Soft modes in magic angle twisted bilayer graphene
Eslam Khalaf, Nick Bultinck, Ashvin Vishwanath, Michael P. Zaletel
SSoft modes in magic angle twisted bilayer graphene
Eslam Khalaf, Nick Bultinck,
2, 3
Ashvin Vishwanath, and Michael P. Zaletel Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Department of Physics, University of California, Berkeley, CA 94720, USA Department of Physics, Ghent university, 9000 Gent, Belgium (Dated: October 28, 2020)We present a systematic study of the low-energy collective modes for different insulating states at integerfillings in magic angle twisted bilayer graphene. In particular, we provide a simple counting rule for the totalnumber of soft modes, and analyze their energies and symmetry quantum numbers in detail. To study the softmode spectra, we employ time dependent Hartree-Fock whose results are reproduced analytically via an effec-tive low-energy sigma model description. We find two different types of low-energy modes - (i) approximateGoldstone modes associated with breaking an enlarged
U(4) × U(4) symmetry and, surprisingly, a secondbranch (ii) of nematic modes with non-zero angular momentum under three-fold rotation. The modes of type(i) include true gapless Goldstone modes associated with exact continuous symmetries in addition to gapped”pseudo-Goldstone” modes associated with approximate symmetries. While the modes of type (ii) are alwaysgapped, we show that their gap depends sensitively on the distribution of Berry curvature, decreasing as theBerry curvature grows more concentrated. For realistic parameter values, the gapped soft modes of both typesare found to have comparable gaps of only a few meV, and lie completely inside the mean-field bandgap. Theentire set of soft modes emerge as Goldstone modes of a different idealized model in which Berry flux is limitedto a solenoid, which enjoys an enlarged U(8) symmetry. In addition, it is shown that certain ground states admitnearly gapless modes despite the absence of a broken symmetry – an unusual property arising from the specialform of the anisotropy terms in the Hamiltonian. Furthermore, we separately discuss the number of Goldstonemodes for each symmetry-broken state, distinguishing the linearly vs quadratically dispersing modes at longwavelengths. Finally, we present a general analysis of the symmetry representations of the soft modes for allpossible insulating Slater determinant states at integer fillings that preserve translation symmetry, independentof the energetic details. The resulting soft mode degeneracies and symmetry quantum numbers provide a fin-gerprint of the different insulting states enabling their experimental identification from a measurement of theirsoft modes.
I. INTRODUCTION
When two graphene sheets are placed on top of each otherwith a small in-plane rotational mismatch close to the so-called magic angle θ M ∼ . ◦ , the dispersion of electronsmoving through the resulting superlattice is strongly reducedand Coulomb interaction effects become important. In partic-ular, when the Fermi level is inside the flat bands, a varietyof interesting states such as correlated insulators [1–6], su-perconductors [2–4, 6–9], orbital ferromagnets and quantumanomalous Hall states [3, 10–16] have been observed. Fur-thermore, tunneling spectroscopy and compressibility mea-surements have revealed that the density of states is signifi-cantly reconstructed by the Coulomb interaction not only atthe integer fillings where correlated insulators are observed,but for almost all fillings inside the flat bands [17–23]. In-terestingly, the physics of twisted bilayer graphene (TBG) isalso found to be very sensitive to external perturbations suchas substrate alignment or small out-of-plane magnetic fields,which can select different ground states and thus completelychange the phase diagram [9–14, 16]. This suggests that TBGis characterized by a small intrinsic energy scale by whichthe different interaction-driven orders at low temperatures areseparated.Previous studies of TBG [18, 24–26] have indeed identifieda large family of closely competing symmetry-broken statesat the different integer fillings. Combining different meth-ods including an exactly soluble parent Hamiltonian, effec-tive sigma model field theory and Hartree Fock calculations, Ref. [27] systematically investigated these competing orders.For integer fillings, a large manifold of low-energy insulatingstates associated with an approximate U(4) × U(4) symmetrywas identified (see also Ref. [28]). For realistic model param-eters, the states in this manifold are split by only a few meV.This hints at the existence of many low-energy bosonic modescorresponding to rotations within the
U(4) × U(4) mani-fold. An explicit non-linear sigma model describing these softmodes was derived by the authors in Ref. [29].In addition to the insulators, numerical studies [18, 24,25, 27] have also identified a C -symmetry breaking nematicsemimetal as one of the competitive states. This suggests thatnematic fluctuations associated with this semimetal will alsobe important for understanding the physics of TBG. Furtherevidence for the importance of nematic fluctuations was pro-vided by recent experiments [5] indicating strong nematicityin both the insulating and superconducting phases close to ν = − , as well as by the observed C symmetry breakingin scanning tunneling spectroscopy measurements [19, 20].In addition, several theoretical works proposed a central rolefor nematic fluctuations in mediating superconducting pair-ing [30–32]. However, the origin of these nematic modes re-mained enigmatic so far. In addition, it was unclear how thesetwo fundamentally different types of collective modes – thoseassociated with the spin-valley flavor symmetry and those ne-matic modes associated with spatial rotation symmetry – canbe unified in a single framework.In this work, we provide such a unifying theoretical frame-work to systematically investigate all possible soft modes in a r X i v : . [ c ond - m a t . s t r- e l ] O c t TBG (and broadly in other Moire systems). We show thatthe soft modes in TBG indeed fall under two categories: (i)approximate Goldstone modes associated with
U(4) × U(4) symmetry breaking and (ii) nematic modes associated withrotational symmetry breaking (possibly combined with flavorsymmetry breaking). We study two different aspects of thesesoft modes. First, we examine their energy spectrum and itsdependence on the different parameters such as twist angleand interaction strength. Second, we study their transforma-tion properties and quantum numbers under different symme-tries. The quantum numbers of the soft modes are decou-pled from the details of their energetics, and provide infor-mation about the symmetry-breaking orders which can be di-rectly accessed experimentally. These quantum numbers alsohave the distinct advantage of being universal, in the sensethat they do not depend on the details of the theoretical modelused. Our results serve to unify and elucidate several findingsfrom theory [27, 28, 30–35], numerics [24, 25, 27] and exper-iment [5, 21, 23, 36, 37] which all point to the existence of alarge number of soft modes that play an important role in thephysics of TBG.The results of our study of the soft modes can be sum-marized as follows. In accordance with the above discus-sion, we find that there are two different types of soft modes.First, there are the approximate Goldstone modes which cor-respond to broken-symmetry generators of the approximate
U(4) × U(4) symmetry identified in Ref. [27]. These in-clude true gapless Goldstone modes associated with brokenexact symmetries as well as gapped pseudo-goldstone modesassociated approximate symmetry generators. The mass of thelatter increases as a function of κ = w /w , the ratio of thesublattice-diagonal inter-layer hopping over the sublattice off-diagonal inter-layer hopping, as this ratio governs the strengthof the U(4) × U(4) -breaking anisotropies in the Hamiltonian[27]. The second type of soft modes are the nematic modes,which carry a non-trivial quantum number associated withthree-fold rotations. As we will see below, the mass of the ne-matic modes decreases as a function of κ , which follows fromthe concentration of the Berry curvature of the flat bands nearthe Gamma point in the mini-Brillouin zone. Interestingly,the realistic value of κ is such that both the pseudo-Goldstoneand the nematic modes have a small mass on the order of − meV.We also introduce a simple and intuitive counting rule forobtaining the number of soft modes on top of a particularsymmetry-breaking state which is summarized in Figs. 1 and2. Our counting rule shows that the total number of soft modesdepends only on the total filling, whereas the relative num-ber of approximate Goldstone and nematic modes depends inaddition on the Chern number of the insulating ground state.Using the results of Refs. [38–42], we also discuss how to sep-arately count the number of true Goldstone modes with linearand quadratic dispersion at long wavelengths. The countingrule and energetics of the soft modes are substantiated viatime-dependent Hartree-Fock (TDHF), which we use to ob-tain the complete soft-mode spectrum for the inter-valley co-herent insulating ground states at charge neutrality ν = 0 andhalf-filling ν = − (cf. Fig. 3). In both cases, we find that the TDHF spectrum produces the expected number of soft modes,and their energies are in good agreement with the anticipatedmass scales.Next, we derive an effective field theory containing only afew parameters which reproduces the soft mode spectrum atlong wavelengths, and which allows us to consider the effectsof different perturbations, such as for example the inter-valleyHund’s coupling. The parameters in the field theory can befixed by input from Hartree-Fock – in particular, by calcu-lating the energy splittings between the different symmetry-breaking orders.In the last section, we switch to a more general symmetryanalysis of the soft modes. We provide a general procedureto extract their symmetry transformation properties which isindependent of the model details. The results for a selectionof experimentally relevant states at different fillings are sum-marized in Tables I, II, and III. II. REVIEW
Let us first recall some basic facts about TBG. It consists oftwo graphene layers twisted relative to each other by a smallangle θ which generates a long wavelength Moir´e potentialon the scale a M ∼ a/θ ∼
10 nm. Since the Moir´e energyscale (cid:126) v F /a M ∼
100 meV is much smaller than the graphenebandwidth, a continuum approximation for the graphene dis-persion can be employed where valleys are taken to be decou-pled. Combined with the weakness of spin-orbit coupling ingraphene, this leads to an effective
U(2) × U(2) correspond-ing to independent
U(1) charge and
SU(2) spin conservationin each valley. As a result, each electron can be labelled bya spin s = ↑ , ↓ and a valley τ = K, K (cid:48) indices. At angles θ ≈ o , a pair of flat bands appears at neutrality for each ofthe four spin-valley flavor which are connected together bytwo Dirac points. This leads to 8 bands in total describing thelow energy physics. A. Band topology and sublattice basis
The connection between the two flat bands within each fla-vor is protected by a combination of spinless time-reversalsymemtry T and two-fold rotation symmetry C which leavethe valley index invariant and ensures the existence of twoDirac points. Three-fold rotation symmetry C further pinsthese Dirac points to the Moire K and K (cid:48) points. The non-trivial topology of the nearly flat bands which is protected by C T symmetry can be inferred in different ways. For a sin-gle spin and valley, the two Dirac points have the same chi-rality - indicating an obstruction to constructing symmetricWannier functions for the two bands [43, 44] . The C T pro-tected band topology is also captured by the so-called Stiefel-Whitney invariant [45–48], which leads to a Wannier obstruc-tion and fragile topology when realized in a two band model.This topology is made explicit as follows [27, 49, 50]. In Ref.[49] it was noticed that in the chiral limit, the flat bands sep-arate into sublattice polarized bands with quantum Hall like FIG. 1.
Schematic illustration of the soft modes in the spinless model : The spinless model consists of four bands in all:
K, A and K (cid:48) , B with Chern number +1 and K, B and K (cid:48) , A with Chern number − . The approximate Goldstone (solid red) and nematic (dashed red) modesare shown for the C = 2 insulator (anomalous quantum Hall), C = 0 insulators (valley polarized, valley Hall or inter-valley coherent) at ν = 0 as well as the ν = 1 insulators which always have | C | = 1 . The total number of modes is − ν with − ν + C nematic modes and − ν − C approximate Goldstone modes.FIG. 2. Soft mode count in MATBG (spinful model) : A schematic illustration of the count of nematic and approximate Goldstone modes forthe different insulating states (including Chern insulators potentially stabilized by an orbital magnetic field) at different filling ν . The insulatingregime in the magnetic field versus filling phase diagram reflects the Chern number. The total soft mode count depends only on the total fillingand is given by − ν . These are split into − ν − C approximate Goldstone modes (solid) and − ν + C nematic modes (dashed) asshown in the figure for the different Chern states. wavefunctions. The flat bands can therefore be viewed as apair of opposite Chern bands with C = ± living on oppo-site sublattices which holds even away from the chiral limit.This Chern basis served as the starting point for investigatinginteraction effects in Ref. [27], following which we will la- bel the bands in this new basis by the sublattice σ = A, B where they have the largest weight. This results in the pic-ture of Fig. 4 [27] consisting of four C = +1 Chern bandsand four C = − with approximate U(4) rotation symmetryin each sector. Note that, in this basis, band dispersion is not E ( m e V ) = 0 k x ( L M ) E ( m e V ) = 2 FIG. 3.
Time-dependent Hartree-Fock (TDHF) spectra : TDHFspectra of the insulating K-IVC states at neutrality ( ν = 0 ), andhalf-filling ( ν = − ) along a BZ cut through x -direction, connect-ing Γ and M points, in units of π/ √ L M , with L M the moir´elattice constant. Goldstone modes are shown in red, while gappedpseudo-Goldstone and nematic modes are shown in blue. For com-parison, we note that the mean-field band gaps at ν = 0 and ν = − are respectively given by and meV, such that all soft modesare well inside these gaps. At ν = 0 , all modes are fourfold de-generate leading to 16 modes in total. At ν = − , there are 12modes including 2 quadratically dispersing and one linearly dispers-ing Goldstone mode in addition to 9 gapped modes with the degen-eracy pattern − − − . The spectra were obtained using an ( N x , N y ) = (18 , grid for ν = 0 and ( N x , N y ) = (18 , at ν = − using a dielectric constant (cid:15) r = 12 . , gate distance d s = 40 nm and κ = 0 . at θ = 1 . o . neglected. Instead, it appears as momentum dependent tun-neling connecting C T related pairs of opposite Chern bandswhich breaks the approximate U(4) × U(4) symmetry downto
U(4) .An important parameter in describing the low energyphysics is the ratio between w , the interlayer tunnelingamplitude within the same sublattice (AA/BB) to w , theinterlayer tunneling amplitude between opposite sublattices(AB/BA) which we denote by κ = w w . This parameter con-trols the amount of sublattice polarization ranging from per-fect sublattice polarization in the limit κ = 0 [49] to a fi- nite but small sublattice polarization for the realistic value κ ≈ . − . [51]. A non-zero value of κ breaks the approx-imate U(4) symmetry discussed above down to the physical
U(2) × U(2) . B. Ground State at ν = 0 , ± , ± , ± The hierarchy of approximate symmetries identified inRef. [27] is captured by a simple energy expression providedin Ref. [29]: E [ Q ] = ρ ∇ Q ) + J Qσ x ) − λ Qσ x τ z ) Q = 1 , [ Q, σ z τ z ] = 0 , tr Q = 2 ν (1)where Q is an × matrix describing the filling of the 8 Chernbands. The eigenvalues of Q are +1 and − denoting full andempty bands, respectively. For an insulator at filling ν , Q sat-isfies tr Q = 2 ν . In addition, Q commutes with the Chernmatrix σ z τ z such that the Chern number C = tr Qσ z τ z iswell defined. It is easy to read off the ground state at differ-ent fillings from (1). At ν = 0 , all three terms are minimizedby the so-called Krammers intervalley coherent (K-IVC) or-der given by Q = σ y τ x,y s or any state related to it via thephysical U(2) × U(2) symmetry [27]. At ν = ± , all termsare minimized by a spin-polarized version of the K-IVC order Q = σ y τ x,y P ↑ + σ τ P ↓ (here P ↑ / ↓ denotes the projector onthe ↑ / ↓ sector) or any state related to it via U(2) × U(2) as explained in Ref. [27]. At odd filling, all insulating groundstates have a finite Chern number [27]. At ν = ± , thereare two degenerate manifolds of states with Chern numbers ± or ± including valley symmetry preserving (valley po-larized) states or valley symmetry breaking states (IVC). Allthese states are degenerate on the level of (1) and their energycompetition is likely determined by smaller anisotropies. Fi-nally, at ν = ± , the ground states have Chern number ± which can be either VP or IVC. III. PHYSICAL PICTURE AND SOFT MODE COUNT
We begin by providing a simple physical picture for the softmodes of MATBG yielding a simple rule for their number,symmetry properties and energetics. This will be substanti-ated by more rigorous analytical and numerical discussionslater in the manuscript.
A. Soft modes in the insulating phases
The soft modes can be understood in a simple manner us-ing the picture of Fig. 4. In the limit of strong interactions,the ground state at any integer filling ν (measure relative tocharge neutrality) is obtained by completely filling ν of theSymmetry Approximation Number ofgapless modesSolenoid fluxBerry phase α → Flat band J → Sublatticepolarized λ → (cid:51) (cid:51) (cid:51) − ν U(4) × U(4) (cid:55) (cid:51) (cid:51) − ν − C U(4) (cid:55) (cid:55) (cid:51) − ν − C U(2) × U(2) (cid:55) (cid:55) (cid:55)
Depends on the state
FIG. 4.
