SO(5) symmetry in the quantum Hall effect in graphene
Fengcheng Wu, Inti Sodemann, Yasufumi Araki, Allan H. MacDonald, Thierry Jolicoeur
SSO(5) symmetry in the quantum Hall effect in graphene
Fengcheng Wu, Inti Sodemann ∗ , Yasufumi Araki, Allan H. MacDonald, † and Thierry Jolicoeur ‡ Department of Physics, University of Texas at Austin, Austin, TX 78712, USA Laboratoire de Physique Th´eorique et Mod`eles statistiques,CNRS and Universit´e Paris-Sud, Orsay 91405, France
Electrons in graphene have four flavors associated with low-energy spin and valley degrees offreedom. The fractional quantum Hall effect in graphene is dominated by long-range Coulombinteractions which are invariant under rotations in spin-valley space. This SU(4) symmetry is spon-taneously broken at most filling factors, and also weakly broken by atomic scale valley-dependentand valley-exchange interactions with coupling constants g z and g ⊥ . In this paper we demonstratethat when g z = − g ⊥ an exact SO(5) symmetry survives which unifies the N´eel spin order parameterof the antiferromagnetic state and the XY valley order parameter of the Kekul´e distortion state intoa single five-component order parameter. The proximity of the highly insulating quantum Hall stateobserved in graphene at ν = 0 to an ideal SO(5) symmetric quantum Hall state remains an openexperimental question. We illustrate the physics associated with this SO(5) symmetry by studyingthe multiplet structure and collective dynamics of filling factor ν = 0 quantum Hall states basedon exact-diagonalization and low-energy effective theory approaches. This allows to illustrate howmanifestations of the SO(5) symmetry would survive even when it is weakly broken. PACS numbers: 73.22.Pr, 73.43.-f
I. INTRODUCTION
Electron-electron interactions in the fractional quan-tum Hall effect (FQHE) regime give rise to a host ofnon-perturbative and unexpected phenomena, includingimportantly the emergence of quasiparticles with frac-tional charge and statistics. In this paper we suggestthat neutral graphene in the FQHE regime could alsoprovide a relatively simple example of the complex many-particle physics that occurs in systems with simultane-ous quantum fluctuations of competing order parame-ters. Because each of its Landau levels has a four-fold spin/valley flavor degeneracy in the absence of Zee-man coupling, large gaps and associated quantum Halleffects are produced by single-particle physics only atfilling factors ν = ± , ± , . . . . The quantum Hall ef-fect nevertheless occurs at all intermediate integer fill-ing factors , and at many fractional filling factors ,usually with a broken symmetry incompressible groundstate. When lattice corrections to the continuum Diracmodel’s Coulomb interactions are ignored the groundstate at neutrality ( ν = 0) is a Slater determinant withall the N = 0 single-particle states of two arbitrarilychosen flavors occupied and, because the Hamiltonian isSU(4) invariant, has four independent degenerate Gold-stone modes. The rich flavor physics of graphene in thequantum Hall regime has already been established by ex-periments which demonstrate that phase transitions be-tween distinct many-electron states with the same fillingfactor ν can be driven by tuning magnetic field strengthor tilt-angle . ∗ Current address: Department of Physics, Massachusetts Instituteof Technology, Cambridge, MA 02139, USA
In graphene the competition between states withKekul´e-distorion(KD), antiferromagnetic(AF), ferromag-netic(F), charge-density wave(CDW), and other types oforder is controlled by Zeeman coupling to the electron-spin, and also by weak atomic-range valley-dependent interactions. A variety of approaches have been used toestimate these short-range corrections to the Coulomb in-teraction . In this paper, we adopt a two-parameterphenomenological model motivated by crystal momen-tum conservation and by the expectation that correctionsto the Coulomb interaction are significant only at dis-tances shorter than a magnetic length l B = (cid:112) (cid:126) c/eB ⊥ .( B ⊥ is the magnetic field component perpendicular tothe graphene plane.) We demonstrate that along a linein this parameter space SU(4) symmetry is reduced onlyto a SO(5) subgroup. In this paper, we take interaction-driven quantum Hall states at ν = 0 as an example toillustrate the physical manifestation of the SO(5) sym-metry. We explicitly derive a low-energy theory at ν = 0that is able to account simultaneously for N´eel anti-ferromagnetism and Kekul´e lattice-distortion order anddemonstrate that along the SO(5) line the four collectivemodes remain gapless in spite of the reduced symme-try. The exact SO(5) symmetry we have identified ingraphene’s quantum Hall regime is analogous to the ap-proximate symmetry conjectured in some models of high- T c superconductivity . Our work demonstrates that anenlarged symmetry like SO(5) can indeed be exactly real-ized in a realistic microscopic Hamiltonian. In the follow-ing, we start with a systematic analysis of Hamiltoniansymmetries and then use both exact-diagonalization andlow-energy effective models at ν = 0 to identify somesymmetry-related properties.Although our work focuses on the properties of thequantum Hall state at neutrality, we demonstrate thatthe SO(5) symmetry is an exact symmetry of the interac- a r X i v : . [ c ond - m a t . s t r- e l ] D ec tion Hamiltonian for the quantum Hall states in the zeroenergy Landau level of graphene. Therefore this symme-try is expected to emerge as well in the phase diagramsat arbitrary filling fractions in this Landau level.The quantum Hall state of graphene at neutrality isbelieved to be a canted antiferromagnet, as indicated bythe behaviour of the edge conductance in experimentswith tilted magnetic fields . However, as we argue be-low, these experiments are not sufficient to determinethe proximity of graphene to the ideal SO(5) symmetricstate. Even if graphene is in the antiferromagnetic side ofthe phase diagram, the presence of a weakly broken SO(5)symmetry would have important consequences, such asthe existence of additional weakly gapped neutral collec-tive modes as we will discuss in detail in Section IV andin Appendix D. II. HAMILTONIAN SYMMETRIES
