Massive Dirac fermions in moir´e superlattices: a route towards topological ﬂat minibands

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-Electrons confined by periodic potential in a crystal can behave very differently from a free particle.Perhaps the most prominent example is graphene in which the time-reversal, inversion, and three-fold rotation symmetries together stabilize a pair of Dirac cones at Brillouin zone corners [1].The massless Dirac fermions can acquire a mass when the time-reversal or inversion symmetry is broken, like in transition metal dichalcogenide (TMD) [2,3].The nontrivial topological properties associated with the exotic quasiparticles enable novel quantum effects such as the Klein tunneling [4], valley Hall effect [5,6], and valley-selective circular dichroism [3,7,8] in these 2D materials which are considered candidates for the next-generation microelectronics.
TMD heterobilayers are another important class of MSL and are being considered platforms to simulate the Hubbard model.Their single-particle physics is modeled by holes with parabolic dispersion subject to a moiré potential that yields topologically trivial moiré minibands [68,69].In this approach, the massive Dirac structure of TMD is neglected by perturbatively dropping the conduction (remote) band, which is far away from the Fermi energy (of the order of 1eV).This theoretical framework can describe the experimentally observed Mott insulator and Wigner crystal in the WSe 2 /WS 2 heterobilayer [70][71][72][73][74][75].
Strikingly, recent experiments report the correlated Chern insulator (CCI) at half filling (ν = 1 hole per moiré unit cell) and quantum valley-spin Hall insulator (QVSHI) at full filling (ν = 2 holes per moiré unit cell) in an ABstacked MoTe 2 /WSe 2 heterobilayer under a vertical electric field [76].The experimental observations suggest valleycontrasting Chern bands in the TMD heterobilayer that cannot be explained by the existing model [68,69].This motivates us to investigate a general problem that whether massive Dirac fermions confined in a moiré potential can give rise to topological minibands.
In this letter, we study the behavior of massive Dirac fermions in a moiré potential.Surprisingly, we show that, no matter how large the Dirac band gap is, topological flat minibands can emerge when the moiré potential has certain symmetries.Our study indicates that the Dirac nature of electrons plays a crucial role in determining the topology of moiré minibands.In particular, we find that the Berry curvature induced by Dirac remote bands stabilizes a topological phase, which is absent if remote bands are ignored.By applying our model to the MoTe 2 /WSe 2 heterobilayer, we demonstrate that the Coulomb interaction can stabilize a CCI at ν = 1 and a QVSHI at ν = 2 in a vertical electric field, which agrees with the recent experiment [76].Furthermore, the potential realizations of our model on the surface of an axion insulator and in a monolayer TMD under spatially periodic modulation are also proposed.Therefore, our work unveils a general route towards topological flat minibands in moiré systems.
Model.-The continuum model describing a massive Dirac fermion in a moiré potential reads where v F is the Fermi velocity, m is the Dirac mass, and σ x,y,z are the Pauli matrices acting on pseudospin.τ = ±1 determines the chirality of the massive Dirac fermon and is dubbed valley index in TMD [3].The Dirac Hamiltonian Here we consider the moiré potential V (r r r) = 2V 0 ∑ 3 j=1 cos(G G G j • r r r + φ ) in TMD heterobilayers [68,69], where  the MSL constant.H τ is invariant under the threefold rotation since 3 , 1) [77] and R 3 are the threefold rotation operator and matrix.
As far as the energy spectrum is concerned, the massive Dirac fermion described by h k k k,τ can be approximated by a free fermion with effective mass m * = ∆/2v 2 F through the second order perturbation theory when the Dirac band gap ∆ v F |k k k| and V 0 .Then Eq. ( 1) is reduced to that is widely adopted to describe the moiré minibands in TMD heterobilayers [68,69].However, as will be shown explicitly below, the topology of minibands can be very different for Eqs.(1) and (2) because H 0 has time-reversal symmetry (TRS) while H τ does not.The TRS in H τ is broken by the massive Dirac femrion, i.e., T h k k k,τ T −1 = h −k k k,−τ where the TRS operator T = K equals the complex conjugate operator K .Therefore the topological moiré minibands can emerge from Eq. (1) but not Eq.( 2).Topological phases.-Due to the C 3 symmetry of H τ , the Chern number C τ of the moiré miniband can be determined by its C 3 eigenvalues η τ (k k k) at the C 3 -invariant points [78] where γ represents the moiré Brillouin zone (MBZ) center and ±κ are the MBZ corners.Here we focus on the top valence band.It is easy to show η τ (γ) = 1, while η τ (±κ) can be evaluated to the leading order by the degenerate perturbation theory in which the coupling among three degenerate Bloch states at ±κ 1,2,3 are considered, as shown in Fig. 1(a).In the basis of the Bloch states of the valence band without a moiré potential, i.e., {|u ±κ 1 ,τ , |u ±κ 2 ,τ , , the matrix representation of the moiré potential operator is whose matrix element w(±φ depends on the dimensionless pa- Namely, s measures the intrinsic Berry curvature from the massvie Dirac fermion. The eigenvalues of Eq. ( 4) are E 0 = 2Re(w) and  1(c).The topological phases with C ± = ∓1 emerge at φ = (2n + 1)π/3 and then expand in a wider range of φ as s increases from zero.This indicates that the intrinsic