Spinful model and hierarchy of symmetries : Schematic illustration of the spinful model in the sublattice basis with 4 C = +1 and4 C = − bands (upper pabel) together with a table summarizing the different approximate symmetries, the corresponding approximation,and the number of ’gapless’ modes arising from each approximation (lower panel). In the limit when the Berry curvature is concentrated ata single momentum point and can be implemented via a solenoid flux in momentum space, the distinction between the ± Chern sectors islost and we have a full
U(8) symmetry. Taking the finite Chern flux density into account but neglecting band dispersion and assuming perfectsublattice polarization leads to a
U(4) × U(4) symmetry [27] which is broken down to
U(4) in the presence of dispersion and further down tothe physical
U(2) × U(2) by including sublattice off-diagonal interaction matrix elements. The parameter α , J , and λ parametrize the strengthof explicit symmetry breaking and are defined in Eq. 46 and plotted in Fig. 6. Each approximate symmetry is associated with a number of softmodes specified in the last column. describing a generalized Chern ferromagnet. Thesoft modes are simply given by the long wavelength limit ofparticle-hole excitations where the particle and the hole livewithin the same Chern sector: ˆ φ γ,γ ; αβ ; q = (cid:88) k φ αβγ,γ, q ( k ) c † γ,α, k c γ,β, k + q , γ = ± (2)where c ± ,α, k denotes the annihilation operator for a flat bandstate in the ± Chern sector labelled by an index α going overthe 4 bands in a given Chern sector and φ α,βγ,γ (cid:48) , q ( k ) are somefunctions describing the momentum space profile of the softmodes. These modes correspond to generators of rotations inthe spin-valley flavor space and thus carry non-trivial spin orvalley quantum numbers. Furthermore, they are gapless gold-stone modes in the limit of unbroken U(4) symmetry withineach Chern sector. Realistic anisotropies which break the
U(4) × U(4) symmetry down to the physical
U(2) × U(2) induce a gap of 1-3 meV depending on the ground state. Inthe following, we will refer to these modes collectively as ap-proximade goldstone (AG) modes. These includes both theperfectly gapless goldstone (G) modes as well as the gappedsoft modes due to the anisotropies which we will refer to aspseudo-goldstone (PG) modes.To determine the number of these modes, we note that aninsulating ground state is described by filling n ± bands in the Note that the approximate
U(4) symmetry allows also for filling arbitrarylinear combinations of bands within the same Chern sector. ± Chern sector such that the total filling ν is n + + n − andthe total Chern number C is n + − n − . Thus, for a given state,the indices α and β in ˆ φ αβ go over the empty and filled states,respectively, yielding a total of n + (4 − n + )+ n − (4 − n − ) softmodes. As an example, consider the possible insulating statesat ν = 0 . If the total Chern number is zero, this correspondsto filling two bands within each sector leading to 4 soft modesper sector; 8 in total. Another example is the zero Chern insu-lating states at ν = 2 with 3 filled bands in each sector leadingto 3 soft modes per sector; 6 in total.What about particle-hole excitations between differentChern sectors? These can be defined very similarly to Eq. 2: ˆ φ γ, − γ ; αβ ; q = (cid:88) k φ αβγ, − γ, q ( k ) c † γ,α, k c − γ,β, k + q , γ = ± (3)To understand their energetics, it is useful to perform aparticle-hole transformation in one of the Chern sectors [52]by defining f + , k = c + , k , f − , k = c †− , − k (4)where we have omitted the U(4) index within each Chern sec-tor. This transformation flips the Chern number in the − sectorso that f + and f − are both living in a +1 Chern band. In ad-dition, it maps a inter-Chern particle-hole excitation c † + , k c − , k into a Cooper pair ∆ k = f † + , k f †− , − k . The energetics of inter-Chern particle-hole excitations can be understood in the trans-formed basis as the energetics of a Cooper pair in a Chernband which is characterized by two main features. First, dueto the band topology, the phase of ∆ k winds by π aroundthe Brillouin zone (BZ) which means that ∆ k has at least two π vortices . Second, the Berry curvature of the bands actslike a magnetic field in momentum space. The energy of suchCooper pair can be estimated as [25, 52] E ∆ ∼ E C (cid:90) d k | ( ∇ k − iA k )∆ k | (5)where E C is of the order of the Coulomb energy scale ∼ meV and A k is the Berry connection. There are two differ-ent regimes for the behavior of this energy. In the limit ofrelatively uniform Berry curvature, the vortices of ∆ k are un-screened yielding a large energy contribution. On the otherhand, if the Berry curvature is strongly concentrated at sin-gle point in the Brillouin zone, we can evade the large energypenalty by placing the two vortices at this point where theyget almost completely screened.The question of energetics of the inter-Chern particle-holeexcitations is then determined by the distribution of the Berrycurvature in momentum space. As shown in Refs. [25, 53],the Berry curvature is relatively uniform in the chiral limit κ = 0 where the wavefunctions resemble the quantum Hallwavefunctions but become more concentrated at the Γ pointwith increasing κ . At the physical value of κ ≈ . − . ,the Berry curvature is very sharply peaked at Γ leading to arelatively small energy for the inter-Chern modes of the orderof a few meVs. The relatively low energy of these modeswill be explicitly verified by the time-dependent Hartree-Fockstudy in Sec. IV A.One important property of the inter-Chern modes is thatthey carry non-zero angular momentum under C . This can beseen by recalling that the action of C in the valley-sublatticespace as C = e πi σ z τ z [43, 48, 54]. Thus, electrons in the ± Chern sector carry a C angular momentum of e ± πi . Asa result, the intra-Chern pseudogoldstone modes (2) alwayscarry zero angular momentum under C whereas the inter-Chern modes (3) always carry a non-zero angular momentumat q = 0 . This can also be seen from the discussion of the en-ergetics where that wavefunction ∆( k ) has a winding of ± π around the Γ point and as a result transforms as e ± πi/ un-der C . Thus, we will henceforth refer to the inter-Chern softmodes as the nematic (N) modes. We note that the conden-sation of these modes yields the nematic semimetal identifiedin previous numerical studies [18, 24, 25, 52] where the twogapless Dirac cones migrate to the vicinity of the Γ point.The total number of soft modes can then be understoodas follows: denoting the number of soft modes between theChern sector γ = ± and the Chern sector γ (cid:48) = ± by N γ,γ (cid:48) , itis easy to see that N γ,γ (cid:48) = n γ (4 − n γ (cid:48) ) (6) Here, we assume a smooth gauge choice Note that there is an ambiguity in defining the phase in intervalley ex-citations since the U V (1) valley charge may in principle transform non-trivially under C . This can be resolved by taking C and U V (1) to com-mute which is assumed throughout this paper Then, the number of approximate goldstone (intra-Chern)modes relating the different insulating states is N ++ + N −− whereas the number of nematic (inter-Chern) modes is N + − + N − + leading to a total of n (8 − n ) = 16 − ν soft modes. It isworth emphasizing that although the number of nematic andAG modes depends on the given state, the total number of softmodes depends only on the total filling.As an example, consider the spinless limit where the prob-lem is reduced to a pair of bands in each Chern sector. In thiscase, the U(4) × U(4) symmetry of the ideal limit reduces to
U(2) × U(2) and the physical symmetry is
U(1) × U(1) cor-responding to total and valley charge conservation. At half-filling, there are several insulating low energy states obtainedby filling two out of the four bands. First, we can fill twobands in the same Chern sector leading to a Quantum anoma-lous Hall state. In this state, there are no AG modes since the
U(2) × U(2) symmetry is not spontaneously broken, but thereare four nematic modes illustrated in Fig. 1. Second, we canfill one band in each sector. This breaks the
U(2) symme-try in each sector to
U(1) × U(1) . As a result, there are twoAG modes in addition to two nematic modes. In the presenceof physical anisotropies which break the symmetries down tothe valley
U(1) (charge
U(1) is always assumed), some of theAG modes acquire a gap. If the filled bands are valley eigen-states, leading to a valley polarized or valley Hall state, thetwo AG modes are are gapped PG modes since the physicalvalley
U(1) symmetry is unbroken. On the other hand, if thefilled band is not a valley eigenstate, e.g. a superposition ofbands in K and K (cid:48) valley, this results in an inter-valley coher-ent state which spontaneously breaks U(1) valley symmetry.As a result, one of the two AG modes will be a gapless gold-stone mode whereas the other will be a gapped PG mode.The same discussion can be applied for the spinful case asshown in Fig. 2. The total number of soft modes depends onlyon the filling ν and is given by − ν . The number of intra-Chern approximate goldstone (AG) and inter-Chern nematicmodes depends in addition on the Chern number C but not onany other detail of the state and is given by n AG = 16 − ν − C , n N = 16 − ν + C (7)In summary, this section presented a unified picture for un-derstanding the number and energetics of the soft modes inany insulating symmetry breaking state which will be sub-stantiated by more rigorous numerical and analytical argu-ments in the following sections. Broadly speaking, we cansplit the soft modes into intra-Chern approximate goldstone(AG) and inter-Chern nematic modes. The energy of the for-mer is controlled by anisotropies in the manifold of insulat-ing states which break the approximate U(4) × U(4) down to
U(2) × U(2) , while the energy of the latter is controlled bythe distribution of the Berry curvature in momentum space. Inthe continuum model of TBG close to the magic angle, twotypes of modes happen to have comparable energies for thephysically realistic value κ ≈ . . ν C State Q Symmetry Type Irrep characters d χ ( C ) χ ( e iϕη z ) χ ( C ) χ ( M y ) χ ( M x ) χ ( η z M y ) γ η z { C , U V (1) , M y , C T }
PG 1 1 e iϕ - 1 - − e iϕ - − - 1N 2 − e iϕ - 0 - 0VH γ z η z { C , U V (1) , M x , T }
PG 2 2 ϕ - - 0 -N 2 − - - 0 -K-IVC γ z η x,y { C , C , iη z M y , iη z T }
G 1 1 - − - - 1PG 1 1 - − - - − N 2 − - − - - 0 T -IVC γ η x,y { C , C , M y , T }
G 1 1 - − − - 1PG 1 1 - − - − N 2 − - 2 0 0 02 QAH γ z η { C , U V (1) , C , M y T }
N 1 e − πi/ - - -1 e − πi/ − - - -2 e − πi/ ϕ - - -1 1 VP-QAH P + η + P − η z { C , U V (1) , M y T }
PG* 1 1 e iϕ - - - -N 1 e − πi/ e − πi/ e iϕ - - - -IVC-QAH P + η + P − η x,y { C , C , M y T }
G 1 1 - − - - -N 1 e − πi/ - − - - -1 e − πi/ - 1 - - -TABLE I. Symmetry representations for the spinless model : Detailed symmetry properties for the bosonic soft modes in the spinless forall possible insulators at integer fillings. The labels ’VP’, ’VH’, ’IVC’, and ’QAH’ denote valley-polarized, valley Hall, intervalley coherentand quantum anomalous Hall states, respectively. The projector P ± projects onto the ± Chern sector and is defined as P ↑ / ↓ = ± γ z . Thesoft modes are divided into true goldstone modes (G) corresponding to continuous symmetry breaking, gapped pseudo-goldstone modes (PG)which only become gapless in the absence of anisotropies (i.e. in the U(4) × U(4) limit) and nematic modes (N) which are gapped andtransform non-trivially under C . PG* denote pseudo-goldstone modes which do not correspond to breaking a continuous physical symmetryyet have a very small gap due to the absence of some symmetry allowed terms in the theory. The irrep character corresponding to a symmetryg is denoted by χ ( g ) with the representation dimension denoted by d = χ ( ) . IV. SOFT MODE SPECTRUMA. Time-dependent Hartree-Fock
In this section we discuss how the soft mode spectrumfor magic angle graphene can be obtained at the mean-fieldlevel via the time-dependent Hartree-Fock (TDHF) formal-ism. Here, we keep the discussion general and we will notrely on any exact or approximate symmetries of the Hamilto-nian. We simply outline how to obtain the TDHF equation,and solve it numerically. The interpretation of the soft modespectrum in terms of the approximate symmetries and the con-nection to the non-linear sigma model will be discussed in thenext section. We also note that TDHF has previously beenused to study collective excitations of quantum anomalousHall states in moir´e systems in Refs. [55–57].
1. Formalism
We start from the following interacting continuum Hamil-tonian for magic angle graphene in the BM band basis: ˆ H = (cid:88) k c † k h ( k ) c k + 12 A (cid:88) q V q δρ q δρ − q (8)Here and throughout, we employ a matrix notation where c k denotes a vector of annihilation operators whose componentsare labelled by the flavor and the band index. In writing thisHamiltonian, we implicitly assume that we have projected thefull Hamiltonian into the subspace where most or all of theremote BM valence bands are completely filled, and most orall of the remote BM conduction bands are completely empty.For our numerics, we project out all but two remote valenceand conduction bands per spin and valley. For the analyti-cal discussion later on, we project out all remote valence andconduction bands.The single-particle term h ( k ) in Eq. (8) contains not onlythe BM band energies, but also contributions from both the re-mote valence bands which have been projected out, and froma subtraction term to avoid double counting of certain inter-action effects [24, 27]. The second term in Eq. (8) corre-sponds to the Coulomb interaction, for which we use a dual-gate screened potential V q = tanh( d s q ) / (cid:15) (cid:15) r q with dielec-tric constant (cid:15) r and gate distance d s . The Fourier componentsof the projected charge density operator are given by δρ q = (cid:88) k c † k Λ q ( k ) c k + q , (9)where Λ q ( k ) is the matrix of form factors in the band andflavor index defined in terms of overlaps between the cell-periodic parts of the BM Bloch states: [Λ q ( k )] α,β = (cid:104) u α, k | u β, k + q (cid:105) (10)with α , β ranging over flavor and band indices. The start-ing point of TDHF is a solution of the Hartree-Fock self-consistency equation, described by the following correlationmatrix P αβ ( k ) = (cid:104) c † β, k c α, k (cid:105) , (11)which projects onto the occupied states in the mean-field bandspectrum. If we write the Hartree-Fock Hamiltonian con-structed from P ( k ) as H SC { P } ( k ) (see App. A for a defi-nition of this Hamiltonian), then self-consistency means thatthe following equation should be true: [ P ( k ) , H SC { P } ( k )] = 0 (12)The self-consistency condition has an intuitive physical inter-pretation, which becomes clear after using Eq. (12) to showthat the following equality holds: (cid:88) λγ P λα ( k ) P ⊥ βγ ( k ) (cid:104) c † λ, k c γ, k ˆ H (cid:105) HF = 0 , (13)where P ⊥ ( k ) = − P ( k ) , and (cid:104)·(cid:105) HF means that we takethe expectation value with respect to the Hartree-Fock groundstate Slater determinant with correlation matrix P ( k ) . Eq.(13) implies that self-consistency is equivalent to the condi-tion that ˆ H should create at least two particle-hole excitationswhen acting on the ground state Slater determinant.Let us now define the following bosonic operators: ˆ φ q = (cid:88) k c † k φ q ( k ) c k + q (14)where φ q ( k ) is a matrix in the flavor and band indices. The goal of TDHF is to find those operators ˆ φ q which (1)are a superposition of creation and annihilation operators ofparticle-hole excitations of the mean-field band spectrum,and (2) which correspond to eigenmodes satisfying i∂ t ˆ φ q =[ ˆ H, ˆ φ q ] = ω q ˆ φ q at the mean-field level. In general, the com-mutator [ ˆ H, ˆ φ q ] will contain both terms with two and fourfermion operators. So working at the mean-field level meansin practice that we map the four-fermion terms in the com-mutator to two-fermion terms by performing all partial Wickcontractions with P ( k ) which leave precisely two fermionoperators uncontracted. As discussed in detail in App. A,the resulting eigenvalue problem for obtaining the soft modespectrum ω q is equivalent to diagonalizing a quadratic bosonHamiltonian. The fact that this mean-field boson Hamiltonianis quadratic is consistent with the fact that ˆ H creates at leasttwo particle-hole excitations.
