Valley Jahn-Teller effect in twisted bilayer graphene

M. Angeli,1 E. Tosatti,1, 2, 3 and M. Fabrizio11International School for Advanced Studies (SISSA), Via Bonomea 265, I-34136 Trieste, Italy

2CNR-IOM Democritos, Istituto Officina dei Material, Consiglio Nazionale delle Ricerche3International Centre for Theoretical Physics (ICTP), Strada Costiera 11, I-34151 Trieste, Italy

(Dated: April 12, 2019)

The surprising insulating and superconducting states of narrow-band graphene twisted bilayershave been mostly discussed so far in terms of strong electron correlation, with little or no attentionto phonons and electron-phonon effects. We found that, among the 33492 phonons of a fully relaxedθ = 1.08◦ twisted bilayer, there are few special, hard and nearly dispersionless modes that resembleglobal vibrations of the moiré supercell, as if it were a single, ultralarge molecule. One of them,doubly degenerate at Γ with symmetry A1 + B1, couples very strongly with the valley degrees offreedom, also doubly degenerate, realizing a so-called E ⊗ e Jahn-Teller (JT) coupling. The JTcoupling lifts very efficiently all degeneracies which arise from the valley symmetry, and may lead,for an average atomic displacement as small as 0.5 mÅ, to an insulating state at charge neutrality.This insulator possesses a non-trivial topology testified by the odd winding of the Wilson loop. Inaddition, freezing the same phonon at a zone boundary point brings about insulating states at mostinteger occupancies of the four ultra-flat electronic bands. Following that line, we further studythe properties of the superconducting state that might be stabilized by these modes. Since the JTcoupling modulates the hopping between AB and BA stacked regions, pairing occurs in the spin-singlet Cooper channel at the inter-(AB-BA) scale, which may condense a superconducting orderparameter in the extended s-wave and/or d± id-wave symmetry.

I. INTRODUCTION

The recent discovery of superconductivity in magicangle (θ ≈ 1.1◦) twisted bilayer graphene (tBLG) [1–4]has stimulated an intense theoretical and experimentalresearch activity. This unexpected phenomenon occursupon slightly doping insulating states found at fractionalfillings, the latter contradicting the metallic behaviourpredicted by band structure calculations. Despite thehuge size of the unit cell, containing more than ≈ 11000atoms, the band structure of tBLG at the first magicangle has been computed with a variety of methods,including tight-binding [5–10], continuum models [11, 12]and DFT [13, 14]. These approaches predict that allthe exotic properties mentioned above arise from fourextremely flat bands (FBs), located around the chargeneutrality point, with a bandwidth of the order of≈ 10−20 meV. Owing to the flatness of these bands, thefractional filling insulators found in [1, 2] are conjecturedto be Mott Insulators, even though rather anomalousones, since they turn frankly metallic above a criticaltemperature or above a threshold Zeeman splitting ina magnetic field, features not expected from a Mottinsulator. Actually, the linear size of the unit cell at themagic angle is as large as ≈ 14 nm, and the effectiveon-site Coulomb repulsion, the so-called Hubbard U ,must be given by the charging energy in this largesupercell projected onto the FBs, including screeningeffects due to the gates and to the other bands. Evenneglecting the latter, the estimated U ∼ 9 meV iscomparable to the bandwidth of the FBs [15]. Since theFBs are reproducibly found in experiments [1–4, 16–20]to be separated from the other bands by a gap ofaround ∼ 30 − 50 meV, the actual value of U should

be significantly smaller, implying that tBLG might notbe more correlated than a single graphene sheet [21].In turn, this suggests that the insulating behaviour atν = ±2 occupancy might instead be the result of aweak-coupling Stoner or CDW band instability driven byelectron-electron and/or electron-phonon interactions,rather than a Mott localization phenomenon.As pointed out in [22], in order to open an insulating gapthe band instability must break the twofold degeneracyat the K-points imposed by the D6 space group symme-try of the moiré superlattice, as well as the additionaltwofold degeneracy due to the so-called valley chargeconservation. This conserved quantity is associated withan emergent dynamical U(1) symmetry that appearsat small twist angles, a symmetry which, unlike spatialsymmetries, is rather subtle and elusive. It is thereforeessential to identify a microscopic mechanism that couldefficiently break this emergent symmetry, hereafterrefereed to as Uv(1). The most natural candidate isthe Coulomb repulsion [22–24], whose Fourier transformdecays more slowly than that of electron hopping,possibly introducing a non negligible coupling amongthe two valleys even at small twist angles. IndeedDFT-based calculations show a tiny valley splitting[13, 14], almost at the limits of accuracy of the method,which is nevertheless too small to explain the insulatingstates found in the FBs.Here we uncover another Uv(1)-breaking mechanisminvolving instead the lattice degrees of freedom, mostlyignored so far. It must be recalled that ab-initioDFT-based calculations fail to predict well defined FBsseparated from other bands unless atomic positions areallowed to relax [10, 14, 25–28], especially out-of-plane.That alone already demonstrated that the effects of

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atomic motions in the lattice are not at all negligiblein tBLG, further supported by the significant phononcontribution to transport [18, 29, 30]. We calculatethe phonon spectrum of the fully-relaxed bilayer at1.08◦ twist angle, which shows the presence, amongthe about thirty thousands phonons, of a small setof very special optical modes, with C-C stretchingcharacter, very narrow and uniquely coherent overthe moiré supercell Brillouin Zone. Among them, wefind a doubly degenerate optical mode that couplesto the Uv(1) symmetry much more efficiently thanCoulomb repulsion seems to do in DFT calculations.A subsequent frozen-phonon tight-binding calculationshows that this mode is able to fully lift the valleydegeneracy even when its lattice deformation amplitudeis extremely small. Remarkably, both electrons andphonons are twofold Uv(1) -degenerate, and the couplingof this mode with the electron bands actually realizes anE ⊗ e Jahn-Teller (JT) effect [31]. This effect is able tostabilize insulating states at integer occupancies of theFBs, both even and odd. Moreover, a surprising andimportant additional result will be that the electron-phonon coupling magnitude controlling this process isextremely large, and not small as one could generallyexpect for a very narrow band. We conclude by studyingthe superconducting state that might be mediated by theJahn-Teller coupling in a minimal tight-binding modelof the FBs that reproduces symmetries and topologi-cal properties of the realistic band structure calculations.

The work is organized as follows. In section II we spec-ify the geometry of the tBLG studied and define usefulquantities that are used throughout the article. In sectionIII we briefly discuss the band structure obtained by arealistic tight-binding calculation of the tBLG with fullyrelaxed atomic positions. The phonon spectrum and itsproperties, especially focusing on special optical modesstrongly coupled with the valley Uv(1) symmetry, arethroughly discussed in section IV. Section VII addressesthe properties of the superconducting state that might bestabilized by the particular phonon mode identified in theprevious section, through a mean-field calculation usinga model tight-binding Hamiltonian of the FBs. Finally,section VIII is devoted to concluding remarks.

II. MOIRÉ SUPERLATTICE ANDSYMMETRIES

In Fig. 1 we show the geometry of the tBLG that weshall use hereafter. We start from a AA stacked bilayerand rotate in opposite directions the two layers aroundthe center of a hexagon by an angle 1.08◦/2, leading to aunit cell with 11164 carbon atoms. The moiré superlat-tice, top left panel, with the reference frame defined inthe top right panel, possesses a full D6 spatial symme-try. Specifically, resolving the action of each symmetryoperation in the indices that identify the two layers, 1

AAAB BA

BABAAB

AB X

Y

Z

AA

B

B

AA

AA

B

B

A

MBZ

G1

G2

M12

-

-'

'+

+ MBZ

MBZ

FIG. 1: Top panel: moiré superlattice formed by two unre-laxed graphene layers (in blue and red) twisted by an angle θ.We indicate the different stacking regions: AA and the twodifferent Bernal regions AB and BA. The domain walls (DWs)separate AB from BA regions and connect different AA re-gions. On the right, a zoom into the AA stacked region isshown: two overlapping hexagons in distinct graphene layersare rotated in opposite directions around the perpendicularz-axis by an angle θ/2. Bottom panel: the folding procedurein tBLG. The original single layer Brillouin zones (in red andblue) are folded into the mini-Brillouin zone (MBZ). Two in-equivalent K points in different layers (K− and K′+ or K′−and K+) are folded into the same points, K1 or K2, of theMBZ. The path K1 → Γ→M→ K2 is also shown.

and 2, the two sublattices within each layer, A and B,and, finally, the two sublattices AB and BA of the moirésuperlattice (see bottom panel of Fig. 1), we have that:

• the rotation C3z by 120◦ degrees around the z-axisis diagonal in all indices, 1 and 2, A and B, AB andBA;

• the C2x rotation by 180◦ degrees around the x-axisinterchanges 1 with 2, A with B, but is diagonal inAB and BA;

• the C2y rotation by 180◦ degrees around the y-axisinterchanges 1 with 2, AB with BA, but is diagonalin A and B;

• finally, the action of a C2z rotation by 180◦ degreesaround the z-axis is a composite symmetry opera-tion obtained by noting that C2z = C2x × C2y.