Berry curvature of massive Dirac fermion measured by s plays a crucial role in determining the topology of moiré minibands.The topological phase boundaries can be obtained analytically by demanding arg[w(±φ )] = (2n + 1)π/3 that yields Significantly, the topological phases at φ = (2n + 1)π/3 persist to arbitrarily large Dirac band gap since s → 0 when ∆ → ∞, as shown in Fig. 1(c).To understand the peculiar behavior, it is noticed that the moiré potential minima for holes form a honeycomb lattice with inversion symmetry P only at these φ s.Then free fermions coupled to the moiré potential, as described by Eq. ( 2), give rise to a pair of Dirac cones at MBZ corners that is stabilized by the PT and C 3 symmetries, same as that in graphene.When the free fermion is replaced by massive Dirac fermion in Eq. ( 1), the PT symmetry is broken as PT h k k k,τ (PT ) −1 = h k k k,−τ that gaps out the Dirac cones and leads to topological minibands.This mechanism to generate topological minibands is protected by the symmetry of moiré potential and is independent on the detailed model parameters.The derivation of φ from (2n + 1)π/3 breaks the P symmetry and induces a staggered potential on the honeycomb lattice that can drive the topological transition as in the Haldane model [79].When φ = 2nπ/3, the moiré potential also has P symmetry but its minima for holes form a triangular lattice that leads to a trivial top valence band [80].
MoTe 2 /WSe 2 heterobilayer.-Toverify the topological phase, we take the MoTe 2 /WSe 2 heterobilayer as an example.MoTe 2 /WSe 2 has the type-I band alignment with a valence band offset about 200∼300 meV.The valence band maximum is from MoTe 2 whose Fermi velocity is v F = 2.526 eV• Å and Dirac band gap is ∆ = 1.017 eV [81].The lattice mismatch is δ ∼ 7% that results in a MSL with a M = a/ √ δ 2 + θ 2 where θ is twist angle and a = 3.565 Å is the lattice constant of MoTe 2 [82].The direct interlayer tunneling is suppressed by the band offset and by the spin-valley locking in the AB-stacking pattern that requires to flip the electron spin.Therefore, the massive Dirac fermion from MoTe 2 coupled to the moiré potential provided by WSe 2 can be described by Eq. (1).By employing the plane wave expansion of the continuum model, we obtain the topological phase diagram in terms of φ and θ in Fig. 1(d).
Here the red dashed lines are the topological phase boundaries predicted by Eq. ( 5) and are consistent with the direct numerical calculation of Eq. ( 1).In Fig. 1(d), only the topological phase around φ = π/3 is shown and other topological phases can be obtained by shifting φ by 2nπ/3.
To compare the moiré minibands for Eqs. ( 1) and ( 2), we choose θ = 1 • and φ = 59 • in the topological phase and set V 0 = 8 meV.In Fig. 2(a), the blue and red energy bands are from Eqs. ( 1) and (2), respectively, and show good agreement with each other.Here only the valence bands from the +K valley are plotted, and those from the −K valley can be obtained by TRS.The Berry curvature of the top blue band is shown in Fig. 2(b) and yields a valley Chern number C + = −1, while that of the top red band is antisymmetric in Fig. 2(c) due to the emergent TRS in Eq. ( 2).The Wannier orbitals of the topological and trivial moiré minibands from different models are compared in the Supplemental Material [83].On the other hand, deep inside the trivial phase, the Berry curvature derived from Eq. ( 2) can be a good approximation to that from Eq. ( 1) [83].
Electric-field-driven topological transition.-According to the first-principle calculation, the phase of the moiré potential in AA-and AB-stacked TMD heterobilayer is unlikely close to φ ∼ (2n + 1)π/3 [68,69,84].Here we show that φ can be tuned by a vertical electric field.It is noted that the two stacking configurations have different lattice corrugations that have been identified in both the STM measurements [85,86] and first-principal calculations [84,87].The electric field couples to the lattice corrugation and modifies the moiré potential as where E ⊥ is the vertical electric field and the topography of the corrugated layer is approximated by the lowest harmonics z(r r r) ≈ z 0 ∑ 3 j=1 cos(G G G j • r r r + φ ).The role of electric field can be described by a modified moiré potential with . As E ⊥ ramps up, the phase of the moiré potential in Eq. ( 6) changes continuously from φ to φ when eE ⊥ z 0 V 0 , which points to an electric-field-driven topological phase transition.
In AA-stacked TMD heterobilayer, z(r r r) is maximal at R M M and minimal at R M X and R X M [85][86][87].In AB-stacked TMD heterobilayer, z(r r r) is maximal (minimal) at R X X R M X , while H M M is in between [84,86].Here M and X refer to the metal and chalcogen, while R and H represent the AA-and AB-stacking.The super-and subscript denote atoms from the top and bottom layer that are aligned locally [83].The variation of z(r r r) in experiments translates into φ ∼ 0 and −π/2 for the AA-and AB-stacked heterobilayer, as shown in Figs.3(a) and 3(b).φ of the moiré potential is usually determined by fitting the continuum model to the first-principal energy bands.It has been reported that φ ∼ π/12 for AB-stacked MoTe 2 /WSe 2 [84] while φ for AA-stacked MoTe 2 /WSe 2 is still unclear.Nevertheless, most AA-stacked TMD heterobilayers have a φ of π/6 ∼ π/4 [69,88] and it is natural to expect AA-stacked MoTe 2 /WSe 2 has φ in the same range.The critical E ⊥ for the topological transition can be obtained from Eq. ( 5) by replacing φ with the phase of the modified moiré potential in Eq. ( 6).
