Nematic topological semimetal and insulator in magic angle bilayer graphene at charge neutrality

Shang Liu, Eslam Khalaf, Jong Yeon Lee, and Ashvin VishwanathDepartment of Physics, Harvard University, Cambridge, MA 02138

(Dated: October 6, 2021)

We report on a fully self-consistent Hartree-Fock calculation of interaction effects on the Moire flat bandsof twisted bilayer graphene, assuming that valley U(1) symmetry is respected. We use realistic band structuresand interactions and focus on the charge neutrality point, where experiments have variously reported eitherinsulating or semimetallic behavior. Restricting the search to orders for which the valley U(1) symmetry remainsunbroken, we find three types of self-consistent solutions with competitive ground state energy (i) insulators thatbreak C2T symmetry, including valley Chern insulators (ii) spin or valley polarized insulators and (iii) rotationC3 symmetry breaking semimetals whose gaplessness is protected by the topology of the Moire flat bands. Wefind that the relative stability of these states can be tuned by weak strains that break C3 rotation. The nematicsemimetal and also, somewhat unexpectedly, the C2T breaking insulators, are stabilized by weak strain. Theseground states may be related to the semi-metallic and insulating behaviors seen at charge neutrality, and thesample variability of their observation. We also compare with the results of STM measurements near chargeneutrality.

I. INTRODUCTION

The discovery of interaction-driven insulating and super-conducting behavior in twisted bilayer graphene (TBG) [1, 2]has inspired intensive efforts to understand this behavior [3–16] and to find related systems which exhibit similar phe-nomenology [17–22]. This work has started to bear fruit withseveral groups announcing similar observations in TBG sam-ples [23–26] as well as other Moire materials [27–31]. Thebasic mechanism underlying the enhancement of correlationin these materials is understood to originate from the long-wavelength Moire pattern leading to quenching of the elec-tron kinetic energy manifested in flat energy bands [32, 33].Nevertheless, the nature of the observed correlated insulatingstates remains under debate [4–6, 8, 9].

Early experiments found clear signature of a correlated in-sulating state at half-filling [34] [1, 23]. Subsequently, in-sulating ferromagnetic states were also observed at quarterand three-quarter fillings [23, 24]. On the other hand, in allthese experiments[1, 2, 23] insulating behavior was absent atcharge neutrality (CN) where signatures of semimetallic be-havior were observed instead. In contrast, a recent experimentsurprisingly found an insulator at CN with a transport gap ex-ceeding those at 1/2, 1/4 and 3/4 fillings [25].

On the theory side, it was realized early on [4, 35] that asimple Mott picture for the insulating phase is complicated bythe band topology which prohibits the construction of local-ized orbitals describing the flat bands while preserving all thesymmetries. Various orders have been proposed to account forthe insulating states. At charge neutrality, a C2T symmetrybreaking insulator, with Chern number ±1 for each spin andvalley flavour was proposed in [9], along with a C2T sym-metry preserving insulator that is believed to require mixingwith remote bands [9]. An intervalley coherent order [4] wasproposed as a candidate for the insulating state at half-fillingwhile nematic orders were discussed in [6, 11, 36] and fer-romagnetic ordering was proposed in [8]. In the presence ofexplicit C2 symmetry breaking induced by a substrate, valleyor spin polarized insulator with valley resolved Chern num-bers have been discussed [17, 37].

In this letter, we perform a self-consistent Hartree-Fockmean field analysis for the screened Coulomb interaction pro-jected onto the flat bands.

We focus on the CN point because of its pivotal role in de-termining the entire phase diagram. We will discuss other fill-ings in subsequent work. For simplicity, we restrict our at-tention to orders that conserve the U(1) valley charge as wellas translation invariance at the scale of the Moire unit cell.Our results include the expected spin-polarized and valley-polarized insulators, which break no other symmetries. In ad-dition we observe a strong tendency to breaking spatial ro-tation symmetries. We find a C2T -breaking insulator andtwo distinctC2T -symmetric semimetallic phases which breakC3-symmetry . The C2T -breaking phase has a Chern num-ber of C = ±1, per flavor (i.e. valley and spin). Differentspin/valley orderings then lead to various ground states rang-ing from quantum anomalous Hall (QAH) to quantum valleyHall (QVH) or quantum spin Hall (QSH) insulators with verysimilar Hartree-Fock energies. Of these, the QVH insulatorbreaks C2 symmetry, allowing for a direct coupling to the C2-breaking hBN substrate, potentially favoring it in situationswhere the sample and substrate are aligned. The gapless C3-breaking phases are obtained by bringing the two Dirac conesfrom the MoireK andK ′ very close to the Γ point. Instead ofmerging and opening a gap as one might normally expect, theDirac points remain gapless since they carry the same chiral-ity [4], a consequence of descending from the Dirac points ofgraphene from the same valley for the two layers. This topo-logical protection prevents them from annihilating, resultingin a gapless semimetallic state. Thus, the metallic nature atCN in this scenario is intimately tied to the topological prop-erties of the magic angle flat bands.

We investigate the effect of small explicit C3 symme-try breaking which can arise in real samples due to strain[21], and show that it strongly influences the competitionbetween different symmetry broken phases, favoring one ofthe C3-breaking semimetallic phases. Surprisingly, the C2T -breaking insulators also exhibits a strong susceptibility to C3

symmetry breaking. The energies of the insulating C2T -breaking and the semimetallic C2T -preserving states are low-

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ered compared to the spin/valley ferromagnets, and approacheach other quickly as the value of the C3-breaking param-eter is increased. Our results suggest that these two statesare candidate ground states in the presence of very small ex-plicit C3 symmetry breaking which is likely to exist in experi-ments. Further competition between these two phases is likelyto be settled by small sample-dependent details, potentiallyexplaining the realization of an insulator in some samples anda semimetal in other samples.

II. PROBLEM SETUP

The single-particle physics is described by the Bistritzer-MacDonald (BM) model [32, 33], which employs a contin-uum approximation close to K and K ′ for a pair of graphenesheets rotated relative to each other by an angle θ. The Hamil-tonian for the K valley is given by:

H+ =∑l

∑k

f†l (k)hk(lθ/2)fl(k)

+

(∑k

3∑i=1

f†t (k + qi)Tifb(k) + h.c.