2. Results
In Fig. 3 we show the TDHF spectra for the Kramers inter-valley coherent (K-IVC) insulators at charge neutrality, and ata flat band filling of two electrons per moir´e unit cell. The K-IVC state was introduced in Ref. [27], and we will discuss itsproperties in more detail below. For now, it suffices to mentionthat both at ν = 0 and ν = − the K-IVC state breaks the U V (1) symmetry, and that the K-IVC state at ν = − is aspin polarized version of the one at neutrality. The mean-fieldband gaps for the K-IVC states at ν = 0 and ν = − arerespectively given by and meV. From Fig. 3 it is clearthat all soft modes lie well below these band gaps.The TDHF spectrum at ν = 0 is shown in the top panel ofFig. 3. Note that the modes are exactly four-fold degenerateat every momentum point, such that there are 16 soft modes intotal, as predicted by the counting rule discussed in Sec. III.To understand the four-fold degeneracy, we assume withoutloss of generality that the K-IVC state at neutrality does notbreak the global spin rotation symmetry (we will come backto this point in more detail in Sec. V B). In this case, the softmodes are labeled by their spin quantum number, and everymode appears both as a spin singlet and a spin triplet, becausemicroscopically it consists of two spin- / fermion operators.For the most general interaction compatible with the symme-tries, this implies that the soft mode spectrum will consist of4 singlet modes and 4 three-fold generate triplet modes (seeSec. VI). However, for a density-density interaction we findthat the singlet and triplet modes are not split, resulting in afour-fold degeneracy. In particular, this implies that there arefour degenerate Goldstone modes, which are shown in red inthe top panel of Fig. 3. For the spin-singlet K-IVC state, theseGoldstone modes are associated with the broken U V (1) sym-metry (spin-singlet mode corresponding to generator τ z ), andwith the broken symmetry of opposite spin rotations in the dif-ferent valleys (spin-triplet mode corresponding to generators τ z s x , τ z s y and τ z s z ).Fig. 3 also shows an additional degeneracy at the Γ pointbetween the two upper branches of the ν = 0 soft mode spec-trum. These modes correspond to the nematic modes, andas we explain in Sec. VI, the additional degeneracy at Γ iscaused by the valley-diagonal mirror symmetry which inter-changes the modes with opposite C angular momenta. Wealso want to point out that beyond the four-fold degeneraciesassociated with the global spin rotation symmetry there are noadditional degeneracies at the M points, even though the twolower and upper soft mode branches are very close in energythere.For the K-IVC state at ν = − , we find 12 different softmodes, which again agrees with the counting rule discussedin Sec. III. Compared to the spectrum at neutrality, which hasfour linearly dispersing Goldstone modes, one of the main dif-ferences is that at ν = − there are three Goldstone modes,two of which are quadratically dispersing at small q , and onewhich has a linear dispersion. The linearly dispersing mode isagain associated with the broken U V (1) symmetry (generator τ z ), while the two quadratically dispersing modes are the re-sult of broken spin rotation symmetry (generators s x and s y ).At the Γ -point, the degeneracies of the gapped soft modes are − − − . We will explain the origin of these degeneraciesin Sec. VI. B. Flat band projection and soft mode energetics
In this section, we provide an analytical understanding forthe soft mode energetics by projecting the Hamiltonian (8)onto the flat bands and employing the approximate
U(4) × U(4) symmetry to understand its energetics. For the analytictreatment, we will find it more convenient to switch from theHamiltonian approach of the previous section to a Lagrangianapproach as explained below.We start by restricting ourselves to the 8 flat bands labelledby a sublattice σ = ± = A/B , valley τ = ± = K/K (cid:48) andspin s = ↑ , ↓ indices, with each band having Chern number στ . It is more convenient to define an alternative basis wherethe bands are labelled by a Chern index γ = ± and a pseu-dospin index η within each Chern sector [29]: γ x,y,z = ( σ x , σ y τ z , σ z τ z ) , η x,y,z = ( σ x τ x , σ x τ y , τ z ) (15)The projector P ( k ) defined in (11) is now an × matrix. Wewill find it convenient to define the matrix Q ( k ) as Q ( k ) =2 P ( k ) − , which can be written directly in terms of the band-projected creation/annihilation operators as Q αβ ( k ) = (cid:104) [ c † α, k , c β, k ] (cid:105) (16)Any Slater determinant state is completely characterized bythe Q matrix which satisfies Q ( k ) = 1 . This conditionmeans that the eigenvalues of Q ( k ) are ± at each k , suchthat there exists a basis where each state is either full (+1) orempty ( − . If we further restrict to insulating or semimetal-lic states at an integer filling ν , then the number of +1 and − eigenvalues is independent of k leading to the additional con-dition tr Q = 2 ν . In the following, we will additionally as-sume that Q is k -independent which is true in the U(4) × U(4) limit and holds to a good approximation in the realistic limit [27]. We will discuss later how this assumption can be liftedwhen considering the effective field theory.We can parametrize the fluctuations around a given Slaterdeterminant state | ψ Q (cid:105) described by the matrix Q by writing | ψ Q ( φ ) (cid:105) = e i (cid:80) q ˆ φ q | ψ Q (cid:105) , (17)where ˆ φ q is defined in (14) in terms of a matrix-valued func-tion φ q ( k ) which anticommutes with Q and acts in the spaceof flat bands. φ q ( k ) satisfies φ † q ( k ) = φ − q ( k + q ) and can beexpanded in terms of the − ν ) generators of U(8) whichanticommute with Q , which we denote by t µ , as φ q ( k ) = − ν ) (cid:88) µ =1 φ µ q ( k ) t µ , { t µ , Q } = 0 (18)where t † µ = t µ . The energy of the state | ψ Q ( φ ) (cid:105) is E ( Q, φ ) = (cid:104) ψ Q ( φ ) | ˆ H | ψ Q ( φ ) (cid:105) (19)where ˆ H is the flat-band projected Hamiltonian. Expand-ing E ( Q, φ ) in powers of φ , we find that the linear termvanishes if and only if Q is a solution to the Hartree-Fockself-consistency equation, which we will assume here. Thequadratic term gives the leading contribution in φE ( Q, φ ) = (cid:88) q , k , k (cid:48) ,µ,ν φ µ q ( k ) ∗ H µ,ν q ( k , k (cid:48) ) φ ν q ( k (cid:48) ) (20)We can expand φ µ q ( k ) in terms of eigenfunctions of H q as φ µ q ( k ) = (cid:88) n a n, q φ µn, q ( k ) , (21) (cid:88) k (cid:48) ,ν H µ,ν q ( k , k (cid:48) ) φ νn, q ( k (cid:48) ) = ε n, q φ µn, q ( k ) (22)Here, n and q are labels for the wavefunctions whereas µ and k are internal indices such that the wavefunction can be un-derstood as a vector in µ and k . Substituting (21) in (20) leadsto E ( Q, φ ) = (cid:88) q ,n ε n, q a ∗ n, q a n, q , (23)The quantum theory is obtained by promoting the fluctuations a n, q to be dynamical and defining the Lagrangian L = (cid:104) ψ ( φ ) | i ddt − ˆ H | ψ ( φ ) (cid:105) (24)The second term yields the energy E ( Q, φ ) (23), whereas thefirst yields (cid:104) ψ ( φ ) | i ddt | ψ ( φ ) (cid:105) = i (cid:88) q ,n,m ρ nm, q a ∗ n, q ∂ τ a m, q , (25)0 ρ nm, q = 12 (cid:88) µ,ν tr Qt µ t ν (cid:88) k [ φ µn, q ( k )] ∗ φ νm, q ( k ) (26)The matrix ρ satisfies ρ T − q = − ρ q and thus defines a sym-plectic structure that pairs up different modes as canonicallyconjugate variables. By going to Fourier space and writing theaction as S = (cid:90) dτ L = (cid:88) n,m, q ,ω a ∗ n, q ,ω M nm, q ,ω a m, q ,ω , (27) M nm, q ,ω = ρ nm, q ω − δ n,m ε n, q (28)The soft mode spectrum is obtained by taking the positive so-lutions ω = ω q of the equation det M q ,ω = 0 [58]. It is worthnoting that up to this point, we have not made any assumptionabout exact or approximate symmetries.Let us now consider the projection of the Hamiltonian (8)onto the flat bands. In this limit, the effect of the remote bandsis included only through the renormalization of the band dis-persion h ( k ) [24, 25, 52, 59]. Following Ref. [27], we canstart by considering the U(4) × U(4) symmetric limit wherethe form factor has the simple form Λ q ( k ) = F q ( k ) e i Φ q ( k ) γ z (29)As a result, the ground states of the Hamiltonian H are Slaterdeterminant states characterized by k -independent Q satisfy-ing [ Q, γ z ] = 0 [27].To make further progress, we switch to a different ba-sis for the generators of U(8) that makes the form of thesoft mode Hamiltonian as simple as possible. This basis,which we denote by { r γγ (cid:48) α } , is labelled by γ, γ (cid:48) = ± and α = 1 , . . . , N γ,γ (cid:48) . The generators { r γ,γ (cid:48) α } correspond toparticle-hole excitations from Chern sector γ to Chern sector γ (cid:48) . We note that the new basis of generators is not hermi-tian since the hermitian conjugate of r γ,γ (cid:48) α belongs to the set { r γ (cid:48) ,γα } . At the end of the calculation, we can transform backto the Hermitian basis { t µ } .As shown in Appendix B, the condition [ Q, γ z ] = 0 impliesthat H q is block diagonal in the (++ , + − , − + , −− ) spacewith blocks denoted by H γ,γ (cid:48) ; q . Furthermore, the generators r γ,γ (cid:48) α can be chosen such that H α,βγ,γ (cid:48) is proportional to δ α,β for all γ and γ (cid:48) (see Appendix B). This means that we canrelabel the soft mode wavefunctions by splitting the index n into ( γ n , γ (cid:48) n , α n , l n ) where γ n , γ (cid:48) n , and α n label the generatorcorresponding to the eignfunction φ n and l n is an integer l n ≥ labelling the set of eigenfunctions corresponding to the samegenerators such that the energy (cid:15) γ,γ (cid:48) ,α,l n ( q ) is an increasingfunction of l n . Due to the simple form of the Hamiltonian(appendix B), these wavefunctions have the form: φ γ ,γ ,βγ ,γ ,α,l, q ( k ) = δ γ ,γ δ γ ,γ δ α,β ψ γ ,γ ; l, q ( k ) (30)where ψ γ,γ (cid:48) ; l, q ( k ) is a scalar wavefunction (with no vectorindices) labelled only by the generator Chern sector γ, γ (cid:48) andan integer l for a given q . Eq. 30 means that φ γ ,γ ,α,l, q which is a vector in the index µ = ( γ , γ , β ) has only one non-vanishing component. As we will show later, the low en-ergy soft modes are obtained by restricting to the lowest lyingeigenfunctions l = 0 .The expressions for H γ,γ (cid:48) can be further simplified in thelimit q = 0 . To simplify the notation, we will drop the q dependence of the Hamiltonian H γ,γ (cid:48) ; q and the wavefunctions ψ γ,γ (cid:48) ; l, q whenever q = 0 . The Hamiltonian H γ,γ (cid:48) is given by: H γ,γ (cid:48) ( k , k (cid:48) ) = 1 A (cid:88) q V q F q ( k ) [ δ k , k (cid:48) − δ k (cid:48) , [ k + q ] e i ( γ − γ (cid:48) )Φ q ( k ) ] (31)where the sum over q extends over all momenta, and, [ k ] equals k modulo a reciprocal lattice vector and lies in the firstBrillouin zone. For the intra-Chern fluctuations, γ = γ (cid:48) , wecan see immediately that a constant function is a zero eigen-function of H γ,γ for γ = ± . This corresponds to the Gold-stone modes of the U(4) × U(4) symmetry breaking. Further-more, we can verify by an explicit calculation that all othereigenvalues of H γ,γ have a relatively large mass ∼ γ = − γ (cid:48) , the situation is different. Due to thephase factor in the second term, the eigenstates of H γ, − γ al-ways have a finite gap. This was shown in the supplementalmaterial of Ref. [27] and we will reproduce this argument be-low. For definiteness, let us focus on the + − sector. We beginby noting that the lowest energy state ∆( k ) = ψ + − ,l n =0 ( k ) is obtained by minimizing the expectation value: E ∆ = (cid:104) ∆ |H + − | ∆ (cid:105) = 1 AN (cid:88) q , k V q F q ( k ) × [∆( k ) − ∆( k )∆( k + q ) e i Φ q ( k ) ] (32)This can be further simplified by assuming the magnitude ofthe form factor decays relatively quickly with the relative mo-mentum q which enables us to expand the expression insidethe sum in q leading to E ∆ = 1 N (cid:88) k E C ( k ) | ( ∇ k − iA k )∆( k ) | (33) E C ( k ) = 1 A (cid:88) q q V q F q ( k ) (34)Here, we used Φ q ≈ q · A k + O ( q ) with A k denoting theBerry connection A k = − i (cid:104) u + , k |∇ k | u + , k (cid:105) where u + , k de-notes the wave-function for any of the bands within the + Chern sector (which are all equal due to
U(4) × U(4) sym-metry). If we further assume E C ( k ) depends weakly on k ,we can pull it out of the k sum and get an expression identi-cal to the energy of a Cooper pair in magnetic field as in Eqn.(5). This energy expression can be understood as the energyof a superconducting vortex if we identify the momentum k with the real coordinate, E C with the superfluid stiffness, andidentify ξ = (cid:104) Ω( k ) k (cid:105) with the area of the vortex corein momentum space (here Ω( k ) is the Berry curvature and (cid:104)·(cid:105) denote BZ average). As a result, we expect the smallest eigen-1 FIG. 5.
Gap of the inter-Chern nematic soft mode : Plot of E ∆ , thesmallest eigenvalue of H + − as defined in Eq. 32 (red) together withthe quantity ξ = π (cid:104) Ω( k ) k (cid:105) (blue) which measures the spread ofBerry curvature, as a function of κ = w /w . Here, Ω( k ) denotesthe Berry curvature and (cid:104)·(cid:105) denote the BZ average with π (cid:104) Ω( k ) (cid:105) =1 . E ∆ decreases with increasing κ as the Berry curvature becomesmore concentrated [25, 53]. Here, we used the dielectric constant (cid:15) = 12 . and gate distance d = 20 nm. values of H + − to decrease with decreasing the vortex area ξ as the Berry curvature becomes more concentrated. This isverified by a numerical calculation of the gap of H + − as afunction of κ , which controls the Berry curvature distribution,shown in Fig. 5. We can also verify that all other eigenstatesof H + − have a relatively large gap and can be integrated out.Thus, for every generator r γ,γ (cid:48) α , we can restrict ourselvesonly to the lowest lying state φ γ,γ (cid:48) ; α,l =0 by integrating out themassive modes with l > . The resulting low energy bosonicmodes can be divided into two categories as anticipated inSec. III: (i) approximate goldstone modes acting within thesame Chern sector which correspond to generators r γ,γα com-muting with γ z and (ii) nematic modes acting between Chernsectors which correspond to the generators r γ, − γα anticommut-ing with γ z .So far the analysis has been restricted to the U(4) × U(4) limit. To see what happens in the realistic limit, we need to in-clude the effect of the sublattice off-diagonal part of the formfactor and the dispersion which were discussed in detail inRef. [27]. The former induces an extra energy cost λ (cid:39) ∼ tr Q T h ( k ) φ q =0 ( k ) which is only non-vanishingfor the inter-Chern component of φ since h ( k ) acts predomi-nantly between Chern sectors, h ( k ) ≈ h x ( k ) γ x + h y ( k ) γ y .Expanding φ q =0 ( k ) into eigenmodes (Eq. 21) and integrat-ing out the massive ones l > leads to an energy contri-bution J (cid:39) h + H − − h − (cid:39) h ± ( k ) = h x ( k ) ± ih y ( k ) . This contribution favors spin and pseu-dospin antiferromagnetic coupling between the Chern sectorand plays an important role in the skyrmion pairing mecha-nism proposed in Ref. [29]. Both corrections are much smaller Strictly speaking, we should remove the lowest mode ψ + − ,l =0 = ∆ whencomputing h + H − − h − . In practice, it makes little difference since theoverlap of h ± ( k ) and ∆( k ) is relatively small than the gap to higher energy modes l > and only induces agap of the order of a few meV, meaning that our restriction tothe l = 0 modes remains valid in the realistic limit.Finally, we can express the matrix ρ defined in (25) in the r -basis where it has a simple q -independent and block-diagonalform with the blocks given by ρ γ,γ (cid:48) αβ = −
12 tr Q [ r γ,γ (cid:48) α ] † r γ,γ (cid:48) β (35)Note that ρ is not antisymmetric in the r -basis since r γ,γ (cid:48) arenot hermitian and are thus related by a non-orthogonal trans-formation to the hermitian generators t µ . To highlight thesymplectic structure of the theory, we can now go back to theHermitian basis t µ (where H is not diagonal) in which ρ isantisymmetric and has the simple form given by ρ µν = −
12 tr Qt µ t ν (36)Thus, ρ defines a symplectic structure of the theory by pairingup different generators as canonically conjugate variables. Asshown in Appendix C, ρ is a full rank matrix, which meansthat all generators are paired in canonically conjugate pairs.Thus, the number of soft modes is equal to half the number ofgenerators yielding − ν . Furthermore, since Q commuteswith γ z , the matrix ρ does not mix the intra-Chern ( [ t µ , γ z ] =0) and inter-Chern ( { t µ , γ z } = 0 ) generators. As a result,the count of approximate goldstone and the nematic modescan be identified with half the number of generators t µ whichcommute or anticommute with γ z , respectively, leading to theexpression in Eq. (7). C. Goldstone mode count
The analysis of the soft modes above does not distinguishthe true gapless Goldstone modes which correspond to break-ing the continuous physical symmetry, given to an excellentapproximation by independent charge and spin rotations in thetwo valleys hence:
U(2) × U(2) symmetry, from the pseudo-Goldstone modes which only break the more approximate
U(4) × U(4) symmetry and have a gap of a few meV.To make this distinction, let us review some recent re-sults related to counting Goldstone modes in systems withoutLorentz invariance [39–41]. These works derived a generalexpression for the count of the Goldstone modes in terms ofthe number of broken symmetry generators n BG and the rankof the matrix ρ defined in (36) given by n G = n BG −
12 Rank ρ (37)This expression also enables us to extract the Goldstone modedispersion by noting that modes corresponding to canonicallyconjugate variable have a linear time derivative term in theeffective action leading to a quadratic dispersion, ω q ∼ q whereas the remaining modes have a quadratic time derivativetime leading to a linear dispersion i.e. ω q ∼ | q | . Thus, we can2identify the number of linearly dispersing and quadraticallydispersing Goldstone modes as: n G - I = n BG − Rank ρ, n G - II = 12 Rank ρ (38)where n G, I and n G, II denote the count of so-called type I andtype II Goldstone modes introduced in Ref. [38] according towhether the leading power in the soft mode dispersion at small q is odd or even, respectively. In our theory, type I and II cor-respond to linearly and quadratically dispersing soft modes,respectively, since higher order dispersions are not possible.Let us first apply these results to the approximate U(4) × U(4) symmetry. In this limit, we can identify the brokensymmetry generators by the generators t µ of U(4) × U(4) which anticommute with Q . The number of such generatorsis − ν − C , where ν is the filling as measured fromcharge neutrality and C is the Chern number of the groundstate. Next, we note that the matrix ρ in this case is alwaysa full rank matrix (see Appendix C) so its rank is equal to itsdimension which is precisely the number of broken genera-tors n BG , app . Thus, using Eq. 37, the number of approximateGoldstone modes is equal to half the number of broken sym-metry generators leading to Eq. 7, i.e. n AG = − ν − C . Inthe U(4) × U(4) limit, all such modes will be type II (quadrat-ically dispersing). Note, for the remaining n N = − ν + C nematic modes, we do not need to discuss the form of theirlow energy dispersion since they are generically gapped.The Goldstone mode count corresponding to the physical U(2) × U(2) symmetry is similar. The main difference is thatwe need to restrict ourselves to the broken symmetry gener-ators corresponding to the physical
U(2) × U(2) symmetrygenerated by s ,x,y,z and η z s ,x,y,z . Denoting these genera-tors by t phys µ , we can define ρ phys as in (36) but using only t phys µ leading to n G = n BG , phys −
12 Rank ρ phys (39)The rank of ρ phys counts the number of physical symme-try generators which are canonically conjugate variables, andwhich thus give rise to a single Goldstone mode.The count of Goldstone modes for different possible statesis provided in Tables I, II, and III. To understand the results ofthese tables, let us start with the spinless limit which serves toillustrate the idea before considering more complicated sce-narios. At neutrality, the K-IVC state spintaneously breaks U(1) valley symmetry generated by η z . In addition, it is easyto verify that ρ phys = 0 leading to a single linearly dispers-ing (type I) mode as seen in Table I. As a more complicatedexample, let us now consider the spinful K-IVC state at neu-trality. In addition to breaking the U(1) valley symmetry, thisstate also breaks independent spin rotations within each val-ley,
SU(2) K × SU(2) K (cid:48) down to a single SU(2) . Thus, thereare 4 broken symmetry generators η z s ,x,y,z . The rank of ρ phys vanishes leading to 4 linearly dispersing (type I) Gold-stone modes as seen in Tables II and IV. Finally, let us con-sider the spin-polarized K-IVC state at ν = 2 . This state breaks U V (1) × SU(2) K × SU(2) K (cid:48) down to the group gen-erated by s z and P ↓ η z (assuming the filled spin flavor is up).Thus, there are 5 broken symmetry generators. The rank of ρ phys in this case is equal to 4 so that the number of Gold-stone modes is − with 2 quadratically dispersing(type II) modes and one linearly dispersing (type I) mode. InTable IV, we also include the count in the presence of interval-ley Hund’s coupling which breaks the U(2) × U(2) explicitlydown to
U(1) × U(1) × SU(2) as explained in the next section.