When projected to the N = 0 Landau level (LL) thegraphene Hamiltonian is H = H C + H v + H Z ,H C = 12 (cid:88) i (cid:54) = j e (cid:15) | (cid:126)r i − (cid:126)r j | ,H v = 12 (cid:88) i (cid:54) = j (cid:0) g z τ iz τ jz + g ⊥ ( τ ix τ jx + τ iy τ jy ) (cid:1) δ ( (cid:126)r i − (cid:126)r j ) ,H Z = − (cid:15) Z (cid:88) i σ iz . (1)In Eq. (1) H C is the valley-independent Coulomb inter-action, (cid:15) is an environment-dependent effective dielectricconstant, H v is the short-range valley-dependent interac-tion, τ α ( α = x, y, z ) are Pauli matrices which act in valleyspace, H Z is the Zeeman energy , (cid:15) Z = µ B B where µ B is the Bohr magneton and B is the total magnetic fieldstrength, and σ α are Pauli matrices which act in spinspace. Note that B can have components both perpen-dicular and parallel to the graphene plane and that wehave chosen the ˆ z direction in spin-space to be alignedwith B . The form used for H v in Eq. (1) was proposedby Kharitonov .The short-range interaction coupling constants g z, ⊥ /l B are estimated to be ∼ a /l B times the Coulomb energyscale e /(cid:15)l B , where a ∼ . l B is the lattice constant ofgraphene. They are therefore weak and physically rel-evant mainly when they lift low-energy Coulomb-onlymodel degeneracies. For later notational conveniencewe define the energy scales u z, ⊥ = g z, ⊥ / (2 πl B ). TheCoulomb interaction H C in Eq. (1) commutes with the fif-teen SU(4) transformation generators which can be cho- sen as follows : S α = 12 (cid:88) i σ iα , T α = 12 (cid:88) i τ iα ,N α = 12 (cid:88) i τ iz σ iα , Π βα = 12 (cid:88) i τ iβ σ iα , (2)where the indices α = x, y, z and β = x, y . S α and T α are respectively the total spin and valley pseudospin. Dueto the equivalence between valley and sublattice degreesof freedom in the N = 0 LL of graphene, N α can beidentified as a N´eel vector. The physical meaning of thesix Π βα operators is discussed below.SU(4) symmetry is broken by the valley-dependentshort range interactions. At a generic point in the( g z , g ⊥ ) plane, H v breaks the SU(4) symmetry down toSU(2) s × U(1) v with the U(1) v symmetry correspondingto conservation of the valley polarization T z and theSU(2) s symmetry corresponding to global spin-rotationalinvariance. Two high-symmetry lines in the ( g z , g ⊥ ) pa-rameter space are evident : (1) for g ⊥ = 0 the systemis invariant under separate spin-rotations in each valleyyielding symmetry group SU(2) K s × SU(2) K (cid:48) s × U(1) v and(2) for g ⊥ = g z there is a full rotational symmetry invalley space yielding symmetry group SU(2) s × SU(2) v .We have discovered that there is even higher symmetryalong the g ⊥ = − g z line where the generic SU(2) s × U(1) v symmetry is enlarged to SO(5) : see Appendix A for anexplicit proof. Along this line the Hamiltonian commuteswith ten ( (cid:126)S , T z , and the six Π operators) of the fifteenSU(4) generators identified in Eq. (2). The other five( T x,y , N x,y,z ) SU(4) generators form a a natural order-parameter vector space on which the SO(5) group acts.As illustrated schematically in Fig. 1, spin operators (cid:126)S generate rotations in the N´eel vector space (cid:126)N , T z gen-erates rotations in the valley XY vector space T x,y , andthe Π operators generate rotations that connect these twospaces. When the Zeeman term is added to the Hamil-tonian the spin-symmetry is limited to invariance underrotations about the direction of the magnetic field. Thesymmetry groups of H C + H v and H and the correspond-ing generators are listed in Table I. NT x T y (cid:80) S T z FIG. 1: Schematic illustration of the five component( T x,y , N x,y,z ) order parameter space, and of rotations in thisvector space produced by the SO(5) generators. As we will demonstrate, the SO(5) symmetry is spon-taneously broken when it is exact. Provided that theZeeman and short-range interaction terms which ex-plicitly breaks SO(5) symmetry is not too strong, the( T x,y , N x,y,z ) vectors can be used to construct a usefulGinzburg-Landau model or quantum effective-field the-ory. The N´eel vector components of the order parametercharacterize the AF part of the order, while the valley XY components capture the KD part of the order.The SO(5) symmetry demonstrates that states which ap-pear quite different at a first glance are close in energyand that they can be continuously transformed into oneanother by appropriate rotations in the SO(5) order pa-rameter space. The 5D vector ( T x,y , N x,y,z ) identifiedhere provides a concrete example for the 56 possible quin-tuplets proposed in graphene . Although we focushere mainly on monolayer graphene, a similar symmetryanalysis applies to the N = 0 LL in bilayer graphene . III. EXACT DIAGONALIZATION
