Chern mosaic and ideal flat bands in equal-twist trilayer graphene

We study trilayer graphene arranged in a staircase stacking configuration with equal consecutive twist angle. On top of the moir\'e cristalline pattern, a supermoir\'e long-wavelength modulation emerges that we treat adiabatically. For each valley, we find that the two central bands are topological with Chern numbers $C=\pm 1$ forming a Chern mosaic at the supermoir\'e scale. The Chern domains are centered around the high-symmetry stacking points ABA or BAB and they are separated by gapless lines connecting the AAA points, where the spectrum is fully connected. In the chiral limit and at a magic angle of $\theta \sim 1.69^\circ$, we prove that the central bands are exactly flat with ideal quantum curvature at ABA and BAB. Furthermore, we decompose them analytically as a superposition of an intrinsic color-entangled state with $\pm 2$ and a Landau level state with Chern number $\mp 1$. To connect with experimental configurations, we also explore the non-chiral limit with finite corrugation and find that the topological Chern mosaic pattern is indeed robust and the central bands are still well separated from remote bands.

The aim of this letter is to study the emergent effect of the superposition of the two moiré patterns in eTTG, see Fig. 1a.The system is the simplest example of a quasiperiodic moiré crystal [67] where the angle θ 12 , between layer 1 (top) and 2 (middle), and θ 23 , between layer 2 and 3 (bottom), are equal θ 12 = θ 23 ≡ θ.In the magic angle region, where θ ≈ 1 • , the two incommensurate periodicity can be decomposed in a fast modulation q j on the moiré scale |q j | ∝ θ and a slow one δq j with much larger periodicity |δq j | ∝ θ 2 [68,71].Applying semiclassical adiabatic approximation [72][73][74][75][76] we define a local Hamiltonian H eTTG (r) = H eTTG (r, φ) where φ depends on the slowly varying supermoiré scale [68].In this picture, we obtain a Chern number versus φ realspace map Fig. 1b that gives rise to a triangular lattice Chern mosaic of regions with ±1 Chern number.The domain walls separating the topological regions close the gaps to the remote bands and form lines connecting the AAA centers.
There are three high-symmetry stacking configurations that are especially significant and indicative of the Chern mosaic pattern: AAA, ABA and BAB.We explore them analytically in the chiral limit to unveil the topological features of the mosaic.We thereby derive analytical expressions for the resulting ideal flat bands emerging at a magic angle.The AAA stacking, considered in the preprint [77], is characterized by a vanishing Berry curvature and a fully connected spectrum protected by C 2z T [78].At the magic angle θ AAA ≈ 0.75 • a fourfold degenerate zero energy flat band sector emerges, connected to a single Dirac cone.The ABA(BAB) stacking, on the other hand, shows a flat band region detached from the remote bands with total Chern number C = 1(−1).The origin of the finite Chern number is readily traced out by the nature of the flat bands at the magic angle θ ABA ≈ 1.69 • which, remarkably, is larger than the one in mirror-symmetric TTG [79].We prove that the flat band sector decomposes into a Chern +2(−2) colorentangled zero mode [54,80,81] and a Chern +1(−1) Landau level like state [36,38,39,47,48,50,82,83]. The resulting imbalance in Chern flux creates a Chern mosaic pattern in real space, which could potentially be detected by measuring the local orbital magnetization in real space [27,28].