). (1)

Here, l = t/b ' ±1 is the layer index, and fl(k) is the K-valley electron originated from layer l. hk(θ) is the mono-layer graphene K-valley Hamiltonian with twist angle θ (seeAppendix for details) and q1 is defined as Kb − Kt with Kl

denoting the K-vector of layer l. q2 = O3q1 is the counter-clockwise 2π/3 rotation of q1, and q3 = O3q2. Finally, theinterlayer coupling matrices are given by

Tj =

(w0 w1 e

−(j−1) 2πi3

w1 e(j−1) 2πi

3 w0

), (2)

with w0 and w1 denoting intrasublattice and intersublatticehopping, respectively. Due to lattice relaxation effects, whichshrink the AA regions relative to the AB regions, the value ofw0 at the magic angle is about 75% of w1 [38, 39]. Through-out this work, we will use the values w1 = 110 meV andw0 = 82.5 meV. Explicit C3 symmetry breaking is imple-mented via the substitution T1 → (1 + β)T1 [40].

To study possible correlated insulating states, we employa momentum-space self-consistent Hartree-Fock mean fieldtheory. The momentum-space description allows us to focuson the pair of flat bands at CN, thereby evading the difficul-ties associated with the real space Wannier obstruction rootedin the fragile topology of these bands [4, 35, 41].Restrictingthe analysis to the flat bands is only justified in the limit whenboth the bandwidth and interaction strength are much smallerthan the bandgap. For realistic interactions, the interactionstrength is of the same order as the bandgap which meansthat symmetry-broken states involving remote bands cannotbe ruled out [9]. However, given the strong angle dependenceand appearance of correlated states only near the magic angle[1, 2, 23], it is likely that the relevant symmetry-broken phasesexperimentally originate mostly from the two flat bands which

justifies our approximation. We note, however, that our ap-proach cannot capture possible changes in the topology of theflat bands arising from mixing with remote bands which wereargued to be relevant in Ref. [9].

The Hartree-Fock (HF) mean field theory is defined interms of the projector

Pα,β(k) = 〈c†α(k)cβ(k)〉, P (k)2 = P (k) = P (k)†, (3)

where cα(k) is the annihilation operator for an electron at mo-mentum k and α = (n, τ, s) is a combined index for band,valley and spin, respectively. For a gapped or semimetallicphase at CN, P satisfies trP (k) = 4 for all k points[42]. TheHF mean field Hamiltonian has the form

HMF =∑k

{c(k)†[h0(k) + hHF(P,k)]c(k)

− 1

2trhHF(P,k)PT (k)}. (4)

Here, c(k) is a column vector in the index α, h0(k) denotesthe single particle Hamiltonian and hHF(k) is given by

hHF(P,k) =1

A

∑G

VGΛG(k)∑k′

trPT (k′)Λ†G(k′)

− 1

A

∑q

VqΛq(k)PT (k + q)Λ†q(k), (5)

where A is the total area. The k′, G, and q summationsrange over the first Brillouin zone, reciprocal lattice vectors,and all momenta, respectively. Vq is the interaction potentialwhich we take to be a single-gate-screened Coulomb interac-tion Vq = e2

2εε0q(1 − e−2qds) with dielectric constant ε = 7

and screening length equal to the gate distance ds ≈ 40 nm.The first term in (5) is the Hartree term while the second

is the Fock term. The matrix Λq(k) contains the form factorsfor the single-particle

[Λq(k)]α,β = 〈uα(k)|uβ(k + q)〉. (6)

where α and β denote a combined index for spin, valley andband. The HF self-consistent analysis starts by proposing anansatz for the projector P (k), then substituting in the Hamil-tonian (4) which is then used to compute the new projector.This procedure is iterated until convergence is achieved.

One important subtlety in the HF approach is that the bandstructure depends on the filling even without symmetry break-ing. This follows from the fact that the form factor matrixΛq(k) is not diagonal in the band index for q 6= 0 since theBloch wavefunctions uα,k from different bands are not or-thogonal at different momenta. In addition, the Hartree termcontains a trace over all filled bands which also affects the dis-persion of the empty bands. This means that the band structureobtained from the BM model is only valid at a specific fillingwhich determines the references point for out analysis. At thispoint, it will be assumed that interaction effects are alreadyincluded in the parameters of the effective model which can

3

FIG. 1. (a) K-valley band structure at empty filling. (b) Normalizeddensity of states at empty filling, valid for both valleys.

be obtained by fitting to ab initio calculations or comparingto STM data away from the magic angles [43–45]. A natu-ral choice of the reference point, which we adopt throughoutthis letter, is the CN point. This means that the single particleHamiltonian h0(k) is given by

h0(k) = hBM(k)− hHF(P0,k) (7)

where hBM(k) is the BM Hamiltonian and P0 is the projectorcorresponding to symmetry unbroken state obtained by fillingthe lower band of the BM Hamiltonian at CN. We note that inthe different approach of Ref. [9] where multiple bands are in-cluded, the reference point was taken in the limit of decoupledlayers. In our complementary approach of projection onto theflat bands it is not possible to implement such a choice.

Using CN as our reference point implies that the bands atempty or full filling should include some HF corrections lead-ing to a modified band structure shown in Fig. 1 together withthe resulting density of states (DOS). We notice that the sep-aration between the two peaks in the DOS is about 10-15meV in agreement with the measured DOS in STM experi-ments [43, 44, 46]. Thus, our approach provides an explana-tion for the discrepancy between the experimentally measuredpeak separation and the expectation based on the BM modelwhose bandwidth close to the magic angle is much smaller(1-3 meV).

