 1,thus realizing the triangular-lattice Hubbard model. In thexy AFM insulator, the half-filled band splits into lower andupper Hubbard bands Eζ

�ðkÞ, separated by the Mott gap U.In the large-U limit, the lower Hubbard band associatedwith hole excitations has the energy dispersion

Eζ−ðkÞ ¼

1

2½EA↑ðk − ζKÞ þ EA↓ðkþ ζKÞ�: ð9Þ

Since the hopping amplitude of holes between adjacentsites is effectively reduced by the noncollinear spinconfiguration, the bandwidth of holes is smaller than thenoninteracting band but remains finite even as U → ∞.This hole dispersion Eζ

−ðkÞ has a single maximum at Γ,which should be contrasted with the noninteracting band,EAσðkÞ, which has two maxima.As the charge transfer energyΔ decreases belowU, the B

sublattice band lies below the upper Hubbard band on the Asublattice. This leads to a charge transfer insulator, in whichlow-energy hole and electron states reside primarily onA andB sublattices, respectively. While the hole band has a uniquemaximum at Γ (after performing the transformationUζ), thelocation of the electron band minimum depends on the spin-orbit coupling parameter ϕB. For ðπ=3Þ < ζϕB < π, thereexist two degenerateminima: a σ ¼↑ state at ζK and a↓ stateat−ζK, both ofwhich are shifted by the transformationUζ toΓ, coinciding with the hole band maximum. In such a case,the charge transfer insulator has a direct gap.We then ask thefollowing question: What happens if Δ is decreased furtherso as to invert the charge transfer gap?To address this question, we develop a low-energy theory

of hole and electron bands around Γ near the gap inversion.Prior to the gap inversion, the B sublattice is largelyunoccupied; hence, the electron band is spin degenerate.In contrast, because of the xy AFM order, the lowerHubbard band associated with holes on the A sublatticeis spin nondegenerate and comprised of a superposition ofσ ¼↑;↓ states. The two bands are coupled by the hybridi-zation term HAB, which takes a p-wave form near the gap.Taking ζ ¼ −1 and ϕB ¼ −½ð2πÞ=3� as in Fig. 2, we have

Tσðkþ sσKÞ ≈ffiffiffi3

p

2tABasσðkx − isσkyÞ

≡ ffiffiffi2

psσλksσ ; ð10Þ

where k� ≡ kx � iky. By projecting the HF Hamiltonianinto this low-energy subspace, we obtain a k · p theory ofquasiparticle band structure in the xy AFM state prior togap inversion:

HeffðkÞ ¼

0BB@

− k22mA

λk− −λeiθkþ

λkþ k22mB

þ δ 0

−λe−iθk− 0 k22mB

þ δ

1CCA; ð11Þ

with mA ¼ ½2=ð3tAa2Þ� in the large U limit, mB ¼ ½1=ð3tBa2Þ�, and where eiθ reflects the direction of in-planeorder on the A sublattice and δ defines the charge trans-fer gap.As the charge transfer gap δ is decreased and eventually

becomes negative (while the charge transfer energy Δremains positive), the occupation of B sublattices increases;hence, the effect of Hubbard repulsion UB betweenelectrons becomes important. The low-energy theory ofour charge transfer insulator, including the one-particleterm and two-body interaction, is

Heff ¼Xk

f†kiHeffij ðkÞfkj þ g

ZdrnB↑ðrÞnB↓ðrÞ; ð12Þ

where f ¼ ðfA; fB↑; fB↓Þ denotes fermion quasiparticles,

nBσ ¼ f†BσfBσ, and the contact interaction g is proportionalto UB. An additional interaction term nAnB appears in theeffective Hamiltonian H when we include the nearest-neighbor interaction between A and B sites. Our interactingHamiltonian H captures the universal aspects of “Hubbardband inversion” in charge transfer insulators, in the samespirit as the Dirac Hamiltonian encapsulates band inversionin narrow gap semiconductors. However, there are funda-mental differences between the two theories. A chargetransfer insulator has an inherent particle-hole asymmetry:Holes associated with the lower Hubbard band are spinnondegenerate, while electrons associated with the chargetransfer band are spin degenerate prior to the inversion. Asa result, new physics arises after inverting the chargetransfer gap, as shown below.We first analyze the quasiparticle energy spectrum at

g ¼ 0, given by HeffðkÞ. At k ¼ 0 where the hybridizationterm vanishes, the spectrum consists of the spin-non-degenerate Hubbard band from the A sublattice and thespin-degenerate band from the B sublattice. Importantly,the twofold degeneracy of the latter is protected by twosymmetries of the xy AFM state: (1) threefold rotation ofthe lattice and electron spin around a hexagon center (C3);(2) time-reversal transformation combined with a π rotationof spin around the z axis (iszΘ). Note that ðiszÞΘ is anantiunitary symmetry operator that squares to identity,effectively acting as a time-reversal operator in spinless

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systems. Thus, the B band at k ¼ 0 provides a real two-dimensional representation of C3.Prior to gap inversion (δ > 0), the B band lies above the

