Weak-field Hall Resistivity and Spin/Valley Flavor Symmetry Breaking in MAtBG

Ming Xie and A. H. MacDonaldPhysics Department, University of Texas at Austin, Austin TX 78712

(Dated: October 16, 2020)

Near a magic twist angle, the lowest energy conduction and valence bands of bilayer graphenemoire superlattices become extremely narrow. The band dispersion that remains is sensitive to themoire’s strain pattern, nonlocal tunneling between layers, and filling-factor dependent Hartree andexchange band renormalizations. In this article we analyze the influence of these band-structuredetails on the pattern of flavor-symmetry-breaking observed in this narrow band system, and on theassociated pattern of Fermi surface reconstructions revealed by weak-field-Hall and Schubnikov-deHaas magneto-transport measurements.

Introduction.—When twisted close to a magic [1] ori-entation angle, the ground state of bilayer graphene ex-hibits [2–5] a rich series of strongly correlated electronicground states. The magic-angle twisted-bilayer graphene(MAtBG) phase diagram is most strongly dependent ontwist angle θ and on moire band filling factor ν = nAM ,where n is the carrier density and AM is the area of themoire pattern unit cell, but is also responsive to otherexternal parameters including the orientation angles ofthe encapsulating hexagonal boron nitride layers, andthe vertical separation between the bilayer and the elec-trical gate or gates used to manipulate the carrier den-sity. Experimental work over the past couple of years[5–15] has established that the spin/valley flavor symme-tries responsible for the four-fold degeneracy of the moirebands are often broken when the flat conduction band ispartially occupied or the flat valence band is partiallyemptied. The flavor symmetry breaking is reminiscentof the behavior of Bernal-stacked bilayer graphene in astrong magnetic field when its flat N = 0 Landau lev-els are partially filled [16–23]. The pattern of symmetrybreaking is however quite distinct in the two cases. In-stead of filling up the eight bands one at a time to min-imize the exchange energy, as observed in the quantumHall case, the guiding principle that describes the flavorsymmetry breaking, illustrated schematically in Fig. 1,is that the flavor-dependent partial filling factors νFSof all valence bands with a Fermi surface normally ex-ceed a critical value νcrv typically ∼ 0.55, and that thepartial filling factors νFS of all conduction bands witha Fermi surface generally lie below νcrc ∼ 0.2. In thisLetter we provide an explanation for this behavior thatis based on the influence of strain, nonlocal tunnelingbetween layers, and filling-factor dependent Hartree andexchange interactions on the band structure. Our anal-ysis is informed by Schubnikov-de Haas and weak-fieldHall magneto-transport data [2, 4–7, 14, 24–29].

MAtBG Bandstructure.—Magic-angle strong correla-tion physics persists over a small range of twist angles,covering perhaps ∼ 0.2◦, within which the typical ve-locity within the flat bands is reduced by an order ofmagnitude or more relative to the band velocity in anisolated graphene layer. We argue here, however, that

1 2 3 4 1234

−4 −3 −2 −1 0 1 2 3 4ν

νF S = 0

νF S = 1

νcrv

νcrc

MAtBG Fermi surface reconstructions

SC

CI

FIG. 1. The flat bands of MAtBG are partially occupied forfilling factor ν ∈ (−4, 4) and exhibit flavor symmetry breakingover much of this range. While not entirely universal, flavorsymmetry breaking tends to follow the following rules. Onthe valence side, flavor symmetry breaking depopulates flatvalence bands so as to keep the partial filling factors νFS ofthe remaining partially occupied flavors, which have a Fermisurface, above νcrv ∼ 0.55 as indicated by the red arrows. Onthe conduction side, flavor symmetry breaking favors com-plete occupation of one, two, or three conduction bands so asto keep νFS of the remaining partially occupied flavors belowνcrc ∼ 0.2. The solid lines plot the band filling factor per par-tially occupied flavor in 1, 2, 3, and 4 Fermi surface states.This pattern of flavor symmetry breaking allows for insulat-ing states at all non-zero ν between -4 and 4, and for Cheninsulators (CI) at odd integer ν. The strongest superconduc-tivity (SC) seems to emerge from states with two valence bandFermi surfaces, and the strongest anomalous Hall effects seemto occur in states with one conduction band Fermi surface.

the band dispersion that survives near the magic twistangle plays a crucial role in establishing the ground statephase diagram.

We specify the single-particle MAtBG flat bands us-ing four phenomenological parameters, wAB - the inter-layer intersublattice tunneling strength, wAA - the in-terlayer intrasublattice tunneling strength, a tunnelingnon-locality parameter wNL, and ε−1, a parameter thatcharacterizes how strongly the moire band Hamiltonianis modified by interactions. The ratio wAA/wAB has thevalue 1 for simplified models [1] in which the graphenelayers are rotated rigidly, but is known [31, 32] to bereduced relative to 1 when strain and corrugation are

arX

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010.