III. SYMMETRY-BROKEN PHASES

The interacting TBG Hamiltonian is characterized by thefollowing symmetries [4]: spinless time-reversal T mappingthe two valleys, SU(2) spin rotation in each valley, U(1) val-ley charge conservation and C6 symmetry as well as a mir-ror symmetry which switches layers and sublattices but actswithin each valley. Of these, only spin rotation, C3, mirrorand C2T act within each valley. At integer filling, differentcorrelated insulating phases can emerge by breaking some ofthese symmetries. Time-reversal symmetry is broken by val-ley polarized (VP) states, where the filling of the two valleys isdifferent. Spin rotation symmetry is broken by spin polariza-tion (SP) leading to ferromagnetic order. U(1) valley chargeconservation is broken in the presence of intervalley coherent(IVC) superposition of states from the two valleys. C2 sym-metry is broken by sublattice polarization which gaps out the

Dirac points at the Moire K and K ′ points.Breaking C3 symmetry alone does not generally lead to a

gapped phase since it only moves the Dirac points away fromthe Moire K and K ′ without gapping them out. Even strongC3 breaking does not result in the merging and gapping of theDirac nodes, in contrast with other familiar band structuressuch as single layer graphene. This follows from the fact thatthe chirality of the Dirac nodes, which is well defined in thepresence of C2T symmetry, is the equal [4, 15], rather thanopposite. Alternately, one can phrase the argument as fol-lows. The topology of the two flat bands is captured by thesecond Stiefel-Whitney invariant w2 [41, 47, 48]. This invari-ant is protected by C2T and only depends on the flat bandeigenstates which are unaffected by any symmetry breakingthat does not involve other bands (which is the main assump-tion in this work). Indeed, the w2 invariant must be trivial fora single isolated band, implying we cannot separate the pairof connected flat bands [49]. Thus the non-trivial w2 invariantimplies that the two Dirac points cannot be removed withoutbreaking C2T .

Before presenting the numerical results, let us make the fol-lowing observations. First, it is relatively easy to show that astate with uniform full spin or valley polarization is always aself-consistent solution to the HF equations at CN for suffi-ciently narrow bands. These two states have the same energyin the absence of intervalley Hund’s coupling [17, 22]. TheIVC state is known to have higher energy than SP/VP statesfor isolated bands with non-vanishing valley Chern number[17, 37]. These arguments do not generalize to TBG wherethe extra symmetries of the problem complicate the discussion(see Appendix and Ref. [50]). Nevertheless, we will excludeIVC orders from our numerical analysis (which is equivalentto assuming unbroken U(1) valley charge conservation) sinceit leads to significant simplification by allowing us to focuson a single flavor (spin and valley). Different diagonal spin-valley orders can then be generated from the single-flavor so-lution by applying different symmetries. The energy compe-tition with U(1) valley symmetry broken phases is consideredin [50].

IV. RESULTS

The results for the self-consistent HF analysis are providedin Fig. 2 showing the energies of the different solutions as afunction of the C3 symmetry breaking parameter β. There aretwo types of gapped solutions corresponding to either flavor-polarized (spin/valley) states or C2T -breaking (C2T I) insu-lators. There are three gapped solutions corresponding tospin-polarized (SP), valley-polarized (VP) and C2T -breaking(C2T I) insulators. The latter does not break C3 when β = 0but develops a large C3 breaking component for β 6= 0. Theextent of C3 symmetry breaking can be quantified by defining

χC3=

1

N

∑k

(1− |〈ψO3k|C3|ψk〉|2), (8)

4

FIG. 2. (a) Energies of the solutions of the self-consistent HF equa-tions as a function of the C3 symmetry breaking parameter β. Allenergies are measured relative to the state with no-broken symmetry.The degree of C3-breaking measured by χC3 (Eq. 8) for the C2T -breaking insulator as a function χC3 for the non-interacting systemfor 0 ≤ |β| ≤ 5× 10−4 are shown in panels (b) and (c) for positiveand negative values of β, respectively.

FIG. 3. 3D band structure for one of the (a) C3-breaking semimetalsand (b) C2T breaking valley Hall insulators.

which vanishes for anyC3-symmetric state. Here, |ψk〉 are theoccupied single-particle eigenstates of the HF Hamiltonianwith momentum k (for a given flavor). The value of χC3

forthe C2T I state is shown in Fig. 2 as a function of χC3

for thecorresponding non-interacting states arising from explicit C3-breaking parameter β 6= 0 in the BM model. We can clearlysee from the figure that a relatively small χC3(BM) ∼ 10−3

in the non-interacting states induces induces a much larger C3

symmetry breaking of almost two orders of magnitude in theC2T I state. This serves to show that the C2T I has very largesusceptibility to C3 symmetry breaking. In the following, wewill refer to this state for positive and negative β as C3C2T Iyand C3C2T Ix, respectively. In addition to this insulatingstate, there are two distinctC2-preserving semimetallic phaseswhich spontaneously break C3 even for β = 0 which we de-note by C3Sx and C3Sy, since they have Dirac points alongkx and ky , respectively. We notice that these ground states aresimilar to the ones obtained within a 10-band model Ref. [43]which used a site-local ansatz for the interactions. Examplesof the band structures for the C3Sy and C3C2T Iy states areshown in Fig. 3.

The C2T I phase obtained here is characterized by a Chernnumber of ±1 for a given spin and valley. The nature of theresulting phase depends on the precise symmetries which are

broken as follows: (i) if T is broken but C2 and spin rotationare preserved, we obtain a quantum anomalous Hall (QAH)insulator with total Chern number C = ±4, (ii) if spin ro-tation is broken we obtain a quantum-spin Hall (QSH) in-sulator with opposite Chern number Cs = ±2 for oppositespins, (iii) if C2 is broken but T and spin rotation symme-try are preserved, then a quantum valley Hall (QVH) insulatorobtains with opposite Chern number Cv = ±2 for oppositevalleys, or (iv) we can additionally break spin rotation sym-metry to obtain a state where opposite spins within the samevalley also have opposite Chern numbers resulting in a quan-tum spin-valley Hall (QSVH) insulator where flipping eitherspin or valley flips the Chern number. There can also be stateswith total Chern number ±2, but these do not lead to any newphysics or a lower energy, therefore we restrict to the abovefour possibilities for simplicity. The QH, QSH and QVSHpreserve either C2 or a combination of C2 and some inter-nal symmetries, thus the only state expected to couple to theC2-breaking hBN substrate is the QVH which is likely to beenergetically favored in aligned samples[51]. The energies ofthe four possible C2T states are very similar differing only byvery small ∼ 10−3 meV Hartree corrections. Similarly, theflavor-polarized state can be spin-polarized, breaking SU(2)spin rotation, valley-polarized breaking T or a spin-valley-locked states which breaks both but preserves a combinationof T and π-spin rotation. These states are degenerate onthe mean field level and are only distinguished by intervalleyHund’s coupling [17, 22] neglected in this study.