A band, and the Fermi level is inside the gap [Fig. 3(a)]. Weremark that at precisely δ ¼ 0, the spectrum ofHeff consistsof a linearly dispersing Dirac cone and a parabolic electronband [Fig. 3(b)]. This critical point has no divergentsusceptibility, and we therefore expect it to be perturba-tively stable to interactions. When δ is tuned to becomenegative, the B band dips below the A band around k ¼ 0.Because of the twofold degeneracy of the B band at k ¼ 0,the spectrum of Heff immediately after band inversion, inthe parameter range −4λ2mB < δ < 0, shows a quadraticband touching at the Fermi level [dashed line in Fig. 3(c)],resulting in finite density of states for both electrons andholes. However, as shown by Sun, Yao, Fradkin, andKivelson [37], this kind of zero-gap state is unstabletowards exciton condensation in the presence of evenarbitrarily weak repulsive interactions. The interactiong ∝ UB on the anions thus plays an essential role afterthe charge transfer gap is inverted. The leading suscep-tibility of such a quadratic band touching is towards theopening of a topological gap [solid lines in Fig. 3(c)],resulting in a QAH state with spontaneous ZB ≠ 0. Thisanalysis, based on the effective field theory, Eq. (12), iscontrolled in the limit of small UB.Our HF calculation confirms the field theory analysis

even beyond the small UB limit. Additionally, because ofthe A − B hybridization, a finite occupation of the Bsublattice is already present at δ > 0. This causes anupward shift in the energy of the charge transfer bandby ðUB=2ÞhnBi, which has the effect of delaying thetransition to the inverted phase from δ ¼ 0 to δc < 0.

More importantly, in the presence of the Hubbard inter-action UB, a spontaneous spin polarization in the �zdirection is found at δ < δc, resulting in a noncoplanarspin structure with canted AFM on the A sublattice andz FM on the B sublattice, as shown in Fig. 1. Note thatwhen UB is small [as in Fig. 3(c)] the gap between the twospin states of the B sublattice, whereas at larger UB the gapis set by hybridization between the A and B sublattices.In the noncoplanar phase, the z FM order parameter

component breaks the effective time-reversal symmetryiszΘ and produces spin splitting of the B band. One of thespin-split bands is pushed to higher energy, while the otherone takes part in the band inversion with the A Hubbardband. Shown in Fig. 2(d) is the k-space Berry curvature ofthe noncoplanar phase, obtained from the self-consistentHF Hamiltonian, which includes both xy AFM and z FMorders. Now, the inversion around Γ between A and BHubbard bands—with removed spin degeneracy andp-wave hybridization—gives rise to a QAH insulator withthe Chern number C ¼ �1 as computed directly from theBerry curvature integration.It is important to note that the appearance of the QAH

phase requires that the cation and anion Hubbard bands aredispersive, so they can be inverted in part of momentumspace near the gap edge before Δ decreases to zero. This issatisfied in our model since magnetic frustration of thecations leads to dispersive quasiparticle bands even forlargeU [Eq. (9)]. As such, the QAH phase is a consequenceof the balance and synergy between electron localizationand itinerancy.Our finding of the QAH phase with a negative charge

transfer gap and noncoplanar magnetism is further con-firmed by DMRG calculations [38,39]. Using the infiniteDMRG algorithm, we study the ground state of theHamiltonian on an infinite cylinder Lx ¼ ∞ of circum-ference Ly ¼ 6 unit cells. The unit cell in x is chosen to becommensurate with the

ffiffiffi3

p×

ffiffiffi3

pAFM order. More details

on the numerical simulations and convergence of DMRG,performed using the TenPy code [40], are provided inAppendix A.Figure 4(a) shows the order parameters as a function of

Δ, for the same set of parameters as before. For each Δ, weperform calculations for both periodic and antiperiodicboundary conditions in the circumferential direction. Thedifference in calculated observables, represented by theerror bars, serves as an indication of finite-size effects(Appendix A). For a range aroundΔ ≈ 5t, both XYα and Zα

are clearly nonzero, showing a canted 120° order on the Asublattice and z polarization on the B sublattice. Moreover,we establish the existence of a QAH effect directly from theevolution of the entanglement spectrum as a h=e fluxquantum is threaded adiabatically through the cylinder(Appendix A, Ref. [35,41]).In Fig. 4(b), we show the response of the QAH phase

to a magnetic Zeeman field, Hz ¼ −ðh=2ÞPiðni↑ − ni↓Þ.

(a) (b) (c)

FIG. 3. Band structure of the effective theory [Eq. (12)] nearinversion. The band colors indicate the sublattice content: blue forA and red for B bands. Panels (a) and (b) show bands before andat inversion. After inversion, the g ¼ 0 bands feature a quadraticband touching (dashed lines), shown in panel (c). A perturbativeinstability then opens a topological gap for g > 0, as illustrated.For δ < −4λ2mB (not shown), Fermi surfaces form, and thesystem is metallic at g ¼ 0.