Chern Mosaic on the supermoiré scale -When the twisting angle is small, non commensurability effects are characterized by a length scale well separated from the moiré scale, |δq j |/|q j | ≈ 0.02 for θ = 1 • .As a result the long wavelength modulation can be treated parametrically, leading to the local Hamiltonian obtained in Ref. [68]: where v F ≈ 10 6 m/s is the graphene velocity, the phase φ = (φ 1 , φ 2 , φ 3 ) defines the local stacking configuration [68,71].Varying φ maps out the supermoiré unit cell in Fig. 1b, σ is the vector of Pauli matrices in the sublattice space and k = −i∇ r .The tunneling between different layers is described by the moiré potential: where T j+1 = w AA σ 0 + w AB [σ x cos 2πj/3 + σ y sin 2πj/3], w AB = 110meV, using complex notation q j+1 = ie 2iπj/3 [78], j = 1, 2, 3, in unit of k θ = θK D with K D = 4π/3a G and a G ≈ 2.46 Å.The moiré lattice is characterized by the reciprocal lattice vectors b 1/2 = q 1 −q 2/3 and primitive vectors a 1/2 .Bloch periodicity takes the form The spectrum is thus invariant upon shifting k by a 1 and a 2 up to a layer dependent phase factor.The Hamiltonian (1) is also invariant under the particle-hole transformation P H eTTG (r, φ)P = −H eTTG (−r, φ) where At low-energy Eq. 1 is characterized by three inequivalent Dirac cones at K, K and Γ of the mini Brillouin zone (BZ) shown in Fig. 2a.The central one at Γ is protected by P while K and K are gapped for generic φ [68].
We now obtain the spectrum of the Hamiltonian (1) and study the topological properties of the nearly-flat bands around charge neutrality.Fig. 1b shows the realspace mosaic pattern obtained by computing the Chern number C(φ) for the two central bands at the magic angle θ ABA = 1.69 • , for finite corrugation w AA = 0.8w AB .The mosaic exhibits a triangular periodic structure, which is generated by the lattice vectors a MM 1/2 = 4πe ∓iπ/3 /3kθ MM , where k MM θ = θ 2 K D .The two central bands are topological everywhere except along lines connecting the AAA centers, where the spectrum is fully connected.Each topological region is centered around r ABA = (a MM 1 − a MM 2 )/3 and r BAB = −r ABA , with opposite Chern numbers of ±1.Therefore, we specifically focus on these two high-symmetry points and consider the chiral limit, where the bands become exactly flat, and an analytical solution can be obtained.
The latter connects ABA to BAB, , explaining the opposite Chern number of the ABA and BAB regions.The combination of C 2x and C 2z T is a symmetry for the model C 2y T which together with P and C 3z protects three Dirac cones at Γ, K and K [71].We henceforth consider the chiral limit w AA = 0 where an inspiring mathematical structure emerges [36].H ABA then anticommutes with the chiral operator Λ z = τ 0 ⊗ σ z with τ 0 the identity in the layer basis.Denoting with ψ l and χ l with l = 1, 2, 3 the wavefunction components polarized in the A and B sublattices, in the basis T the Hamiltonian H ABA reads: we look for zero mode solutions: with boundary conditions: In Eq. 4 the operator is: with We focus on the first magic angle θ ABA ≈ 1.69 • where the renormalized velocity v * vanishes, see blue line in Fig. 2b, and correspondingly the bands around charge neutrality becomes perfectly flat as shown in Fig. 2c.Interestingly, the single particle gap that separates the flat bands from remote ones is E gap ≈ 130meV quite large if compared with the typical value of the Coulomb interaction screened by metallic gates [41].The C 3z symmetry yields χ Γ1 (0) = χ Γ3 (0) = 0, while χ Γ2 (0) is usually non-zero.The magic angle θ ABA is exactly defined by χ Γ,2 (0) = 0.As the spinor χ Γ (0) then fully vanishes at θ ABA , the B-polarized flat band has an analytical expression where the antiholomorphic ηk (z) = η * k (−z) is related to the meromorphic function with the notation k 1 = k • a 1 and ϑ 1 (z, ω) is the Jacobi theta-function [71], which vanishes at z = 0 and results in a Bloch periodicity Eq. ( 6).The Bloch wavefunction associated with Eq. ( 8) is k-antiholomorphic χ Γ (r) (10) corresponding