At β = 0, the energies of the insulating VP/SP and C2T Istates are very close and smaller by about 1 meV per particlethan the energies of the two semimetallic C3 breaking states.An explanation for this fact is provided in the next sectionin the limit where the intrasublattice hopping is switched off[52]. In this limit, we can establish a rigorous bound for theFock energy which plays the dominant role in the energy com-petition. We will show that the SP/VP and the C2T I statessaturate the bound in this limit and should thus be approxi-mately degenerate. The energies of the C3S states are higherthan these two. This analysis explains why these states are ex-pected to be close in energy, but it is generally insufficient tocapture the details of the energy competition which dependssensitively on the intrasublattice hopping w0 and can only bedetermined numerically. A more detailed analysis of the ef-fects of dispersion and finite sublattice symmetry breaking w0

is given in Ref. [50].Once β becomes non-zero, the energy of the C2T breaking

state is reduced relative to the VP/SP state. Furthermore, oneof the two C3-breaking states (depending on the sign of β)goes down in energy becoming more energetically favorableto the VP/SP state around β = ±3 × 10−4. For larger valuesof β & 4×10−4, the energies of the insulating C2T -breakingphase and the semimetallic C2T -preserving phase approachthe same value. This indicates that even very small explicitC3-breaking picks out these two states as the main candidatesfor the ground state at CN. Assuming that such small explicitC3 symmetry breaking exists in real samples due to strain, ouranalysis leads to the conclusion that the ground state of TBGat CN is either a C2T -breaking insulator or a C2T -symmetric

5

FIG. 4. (a) Normalized DOS of the empty filling band structure. Thetwo vertical dashed lines indicate the band bottom and top whichis chosen to be at E = 0. (b) Normalized DOS and (c) local fill-ing fraction for the upper layer defined in (24) for the four potentialground states for strain parameter β = ±4×10−4. For positive (neg-ative) beta: the two competing ground states are a C2T -preservingsemimetal C3Sy (C3Sx) and a C2T -breaking valley Hall insulatorC3C2T Iy (C3C2T Ix), both strongly breaking C3. The red dots in(c) indicates the AA position. In (c) the breaking of C2 and C3 rota-tion symmetries are clearly visible.

semimetal. Both states strongly break C3 symmetry and arevery close in energy. It is worth noting that, in the presenceof disorder, massless Dirac fermions may also emerge fromthe spatial domain walls of locally C2T breaking insulatingregions [53].

V. ENERGY COMPETITION IN CHIRAL LIMIT

The existence of several states close in energy can be ex-plained by considering the simplified chiral model of Ref. [52]where the Moire intrasublattice hopping is switched off. Insuch limit, the band becomes strictly flat at charge neutral-ity in the absence of interaction and the model has an extrasublattice symmetry i.e. the single-particle Hamiltonian an-ticommutes with the sublattice operator σz . As a result, theflatband wavefunctions can be chosen to be eigenfunctions ofσz such that the wavefunction σzuA/B,τ,k = ±uA/B,τ,k. Inthis limit, we can show (see Appendix for details) that theform factor matrix has the simple form

Λq(k) = Fq(k)eiΦq(k)σzτz , (9)

where Fq(k) and Φq(k) denoting the magnitude and phase ofthe form factor for the flatband wavefunction in sublattice Aand valley K.

Let us now compare the mean field energy for different

phases. First, we notice that, even in the chiral limit where thenon-interacting band dispersion vanishes, a sizable dispersionwould be generated by the interaction [50]. It will be instruc-tive, however, to start by ignoring the interaction generatedband dispersion and focusing on the Hartree-Fock energy. Ata fixed integer filling, the Hartree term does not play a rolein the energy competition between phases so we can focus onthe Fock term. We will find it convenient to write the projectorP (k) as

P (k) =1

2[1 +Q(k)], Q(k)2 = 1, trQ(k) = 0 (10)

In terms of Q, the Fock energy is given by

EF = − 1

8A

∑k,q

Vq trQ(k)Λ†q(k)Q(k + q)Λq(k). (11)

up to a constant. We note that 〈A,B〉 = trAB defines a posi-tive definite inner product on the space of hermitian matrices.Using Cauchy-Schwarz inequality, we get

EF ≥ EF,min = − 1

A

∑k,q

VqF2q (k). (12)

This inequality is satisfied if and only if Q(k + q) is parallelto Λq(k)Q(k)Λ†q(k) for every k and q which implies

Q(k + q) = eiΦq(k)σzτzQ(k)e−iΦq(k)σzτz , (13)

A k-independent flavor (spin,valley) polarized state obvi-ously satistfies this constraint since σzτz is diagonal in fla-vor, thereby saturating the Fock bound. This state can eitherbe spin-polarized Q = sz , valley polarized Q = τz or spin-valley locked Q = τzsz .

Next let us discuss C2T breaking insulating solutions. AC2T breaking solution to Eq. 13 is obtained by taking Qk =σz (since C2T is off diagonal in the sublattice index). The re-sulting state saturates the Fock bound and is characterized bya Chern number ±1 in the lower band. Its energy competitionwith the flavor-polarized states is settled by the effects of thesingle-particle term hk and w0 which are neglected here. Inour numerics, we found that these states are very close in en-ergy for the realistic model with their energy difference onlysensitive to the strain parameter β. Depending on its structurein flavor space, the C2T -breaking insulator may correspondto one of four states: (i) Q = s0τ0σz breaks C2 but not Tand corresponds to a valley Hall state, (ii) Q = s0τzσz breaksT but not C2 and corresponds to a quantum Hall state withChern number ±4, (iii) Q = szτ0σz corresponds to a valley-spin-Hall state where the Chern number is invariant under flip-ping spin and valley, and (iv) Q = szτzσz corresponds to aspin-Hall state.