For θ = 0 • , V 0 = 4.3 meV, and z 0 = 0.024 nm, the topological phase diagrams in terms of φ and E ⊥ are displayed in Figs.3(c) and 3 (d) for AA-and AB-stacked MoTe 2 /WSe 2 , respectively.The former shows a topological phase for φ around π/6 ∼ π/4 in negative E ⊥ , while the later exhibits two topological phases for φ ∼ π/12 in both positive and negative E ⊥ .Note that the positive (negative) E ⊥ reduces (enlarges) the valence band offset and is applied in the experiment.Therefore, we can focus on E ⊥ > 0 and there is no topological phase for AA-stacked MoTe 2 /WSe 2 as observed in the experiment [89].For AB-stacked MoTe 2 /WSe 2 , a topological phase appears for E ⊥ within 0.66 ∼ 0.73 V/nm that agrees well with the experimental result of 0.68 ∼ 0.70 V/nm [76].CCI and QVSHI.-Tostabilize a Chern insulator, it is required to break the TRS, which can be achieved by the Coulomb interaction.The Coulomb interaction projected onto the moiré minibands reads q q q ρ(q q q)V q q q ρ(−q q q), (7) where c n,k k k,τ is the annihilation operator of the eigenstate given by A is the area of the system, and µ is the chemical potential.V q q q = e 2 tanh(qd ⊥ )/2ε 0 εq is the screened Coulomb interaction in a dual-gated setup whose gate distant is d ⊥ ∼ 10 nm [76].Here ε is the dielectric constant and ε 0 is the vacuum permittivity.The density operator where the form factor Λ (τ) n,n (k k k, k k k , q q q) = ψ n,k k k,τ | e iq q q•r r r |ψ n ,k k k ,τ encodes the correlation between states in different bands and at different momenta.
The interacting Hamiltonian Eq. ( 7) can be solved selfconsistently by using the standard Hartree-Fock approximation [83].To identify the CCI at ν = 1 and QVSHI at ν = 2 in AB-stacked MoTe 2 /WSe 2 , we calculate the Hall conductance G H and spin Hall conductance G SH as a function of ε under the electric field E ⊥ = 0.69 V/nm at which the CCI was observed in the experiment [76].At ν = 1, the Hall conductance drops from e 2 /h to 0 at ε ∼ 21, as shown in Fig. 4(a).When ε < 21, the system becomes a valley-polarized CCI whose energy bands are shown in Fig. 4(b).Here the blue and red bands are from the ±K valleys, respectively, and the top valence band from the +K valley with C + = −1 is empty [90].
The energy gap ∆ g decreases with ε and vanishes with G H at ε ∼ 21 above which the valley polarization disappears and the system becomes a normal metal.The energy gap for ε = 8 in Fig. 4(b) is ∆ g = 2.71 meV that agrees well with the experimental data ∼2.5 meV extracted from the capacitance measurement [76].At ν = 2, the spin Hall conductance jumps from 0 to 2e 2 /h at ε ∼ 6 above which the system becomes a QVSHI, as shown in Figs.4(c) and 4(d).In this case, the top valence bands from ±K valleys with opposite Chern numbers C ± = ∓1 are empty.The energy gap decreases with ε.
The valley polarization only appears at strong interaction for ε < 6, and the top two valence bands from either +K or −K valley are empty.Because the two bands from the same valley carry opposite Chern numbers, the system becomes a valleypolarized trivial insulator.Discussion and summary.-It is noted that the minimum requirement to realize topological minibands in our study is some nonzero Berry curvature, which appears naturally due to the Dirac structure of TMD.This is different from the other proposals that require the inclusion of interlayer tunneling [84,91] or pesudomagnetic field [92] in the TMD heterobilayer.In particular, we show that the topological phase, when protected by the symmetry of moiré potential, survives to arbitrarily large Dirac band gap that cannot be captured by the existing model Eq. ( 2).This mechanism is also verified for a different moiré potential with C 4 symmetry that forms a square MSL [83].Besides TMD heterobilayers, the massive Dirac fermions on the surface of an axion insulator and in the bulk of a monolayer TMD can also couple to a modulating potential and give rise to topological flat mininbands, as explicitly elucidated in the Supplemental Material [83].In these systems, the modulating potential can be generated by the spatially periodic modulation of magnetic proximity coupling and dielectric screening [93].
A single Dirac cone is generally not allowed to appear in 2D systems with TRS according to the Nielsen-Ninomiya theorem [94].In MSL, the emergent valley charge conservation allows to assign a valley flavor to fermions.With the Coulomb interaction, one valley flavor is populated while the other valley remains empty, which results in the spontaneous breaking of TRS.Therefore, the TMD heterobilayer becomes an exciting platform to study the physics of a single massive Dirac cone, where topological minibands can be realized.A similar situation for massless Dirac cone can be realized on the surface of 3D topological insulators [95,96].
In summary, we spotlight the topological flat minibands that can emerge from the massive Dirac ferminons confined in a moiré potential.The topological phase is enabled by the Dirac structure and can be protected by the symmetry of moiré potential, which provides a paradigm to study the interplay between electric correlation and nontrivial topology.We take the MoTe 2 /WSe 2 heterobilayer as an example and show that the CCI and QVSHI can be stabilized by the Coulomb interaction.Our work provides a mechanism to the topological states observed in the TMD heterobilayer and points a direction to design topological moiré materials.
Supplemental Material: Massive Dirac fermions in moiré superlattices: a route toward correlated Chern insulators