to an ideal flat band [39,47].The Chern number of the band can be readily read off from the kspace boundary conditions where and φk , b1 = π which implies C B = −1 where the Chern number has been computing employing [47,54]: We turn to the exact solution for the A-polarized wavefunction ψ.C 3z yields again ψ Γ1/3 (0) = 0 at Γ and ψ K/K 2 (0) = 0 at K (K ), however ψ Γ2 (0) does not vanish at the magic angle.Nevertheless, we numerically find that ψ K,1 (0) = −ψ K,3 (0), right at the magic angle θ ABA which, combined with particle-hole symmetry P , can be used to prove that the two spinors at θ ABA .This remarkable identity allows us to exhibit an exact analytical expression for the A-polarized flat band (up to k-dependent prefactor) satisfying the Bloch periodicity Eq. ( 6), with the holomorphic function defined in Eq. ( 9) and For more details on the symmetry conditions leading to Eq. ( 14) we refer to SM [71].In Eq. ( 14), we set the K and K points at ±q 1 respectively.Thanks to the following property of the theta-function , it is readily checked that the poles of η k±q1 (z) at z = 0 cancel each other in Eq. ( 14), as a result of Eq. ( 13), and the wavefunction is finite everywhere.We note that, the corresponding unnormalized Bloch function u k (r) = ψ k (r)e −ik•r is k-holomorphic and thus constitutes an ideal flat band [39,47].In addition, momentum space boundary conditions 11 give φ k,b1 = 4πk/b 2 + 2πb 1 /b 2 and φ k,b2 = 0 leading to a Chern number C A = 2 which gives the total Chern number C = C A + C B = +1 characteristic of the triangular regions centered at the ABA sites of the real-space pattern in Fig. 1b.Remarkably, the Chern 2 band of Eq. ( 14) describes a color-entangled wavefunction [54] and, upon translation of a lattice vector r 0 → r 0 + a 1 , the k-space zeros of |ψ k2 (r)| get swapped, see Fig. 3.The emergence of the Chern bands +2 and −1 can be intuitively understood as a direct consequence of the original three Dirac cones of each layer, similar to twisted monolayer-bilayer graphene [84,85].AAA-stacking and domain wall lines-The local Hamiltonian describing the AAA points is obtained by setting φ = 0 in Eq. (1).It satisfies all symmetries [78]: C 2z T , C 2x , C 3z and particle-hole symmetry P , protecting the Dirac cones at K, K and Γ. C 2z T furthermore enforces [78] a fully connected spectrum as an odd number of Dirac cones cannot form isolated minibands [68,86,87].In the chiral limit (w AA = 0) the Hamiltonian H AAA in the basis Ψ = ψ 1 ψ 2 ψ 3 χ 1 χ 2 χ 3 T takes the form: where the operators reads: and U ω * (r) = U * ω (−r).At the magic angle taking place at θ AAA ≈ 0.75 • , see red line in Fig. 2b, the spectrum shown in Fig. 2d is composed by a fourfold degenerate zero mode subspace and a renormalized Dirac cone located at Γ. Interestingly, the wavefunction of the zero modes can be exactly expressed in terms of meromorphic functions as shown in Ref. [77].We here focus on the topological properties of the fourfold degenerate flat band sector.Away from the Γ point the flat bands are isolated and the degeneracy can be partially resolved by Λ z which gives two dimensional subspaces with opposite sublattice polarization Λ z = ±1.For a given sublattice the topological properties are characterized by the non-Abelian quantum geometric tensor Q ab nm (k) = D a u nk |D b u mk with D a the covariant derivative [88].The non-Abelian trace condition [35] reads as Tr tr g = tr Ω with Tr trace over space directions and tr over the 2D subspace.Fig. 4a shows the Berry curvature tr Ω A , we check numerically that the trace condition is satisfied everywhere at the exclusion of the Γ point where the Berry curvature is ill-defined.C 2z T imposes that the two sublattice sectors yield opposite Berry curvature trΩ A = −trΩ B .The points AAA are however singular.The fourfold degeneracy is lifted by any small but finite φ and the flat band sector with Chern number ±1 is recovered, see Fig. 4b.The Chern mosaic of Fig. 1b is thus largely governed by the topology of the ABA and BAB points.