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The Hall conductivity σxy changes sign abruptly at h ¼ 0.There is a discontinuity in Z at h ¼ 0 due to brokensymmetry, after which total jZj increases smoothly with h.This is possible via an increase in canting of the Asublattice, and it is a signature of our QAH phase with apartial spin sz polarization—as opposed to fully saturated—at zero field.We also comment on the stability of the QAH phase

against nearest-neighbor repulsion, HV ¼ VP

hi;ji ninj. Atsmall V, the QAH remains present, albeit in a narrowerrange of Δ (Appendix A). When V is sufficiently large, anabrupt transition between A- and B-sublattice polarizedMott insulators is found around Δ ¼ 0, without theintervening QAH phase.Let us now apply our theory to TMD bilayers and, in

particular, AB-stacked MoTe2=WSe2 heterobilayers. Ourtheory provides a direct explanation for the observedtransition from a Mott insulator to a QAH state inMoTe2=WSe2 at n ¼ 1 filling of holes, driven by theapplied displacement field [32]. Our tight-binding modelcaptures the topology and essential features of the topmostvalence bands from the two layers after a particle-holetransformation. The role of the displacement field is todecrease the band offset between the two layers or,equivalently, reduce the charge transfer energy Δ. For Δbelow a critical value Δc > 0, the quasiparticle gapbetween MoTe2 and WSe2 Hubbard bands is inverted,leading to a QAH insulator.Figure 5(a) shows the HF phase diagram calculated using

realistic parameters for MoTe2=WSe2 (Appendix B), as afunction of Δ and U near band inversion. As Δ is reduced,we find that the Mott insulating phase transitions into thenoncoplanar QAH phase, which further transitions into ametal for U ≲ 160 meV. In this metallic phase, the bands

are deeply inverted beyond the UB ¼ 0 quadratic band-touching regime (δ < −4λ2mB in our effective theory), andU is not large enough to spin polarize the B band. Theresulting quasiparticle band structure, shown in the inset ofFig. 5(a), features a nearly spin-degenerate hole pocket onthe WSe2 layer and a spin-nondegenerate electron pocketon the MoTe2 layer. Thus, this metal phase is a compen-sated semimetal with xy magnetic order and small quasi-particle Fermi surfaces. Our phase diagram showing theMott insulator, QAH state, and compensated semimetal as afunction of displacement field agrees with the experimen-tally observed phases in MoTe2=WSe2 [32].Our theory further predicts that (1) at small displacement

field, the Mott insulator on the MoTe2 layer is anintervalley coherent (xy-ordered) state; (2) the QAH statedisplays partial valley z polarization on both layers and,simultaneously, intervalley coherence on the MoTe2 layer.The z and xy components of the valley order parameterincrease and decrease with the displacement field, respec-tively. The spontaneous valley z polarization predicted inthe QAH phase (but not in the Mott insulator) and itsincrease with displacement field can be detected bymagnetic circular dichroism from exciton spin splittingat zero field. The existence of intervalley coherence,predicted for both the Mott and QAH phases, can beestablished through gapless spin-wave transport [42],which can be detected by optical means as demonstratedin other TMD heterobilayers [43].In the lightly inverted regime, our QAH state features a

predominantly xy magnetic order, with only a small z

(a) (b)

FIG. 4. (a) Order parameters obtained from DMRG as afunction of Δ, showing qualitatively similar results as the self-consistent HF Hamiltonian. The small XYB ≠ 0 is likely due tothe cylindrical geometry (Appendix A). (b) Response to anapplied Zeeman field h. There is a discontinuity at h ¼ 0 due tobroken symmetry. While ZB is quickly saturated, ZA continues toincrease with h while XYA decreases, indicating a smoothvariation in the canting. The Hall conductivity σxy changes signdiscontinuously at h ¼ 0. Hysteresis is absent as we obtain theground state independently for each value of h.

(a) (b)

FIG. 5. (a) HF phase diagram using the realistic model param-eters ðtA; tB; tABÞ ¼ ð4.5; 9; 2Þ meV (Appendix B) describingholes in MoTe2=WSe2. Color indicates the charge gap. We findMott, QAH, andmetal phases near band inversion. The inset showsthe quasiparticle bands near the Fermi energy in the metal phase atΔ ¼ 65 meV and U ¼ 100 meV. Note that our tight-bindingmodel describes holes in this system; hence, these bands are minusthe electron bands. (b) Illustration of the moire superlattice inMoTe2=WSe2. Low-energy hole states on the MoTe2 layer arelocalized on the MM (red), and WSe2 on the XX (blue) regions.Together, they form an effective honeycomb lattice.

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component. It differs from the QAH state in magneticallydoped topological insulator films [44], where the magneticmoments spontaneously polarize along the z direction. Ourcase should also be contrasted with a fully valley-polarizedQAH state that arises from topological flat bands withvalley-contrasting Chern numbers, as widely discussed formagic-angle graphene [45–49] and recently proposed forslightly twisted TMD homobilayers [35]. This scenario wasalso proposed for MoTe2=WSe2 [50,51]. In the casesdiscussed above, full valley polarization would be expectedthroughout the QAH phase. In contrast, we predict that thespontaneous valley polarization is zero prior to inversionand develops smoothly in the QAH phase after inversion.Our work therefore uncovers a general mechanism bywhich QAH can emerge in the absence of flat bands.Our mechanism of QAH from inverted Hubbard bands in

charge transfer insulators is robust and does not rely onfine-tuning. The effective theory (12), which only involveslow-energy quasiparticles, is universally applicable in thevicinity of gap inversion, provided that prior to inversion(1) the charge transfer insulator has a direct quasiparticleband gap and (2) its electron and hole states at the gap edgehave different symmetry eigenvalues. Note that theserequirements are for the quasiparticle band structure ofan interaction-induced insulator not the noninteractingband structure.The central idea of this work, creating magnetic topo-

logical states by inverting the charge transfer gap, ispotentially applicable to a broad range of materials.Besides MoTe2=WSe2, twisted TMD homobilayers undera displacement field also realize a two-band Hubbardmodel with a tunable charge transfer energy and thereforemay display a similar QAH phase without requiring magic-angle flat bands. Another promising platform is hetero-structures between two-dimensional semiconductors andmagnetic insulators. We also note the possibility of anegative charge transfer gap in transition metal oxides[52,53] and perovskite nickelates [54], which may providea new venue for topological physics.