V. EFFECTIVE FIELD THEORY OF SOFT MODES
The energetics of the soft modes can be conveniently cap-tured by deriving an effective field theory in the form of anon-linear sigma model. Such field theory reproduces the softmode Hamiltonian derived in the previous section but it canalso allow us to go beyond the quadratic approximation andinclude interactions between the soft modes. We note that anon-linear sigma model describing the intra-Chern soft modeswas already derived in Ref. [29]. The main difference here isthe additional inclusion of the inter-Chern nematic modes.
A. Non-linear sigma model
To derive a non-linear sigma model, we need to identifya large manifold of soft modes related by symmetries whichare weakly broken. For the intra-Chern soft modes, this is the
U(4) × U(4) symmetry identified in Ref. [29] which is brokenby the dispersion and the sublattice off-diagonal form factor.The approximate symmetry related to the nematic inter-Chernmodes is more subtle. To understand this symmetry, we no-tice that these modes are only low in energy when the Berrycurvature is strongly concentrated at a point. In this case, wecan approximate the Berry curvature by a delta function rep-resenting the flux of a solenoid at the Γ point such that theapproximate energy expression in (32) vanishes for an appro-priate choice of ∆( k ) . One such choice is A k = ( − k y , k x ) | k | , ∆( k ) = e iϕ k , ϕ k = arg( k x + ik y ) (40)In this limit, the inter-Chern nematic modes become trueGoldstone modes. To see the corresponding symmetry explic-itly, we note that under the gauge transformation c k (cid:55)→ ˜ c k = e iϕ ( k ) γ z c k , the sublattice-diagonal form factor (29) changesas Λ q ( k ) (cid:55)→ ˜Λ q ( k ) = e iϕ ( k ) γ z Λ q ( k ) e − iϕ ( k + q ) γ z ≈ F q ( k ) e i q · ( A k −∇ k ϕ k ) = F q ( k ) (41)Thus, under such an approximation, the symmetry of thesublattice-diagonal part of the interaction is enhanced to U(8) .Physically, this can be understood as follows. Recall, the ini-tial
U(4) × U(4) symmetry does not permit rotation betweenopposite Chern sectors since the single particle wavefunctionsof the opposite Chern bands are rather different. However, if3all the Chern flux is concentrated into a solenoid, it can beeliminated by a singular gauge transformation, and this ob-stacle is circumvented. The extra symmetry generators corre-spond to the nematic modes and relate the insulating quantumHall ferromagnets to the nematic semimetals. To see this, wenote that any k -independent ˜ Q matrix (in terms of the trans-formed variable ˜ c ) is a ground state for the U(8) symmetricinteraction. The corresponding states in the original basis aregiven by the k -dependent Q matrix: Q k = e iϕ ( k ) γ z ˜ Qe − iϕ ( k ) γ z (42)For ˜ Q commuting with γ z , this transformation does nothingand we get the same insulating states as before Q = ˜ Q . Onthe other hand, for ˜ Q anticommuting with γ z , we get Q k = ˜ Q cos 2 ϕ ( k ) − i ˜ Qγ z cos 2 ϕ ( k ) (43)which describes a semimetal where Q k winds around the ze-ros of ϕ ( k ) . For example, ˜ Q = γ x yields the order parameter Q k = γ x cos 2 ϕ ( k ) + γ y sin 2 ϕ ( k ) which describes the ne-matic semimetal identified in Ref. [25].This naturally leads to a U(8) sigma model descriptionwhich unifies the insulating and semimetallic order parame-ters incorporating both nematic and approximate Goldstonemodes. The sigma model Lagrangian can be written as L [ ˜ Q ] = 12 tr T ( r ) † ˜ Q∂ t T ( r ) − ρ ∇ ˜ Q ( r )] − E [ ˜ Q ( r )] (44)where the spatially dependent ˜ Q ( r ) is obtained from the spa-tially constant ground state by applying the unitary rotation T ( r ) = e iφ ( r ) through ˜ Q ( r ) = T † ( r ) ˜ QT ( r ) , T ( r ) = e iφ ( r ) ,φ ( r ) = (cid:88) q , k e i q · r φ q ( k ) (45) E [ ˜ Q ] contains the anisotropy terms which break the U(8) symmetry down to the physical
U(2) × U(2) which can besummarized as follows. First, there is the energy penalty as-sociated with deviations from the approximations leading to(41) which disfavors the inter-Chern states (semimetals whichanticommute with γ z ) relative to intra-Chern states (insulatorswhich commute with γ z ). This can be captured by a term ofthe form − tr( Qγ z ) . In addition, there is the anisotropy termdue to the sublattice off-diagonal form factor − tr( γ x,y η z Q ) and the antiferromagnetic coupling obtained from the disper-sion + tr( γ x,y Q ) already derived in Refs. [27, 29]. Thisleads to the energy expression E [ ˜ Q ] = − α Qγ z ) + J Qγ x ) + ( ˜ Qγ y ) ] − λ Qγ x η z ) + ( ˜ Qγ x η z ) ] (46)The parameter J and λ were derived from the microscopictheory in Ref. [29]. The new parameter α can be identified FIG. 6.
Field theory parameters : Parameters of the non-linearsigma model defined by Eqs. (44) and (46) as a function of thetwist angle θ . The parameters are extracted using the energy split-ting between the different self-consistent Hartree-Fock solutions inaccordance with Eq. 47 at κ = 0 . with a grid size of × , 6bands included per spin and valley and with interaction parameters (cid:15) = 12 . and screening length d = 20 nm. with E ∆ which can be seen by substituting (45) in (46) andexpanding to quadratic order in φ ignoring the J and λ terms.An alternative approach is to consider these as phenomeno-logical parameters that can be fit to numerics. This is done bynoting that the energy per particle obtained from E [ ˜ Q ] for thedifferent states relative to the minimum K-IVC state at ν = 0 is ∆ E VP = J, ∆ E VH = λ, ∆ E SM = α + J + λ β (47)where ’VP’, ’VH’, and ’SM’ denote the valley polarized ˜ Q = η z , valley Hall ˜ Q = γ z η z and semimetal ˜ Q = γ z , respec-tively. Comparing to the Hartree-Fock numerics these param-eters can be extracted as shown in Fig. 6. It is worth notingthat the field theory derived in Ref. [29] can be obtained from(44) and (46) in the limit of large α .A notable aspect about the field theory (44) is that it doesnot include all possible symmetry allowed terms. In par-ticular, notice the absence of terms of the form tr( Qη z ) and tr( Qγ z η z ) which, despite being symmetry allowed, turnout to be significantly smaller than the other terms and canbe neglected. To understand what these terms represent, itis instructive to focus on the insulating states [ Q, γ z ] = 0 for which Q can be split into Q ± corresponding to the ± Chern sectors. In this limit, the λ term above takes theform tr Q ± η z Q ∓ η z whereas the neglected symmetry allowedterms take the form tr Q ± η z Q ± η z . Clearly both types ofterms represent anisotropy terms of the pseudospin ( η ) favor-ing easy axis or easy plane orientation. However, the λ termis an inter-Chern term which only yields a non-vanishing con-tribution when both Q + and Q − are non-trivial (not equal toa multiple of the identity) whereas the other terms are intra-Chern terms. This can be seen more clearly in spinless limitwhere Q ± = n ± · η such that the λ term has the form n + ,z n − ,z whereas the other terms have the form n ± ,z . Thishas important implications for the soft mode dispersion andoccasionally leads to cases where the sigma model (44) yields4a gapless mode which is not associated with any continuoussymmetry breaking (and is thus not a true Goldstone mode).As a simple example consider the spinless model at ν = 1 ,and C = 1 . In this case, we fill both bands in C = 1 and asingle band in the opposite Chern sector, which we will taketo be valley polarized (VP). Now, the physical valley symme-try would not predict any gapless modes, however the mech-anism described above implies the anisotropy λ that usuallylifts the SU(2) valley rotation symmetry is absent in this par-ticular case leading to a nearly gapless mode which we denoteby PG*. In the TDHF numerics, these correspond to modeswith a very small gap ∼ . − . meV. In tables I, II, and III,we see several instances of PG* which are pseudo-Goldstonemodes which are approximately gapless.Another type of symmetry allowed term not included in thefield theory (44) are possible anisotropies in the stiffness ρ .Such terms will assign slightly different energy cost to spatialdeformations of ˜ Q in different directions. These likely affectthe overall soft mode dispersion but will be unimportant if weare only interested in leading contribution to the soft modedispersion at small q . B. Soft mode spectrum from the field theory
In this section, we explain how the soft mode spectrumat small q can be obtained from the sigma model given byEqs. 44 and 46.
1. Spinless model
We start by considering the spinless limit which serves asan illustration for the computation. At ν = 0 , the groundstate minimizing all three terms in (46) can be chosen withoutloss of generality to be Q = γ z η x Which is nothing but theK-IVC of reference [27]. A basis of hermitian generators an-ticommuting with Q can be chosen as t µ = γ x,y η ,x , γ ,z η y,z .Substituting Q = e − i (cid:80) µ φ µ t µ Q e i (cid:80) µ φ µ t µ and expandingto quadratic order in φ yields an action of the form (27)from which we can extract the matrix M . The dispersionof the Goldstone modes is obtained by solving the equation det M ( q , ω ) = 0 which has four positive solutions: ω G , I ( q ) = 2 (cid:112) Jρ | q | + O ( q ) ω PG ( q ) = 4 (cid:112) λ ( J + λ ) + O ( q ) ω N;1 , ( q ) = 4 β + O ( q ) (48)We see that ω G , I ( q ) represents the linearly dispersing Gold-stone mode, ω PG ( q ) represents the gapped pseudo-Goldstonemode which becomes gapless in the limit of perfect sublatticepolarization λ = 0 , and ω N ( q ) represent the two gapped ne-matic modes which are exactly degenerate at q = 0 . The ex-istence of a single Goldstone mode is consistent with Eq. (37)with a single broken symmetry generator η z and with the rankof ρ phys equal to 0. In Fig. 7, we compare the gaps of thepseudo-Goldstone mode ∆ PG = ω PG (0) and the nematic FIG. 7.
Comparison of TDHF and NLSM results : The gap for thepseudo-Goldstone mode(s) (lower branch) and the nematic modes(upper branch) for the K-IVC state at neutrality as a function ofthe twist angle θ computed from the time-dependent Hartree-FockTDHF (red) and the non-linear sigma model NLSM (blue). TheNLSM gaps are given by (cid:112) λ ( J + λ ) and β (cf. Eq. 48) with J , λ , and β extracted from the energy spliting between different self-consistent Hartree-Fock states (cf. Eq. 47). To calculate both thefield theory parameters and the TDHG gaps, we used a × mo-mentum space grid with 6 bands per spin and valley with at κ = 0 . , (cid:15) = 12 . and gate distance d = 20 nm. modes ∆ N = ω N (0) obtained from the TDHF with the fieldtheory result (48) and we can see that the two agree reason-ably well. This serves as a justification for the field theorydescription in the limit of small q .
2. Spinful model at ν = 0 The spectrum of the spinful model at charge neutrality isthe same as the for the spinless model with four copies ofeach mode which correspond to a spin-singlet and three spin-triplet modes for each mode of the spinless model. To see this,we note that the spin unpolarized K-IVC state of the spinfulmodel, Q = γ z η x s breaks the U(2) × U(2) symmetry downto
U(2) which corresponds to overall spin and charge con-servation. This means that there is a manifold of degenerateK-IVC ground states given by
U(2) × U(2)U(2) (cid:39)
U(2) . This man-ifold includes both spin singlet and spin triplet K-IVC statesand can be generated by acting on the singlet K-IVC statewith different spin rotations in the two valleys. Since all K-IVC states are symmetry related, their spectra are identicaland we can focus on a single representative which we taketo be the spin-singlet K-IVC state. Since this state preservesthe overall
SU(2) symmetry, we can label the soft modes bya spin quantum number such that each mode in the spinlessmodel corresponds to a singlet and three triplet modes whichare degenerate. The existence of 4 linearly dispersing Gold-stone modes arises due to breaking the continuous symmetriesgenerated by the 4 generators η z and s x,y,z η z , which result ina matrix ρ phys that vanishes. The detailed symmetry proper-ties of the different soft modes will be discussed in detail inthe next section.We can distinguish the spin-singlet and triplet K-IVC state5by including an intervalley Hund’s coupling term [60] whichhas the form: L H = J H (cid:88) i = x,y,z (cid:26) tr 1 + η z s i Q tr 1 − η z s i Q − tr 1 + η z s i Q − η z s i Q (cid:27) (49)where | J H | is given roughly by the Coulomb scale divided bythe ratio of the Moir´e to lattice length scale yielding a value of0.1-0.2 meV. This term breaks the separate SU(2) spin sym-metry in each valley down to a single overall
SU(2) .The ground state for finite J H depends on the sign of J H with J H > favoring the spin-singlet state Q = γ z η x s and J H < favoring the spin-triplet state Q = γ z η x s z [27]. Thesoft mode spectrum at ν = 0 for finite J H is summarized inTable IV. For J H > , the three triplet Goldstone modes aregapped leaving the singlet mode gapless. This arises sincethe symmetries corresponding to the generators s x,y,z η z areexplicitly broken, leaving a single broken symmetry genera-tor corresponding to U v (1) which gives rise to a single lin-early dispersing Goldstone mode. In contrast, for J H < ,the overall SU(2) spin symmetry is broken (down to
U(1) ),in addition to the broken U v (1) symmetry. As a result, threeof the four Goldstone modes remain gapless, corresponding tothe broken symmetry generators η z and s x,y (since ρ phys = 0 also for the spin-triplet K-IVC state).
3. Spinful model at ν = 2 At ν = 2 , the ground state is a spin-polarized K-IVC state.Similar to the case of ν = 0 , there is a manifold of statesrelated by the action of U(2) × U(2) symmetry on the sim-ple K-IVC ferromagnet Q = P ↑ + P ↓ γ z η x . This manifoldconsists of states where an arbitrary spin orientation is cho-sen independently for each valley. Since all these states aresymmetry related, we can restrict ourselves to the K-IVC fer-romagnet where the same spin direction is chosen in both val-leys. Following the same procedure as in the spinless case, wecan extract the soft mode spectrum which is given by ω G , I ( q ) = 2 (cid:112) ρJ | q | , ω G , II;1 , ( q ) = ρ q ω PG;1 ( q ) = 4 (cid:112) λ ( λ + J ) , ω PG;2 , ( q ) = 4 λ,ω N;1 , ( q ) = 4 β, ω N;3 , , , ( q ) = 4 β − J (50)The spectrum consists of 3 Goldstone, 3 pseudo-Goldstoneand 6 nematic modes whose dispersion. Of the three Gold-stone modes, only one is linearly dispersing and can be asso-ciated with the broken U(1) valley symmetry while the othertwo are quadratically dispersing and can be associated withspin symmetry breaking. This is compatible with Eqs. (37)and (38) since there are 5 broken symmetry generators cor-responding to s x,y and η z s ,x,y with Rank ρ phys = 4 , lead-ing to 2 type-II Goldstone modes and one type-I mode. Thepseudo-Goldstone modes are split into 1 + 2 whereas the ne-matic modes are split into 2 + 4. These degeneracies match the ones seen in the numerical TDHF spectrum in Fig. 3 andwill be explained using the symmetry analysis of Sec. VI.In the presence of intervalley Hund’s coupling, the inde-pendent SU(2) spin rotation symmetry in each valley is ex-plicitly broken down to a single overall
SU(2) spin rotationsymmetry. As a result, there are only two spontaneously bro-ken spin symmetry generators s x,y , in addition to the sponta-neously broken U(1) valley symmetry. For a ferromagneticHund’s coupling J H < , the ground state is the ferromag-netic K-IVC state with Rank ρ phys = 2 leading to one type-Imode and one type-II mode. In contrast, the antiferromagneticHund’s coupling J H > selects a K-IVC state with oppositespin orientation in the two opposite valleys. This state has Rank ρ phys = 0 , leading to three type-I Goldstone modes.The modified dispersion for the soft modes in the presence ofintervalley Hund’s coupling is given in Table IV.