Finally, the different topological regions extended around ABA and BAB meet along lines where the gaps to the remote bands vanish.These lines form a triangular lattice originating from the AAA lattice sites as shown in Fig. 1b.One can prove that, along these lines, the C 2x symmetry, combined with C 3z yields a fully connected band structure for Eq. ( 1), as seen in Fig. 4c.The proof, which follows Ref. [78], is given in the SM [71].However, breaking these symmetries can move these domain walls but not suppress them since the distinct topological domain must be separated by gap-closing contours.
Stability away from the chiral limit -Our predictions formally derived in the chiral limit are stable and persists for finite value of w AA .We see that at finite w AA the low-energy bands at ABA regions acquire a finite dispersion as shown in Fig. 5b.However, the two flat bands highlighted in red in Fig. 5b are characterized by a total Chern number 1 as shown by the winding 2π of the center of mass of the Wilson loop W(k 2 ) [89], gray dots in Fig. 5c while red and blue dots show the evolution of the two eigenvalues.The Chern number mosaic pattern over the entire supermoiré space was previously presented in Fig. 1b with w AA = 0.8w AB .
Conclusions -In summary, we showed that in equaltwist angle trilayer graphene separation between length scale leads to an interesting concept of supermoiré lattice where the local registry corresponds to twisting around AAA or ABA stacking but has a local-to-local variation over long range.The local adiabatic Hamiltonian which depends parametrically on the supermoiré lattice coordinate shows topologically distinct regions where the lowenergy flat bands have finite and opposite Chern number.Remarkably, the non-vanishing Chern number at ABA regions originates from a zero modes composed by a Chern +2 color-entangled wavefunction and a Chern −1 Landau level like state.The intermediate domain wall regions originating from the AAA stacking configurations are characterized by a fully connected spectrum which is gapless at all energies.We conjecture that similar properties can be realized also for unequal twist angle configurations once the decoupling between slow and fast length scales is performed.The large energy gap between flat and remote bands compared to the typical Coulomb energy scale makes ABA stacking eTTG an ideal playground for studying Fractional Chern insulators in higher Chern number bands.
Acknowledgments -We are grateful to Jie Wang, Andrei Bernevig, Jed Pixley, Nicolas Regnault and Raquel Queiroz for insightful discussions.D.G. also acknowledge discussions held at the 2023 Quantum Geometry Working Group meeting that took place at the Flatiron institute where he was introduced to some of the concepts presented here.We acknowledge support by the French National Research Agency (project TWIST-GRAPH, ANR-21-CE47-0018).The Flatiron Institute is a division of the Simons Foundation.
Note added: after the completion of the draft we become aware of the work by Trithep Devakul et al. [90] which overlapped with part of our results.
Appendix C: Symmetries and protected Dirac cones for ABA stacking In the following we discuss the symmetry properties of eTTG with respect to the ABA stacking configuration.The Hamiltonian representative of the ABA region is obtained setting the values of the phases φ ABA = (φ 1 , φ 2 , φ 3 ) = (0, ϕ, −ϕ) with ϕ = 2π/3, notice that one of the phases say φ 1 can be gauged away by global phase shift φ j → φ j + φ 0 .The Hamiltonian in the basis B4 reads: where the tunneling matrices are: The C 3z symmetry takes the form: and We observe that C 2x is defined as: is broken, since under C 2x we map the ABA to the BAB stacking.This can be readily realized observing that: Similarly, we find: Symmetries of the local Hamiltonian for the two different high-symmetry stacking configurations for the K-valley.Original and chiral shows the matrix expression of the symmetry in the two different basis.Finally, last columns show how the symmetry acts on k± = kx ± iky and r± = x ± iy.We further notice that K is the complex conjugation operator and the ABA stacking breaks C2zT and C2x symmetries.The combination C2xC2zT is still a symmetry of the model.