Finally, let us consider semimetallic states which break nei-ther flavor nor C2T symmetry. These can be generally de-scribed (within each flavor) by the order parameter

QSM(k) = σxeiαkσzτz . (14)

6

Substituting in the Fock energy (11) yields

EF = − 1

A

∑k,q

VqF2q (k) cos[αk − αk+q + 2Φq(k)]. (15)

To simplify further, we make the reasonable assumption thatFq(k) decays quickly with the relative momentum q whichenables us to expand Φq(k) and αk+q in q. Noting thatΦq(k) = q · Ak + O(q2) with Ak the Berry connection−i〈uA,K,k|∇k|uA,K,k〉, we get

EF = EF,min +1

2A

∑k,q

Vqq2F 2

q (k)(∇kαk − 2Ak)2. (16)

The second term in the energy is non-negative and impliesthat the semimetal does not generally satisfy the bound. Infact, this term has the form of the energy of a superconductorin a magnetic field in momentum space. Due to the non-trivialChern number of the sublattice-polarized bands [50, 52], theBloch wavefunctions cannot be chosen to be simultanuouslysmooth and periodic over the Brillouin zone. If we choosethem to be smooth, the Berry connection Ak will be smoothbut the phase αk will necessarily wind by 4π around the Bril-louin zone, leading to at least two vortices in the Brillouinzone where the projector is undefined due to the vanishingof the gap. These will yield a finite contribution to the sec-ond term which implies that the semimetal never satisfies theFock bound in the ideal limit [54]. The location of the vorticeswhich minimize this term is determined by the competitionbetween the weak (logarithmic) repulsion between the vor-tices in 2D and the tendency of the vortices to sit wherever themagnetic field corresponding to A, i.e. the Berry curvature,is maximal. As shown in Fig. 5, in the chiral limit w0 = 0,the Berry curvature is relatively uniform so that vortices pre-fer to sit far apart at the K and K′ point, thereby preserving C3

symmetry. However, for finite w0/w1, the Berry curvature (inthe basis which diagonalizes the sublattice operator σz [50])becomes strongly peaked at the Γ point and for the realisticparameter w0/w1 ≈ 0.75, the vortices prefer to sit close tothe Γ point. This explains the energetic advantage of C3 sym-metry breaking in the realistic parameter regime considered inthe numerics.

Finally, let us consider the effect of the dispersion (whichis mostly generated by the interaction). Using C2T , valleyand particle-hole symmetries, the form of the dispersion canbe reduced to [50]

h0(k) = σxf(k)eiφ(k)σzτz (17)

where f(k) is a positive real function which vanishes at thetwo Dirac points and φ(k) winds by 2π around each of theDirac points. These points lie at the Moire KM and K ′M forβ = 0 but move away for finite β. The energy contribution ofh0(k) for a state described by the matrix Q is

Eh[Q] =1

2

∑k

trh0(k)Q(k) (18)

We notice that this term favors the semimetal order parameterwith Q given by (14) but vanishes for all the order parameterswhich satisfy the Fock bound. This might suggest none ofthese states benefit energetically from dispersion but this turnsout not to be true as we will show below.

We start by writing the total energy functional in terms ofQ (ignoring the Hartree term which does not play a role in thecompetition between phases)

E[Q] = Eh[Q] + EF [Q] (19)

with Eh[Q] given by (18) and EF [Q] given by (11). TheHartree-Fock self consistency equation is taken by setting thevariation of E[Q] relative to Q to zero (subject to the con-straint Q2 = 1) which leads to the equation

[Q(k),1

2h0(k)− 1

4A

∑q

Vq(k)Λq(k)∗Q(k+q)Λq(k)T ] = 0

(20)From this equation, we see that the flavor polarized stateswhich are characterized by [Q, h0] = 0 remain solution of theself-consistency equation for non-zero h0 whereas the C2Tbreaking insulators which are characterized by {Q, h0} = 0are no longer solution to the self-consistency equation. Towrite a C2T breaking solution, we write

Q(k) = QC2T I(k) cos θ(k) +QSM(k) sin θ(k) (21)

which satisfies the condition Q2 = 1 due to the anticommu-tation of QC2T I and QSM. The k dependence of θ can be de-termined from the self-consistency condition (20). In the fol-lowing, we will assume for simplicity that θ is k independentand choose it to minimize the the total energy (19) leading to

| sin θ| = Eh[QSM]

2(EF [QSM]− EF,min)(22)

where we used the fact that QC2T I saturates the Fock bound.We notice that the denominator is the same as the second termon the right hand side of (16) which is guaranteed to be finiteand roughly of the order of the interaction scale. This impliesthat θ is small whenever the interaction dominates the disper-sion. In this case, the order parameter described by (21) isvery close to the pure insulator order parameter. The corre-sponding energy is given by

E[Q] = EF,min −Eh[QSM]2

4(EF [QSM]− EF,min)(23)

Thus, the C2T -breaking insulators actually benefit from thedispersion in quadratic order by developing a small compo-nent proportional to the semimetal order parameter. Thisexplains why their energy is decreased in response to ex-plicit C3-breaking which energetically favors the semimetal(thus increasing |Eh[QSM]|). It also explains their high C3-breaking susceptibility which mainly arises from their QSM

component.

7

FIG. 5. Berry curvature in the sublattice polarized basis for differentvalues of the ratio w0/w1.

VI. CONSEQUENCES FOR EXPERIMENT

The possibility of C3 breaking at CN is consistent with sev-eral recent reports [43–46] which observed direct evidence ofC3 breaking in STM measurements. To check the compat-ibilty of these measurements with our mean-field solutions,we compute the DOS for the four possible C3-breaking states(arising for positive or negative values of β) in Fig. 4b. Wesee that in all cases the global DOS consists of two broadpeaks (which are sometimes further split into two) separatedby about 60-80 meV, in agreement with the STM measure-ments [43, 45]. The DOS alone however, is insufficient todistinguish the insulating and semi-metallic state, since bothhave very low DOS close to zero energy. One way to distin-guish the two is to compute the local filling fraction definedas [44]

ν(r) = 8

(ρLB(r)

ρLB(r) + ρUB(r)− 1

2

)∈ [−4, 4], (24)

where ρLB/UB(r) denote the integrated local DOS from theupper layer for the lower/upper band. ν(r), shown in Fig. 4,exhibits clear C3-symmetry breaking pattern for all the four

phases. The patterns for the C3S and C3C2T I QVH phasescan be distinguished by C2 symmetry which is visibly presentin the former but absent in the latter (the pattern of the otherC3C2T I is obtained by symmetrizing the QVH result rela-tive to C2 with the result being very similar to the one for theC3S breaking states up to an overall factor). The C3S pat-tern is qualitatively similar to that measured in Ref. [44], withthe density vanishing at the Kagome lattice site lying on themirror plane and changing sign twice around it, while hav-ing non-vanishing magnitude with opposite signs on the othertwo Kagome lattice sites. Such structure is generic for anyC3-breaking phase which preserves mirror and particle-holesymmetries [55] and a combination ofC2 and some local sym-metry. [56]

VII. CONCLUSION

In conclusion, we have performed a momentum-space self-consistent Hartree-Fock analysis to uncover the nature of thesymmetry-broken phase in twisted bilayer graphene at chargeneutrality. In addition to insulating states corresponding tospin, valley or sublattice polarization, we found two C3-breaking C2T -symmetric semimetallic solutions. Our mainfinding is that the existence of very small explicit C3-breakingenergetically favors one of these C2T -symmetric metallicstate together with C2T -breaking insulating states. Both setsof states have similar energy within HF, strongly break C3

symmetry and are consistent with the density of states mea-sured in STM experiments. They can be experimentally dis-tinguished in transport measurements or by comparing space-resolved local filling fractions in STM. We propose these twostates as candidates for the insulating and conducting statesobserved in different experiments at the CNP and suggest thatthe competition between the two is settled by small details thatare likely sample-dependent.