We have performed exact diagonalization (ED) studiesfor the Hamiltonian specified in Eq. (1) acting in a ν = 0torus-geometry Hilbert space with up to N φ = 8 orbitalsper flavor. When only Coulomb interactions are included,we verify that the ground state is a single Slater deter-minant with two occupied and two empty flavors . TheSU(4) multiplet structure of this broken-symmetry stateis discussed in Appendix B. We specify the ratio of g z to g ⊥ by the angle θ g = tan − ( g z /g ⊥ ) and fix the valley-dependent interaction strength g/l B = (cid:112) g ⊥ + g z /l B at0 . e / ( (cid:15)l B ). Because gN φ /l B is small compared to theCoulomb model charge-neutral energy gap that separatesthe ground state multiplet from the first excited multi-plet at zero momentum, the role of the valley-dependentinteractions is simply to lift the Coulomb model degen-eracy and split the corresponding SU(4) ground statemultiplet. Over the angle ranges θ g ∈ [ − π/ , π/
2] and θ g ∈ [5 π/ , π/
4] the exact ground states of H C + H v are single-Slater determinants, with F and CDW orderrespectively. For other values of θ g valley-dependent in-teractions are non-trivial.Fig. 2 illustrates the θ g -dependence of the Hamilto-nian spectrum for N e = 16 electrons in N = 0 Landaulevels with N φ = 8 over the θ g ∈ [ π/ , π/
4] interval.Fig. 2(a) plots ground state energies in various ( S z , T z )sectors and demonstrates that the overall ground statehas total valley polarization T z = 0 and total spin S = 0at all θ g values in this range. Note that the dependenceof energy on T z is suppressed as the CDW state is ap-proached ( θ g → π/
4) and that the dependence of en-ergy on S is suppressed as the F state is approached( θ g → π/ T z = 0 sec-tor of the SU(4) Coulomb ground-state multiplet is splitby H v . Since H v preserves SU(2) s spin symmetry, allenergies in Fig. 2(a,b) occur in SU(2) s multiplets. At θ g = 3 π/ S mergeto form SO(5) multiplets, each forming an irreduciblerepresentation of the SO(5) group. (A geometric repre- θ g S=0 / g E v / 2 T z = 8 7 6 5 4 3 2 1 0 T z =0 8 7 6 5 4 3 2 1 0 S (a) F AF KD CDW z g g -10123 π/ π/ π π/ E v / g (b) θ g T z = 0 S = 012345678 FIG. 2: Low-energy spectrum on the torus geometry for zerototal momentum, filling factor ν = 0, and orbital Landau leveldegeneracy N φ = 8 as a function of θ g in the range [ π/ , π/ E v is defined as the difference between the eigenvalues of H C + H v and the Coulomb-only ground state energy. All plottedeigenvalues are degenerate in the absence of H v . (a) Groundstate energies in a series of ( T z , S ) sectors. The solid linesshow the lowest T z = 0 energies for different total spin S values. Similarly, the dashed lines show the lowest spin singlet( S = 0) energies in different T z sectors. The ground state has S = 0 and T z = 0 throughout the plotted θ g range. The insetshows the mean-field phase diagram over the full θ g rangefrom Ref. 20. (b) Low-energy states in the T z = 0 sector fora series of total spin S quantum numbers. Note that at θ g =3 π/ S values are degenerate because ofthe hidden SO(5) symmetry. sentation of the SO(5) multiplet structure is provided inAppendix B.) All eigenstates have a definite value of theSO(5) Casimir operator Γ = S + T z + Π = l ( l + 3),with integer l = 0 , ...N φ . The low-energy spectrum at θ g = 3 π/ H effv ( θ g = 3 π u z (cid:16) N φ + 1 − N φ ( N φ + 5) N φ + 1 (cid:17) , (3)implying that the ground state, | G (3 π/ (cid:105) , is a SO(5) sin- TABLE I: Expanded symmetries along high-symmetry lines in the ( g z , g ⊥ ) plane. At a generic point in the ( g z , g ⊥ ) plane H C + H v has SU(2) s × U(1) v symmetry and H = H C + H v + H Z has U(1) s × U(1) v symmetry.Symmetry of H C + H v generators Symmetry of H generators g ⊥ = 0 SU(2) K s × SU(2) K (cid:48) s × U(1) v S α , N α , T z U(1) K s × U(1) K (cid:48) s × U(1) v S z , N z , T z g ⊥ = g z SU(2) s × SU(2) v S α , T α U(1) s × SU(2) v S z , T α g ⊥ + g z = 0 SO(5) S α , T z , Π xα , Π yα U(1) s × SU(2) S z , T z , Π xz , Π yz glet with Γ = 0. It follows that the 5D order parametervector ( T x,y , N x,y,z ) is maximally polarized : (cid:104) T x + T y + N (cid:105) π/ = (cid:104) C − Γ (cid:105) π/ = (cid:104) C (cid:105) π/ ≈ C ∗ , (4)where (cid:104)· · · (cid:105) π/ denotes expectation values in the groundstate | G (3 π/ (cid:105) and C ∗ = N φ ( N φ + 4) is the value of theSU(4) Casimir operator C in the Coulomb model SU(4)multiplet. The approximation leading to C ∗ in Eq. (4) isvalidated by numerical calculation, and also follows fromthe argument that | G (3 π/ (cid:105) is adiabatically connectedto a state in the SU(4) multiplet. Because | G (3 π/ (cid:105) doesnot break SO(5) symmetry, (cid:104) N α (cid:105) π/ = (cid:104) T β (cid:105) π/ ≈ C ∗ / α = x, y, z and β = x, y .Eq. (3) predicts that in the thermodynamic limit N φ →∞ , small l multiplets will approach degeneracy. By mak-ing an analogy with the quantum rotor model, we can seethat this property signals spontaneous SO(5) symmetrybreaking. The energy in Eq. (3) can be interpreted asthe kinetic energy of a generalized rotor model in the 5D( T x,y , N x,y,z ) space with the SO(5) generators playing therole of angular momenta. In the thermodynamic limit N φ → ∞ , the moments of inertia of the rotors divergeand it can be stuck in a spontaneously chosen direction,resulting in symmetry breaking. The absence of ground state level crossings along the θ g = 3 π/ IV. LOW-ENERGY EFFECTIVE THEORY ANDCOLLECTIVE MODES