We are grateful to Yang Zhang, Valentin Crepel, Kin FaiMak, Jie Shan, Shengwei Jiang, and Tingxin Li for helpfuldiscussion on this work and related collaborations. Thiswork is funded by the Simons Foundation through aSimons Investigator Award and the U.S. Department ofEnergy, Office of Science, Basic Energy Sciences, underAward No. DE-SC0020149. L. F. is partly supported by theDavid and Lucile Packard Foundation.

APPENDIX A: DETAILS OF NUMERICALCALCULATION

1. Hartree-Fock calculation

In this Appendix, we present more details on our HFcalculations. We employ two different approaches. In thefirst approach, we perform the self-consistent HF

calculation using a six-site unit cell commensurate withthe expected

ffiffiffi3

p×

ffiffiffi3

porder. In the second approach, we

use Uζ to transform the Hamiltonian and perform the self-consistent HF calculation assuming translation invariance(a two-site unit cell) and pick the one ζ ¼ 0;� with thelowest energy. These calculations transform any potentialxy AFM order into xy FM order.The advantage of the second approach is that it does not

reduce the BZ and is conceptually simpler, with only asingle filled quasiparticle band, whereas in the firstapproach, one must work in a reduced BZ with three filledbands. The band structure and Berry curvature in Figs. 2(c)and 2(d) of the main text are computed from this secondapproach.The disadvantage of the second approach is that it is not

able to capture all types of spin orders. For example, itcannot describe a state with different wave vectors on thetwo sublattices, such as a state with 120° xy AFM on the Asublattice and xy FM on the B sublattice. However, in therange of parameters we have examined, we find that thesetwo approaches converge to the same result, indicating thatsuch spin configurations do not appear.We obtain the self-consistent HF solution by iteration.

We consider initial starting values for density hnii and spinhsii expectation values. In the first approach, we start with az FM, xy FM, and ζ ¼ �1 xy AFMs, and in the secondapproach, we consider z and xy FM phases, all of which aresublattice balanced, nA ¼ nB ¼ 1

2. In addition, we add a

small random noise of order about 0.01 to the initial startingexpectation values. Using these expectation values, theHubbard term in the Hamiltonian is then replaced by

HHFHub ¼

U2

Xi

�nihnii − si · hsii −

1

2hnii2 þ

1

2jhsiij2

�

ðA1Þ

and diagonalized, in anNk × Nk momentum space grid. Weuse Nk ¼ 180. The new expectation values hnii0,hsii0 arethen calculated at filling n ¼ 1. The calculation is thenrepeated using these new expectation values to constructthe HF Hamiltonian. We repeat the calculation until thenorm of the difference between consecutive iterations,D ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPi jhnii0 − hniij2 þ jhsii0 − hsiij2

p, falls below the

threshold D ≤ 10−10. In the case where different initialvalues converge to different states, the one with the lowestenergy is chosen.Throughout the parameter range shown in Figs. 2(a)

and 2(b), the ground state can bewell captured by a statewithζ ¼ −1. We may directly compare the energy of this statewith the states of the Hamiltonian restricted to ζ ¼ 0;þ1, bytransforming the Hamiltonian using Uζ and obtaining thelowest-energy self-consistent HF solution with enforcedtranslation invariance. The energy differences are shownin Fig. 6 forU ¼ 30t andU ¼ 50t. As can be seen, the state

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with ζ ¼ −1 is always lowest in energy throughout theentire range.For U ¼ 50t, as Δ is decreased below 5t, a fully Z

polarized state becomes favored in the HF solution. This isan artifact of the HF method: The HF solution captures theenergy of the fully polarized state exactly (as the Hubbardinteraction energy is identically zero for the fully polarizedstate), while it merely provides an upper bound for theenergy of the strongly correlated AFM state. Thus, the HFmethod overestimates the favorability of the fully polarizedstate. This is corroborated by the fact that DMRG does notobserve any tendency towards the fully polarized statewithin this parameter regime.