In Table I we list the irreducible representations (irreps)

3

of the D6 space group and the action on each of them ofthe symmetry transformations C3z, C2x and C2y.

C3z C2x C2yA1(1) +1 +1 +1A2(1) +1 -1 -1B1(1) +1 +1 -1B2(1) +1 -1 +1

E1(2)

(cosφ − sinφsinφ cosφ

) (+1 0

0 −1

) (−1 00 +1

)

E2(2)

(cosφ − sinφsinφ cosφ

) (1 0

0 −1

) (1 0

0 −1

)

TABLE I: Nontrivial irreducible representations of the spacegroup D6. Each representation has the degeneracy shown inparenthesis. We also list the action of the symmetry opera-tions for each representation, where φ = 2π/3.

A. Uv(1) valley symmetry

The mini Brillouin zone (MBZ) that corresponds tothe real space geometry of Fig. 1 and its relationshipwith the original graphene Brillouin zones are shown inthe bottom panels of that figure (for better readability,at a larger angle than the actual 1.08◦ which we use). Be-cause of the chosen geometry, the Dirac point K+(K ′+)of the top layer and K ′−(K−) of the bottom one foldonto the same point, K1 or K2, of the MBZ, so that afinite matrix element of the Hamiltonian between themis allowed by symmetry. Nevertheless, as pointed out by[11], the matrix element of the one-body component ofthe Hamiltonian is negligibly small at small twist angles,so that the two Dirac points, hereafter named valleys, re-main effectively independent of each other. This impliesthat the operator

∆Nv = N1 −N2 , (1)

where N1 and N2 are the occupation numbers of eachvalley, must commute with the non-interacting Hamilto-nian of the tBLG at small angles. That operator is in factthe generator of the Uv(1) symmetry. As we will see inthe following (see IIIA), the interplay between the valleysymmetry and C2y is responsible for the additional two-fold degeneracies beyond those of Table I in both electronband structure and phonon spectra along all the pointsin the MBZ invariant under that symmetry.

B. Wannier orbitals

We shall assume in the following Wannier orbitals(WOs) that are centered at the Wyckoff positions 2c ofthe moiré triangular superlattice, i.e., at the AB and BAregion centers, even though their probability distribution

has a very substantial component in other regions suchas AA. The large size of the unit cell has originated sofar an intense effort to find a minimal model faithfullydescribing the FBs physics. While a variety of WOs cen-tered at different Wyckoff positions has been proposed[13, 22, 27, 32–37], our simple assumption is well suitedfor our purposes. The site symmetry at the Wyckoff posi-tions 2c is D3, and includes only C3z and C2x with irrepsA1, A2 and E. Inspired by the symmetries of the Blochstates at the high-symmetry points [10], see Sec. III),we consider two A1 and A2 one-dimensional irreps (1d-irreps), both invariant under C3z and eigenstates of C2xwith opposite eigenvalues c2x = ±1. In addition, weconsider one two-dimensional irrep (2d-irrep) E, whichtransforms under C3z and C2x as the 2d-irreps in Ta-ble I, hence comprises eigenstates of C2x, with oppositeeigenvalues c2x = ±1, which are not invariant under C3z.We define on sublattice α = AB,BA the spin-σ WO an-nihilation operators Ψα,Rσ and Φα,Rσ corresponding tothe two 1d-irreps ( Ψ) or the single 2d-irrep (Φ), respec-tively

Ψα,Rσ =

Ψα,1,s,RσΨα,2,s,RσΨα,1,p,RσΨα,2,p,Rσ

, Φα,Rσ =

Φα,1,s,RσΦα,2,s,RσΦα,1,p,RσΦα,2,p,Rσ

,(2)

where the subscript s refers to c2x = +1, and p toc2x = −1, while the labels 1 and 2 refer to the two val-leys. It is implicit that each component is itself a spinorthat includes fermionic operators corresponding to dif-ferent WOs that transform like the same irrep. We shallcombine the operators of different sublattices into a singlespinor

ΨRσ =

(ΨAB,RσΨBA,Rσ

), ΦRσ =

(ΦAB,RσΦBA,Rσ

). (3)

We further introduce three different Pauli matrices σathat act in the moiré sublattice space (AB, BA), µa inthe c2x = ±1 space (s, p), and τa in the valley space (1,2), where a = 0, 1, 2, 3, a = 0 denoting the identity.With these definitions, the generator (1) of the valleyUv(1) symmetry becomes simply

∆Nv =∑Rσ

(Ψ†Rσ σ0 τ3 µ0 ΨRσ + Φ

†Rσ σ0 τ3 µ0 ΦRσ

).

(4)It is now worth deriving the expression of the space

group symmetry operations in this notation and repre-sentation.By definition, the C2x transformation corre-sponds to the simple operator

C2x(

ΨRσ

)= σ0 τ0 µ3 ΨC2x(R)σ ,

C2x(

ΦRσ

)= σ0 τ0 µ3 ΦC2x(R)σ .

(5)

The 180◦ rotation around the z-axis that connects sub-lattice AB with BA of each layer (C2z) is not diagonal in

4

K1 Γ M K2

-200

-100

0

100

200

300

K1 Γ M K2-5

0

5

10

E (meV) E (m

eV)π

π-π

0

0

FIG. 2: (a) Electronic band structure of twisted bilayergraphene at the angle θ = 1.08 after full atomic relaxation.The charge neutrality point is the zero of energy. The irreps atthe Γ point are encoded by colored circles, where blue, greenand red stand for the A1 +B1, A2 +B2 and E1 +E2 irreps ofthe D6 space group, respectively. At the K2 point the E andA1 +A2 irreps of the little group D3 are represented by greenand violet triangles, respectively. b) Zoom in the FBs region.c) Wilson loop of the four FBs as function of k = G2/π.

the valley indices and can be represented by [13, 33]

C2z(

ΨRσ

)= σ1 τ1 µ0 ΨC2z(R)σ ,

C2z(

ΦRσ

)= σ1 τ1 µ0 ΦC2z(R)σ .

(6)

Finally, since C2y = C2z × C2x, then

C2y(

ΨRσ

)= σ1 τ1 µ3 ΨC2y(R)σ ,

C2y(

ΦRσ

)= σ1 τ1 µ3 ΦC2y(R)σ .