Following Refs. , we can derive a low-energy ef-fective field theory for ν = 0 quantum Hall states byconstructing the Lagrangian, L = (cid:104) ψ | i∂ t − H | ψ (cid:105) = (cid:90) d r πl B (cid:2) B − H (cid:3) , (5)where | ψ (cid:105) is a Slater-determinant state in which two or-thogonal occupied spinors χ , are allowed to vary slowlyin space and time. The Lagrangian density L = B − H has kinetic Berry phase ( B = i ( χ † ∂ t χ + χ † ∂ t χ )) and en-ergy density H contributions. As detailed in AppendixC we find that : H = − u ⊥ − (cid:15) Z s z + ( u z + u ⊥ )( t z − (cid:88) α = x,y,z s α ) + 2 u ⊥ (cid:88) β = x,y t β + ( u ⊥ − u z ) (cid:88) α = x,y,z n α + l B (cid:2) ρ z ( ∇ t z ) + ρ ⊥ (cid:88) β = x,y ( ∇ t β ) + (cid:88) α = x,y,z ρ s ( ∇ s α ) + ρ n ( ∇ n α ) + ρ π (cid:0) ( ∇ π xα ) + ( ∇ π yα ) (cid:1)(cid:3) . (6)The stiffness coefficients ρ z = ρ − (3 u z + 2 u ⊥ ) / ρ ⊥ = ρ − ( u z + u ⊥ ) / ρ s = ρ + ( u z + 2 u ⊥ )4, ρ n = ρ + ( u z − u ⊥ ) / ρ π = ρ − u z /
4, are dominated bythe common Coulomb contribution ρ = √ πe / (16 (cid:15)l B ).It is easy to check that the energy density function H hasthe same symmetries as the Hamiltonian H . The mean-field theory ground state is determined by assuming thatall fields are static and spatially uniform. The energycompetitions behind the mean-field phase diagram pre-viously derived by Kharitonov are transparent whenEq. (6) is combined with the normalization constraint (cid:80) α ( t α + n α + s α +( π xα ) +( π yα ) ) = 1 (see Appendix C). In the absence of a Zeeman field the four mean field phasesare the F state ( (cid:80) s α = 1), the AF state ( (cid:80) n α = 1),the KD state ( t x + t y = 1), and the CDW state ( t z = 1).The phase boundaries between these states, shown in theinset of Fig. 2(a), lie along the high symmetry lines iden-tified in Table I.We now concentrate on physics near u z + u ⊥ = 0 wherea first order phase transition occurs between KD and AFstates and the system exhibits SO(5) symmetry. The u z + u ⊥ = 0 line in graphene is analogous to the J xy = J z line in a XXZ spin model, along which a phase transi-tion occurs between Ising and XY ground states and thesystem exhibits expanded O(3) symmetry. One physicalmanifestation of SO(5) symmetry along the transitionline is the response to an external Zeeman field, whichinduces a finite z direction spin polarization s z . It followsfrom orthogonality constraints on the fields discussed inAppendix C that when among the ten SO(5) generatorsonly s z has a finite expectation value, t x,y and n z mustvanish. A finite Zeeman energy therefore favors the AFstate over the KD state because the AF state can dis-tort to a canted AF with a finite s z and a N´eel vectorlying in the xy plane. A sufficiently strong Zeeman fieldeventually favors the F state. Because experiments de-tect what appears to be a continuous phase transition asa function of Zeeman coupling strength , they suggestthat the ground state in the absence of Zeeman couplinglies in the AF region of the phase diagram.Close to the u z + u ⊥ = 0 line, the system retains crucialSO(5) properties in the presence of a small Zeeman term.Approximate SO(5) symmetry is revealed in the collec-tive mode spectra of both KD and AF states. The KDphase spontaneously breaks the valley U(1) v symmetry.Chosing the ground state to have valley polarization t x with a spontaneous non-zero value, we see that infinites-imal SU(4) rotations give rise to infinitesimal values ofeight fields, { t y,z , n x,y,z , π yx,y,z } , and leave the remainingsix fields, { s x,y,z , π xx,y,z } at zero. The eight dynamicalfields parametrize the tangent manifold of the mean-fieldground state. By evaluating the Berry phase we findthat for small fluctuations the valley pseudospin fields t y and t z are canonically conjugate, and that the N´eelvector field n α is conjugate to π yα . The valley pseudospinand N´eel vector collective modes therefore decouple. Thevalley collective mode is gapless because of the Kekul´estate’s broken U(1) symmetry and has dispersion : ω (KD) = 2 k (cid:112) ρ ⊥ ( u z − u ⊥ + ρ z k ) , (7)where k is wave vector and lengths are in units of l B . Thethree additional collective modes are kinetically coupledN´eel- π modes and have energy : ω , , (KD) = 2 (cid:112) ( | u z + u ⊥ | + ρ n k )(2 | u ⊥ | + ρ π k ) . (8)Note that these modes become gapless as the SO(5) sym-metry line is approached and the energy cost of N´eel fluc-tuations away from the KD state vanishes, and that theZeeman field does not influence collective mode energiesin the KD phase because s z is not a dynamical field. Sim-ilarly the AF state spontaneously breaks the spin SU(2) s symmetry. When the N´eel vector is chosen to lie alongthe x-axis, the dynamical fields generated by infinitesimalSU(4) rotations are { s y,z , n y,z , t x,y , π x,yx } . Evaluating theBerry phase we find that s y is conjugate to n z and s z to n y , as in a standard antiferromagnet. The spin-collectivemodes are : ω , (AF) = 2 k (cid:112) ρ n (2 | u ⊥ | + ρ s k ) . (9)In the AF state ( t x , π yx ) and ( t y , π xx ) fluctuations formkinetically coupled conjugate pairs and give rise to the sublattice/ π collective mode energies : ω , (AF) = 2 (cid:112) ( u z + u ⊥ + ρ ⊥ k )( u z − u ⊥ + ρ π k ) . (10)Note that all four collective modes are gapless and de-generate along the u z + u ⊥ = 0 line. The degeneracyarises from the SO(5) symmetry. Appendix D describeshow the collective modes in Eq. (9) and (10) are modifiedby the Zeeman field. V. DISCUSSION AND SUMMARY