We now notice that the combination is a symmetry of the model: We summarize the symmetries in Table I.
To the aim of showing the protection of three Dirac cones at Γ, K and K we further discuss the spatial symmetries in the ABA configuration and the symmetry group they generate.As detailed above, the model exhibits two spatial symmetries in that case, C 3z and C 2y T .They form the magnetic space group P 32 1 (#150.27 in the BNS notation).The irreducible representations at the high-symmetry momenta Γ and K (K ) verify the C 3 point group character table and are all one-dimensional.The absence of two-dimensional representations noticeably prevents the stabilization of Dirac cones in the spectrum.
It can be understood more directly by considering an eigenstate of C 3z , C 3z |ψ + = ω|ψ + .From the relation we obtain C 3z C 2y T |ψ + = ω C 2y T |ψ + .C 2y T thus does not circulate between the eigenstates of C 3z and cannot protect a twofold degeneracy.The three zero-energy Dirac cones arising in the band spectrum of ABA trilayer graphene are therefore only stable in the presence of the additional particle-hole symmetry P .Since P and C 3z commute, applying P on a given state does not change its C 3z eigenvalue.Therefore, if the spectrum, at Γ or K/K , hosts two states with C 3z eigenvalues ω, ω * near charge neutrality (and no other states), particle-hole symmetry P automatically pins these two states at zero energy.This is proven by contradiction: if we assume that the two states sit at opposite non-vanishing energies, then P permutes them.This is however impossible since P cannot change the C 3z eigenvalue which completes the proof.In fact, P restricted to these two states must be the identity as it commutes with C 3z .It further shows that breaking C 3z does not lift the Dirac crossings as the trivial (identity) representation of P = I 2 cannot deform continuously to the traceless σ x matrix permuting states with opposite non-zero energies.
In summary, the Dirac cone at Γ is explained using C 3z and P .K and K are however not stable under P -which permutes K and K -and the stability of their Dirac cones must come from a different operator.Since the symmetry C 2y T also permutes K and K , the combination P = P C 2y T leaves them invariant and acts as a (anti-unitary) particle-hole operator.Using that one verifies that P cannot permute the C 3z eigenvalue between ω and ω which implies that an isolated pair of states at K or K with C 3z eigenvalues (ω, ω * ) is degenerate and pinned at zero energy.In this subspace, P = K I 2 and the Dirac cone survives a breaking of C 3z .
1. Symmetry constraints on the Chern 2 flat band Here we provide the symmetry constraint on the C = 2 zero mode that leads to the analytical expression 14 given in the main text.