(7)

III. LATTICE RELAXATION AND SIMMETRYANALYSIS OF THE BAND STRUCTURE

Since the tBLG must undergo , relative to the idealsuperposition of two rigid graphene layers, a substan-tial lattice relaxation , whose effects have also been ob-served in recent experiments [16, 17, 38, 39], we per-formed lattice relaxations via classical molecular dynam-ics techniques using state-of-the-art force-fields, allowingfor both in-plane and out -of-plane deformations. Thedetails about the relaxation procedure are in AppendixA and are essentially those in Ref.[10]. It is well known[10, 14, 25, 28, 40–44] that, after full relaxation, the en-ergetically less favourable AA regions shrink while theBernal-stacked ones, AB and BA, expand in the (x, y)plane. In addition, the interlayer distance along z of theAA regions increases with respect to that of the AB/BA

zones, leading to significant out-of-plane buckling defor-mations, genuine ’corrugations’ of the graphene layers,which form protruding AA bubbles. The main effect ofthe layer corrugations is to enhance the bandgaps be-tween the FBs and those above and below [10, 14, 45],important even if partially hindered by the h-BN en-capsulation of the samples during the experiments [1].Moreover, since both the AB and BA triangular domainshave expanded, the initially broad crossover region be-tween them sharpens into narrow domain walls (DWs)that merge at the AA centers in the moiré superlattice(see 1). The electronic structure shown in Fig. 2 is ob-tained with standard tight-binding calculations, see A forfurther details, with the relaxed atomic positions, usinghopping amplitudes tuned so to reproduce ab-initio cal-culations [5]. The colored circles and triangles at the Γand K points, respectively, indicate the irreps that trans-form like the corresponding Bloch states. For instance, atΓ the FBs consist of two doublets, the lower correspond-ing to the irreps A1 + B1, and the upper to A2 + B2.Right above and below the FBs, we find at Γ two quar-tets, each transforming like E1 + E2. At K, the FBsare degenerate and form a quartet E + E. Consistentlywith the D3 little group containing C3z and C2y, at K wefind either quartets, like at the FBs, made of degeneratepairs of doublets, each transforming like E, or doubletstransforming like A1 +A2, where A1 and A2 differ in theparity under C2y. This overall doubling of degeneraciesbeyond their expected D6 space group irreps, reflects thevalley Uv(1) symmetry of FBs that will be discussed be-low, and whose eventual breaking will be addressed laterin this paper. We end by remarking that the so-called’fragile’ topology [13, 22, 35, 46], diagnosed by the oddwinding of the Wilson loop (WL) [13], is actually robustagainst lattice corrugations [45] and relaxation, as shownby panel (c) in Fig. 2.

A. SU(2) symmetry and accidental degeneracyalong C2y invariant lines

We note that along all directions that are invariantunder C2y, which include the diagonals as well as all theedges of the MBZ, the electronic bands show a twofolddegeneracy between Bloch states that transform differ-ently under C2y. These lines corresponds to the domainwalls (DWs) in real space. This "accidental" degeneracyis a consequence of the interplay between Uv(1) and C2ysymmetries. Indeed, along Γ → K1,2 and M → K1,2 inthe MBZ we have that:

C2y(

Ψkσ

)= σ1 τ1 µ3 Ψkσ , (8)

and similarly for Φkσ. It follows that the generator ofUv(1), i.e., the operator σ0 τ3 µ0, anticommutes with theexpression of C2y along the lines invariant under thatsame symmetry, namely the operator σ1 τ1 µ3, and bothcommute with the Hamiltonian. Then also their product,

5

σ1 τ2 µ3, commutes with the Hamiltonian and anticom-mutes with the other two. The three operators

2T3 = σ0 τ3 µ0 , 2T1 = σ1 τ1 µ3 , 2T2 = σ1 τ2 µ3 ,(9)

thus realise an SU(2) algebra and all commute withthe Hamiltonian Ĥk in momentum space for any C2y-invariant k-point. This emergent SU(2) symmetry istherefore responsible of the degeneracy of eigenstateswith opposite parity under C2y. We note that C2x in-stead commutes with Uv(1), so that there is no SU(2)symmetry protection against valley splitting along C2xinvariant lines (Γ→M).

IV. PHONONS IN TWISTED BILAYERGRAPHENE

The Uv(1) valley symmetry is an emergent one since,despite the fact that its generator (1) does not commutewith the Hamiltonian, the spectrum around charge neu-trality is nonetheless Uv(1)–invariant. It is therefore notobvious to envisage a mechanism that could efficientlybreak it.However, since the lattice degrees of freedom play an im-portant role at equilibrium, as discussed in Section III,it is possible that they could offer the means to destroythe Uv(1) valley symmetry. In this Section we shall showthat they indeed provide such a symmetry-breaking tool.

A. Valley splitting lattice modulation: the key roleof the Domain Walls

In Section IIA we mentioned that the valley symme-try arises because, even though inequivalent Dirac nodesof the two layers should be coupled to each other by theHamiltonian after being folded onto the same point of theMBZ (see bottom panel in Fig. 1), at small angle thesematrix elements are vanishingly small and thus the val-leys are effectively decoupled. That is true if the carbonlattice, although mechanically relaxed, is unperturbed bythe presence of the electrons. Once coupling with elec-trons is considered, we cannot exclude that for example alattice distortion modulated with the wave vectors con-necting the inequivalent Dirac nodes of the two layers,K+ with K ′− and K ′+ with K− in Fig. 1, might insteadyield a significant matrix element among the valleys. Toinvestigate that possibility we build an ad-hoc distortioninto the bilayer carbon atom positions. We define thevector qij = K+,i − K′−,j , where i, j = 1, 2, 3 run overthe three equivalent Dirac points of the BZ of each layer,and the D6 conserving displacement field

η(r) =∑a

3∑i,j=1

sin(qij · ra

)ua,ij δ

(r− ra

), (10)

where a runs over all atomic positions, and ua,ij cor-responds to a displacement of atom a in direction qij ,

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Rx [nm]

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0

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0

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R y [nm]

Rx [nm

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0

R y [nm]

Rx [n

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1.5e−02

2.0e−02

5.0e−03

1.0e−02

1.5e−02

2.0e−02

40

Ry

[nm

]

40

m]

40

1

2.0e

−02

−40

−20

0

20

40

−40 −20 0 20 40

R y[n

m]

Rx [nm]

0.0e

5.0e

1.0e

1.5e

2.0e

0

5

10

15

20

0 2 4-4 -2

-4

-2

0

4

2

AA

AB BA

DWa) b)

K1 Γ M K2E

(meV

)-20

-10

0

10

20

30c)

FIG. 3: Panel a): the displacement field η(r), Eq. (10), re-stricted on the DWs: the direction of the atomic displacementis depicted by a small arrow, while its magnitude in mÅ is ex-pressed in colors. Panel b): displacement along the DW areahighlighted by a black dashed line in panel a). The overalleffect of the distortion is a narrowing of the DW. Panel c):low energy band structure of tBLG at θ = 1.08 after the dis-tortion in panel a). The two-fold degeneracies protected bythe valley symmetry are completely lifted.

whose (tuneable) magnitude is the same for all atoms.In proximity of the AA regions the distortion is locallysimilar to the graphene breathing mode, the in-planetransverse optical phonon at K [47] with A1 symmetry.Since by construction qij is a multiple integer of thereciprocal lattice vectors, η(r) has the same periodicityof the unit cell, i.e., the distortion is actually at the Γpoint. Moreover, the distortion’s D6 invariance impliesno change of space group symmetries.

Since the most direct evidence of the Uv(1) symme-try is the accidental degeneracy in the band structurealong all C2y–invariant lines, corresponding just to theDWs directions in real space, we further assume the ac-tion of the displacement field η(r) to be restricted toa small region in proximity of the DWs, affecting only≈ 1% of the atoms in the moiré supercell, see top panelsin Fig. 3. The modified FBs in presence of the displace-ment field η(r) with

∣∣ua,ij∣∣ = 20 mÅ are shown in thebottom panel of Fig. 3. Remarkably, despite the minutedistortion magnitude and the distortion involving only

6

0 50 100 150 200ω (meV)

0

2

4

6

8F(

ω)

(meV

-1nm

-1)

FIG. 4: Phonon density of states F (ω) for Bernal stackedbilayer graphene at zero twist angle (red dashed line) andfully relaxed tBLG at θ = 1.20, 1.12, 1.08 (blue, green andblack lines).

the minority of carbon atoms in the DWs, the degener-acy along Γ → K1 and M → K2 is lifted to such anextent that the four bands split into two similar copies.This is remarkable in two aspects. First, we repeat, be-cause there is no space symmetry breaking. The onlysymmetry affected by the distortion is Uv(1), since, byconstruction, η(r) preserve the full space group symme-tries. Second, the large splitting magnitude reflects anenormous strength of the effective electron-phonon cou-pling, whose origin is interesting. Generally speaking, infact, broad bands involve large hoppings and large ab-solute electron-phonon couplings, while the opposite isexpected for narrow bands. The large electron-phononcouplings which we find for the low energy bands of tBLGsuggests a possible broad-band origin of the FBs, as weshall discuss in Sec. VII. A consequence of this e-p cou-pling magnitude is that all potential phenomena involv-ing lattice distortions, either static or dynamic, shouldbe considered with a much larger priority than done sofar.