In ordered systems a Landau-Ginzburg or quantumeffective model which includes a single-order parame-ter, for example a complex pair-amplitude order param-eter for a superconductor or a magnetization directionorder parameter for a magnetic system, is often ableto describe thermodynamic, fluctuation, and responseproperties over wide ranges of temperature and exper-imentally tunable system parameters. These theoriescan be powerfully predictive even when their parame-ters cannot be reliably calculated from the underlyingmicroscopic physics. The naive effective-field-theory ap-proach sometimes fails however. A notable example is thecase of high-temperature superconductors in which ex-periments indicate that charge-density, spin-density, andpair-amplitude order parameters have correlated quan-tum and thermal fluctuations that must be treated si-multaneously. Unlike the case discussed in the presentpaper in which an N=5 component effective theory canbe motivated and its parameters estimated on the basis ofmicroscopic physics, large- N field theories are typ-ically constructed on the basis of hints from experimentaldata, for example from observed correlations in the tem-perature and parameter dependence of the fluctuationamplitudes of different observables. In these theories, itis often difficult to be certain that all relevant fields havebeen identified, and to identify constraints imposed onthe fluctuations of these fields by the underlying micro-scopic physics. As discussed below, the remarkably sim-ple example of ordered states in graphene quantum Hallsystems, particularly ordered states at ν = 0, suggestscriteria which can be tested experimentally to validatelarge-N unified theories of systems with competing or-ders. TABLE II: Comparison between the Kekul´e-distortion statein graphene and the d -wave state in high temperature super-conductors. Parameter Kekul´e-distortion state d -wave stateOrder Parameter ( T x , T y ) (∆ x , ∆ y )U(1) generator T z Charge Q External Potential Staggered potential (cid:15) v Chemical potential µ As summarized in Table II, there is a close analogybetween SO(5) symmetry in the quantum Hall effect ofgraphene and SO(5) symmetry in some theories of high- T c superconductivity (HTS) . The SO(5) theory ofHTS theory unifies antiferromagnetism and d -wave su-perconductivity (dSC). The analog of d -wave supercon-ductivity in the graphene quantum Hall case is Kekul´edistortion order. The order parameters of both theo-ries involve a sublattice degree of freedom, the honey-comb sublattice degree-of-freedom in the case of grapheneand the sublattice degree of freedom of the magneti-cally ordered state in HTS SO(5) theory case. Thegraphene analog of the chemical potential µ term whichtunes transitions between antiferromagnetic and d -wavesuperconducting states in the HTS case, is a sublattice-staggered potential (cid:15) v . Interestingly this field is eas-ily tunable experimentally in the bilayer graphenecase. SO(5) symmetry in HTS is conjectured to emergein low-energy effective theory , and can be exactly re-alized in extended Hubbard model with artificial long-range interactions ; however, it never becomes exactfor commonly used models like t − J or Hubbard model.In contrast, SO(5) symmetry and its explicit symmetrybreaking naturally appear in the microscopic Hamilto-nian (Eq. (1)) for the quantum Hall effect in graphene atany filling factors within N = 0 LL. We note that genericSO(5) symmetry without any fine-tuning parameters canappear in spin-3/2 ultracold femionic system .The SO(5) symmetry in graphene is manifested bymultiplet structure in exact diagonalization spectra, andby the appearance of soft collective modes beyond thoseassociated with Kekul´e or antiferromagnetic order. Inparticular, the antiferromagnetic state of graphene has π -operator fluctuation collective modes. The observationof the analogous collective modes in the antiferromag-netic state of high temperature superconductors wouldprovide powerful evidence for the applicability of an ef-fective theory which unifies antiferromagnetism and su-perconductivity only. On the other hand their absencewould likely indicate that an effective theory of this typeis not adequate over a useful range of the tunable doping-level parameter of HTSs. Similarly a recently proposedalternate N=6 parameter theory which unifies charge-density-wave and d -wave superconducting order, also hasimplications for collective mode structure which, if veri-fied, would provide powerful validation.Finally we would like to comment on the relevance ofour study to the understanding of the highly insulat-ing quantum Hall state found in graphene at neutral-ity. Experiments with tilted magnetic fields are con-sistent with the view that the state at neutrality is acanted antiferromagnet. Since the transition betweencanted antiferromagnet and the spin polarized state iscontrolled solely by the ratio of the Zeeman term to the u ⊥ interaction strength , these very experiments serveto estimate the value of u ⊥ , which is found to be about u ⊥ ∼ − (cid:15) Z11,12 . This experiment however does notserve to estimate the value of u z , but simply to con-strain it to satisfy u z (cid:38) | u ⊥ | , from the requirement thatthe system is in the canted Antiferromagnet phase. The determination of the value of u z relevant for monolayergraphene, and hence of its proximity to the ideal SO(5)symmetric state is therefore an open experimental prob-lem. The presence of a weakly broken SO(5) symmetrywould have important physical consequences, such as theexistence of additional weakly gapped neutral collectivemodes as we illustrated in Section IV and in Appendix D. VI. ACKNOWLEDGMENTS
This work was supported by the DOE Division of Ma-terials Sciences and Engineering under grant DE-FG03-02ER45958, and by the Welch foundation under grantTBF1473. We gratefully thank Texas Advanced Com-puting Center(TACC) and IDRIS-CNRS project 100383for providing technical assistance and computer time al-locations.