I. WANNIER ORBITALS OF MOIR É MINIBANDS
We compare the Wannier orbitals of moiré minibands from the two different continuum models in Eq. ( 1) and ( 2 For the top trivial miniband given by the continuum model Eq. ( 2), the Wannier orbital can be obtained from the Fourier transform of the Bloch wave as where R R R is the moiré superlattice vector and N is number of moiré unit cells.In Fig. S2, we show the Wannier orbital of the top trivial band for R R R = 0.Here we choose a gauge in which the Bloch wave is real at H M M at the origin.The Wannier center at H M M forms a triangular lattice, as shown in Fig. S2.Therefore, the Coulomb interaction in the trivial moiré miniband can simulate the Hubbard model on a triangular lattice.
On the other hand, the moiré minibands are topologically nontrivial according to the continuum model Eq. ( 1), and Eq. ( S1) is inapplicable to the Chern bands due to the Wannier obstruction.In this case, we need to consider the two topological minibands with opposite Chern numbers in Fig. S1.For the two-band system, the Wannier orbitals can be prepared as  M and H M X form a honeycomb lattice and each Wannier center is surrounded by three peaks of wavefunction amplitude maxima.The peculiar three-peak structure of the Wannier orbitals is similar to that in twisted bilayer graphene [S1, S2].The two moiré minibands with opposite Chern numbers together can realize the Haldane model as that proposed in TMD homobilayers [S3].Further considering the time-reversal counterparts from two distinct valleys as well as the Coulomb interaction, the system can simulate the Kane-Mele-Hubbard model.