To start with we observe that the C 3z symmetry implies ψ Γ1/3 (r) = ωψ Γ1/3 (C 3z r) and ψ Γ2 (r) = ψ Γ2 (C 3z r) resulting in the condition ψ Γ1/3 (0) = 0.However, differently from the B-sublattice component we find ψ Γ2 (0) = 0 at the magic angle.To proceed further we now observe that MD † 1 (−r)M = −D † 1 (r) with M defined in Eq. 3 of the manuscript.At the Γ point, it leads to ψ Γ2 (r) = ψ Γ2 (−r), and ψ Γ1 (r) = ψ Γ3 (−r) which connects the top and bottom components at inversion-symmetric positions.Looking at K and K = −K points the C 3z symmetry leads to the identities ψ ±K1/3 (r) = ψ ±K1/3 (C 3z r) and ψ ±K2 (r) = ω * ψ ±K2 (C 3z r) consistent with ψ K = (1, 0, 0) and ψ K = (0, 0, −1) at w AB = 0.A direct consequence is that ψ ±K2 (0) = 0.In contrast to C 3z , the particle-hole symmetry M couples K and K we find ψ K1/3 (r) = −ψ −K1/3 (−r) and ψ K2 (r) = ψ −K2 (−r) resulting in ψ ±K1 (0) = −ψ ∓K3 (0).Moreover and remarkably, we numerically find that, right at the magic angle θ ABA which results in the fact that the two spinors ψ K (r = 0) and ψ K (r = 0) become in fact identical.We now generalize the argument in Ref. [38,50] defining the Wronskian for the case of three layers.To this aim we consider three zero mode solutions ψ kj with momenta k 1,2,3 and we introduce the triple product which satisfies ∂W (r) = 0, see Sec.G, proving the holomorphy of W (r). Since W (r) also satisfies Bloch periodicity (being the product of three Bloch-periodic functions) and cannot have any pole in the complex plane, by Lioville's theorem it must be constant in space.Taking k 2 = K and k 3 = K and an arbitrary momentum k 1 , the quantity W (r = 0) = 0 since ψ K (0) and ψ K (0) are collinear as discussed above.Therefore the triple product W (r) vanishes in this case and the three spinors ψ k1 (r), ψ K (r) and ψ K (r) are coplanar for all r.Since this is true for arbitrary k 1 , it implies that all zero-energy states belong to the same plane generated by ψ K (r) and ψ K (r) and the triple product W (r) is identically vanishing irrespective of the values of r and k 1,2,3 .Thus, an analytical expression also emerges from the previous analysis and the wavefunction for the A-polarized flat band takes the form (up to k-dependent prefactor) in Eq. 14 of the manuscript.
Appendix D: Symmetries for AAA stacking In the following we discuss the symmetry properties of eTTG with respect to the AAA stacking configuration.
The AAA stacking configuration φ AAA = (0, 0) recently discussed in the preprint [77] is described by the Hamiltonian [78]: In this case the tunneling matrix simply reads: T j e −i qj •r , (D2) under C 3z it transforms as: e −iϕσz T 0 (C 3z r)e iϕσz = T 0 (r).(D3) We readily understand that the three layers in this case are characterized by the same C 3z eigenvalue.Thus, the C 3z operator reads: and For the sake of completeness we list the remaining symmetries: so that

FIG. 1 .
FIG.1.a) Equal-twist angle trilayer graphene lattice in real space.b) Real-space Chern mosaic in the moiré of moiré lattice scale.We distinguish three high-symmetry regions of character AAA, ABA and BAB.The former shows a fully connected spectrum while the latter is characterized by flat bands with total Chern number +1 and −1, respectively.Domain walls regions marking topological transition between Chern ±1 states show fully connected spectrum.

FIG. 3 .
FIG. 3.Each figure shows |ψ k2 (r)| as a function of k for a fixed r.From left top to right bottom the position r evolves from r0 = (0.1, 0.3) (unit of 1/k θ ) to r0 + a1.There are in total CA = 2 zeros in the Brillouin zone at fixed position r, and their pattern is lattice translational invariant but indices are be exchanged going from r0 to r0 + a1.The effect resembles Thouless charge pumping but in reciprocal space.The solid gray line denotes the diamond shaped BZ.

FIG. 5 .
FIG. 5. Results away from chiral limit (wAA/wAB = 0.7) for twist angle θ = 1.69 • corresponding to the magic angle for ABA stacking in the chiral limit.a) Fully connected spectrum for AAA stacking.b) Spectrum for ABA stacking, the flat bands separated from remote ones are highlighted in red.c) Wilson loop W(k2) spectrum for the flat bands as a function of k2 = k • a2/2π.Red and blue dots shows the eigenvalues for the two flat bands, grey dots, instead, show the evolution of the center of mass characterized by a 2π winding corresponding to a total Chern number 1.

TABLE II .
Character table of C3v at Γ and M .E, C3 and C2 represent the conjugation classes of the identity, C3z and C2x, respectively.