4. Spinful model at ν = 1 and C = 3 : Correlated Chern insulator Let us now consider the odd filling ν = 1 . In this case,there are several possible states which minimize the energyfunctional (46). These are divided into two categories: (i)we can fill 3 bands in one Chern sector and 2 in the otherChern sector leading to a state with | C | = 1 , or (ii) we canfill 4 bands in one Chern sector and 1 in the other leading toa state with | C | = 3 . These states are degenerate on the levelof the energy functional (46) but we expect a small orbitalmagnetic field to favor the | C | = 3 state since it lowers theenergy of one Chern sector relative to the other. Indeed, suchstates have recently been observed in experiments [12, 13, 16].For the | C | = 3 , one Chern sector is fully filled and the other isquarter filled where we have to pick one spin-valley flavor outof the four flavors within this sector to fill. This necessarilybreaks SU(2) spin rotation symmetry but it may or may notbreak
U(1) valley rotation. The latter depends on whetherwe choose the filled state to be a valley eigenstate (VP) with Q = P + + P − ( P ↑ η z − P ↓ ) or to be an equal superpositionof the two valleys (IVC) with Q = P + + P − ( P ↑ η x,y − P ↓ ) ,where P ± and P ↑ / ↓ denote the projectors unto the differentChern and spin sectors, respectively. The two possibilities areconsidered below.First, we can consider the VP case where the spectrum canbe computed leading to ω G;1 , , ( q ) = ρ q , ω N;1 , , ( q ) = 2(2 β − J − λ ) , ,ω N ;3 ,..., ( q ) = 4( β − λ ) (51)Note that the expression of the Goldstone mode count (38)yields a single type II goldstone mode corresponding to thebreaking of SU(2) spin symmetry in the K valley down to U(1) . On the other hand, the spectrum obtained above seemsto have three quadratically dispersing gapless modes. To re-solve this issue, recall the discussion of Sec. V A that thesigma model does not include all possible symmetry allowedterms and thus can have ’accidential’ gapless modes not as-sociated with any continuous symmetry breaking. In the caseconsidered here, the issue arises because the pseudospin ’easy6plane anisotropy’ term λ which prevents us from performingan arbitrary SU(2) valley rotation vanishes if one Chern sec-tor is completely filled or completely empty. This can be ver-ified by adding the symmetry allowed intra-Chern easy planeanisotropy term κ tr( Qη z ) to (46). This perturbation splitsthe degeneracy between the VP and the IVC state by favor-ing the VP state for κ > and the IVC state for κ < . Ifwe now recompute the soft mode dispersion for the VP statewith κ > , we find that the two of the three quadratically dis-persing modes acquire a gap of κ verifying that these are nottrue gapless goldstone modes. We will denote these modes byPG* which means they are pseudo-goldstone modes that arenot associated with a broken physical symmetry yet is almostgapless due to the form of the Hamiltonian.Next, let us consider the IVC state with Q = P + + P − ( P ↑ η x,y − P ↓ ) . The soft mode spectrum for this state isgiven by ω G;1 , , ( q ) = ρ q , ω N;1 ,..., ( q ) = 4( β − λ ) ,ω N ;7 , , ( q ) = 2(2 β − λ ) , ω N ;7 , , ( q ) = 2(2 β − λ − J ) (52)Here, we also see a discrepancy with the Goldstone modecount (38) which yields two quadratically dispersing modesand one linearly dispersing mode rather than three quadrat-ically dispersing modes as seen above. This again can beresolved by including the extra anisotropy term κ tr( Qη z ) (with κ < in this case). Redoing the calculation, we findthat one of the three gapless quadratic modes acquire lineardispersion given by √− κρ | q | as expected. VI. SYMMETRY REPRESENTATIONS OF THE SOFTMODES
In this section, we present an analysis of the symme-try representations of the soft modes. The analysis will bedone in full generality without assuming a particular groundstate. Each symmetry breaking state, regardless of the de-tailed anisotropies or perturbations which selects for it, yieldsa set of soft modes which transform as a representation un-der the group of symmetries which leave this state invariant.The decomposition of this representation into irreducibles (ir-reps) only depends on the state in question and encodes thedegeneracies and symmetry quantum numbers of the modes,representing a fingerprint of insulating state. Such fingerprintcan be very useful in identifying the broken symmetry statein experiments by measuring the soft mode spectrum and re-sponse to different perturbations.
A. Formalism
The symmetry group of MATBG, which we denote by G , isgenerated by the following symmetries: U V (1) valley chargeconservation, SU(2) K × SU(2) K (cid:48) corresponding to indepen-dent SU(2) spin conservation in each valley, time-reversalsymmetry T and two-fold rotation symmetry C z which ex- changes valleys, in addition three-fold rotation C and mirrorsymmetry M y which act within a given valley (the latter ex-changes layers and is sometimes denoted by C x ). A givenstate described by the matrix Q breaks G down to a subgroup G Q ⊂ G . The generators of G Q for a selection of states atdifferent integer fillings are given in Table I for the spinlessmodel and Tables II and III for the spinful model.To obtain the symmetry representations of the soft modes,we start by considering a general symmetry S whose actionon the band-projected operator c k is given by S c k S † ≡ U k c O k , U k U † O k = 1 (53)where O is an element of the O (2) group corresponding to thespatial action of the symmetry S . The soft modes are definedthrough the operator ˆ φ n, q which creates a soft mode charac-terized by the eigenfunction φ n, q at energy ε n, q (cf. Eq. 22)given explicitly by ˆ φ n, q = (cid:88) k c † k φ µn, q ( k ) t µ c k + q (54)The soft modes ˆ φ n, q transform as a representation of G Q givenexplicitly by S ˆ φ n, q S − = (cid:88) m S nm ( q ) ˆ φ m,O q (55)The matrix representation S ( q ) can be obtained from sym-metry action on the operators c k in Eq. 53 in addition to theknowledge of the soft mode wavefunctions φ n, q as explainedin detail in appendix (D).So far the discussion has been completely general. To makefurther progress, we restrict ourselves to the flat bands wherethe action of the different symmetries in G in the Chern-pseudospin-spin basis was derived in Refs. [27, 29] (see ap-pendix D for details). Furthermore, we restrict our attentionto the − ν lowest energy bosonic modes which are sepa-rated by a large gap ( ∼ − meV) from the remaining highenergy modes (See discussion of Sec. IV B). One simplifica-tion which will enable us to obtain the symmetry representa-tions explicitly is to consider U(4) × U(4) limit discussed inSec. IV B where the eigenfunctions φ n, q for the soft modeshave a particularly simple form. Since the symmetry repre-sentations cannot change continuously, these results shouldequally apply for the realistic limit with all anisotropies in-cluded as long as there is no mixing with the higher energymodes.Thus, given a state Q which is invariant under a symmetrygroup G Q ⊂ G , the full symmetry action of G Q on the softmodes is specified by knowing the symmetry representations S ( q ) and the symplectic matrix ρ in a certain basis which en-ables us to construct them in any other basis. In a hermi-tian basis of generators, ρ is antisymmetric and it pairs differ-ent generators as canonically conjugate position-momentumvariables. However, it is more convenient to choose a non-hermitian basis which diagonalizes ρ . The corresponding ba-sis transformation is non-orthogonal and corresponds to go-7ing from a pair of position-momentum canonical variables tocreation-annihilation operators. Since ρ in invariant under anysymmetry S ∈ G G , we can restrict the symmetry action to the +1 eigenvalue sector of ρ (corresponding to, let’s say, annihi-lation operators) which we will denote by S + . Furthermore,at a given momentum q , we should only consider the ’littlegroup’ G Q, q ⊆ G Q which leaves the point q invariant, leadingto a − ν dimensional matrix representation for G Q, q whichcan be decomposed into irreps. B. Results
In the following, we will restrict our attention to the Γ point, q = 0 , since many experimental probes only coupleto the small momentum modes but our analysis can be ex-tended to any momentum. At the Γ point G Q, q is equal tothe full group G Q and the irreps are obtained by simultane-ously block-diagonalizing the matrices S + for all symmetriesin G Q . Such process can be automated yielding the results oftables I, II, and III. Table I includes the results for all pos-sible insulating state for the spinless model which serves toillustrate the concept. In Table II, we include a selection ofphysically relevant states for the spinful model at even integerfillings ν = 0 and where C = 0 insulators have been exper-imentally observed [1–3]. In addition, Table III includes theChern insulators with maximal Chern number | C | = 4 − | ν | which have been recently observed to be stabilized for smallout-of-plane [12–14, 16].The tables specify the symmetry content of the soft modesas follows. The first columns specify the state Q and the gen-erators of the symmetry group G Q at a given filling ν andChern number C . The soft modes are divided into Goldstonemodes (G) which correspond to breaking an exact continuoussymmetry, pseudo-Goldstone modes (PG) which break the ap-proximate U(4) × U(4) symmetry but none of the continuousphysical symmetries. Both G and PG modes correspond toexcitations within the same Chern sectors. Then there are ne-matic modes (N) which correspond to inter-Chern modes. Thegoldstone modes can be further divided into type I and type IIdepending on whether they have linear or quadratic disper-sion at small momenta (see Sec. IV C). The last few columnsspecify the decomposition of the soft modes into irreduciblerepresentations of the symmetry group G Q which are speci-fied by their characters. Recall that a representation characterdenotes the traces of the symmetry elements in the given irrepproviding a basis-independent characterization of the irrep.Given the characters, it is straightforward to deduce thesymmetry quantum numbers of the soft modes. Let us be-gin with the U V (1) symmetry generated by e iϕη z . A d -dimensional representation with character de ± imϕ means thatall the d modes in the representation carry the same valleycharge of m . A character of d cos mϕ describes a representa-tion with d/ modes of valley charge + m and d/ with valleycharge − m . In general, we can extract the number of modes n m with valley charge m within a given representation by pro- jecting onto this charge sector using n m = (cid:90) π dϕ π e − imϕ χ ( e iφη z ) (56)Similarly for C symmetry, if all the soft modes carry thesame ”angular momentum” l under C , we get a charac-ter of de πi l . On the other hand, a character of − d/ d ( e πi + e − πi ) corresponds to an irrep with an equal num-ber of l = +1 and l = − modes. Generally, the number ofmodes with angular momentum l = 0 , ± is n = d + 2Re χ ( C )3 , n ± = d − Re χ ( C )3 ± Im χ ( C ) √ (57)This enables us to read of the symmetry properties of thesoft modes in the spinless limit from Table I. For example, letus start with the valley-polarized (VP) state where one band isfilled in each Chern sector corresponding to the same valley.There are two intra-Chern PG and two inter-Chern N modes.Since all modes create electron-hole excitations between op-posite valleys, they all carry the same valley quantum num-ber +2 . The two PG modes transform trivially under C andeach form a 1D irrep while the two N modes transform withopposite angular momenta l = ± which map to each otherunder M y forming a 2D irrep. Another example is the valley-polarized quantum anomalous Hall state (VP-QAH) at ν = 1 where one Chern sector is completely filled whereas the otherhas one valley completely filled. There is one PG mode whichcarries a valley charge of − within the half-filled Chern sec-tor. This mode is not associated with any continuous symme-try breaking but it will be gapless according to the field theorydescibed by Eqs. (44) and (46) due to the absence of symme-try allowed anisotropy terms within each Chern sector sepa-rately (see Sec. V A). In numerics, such mode will have verysmall gaps and we denote them by PG*. In addition, there istwo nematic modes which carry the same angular momentum l = − and transform as a 1D irrep each.The representation content for the spinful case is more com-plicated due to the existence of the SU(2) spin symmetries ineach valley. As usual, the irreps of
SU(2) are labelled by ahalf-integer S with the corresponding character given by χ ( e iαs z ) = S/ (cid:88) l = − S/ e iαl (58)which yields , α , and α for S = 0 , / , respectively. The S z charge can be extracted from χ ( e iαs z ) asin Eq. 56. Using this information, we can read the symmetryproperties of the soft modes from Tables II and III. C. Examples
Let us now consider a few examples:
Spin-singlet K-IVC at ν = 0 : As an example, let us con-sider the spin-singlet K-IVC state. Here, individual spin ro-8tation in each valley is broken but the overall spin symmetryremains unbroken. There are four Goldstone and four pseudo-Goldstone modes which split each into a singlet S = 0 and atriplet S = 1 . In addition, there are 8 nematic modes whichsplit into a 2D irrep and a 6D irrep. The former (latter) con-sists of two singlets (triplets) with opposite C angular mo-menta tied together with η z M y . Note that from the point ofview of the unbroken symmetry group G Q , there is no reasonto expect the spin-singlet and spin-triplet modes to be degen-erate as seen in the TDHF spectrum (Fig. 3). This can beseen from the addition of inter-valley Hund’s coupling whichdoes not break any symmetry in G Q but lifts the degeneracybetween the singlet and triplet modes (Table IV). Thus, it ispossible sometimes for the soft mode spectrum to have moresymmetry than that of the state Q . Since in this section we aretaking the most general viewpoint allowing for perturbationsto the Hamiltonian which may break some of its symmetriesto select a specific ground state, we are not going to considersuch symmetries, i.e. we are going to assume the presence ofthe most general perturbation compatible with the symmetrygroup G Q of the state. One should keep in mind though thatthere may be some extra unaccounted for degeneracies in thespectrum in the absence of these perturbations. Spin-singlet VP at ν = 0 : Another example at ν = 0 is the valley-polarized state which is invariant under the full SU(2) K × SU(2) K (cid:48) (cid:39) SO(4) . Here, there are 8 pseudo-Goldstone modes which split into two 4D irreps correspond-ing to ( S K , S K (cid:48) ) = (1 / , / (this is equivalent to thefundamental of SO(4) ) carrying the same valley charge +2 and angular momentum l = 0 . In addition, there are 8 ne-matic modes transforming as an 8D irrep consisting of two4D SU(2) K × SU(2) K (cid:48) irreps with opposite C angular mo-menta l = ± tied together by M y . Spin polarized K-IVC at ν = 2 : At Half-filling ν = 2 , wecan consider the spin-polarized (SP) K-IVC which is found tobe the minimum energy state within Hartree-Fock. In additionto the linearly dispersing and the pair of quadratically dispers-ing Goldstone modes, the dispersion for the gapped, shown inFig. 3, exhibits the pattern of degeneracies − − − at Γ .This is in agreement of the symmetry irreps in Table II withthree PG modes split into a 1D and 2D irrep and 6 nematicmodes split into a 2D and a 4D irrep. To understand the ori-gin of these, we note that for the SP K-IVC, U V (1) is brokenonly in the filled spin species, which we take to be ↑ . Thus,the state is invariant under the U(1) symmetry generated byvalley rotation acting only on the down spin, P ↓ η z . The 2Dirrep for the PG mode corresponds to a pair of modes trans-forming with opposite charge under this U(1) symmetry tiedtogether with the combination η z T . For the nematic modes,there is an extra degeneracy due to the opposite C angularmomenta being tied together via M y which leads to 2D and4D irreps. Correlated C = 3 insulators at ν = 1 : As a final example,let us consider the C = 3 insulating states at ν = 1 discussedin Sec. V B 4. For the VP state Q = P + + P − ( P ↑ η z − P ↓ ) , the SU(2) spin rotation symmetry in the K valley is broken downto U(1) leading to a single quadratically dispersing goldstonemode. In addition, there is a pair of almost gapless pseudo- goldstone modes (PG*) which transform as a 2D irrep trans-forming as S = 1 / under SU(2) K (cid:48) with both modes carry-ing the same valley and S zK charges. In addition, there are 12nematic modes which all carry the same angular momentumunder C and are split into 3 1D irreps, 2 2D irreps, and 1 3Dirrep corresponding to S = 0 , / , and representations un-der SU(2) K (cid:48) . For the IVC state Q = P + + P − ( P ↑ η x − P ↓ ) , U V (1) × SU(2) K × SU(2) K (cid:48) is broken down to U(1) × U(1) generated by s z and P ↓ η z . This results in one linearly dispers-ing and two quadratically dispersing goldstone modes whichform a 2D irrep of opposite e iθP ↓ η z eigenvalues paired via M y T . In addition, there are 12 nematic modes with the same C angular momentum which are split into 4 1D irreps and4 2D irreps which all correspond to the doublet of the groupgenerated by { e iθP ↓ η z , M y T } similar to the type-II Goldstonemode above. It is worth noting that symmetry irreps for thespin-valley polarized state at ν = 3 (Table III) are compati-ble with earlier time-dependent Hartree-Fock studies at ν = 3 [61, 62]. VII. DISCUSSION
Let us now summarize our main findings. We began byproviding a simple criterion to determine the count of softmodes based on the picture of Fig. 4 leading to − ν − C approximate Goldstone modes associated with the enlarged U(4) × U(4) symmetry and − ν + C nematic modes whichcorrespond to excitations between Chern sectors. We then ob-tained the soft mode spectrum numerically in Sec. IV A for thecandidate ground states at ν = 0 and ν = 2 . In Sec. IV B, weprovided a detailed analysis of the energetics based on approx-imate symmetries which identified the ratio κ = w /w as themain parameter in determining the soft mode gaps. In partic-ular, we find that increasing κ has an opposite effect on thegaps of the approximate Goldstone and the nematic modes;Whereas the gap of the former increases with κ which actsas an anisotropy term breaking the approximate U(4) × U(4) symmetry, the gap to the latter decreases with κ as a result ofthe increased concentration of the Berry curvature in momen-tum space close to the Γ point. This motivated the discussionof a simplified solenoid flux model for the Berry curvaturewhere the symmetry is enhanced to U(8) and where the effectsof different symmetry breaking anisotropies can be systemat-ically included. The resulting
U(8) non-linear sigma model isused to compute the soft mode gaps and shown to agree verywell with the numerics (cf. Fig. 7).In Sec. VI, we switch to a more general discussion of thesoft modes where we assumed by assuming the most gen-eral Slater determinant translationally symmetric insulatingstate and deriving the degeneracies and the symmetry quan-tum numbers for the soft modes. This, combined with thegeneral discussion of the properties of the Goldstone modesin Sec. IV C, provides a complete picture of the soft modes inany given state which is independent on details of the energet-ics and model parameters.We now discuss a few implications of our results in lightof some recent works. Two very recent experimental works9[36, 37] have provided evidence for a Pomeranchuk Ising-typetransition at finite temperature in twisted bilayer grapheneclose to ν = ± . In both cases, it was proposed that the tran-sition is driven by a large entropic contribution arising fromthe existence of a large number of soft bosonic modes. Inparticular, Ref. [37] argued that such finite temperature stateis the same state stabilized by a finite in-plane magnetic fieldpointing to ferromagnetic order. Similarly, Ref. [36] also sug-gested a magnetic origin for the soft modes by showing thatthe large entropy gain at the ’cascade transition’ at ν = 1 issuppressed via an in-plane magnetic field. In principle, suchscenarios are compatible with our finding of a large numberof soft bosonic modes for the symmetry breaking insulatorswhich become activated at a temperature of about 10-30 K.However, to make a more quantitative statement, we need adetailed quantitative comparison to the entropic contributionof the metallic state arising from filling the charge neutral-ity band structure at ν = 1 . This computation is more com-plicated than the corresponding one for the gapped insulatingphases since we need to carefully distinguish between the con-tributions to the entropy coming from the particle-hole contin-uum and the soft modes. Thus, we leave such an analysis tofuture work.The results presented here suggests an experimentally fea-sible way to investigate the properties of the correlated Cherninsulators seen in recent experiment in the presence of smallout-of-plane magnetic field [12, 13, 16]. This can be achievedby employing the setting of Ref. [63] which used the edgemodes to measure the magnon gap in graphene based quan-tum Hall ferromagnets. The results of such experiment can bedirectly compared to the predictions of Table III. For exam-ple, at ν = 1 , there are 3 gapless (or almost gapless modes),two of which are magnons, i.e. carry S z quantum numbers. Athigher energies of a few meV, there are 12 more nematic softmodes, half of which carry non-vanishing S z . It is worth not-ing that the soft modes can also be directly probed via opticalexcitations [64].On the theory side, coupling between electrons and softmodes have been proposed as a pairing mechanism for super-conductivity in several works [33–35, 43]. Refs. [33] and [34]focused on soft modes associated with broken valley symme-try for intervalley coherent orders whereas Ref. [35] consid-ered the soft modes associated with U(4) spin-valley flavorrotation symmetry. Our current work suggests an even largerset of soft modes which includes pseudo-Goldstone modes as-sociated with an enlarged
U(4) × U(4) symmetry in additionto a large number of nematic modes which transform non-trivially under three-fold rotation. The importance of the lat-ter was pointed out in a recent experiment which observedpronounced nematicity in the insulator and superconducting states [5]. Our symmetry analysis summarized in Tables IIand III imposes important restrictions on the coupling of thesoft modes to the electrons at small momenta which ultimatelydetermines the favored pairing channels. This can be used asthe basis of systematic study of superconductivity induced bycoupling to the soft modes which will be the subject of futurestudy.Finally, the field theory calculation whose results are sum-marized in Table IV enables us to bridge theory and experi-ment by providing a direct experimental probe to measure thefield theory parameters. The field theory given by Eqs. (44)and (46) can be seen as the simplest theory which capturesthe essentials of the symmetries and energetics of the differ-ent competing phases in terms of a few parameters. Althoughthese parameters can be computed microscopically or numeri-cally, they are likely to be sensitive to microscopic details suchas strain and substrate alignment. Thus, a better approach isto view them as phenomenological fitting parameters to be di-rectly compared to experiment. Thus, by using the soft modegaps and Goldstone modes computed in Sec. V A and summa-rized in Table IV, we can use experimental data about the softmodes to directly access the field theory parameters. This hasimportant implications for the behavior of this field theory, inparticular, in relation to the proposed skyrmion mechanism ofsuperconductivity [29, 65] which depends sensitively on theratio between the parameters J and λ . Note added –
During the final stages of this work, Ref. [66]appeared, which employed a promising RG approach andcomputed the soft mode spectrum in the strong coupling limitat charge neutrality. In addition, we note an independent re-lated work by Kumar, Xie, and Macdonald [62] which alsodiscusses collective modes in TBG. The results of both worksare consistent with ours.