B. The phonon spectrum

We compute the phonon eigenmodes in the relaxed bi-layer structure, by standard methods, see Appendix Afor details of the calculation. Fig. 4 shows the phonondensity of states F (ω) for tBLG at three different twistangles in comparison with the Bernal AB-stacked bi-layer. As previously reported [28, 48], F (ω) is almostindependent of the twist angle, which only affects theinter-layer van der Waals forces, much weaker than thein-plane ones arising from the stiff C − C bonds. As aconsequence, phonons in tBLG are basically those of theBernal stacked bilayer. This is true except for a small setof special phonon modes, clearly distinguishable in Fig. 5that depicts the phonon spectrum zoomed in a very nar-

K1 Γ M K2

206.97

206.98

206.9 9

2 0 7

ω (m

eV)

FIG. 5: Zoom in the optical region of the phonon spectrumof the fully relaxed tBLG at θ = 1.08. Among many scat-tered energy levels of highly dispersive branches (not resolvedin this narrow energy window) a set of 10 narrow continu-ous branches stands out (no line drawn through data points,which just fall next to one another). The degeneracy of thesemodes is twice that expected by D6 space symmetry, similarin this to electronic bands. The two-fold degenerate modewith the highest overlap with the deformation η(r), drawn inFig. 6, is marked by a red arrow, while the mode at M usedin section VI by a green arrow. The avoided crossing whichoccurs close to Γ is encircled by a blue dashed line.

row energy region ≈ 0.04 meV around the high frequencygraphene K-point peak of the phonon density of states.Specifically, within the large number of energy levels ofall other highly dispersive phonon bands, unresolved inthe narrow energy window, a set of 10 almost dispersion-less modes emerges. We note that these special modesshow the same accidental degeneracy doubling along theC2y invariant lines as that of the electronic bands aroundthe charge neutrality point.

Similar to the electronic degeneracy, whose underlyingUv(1) symmetry arises from vanishingly weak hoppingmatrix elements between interlayer K-K’ points in theHamiltonian, the mechanically weak van der Waals inter-layer coupling here leads to an effective Uv(1) symmetryfor this group of lattice vibrations. Their poor dispersionis connected with a displacement which is non-uniformin the supercell, and is strongly modulated on the moirélength scale, a distinctive feature of these special modesthat we shall denote as ‘moiré phonons’. In particular,it is maximum in the center of the AA zones, finite inthe DWs, and negligible in the large AB and BA Bernalregions. The overlap between the displacement η(r) in(10) and the 33492 phonons of the θ = 1.08 tBLG at theΓ point is non-negligible only for those moiré phonons.In particular, we find the highest overlap with the dou-bly degenerate mode marked by a red arrow in Fig. 5,and which transforms like A1 +B1. In Fig. 6a) we showthe real space distortion corresponding to the A1 com-ponent of the doublet, where the displacement direction

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−40 0 40 80Rx [nm]

000.0e

2.0e

4.0e

6.0e

8.0e

1.0e

1.2e

1.4e

m1.6

-4-8 0 4 8

-8

4

0

4

8

0

0.2

0.4

0.6

0.8

1

1.2

1.4

AA DW

a)

b) c)

AB BA

AA

FIG. 6: Panel a): atomic displacements on one of the twolayers corresponding to the A1-symmetry moiré mode of thephonon doublet marked by a red arrow in Fig. 5. The direc-tion of displacement is represented by a small arrow centeredat each atomic position, while its modulus is encoded in col-ors. The mean displacement per atom is 0.57 mÅ. Panel b):zoom in the center of an AA region, shown for both layers.Panel c): zoom along one of the domain walls. Note the sim-ilarity with the ad-hoc displacement in Fig. 3.

is represented by small arrows, while its intensity is en-coded in colors. This inspection of the eigenvectors ofthese modes at the atomistic level reveals a definite un-derlying single-layer graphene character, specifically thatof the A1 symmetry transverse optical mode at K. Sincethe graphene K-point does not fold into the bilayer Γ-point, their appearance along the whole Γ → K line inthe spectrum of the fully relaxed bilayer must be merelya consequence of relaxation, a relaxation that is particu-larly strong precisely in the AA and DW regions.

V. INSULATING STATE AT CHARGENEUTRALITY

We will next focus on the effect on the electronic bandstructure of a carbon atom displacement correspondingto the two degenerate phonon modes A1 and B1 at Γ,which should affect the valley symmetry as the displace-ment in Fig. 3. In order to verify that, we carried out afrozen phonon calculation of the modified FB electronicstructure with increasing intensity of the deformation

K1 Γ M K2

-10

0

10

20

E (m

eV)

K1 Γ M K2-5

0

5

10

E (m

eV)

a) b)

c) d)

K1 Γ M K2

2 M K

-5

0

5

1 0

E (m

eV)

2

-101

K1 Γ M K2-10

-5

0

5

10

15

E (m

eV)

-π

π

0

π0

FIG. 7: Evolution of the FBs when the lattice is distortedwith increasing intensity along one of the two modes indi-cated by a red arrow in Fig. 5. Panel (a) undistorted. Panel(b) mean displacement per atom 0.07 mÅ. In the inset weshow the avoided crossing along M → K2. Panel (c) meandisplacement 0.19 mÅ. Now the avoided crossing appears asa genuine crossing protected by C2x symmetry along Γ→M,which actually leads to Dirac points. Panel (d) mean dis-placement 0.57 mÅ, opening a gap between the FBs. Inset:Wilson loop of the lowest two bands in panel d).

[49]. Remarkably, despite transforming as different ir-reps (A1 or B1), both frozen phonon distortions are notonly degenerate, but have exactly the same effect on thebands. As soon as the lattice is distorted, see Fig. 7 b),the fourfold degeneracy at K1 and K2, and the twofoldone at Γ and M is lifted, and small avoided crossings ap-pear and start to move from K1(2) towards the M points.Once they cross M, they keep moving along M→ Γ, seeFig. 7 c). However, along these directions, the C2x sym-metry prevents the avoided crossings, and thus leads tosix elliptical Dirac cones. Finally, once they reach Γ ata threshold value of the distortion, the six Dirac pointsannihilate so that a gap opens at charge neutrality, seeFig. 7 d). This gap-opening mechanism is very efficient,with large splittings even for small values of the atomicdisplacement amplitude shown in Fig. 7. For instance, anaverage displacement as small as ≈ 0.5 mÅ per atom isenough to completely separate the four FBs and to opena gap at charge neutrality.We emphasize that this occurs without breaking any spa-tial symmetry of the tBLG, just Uv(1). As a consequence,the insulator state possesses a non trivial topology, ashighlighted by the odd-winding of the Wilson loop of thelowest two bands, shown in the inset of the same figure 7

8

0 π 2π

-50

0

50E

(meV

)

FIG. 8: The band structure of twisted bilayer graphene atθ ≈ 1.08◦ in a ribbon geometry with open boundary condi-tions. The atomic structure inside the unit cell (replicated 7times along the x direction) has been deformed with the A1symmetric moiré phonon mode. With a mean deformationamplitude of ≈ 1.14 mÅ, that mode is so strongly coupled toopen a large electronic gap at CNP of ≈ 25 meV . The bulkbands have been highlited in blue to emphasize the presenceof edge states both within the phonon-driven FBs gap andthe gaps above and below them.

d). In turn, the non trivial topology of the system impliesthe existence of edge states within the gap separating thetwo lower flat bands from the two upper ones. We thus re-calculated the band structure freezing the moiré phononwith A1 symmetry at Γ in a ribbon geometry, which isobtained by cutting the tBLG along two parallel domainwalls in the y-direction at a distance of 7 supercells. Theribbon has therefore translational symmetry along y, butit is confined in the x-direction. In Fig. 8 we show thesingle-particle energy levels as function of the momen-tum ky along the y-direction. Edge states within the gapat charge neutrality are clearly visible. In particular, wefind for each edge two counter-propagating modes.