0792

8v1

[co

nd-m

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es-h

all]

15

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0

2

FIG. 2. Band-structures in the full and empty moire bandlimits for band parameters ~vkθ = 1.69wAB , corresponding toa twist angle of about θ ∼ 1.1◦. wAA/wAB = 0.6, wNL = 20meV and ε−1 = 0.03. Interaction effects strongly weakendispersion when the Fermi level moves toward γ on eitherelectron or hole sides.

taken into account. The non-locality of the interlayertunneling is known [32] to be principally responsible forparticle-hole asymmetry in MAtBG, which we model by

T (r, r′) =

2∑j=0

∑p

Tj(p)e−iqj ·(r+r′)/2eip·(r−r′) (1)

where Tj(p) = t(p)TBMj and the momentum dependenttunneling amplitude t(p) = tkD + (dt/dp)p=kD (kD − p).We parametrize the slope of the momentum dependencein t(p) by wNL ≡ (dt/dp)p=kD |b|. TBMj is the interlayertunneling matrix in the BM model (see SupplementalMaterial for details). ε−1 accounts for the filling factordependence of the moire bands due to modifications fromelectron-electron interactions. Below we explain whatmagneto-transport measurements tell us about the typi-cal values of these band parameters.

The flat bands of MAtBG have a simple and system-atic dependence on band filling, one that we argue playsan important role in the flat-band phase diagram. InFig. 2 we plot the flat bands when they are completelyfull and when they are empty. These calculations ne-glect mixing between flat and remote bands, an approx-imation that is justified by flat-band spectral isolation.In this approximation the many-electron ground state isfully determined by the Pauli exclusion principle in bothlimits, and is a single Slater determinant in which single-particle-state occupations numbers are either 0 or 1. Itfollows that the electron self-energy is given exactly bythe Hartree-Fock operator:

ΣHF = ΣH(δρ) + ΣF (δρ), (2)

where δρ =∑′α,n,k |α, n,k〉〈α, n,k| − ρiso is the ground

state density matrix defined relative to neutral isolatedgraphene states, α is a composite label for valley and

spin, n is a single-particle band label, the prime restrictsthe summation to filled bands, and ΣH and ΣF are theusual Hartree and Fock self-energies (See SupplementalMaterial for further detail.)

The bands plotted in Fig. 2 are eigenvalues of the bandHamiltonian HB = Hsp + ΣHF , where Hsp is the single-particle moire band Hamiltonian. The difference betweenthe quasiparticle dispersions in the empty and full bandlimits is due to the differences between the self-energyoperator when the flat bands are full, ΣHFf , and when

the flat bands are empty, ΣHFe . We see in Fig. 2 that theeffect of the self-energy is to lower (raise) the energy nearthe center of the moire Brillouin zone (k = γ), relative tothose near the corners of the Brillouin zone (k = κ,κ′)as the flat conduction (valence) bands are filled (emp-tied). This behavior has been explained [33, 34] in termsof Hartree interactions and the concentration of flat bandstates at AA positions in the bilayer moire pattern, wherethe wavefunctions of flat band states near γ have lowerweight. Exchange interactions also play a role in reshap-ing the bands, and a more critical role in breaking flavorsymmetries (see Supplemental Material). These interac-tion effects have a smooth dependence on filling factorwhich justifies the approximation

HB = Hsp +1

2

[ΣHFf + ΣHFe +

ν

4(ΣHFf − ΣHFe )

]. (3)

The end result is that Fermi velocities are extremelysmall when the bands are nearly empty and nearly full,in contrast to the case of single-particle band models forwhich Fermi velocities are maximized for nearly full andnearly empty bands. This property is captured only whenself-energies from frozen remote bands are included in thetheory. We argue below that it also plays a crucial rolein determining the pattern of flavor symmetry breaking.

Schubnikov-de Haas oscillations.—Oscillations inphysical properties associated with periodic filling andemptying of nascent Landau levels in weak magneticfields have long [35] been used to measure the Fermi sur-faces of metals. In two-dimensional materials the mostaccessible oscillations are normally those of longitudinalresistance referred to as Schubnikov-de Haas (SdH) os-cillations. SdH oscillations in MAtBG [2, 4–7, 14, 24–29]are generally speaking observable only near neutralityfor ν ∈ (−1.6, 0.8) and for |ν| ∈ (2, 3). We attribute thisproperty to the interaction induced reductions in Fermivelocity when the bands are nearly empty or nearly fullas explained above. Comparison of Fermi surface areato carrier density suggests that all four flavors haveequivalent Fermi surfaces near neutrality. For |ν| ∈ (2, 3)the same comparison suggests that only two of the fourflavors have Fermi surfaces, the other two flat bandshaving apparently been depopulated on the valence bandside and completely filled on the conduction band side.These observations are consistent with the interpretationof weak-field Hall observations discussed below, which

3

(I)

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(II)

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(III)

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(IV)

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FIG. 3. Hall filling factor νH = nHAM vs. band filling factorν of a single flavor for the band models (I-IV) defined in themain text. In numerical order the four models improve theBM model (I) by sequentially adding corrections for strain(II), nonlocal tunneling (III), and interactions (IV). The bluedashed lines are the Hall densities νH = ν + 1, ν, ν − 1 forisotropic Fermi surfaces near v = −1, 0, 1 respectively.

are able to provide valuable information over the fullrange of flat-band filling factors because they do not relyon an adequate Landau level spacing.