ACKNOWLEDGMENTS

We thank Eva Andrei, Allan MacDonald, Adrian Po, A.Thomson and M. Xie for helpful discussions. S. L., E. K., J. Y.L. and A. V. were supported by a Simons Investigator Fellow-ship and by NSF-DMR 1411343. E. K. was supported by theGerman National Academy of Sciences Leopoldina throughgrant LPDS 2018-02 Leopoldina fellowship.

Appendix A: Single-particle physics: Bistritzer-Macdonald model

Our starting point is the Bistritzer-MacDonald (BM) model of the TBG band structure [32], which we now briefly review. Webegin with two layers of perfectly aligned (AA stacking) graphene sheets extended along the xy plane, and we choose the frameorientation such that the y-axis is parallel to some of the honeycomb lattice bonds. Now we choose an arbitrary atomic site andtwist the top and bottom layers around that site by the counterclockwise angles θ/2 and −θ/2 (say θ > 0), respectively. Whenθ is very small, the lattice form a Moire pattern with very large translation vectors; correspondingly, the Moire Brillouin zone

8

(MBZ) is very small compared to the monolayer graphene Brillouin zone (BZ), as illustrated in Fig. 6. In this case, couplingbetween the two valleys can be neglected. If we focus on one of the two valleys, say K, then the effective Hamiltonian is givenby:

H+ =∑l

∑k

f†l (k)hk(lθ/2)fl(k) +

(∑k

3∑i=1

f†t (k + qi)Tifb(k) + h.c.

). (A1)

Here, l = t/b ' ±1 is the layer index, and fl(k) is the K-valley electron originated from layer l. The sublattice index σ issuppressed, thus each fl(k) operator is in fact a two-component column vector. hk(θ) is the monolayer graphene K-valleyHamiltonian with twist angle θ:

hk(θ) = ~vF(

0 (kx − iky)eiθ

(kx + iky)e−iθ 0

), (A2)

where vF = 9.1× 105 m/s is the Fermi velocity. Let Kl be the K-vector of layer l, then q1 is defined as Kb −Kt. q2 = O3q1

is the counterclockwise 120◦ rotation of q1, and q3 = O3q2. Finally, the three matrices Ti are given by

T1 = (1 + β)

(w0 w1

w1 w0

), T2 =

(w0 w1e

−2πi/3

w1e2πi/3 w0

), T3 =

(w0 w1e

2πi/3

w1e−2πi/3 w0

), (A3)

where we introduced an explicitC3-breaking parameter β. We takew1 = 110 meV andw0 = 82.5 meV. The difference betweenw0 and w1 reflects the effect of lattice relaxation. Note that the argument of fl(k) is measured from Kl, i.e. the monolayer UVmomentum associated to fl(k) is in fact Kl + k. It is convenient to also choose a common momentum reference point for thetwo layers. For example, we can define ft(k) = ft(k− q3) and fb(k) = fb(k+ q2), such that the arguments for both ft and fbare measured from Kt − q3, indicated as γ in the right panel of Fig. 6.

One intuitive way of thinking about this effective Hamiltonian is to imagine a honeycomb lattice of Dirac points in themomentum space, as shown in Fig. 7b, where the two sublattices correspond to the two layers. A Dirac point at momentum qcontributes a diagonal block hk−q(±θ/2) to the Hamiltonian for the MBZ momentum k, where the sign is determined by thesublattice that q belongs to. The off-diagonal blocks Ti are nothing but the nearest-neighbor couplings of these Dirac points.

FIG. 6. Real space (left) and momentum space (right) structures for both monolayer and Moire lattices of twisted bilayer graphene. In the realspace (left) panel, we also show the underlying Moire pattern structure.

When the twist angle is near the magic angle θ = 1.05◦, two isolated flat bands per spin and valley appear near the chargeneutrality (CN) Fermi energy, shown in Fig. 7a. These two bands are the focus of the current work.

The single particle Hamiltonian within each valleyH± is invariant under the following symmetries

C3fkC−13 = e−

2π3 iτzσz fC3k, (C2T )fk(C2T )−1 = σxfk, My fkM−1

y = σxµxfMyk, (A4)

In addition, the two valleys are related by time-reversal symmetry given by

T fkT −1 = τxf−k. (A5)

Here, σ, τ and µ denote the Pauli matrices in sublattice, valley and layer spaces, respectively.

9

FIG. 7. (a) Band structure of magic-angle TBG obtained from the BM model. The sampling path is shown by the cyan arrows in the rightpanel. (b) Dirac point lattice at the MBZ.