II. COMPARISON BETWEEN THE TWO CONTINUUM MODELS IN THE TRIVIAL PHASE
In this section, we focus on the topologically trivial phase and compare the energy bands and Berry curvatures from the two continuum models.Here the blue solid and red dashed bands are from Eqs.

FIG. 1 .
FIG. 1.(a) Schematic MBZ in a TMD heterobilayer.The green lines denote the coupling among three degenerate Bloch states at MBZ corners.(b) Eigenvalues of Eq. (4) as a function of arg(w).(c) Topological phase diagram of the continuum model Eq.(1) in terms of φ and s.(d) Topological phase diagram of the MoTe 2 /WSe 2 heterobilayer in terms of φ and θ .In (c) and (d), the yellow regions are the topological phase with valley Chern numbers C ± = ∓1, while the blue regions are the trivial phase with zero valley Chern numbers.The red dashed lines in (d) are the topological phase boundaries predicted by Eq. (5).
w), and the corresponding eigenstates have the C 3 eigenvalues C 3 |E j = e i 2π j 3 |E j .In Fig. 1(b), the three eigenvalues are shown as a function of arg(w) and the top valence band at ±κ changes among E 0 and E ±1 through the band crossing at arg(w) = (2n + 1)π/3 with n ∈ Z where the topological transition can occur.In this way, we can identify η τ (±κ) and hence the valley Chern number C τ according to Eq. (3).Moreover, the TRS guarantees η + (±κ) = η − (∓κ) * and C + = −C − .A global phase diagram in terms of φ and s is constructed in Fig.

4 −FIG. 3 .
FIG. 3. (a) and (b) Topography of the corrugated AA-and ABstacked TMD heterobilayer.The white parallelogram encloses the moiré unit cell.(c) and (d) Topological phase diagrams of the AAand AB-stacked MoTe 2 /WSe 2 heterobilayer in terms of φ and E ⊥ .The yellow regions are the topological phase with valley Chern numbers C ± = ∓1, while the blue regions are the trivial phase with zero valley Chern numbers.The red dashed lines are the topological phase boundaries predicted by Eq. (5).The green dashed lines in (d) are for φ = π/12 and E ⊥ = 0.69 V/nm.

FIG. 4 .
FIG. 4. (a) Hall conductance and band gap of the AB-stacked MoTe 2 /WSe 2 heterobilayer at ν = 1 and under a vertical electric field E ⊥ = 0.69 V/nm as a function of ε.(b) The corresponding energy bands given by the Hartree-Fock approximation for ε = 8.The blue solid and red dashed bands are originated from the ±K valleys and the chemical potential is set at zero energy.Besides replacing the Hall conductance by the spin Hall conductance, (c) and (d) display the same as those in (a) and (b) at ν = 2.
) of the main text.In Fig S1, we extract the top four minibands from Fig. 2(a) of the main text.The blue solid bands are from the continuum model Eq.(1) and have valley Chern numbers C + = ∓1, while the red dashed bands are from the continuum model Eq.(2) and have zero valley Chern numbers.

FIG. S1 .
FIG. S1.The top two valence bands extracted from Fig. 2(a) of the main text.The blue solid and red dashed bands are from the continuum models in Eq. (1) and (2) of the main text, respectively.The valley Chern numbers of the valence bands are marked in the figure.

FIG. S2 .
FIG. S2.The Wannier orbital of the top valence band (red dashed line) in Fig. S1 whose valley Chern number is C + = 0.The Wannier center at H M M is connected by the black triangular lattice.
(1) and (2) of the main text.We choose θ = 1 • and φ = 40 • in the trivial phase in Fig.1(d), and set V 0 = 8 meV.The other model parameters are same as those specified in the main text.The moiré minibands from the two different models exhibit good agreement with each other, as shown in Fig.S4(a).The Berry curvatures of the top va-

1 )
FIG. S6. (1) Moiré potential for holes as described by Eq. (S5).The potential minima form a square lattice with two sublattices.The black dashed hexagon encloses the MSL unit cell.(b) The blue solid and red dashed energy bands are from the continuum models Eqs.(1) and (2), respectively.The top blue solid band has a valley Chern number C + = −1.The insets show the schematic MBZ and zoom in around the quadratic band-touching point.(c) The Berry curvature of the top blue solid band in (b).