Weak-field Hall resistivity.—When mean-free-pathsexceed Fermi wavelengths, the transport properties oftwo-dimensional Fermi liquids can be described usingBoltzmann transport theory. Employing a relaxationtime approximation, a practical necessity when thesource of scattering is unknown, the conductivity ten-sor of a system with C6 symmetry is given [36] to leadingorder in magnetic field B by

σxx = e2τ∑k

(−∂fk∂E

)v2x,

σyx =2e3τ2B

~∑k

(−∂fk∂E

)vx(v × z) ·∇vy, (4)

with σxx = σyy and σyx = −σxy. It follows that to firstorder in B the Hall resistivity

ρxy =σyxσ2xx

≡ B

nHec. (5)

In Eq. 5 the coefficient of the B-linear term in ρxy is apure bandstructure property because the scattering timeτ cancels between numerator and denominator. As sug-gested by the final form on the right-hand-side, it is con-venient to characterize this quantity by the Hall den-sity, nH , which is defined by this equation. For isotropicelectron-like Fermi surfaces (occupied states inside cir-cle), nH = n, whereas for isotropic hole-like Fermi sur-faces (occupied states outside circle), nH = −p, where nand p are the corresponding electron and hole densities.

In Fig. 3 we plot Hall filling factor νH = nHAMvs. filling factor ν for partially occupied MAtBG bands

at wAB/~vK = 1.69 for four different bandstructuremodels: (I)wAA = wAB , wNL = 0, (II)wAA/wAB =0.6, wNL = 0, (III)wAA/wAB = 0.6, wNL = 20 meV,and (IV)wAA/wAB = 0.6, wNL = 20 meV with ε−1 =0.03. The first model is the Bistritzer-MacDonald (BM)model specified in [1], which is improved in models (II),(III), and (IV) by sequentially adding corrections forstrain and corrugation, nonlocal tunneling, and interac-tions. In all cases νH ∼ ν close to neutrality, expectedfor isotropic Fermi surfaces. Away from neutrality, Fermisurfaces are more anisotropic for larger ωAA/ωAB andthis leads to larger deviations [36] of the Hall density fromthe corresponding Fermi surface area curves marked inblue. (The sensitivity of the Hall density to band param-eters is discussed in detail in the Supplemental Material.)

The most prominent features in Fig. 3 are the switchesbetween large νHs of opposite signs that occur once forpositive and once for negative filling factors. The fillingfactors at which these sign changes occur are close tothe filling factors at which the topology of the Fermi sur-faces changes from electron-like to hole-like. As explainedin the Supplemental Material, the sign changes do notprecisely match the van Hove singularities (VHS) [37]at which the flat band density of states (DOS) divergeslogarithmically yielding a feature that is prominent intunneling spectroscopy measurements [8–12]. We see inFig. 3 that the positions of these sign changes are sensi-tive to the bandstructure model details. They move veryclose to neutrality when the strain/corrugation correc-tions are added. Including nonlocal tunneling shifts theposition closer to (further from) neutrality on the conduc-tion (valence) band side, strongly violating particle-holesymmetry. Interaction renormalizations move the posi-tions on both sides away from neutrality by an amountdetermined by the strength of interaction.

Hall Density, Fermi Surface Reconstruction, and Lift-shitz Transitions.—The Hall densities in Fig. 3 differqualitatively from experimental data, which typically ex-hibit around five jumps in value for ν ∈ (−4, 4) comparedto the two jumps present in Fig. 3 assuming no flavorsymmetry breaking. Hall density jumps, evidently alsooccur because of spin/valley flavor symmetry breakingphase transitions. These transitions sequentially maxi-mize hole densities for one, two, or three flavors on thehole side, and electron densities of one, two, or three fla-vors on the conduction band side. The guiding principlefor these phase transitions seems to be to place the Fermilevels of the partially occupied bands, whenever possible,on the side of the van Hove singularity closest to theDirac point at neutrality.