Appendix B: Projecting the interaction onto the flat bands

In the following, we derive the form of the interaction when projecting onto the two flat bands. Since these two bands havea Wannier obstruction, we can only write such projected interaction in k-space. Let c†α(k) be the creation operator for theenergy eigenstate in the band structure with internal flavor µ and band index n, where µ = (τ, s) is a collective index includingboth valley τ = ± and spin s =↑ / ↓, and n = 1, 2 represents the lower and upper bands, respectively. Also let f†µ,I(q) bethe “elementary” continuous fermion with monolayer momentum q, flavor µ = (τ, s) and I = (l, σ) representing layer andsublattice, then c† and f† are related to each other by the k-space wave functions as follows:

c†µ,n(k) =∑G,I

uτ,n;G,I(k)f†µ,I(k +G), (B1)

where G is a Moire reciprocal lattice vector. In the above expression, we are already using the fact that the wave functionsare spin-independent. Once we choose a gauge of uτ,n;G,I(k) for all k in some MBZ, c†(k) are defined in terms of the f†(q)for those k, and whenever necessary, we define c†(k + G) = c†(k) for any reciprical lattice vector G, which is equivalentto defining uτ,n;G,I(k + G0) = uτ,n;G+G0,I(k). Note that the momentum argument for f† is unconstrained since we areusing the continuum theory for monolayers of graphene. We choose the normalization {fµ,I(q), f†µ′,I′(q

′)} = δµµ′δII′δqq′

(suppose the system size is finite), and 〈uτ,n(k)|uτ ′,n′(k)〉 :=∑

G,I u∗τ,n;G,I(k)uτ ′,n′;G,I(k) = δττ ′δnn′ , which imply

{cµ,n(k), c†µ′,n′(k′)} = δµµ′δnn′δkk′ when k,k′ are confined in the MBZ. For the purpose of projecting the interaction intothese two bands, it is convenient to introduce the form factor notation:

λmn,τ ;q(k) := 〈uτ,m(k)|uτ,n(k + q)〉 (B2)

where q is not restricted to the first Brillouin zone. The form factors satisfy

λmn,τ ;q(k) = λnm,τ ;−q(k + q)∗ (B3)

just from the definition, and also has the property

λmn,τ ;q(k) = λmn,−τ ;−q(−k)∗ (B4)

due to the time-reversal symmetry.

The interaction Hamiltonian is given by

Hint =1

2A

∑σ,σ′,τ,τ ′

∑q

V (q) : ρσ,τ,qρσ′,τ ′,−q :, (B5)

where A is the total area of the system and V (q) is the momentum space interaction potential, related to the real-space one by

10

V (q) :=∫d2rV (r)e−iq·r. Depending on the number of gates, V (q) takes the following form in the SI units:

V (q) =e2

2εε0q

{(1− e−2qds), (single-gate)

tanh(qds), (dual-gate)(B6)

where the screening length ds is nothing but the distance from the graphene plane to the gate(s). Projecting onto the two narrowbands, this Hamiltonian has the form

Hint =1

2A

∑σ,σ′,τ,τ ′

∑q,n1,n2,n3,n4

∑k1,k2∈BZ

λn1,n2;τ,q(k1)V (q)λ∗n4,n3;τ ′,q(k2)

× c†n1,σ,τ (k1)c†n3,σ′,τ ′(k2 + q)cn4,σ′,τ ′(k2)cn2,σ,τ (k1 + q). (B7)

Appendix C: Hartree-Fock analysis in the chiral limit

The chiral limit of the BM model is obtained by switching off the w0 term in (A3) [52]. In this limit, the bands become exactlyflat at the magic angle and the eigenstates of the Hamiltonian have a simple form similar to the Landau levels on a torus In thechiral limit, the single particle Hamiltonian anticommutes with the chiral (sublattice) symmetry operator given by Γ = σz . Inaddition, it is invariant under the particle-hole symmetry P fkP−1 = iσxµy f

†−k (modulo a small basis rotation gauging away

the θ dependence). This means that we can choose the wavefunctions for the flat bands to be eigenfunctions of the sublatticeoperator σz i.e. completely sublattice polarized [50]. The wavefunctions can then be labelled by their sublattice index σ =A/B.This means that the sublattice off-diagonal components of the form factor vanishes, i.e. λAB;τ,G(k,k′) = λBA;τ,G(k,k′) = 0.Furthermore, the action of C2T is given by

C2T uA,τ,k = eiφA,τ,ku∗B,τ,k, C2T uB,τ,k = eiφB,τ,ku∗A,k (C1)

which implies that λAA;τ,q(k) = λBB;τ,q(k)∗. Finally, we can use PT symmetry to restrict the form factors further by notingthat it exchanges positive and negative energy eigenstates in opposite valleys and sublattices

PT uA,τ,k = eiηA,τ,kuB,−τ,k, PT uB,τ,k = e−iηB,τ,kuA,−τ,k (C2)

which implies that λA/B,τ,q(k) = λB/A,−τ,q(k). Summarizing these conditions, we can write

λσσ′;τ,q(k) = δσσ′Fq(k)eiΦq(k)στ (C3)

where Fq(k) = |λAA,+,q(k)| and φq(k) = arg λAA,+,q(k)

We now investigate the Hartree-Fock solutions by looking for the minima of the Hartree-Fock energy. The Hartree-Fockenergy is defined in terms of the order parameter

Pαβ(k) = 〈c†α(k)cβ(k)〉 (C4)

where α, β range over spin, valley and band indices. For an insulator or a semimetal, the number of filled states is k independentand equal 4. This means that the order parameter Pk is a projector satisfying

P (k)2 = P (k) = P †(k), trP (k) = 4 (C5)

The Hartree-Fock energy can then be written as (using properties of the chiral limit)

EHF = EH + EF , (C6)

EH =1

2A

∑G,k,k′

VG trP (k)Λ†G(k) trP (k′)ΛG(k′), (C7)

EF = − 1

2A

∑q,k

Vq trP (k)Λ†q(k)P (k + q)Λq(k) (C8)

11

where we defined the form factor matrix Λq(k) in terms of the combined index α = (s, τ, n) for spin, valley, and band as

Λq(k) = Fq(k)eiΦq(k)σzτz (C9)

The form factors decay in the separation q with a characteristic scale which is typically smaller than the size of the Brillouinzone. This means that the sums in the Hartree and Fock terms are dominated by the G = 0 term. For the Hartree term, thisequals 4V (0)N and is independent of the order parameter P (k). Thus, the Hartree term has little effect on the competitionbetween different phases and and will be neglected in the following.