As a matter of fact, recent experiments do report theexistence of a finite gap also at charge neutrality [4, 50],which is actually bigger than at other non zero integer fill-ings, and appears without a manifest breakdown of timereversal symmetry T [4] or C2z symmetry [50], and withthe FBs still well separated from other bands [50]. Theseevidences seem not to support an interaction driven gap,which would entail either T or TC2z symmetry break-ings [4, 51], or else the FBs touching other bands at theΓ point [4]. Our phonon-driven insulator at charge neu-trality breaks instead Uv(1) and, eventually, C2y if thefrozen phonon has B1 character, both symmetries exper-imentally elusive. However, the edge states that we pre-dict could be detectable by STM or STS, thus providing

support or disproving the mechanism that we uncovered.We end mentioning that near charge neutrality there arecompelling evidences of a substantial breakdown of C3zsymmetry [16, 17, 50], which, although is not expectedto stabilise on its own an insulating gap unless the latterwere in fact just a pseudo gap [51], still it is worth beingproperly discussed, which we postpone to Section VIA.

A. E⊗ e Jahn-Teller effect

The evidence that the two degenerate modes markedby a red arrow in Fig. 5 produce the same band struc-ture in a frozen-phonon distortion is reminiscent of a E⊗eJahn-Teller effect, i.e., the coupling of a doubly degener-ate vibration with a doubly degenerate electronic state[31].Let us consider the action of the two Γ-point A1 and B1phonon modes, hereafter denoted as q1 and q2, along theC2y invariant lines. We note that the A1 mode, q1, al-though invariant under the D6 group elements, is able tosplit the degeneracy along those lines. Therefore it mustbe coupled to the electrons through a D6 invariant oper-ator that does not commute with the Uv(1) generator τ3.That in turn cannot but coincide with C2y itself, which,along the invariant lines, is the operator T1 = σ1 τ1 µ3/2in Eq. (9). On the other hand, the B1 mode, q2, is oddunder C2y, and thus it must be associated with an oper-ator that anticommutes with C2y and does not commutewith τ3. The only possibility that still admits a U(1) val-ley symmetry is the operator T2 = σ1 τ2 µ3/2 in Eq. (9).Indeed, with such a choice, the electron-phonon Hamil-tonian is

Hel-ph = −g(q1 T1 + q2 T2

), (11)

with g the coupling constant. This commutes with theoperator

J3 = T3 + L3 =τ32

+ q ∧ p , (12)

(where p = (p1, p2) the conjugate variable of the displace-ment q = (q1, q2)), the generator of a generalized Uv(1)symmetry that involves electron and phonon variables.As anticipated, the Hamiltonian (11) describes preciselya E⊗ e Jahn-Teller problem [31, 52, 53].Since the phonon mode q is almost dispersionless, seeFig. 5, we can think of it as the vibration of a moiré su-percell, as if the latter were a single, though very large,molecule, and the tBLG a molecular conductor. In thislanguage, the band structures shown in Fig. 7 would cor-respond to a static Jahn-Teller distortion. However, sincethe phonon frequency is substantially larger than thewidth of the flat bands, we cannot exclude the possibilityof a dynamical Jahn-Teller effect that could mediate su-perconductivity [52, 54], or even stabilize a Jahn-TellerMott insulator [53] in presence of a strong enough inter-action.

9

0 4 8

-8

-4

0

4

8

0−80

−40

0

40

80

120

160

200

0 40 80 120 160 200 240

R y[n

m]

Rx [nm]

5.0e−04

1.0e− 3

1.5e−03

2.0e−03

2.5e−03

3.0e−03

−2

0

2

−4 −2 0 2

R y[n

m]

Rx [nm]

0.0e+00

5.0e−04

1.0e−03

1.5e−03

2.0e−03

2.5e−03

3.0e−03AA

12

12 16 20 2480 120 160 200 240Rx [nm]

5. e

1.5 e

1.5e

2.0e

2.5e

3.0e

1

1.5

2

0.5

2.5

AA

AA

AA

FIG. 9: Atomic displacements on one of the two layers cor-responding to one of the moiré modes at M marked by agreen arrow in Fig. 5. The direction of displacement is rep-resented by a small arrow centered at each atomic position,while its modulus is encoded in colors. The mean deforma-tion is 1.8 mÅ and leads to the DOS in Fig. 10. The insetshows a zoom in the AA region close to the origin. The rect-angular unit cell, now containing twice the number of atoms,is highlighted by a black dashed line.

The Jahn-Teller nature of the electron-phonon couplingentails a very efficient mechanism to split the accidentaldegeneracy, linear in the displacement within a frozenphonon calculation. However, this does not explain whyan in-plane displacement as small as 0.57 mÅ is able tosplit the formerly degenerate states at Γ by an amountas large as 15 meV, see Fig. 7 d), of the same order asthe original width of the flat bands. To clarify that,we note that this displacement would yield a changein the graphene nearest neighbour hopping of aroundδt ' 3.6 meV, see Eq. (A1), which in turn entails a split-ting at Γ of 6 δt ∼ 21 meV, close to what we observe. Webelieve that such correspondence is not accidental, butindicates that the actual energy scale underneath the flatbands is on the order of the bare graphene bandwidth,rather than the flat bandwidth itself. We shall return onthis issue later in Section VII.

VI. INSULATING STATES AT OTHERCOMMENSURATE FILLINGS

The Γ-point distortion described above can lead to aninsulating state at charge neutrality, possibly connectedwith the insulating state very recently reported [4, 50].The same phonon branch might also stabilize insulat-ing states at other integer occupancies of the mini bandsbesides charge neutrality. However, this necessarily re-quires freezing a mode at a high-symmetry k-point dif-ferent from Γ in order to get rid of the band touch-ing at the Dirac points, K1 and K2, protected by the

-30 -20 -10 0 10 20 300

5

10

15

20

DO

S (e

V-1

nm

-2)

(meV)E

-4 -3 -2 -1 0 1 2 3 4ν

FIG. 10: Density of states in the flat bands region after thetBLG has been distorted by one of the two degenerate modesat M marked by a green arrow in Fig. 5. A mesh of 40× 40points in the MBZ has been used. The DOS is expressed asfunction of energy, the zero corresponding to the CNP, andcorresponding occupancy ν of the FBs. The band gaps atν = ±2,±4 are highlighted in violet. The mean displacementper atom is 1.8 mÅ.

C6z symmetry. We consider one of the two degeneratephonon modes at the M-point marked by a green arrowin Fig. 5. This mode has similar features as the Jahn-Teller one at Γ, even if it belongs to an upper branch dueto an avoided crossing along Γ→M (blue dashed line inFig. 5). In Fig. 9 we depict this mode, which still trans-forms as the A1 of graphene K-point on the microscopicgraphene scale, but whose long-wavelength modulationnow forms a series of ellipses elongated along some of theDWs, thus macroscopically breaking the C3z symmetryof the moiré superlattice.In Fig. 10 we show the DOS of the FBs obtained by afrozen-phonon realistic tight-binding calculation. Besidesthe band gaps at ν = ±4, which separates the FBs fromthe others bands, now small gaps opens at ν = ±2 withan average atomic displacement of 1.8 mÅ induced bythe mode at M.Finally we also considered a more exotic multi-component distortion induced by a combination of themodes at the three inequivalent M points, which quadru-ples the unit cell (see Fig. 11). The resulting DOS of theFBs is shown in Fig. 12, and displays small gaps at theodd integer occupancies ν = 1 and ν = ±3.