The number of additional Hall density jumps and theirprecise filling-factor positions are somewhat sample de-pendent. Fig. 4 shows calculated Hall density that ac-counts for flavor symmetry breaking transitions. Thesecond Hall density jump on the valence band side is con-sistent with traversal of VHSs in doubly degenerate Fermi

4

4 3 2 1 0 1 2 3 410

5

0

5

10H

42 3 2 1

FIG. 4. Hall filling factor as a function of band filling factorallowing spin/valley flavor symmetry breaking, calculated us-ing interaction renormalized bands and the pattern of flavorsymmetry explained in the main text. The shading specifiesthe number of partially occupied bands, i.e. the number ofFermi surfaces in each region of filling factor. The solid linesmark the Hall density that would be calculated if the Fermisurfaces were approximated as isotropic.

surfaces. On the conduction band side the first Hall den-sity jump occurs already near ν ≈ 4νcrc . We attribute thedifference between electrons and holes to the differencein the position at which the VHS occurs. Two additionaljumps in Hall density typically occur on the conductionband side, and each seems to be associated with a flavordepopulation event. One important consequence is thatstates with an odd number of Fermi surfaces are morecommon on the conduction band side. The theory curvein Fig. 4 is constructed from Fig. 3, the assumption ofone flavor symmetry breaking transition on the hole sideat νFS ≈ νcrv (or equivalently ν ≈ −4(1 − νcrv )) froma four-Fermi surface state to a two-Fermi surface state,the assumption of three flavor-symmetry-breaking tran-sitions on the electron side, at ν ≈ 4νcrc , 1+3νcrc , 2+2νcrc ,to three, two, and one Fermi surface states, and flavor-dependent Hall density calculations discussed in the Sup-plemental Material that reflect broken symmetries, butare still similar to the results plotted in Fig. 3.

The flavor symmetry breaking behavior evident in theweak-field Hall and SdH data is understandable in termsof mean-field considerations. States that have a Fermilevel on the neutrality side of the VHS are favored by alow DOS close to the Fermi level. Particle-hole asym-metry can be explained by the closer proximity of VHSto neutrality on the conduction band side, which forcesflavor symmetry breaking earlier in the conduction bandfilling process than in the valence band emptying pro-cess, and therefore favors states with an odd numberof Fermi surfaces that can host topologically non-trivialstates [6, 7, 38]. The even lower DOS interval that occursfor nearly full and nearly empty bands when interactionsare neglected does not share the enhanced stability of itsDirac point cousin because of interaction renormaliza-tions that reduce Fermi velocities when bands are nearlyfull or nearly empty.

Discussion.—This Letter addresses the implicationsof weak-field magneto-transport data for the correlationphysics of MAtBG. The moire filling factor ranges overwhich SdH oscillations are visible identify where Fermilevel quasiparticles have the largest velocities. The factthat SdH is most visible near neutrality is at first sightsurprising since the independent electron flat bands aremost dispersive in precisely the opposite limit, namely fornearly full or nearly empty bands. We have interpretedthis behavior as a consequence of interaction-inducedband renormalizations that flatten the conduction bandtop and the valence band bottom when the Fermi levelis near these band edges.

Weak-field Hall effect measurements provide additionalinformation since this quantity is observable at largerquasiparticle masses and stronger disorder. We haveconcluded that the many jumps in Hall density thattypically occur as a function of band filling cannot beexplained on the basis of single-particle physics, sincethese allow for only one jump each side of the neutral-ity point. We attribute the additional jumps to a seriesof flavor-symmetry breaking phase transitions that re-construct Fermi surfaces by redistributing the occupancyamong flavors. These reconstructions favor partially oc-cupied bands whose Fermi surfaces are on the neutralityside of the VHS present in each band. A similar con-clusion was reached on the basis of thermodynamic com-pressibility measurements in Ref [13]. Our analysis ar-gues that filling-factor dependent band renormalizationsplay an essential role in the robustness of this effect, andthat nonlocal interlayer tunneling controls its particle-hole asymmetry which is substantial in most cases. Ifindeed mean-field considerations are able to describe theintermediate energy scale physics responsible for flavorpolarized states at least qualitatively, it is appropriate toask why given the small ratio of band to interaction en-ergy scales. We attribute this behavior to the entanglednature of the band wavefunctions that precludes Wannierfunction construction, combined with the long-range ofthe Coulomb interaction between electrons.

The full Hamiltonian of MAtBG, including all interac-tion terms is, in the absence of external fields, nearlyexactly spin-independent, because of graphene’s weakspin-orbit coupling. The long-range Coulomb interac-tion effects that dominate are fully SU(4) invariant inspin/valley space. It follows that the flavor spinors as-sociated with the Fermi surfaces discussed in this MSare fixed by smaller anisotropy energy scales, much likethe magnitude of the spin-polarization of a ferromagneticmetal is fixed on much higher energy scales than the spin-orientation. Unlike the metal case, in which the spin-dependence of electron-electron interactions is normallyignored, the valley and orbital dependence of interactionscould [39, 40] play as important a role as the band Hamil-tonian in the anisotropy energy scales, and could play arole in the superconducting state that emerges at the low-

5

est temperatures. Surveying the experimental literatureand assigning Fermi surface degeneracies via considera-tions like those discussed in this MS, it seems that super-conductivity occurs with similar transition temperaturesin states with two and four Fermi surfaces, but has muchlower transition temperatures when seen in states withan odd number of Fermi surfaces.