2. DMRG

In this Appendix, we present a more detailed descriptionof our numerical DMRG calculation. As stated in the maintext, we employ the infinite DMRG (IDMRG) algorithm onan infinite cylinder. We take the XC geometry, in which oneof the nearest-neighbor bonds is oriented in the x (infinite)direction. We utilize a 1 × Ly unit cell with boundaryconditions commensurate with the

ffiffiffi3

p×

ffiffiffi3

porder. The

sites are ordered in DMRG starting with A sublattice sitesin order of increasing y, and then again for the B sublattice.We begin with a random product state of fermions in the Sz

basis, at the desired density of n ¼ 1 fermions per unit cell,and total Sz ¼ 0.The IDMRG algorithm is performed with conserved

quantum numbers corresponding to total particle numberN ¼ N↑ þ N↓ and spin Sz parity ð−1ÞN↑−N↓. Although theHamiltonian has spin-Uð1Þ symmetry, and therefore con-served total Sz, we choose to only conserve the paritybecause states that spontaneously break spin-Uð1Þ sym-metry, such as xy ordered states, can be represented and

diagnosed explicitly. Thus, xy long-range ordered statescan be diagnosed simply via a nonzero expectation valuehSx;yi ≠ 0, rather than spin-spin correlation functions.Furthermore, this allows us to access states with Sz densitythat does not correspond to a particular choice ofN↑; N↓, inthe 1 × Ly unit cell. This is important as the noncoplanarQAH phase has smoothly varying Sz as a function of Δ.We also consider applying flux ψ through the cylinder.

This is modeled by modifying the hopping terms such that afermion picks up an additional phase factor eiψ upon goingaround the circumference of the cylinder. We compute theorder parameters in Fig. 4 for fluxes ψ ¼ 0; π. In the 2Dlimit, Ly → ∞, all observables should be independent of ψ .Thus, the difference of observables between ψ ¼ 0 andψ ¼ π is an indication of finite circumference effects. InFig. 4 of the main text, we plot the average of the orderparameters obtained for ψ ¼ 0; π, and the error bar indi-cates the difference, on an Ly ¼ 6 cylinder with maximumbond dimension χ ¼ 1600.In Figs. 7(a) and 7(b), we show the order parameters for

ψ ¼ 0; π as a function of Δ, for bond dimensions χ ¼ 800,1200, 1600. Unless stated otherwise, we use the parameterstA ¼ tB ¼ 1

2tAB ≡ t and UA ¼ UB ¼ U ¼ 50t. As can be

seen, there is only a small difference in the value of theorder parameters as χ is increased. In Fig. 7(c), we focus onΔ ¼ 5t, and the dependence of various quantities on bonddimensions from χ ¼ 400–1800 is shown. Importantly, wefind that IDMRG converges to a state with canted AFMorder: finite hSzi and hSx;yi in the 120° configuration withζ ¼ −1, as defined in the main text.In Figs. 8(a) and 8(b), we show the effect of nearest-

neighbor repulsion V. The QAH phase, identified by anonzero Z ¼ ZA þ ZB expectation value, persists in a finitewindow of Δ. The leading effect of a small V is to narrowthe range in which this phase appears. For large V, there is a

FIG. 6. Plots of the energy of the self-consistent Hartree-Fock solutions with ζ ¼ 0;þ1, compared to the ground-state energy EGS,which has ζ ¼ −1.

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first-order transition directly from an A-sublattice polarizedMott state to a B-sublattice polarized state.To confirm the nontrivial topology of this phase, we

examine the entanglement spectrum under an adiabaticthreading of 2π flux through the cylinder [41,55–57].Considering a cut in the cylinder, the ground state com-puted in IDMRG is naturally represented by the Schmidtdecomposition jψi ¼ P

i λijiLi ⊗ jiRi, where jiLðRÞi formsan orthonormal basis for states on the left (right) side of thecut, and λ2i ≡ e−εi > 0 is the entanglement spectrum. As weexplicitly conserve particle number, each εi can be labeledby the integer particle number NL;i associated with the leftstate jiLi. In Fig. 8(c), we show the evolution of theentanglement spectrum εi in the ground state at Δ ¼ 5t asflux is threaded through the cylinder, withNL;i indicated bycolor. As can be seen, the spectrum comes back to itself

after ψ ¼ 2π flux is threaded; however, the associatedparticle number NL;i only comes back to itself minus one.This indicates that, upon threading 2π flux, one particle ofcharge is pumped from the left of the cut to the right. Thischarge pumping is direct proof of the nontrivial Chernnumber and QAH effect in the ground state.We comment on the small but nonzero XYB order

parameter in the QAH phase, which is found in DMRGbut is absent in the HF method. This corresponds to a smallXY component on a B site, which is aligned with the XYcomponent of one of its neighboring A sites along thelength of the cylinder. Taken seriously, this would corre-spond to a nematic order when extrapolated to the infinitehoneycomb lattice. However, we believe this is likely anartifact arising from the intrinsic anisotropy of the cylin-drical geometry used in DMRG. The geometry explicitlybreaks rotation symmetry, which may explain the smallnematic component. Additional numerical work for largercylinder circumferences is necessary to ascertain the sourceof this apparent nematicity.In the magnetic field calculation in Fig. 4(b) of the main

text, each data point represents an independent DMRGcalculation. Thus, we do not see hysteresis, which would beexpected if one slowly swept h.We have also performed IDMRG calculations with

boundary conditions commensurate with a 2 × 2 unit cell(which can capture tetrahedral or stripe magnetic order, forexample). The resulting state attempts to form the 120°order, but it is unable to because of incommensuration withthe IDMRG unit cell. The resulting energy is higher thanthat of the

ffiffiffi3

p×

ffiffiffi3

pcommensurate phase.