The very qualitative conclusion of this exploration isthat frozen phonon distortions with various k-vectors canvery effectively yield Peierls-like insulating states at inte-ger hole/electron fillings. Of course all distortions at k-points different from Γ also represent super-superlattices,with an enlargement of the unit cell that should be veri-fiable in such a Peierls state, even if the displacement isa tiny fraction of the equilibrium C–C distance. We notehere that the the zone boundary phonons are less effec-

10

FIG. 11: Atomic displacements on one of the two layers cor-responding to multicomponent deformation obtained combin-ing moiré JT phonons at three inequivalent M points. Themean deformation is 1.7 mÅ and leads to the DOS in Fig. 12.The intensity of displacement is encoded in colors. The insetshows a zoom in one of the AA regions when both layers areconsidered. The unit cell, denoted by a black dashed line, isfour times larger than the original one.

-40 -30 -20 -10 0 10 20 30 40(meV)E

0

5

10

DO

S (e

V-1

nm

-2)

-4 -3 -2 -1 0 1 2 3 4ν

FIG. 12: Electronic density of states with a multicomponentlattice distortion obtained by freezing a combination of themodes at the three inequivalent M points. Band gaps at ν =+1,±3,±4 are highlighted in violet. The mean displacementper atom is 1.7 mÅ.

tive in opening gaps at non zero integer fillings than thezone center phonons are at the charge neutrality point.The reason is that away from charge neutrality the Jahn-Teller effect alone is no longer sufficient; one needs to in-voke zone boundary phonons that enlarge the unit cell,and thus open gaps at the boundaries of the folded Bril-louin zone. The efficiency of such gap opening mechanismis evidently lower than that of the Jahn-Teller Γ-point

mode.

A. C3z symmetry breaking

We observe that the multicomponent distortion withthe phonons frozen at the inequivalent M points leads toa width of the FBs five times larger than in the undis-torted case, see Fig. 12 as opposed to Fig. 7 a), which isnot simply a consequence of the tiny gaps that open atthe boundary of the reduced Brillouin zone. Such sub-stantial bandwidth increase suggests that the tBLG maybe intrinsically unstable to C3z symmetry breaking, es-pecially near charge neutrality. Electron-electron inter-action treated in mean-field does a very similar job [17].Essentially, both interaction- and phonon-driven mecha-nisms act right in the same manner: they move the twovan Hove singularities of the fully symmetric band struc-ture away from each other, and split them into two, withthe net effect of increasing the bandwidth. As such, thosetwo mechanisms will cooperate to drive the C3z symme-try breaking, or enhance it when explicitly broken bystrain, not in disagreement with experiments [16, 17, 50].The main difference is that the moiré phonons also breakUv(1), and thus are able to open gaps at commensuratefillings that C3z symmetry breaking alone would not do.

VII. PHONON MEDIATEDSUPERCONDUCTIVITY

Here we indicate how the phenomena described abovecan connect to a superconducting state mediated byelectron-phonon coupling. For that, we need to builda minimal tight-binding model containing a limited setof orbitals. For the sake of simplicity we shall not re-quire the model Hamiltonian to reproduce precisely theshape of all the bands around charge neutrality, espe-cially those above or below the FBs, but only the correctelementary band representation, topology and, obviously,the existence of the four flat bands separated from theall others. Considering for instance only the 32 states atΓ closest to the charge neutrality point, and maintainingour assumption of WOs centred at the Wyckoff positions2c, those states (apart from avoided crossings allowed bysymmetry) would evolve from Γ to K1 in accordance withthe D6 space group as shown in Fig. 13. Once we allowsame-symmetry Bloch states to repel each other alongΓ→ K1, the band representation can look similar to thereal one (Fig. 2), including the existence of the four FBsthat start at Γ as two doublets, A1+B1 and A2+B2, andend at K1 as two degenerate doublets, each transformingas the 2d-irrep E, see the two solid black lines in Fig. 13.While this picture looks compatible with the actual bandstructure, we shall take a further simplification and justconsider the thicker red, blue and green bands in Fig. 13,which could still produce flat bands with the correct sym-metries. This oversimplification obviously implies giving

11

Γ K1

K1 Γ M K2-10

-5

0

5

10

E (m

eV)

a) b)

c)

0 π-π

0

π

FIG. 13: Panel a): sketch of the elementary band repre-sentation along Γ → K1 taking into account the 32 bandsclosest to the charge neutrality point and without allowingavoided crossing between same symmetry Bloch states. Blue,red and green lines refer to Bloch states that at Γ transformlike A1+B1, A2+B2 and E1+E2, respectively. Should we al-low for avoided crossings, close to charge neutrality we wouldobtain the four flat bands shown as black lines surroundingthe shaded region. Panel b): the FBs obtained by a modeltight-binding Hamiltonian that includes only the solid bandsof panel a). The details of this Hamiltonian are given in Ap-pendix B. Panel c): Wilson loop corresponding to the fourbands in panel b) fully occupied.

up the possibility to accurately reproduce the shape ofthe FBs - but it makes the algebra much simpler.

Within this approximation the components of thespinor operators in Eq. (2) are actually single fermionicoperators, so that we limit ourselves to just four WOsfor each sublattice, AB or BA, and valley, 1 and 2. Twoof such WOs transform like the 1d-irreps, one even, A1,and the other odd, A2, under C2x. The other two in-stead transform like the 2d-irrep E. We build a minimaltight-binding model

Hel = −∆∑Rσ

Ψ†Rσσ0τ0µ3ΨRσ + Tel, (13)

where ∆ splits the s from the pWOs of the 1d-irreps, andTel includes first and second neighbour hopping betweenAB and BA regions in the moiré superlattice compati-ble with all symmetries, see Appendix B for details. InFig. 13 we show the resulting FBs as well as their Wilsonloop. We emphasise that the FBs arise in this picturefrom a sequence of avoided crossings between a large setof relatively broad bands that strongly repel each otheraway from the high symmetry points, rather than fromtruly localized’ WOs. This mechanism is also compati-ble with, and in fact behind, the strong electron-phononcoupling strength, and the consequently large effects onthe FBs of the Jahn-Teller phonons that, according toEq. (11), simply modulate the AB–BA hopping.

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�1 ⌧1 µ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� g2

2!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

� g2

2!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

FIG. 14: Phonon mediated attraction. The two scatteringchannels corresponds to the two phonons and have the sameamplitude g2/2ω.

A. Mean-field superconducting state

Neglecting the extremely small dispersion of the moiréJahn-Teller phonons, we can write their Hamiltonian sim-ply as

Hph =ω

2

∑R

(pR · pR + qR · qR

), (14)

with ω ' 207 meV.Rather than trying to model more faithfully the Jahn-Teller coupling (11), we shall follow a simplified approachbased just on symmetry considerations.In general, we could integrate out the phonons to obtaina retarded electron-electron attraction that can mediatesuperconductivity. However, since here the phonon fre-quency is much larger that the bandwidth of the FBs,where the chemical potential lies, we can safely neglectretardation effects making a BCS-type approximationvirtually exact. The attraction thus becomes instanta-neous and can be represented as in Fig. 14. The phononcouples electrons in nearest neighbor AB and BA re-gions, giving rise to an inter-moiré site spin-singlet pair-ing, a state which we expect to be much less affected byCoulomb repulsion than an on-site one. Therefore, ne-glecting Coulomb repulsions we can concentrate on thepairing channel between nearest neighbor AB and BA re-gions. The scattering processes in Fig. 14 imply that thepairing channels are only τ1 µ0 and τ1 µ3, correspondingto inter valley pairing, as expected because time reversalinterchanges the two valleys.Having assumed pairing between nearest neighbor ABand BA regions, we must identify pair functions in mo-mentum space that connect nearest neighbor unit cells,and transform properly under C3z. These functions are

γ(k) = eik·(a+b)/3(

1 + e−ik·a + e−ik·b),

γ+1(k) = eik·(a+b)/3(

1 + ω e−ik·a + ω∗ e−ik·b),

γ−1(k) = eik·(a+b)/3(

1 + ω∗ e−ik·a + ω e−ik·b),

(15)

12

5 0 5 10Rx [nm]

−π

−π/2

0

π/2

−10

−5

0

5

−10 −5 0 5

R y[n

m]

Rx [nm]

−5

0

5

10

FSdelta4(k)R y

[nm

]