Appendix A: Proof of SO(5) Symmetry for g z + g ⊥ = 0 Let us first briefly review how SO(5) arises naturallyas a subgroup of SU(4). The fifteen generators of SU(4)can be chosen to be the Pauli matrices in spin and val-ley space and their direct products: { σ α , τ β , σ α τ β } . TheClifford algebra, { γ µ , γ ν } = 2 δ µν , is realized by a subsetof these generators, namely the 4x4 γ matrices, whichcan be chosen as: γ = τ x , γ = τ z σ x , γ = τ z σ y , γ = τ z σ z , γ = τ y . (A1)SO(5) can be shown to be generated by the commuta-tors of these γ matrices: [ γ µ , γ ν ]. More specifically, wehave the following ten generators of SO(5): γ ab = − i γ a , γ b ] (A2)which can be thought of as a 5 × γ ab = τ y σ x τ y σ y − σ z τ y σ z σ y − σ x − τ z τ x σ x τ x σ y τ x σ z . (A3)These matrices satisfy the following commutation rela-tions :[ γ ab , γ cd ] = 2 i ( δ ac γ bd + δ bd γ ac − δ ad γ bc − δ bc γ ad ) , (A4)[ γ ab , γ c ] = 2 i ( δ ac γ b − δ bc γ a ) . (A5)Eq. (A4) shows that the ten independent γ ab matricesobey a set of closed commutation relations, which is theSO(5) Lie algebra. Additionally according to Eq. (A4)and (A5), when the group is viewed as acting on γ ab and γ a by matrix conjugation, we have respectively a tensorand a vector representation of SO(5) .We will now demonstrate explicitly that SO(5) is anexact symmetry of the Hamiltonian in the absence ofZeeman coupling for g z + g ⊥ = 0. From among the fifteengenerators of SU(4) identified in the main text, the spinoperator S α , the valley polarization operator T z and theΠ βα operators are the ten generators of the SO(5) group. S α and T z automatically commute with H v for any valuesof g z and g ⊥ . Thus, SO(5) will be a symmetry group if the six Π βα operators also commute with H v . To simplifythe calculation of these commutators, we define the Πladder operators :Π λλ (cid:48) = (cid:88) i τ iλ σ iλ (cid:48) , Π λz = (cid:88) i τ iλ σ iz , (A6)where λ and λ (cid:48) can be + or − . τ ± = ( τ x ± iτ y ) / σ ± are similarly defined. We work out the com-mutator (cid:2) Π ++ , H v (cid:3) in detail below : (cid:2) Π ++ , H v (cid:3) = 2 (cid:88) i (cid:54) = j (cid:0) − g z τ jz τ i + σ i + + g ⊥ τ j + τ iz σ i + (cid:1) δ ( (cid:126)r i − (cid:126)r j )= 2 (cid:88) v,s (cid:88) p p p p τ vvz D p p p p (cid:0) − g z c † p K ↑ c † p vs c p vs c p K (cid:48) ↓ + g ⊥ c † p v ↑ c † p Ks c p K (cid:48) s c p v ↓ (cid:1) = 2 ( g z + g ⊥ ) (cid:88) p p p p D p p p p (cid:0) c † p K ↑ c † p K (cid:48) ↑ c p K (cid:48) ↑ c p K (cid:48) ↓ + c † p K ↑ c † p K ↓ c p K (cid:48) ↓ c p K ↓ (cid:1) . (A7)The second line of Eq. (A7) is the Landau gauge sec-ond quantized form of the first line. c † pvs ( c pvs ) is anelectron creation (annihilation) operator, p denotes theorbital index within the N = 0 Landau level, v = K, K (cid:48) labels valley, and s = ↑ , ↓ labels spin. D p p p p is theorbital two-particle matrix element for the δ function in-teraction : D p p p p = (cid:90) (cid:90) d(cid:126)r d(cid:126)r φ ∗ p ( (cid:126)r ) φ ∗ p ( (cid:126)r ) δ ( (cid:126)r − (cid:126)r ) φ p ( (cid:126)r ) φ p ( (cid:126)r )= (cid:90) d(cid:126)r φ ∗ p ( (cid:126)r ) φ ∗ p ( (cid:126)r ) φ p ( (cid:126)r ) φ p ( (cid:126)r ) , (A8)where φ p ( (cid:126)r ) is the wave function for orbital p . In the sim-plification leading to the last line of Eq. (A7), we used (1)fermion anticommutation relations, and (2) the identity D p p p p = D p p p p , which is a special property of δ function interaction. Eq. (A7) shows that (cid:2) Π ++ , H v (cid:3) = 0at g z + g ⊥ = 0. In a similar fashion, it can be shownthat the other Π operators also commute with H v at g z + g ⊥ = 0. Thus, H v has exact SO(5) symmetry for g z + g ⊥ = 0 independent of filling factors. The sym-metry follows from the short-range nature of the valley-symmetry breaking interaction combined with the Pauliexclusion principle for electrons. Note that in Eq.(A7),we did not make use of the explicit form of the wavefunction φ p ( (cid:126)r ). The same Hamiltonian in Eq.(1) hasalso been used to describe physics in N = 0 LL of bi-layer graphene(BLG) . There is a similar equivalenceamong valley, sublattice and layer degrees of freedom within N = 0 LL in BLG. The main difference is that N = 0 LL in BLG contains both n = 0 and n = 1 mag-netic oscillator states. Since the SO(5) symmetry iden-tified for Hamiltonian in Eq.(1) is independent of single-particle wave function basis, it can also be applied to thecase of BLG. Appendix B: Exact Diagonalization Results
Our ED results for finite-size systems with up to 16electrons verify that the ground state at ν = 0 forCoulomb interactions only ( H = H C ) is given exactlyby mean field theory. The ground state wave functionsat ν = 0 are single Slater determinants with filled Lan-dau levels for two of four flavors. This property is ageneralization of simple, quantum Hall ferromagnetism,the occurrence of a spontaneously spin-polarized statesat odd filling factors when the spin degree-of-freedom isadded to the physics of a parabolic band system Landaulevels. We have used periodic boundary conditions andclassified many-body states by their magnetic translationsymmetries . In graphene the ν = 0 ground states occurat zero momentum and form an irreducible representa-tion of SU(4).The ν = 0 F, AF and CDW states are included in theground state multiplet and can be expressed in the form: | χ , (cid:105) = N φ (cid:89) p =1 c † pχ c † pχ | (cid:105) , (B1)where χ , are the two spinors defining the state and p is the index of the LL orbital. When considered as atensor representation of SU(4), this formula implies thatthe states in this multiplet are tensors with 2 N φ indicesin two symmetric sets each with N φ indices i.e. they aredescribed by the Young tableau:. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .with N φ columns and two rows. Fig. S1(a) representsthe SU(4) multiplet structure geometrically in terms ofan octahedron in ( S z , N z , T z ) space . The octahedralshape is understood to bound a tetrahedral lattice ofpoints in which each point designates the states withinthe multiplet with common S z , N z , T z quantum numbers.Fig. S1(b) shows a slice of this lattice with T z = N φ − C = S + N + T + Π , (B2)where S = (cid:80) α = x,y,z S α , N and T are similarly de-fined, and Π = (cid:80) α = x,y,z (Π xα ) + (Π yα ) . C takes value N φ ( N φ + 4) for the Coulomb ground state multiplet at ν = 0. Fig. S1(b) demonstrates that there can be morethan one state in the multiplet at a given ( S z , N z , T z )point. Hence, an additional quantum number, such as S + N , is needed to uniquely label a state within theSU(4) multiplet of interest . S + N is one of thequadratic Casimir operator of the SU(2) K s × SU(2) K (cid:48) s sub-group of SU(4). We note that SU(2) K s × SU(2) K (cid:48) s grouphas another quadratic Casimir operator (cid:80) α = x,y,z S α N α ,which is identical to 0 for Coulomb ground states at ν = 0.SU(4) symmetry is lifted by the valley-symmetrybreaking interaction H v , and the octahedral multipletis split. At θ g = 3 π/
4, SU(4) symmetry is reduced to T z S z T z S z T z S z S S S level 0 level 1 level 2 CDW AF F T z N z S z (a) (b) S z N z (c) Fig. S 1: Geometric representation of SU(4) multiplet struc-tures. (a) The octahedron in ( S z , N z , T z ) space representsthe SU(4) multiplet structure of Coulomb ground states at ν = 0. (b) A T z -constant plane in the octahedron displayedfor T z = N φ − T z = N φ . The size of the symbols indi-cates the degeneracy at each point in the ( S z , N z ) plane. (c)Multiplet structures of the first three levels of SO(5). SO(5) symmetry. Fig. S1(c) shows the SO(5) multipletstructure of the three lowest energy states, which coin-cide with the lowest degeneracies. Within a level, statesare distinguished by T z , S z and total spin S quantumnumbers, and share the same value of the SO(5) Casimiroperator Γ = S + T z + Π = l ( l + 3), l being a nonneg-ative integer . We note that the same SO(5) multipletstructure has arisen previously in numerical studies ofthe t − J model . Interestingly, along the SO(5) line, i.e. for θ g = 3 π/
4, we find numerically that the eigenen-ergies are linear in Γ , as illustrated in Fig. S2(a). Thelow-energy part of the spectrum along the SO(5) line isaccurately fit by the following equation: H effv ( θ g = 3 π u z (cid:16) N φ + 1 − N φ ( N φ + 5) N φ + 1 (cid:17) . (B3)The ground state at θ g = 3 π/
4, is an SO(5) singlet withΓ = 0.Away from θ g = 3 π/
4, SO(5) symmetry is explicitlybroken, leading to anisotropy in the 5D space. Interest-ingly, the spectrum can also be fit by a linear form inthe appropriate quadratic Casimir operators along otherhigh symmetry lines. For example, at θ g = π/
2, theCasimir operators of the corresponding symmetry groupSU(2) K s × SU(2) K (cid:48) s × U(1) v are S + N and T z . For a T z − constant plane shown in Fig. S1(b), S + N takesvalues f ( f + 2), with nonnegative f = N φ − | T z | , N φ − E v / g N ϕ = 4 5 6 7 8 Γ θ =3 π/4 g I Γ N ϕ S
5 6 7 8 E v / g I S T z =0, θ =4 π /7 g T z E v / g
5 6 7 8 S =0 θ = π g I T (a) (b) (c) N ϕ N ϕ N ϕ = 4 N ϕ = 4 Fig. S 2: Finite size scaling analysis. (a) E v /g at θ g = 3 π/ for N φ ranging from 4 to 8. (b) In a given( T z = 0 , S ) sector, the lowest energy at θ g = 4 π/ ∈ [ π/ , π/
4] as a function of S . (c) In a given ( S = 0 , T z ) sector, the lowest energy at θ g = π as a function of T z . The inset in each figure shows the inverse of slope versus N φ . See text for a moredetailed description. | T z | − , · · · . In analogy with the θ g = 3 π/ θ g = π/ H effv ( θ g = π u z T z − ( S + N ) + N φ N φ + 1 . (B4) By interpolating between Eq. (B3) and (B4), we arrive atan expression which describes the low-energy spectrumof the SU(4) ground state manifold over the full θ g ∈ ( π/ , π/
4) interval: H effv = 1 N φ + 1 (cid:16) − u ⊥ Γ + ( u z + u ⊥ )( T z − S − N ) + u z N φ + u ⊥ N φ ( N φ + 6) (cid:17) . (B5)Eq. (B5) is limited in two ways: (1) it describes only thelow-energy part of the spectrum which evolves adiabati-cally from the SU(4) ground state multiplet; and (2) it isobtained by fitting numerical data at the high-symmetrypoints θ g = π/ π/
4. The SO(5) symmetry-breaking states at θ g = 3 π/ θ g = 3 π/ θ g ∈ ( π/ , π/