APPENDIX B: RELATION TO MOIRÉTMD BILAYERS

In this Appendix, we discuss in detail the connectionbetween our tight-binding model and moire TMD bilayers,

(a) (b) (c)

FIG. 7. Plot of the ordered parameters as a function of Δ obtained from DMRG for (a) zero and (b) π flux threaded through thecylinder. In panel (c), we fix Δ ¼ 5t and show the order parameters as a function of bond dimension χ. The inset shows the DMRGtruncation error.

(a)

(b) (c)

FIG. 8. Total Z ¼ ZA þ ZB as a function Δ shown for nearest-neighbor repulsion (a) V ¼ 0 and (b) V ¼ t. The physics isqualitatively similar, except it occurs within a smaller range of Δ.In panel (c), we verify the nontrivial topology of this phase inDMRG by computing the particle-number-resolved entanglementspectrum adiabatically as 2π flux is threaded through the cylinder.We use a maximum bond dimension χ ¼ 800.

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and especially the origin of the phase factors ϕα mentionedin the main text. Specifically, our tight-binding modelcaptures the main qualitative features of the first bands ofeach layer and valley in K-valley derived moire TMDbilayer systems with an effective honeycomb latticedescription. As we shall show, the two sublattices corre-spond to the two layers, and the spin corresponds to thevalley �K degrees of freedom.To motivate the tight-binding model, we begin with a

single layer α, in which the low-energy degrees of freedomare spin polarized at the �Kα points of the original BZ. Inthe presence of a second layer, effects such as latticerelaxation result in an effective potential in the first layer.The moire bands can be well described by a continuummodel description [24,58]. Neglecting interlayer tunnelingfor now, the α layer Hamiltonian can be well described byan effective mass description of the electron (or hole)dispersion about the �Kα points in the presence of aperiodic potential with the moire period,

Hcontα ¼ ðk − σKαÞ2

2mαþ VαðrÞ; ðB1Þ

where σ ¼ � encodes the �K valley degree of freedom.Without fine-tuning, the potential VαðrÞ will genericallyhave a minimum at one of the high-symmetry stackingregions of the moire structure, forming a triangular super-lattice. The resulting bands in the reduced moire BZ can bedescribed by a triangular-lattice tight-binding model interms of localized Wannier orbitals centered at thesepotential minima with valley pseudospin internal degreesof freedom. The Hamiltonian Hα in the main text containsonly the nearest-neighbor hopping term. Incorporating thetriangular lattices of both layers then results in an effectivehoneycomb lattice, as long as the potential minima ofthe two layers lie at different high-symmetry positions inthe moire unit cell. The interlayer tunneling gives rise to thehopping term HAB. The effective tight-binding modelincorporates the first band of each valley and layer, whichis sufficient in describing the physics at filling n ¼ 1 thatdoes not involve any higher bands. Additional terms arisingfrom neglected higher bands or strain [50,59–61] may bepresent, although large-scale DFT on fully relaxed struc-tures indicates that such terms are small compared to thepotential term [33]. Figure 9 shows the continuum modeldescription from Ref. [33] and the tight-binding approxi-mation used in the main text. Although many MoTe2 (red)bands below −40 meV have been neglected, the keyphysics involves only the first MoTe2 band and WSe2states at the valence-band maximum, which are unaffectedby the neglected bands.In order to discuss the finer details of the mapping to the

tight-binding model, we must first address two things: thefolding to the moire BZ and the C3 eigenvalues at high-symmetry momenta.

First, we discuss the folding: specifically, where thepointsKA andKB, which determine the position of the bandminimum, fold to in the moire BZ. In general, the truefolding will depend on the precise commensurate structureof the bilayer. For example, the moire structure ofMoTe2=WSe2 is close to the commensurate approximationof 13 × 13 MoTe2 (A) unit cells and 14 × 14WSe2 (B) unitcells [33]. In this case, the folding to the moire BZ isKAð¼ 13KÞ ≅ K and KB ≅ −K. On the other hand,another close approximation is 14 × 14 A and 15 × 15 Bunit cells, in which case KA ≅ −K and KB ≅ Γ. However,the precise folding should not affect any physical observ-ables on the moire scale (since, in general, the structureneed not even be commensurate). In this sense, there is afreedom of choice in selecting a folding scheme. We definefolding schemes by ξ ¼ 0;�1, such that KA ≅ ðξþ 1ÞKand KB ≅ ðξ − 1ÞK. The two folding schemes mentionedabove for MoTe2=WSe2 correspond to ξ ¼ 0 and ξ ¼ 1,respectively.Second, we discuss the C3 eigenvalues. We define C3 to

be a 2π=3 counterclockwise rotation about the z axiscentered at the MM region where two metal atoms fromboth layers lie on top of each other. The C3 eigenvalues aredetermined by the wavefunction of the monolayer at K andthe position of the Wannier center. Let us denote the α layermonolayer C3 eigenvalue at σK as e½ð2πiÞ=3�σjα (jα is half-integer due to spin-1

2). For the TMD heterobilayer, we label

the three high-symmetry stacking positions in a moire unitcell as Rn ¼ ðaM=

ffiffiffi3

p Þð0; nÞ for n ¼ 0, 1, 2, correspondingto MM [33], XX [33], and MX [33] stacking regions,respectively. For the folding scheme ξ, the C3 eigenvaluesof the first band of valley σ and layer α at momentumlK (l ¼ 0;�1) is given by

ΘσαðlKÞ ¼ exp

�2πi3

½σjα þ ðl − σ½ξþ τα�Þnα��; ðB2Þ

FIG. 9. Left panel: continuum-model noninteracting bands forAB-stacked MoTe2=WSe2 valence bands [33]. Right panel: tight-binding model approximation, near noninteracting band inver-sion. The tight-binding model captures the qualitative features ofthe first bands of each valley and layer. Color indicates layercontent: MoTe2 is red, and WSe2 is blue.