−π/2

0

π/2

π

−10

−5 0 5

10−

10

−5

05

10

FS

d elta

1(k

)

Ry [nm]

Rx

[nm

]

−π

−π/2

0 π/2

π

a)

c) d)

b)

FIG. 15: Pair amplitudes ∆(k)† of the leading superconduct-ing channels: d− id-wave (a-b) and extended s-wave (c-d). Ineach panel we show the hexagonal MBZ and the phase ampli-tudes restricted to a narrow region close to the Fermi surfacescorresponding to the occupancies ν ∼ −2 (a-c) and ν ∼ −1(b-d). The phase of the superconducting order parameter isexpressed in color. Note that the double line are due to thefact that two bands cross the chemical potential at differentk-points.

where ω = ei2π/3. Specifically, γ(k) ∼ A1 is invariantunder C3z, while

γ±1

(C3z(k)

)= ω±1 γ±1(k) . (16)

In other words,(γ+1(k), γ−1(k)

)form a representation

of the 2d-irrep E =(E+1, E−1

)in which C3z is diagonal

with eigenvalues ω and ω∗.Here it is more convenient to transform the spinor Φkσ(

Φ+1,kσΦ−1,kσ

)=

1√2

(1 −i1 +i

) (Φs,kσΦp,kσ

), (17)

so that Φ±1,kσ is associated with a WO that transformslike E±1. Under the assumption of pairing diagonal in theirreps, we can construct the following spin-singlet Cooperpairs:∑

σ

σΦ†+1,AB,kσ τ1 Φ†+1,BA,−k−σ ∼ E−1,k , (18)∑

σ

σΦ†−1,AB,kσ τ1 Φ†−1,BA,−k−σ ∼ E+1,k , (19)

1√2

∑σ

σ(

Φ†+1,AB,kσ τ1 Φ†−1,BA,−k−σ (20)

+(−)Φ†−1,AB,kσ τ1 Φ†+1,BA,−k−σ

)∼ A1(2),k , (21)

1√2

∑σ

σΨ†AB,kσ τ1 µ0(3) Ψ†BA,−k−σ ∼ A

′1(2),k , (22)

which can be combined with the k-dependent functions in(15) to give pair operators that transform like the irrepsof D3. For instance, multiplying (18) by γ+1(k), (19) byγ−1(k), (21) with the plus sign by γ(k), or (22) with µ0by γ(k), we obtain pair operators that all transform likeA1. We shall denote their sum as A

†1k, and, similarly,

all other symmetry combinations as A†2k and E†±1,k. Ev-

idently, since in our modeling the FBs are made of 1dand 2d irreps, the gap function will in general involve acombination of γ(k) and γ±1(k), namely, it will be a su-perposition of s and d±id symmetry channels. The dom-inant channel will depend on which is the prevailing WOcharacter of the Bloch states at the chemical potential,as well as on the strength of the scattering amplitudes inthe different pairing channels, A†1k, A

†2k and E

†±1,k.

In general, we may expect the totally-symmetricA1 chan-nel to have the largest amplitude, thus we assume thefollowing expression of the phonon-mediated attraction

Hel-el ' −λ

V

∑kp

A†1k A1p , (23)

that involves a single parameter λ ∼ g2/2ω. We treat thefull Hamiltonian (13) plus (23) in mean field allowing fora superconducting solution, which is always stabilized bythe attraction provided the density of states is finite atthe chemical potential. We find that superconductivityopens a gap everywhere in the Brillouin zone. Since

〈A†1k 〉 = γ+1(k) 〈E−1,k 〉+ γ−1(k) 〈E+1,k 〉+ γ(k) 〈A1,k 〉+ γ(k) 〈A′1,k 〉 ,

(24)

the order parameter may have finite components withdifferent symmetries, E±1 and A1. In the model calcula-tion all components acquire similar magnitude, implyinga mixture of s and d ± id wave symmetries. In Fig. 15we show 〈A′1,k 〉 and 〈E−1,k 〉 at the Fermi surface cor-responding to densities ν ≈ −1 and ν ≈ −2 with respectto charge neutrality. We conclude by emphasizing thatthe Cooper pair is made by one electron in AB andone in BA, thus leading, in the spin-singlet channel, toextended s and/or d ± id symmetries. That is merely aconsequence of the phonon mode and electron-phononproperties, hence it does not depend on the abovemodeling of the FBs.

There are already in literature several proposals aboutthe superconducting states in tBLG. Most of them, how-ever, invoke electron correlations as the element respon-sible, or strongly effecting the pairing [22, 24, 55–72].There are few exceptions [29, 73, 74] that instead pro-pose, as we do, a purely phonon mediated attraction. Inboth works [73] and [74] the tBLG phonons are assumedto coincide with the single layer graphene ones, as if theinterlayer coupling were ineffective in the phonon spec-trum, which is not what we find for the special modesdiscussed above. Moreover, they both discuss the ef-fects of such phonons only in a continuum model for

13

the FBs. In particular, the authors of [73] consider afew selected graphene modes, among which the trans-verse optical mode at K that has the largest weight inthe tBLG phonon that we consider. They conclude thatsuch graphene mode mediates d ± id pairing in the A2channel, τ1 µ3 in our language, leading to an order pa-rameter odd upon interchanging the two layers. On thecontrary, the authors of [74] focus just on the acousticphonons of graphene, and conclude they stabilize an ex-tended s-wave order parameter.

VIII. CONCLUSIONS

In this work, we uncovered a novel and strong electron-phonon coupling mechanism and analysed its potentialrole in the low temperature physics of magic angle twistedbilayer graphene, with particular emphasis on the insu-lating and superconducting states. By working out thephonon modes of a fully relaxed tBLG, we found a specialgroup of few of them modulated over the whole moiré su-percell and nearly dispersionless, thus showing that themoiré pattern can induce flat bands also in the phononspectrum. In particular, two of these modes, which aredegenerate at any k-point invariant under 180◦ rotationaround the y-axis and its C3z equivalent directions, arefound to couple strongly to the valley Uv(1) symme-try, that is responsible for the accidental degeneracy ofthe band structure at the same k-points where the twomodes are degenerate. This particular phonon doublet isstrongly Jahn-Teller coupled to the valley degrees of free-dom, realizing a so-called E⊗ e Jahn-Teller model. Thismechanism, if static, would generate a filling-dependentbroadening of the flat bands and eventually insulatingphases at all the commensurate fillings. Interestingly,freezing the modes at Γ can stabilize a topologically non-trivial insulator at charge neutrality that sustains edgemodes. We also investigate the symmetry properties ofa hypothetical superconducting state stabilized by thisJahn-Teller mode. We find that the phonon mediatedcoupling occurs on the moiré scale, favoring spin-singletpairing of electrons in different Bernal (AB/BA) regions,which may thus condense with an extended s- and/ord ± id-wave order parameter. In a mean-field calcula-tion with a model tight-binding Hamiltonian of the flatbands, we find that the dominant symmetry depends onthe orbital character that prevails in the Bloch statesat the Fermi energy, as well as on the precise values ofscattering amplitudes in the different Cooper channelsallowed by the C3z symmetry. We cannot exclude thata nematic component might arise due to higher orderterms not included in our mean-field calculation, as dis-cussed in [75] and [76]. These results herald a role ofphonons and of lattice distortions of much larger impactthan supposed so far in twisted graphene bilayers, whichdoes not exclude a joint action of electron-phonon andelectron-electron interaction. Further experimental andtheoretical developments will be called for in order toestablish their actual importance and role.

Acknowledgments

The authors are extremely grateful to D. Mandelli forthe technical support provided during the lattice relax-ation procedure. We acknowledge useful discussions withP. Lucignano, A. Valli, A. Amaricci, M. Capone, E. Ku-cukbenli, A. Dal Corso and S. de Gironcoli. M. F. ac-knowledges funding by the European Research Coun-cil (ERC) under H2020 Advanced Grant No. 692670

14

“FIRSTORM”. E. T. acknowledges funding from the Eu-ropean Research Council (ERC) under FP7 AdvancedGrant No.320796 “MODPHYSFRICT” , later continuingunder Horizon 2020 Advanced Grant No. 824402 ”UL-TRADISS”.