VIII. ACKNOWLEDGMENTS
We acknowledge stimulating discussions with Shahal Ilani,Pablo Jarillo-Herrero and Andrea Young regarding theirexperimental data. NB would also like to thank SidParameswaran for helpful discussions. AV was supported bya Simons Investigator award and by the Simons Collabora-tion on Ultra-Quantum Matter, which is a grant from the Si-mons Foundation (651440, AV). EK was supported by a Si-mons Investigator Fellowship, and by the German NationalAcademy of Sciences Leopoldina through grant LPDS 2018-02 Leopoldina fellowship. MZ was supported by the Director,Office of Science, Office of Basic Energy Sciences, MaterialsSciences and Engineering Division of the U.S. Department ofEnergy under contract no. DE-AC02-05-CH11231 (van derWaals heterostructures program, KCWF16) [1] Y. Cao, V. Fatemi, A. Demir, S. Fang, S. L. Tomarken, J. Y. Luo,J. D. Sanchez-Yamagishi, K. Watanabe, T. Taniguchi, E. Kaxi-ras, et al. , Nature , 80 (2018). [2] M. Yankowitz, S. Chen, H. Polshyn, Y. Zhang, K. Watanabe,T. Taniguchi, D. Graf, A. F. Young, and C. R. Dean, Science ,1910 (2019). [3] X. Lu, P. Stepanov, W. Yang, M. Xie, M. A. Aamir, I. Das,C. Urgell, K. Watanabe, T. Taniguchi, G. Zhang, A. Bachtold,A. H. MacDonald, and D. K. Efetov, Nature , 653– (2019).[4] P. Stepanov, I. Das, X. Lu, A. Fahimniya, K. Watanabe,T. Taniguchi, F. H. Koppens, J. Lischner, L. Levitov, and D. K.Efetov, arXiv preprint arXiv:1911.09198 (2019).[5] Y. Cao, D. Rodan-Legrain, J. M. Park, F. Noah Yuan, K. Watan-abe, T. Taniguchi, R. M. Fernandes, L. Fu, and P. Jarillo-Herrero, arXiv e-prints , 2004.04148 (2020).[6] X. Liu, Z. Wang, K. Watanabe, T. Taniguchi, O. Vafek, andJ. I. A. Li, arXiv e-prints , 2003.11072 (2020).[7] Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxi-ras, and P. Jarillo-Herrero, Nature , 43 (2018).[8] Y. Saito, J. Ge, K. Watanabe, T. Taniguchi, and A. F. Young,Nature Physics , 926 (2020).[9] H. S. Arora, R. Polski, Y. Zhang, A. Thomson, Y. Choi,H. Kim, Z. Lin, I. Z. Wilson, X. Xu, J.-H. Chu, K. Watanabe,T. Taniguchi, J. Alicea, and S. Nadj-Perge, Nature , 379(2020).[10] A. L. Sharpe, E. J. Fox, A. W. Barnard, J. Finney, K. Watanabe,T. Taniguchi, M. A. Kastner, and D. Goldhaber-Gordon, Sci-ence , 605 (2019), arXiv:1901.03520 [cond-mat.mes-hall].[11] M. Serlin, C. L. Tschirhart, H. Polshyn, Y. Zhang, J. Zhu,K. Watanabe, T. Taniguchi, L. Balents, and A. F. Young, Sci-ence , 900 (2020).[12] K. P. Nuckolls, M. Oh, D. Wong, B. Lian, K. Watanabe,T. Taniguchi, B. A. Bernevig, and A. Yazdani, arXiv e-prints ,2007.03810 (2020).[13] S. Wu, Z. Zhang, K. Watanabe, T. Taniguchi, and E. Y. Andrei,arXiv eprints , 2007.03735 (2020).[14] I. Das, X. Lu, J. Herzog-Arbeitman, Z.-D. Song, K. Watanabe,T. Taniguchi, B. A. Bernevig, and D. K. Efetov, arXiv e-prints, 2007.13390 (2020).[15] C. L. Tschirhart, M. Serlin, H. Polshyn, A. Shragai, Z. Xia,J. Zhu, Y. Zhang, K. Watanabe, T. Taniguchi, M. E. Huber, andA. F. Young, arXiv e-prints , 2006.08053 (2020).[16] Y. Saito, J. Ge, L. Rademaker, K. Watanabe, T. Taniguchi, D. A.Abanin, and A. F. Young, arXiv e-prints , 2007.06115 (2020).[17] A. Kerelsky, L. J. McGilly, D. M. Kennes, L. Xian,M. Yankowitz, S. Chen, K. Watanabe, T. Taniguchi, J. Hone,C. Dean, A. Rubio, and A. N. Pasupathy, Nature , 95(2019).[18] Y. Choi, J. Kemmer, Y. Peng, A. Thomson, H. Arora, R. Polski,Y. Zhang, H. Ren, J. Alicea, G. Refael, F. von Oppen, K. Watan-abe, T. Taniguchi, and S. Nadj-Perge, Nature Physics (2019),10.1038/s41567-019-0606-5.[19] Y. Jiang, X. Lai, K. Watanabe, T. Taniguchi, K. Haule, J. Mao,and E. Y. Andrei, Nature , 91 (2019).[20] Y. Xie, B. Lian, B. J¨ack, X. Liu, C.-L. Chiu, K. Watanabe,T. Taniguchi, B. A. Bernevig, and A. Yazdani, Nature , 101(2019).[21] D. Wong, K. P. Nuckolls, M. Oh, B. Lian, Y. Xie, S. Jeon,K. Watanabe, T. Taniguchi, B. A. Bernevig, and A. Yazdani,Nature , 198 (2020).[22] S. L. Tomarken, Y. Cao, A. Demir, K. Watanabe, T. Taniguchi,P. Jarillo-Herrero, and R. C. Ashoori, Phys. Rev. Lett. ,046601 (2019).[23] U. Zondiner, A. Rozen, D. Rodan-Legrain, Y. Cao, R. Queiroz,T. Taniguchi, K. Watanabe, Y. Oreg, F. von Oppen, A. Stern,E. Berg, P. Jarillo-Herrero, and S. Ilani, Nature , 203(2020).[24] M. Xie and A. H. MacDonald, Phys. Rev. Lett. , 097601(2020). [25] S. Liu, E. Khalaf, J. Y. Lee, and A. Vishwanath, arXiv preprintarXiv:1905.07409 (2019).[26] T. Cea and F. Guinea, Phys. Rev. B , 045107 (2020).[27] N. Bultinck, E. Khalaf, S. Liu, S. Chatterjee, A. Vishwanath,and M. P. Zaletel, Phys. Rev. X , 031034 (2020).[28] J. Kang and O. Vafek, Phys. Rev. Lett. , 246401 (2019).[29] E. Khalaf, S. Chatterjee, N. Bultinck, M. P. Zaletel, andA. Vishwanath, arXiv preprint arXiv:2004.00638 (2020).[30] V. Kozii, H. Isobe, J. W. F. Venderbos, and L. Fu, Phys. Rev. B , 144507 (2019).[31] R. M. Fernandes and J. W. F. Venderbos, Sci-ence Advances (2020), 10.1126/sciadv.aba8834,https://advances.sciencemag.org/content/6/32/eaba8834.full.pdf.[32] D. V. Chichinadze, L. Classen, and A. V. Chubukov, Phys. Rev.B , 224513 (2020).[33] Y.-Z. You and A. Vishwanath, npj Quantum Materials , 16(2019).[34] V. Kozii, M. P. Zaletel, and N. Bultinck, arXiv e-prints ,2005.12961 (2020).[35] Y. Wang, J. Kang, and R. M. Fernandes, arXiv e-prints ,2009.01237 (2020).[36] A. Rozen, J. M. Park, U. Zondiner, Y. Cao, D. Rodan-Legrain, T. Taniguchi, K. Watanabe, Y. Oreg, A. Stern, E. Berg,P. Jarillo-Herrero, and S. Ilani, arXiv e-prints , 2009.01836(2020).[37] Y. Saito, J. Ge, K. Watanabe, T. Taniguchi, E. Berg, and A. F.Young, arXiv e-prints , 2008.10830 (2020).[38] H. Nielsen and S. Chadha, Nuclear Physics B , 445 (1976).[39] H. Watanabe and H. Murayama, 10.1103/Phys-RevLett.108.251602.[40] H. Watanabe and H. Murayama, (2013), 10.1103/Phys-RevLett.110.181601.[41] H. Watanabe and H. Murayama, (2014), 10.1103/Phys-RevX.4.031057.[42] A. Matsugatani, Y. Ishiguro, K. Shiozaki, and H. Watanabe,Phys. Rev. Lett. , 096601 (2018).[43] H. C. Po, L. Zou, A. Vishwanath, and T. Senthil, Phys. Rev. X , 031089 (2018).[44] L. Zou, H. C. Po, A. Vishwanath, and T. Senthil, Phys. Rev. B , 085435 (2018).[45] J. Ahn, S. Park, and B.-J. Yang, Phys. Rev. X , 021013 (2019).[46] Z. Song, Z. Wang, W. Shi, G. Li, C. Fang, and B. A. Bernevig,Phys. Rev. Lett. , 036401 (2019).[47] H. C. Po, L. Zou, T. Senthil, and A. Vishwanath, Physical Re-view B (2019), 10.1103/physrevb.99.195455.[48] B. Lian, F. Xie, and B. A. Bernevig, Phys. Rev. B , 041402(2020).[49] G. Tarnopolsky, A. J. Kruchkov, and A. Vishwanath, Phys. Rev.Lett. , 106405 (2019).[50] J. Liu, J. Liu, and X. Dai, Phys. Rev. B , 155415 (2019).[51] S. Carr, S. Fang, Z. Zhu, and E. Kaxiras, An exact contin-uum model for low-energy electronic states of twisted bilayergraphene , Tech. Rep. (2019) arXiv:1901.03420v4.[52] N. Bultinck, S. Chatterjee, and M. P. Zaletel, Phys. Rev. Lett. , 166601 (2020).[53] P. J. Ledwith, G. Tarnopolsky, E. Khalaf, and A. Vishwanath,Phys. Rev. Research , 023237 (2020).[54] K. Hejazi, C. Liu, H. Shapourian, X. Chen, and L. Balents,Phys. Rev. B , 035111 (2019).[55] Y. H. Kwan, Y. Hu, S. H. Simon, and S. A. Parameswaran,arXiv e-prints , 2003.11560 (2020).[56] F. Wu and S. Das Sarma, Phys. Rev. Lett. , 046403 (2020).[57] F. Wu and S. Das Sarma, Phys. Rev. B , 155149 (2020). [58] A. Auerbach, Interacting electrons and quantum magnetism (Springer Science & Business Media, 2012).[59] C. Repellin, Z. Dong, Y.-H. Zhang, and T. Senthil, Phys. Rev.Lett. , 187601 (2020).[60] Y.-H. Zhang, D. Mao, Y. Cao, P. Jarillo-Herrero, and T. Senthil,Phys. Rev. B , 075127 (2019).[61] A. Macdonald, “Modeling moir´e superlattices,” (2020), aspenWinter conference: quantum matter.[62] A. Kumar, M. Xie, and A. MacDonald, arXiv preprintarXiv:2010.05946 (2020).[63] D. S. Wei, T. van der Sar, S. H. Lee, K. Watanabe, T. Taniguchi,B. I. Halperin, and A. Yacoby, Science , 229 (2018),https://science.sciencemag.org/content/362/6411/229.full.pdf.[64] G. Wang, A. Chernikov, M. M. Glazov, T. F. Heinz, X. Marie,T. Amand, and B. Urbaszek, Rev. Mod. Phys , 21001 (2018).[65] S. Chatterjee, M. Ippoliti, and M. P. Zaletel, arXiv preprintarXiv:2010.01144 (2020).[66] O. Vafek and J. Kang, arXiv preprint arXiv:2009.09413 (2020). ν State Symmetry Type n Irrep characters d χ ( e iαs z ) χ ( e iβP K s z ) χ ( e iϑP ↓ s z ) χ ( C ) χ ( e iϕη z ) χ ( e iζP ↓ η z ) γ z η x s { C , SU(2) S , η z M y ,iη z T }
G-I 1 1 1 - - 1 - -1 3 α - - 3 - -PG 1 1 1 - - 1 - -1 3 α - - 3 - -N 1 2 2 - - − - -1 6 α - - − - -VP γ η z s { C , SU(2) K , SU(2) K (cid:48) ,U V (1) , M y , C T }
PG 2 4 α β ϑ e iϕ (1 + e iζ ) N 1 8 α β ϑ − e iϕ e iζ ) VH γ z η z s { C , SU(2) K , SU(2) K (cid:48) ,U V (1) , M x , T }
PG 1 8 α β ϑ ϕ ζ cos ζ N 1 2 2 2 2 − α β ϑ − ζ SP γ η s z { C , S zK , S zK (cid:48) ,U V (1) , C , M y , T }
G-II 1 2 e iα e iβ e iϑ ζ PG 1 2 e iα e iβ e iϑ ζ e iα e iβ e iϑ ϕ ζ N 1 4 e iα e iβ e iϑ − ϕ ζ e iα e iβ ) 2(1 + e iϑ ) − ζ T -IVC γ η x s { C , SU(2) S , M y , T }
G-I 1 1 1 - - 1 - -1 3 α - - 3 - -PG 1 1 1 - - 1 - -1 3 α - - 3 - -N 1 2 2 - - − - -1 6 α - - − - -2 SP K-IVC γ η P ↓ + γ z η x P ↑ { C , S z , η z M y ,e iζP ↓ η z , η z T }
G-I 1 1 1 - - 1 - 1G-II 1 2 e − iα - - 2 - ζ PG 1 1 1 - - 1 - 11 2 e − iα - - 2 - ζ N 1 2 2 - - − - 21 4 e − iα - - − - ζ SP VP γ η P ↓ + γ η z P ↑ { C , SU(2) K , S zK (cid:48) ,U V (1) , M y , T }
G-II 1 1 e − iα e − iϑ e − iζ PG 1 1 e − iα e − iϑ e − iζ e − iα β e − iϑ e iϕ e iζ N 1 2 e − iα e − iϑ − e − iζ e − iα β e − iϑ − e iϕ e iζ ) SP VH γ η P ↓ + γ z η z P ↑ { C , S zK , S zK (cid:48) ,U V (1) , M x , T }
G-II 1 2 e − iα e − iβ e − iϑ ζ PG* 1 2 e − iα e − iβ e − iϑ ϕ ζ PG 1 2 β ϑ ϕ
2N 1 2 2 2 2 − e − iα e − iβ e − iϑ − ϕ ζ e − iα e − iβ e − iϑ − ζ SP T -IVC γ η P ↓ + γ η x P ↑ { C , S z , M y ,e iθP ↓ η z , T }
G-I 1 1 1 - - 1 - 1G-II 1 2 e − iα - - 2 - ζ PG 1 1 1 - - 1 - 11 2 e − iα - - 2 - ζ N 1 2 2 - - − - 21 4 e − iα - - − - ζ TABLE II.