We now consider possible types of symmetry-broken orders at charge neutrality. The order parameter P (k) can generally bewritten as

P (k) =1

2[1 +Q(k)], trQ(k) = 0 (C10)

Since the Fock term is the largest contribution to the mean-field energy, let us now neglect the Hartree term as well as the single-particle term h0. We note that 〈A,B〉 = trAB defines a positive definite inner product on the space of hermitian matrices.Using Cauchy-Schwarz inequality, we get

EF ≥ −1

2A

∑k,q

Vq

√trP (k) tr[Λ†q(k)P (k′)Λq(k)]2 = − 2

A

∑k,q

VqF2q (k) (C11)

This inequality is satisfied if and only if P (k + q) is parallel to Λq(k)P (k)Λ†q(k) for every k, q. This means

P (k + q) = eiΦq(k)σzτzP (k)e−iΦq(k)σzτz ⇒ Q(k + q) = eiΦq(k)σzτzQ(k)e−iΦq(k)σzτz (C12)

1. Intervalley coherent states

Let us now briefly discuss states which break U(1) valley symmetry. These states were argued to be energetically unfavorablein the context of twisted bilayer graphene with an aligned hBN substrate [37] and other Moire materials which lackC2 symmetry[17, 22]. Here, we will show that a similar argument fails [57] inC2-symmetric twisted bilayer graphene due to the extra particle-hole symmetry P which is exact in the chiral limit. A more detailed discussion of such phases is provided in Ref. [50]. To clarifythe importance of P , we will assume that the symmetry is not present in which case, the form factors in the two opposite valleysat the same momentum k are not related, i.e. λA/B,+,q(k) 6= λB/A,−,q(k). Furthermore, we will assume unbroken time-reversalsymmetry. The case of time-reversal symmetry breaking can be addressed similarly. The projector for an IVC state can be splitinto a diagonal and off-diagonal component in valley space

P (k) = Pd(k) + Po(k), Pd(k)2 + Po(k)2 = Pd(k), (C13)

In terms of valley resolved blocks of P (k), i.e.

P =

(P+ P12

P21 P−

), (C14)

the second condition can be written as

P 2+ + P12P21 = P+, P 2

− + P21P12 = P−. (C15)

12

Since the form factors are diagonal in valley space, the Fock energy can be written as a sum of a term with only diagonal partand one with only off-diagonal parts as

EF = − 1

2A

∑k,q

Vq tr[Pd(k)Λ†q(k)Pd(k + q)Λq(k) + Po(k)Λ†q(k)Po(k + q)Λq(k)]

≥ − 1

2A

∑k,q

Vq[|λA,+,q(k)|2√

trP+(k)2 trP+(k + q)2 + |λA,−,q(k)|2√

trP−(k)2 trP−(k + q)2

+ |λA,+,q(k)||λA,−,q(k)|√

trPo(k)2 trPo(k + q)2]

≥ − 1

2A

∑k,q

Vq[|λA,+,q(k)|2 trP+(k)2 + |λA,−,q(k)|2 trP−(k)2 + |λA,+,q(k)||λA,−,q(k)| trPo(k)2]

= − 1

2A

∑k,q

Vq[|λA,+,q(k)|2 tr(P+(k)− P12(k)P21(k)) + |λA,+,q(k)|2 tr(P−(k)− P21(k)P12(k))

+ 2|λA,+,q(k)||λA,−,q(k)| tr(P12(k)P21(k))]

= − 1

A

∑k,q

Vq(|λA,+,q(k)|2 + |λA,−,q(k)|2) +1

4A

∑k,q

Vq(|λA,+,q(k)| − |λA,−,q(k)|)2 trPo(k)2, (C16)

Here, we used the Cauchy-Shwarz inequality to go from the first to the second line. We then used the geometric-arithmetic meaninequality

√xy ≤ x+y

2 to go from the second to the third. We now see that whenever the second term in the last line is non-zero,we would conclude that the Fock energy for the IVC is larger than the energy bound for the valley polarized or unpolarizedphases by an amount which is proportional to the valley off-diagonal part of the order parameter. This energy difference in (C16)is generally not expected to be small unless there is a symmetry relating the same mometa in the two valleys (or equivalentlya symmetry relating opposite momenta in the same valley). This is precisely the reason why the existence of particle-hole Psymmetry forces the second term to vanish which makes the bound (C16), while correct, inconclusive to rule out IVC states.

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[41] Hoi Chun Po, Haruki Watanabe, and Ashvin Vishwanath,“Fragile topology and wannier obstructions,” Phys. Rev. Lett.121, 126402 (2018).

[42] For semimetals, this is true everywhere except for the gaplesspoints where the projector is not defined.

[43] Youngjoon Choi, Jeannette Kemmer, Yang Peng, Alex Thom-son, Harpreet Arora, Robert Polski, Yiran Zhang, Hechen Ren,Jason Alicea, Gil Refael, Felix von Oppen, Kenji Watanabe,Takashi Taniguchi, and Stevan Nadj-Perge, “Electronic corre-lations in twisted bilayer graphene near the magic angle,” Na-ture Physics 15, 1174–1180 (2019).

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[48] Biao Lian, Fang Xie, and B Andrei Bernevig, “The landau level

14

of fragile topology,” arXiv preprint arXiv:1811.11786 (2018).[49] In general, the total second Stiefel-Whitney number of two

isolated bands is not equal to the sum of the second Stiefel-Whitney numbers separately for the two bands; there is an ad-ditional contribution from the first Stiefel-Whitney classes. Inthe special case we considered here, the C3 symmetry of thetwo-band subspace guarantees this additional term vanishes.

[50] Nick Bultinck, Eslam Khalaf, Shang Liu, Shubhayu Chatter-jee, Ashvin Vishwanath, and Michael P. Zaletel, “Ground stateand hidden symmetry of magic angle graphene at even integerfilling,” arXiv preprint arXiv:1911.xxxxx (2019).

[51] We thank A. Thomson for correspondence on this point.[52] Grigory Tarnopolsky, Alex Jura Kruchkov, and Ashvin Vish-

wanath, “Origin of magic angles in twisted bilayer graphene,”Phys. Rev. Lett. 122, 106405 (2019).

[53] Alex Thomson and Jason Alicea, “Recovery of massless Diracfermions at charge neutrality in strongly interacting twisted bi-layer graphene with disorder,” arXiv preprint arXiv:1910.11348(2019).

[54] Notice that alternatively, if we choose a periodic gauge, αmk

can be made continuous but the Berry connection will be sin-gular leading also to finite energy contribution.

[55] The BM Hamiltonian has an approximate particle-hole symme-try at small angles.

[56] Kasra Hejazi, Chunxiao Liu, Hassan Shapourian, Xiao Chen,and Leon Balents, “Multiple topological transitions in twistedbilayer graphene near the first magic angle,” Physical Review B99, 035111 (2019).

[57] We thank Mike Zaletel for discussions on this point.