Appendix A: Details on the lattice relaxation, bandstructure and phonon calculations

The lattice relaxation and bandstructure calculationprocedures are the same as those thoroughly described inRef. [10], with the exception that now the carbon-carbonintralayer interactions are modeled via the Tersoff poten-tial [77]. The interlayer interactions are modelled via theKolmogorov-Crespi (KC) potential [78], using the recentparametrization of Ref. 79. Geometric optimizations areperformed using the FIRE algorithm [80]. The hoppingamplitudes of the tight-binding Hamiltonian are:

t(d) = Vppσ(d)

[d · ezd

]2+ Vppπ(d)

[1−

(d · ezd

)2],

(A1)where d = ri − rj is the distance between atom i andj, d = |d|, and ez is the unit vector in the directionperpendicular to the graphene planes. The out-of-plane(σ) and in-plane (π) transfer integrals are:

Vppσ(x) = V0ppσe

− x−d0r0 Vppπ(x) = V0ppπe

− x−a0r0 (A2)

where V 0ppσ = 0.48 eV and V 0ppπ = −2.8 eV are valueschosen to reproduce ab-initio dispersion curves in AAand AB stacked bilayer graphene, d0 = 3.344Å isthe starting inter-layer distance, a0 = 1.3978Å is theintralayer carbon-carbon distance obtained with theTersoff potential, and r0 = 0.3187 a0 is the decay length[5, 25].

We compute phonons in tBLG using the force con-stants of the non-harmonic potentials U = UTERSOFF +UKC that we used to relax the structure:

Cαβ(il, js) =∂2U

∂Rαil∂Rβjs(A3)

where (i, j) label the atoms in the unit cell, (l, s) themoiré lattice vectors and α, β = x, y, z. Then, we definethe Dynamical Matrix at phonon momentum q as:

Dαβij(q) =1

MC

∑l

Cαβ(il, j0)e−iqRl (A4)

whereMC is the carbon atom mass. Using the above rela-tion we determine the eigenvalue equation for the normalmodes of the system and the phonon spectrum:∑

jβ

Dαβij(q)�jβ(q) = ω2q�iα(q) (A5)

where ωq denote the energy of the normal mode�(q). The phonon dispersion of Bernal stacked bilayergraphene obtained with this method is shown Fig. 16. Ascan be seen, the transverse optical (TO) modes at K arefound at ω ≈ 207 meV. As a consequence, the Jahn-Tellermodes discussed in the main text, which vibrate in thesame way on the graphene scale, have similar frequency.

15

Γ M K Γ0

50

100

150

200

ω (

meV

)

FIG. 16: Phonon dispersion obtained with our choice of in-tralayer and interlayer potentials in Bernal stacked bilayergraphene.

However, the frequency of the TO modes in graphene isstrongly sensitive to the choice of the intralayer poten-tial used [81], so that these modes can be predicted tohave frequencies as low as ≈ 170 meV [82, 83]. This im-plies that also the JT modes may be observed at a lowerfrequencies than ours.

Appendix B: Hamiltonian in momentum space

In order to make the invariance under C3z more ex-plicit, we shall use the transformed spinors(

Φ+1,kσΦ−1,kσ

)=

1√2

(1 −i1 +i

) (Φs,kσΦp,kσ

), (B1)

for the 2d-irreps, which correspond to WOs eigenstatesof C3z. Moreover, it is convenient to transform also thespinors Ψkσ of the 1d-irreps in the same way, i.e.,(

Ψ+1,kσΨ−1,kσ

)=

1√2

(1 −i1 +i

) (Ψs,kσΨp,kσ

), (B2)

which correspond to WOs still invariant under C3z butnot under C2x, whose representation both in Ψkσ andΦkσ becomes the Pauli matrix µ1. In conclusion thespinor operators defined above satisfy

C3z(

Φkσ

)=

(ω 0

0 ω∗

)ΦC3z(k)σ ,

C3z(

Ψkσ

)= ΨC3z(k)σ ,

C2x(

Φkσ

)= µ1 ΦC3z(k)σ ,

C2x(

Ψkσ

)= µ1 ΨC3z(k)σ .

(B3)

We shall, for simplicity, consider only nearest and nextnearest neighbor hopping between AB and BA region,

which correspond to the following functions in momen-tum space:

γ1(k) = αk

(1 + e−ik·a + e−ik·b

),

γ1,+1(k) = αk

(1 + ω e−ik·a + ω∗ e−ik·b

),

γ1,−1(k) = αk

(1 + ω∗ e−ik·a + ω e−ik·b

),

(B4)

for first neighbors, and

γ2(k)= αk

(eik·(a−b) + e−ik·(a−b) + e−ik·(a+b)

),

γ2,+1(k)= αk

(ω eik·(a−b)+ ω∗e−ik·(a−b)+ e−ik·(a+b)

),

γ2,−1(k)= αk

(ω∗eik·(a−b)+ ω e−ik·(a−b)+ e−ik·(a+b)

),

(B5)for second neighbors, where ω = ei2π/3, αk = eik·(a+b)/3,and the lattice constants a = (

√3/2,−1/2) and b =

(√

3/2, 1/2). Since

C3z(a)

= b− a , C3z(b)

= −a ,C2x

(a)

= b , C2x(b)

= a ,(B6)

then, for n = 1, 2,

γn

(C3z(k)

)= γn(k) ,

γn,±1

(C3z(k)

)= ω±1 γn,±1(k) ,

γn

(C2x(k)

)= γn(k) ,

γn,±1

(C2x(k)

)= γn,∓1(k) ,

(B7)

which shows that γn,±1(k) transform like the 2d-irrepE.

We assume the following tight-binding Hamiltonian forthe 1d-irreps

H1d−1d =∑kσ

[−∆ Ψ†kσ σ0 µ1 τ0 Ψkσ (B8)

−∑n=1,2

t(n)11

(γn(k) Ψ

†kσ σ

+ µ0 τ0 Ψkσ +H.c.)],

where t(1)11 and t(2)11 are the first and second neighbor hop-

ping amplitudes, respectively, which we assume to bereal.The 2d-irreps have instead the Hamiltonian

H2d−2d = −∑kσ

2∑n=1

[t(n)22 γn(k) Φ

†kσ σ

+ µ0 τ0 Φkσ

+g(n)22 Φ

†kσ σ

+ γ̂n(k)µ1 τ0 Φkσ +H.c.

], (B9)

16

with real hopping amplitudes, where

γ̂n(k) =

(γn,+1(k) 0

0 γn,−1(k)

). (B10)

Finally, the coupling between 1d and 2d irreps is repre-sented by the Hamiltonian

H1d−2d = −∑kσ

2∑n=1

t(n)12

[Φ†kσ σ

+ µ1 γ̂n(k)µ1 τ0 Ψkσ

+Ψ†kσ σ+ γ̂n(k) τ0 Φkσ

+iΦ†kσ σ+ µ1 γ̂n(k) τ3 Ψkσ (B11)

+iΨ†kσ σ+ µ1 γ̂n(k) τ3 Φkσ +H.c.

],

with real t(n)12 .The Hamiltonian thus reads

H = H1d−1d +H2d−2d +H1d−2d , (B12)

which, through the equations (B3) and (B7), can be read-ily shown to be invariant under C3z and C2x, and is ev-idently also invariant under the Uv(1) generator τ3. Inaddition, the Hamiltonian must be also invariant underTC2z, where T is the time reversal operator. Noting that

TC2z(

Φkσ

)= σ1 µ1 Φk−σ ,

TC2z(

Ψkσ

)= σ1 µ1 Ψk−σ ,

(B13)

one can show that H in (B12) is also invariant underthat symmetry.

The model Hamiltonian thus depends on eight param-eters. The FBs shown in Fig. 13 have been obtainedchoosing: ∆ = 10 , t111 = 2 , t122 = 5 , g122 = 10 ,t222 = g

222 = −t211 = 1.2 , t112 = 2 and t212 = 0.5.

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