Acknowledgment.—The authors acknowledge helpfulinteractions with Eva Andrei, Dmitry Efetov, WeiQin, Petr Stepanov, Shuang Wu, Andrea Young, andZhenyuan Zhang. We thank Stephen Carr for a help-ful conversation explaining the relationship betweenparticle-hole asymmetry in the flat bands and tunnel-ing non-locality. This work was supported by DOE BESunder Award DE-FG02-02ER45958.

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[26] A. Uri, S. Grover, Y. Cao, J. A. Crosse, K. Bagani,D. Rodan-Legrain, Y. Myasoedov, K. Watanabe, T.Taniguchi, P. Moon, M. Koshino, P. Jarillo-Herrero, andE. Zeldov, Nature 581, 47 (2020)

[27] I. Das, X. Lu, J. Herzog-Arbeitman, Z.-D. Song, K.Watanabe, T. Taniguchi, B. A. Bernevig, and D. K. Efe-tov, arXiv:2007.13390

[28] Y. , J. Ge, L. Rademaker, K. Watanabe, T. Taniguchi,D. A. Abanin, and A. F. Young, arXiv:2007.06115

[29] Y. Saito, J. Ge, K. Watanabe, T. Taniguchi, E. Berg andA. F. Young, arXiv:2008.10830

[30] G. Tarnopolsky, A. J. Kruchkov, and A. VishwanathPhys. Rev. Lett. 122, 106405 (2019)

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dle points of the energy-momentum surfaces throughoutthis work.

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6

Supplemental Material for“Weak-field Hall Resistivity and Spin/Valley Flavor Symmetry Breaking in MAtBG”

Effective nonlocal tunneling model

We generalize the continuum BM model to account for the nonlocal nature of interlayer tunneling in twisted bilayergraphene, which is shown by ab initio calculations to be responsible for the particle-hole asymmetry. In the originalBM model, the interlayer tunneling term is taken to be momentum independent which means that in real spaceelectrons can only tunnel vertically between the same in-plane locations of the top and bottom layers. In other words,the interlayer tunneling term between electrons at r in the top layer and at r′ in the bottom layer is diagonal in acoordinate representation: T (r, r′) = T (r)δ(r, r′). We generalize this tunneling model to allow tunneling which isoff-diagonal in in-plane coordinates, i.e. tunneling that is nonlocal. The spinless single-particle Hamiltonian projectedonto valley K takes the form:

HKsp =

(hθ/2(k) T (r, r′)T †(r, r′) h−θ/2(k′)

)(S1)

where h±θ/2 are the Dirac Hamiltonians for isolated rotated graphene top (+) and bottom (-) layers,

hθ(k) = −~vD|k|(

0 ei(θk−θ)

e−i(θk−θ) 0

), (S2)

θk is the orientation angle of momentum measured from the Dirac point of corresponding layer k = k − K±θ/2.K±θ/2 is the Dirac momentum of top/bottom layer. We choose the convention in which k and k′ are measured froma common origin of momentum.

The interlayer tunneling Hamiltonian takes the form

T (r, r′) =

2∑j=0

∑p

Tj(p)e−iqj ·(r+r′)/2eip·(r−r′). (S3)

The average coordinate experiences the moire periodic modulation which is accompanied by momentum boosts qj = 0,b1 or b2 for j = 0, 1, 2 respectively. b1,2 = (±1/2,

√3/2)4π/(

√3aM ) are the basis vectors of moire reciprocal lattice,

where aM = a/(2 sin(θ/2)) is the lattice constant of moire pattern and a the lattice constant of monolayer graphene.The dependence on the relative coordinate is captured by the momentum dependent tunneling matrices:

Tj(p) = t(p)TBMj , (S4)

t(p) = tkD +dt

dp|p=kD (p− kD), (S5)

TBMj =ωAAωAB

σ0 + [cos(jφ)σx + sin(jφ)σy] . (S6)

In the BM model, t(p) ≡ tkD . Here we retain momentum dependence in t(p) to first order in p and assume thatit is independent of the orientation of p. We parameterize the first derivative using ωNL ≡ dt/dp · |b| where |b| isthe length of basis vectors in moire reciprocal space. The average magnitude of interlayer tunneling is taken to beωAB ≡ tkD = 110 meV and ωAA/ωAB = 0.6 if not otherwise specified.