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where nα is such that Rnα is the position of the Wanniercenters, and τA ¼ −τB ¼ 1.The role played by the C3 eigenvalues is crucial: The

interlayer tunneling only couples states at the samehigh-symmetry momentum if they have the same C3

eigenvalue. Our tight-binding model describes the foldingchoice ξ0 in which the C3 eigenvalues match at Γ,

ξ0 ¼ ðnB − nAÞðjB − jA þ nA þ nBÞ mod 3: ðB3Þ

Note that nB − nA ≠ 0 since we assume a honeycomblattice structure. Other folding choices can be describedby the shifted Hamiltonians, UζHU†

ζ , in which the nearest-neighbor hoppings are direction and spin dependent.Next, the phase factors ϕα in the tight-binding model

should be chosen to describe the correct band dispersion. Ifthere was no momentum offset in Eq. (B1), then we wouldhave ϕ ¼ 0. A momentum shift to σKα ≅ σðξþ ταÞKcorresponds to ϕα ¼ ½ð2πÞ=3�ð1þ ταξÞ.We also note that in TMD bilayers, ϕα is not strictly fixed

to be a multiple of 2π=3. The above analysis gives anestimate of ϕα in order to match the topology and positionsof the maxima and minima. However, a small deviation ofϕα from this value is expected in real systems (see, forexample, Ref. [35]).Let us take AB-stacked MoTe2=WSe2 as an example.

For the MoTe2 layer, the C3 eigenvalue of the state at K ise−½ðiπÞ=3�, and the moire bands are localized at the MMregion [33]; thus, jA ¼ − 1

2and nA ¼ 0. Similarly for the

WSe2 layer, because ofAB stacking,we have jB ¼ 12, and the

XX localized wavefunction corresponds to nB ¼ 1. Directcalculation of C3 eigenvalues from large-scale DFT is inagreement with Eq. (B2), with the folding choice ξ ¼ 0 [33](note our definition ofK is opposite to that of Ref. [33]). Ourtight-binding model describes the folding ξ0 ¼ −1, in whichKA ≅ Γ and KB ≅ K. This corresponds to the phase param-eters ϕA ¼ 0 and ϕB ¼ −½ð2πÞ=3� (mod 2π), as used in themain text. In homobilayer systems, such as small-angletwistedWSe2=WSe2, as long as there is a honeycomb latticedescription at small angles, ξ0 ¼ 0 is fixed by symmetry,and we have ϕα ≈ ½ð2πÞ=3�. Tight-binding models forother folding choices ξ ≠ ξ0 are described by the shiftedHamiltonians UζHU†

ζ , with ζ ¼ ξ − ξ0.For either of ξ0 ¼ 0;−1, the band inversion at positive Δ

is topological, and the physics discussed in the main textapplies. For ξ0 ¼ þ1, the C3 eigenvalues match at bandinversion, and there is no topological band inversion. Thereis another band inversion starting from the fully occupied Bsublattice at Δ → −∞ and reducing jΔj, which is topo-logical for ξ0 ¼ 0;þ1 but nontopological for ξ0 ¼ −1.The magnitude of the hopping terms, tA, tAB, and tAB,

can be fit to best match the band structure from large-scaleDFT. In Fig. 9, we show the continuum model bandsfor MoTe2=WSe2 with parameters from Ref. [33] and the

tight-binding model bands with parameters tA ¼ 4.5 meV,tB ¼ 9 meV, and tAB ¼ 2 meV, which roughly matchesthe band widths and the magnitude of the interlayertunneling. The bands of −H are plotted (due to particle-hole transformation), and the folding choice ξ ¼ 0 is used,to conform with Ref. [33]. Matching the exact shape of theband requires further range hoppings but should notqualitatively affect the universal physics near band inver-sion as described in the main text.Finally, let us briefly discuss an alternate explanation for

the QAH phase in MoTe2=WSe2 presented in Refs. [50,51].The mechanism for topology suggested by Ref. [50] is that astrain-induced pseudomagnetic field [a term left out ofEq. (B1)] may cause the noninteracting first moire bandof the MoTe2 layer to carry nontrivial valley-contrastingChern number. Interactions then induce a fully valley-polarized state, resulting in QAH. However, the strain-induced topology is not supported by fully relaxedlarge-scale DFT [33], which shows topologically trivial,firstMoTe2 bands.Also, the self-consistentHF calculation inRef. [50], used to argue for a fully valley-polarized state,assumes translation invariance and therefore misses the 120°ordered state, which we find is significantly more energeti-cally favorable. Reference [51] performs HF calculations onthe continuum model and also finds a fully valley-polarizedQAH phase in a large displacement field. However, theinterlayer tunneling strength w used in Ref. [51] is muchlarger than predicted from first-principles calculations inAB-stacked MoTe2=WSe2 [33]. A large interlayer tunnelingallows for a fully valley-polarized phase through flat-bandferromagnetism, discussed in the main text, which doesnot occur for smaller, more realistic values of interlayertunneling.