Symmetry representations for the soft modes for the C = 0 insulators at even integer fillings : The labels ’VP’, ’VH’, and’IVC’ have the same meaning as in Table I with the additional label ’SP’ denoting spin-polarized states. The projector P ↑ / ↓ projects on thespin ↑ / ↓ sector. As in Table I, the soft modes are divided into goldstone (G) which can be of type I (linear dispersion) or type II (quadraticdispersion), pseudo-goldstone (PG), and nematic modes (N). PG* denotes pseudo-goldstone modes which do not correspond to breaking acontinuous physical symmetry yet are gapless to a very good approximation. The irrep characters χ ( g ) corresponding to S z , S zK,K (cid:48) , U V (1) as well as C are given, with n denotes the count of irreps with these characters. The total number of modes is − ν , half of which arenematic, in agreement with the expression derived in the main text. ν C State Symmetry Type n Irrep characters d χ ( e iαs z ) χ ( e iβP K s z ) χ ( e iϑP K (cid:48) s z ) χ ( C ) χ ( e iϕη z ) χ ( e iζP ↓ η z ) γ z { C , SU(2) K , SU(2) K (cid:48) ,U V (1) , C , M y T }
N 2 1 1 1 1 e − πi α β ϑ e − πi ζ α β ϑ e − πi ϕ ζ cos ζ P + + P − ( P ↑ η z − P ↓ ) { C , S zK , SU(2) K (cid:48) , U V (1) , M x T }
G-II 1 1 e iα e iβ e − iζ PG* 1 2 e iα e iβ ϑ e iϕ e iζ N 2 1 1 1 1 e − πi e iα e iβ e − πi e − iζ e iα e iβ ϑ e − πi e − iϕ e − iζ (1 + e − iζ ) e iα e iβ ϑ e − πi e iϕ e − iζ e − iα e − iβ ϑ e − πi e iϕ e − iζ (1 + e − iζ ) α ϑ e − πi ζP + + P − ( P ↑ η x − P ↓ ) { C , S z , e iθP ↓ η z , M y T }
G-I 1 1 1 - - 1 - 1G-II 1 2 e iα - - 2 - ζ N 4 1 1 - - e − πi - 11 2 2 - - e − πi - ζ e iα - - e − πi - ζ e − iα - - e − πi - ζ P + + P − η z { C , SU(2) K , SU(2) K (cid:48) , U V (1) , M x T }
PG* 1 4 α β ϑ e iϕ (1 + e iζ ) N 1 1 1 1 1 e − πi α ϑ e − πi ζ α β ϑ e − πi e iϕ (1 + e iζ ) P + + P − s z { C , S zK , S zK (cid:48) , U V (1) , C , M x T }
G-II 1 2 e iα e iβ e iϑ ζ PG* 1 2 e iα e iβ e iϑ ϕ ζ N 2 1 1 1 1 e − πi β ϑ e − πi ϕ ζ e iα e iβ e iϑ e − πi ζ e iα e iβ e iϑ e − πi ϕ ζP + + P − η x { C , SU(2) , M y T }
G-I 1 1 1 - - 1 - -PG 1 3 α - - 3 - -N 2 1 1 - - e − πi - -2 3 α - - e − πi - -3 1 P + + P − ( P ↑ + P ↓ η z ) { C , S zK (cid:48) , SU(2) K , U V (1) , M x T }
G-II 1 1 e − iα e − iϑ e − iζ PG* 1 2 e − iα β e − iϑ e iϕ e iζ N 1 1 1 1 1 e − πi e − iα e − iϑ e − πi e − iζ e − iα β e − iϑ e − πi e iϕ e iζ P + + P − ( P ↑ η x + P ↓ ) { C , S z , e iθP ↓ η z ,M y T }
G-I 1 1 1 - - 1 - 1G-II 1 2 e − iα - - 2 - ζ N 1 2 e − iα - - e − πi - ζ e − πi - 1 TABLE III.
Symmetry representation of the soft modes for the Chern insulators : Insulators with maximal Chern number C = | − ν | at integer filling ν , which are expected to be stabilized at finite out-of-plane field, are considered. The state Q is described in terms of theprojectors P ± = ± γ z and P ↑ / ↓ = ± s z . As in Table I, the soft modes are divided into goldstone (G), pseudo-goldstone (PG) which canbe of type I (linear dispersion) or type II (quadratic dispersion), and nematic modes (N). PG* denotes pseudo-Goldstone modes which do notcorrespond to breaking a continuous physical symmetry yet are gapless to a very good approximation. The irrep characters χ ( g ) correspondingto S z , S zK,K (cid:48) , U V (1) as well as C are given, with n denotes the count of irreps with these characters. The total number of modes is − ν with − ν ) nematic modes in agreement with the expression derived in the main text. ν Type J H = 0 J H > J H < n ω q n ω q n ω q √ ρJ | q | (cid:112) ρ (4 J + 3 J H ) | q | (cid:112) ρ (4 J − J H ) | q | (cid:112) J H (4 J − J H ) (cid:112) ρ ( J + 3 J H ) | q | (cid:112) − J H (4 J − J H ) PG 4 (cid:112) λ ( J + λ ) (cid:112) (4 λ + 3 J H )( J + λ ) (cid:112) (4 λ − J H )( J + λ − J H ) (cid:112) (4 λ + 3 J H )( J + λ + J H ) (cid:112) (4 λ − J H )( J + λ ) N 8 β (cid:112) β (4 β + 3 J H ) (cid:112) β − J H )(4 β − J H ) (cid:112) ( β + J H )(4 β + 3 J H ) (cid:112) β (4 β − J H ) √ ρJ | q | (cid:112) ρ (4 J + J H ) | q | (cid:112) ρ (4 J − J H ) | q | ρ q √ ρJ H | q | ρ q − J H + ρ q PG 1 (cid:112) λ ( λ + J ) (cid:112) (4 λ − J H )( λ + J ) (cid:112) (4 λ + J H )( λ + J ) λ (cid:112) λ (2 λ + J H ) λ λ − J H N 2 β (cid:112) β (4 β − J H ) (cid:112) β (4 β + J H ) β − J (cid:112) (2 β − J )(2 β − J + J H ) β − J ) β − J − J H ) TABLE IV.
Soft mode dispersion : Dispersion of the bosonic soft modes for the spinful model at charge neutrality ν = 0 and half-filling ν = 2 obtained from the sigma model defined in Eqs. 44 and 46 to leading order in the momentum q in terms of the sigma model parameter J , λ , and β = α − J + λ . At ν = 0 and in the absence of intervalley Hund’s coupling J H , there are 4 degenerate linearly dispersing gaplessgoldstone (G) modes, four degenerate gapped psuedo-goldstone (PG) modes, and 8 degenerate nematic (N) modes. For J H > , three ofgapless modes acquire a gap ∼ √ J H J whereas for J H < , three modes remain gapless and one acquires a gap ∼ (cid:112) | J H | J . At ν = 2 and inthe absence of intervalley Hund’s coupling J H , there are 3 gapless goldstone (G) modes (one with linear and two with quadratic dispersion),3 gapped psuedo-goldstone (PG) modes, and 6 nematic (N) modes. For J H > , the three goldstone modes remain gapped but the dispersionof two of them become linear rather than quadratic whereas for J H < m one of the two quadratic modes acquires a gap ∼ | J H | . Appendix A: Time-dependent Hartree-Fock
In this appendix we provide more details about the derivation of the TDHF equation. For convenience, we repeat here theinteracting continuum model for MATBG: ˆ H = (cid:88) k c † k h ( k ) c k + 12 A (cid:88) q V q δρ q δρ − q (A1)Before discussing the TDHF equation, we first define for future use the following generalized Hartree Hamiltonian functionalconstructed from ˆ H : H H { φ q } ( k ) = 1 A (cid:88) G V q + G (cid:34)(cid:88) k (cid:48) tr (Λ − q − G ( k (cid:48) ) φ q ( k (cid:48) )) (cid:35) Λ q + G ( k ) . We also similarly define a corresponding generalized Fock Hamiltonian functional: H F { φ q } ( k ) = − A (cid:88) q (cid:48) V q (cid:48) Λ q (cid:48) ( k ) φ q ( k + q (cid:48) )Λ − q (cid:48) ( k + q + q (cid:48) ) (A2)Note that H H { φ q } and H F { φ q } respectively become the conventional Hartree and Fock Hamiltonians if we take q = 0 . Thesum of the generalized Hartree and Fock Hamiltonians we write as H HF { φ q } ( k ) = H H { φ q } ( k ) + H F { φ q } ( k ) (A3)Let us now consider a solution of the self-consistent Hartree-Fock equation described by the correlation matrix [ P ( k )] αβ = (cid:104) c † β, k c α, k (cid:105) . If we define the corresponding Hartree-Fock Hamiltonian as H SC { P } ( k ) = h ( k ) + H HF { P } ( k ) , (A4)then self-consistency implies that [ P ( k ) , H SC { P } ( k )] = 0 .To derive the TDHF equation, we define the following bosonic operator: ˆ φ q = (cid:88) k c † k φ q ( k ) c k + q , (A5)and evaluate its commutator with ˆ H . As discussed in the main text, this commutator has to be evaluated at the mean-field levelby reducing the four-fermion terms to two-fermion terms by partial Wick contractions with P ( k ) . If we write the resultingpartially contracted commutator as (cid:104) [ ˆ H, ˆ φ q ] (cid:105) HF , then we find (cid:104) [ ˆ H, ˆ φ q ] (cid:105) HF = (cid:88) kk (cid:48) (cid:88) αβλγ c † α, k c β, k + q ˜ L αβ,λγ q ( k , k (cid:48) ) φ λγ q ( k (cid:48) ) ≡ (cid:88) k (cid:88) λγ c † α, k c β, k + q ˜ L αβ [ φ q ]( k ) (A6)In the first line of Eq. (A6) we have written ˜ L q as a matrix acting on the vector φ q ( k ) , while in the second line we have written ˜ L as an operator acting on matrices. Both notations will be used below. The explicit expression for ˜ L is most conveniently writtenvia its action on matrices, and it takes the following form: ˜ L [ φ q ]( k ) = [ H SC , φ q ]( k ) + H HF { [ P, φ q ] } ( k ) , (A7)where we have used the Hamilonians H SC and H HF respectively defined in Eqs. (A4) and (A3).At this point, it is important to note that we have defined ˆ φ q using a general matrix φ αβ q ( k ) with both α and β running overall band and flavor indices. However, we want to restrict ourselves to only those ˆ φ q which create or annihilate particle-holeexcitations of the mean-field band spectrum. To this end, we define the projector P PH P αβ,λγ PH ( k , k (cid:48) ) = δ k , k (cid:48) (cid:2) P ⊥ αλ ( k ) P βγ ( k ) + P αλ ( k ) P ⊥ βγ ( k ) (cid:3) (A8)6Acting with P PH on φ q projects out all contributions in ˆ φ q which do not create or annihilate a particle-hole excitation. Corre-spondingly, we are also only interested in the part of ˜ L q which acts within the particle-hole subspace, so we define L q = P P H ˜ L q P P H (A9)To find the soft mode spectrum ω q , we numerically solve for the smallest eigenvalues of L q to obtain the bosonic operatorswhich satisfy i∂ t ˆ φ q = [ ˆ H, ˆ φ q ] = ω q ˆ φ q at the mean-field level. However, note that L q is not hermitian. To investigate theproperties of the eigenvalue problem at hand, we first define the following matrix Z αβ,λγ ( k , k (cid:48) ) = δ k , k (cid:48) [ P αλ ( k ) δ γβ − δ αλ P γβ ( k )] (A10)As an operator acting on matrices, the action of Z can be written as a simple commutator: Z φ q = [ P, φ q ] . It is readily verifiedthat this matrix satisfies Z = P P H and has eigenvalues +1 ( − ) if α, λ lie in the subspace of occupied (unoccupied) mean-fieldstates and β, γ lie in the unoccupied (occupied) subspace. Let us now define the matrix H q via the relation L q = ZH q (A11)In contrast to L q , the matrix H q is hermitian. In fact, H q has one more important property – namely, it has a particle-holesymmetry: X † q H ∗ q X q = H − q with X αβ,λγ q ( k , k (cid:48) ) = δ [ k + q ] , k (cid:48) δ αγ δ βλ , (A12)where [ k + q ] lies in the first mini-BZ, and is equal to k + q modulo a moire reciprocal lattice vector. From X q Z = −Z X q ,we also immediately see that L q satisfies X † q L ∗ q X q = −L − q (A13)From this we conclude that solving the TDHF equation is equivalent to solving the equation of motion of an effective quadraticboson Hamiltonian H q obtained from ˆ H at the mean-field level. As discussed in the main text, this interpretation is consistentwith the fact that the self-consistency condition implies that ˆ H creates at least two particle-hole excitations. Appendix B: Derivation of the soft mode Hamiltonian H q In this appendix, we will provide details for the derivation of the soft mode Hamiltonian H q in Sec. IV B. The derivationfollows closely the related derivation in Ref. [29] and employ the same notation and conventions. H q is obtained by expandingthe energy defined in Eq. 23 to second order in φ . Written more explicitly E ( Q, φ ) = (cid:104) ψ ( φ ) | ˆ H | ψ ( φ ) (cid:105) = (cid:104) e − i (cid:80) q ˆ φ q ˆ He i (cid:80) q ˆ φ q (cid:105) (B1)The second order term in φ is given by E = − A (cid:88) q , q (cid:48) V q (cid:48) (cid:104) [ ˆ φ q , ρ q (cid:48) ][ ˆ φ − q , ρ − q (cid:48) ] (cid:105) = − A (cid:88) q , q (cid:48) V q (cid:48) Tr ˜ P T [ ˜ φ q , ˜Λ q (cid:48) ][ ˜ φ − q , ˜Λ − q (cid:48) ] (B2)The second equality is obtained by evaluating the expectation value of the product of commutators of the two bilinear operatorsas explained in Ref. [29]. Here, the trace with capital T includes momentum summation in addition to tracing over the matrixindex and the symbols with a tilde are matrices in momentum as well as internal indices defined as [ ˜ φ q ] k , k (cid:48) = φ q ( k ) δ k (cid:48) , k + q , [˜Λ q ] k , k (cid:48) = Λ q ( k ) δ k (cid:48) , k + q , [ ˜ P ] k , k (cid:48) = P δ k (cid:48) , k (B3)The projector P is related to Q via P = (1 + Q ) . We notice that since φ q anticommutes with Q while Λ q commutes with Q ,the term containing Q in the energy vanishes since (cid:88) q , q (cid:48) V q (cid:48) Tr ˜ Q T [ ˜ φ q , ˜Λ q (cid:48) ][ ˜ φ − q , ˜Λ − q (cid:48) ] = − (cid:88) q , q (cid:48) V q (cid:48) Tr ˜ Q T [ ˜ φ − q , ˜Λ − q (cid:48) ][ ˜ φ q , ˜Λ q (cid:48) ] = − (cid:88) q , q (cid:48) V q (cid:48) Tr ˜ Q T [ ˜ φ q , ˜Λ q (cid:48) ][ ˜ φ − q , ˜Λ − q (cid:48) ] = 0 (B4)7Here, we have used the cyclic property of the trace in the first line and made the replacements q (cid:55)→ − q , q (cid:48) (cid:55)→ − q (cid:48) in the secondline while also using V − q = V q . Thus, Eq. B2 simplifies to E = − A (cid:88) q , q (cid:48) V q (cid:48) Tr[ ˜ φ q , ˜Λ q (cid:48) ][ ˜ φ − q , ˜Λ − q (cid:48) ]= − A (cid:88) q , q (cid:48) , k V q (cid:48) [ φ q ( k )Λ q (cid:48) ( k + q ) − Λ q (cid:48) ( k ) φ q ( k + q (cid:48) )][ φ − q ( k + q + q (cid:48) )Λ − q (cid:48) ( k + q (cid:48) ) − Λ − q (cid:48) ( k + q (cid:48) + q ) φ − q ( k + q )]= − A (cid:88) q , q (cid:48) , k V q (cid:48) [ φ q ( k )Λ q (cid:48) ( k + q ) − Λ q (cid:48) ( k ) φ q ( k + q (cid:48) )][ φ † q ( k + q (cid:48) )Λ † q (cid:48) ( k ) − Λ † q (cid:48) ( k + q ) φ † q ( k )] (B5)where we used the relations φ q ( k ) = φ †− q ( k + q ) and Λ q ( k ) = Λ †− q ( k + q ) . To simplify this expression further, we expand φ into a sum of generators φ q ( k ) = (cid:88) γ,γ (cid:48) = ± N γ,γ (cid:48) (cid:88) µ =1 φ αγ,γ (cid:48) ; q ( k ) r γ,γ (cid:48) α , tr[ r γ ,γ (cid:48) α ] † r γ ,γ (cid:48) β = 2 δ α,β δ γ ,γ δ γ (cid:48) ,γ (cid:48) (B6)where we have split the generators into those connecting the Chern sector γ = ± to γ (cid:48) = ± . In the Chern basis, they have thesimple forms r ++ ∝ (cid:18) (cid:19) , r + − ∝ (cid:18) (cid:19) , r − + ∝ (cid:18) (cid:19) , r −− ∝ (cid:18) (cid:19) (B7)For Λ q ( k ) = F q ( k ) e i Φ q ( k ) γ z , it is easy to see that Λ q ( k ) r γ,γ (cid:48) α = F q ( k ) e iγ Φ q ( k ) r γ,γ (cid:48) α , r γ,γ (cid:48) α Λ q ( k ) = F q ( k ) e iγ (cid:48) Φ q ( k ) r γ,γ (cid:48) α (B8)Substituting (B6) in (B5) and using (B8), we get E = 1 A (cid:88) q , q (cid:48) , k ,α,γ,γ (cid:48) V q (cid:48) F q (cid:48) ( k ) φ αγ,γ (cid:48) , q ( k ) (cid:110) F q (cid:48) ( k )[ φ αγ,γ (cid:48) , q ( k )] ∗ − F q (cid:48) ( k + q )[ φ αγ,γ (cid:48) , q ( k + q (cid:48) )] ∗ e i [ γ Φ q (cid:48) ( k + q ) − γ (cid:48) Φ q (cid:48) ( k )] (cid:111) (B9)Comparing with the definition of H q in Eq. (20), this implies that, as anticipated, H q has a block diagonal form in the γ indicesand is proportional to unity in the remaining internal indices with its explicit form given by H µ,νγ,γ (cid:48) ; q ( k , k (cid:48) ) = δ µν A (cid:88) q (cid:48) V q (cid:48) F q (cid:48) ( k ) (cid:110) δ k (cid:48) , k F q (cid:48) ( k ) − δ k (cid:48) , [ k + q (cid:48) ] F q (cid:48) ( k + q ) e i [ γ Φ q (cid:48) ( k + q ) − γ (cid:48) Φ q (cid:48) ( k )] (cid:111) (B10)which reduces to Eq. (31) in the main text for q = 0 . Appendix C: Properties of ρ In this appendix, we investigate the properties of the matrix ρ defined as ρ µν = 14 tr Q [ t µ , t ν ] (C1)In the following, we will restrict ourselves to generators which anticommute with Q for which tr Q [ t µ , t ν ] = 2 tr Qt µ t ν leadingto Eq. (36) in the main text. In the following, we are going to show that when t µ goes over all generators which anticommutewith Q , ρ is a full rank matrix with an equal number of +1 and − eigenvalues equal to − ν . To this end, we start bychoosing a basis where Q is diagonal with +1 eigenvalues followed by − eigenvalues. Using the relation tr Q = 2 ν , it is easyto see that the number of ± eigenvalues is ± ν . Now let us define the matrices X ij whose elements are given by [ X ij ] mn = δ i,m δ j,n (C2)8It is easy to show that X ij satisfy the following identities X Tij = X ji , tr X Tij X nm = 2 δ in δ jm , QX ij = X ij (cid:40) +1 : 1 ≤ i ≤ ν − ν < i ≤ , X ij Q = X ij (cid:40) +1 : 1 ≤ j ≤ ν − ν < j ≤ (C3)We can then choose a basis of hermitian generators by restricting ourselves to X ij with i = 1 , . . . , ν and j = 4+ ν +1 , . . . , .Labelling such ordered pairs ( i l , j l ) by an index ν , we can define t ν − = X i ν ,j ν + X Ti ν ,j ν , t ν = i ( X i ν ,j ν − X Ti ν ,j ν ) (C4)which satisfy tr t µ t ν = 2 δ µ,ν , Qt ν − = − it ν , Qt ν = it ν − (C5)Thus, in this basis ρ has a block diagonal form with × blocks given by σ y . It follows that ρ is a full rank matrix with an equalnumber of +1 and − eigenvalues equal to − ν . Appendix D: Symmetry representations on the soft modes
In this appendix, we provide details for the procedure to explain the symmetry representations for the soft modes. We start byconsidering a general symmetry S with action, given by Eq. 53. The soft modes are defined through the operator ˆ φ n, q given byEq. 54. The symmetry S acts on φ n, q via the matrix representation S mn ( q ) (cf. Eq. 55). This matrix can be obtained explicitlyby first noting that the soft mode Hamiltonian H q transforms under S as [ SHS − ] µν q ( k , k (cid:48) ) ≡ (cid:88) λ,ρ [Γ λµ q ( k )] ∗ H λρO q ( O k , O k (cid:48) )Γ ρν q ( k (cid:48) ) (D1)where Γ is defined through U k t µ U † k + q = (cid:88) ν Γ νµ q ( k ) t ν (D2)For any S which leaves Q invariant, i.e. S ∈ G Q , H satisfies SHS − = H which means that φ n, q transforms as a representationunder the action of S : [ Sφ ] n, q ( k ) ≡ Γ q ( k ) φ n, q ( k ) = (cid:88) m S nm ( q ) φ m,O q ( O k ) (D3)The matrix S mn ( q ) can be obtained from the knowledge of the wavefunctions φ n, q ( k ) . These have a particularly simple formin the U(4) × U(4) limit as explained in the main text. In the following, we will employ this form to obtain the matrices S explicitly.We start by recalling the form of the symmetries when projected onto the flat bands which act in the Chern-pseudospin-spinbasis as [27, 29] U V c k U † V = e iφη z c k , S K/K (cid:48) c k S − K/K (cid:48) = e iP K/K (cid:48) n · s c k C c k C † = η x e iθ ( k ) c − k , T c k T − = η x γ x c − k C c k C † = e iθ ( k ) γ z c c k , M y c k M † y = γ x e iθ y ( k ) γ z c m y k (D4)where we have defined P K/K (cid:48) = ± η z . Notice the k -dependent phase factors θ , ,y ( k ) which cannot be removed due to theband topology of the bands in the sublattice basis. In particular, the phase θ ( k ) is equal to 0 at Γ and π at K and K (cid:48) points.In the U(4) × U(4) limit, there is a one-to-one correspondence between the lowest energy eigenstates of the soft modesHamiltonian H q and the generators r γ,γ (cid:48) α . To make correspondence manifest, we label the generator corresponding to theeigenstate φ n, q , n = 1 , . . . , − ν ) , by the indices ( γ n , γ (cid:48) n , α n ) . In this case, we can write the eigenstates as φ α,γ,γ (cid:48) n, q ( k ) = φ α,γ,γ (cid:48) α n ,γ n ,γ (cid:48) n , q ( k ) = δ α,α n δ γ,γ n δ γ (cid:48) ,γ (cid:48) n φ γ,γ (cid:48) ; q ( k ) (D5)9 FIG. 8. Magnitude and phase for the lowest eigenfunction of the inter-Chern soft mode Hamiltonian H + − (at q = 0 ) denoted by ∆( k ) in theperiodic gauge for κ = 0 . and κ = 0 . The phase winds by π around Γ and by − π at M , M (cid:48) and M (cid:48)(cid:48) . The phase winding at the M pointsis not associated with vanishing of | ∆( k ) | and thus can be removed by a singular gauge transformation. where φ γ,γ (cid:48) ; q ( k ) is the lowest energy eigenfunction of the Hamiltonian H γ,γ (cid:48) ; q given in (B9). For q = 0 , this has the simpleform φ γ,γ (cid:48) ; q =0 ( k ) = √ N : γ = γ (cid:48) ∆( k ) : γ = + , γ (cid:48) = − ∆( k ) ∗ : γ = − , γ (cid:48) = + (D6)where N is the number of points in the BZ and we assumed the normalization (cid:80) k | φ γ,γ (cid:48) ; q ( k ) | = 1 . ∆( k ) is inter-Chern softmode wavefunction introduced in the main text. The form of ∆( k ) depends on the gauge chosen for the flat band wave-functions.In the gauge chosen in Eq. 40 in the main text, the phase of ∆( k ) winds by π around the BZ due to a pair of vortices centeredat Γ . On the other hand, in a periodic gauge, the phase of ∆( k ) cannot wind around the BZ and extra vortices must appearelsewhere in the BZ. These extra vortices can be removed by a singular gauge transformation at the expense of making the gaugenon-periodic. In numerics, it is usually a lot easier to construct the wavefunctions in the periodic gauge. One such gauge choiceis shown in Fig. 8. We can see that the phase of the wavefunctions winds by π rather than π around Γ in addition to thepresence of π vortices at the three M points to cancel the total winding. By looking at the magnitude of the wavefunctions,we can see that the winding at the M points is not associated with a vanishing of the magnitude of ∆( k ) (except possibly at asingle point) and thus can be removed by a singular gauge transformation. On the other hand, the vortices at Γ are true vorticesassociated with the vanishing of the | ∆( k ) | . This can be more clearly illustrated by choosing a smaller value of κ where thevanishing of the magnitude at Γ becomes slower.The symmetry representation matrix can now be expressed in the basis of the generators r γ,γ (cid:48) µ using Eq. D3 Γ γ ,γ (cid:48) ,α ; γ ,γ (cid:48) ,β q ( k ) φ γ ,γ (cid:48) ; q ( k ) = S γ ,γ (cid:48) ,β ; γ ,γ (cid:48) ,α ( q ) φ γ ,γ (cid:48) ; O q ( O k ) (D7)0Note that there are no sums in the above equation. This equation can be solved for S ( q ) as S γ ,γ (cid:48) ,β ; γ ,γ (cid:48) ,α ( q ) = (cid:88) k φ γ ,γ (cid:48) ; O q ( O k ) ∗ Γ γ ,γ (cid:48) ,α ; γ γ (cid:48) ,β q ( k ) φ γ γ (cid:48) ; q ( k ) (D8)Let us start with the global symmetries: U(1) valley charge conservation or
SU(2) K × SU(2) K (cid:48) spin rotation. These symme-tries are characterized by a k -independent action U and do not act spatially, i.e. O = 1 . Furthermore, they act trivially on theChern index γ . As a result, their action on the generators r γ,γ (cid:48) α has the form Γ γ ,γ (cid:48) ,α ; γ ,γ (cid:48) ,β q ( k ) = δ γ ,γ δ γ (cid:48) ,γ (cid:48) U r γ γ (cid:48) β U † [ r γ γ (cid:48) α ] † (D9)Substituting in Eq. D8, we find S γ ,γ (cid:48) ,α ; γ ,γ (cid:48) ,β ( q ) = δ γ ,γ δ γ (cid:48) ,γ (cid:48) U r γ γ (cid:48) α U † [ r γ γ (cid:48) β ] † = 12 tr U r γ γ (cid:48) α U † [ r γ γ (cid:48) β ] † (D10)where we used the fact that U leaves the Chern index invariant. This can be now expressed back in the hermitian basis t µ (with tr t µ t ν = 2 δ µ,ν ) as S µν ( q ) = 12 tr U t µ U † t ν (D11)Next, let us consider C symmetry with U k = η x e iθ ( k ) and O k = − k . The symmetry action on the generators r γ,γ (cid:48) α is givenby Γ γ ,γ (cid:48) ,α ; γ ,γ (cid:48) ,β q ( k ) = δ γ ,γ δ γ (cid:48) ,γ (cid:48) e i [ θ ( k ) − θ ( k + q )] tr η x r γ γ (cid:48) β η x [ r γ γ (cid:48) α ] † (D12)We now restrict ourselves to the Γ point q = 0 and substitute in (D8) to get S γ ,γ (cid:48) ,α ; γ ,γ (cid:48) ,β (Γ) = δ γ ,γ δ γ (cid:48) ,γ (cid:48) η x r γ γ (cid:48) α η x [ r γ γ (cid:48) β ] † (cid:88) k φ γ ,γ (cid:48) ;Γ ( − k ) ∗ φ γ γ (cid:48) ;Γ ( k ) (D13)For the periodic gauge choice of Fig. 8, ∆( − k ) = − ∆( k ) leading to (cid:80) k φ γ ,γ (cid:48) ;Γ ( − k ) ∗ φ γ γ (cid:48) ;Γ ( k ) = γ γ (cid:48) . Substituting in(D17), we get S γ ,γ (cid:48) ,α ; γ ,γ (cid:48) ,β (Γ) = γ γ (cid:48) δ γ ,γ δ γ (cid:48) ,γ (cid:48) η x r γ γ (cid:48) α η x [ r γ γ (cid:48) β ] † = tr η x γ z r γ γ (cid:48) α η x γ z [ r γ γ (cid:48) β ] † (D14)Going back to the t µ basis, we find S C µν (Γ) = 12 tr η x γ z t µ η x γ z t ν (D15)Next, we consider C symmetry with U k = e iθ ( k ) γ z and O = c . The symmetry action on the generators is Γ γ ,γ (cid:48) ,α ; γ ,γ (cid:48) ,β q ( k ) = δ γ ,γ δ γ (cid:48) ,γ (cid:48) e i [ γθ ( k ) − γ (cid:48) θ ( k + q )] δ αβ (D16)Resticting to the Γ point, we get S γ ,γ (cid:48) ,α ; γ ,γ (cid:48) ,β (Γ) = δ γ ,γ δ γ (cid:48) ,γ (cid:48) δ αβ (cid:88) k e i ( γ − γ (cid:48) ) θ ( k ) φ γ ,γ (cid:48) ;Γ ( c k ) ∗ φ γ γ (cid:48) ;Γ ( k ) (D17)For γ = γ (cid:48) , it is easy to see that the k integral yields 1. For γ = − γ (cid:48) , we can verify by a direct calculation that the k integralyields e − πi γ which gives S γ ,γ (cid:48) ,α ; γ ,γ (cid:48) ,β (Γ) = δ γ ,γ δ γ (cid:48) ,γ (cid:48) δ αβ e πi ( γ − γ (cid:48) ) (D18)1Any easier way is to note that Eq. D7 holds for every k and thus can be used to determine S ( q ) by choosing a C invariantmomentum k for which φ Γ ( k ) does not vanish. Choosing k = K and noting that there is no phase winding at this pointfor the gauge of Fig. 8, we can immediately read off S (Γ) = δ γ ,γ δ γ (cid:48) ,γ (cid:48) δ αβ e θ ( K )( γ − γ (cid:48) ) which is the same as (D18) since θ ( K ) = π . If we now go back to the t µ basis, we find S C µν (Γ) = 12 tr e π γ z t µ e − π γ z t ν (D19)We next consider M y which flips the Chern sector so its action on the generators is Γ γ ,γ (cid:48) ,α ; γ ,γ (cid:48) ,β q ( k ) = δ γ , − γ δ γ , − γ e − i [ γ θ y ( k ) − γ (cid:48) θ y ( k + q )] tr γ x r γ ,γ (cid:48) β γ x [ r − γ , − γ (cid:48) β ] † (D20)Substituting in (D8), we get S γ ,γ (cid:48) ,α ; γ ,γ (cid:48) ,β (Γ) = δ γ ,γ δ γ (cid:48) ,γ (cid:48) γ x r γ γ (cid:48) α γ x [ r γ γ (cid:48) β ] † (cid:88) k e − i ( γ − γ (cid:48) ) θ y ( k ) φ − γ , − γ (cid:48) ;Γ ( m y k ) ∗ φ γ γ (cid:48) ;Γ ( k ) (D21)We can verify by direct evaluation that the k summation yields 1 regardless of γ and γ (cid:48) which yields S γ ,γ (cid:48) ,α ; γ ,γ (cid:48) ,β (Γ) = δ γ ,γ δ γ (cid:48) ,γ (cid:48) γ x r γ γ (cid:48) α γ x [ r γ γ (cid:48) β ] † = 12 tr γ x r γ γ (cid:48) α γ x [ r γ γ (cid:48) β ] † (D22)In the hermitian basis t µ , this becomes S M y µ,ν (Γ) = 12 tr γ x t µ γ x t n u (D23)Finally, we consider time-reversal symmetry whose action on the generators is Γ γ ,γ (cid:48) ,α ; γ ,γ (cid:48) ,β q ( k ) = δ γ , − γ δ γ , − γ γ x η x [ r γ ,γ (cid:48) α ] ∗ γ x η x [ r − γ , − γ (cid:48) β ] † (D24)Substituting in (D8), we get S γ ,γ (cid:48) ,α ; γ ,γ (cid:48) ,β (Γ) = δ γ ,γ δ γ (cid:48) ,γ (cid:48) γ x η x r γ γ (cid:48) α γ x η x [ r γ γ (cid:48) β ] † (cid:88) k φ − γ , − γ (cid:48) ;Γ ( − k ) φ γ γ (cid:48) ;Γ ( k ) = 12 tr γ x η x r γ γ (cid:48) α γ x η x [ r γ γ (cid:48) β ] † (D25)This becomes in the t µ basis S T µ,ν = 12 tr γ x η x t µ γ x η x t ∗ νν