It is important to note that p is momentum measured from the Γ point and implicitly depends on the directionof momentum boost or the index j as detailed below. In the extended zone scheme, interlayer tunneling preservesmomentum p = k +Gj = k′ +G′j given that k and k′ are also measured from Γ. Gj and G′j are reciprocal latticevectors of the rotated top and bottom graphene layers which satisfy G0 = G′0 = 0, G′2−G2 = b2 and G′1−G1 = b1.Because we are interested in a small region with a size on the order of mBZ in momentum space near the Dirac points,|k −K±θ/2| ∼ |b|/|K±θ/2| ∼ θ, the relevant reciprocal vectors G = mG1 + nG2 are only G0, G1, and G2.

The value of ωNL can be determined experimentally, for example, by measuring the single-particle energy gapbetween the flat bands and the remote bands. As we detail in the section below, the difference between the gaps toremote conduction bands and remote valence bands is directly related to ωNL. We take ωNL to be ∼ 20meV, whichapproximately produces the correct single-particle gap sizes. In the next section we discuss the effect of nonlocaltunneling on the energy bandstructure.

7

Symmetry of the nonlocal tunneling Hamiltonian and particle-hole asymmetry

The original single-particle continuum model has an approximate particle-hole symmetry that becomes exact whenwe ignore the phase factor e±iθ/2 due to the rotation of top and bottom layers’ momentum space or when the interlayerintra-sublattice tunneling amplitude set to zero (ωAA = 0), as in chiral limit models. In reality, ωAA is neither notequal to 0, or even close to 0. Instead, ab initio calculations estimate that ωAA/ωAB is in the range 0.6 − 0.7.Therefore we consider the more general and realistic case with nonzero ωAA. Because the symmetry of the intralayerDirac Hamiltonian remains intact, we focus on the symmetry of the interlayer tunneling Hamiltonian when it isgeneralized to become nonlocal.

The real space representation D of a symmetry transformation S involving space group element g in general dependson spatial coordinates, DS ≡ DS(r). For nonlocal operators, the transformation works as

O′(r, r′) = DS(r)O(gr, gr′)D−1

S(r′) (S7)

The nonlocal tunneling Hamiltonian in Eq. S1 preserves the original model’s C2T symmetry, where C2 is two-foldrotation around z axis and T is the spinless time-reversal transformation:

DC2T T (C2r, C2r′)D−1

C2T= σxT

∗(−r,−r′)σx = T (r, r′), (S8)

where DC2T = σxK is independent of r and layer, and K is the complex conjugation operator. The last equality can

be obtained using Eq. S3 keeping in mind that TBMj is invariant under DC2T .

Three-fold rotational symmetry C3 (around z axis) is also retained because

DtC3(r)T (C3r, C3r′)(DbC3(r′))−1 = T (r, r′). (S9)

where DtC3(r) = ei2π3 σzeiq1·r for the top layer and DbC3(r′) = ei

2π3 σzei(q1−q2)·r for the bottom layer. The tunneling

amplitude t(C3p) = t(p) is invariant under C3.In addition, the nonlocal tunneling Hamiltonian also preserves the two-fold rotational symmetry around the x axis,

DMx

(0 T (r, r′)

T †(r, r′) 0

)D−1

Mx=

(0 T (r, r′)

T †(r, r′) 0

). (S10)

where Mx changes (x, y) to (x,−y) and swaps both sublattice and layer as DMy= σxτx. τ is Pauli matrix for layer

pseudo-spin. In deriving this equality, we have used Mxq1,2=−q2,1.

Finally we come to the original model’s particle-hole symmetry under the transformation P that applies when thephase factors ei±θ/2 are ignored. P operates on the tunneling Hamiltonian as

DtP(r)T (Mxr)(DbP(r))−1 = −T (r) (S11)

where Dt/bP

(r) = ±σxe2i(q1−q2)·r and My changes (x, y) to (−x, y). Eq. S11 follows directly from the relation

Myq0 = q0 and Myq1,2 = q2,1. When generalized to include a nonlocal contribution, the tunneling Hamiltonian isno longer particle-hole symmetric and instead satisfies

DtP(r)T (Mxr,Mxr′)(DbP(r′))−1 = −T (r, r′)e2i(q1−q2)·(r−r′). (S12)

In summary, the nonlocal tunneling Hamiltonian preserves the original model’s C2T symmetry, three-fold rotationsymmetry and mirror symmetry. However, it violates the particle-hole symmetry. In the next section, we discussquantitative changes in the single-particle energy dispersion due to tunneling non-locality.