APPENDIX C: ADDITIONAL DETAILSOF HF BANDS

In this Appendix, we discuss additional details and HFband structures related to the discussion in the main text.We use a representative set of parameters tA ¼ tB ¼ 1

2tAB ≡

t and ϕA ¼ 0, ϕB ¼ −½ð2πÞ=3�, as in the main text.Figure 10 shows the HF band structure at various inter-actions UA, UB, and charge transfer energies Δ.First, in Figs. 10(a) and 10(b), we show the noninteract-

ing band structure (UA ¼ UB ¼ 0) of H, prior to the spin-dependent shift Uζ, before and after band inversion.Next, we apply the shift Uζ. The noninteracting band

structure of the shifted Hamiltonian, Hζ ¼ UζHU†ζ , is

given by

HζσðkÞ ¼

�EζAσðkÞ Tζ

σðkÞT†σðkÞ EBσðkÞ

�ðC1Þ

for spin σ, where

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EζασðkÞ¼−2tα

Xn

cos

�k ·anþ

2πζsσ3

þ sσταϕα

�−1

2ταΔ;

ðC2Þ

TζσðkÞ ¼ −tABðe½ð2πiζsσÞ=3�e−ik·b1 þ e−½ð2πiζsσÞ=3�e−ik·b2

þ e−ik·b3Þ: ðC3Þ

Notice that the nearest-neighbor hopping term of the shiftedmodel, Tζ

σ, is now both spin and direction dependent. Thebands in Figs. 10(c)–10(h) are plotted with the shiftζ ¼ −1, which transforms the observed A sublattice xyAFM to an xy FM.First, in Figs. 10(c) and 10(d), we show the noninteract-

ing band structure from Figs. 10(a) and 10(b), but with theshift Uζ¼−1 applied. This shifts both band minima of the Bsublattice bands to Γ. Without interactions, the ground stateis a metal with partial filling of both σ bands on the Asublattice.Next, Figs. 10(e) and 10(f) show the HF band structure

withUA ¼ 30t andUB ¼ 0, before and after band inversion.The A bands (red) are split into lower and upper Hubbardbands, separated by a Mott gap of about U. Before bandinversion, the ground state is insulatingwith full filling of thelower Hubbard band. Figure 10(e) shows Δ ¼ 12t, wherethe B bands lie in between the lower and upper Hubbardbands, resulting in a charge transfer insulator. As Δ isreduced, the charge transfer gap becomes negative. Rightafter band inversion, as shown in Fig. 10(f), the resultingband structure exhibits a quadratic band touching at theFermi energy (the inset shows magnified band structure).Finally, Figs. 10(g) and 10(h) show the effect of

interactions UB ¼ 15t on the B sublattice, before and after

inversion. Prior to inversion, UB does not play an importantrole because of the small density on the B sublattice. Afterinversion, the B sublattice develops a spontaneous zpolarization due to nonzero UB. The resulting filled lowerHubbard band, after inversion, carries nontrivial Chernnumber C ¼ �1. We remark that other effects, such as anapplied Zeeman field in the z direction, can also inducepolarization in the B sublattice, resulting in Chern bandseven at UB ¼ 0.We end with a discussion on the Mott state at large Δ. In

our tight-binding model, ϕA ¼ 0 is special in that there isan emergent SUð2Þ symmetry in the limit Δ → ∞when theB sublattice can be ignored. The two xy AFMs with ζ ¼ �are degenerate in this limit, along with any in-plane orderedphases beyond xy. This symmetry can be broken in twoways: (1) AsΔ is reduced, the effect of the B sublattice withϕB ¼ �½ð2πÞ=3� favors the xy AFM with ζ ¼ �, or (2) itmay be that ϕA is nonzero but small, which will favor the xyAFM with ζ ¼ −signðϕAÞ. An interesting scenario ariseswhen these effects favor states with different ζ: Forexample, ϕB ¼ −½ð2πÞ=3� and ϕA ¼ −ϵ for small ϵ > 0.For largeΔ, the ground state is an xyAFMwith ζ ¼ þ, dueto the sign of ϕA. For this state, the quasiparticle gap isindirect, and there is no QAH phase. However, as Δ isreduced, at some point, there is a first-order phase transitionto the xy AFM with ζ ¼ − due to the coupling to the Blayer, for which our mechanism for QAH is possible.Exactly where this transition occurs depends on nonuni-versal details such as the values of the hoppings and ϵ.Thus, even though the quasiparticle gap at Δ → ∞ isindirect, QAH may still be possible by our mechanismthrough a transition to a competing ordered state (in whichthe quasiparticle gap is direct) as Δ is reduced.

(a) (c)

(d)

(e)

(f) (h)

(g)

(b)

FIG. 10. Plot of the HF band structure (see accompanying discussion) at various interactions UA, UB, before and after band inversion.The black dashed line indicates the Fermi energy at filling n ¼ 1. Color indicates the sublattice content: The A sublattice is in red, and Bin blue.

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