Bandstructure of the nonlocal tunneling model

In this section, we show how the nonlocal tunneling modifies the single-particle bandstructure. Fig. S1(a) plotsthe typical bandstructure with ωNL = 20meV. The bands shift upward in energy relative to those of the originalmodel, which is approximately symmetric around zero energy with a small asymmetry due to the e±iθ/2 factor. Thesingle particle gap is reduced on the valence band side but increased on the conduction band side. The flatband

8

(a)

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(c)

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(b)

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FIG. S1. Bandstructure of the nonlocal tunneling model. (a) Comparison of bandstructures between local ωNL = 0 (blackdashed) and nonlocal ωNL = 20meV (blue solid) models. θ = 1.1 and ωAA/ωAB = 0.6 are used in both cases. The red arrowsmark the single-particle gaps Evg and Ecg. (b) Flat-band zoom for the bandstructure in (a). For comparison purposes, the bluesolid curve is shifted in energy so that the two cases match in energy at the Dirac points. (c) The single-particle gaps Evg and

Ecg as a function of the strength of nonlocality ωNL. The small e±iθ/2 phase factor is included in both cases.

dispersion changes as the valence band becomes flatter while the conduction band becomes wider. The detailedchange in flatband dispersion is also manifested in its Fermi surface topology as discussed in next section. As thenonlocality parameter ωNL is increased, the single particle gap to the remote bands on the conduction/valence bandside increases/decreases almost linearly, as shown in Fig. S1(c). The reversal of the relative size of Ecg and Evg indicates

that e±iθ/2 factor causes particle-hole asymmetry of the opposite sense compared to the larger nonlocal tunneling.

Fermi surface topology and weak-field Hall resistivity

The topology of the flat-band Fermi surface is directly tied to many experimental observations including SdHoscillations and magneto-transport measurements. It is also crucial for understanding the pairing mechanism ofsuperconductivity. Up to the present, knowledge of the flat-band Fermi surface is still very limited. In this section,we start by studying how nonlocal tunneling influences the Fermi surface topology. We ignore interaction effects forthe moment so the energy dispersion is independent of filling factor.

Fig. S2 plots Fermi surface contours, i.e equal-energy contours of the single-particle continuum model with andwithout nonlocal tunneling corrections. The flat-band Fermi surface without nonlocal tunneling (Fig. S2 (a,b)) ischaracterized by a single electron (hole) pocket centered around the γ point in momentum space, when the Fermienergy is near the bottom (top) of the valence (conduction) band. As the size of the Fermi surface increases, its shapechanges gradually from circular-like to triangular-like. On the other hand, when the Fermi energy is slightly below(above) the Dirac energy, the Fermi surface has two hole (electron) pockets centered around the two Dirac pointsand their shapes also changes from circular-like to triangular-like. The two sides meet at a saddle-point van Hovesingularity (VHS) point, at which γ centered triangle-like Fermi pockets and κ, κ′ centered Fermi pockets touch..

With nonlocal tunneling (Fig. S2 (c,d)), the symmetry between conduction and valence flat bands is noticeablybroken. Although the valence band Fermi surface contours do not change much, those of the conduction flat bandchange qualitatively. At the conduction band bottom, in addition to electron pockets near the two Dirac points,additional pockets emerge near m as a result of the distortion of γ pockets which reaches the mBZ boundary beforebecoming connected with the Dirac pockets.

As mentioned in the main text, the filling factor position at which Hall density changes sign does not coincideexactly with the VHS. Fig. S3(a,b) shows two typical examples of Hall density and DOS of the single particle bandwithout and with nonlocal tunneling as the filling factor is varied. The sign change position and the VHS pointdeviates slightly from each other. Fig. S3(c) plots the sign change and VHS positions on both valence and conductionband sides as a function of the nonlocal tunneling parameter.

9

(a)

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(b)

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(c)

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(d)

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FIG. S2. Fermi surface contours of single-particle models (a, b) without ωNL = 0 and (c, d) with nonlocal tunneling termωNL = 20meV . (a, c) are for flat valence bands and (b, d) for flat conduction bands. Same parameters are used as in Fig. S1(a,b). The color of each contour line represents its Fermi energy and the energy difference between neighboring contours is constantso the density of contour lines indicates the steepness of the dispersion.

Hartree and Fock self-energy

The Hartree and Fock self-energies are calculated using unscreened and SU(4) flavor symmetric Coulomb interac-tions, Vαβ(q) = 2πe2/(ε|q|) exp(−qdαl,βl), where α, β are composite indices of layer (l) and sublattice (s). ε is theeffective dielectric constant which we take as a control parameter for interaction strength. dαl,βl = d(1 − δαl,βl) ac-counts for the vertical distance between α and β sublattices with layer index αl and βl respectively. d is the interlayer

(a)

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(b)

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(c)

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FIG. S3. Weak-field Hall density (red solid) and density of states (blue dashed) of single-particle flat bands without (a) andwith (b) nonlocal tunneling it vs. band filling factor ν. The parameters are the same as in Fig. S1(a). (c) Locations of Halldensity sign changes and of VHSs as a function of nonlocal tunneling parameter ωNL.

10

spacing. In layer and sublattice basis, the Hartree and Fock self-energy operators take the forms

ΣH,s,µα,G;β,G′(k) =1

A

∑α′

Vα′α(G′ −G)δρα′α′(G−G′)δαβ , (S13)

ΣF,s,µα,G;β,G′(k) = − 1

A

∑G′′,k′

Vαβ(G′′ + k′ − k)δρs,µα,G+G′′;β,G′+G′′(k′). (S14)

Here k is understood to be restricted to the first moire Brillouin zone (mBZ) (k ∈ mBZ). G = mb1 + nb2 is moirereciprocal vectors with integers m,n. s and µ are labels of spin and valley. δρ ≡ ρ− ρiso is the density matrix definedrelative to that of isolated rotated graphene layers each filled up to the charge neutrality point. Because we do notconsider coherence between different valleys or spins, δρ =

∑s,µ δρ

s,µ. As in the main text, δρe is the density matrixof filling all the remote valence bands and keeping the flat bands empty. δρf is the density matrix of filling all theremote valence and flat bands.

We then project the self-energy onto the flatband Hilbert space defined by the eigenvectors,

|ψs,µ,n,k〉 =∑α

zn,s,µα,G,k|α,G,k〉, (S15)

where ± represents valley K/K ′ and n = c, v for conduction and valence bands. The self-energy operator projectedonto the flatband Hilbert is therefore

ΣH(δρ) =∑

s,µ,n,k

〈ψs,µ,n,k|ΣH,s,µ(k)|ψs,µ,n,k〉c†s,µ,n,kcs,µ,n,k, (S16)

ΣF (δρ) =∑

s,µ,n,k

〈ψs,µ,n,k|ΣF,s,µ(k)|ψs,µ,n,k〉c†s,µ,n,kcs,µ,n,k. (S17)

The Hartree self-energy of hole (electron) doped state lowers (raises) the quasiparticle energy at κ, κ′ relative to thatat γ. As shown in Fig. S4(a), the γ point energy is kept unchanged because the electron wavefunction has vanishingweight on AA position in real space where the Hartree self-energy attracts (repels) the most. On the other hand Fockself-energy shifts the whole bands up when doped with holes and down when doped with electrons (Fig. S4(b)). Inaddition, it also increase the quasiparticle band width.

Because the interaction self-energy is filling-factor dependent, the quasiparticle dispersion changes as the flat bandis filled or emptied. In Fig. S5, we plot the quasiparticle dispersions of the ferromagnetic state at ν = −2 and of thespin-valley polarized state at ν = −3, which are both single-Slater determinant states.

(a)

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(b)

<latexit sha1_base64="zVpXaXYpCy/5Krr2sSqRENfT7FA=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoMQL2FXInoMevEY0TwgWcLspDcZMju7zMwKIeQTvHhQxKtf5M2/cZLsQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGtzO/9YRK81g+mnGCfkQHkoecUWOlh3Jw3iuW3Io7B1klXkZKkKHeK351+zFLI5SGCap1x3MT40+oMpwJnBa6qcaEshEdYMdSSSPU/mR+6pScWaVPwljZkobM1d8TExppPY4C2xlRM9TL3kz8z+ukJrz2J1wmqUHJFovCVBATk9nfpM8VMiPGllCmuL2VsCFVlBmbTsGG4C2/vEqaFxWvWrm8r5ZqN1kceTiBUyiDB1dQgzuoQwMYDOAZXuHNEc6L8+58LFpzTjZzDH/gfP4AjMiNUw==</latexit>

FIG. S4. Effects of Hartree (a) and Fock (b) self-energy renormalization on quasiparticle dispersions. The parameters used aresame as in Fig. 2 of main text. See Fig. 1 for the total effect of Hartree and Fock terms.

11

(a)

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(b)

<latexit sha1_base64="zVpXaXYpCy/5Krr2sSqRENfT7FA=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoMQL2FXInoMevEY0TwgWcLspDcZMju7zMwKIeQTvHhQxKtf5M2/cZLsQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGtzO/9YRK81g+mnGCfkQHkoecUWOlh3Jw3iuW3Io7B1klXkZKkKHeK351+zFLI5SGCap1x3MT40+oMpwJnBa6qcaEshEdYMdSSSPU/mR+6pScWaVPwljZkobM1d8TExppPY4C2xlRM9TL3kz8z+ukJrz2J1wmqUHJFovCVBATk9nfpM8VMiPGllCmuL2VsCFVlBmbTsGG4C2/vEqaFxWvWrm8r5ZqN1kceTiBUyiDB1dQgzuoQwMYDOAZXuHNEc6L8+58LFpzTjZzDH/gfP4AjMiNUw==</latexit>

FIG. S5. Quasiparticle bands of (a) ν = −1 state with both spin and valley polarization and (b) ν = −2 ferromagnetic state.The nonlocal parameter is ωNL = 20meV. The pattern of broken symmetries and Hall density jumps can be explained byretaining strain, non-locality, and interaction band modifications.