4) range is singlydegenerate and has S = 0 and T z = 0. Therefore, onthe θ g < π/ − ( u z + u ⊥ ) N in Eq. (B5) is an easy-plane anisotropy in the 5Dspace with the N´eel vector space being the easy-plane;N´eel order is favored over Kekul´e order for θ g < π/ θ g > π/ T x,y vectors lie in the easy-plane. Weconclude that there is a spin-flop phase transition in the5D space across the SO(5) point. The phase transitionis of first order. Our analysis is in agreement with themean-field prediction of a zero temperature first-orderphase transition and places it on rigorous grounds.We will now describe how the finite size scaling demon-strates the existence of spontaneous symmetry-breakingaway from the SO(5) point. In Fig. S2(b), we plot the lowest energy at a representative angle θ g = 4 π/ T z = 0 , S ) sectors as a function of S for N φ from4 up to 8. There is good linear relationship betweenthe plotted energy and S . The quantity I S , definedas the inverse of the slope, increases linearly as N φ in-creases. This quantity is a generalized moment of inertiaand its divergence indicates spontaneous SU(2) s symme-try breaking in the thermodynamic limit at θ g = 4 π/ θ g = 3 π/
4. In Fig. S2(c), a similar scalinganalysis is applied to the spin singlet sector with varying T z numbers at θ g = π . In this case, the analysis signalsa spontaneous U(1) v symmetry breaking in the thermo-dynamic limit. We remark that the finite-size scalingbehavior in our system is very similar to that in the two-dimensional antiferromagnetic Heisenberg model. Theground state of the latter model is a spin singlet in anyfinite size system. However, low-lying energy levels col-lapse to the ground state in the thermodynamical limit,resulting in spontaneous symmetry breaking . Thisset of low-lying states is often referred to as a tower ofstates .So far, the Zeeman field has been neglected. Since S z has been chosen as a good quantum number in our ex-act diagonalization calculations, the Zeeman field simplyshifts the energy of a state by an amount proportional to0its S z value. We found that the mean-field phase bound-ary between canted antiferromagnetic state and KD inthe presence of a Zeeman field is in quantitative agree-ment with exact diagonalization results for N φ = 8. Appendix C: Low-energy effective theory
The continuum model Lagrangian L = (cid:104) ψ | i∂ t − H | ψ (cid:105) = (cid:90) d r πl B (cid:2) B − H (cid:3) , (C1)where | ψ (cid:105) is a Slater-determinant state in which two or-thogonal occupied spinors χ , are allowed to vary slowlyin space and time. The Lagrangian density L = B − H has a Berry phase part : B = i ( χ † ∂ t χ + χ † ∂ t χ ) , (C2)and an energy density contribution : H = l B E ( ∇ P ) + E v ( P ) − l B E v ( ∇ P ) + E Z ( P ) . (C3)where P is the local density matrix, P = χ χ † + χ χ † and E ( ∇ P ) is the contribution from the SU(4) symmetricCoulomb interaction which is non-zero only when P isspace-dependent : E ( P ) = ρ Tr[ ∇ P ∇ P ] , (C4)with stiffness ρ = √ πe / (16 (cid:15)l B ). The next two termsare contributed by the valley-dependent interactions : E v ( P ) = 12 (cid:88) α = x,y,z u α ξ α ( P ) , (C5)where u x,y = u ⊥ = g ⊥ / (2 πl B ), u z = g z / (2 πl B ), and ξ α ( P ) = Tr[ τ α P ] Tr[ τ α P ] − Tr[ τ α P τ α P ]. E v ( ∇ P ) is agradient term, and has a similar expression as E v ( P ).The last term is the Zeeman energy : E Z ( P ) = − (cid:15) Z Tr[ σ z P ] . (C6)The position-dependent density matrix P has the follow-ing properties : P † = P, Tr P = 2 , P = P. (C7) It is convenient to reparametrize the state with a matrix R , where P = (1 + R ). R is Hermitian, traceless, and R = 1. Thus, R can be expressed in terms of SU(4)generators : R = (cid:88) a l a γ a + (cid:88) a>b l ab γ ab , (C8)where l a and l ab are classical real fields. The condition R = 1 gives rise to constraints on these fields. One typeis normalization constraint enforcing Tr[ R ] = 4 : (cid:88) a l a + (cid:88) a>b l ab = 1 , (C9)Another type are orthogonality constraints : (cid:15) abcde l cd l e = 0 , (cid:15) abcde l bc l de = 0 , (C10)where (cid:15) abcde is the fully antisymmetric Levi-Civita sym-bol in five dimensions. The orthogonality constraint isgiven by Tr[ R γ ab ] = 0 and Tr[ R γ a ] = 0.The SO(5) theory of high- T c superconductivity re-quires a similar orthogonality constraint, which plays anessential role in predicting the phase transition betweenAF and dSC phases. There, it was proposed based on ageometric interpretation of rotations in 5D , and sepa-rately based on maximum entropy considerations. Inour theory, the orthogonality constraint naturally ap-pears because of the assumption that at each LL orbitaltwo spinors are occupied, i.e. that charge fluctuationsare quenched. To make the physical meaning of the fif-teen fields { l a , l ab } transparent, we rename them usingspin and valley language : l , , = s x,y,z , l , = t x,y , l = t z ,l , , = n x,y,z ,l , , = π xx,y,z , l , , = π yx,y,z . (C11) s α , t α and n α with α = x, y, z are respectively spin, valleyand N´eel fields, and there are six π fields. The explicitform of the energy density H expressed in terms of theseclassical fields is given in Eq. (6) of the main text. Appendix D: Collective Modes in the Presence of a Zeeman Field
In the presence of Zeeman field, the AF is transformed to a canted antiferromagnetic (CAF) state in which thespin-polarizations on opposite sublattices are not collinear. In the CAF state, the density matrix P (CAF) = (1 +sin θ s τ z σ x + cos θ s σ z ) where the canting angle cos θ s = (cid:15) Z / | u ⊥ | . One of the spin wave mode remains gapless in theCAF state : ω (CAF) = 2 (cid:113) ρ n (2 | u ⊥ | sin θ s + ( ρ n cos θ s + ρ s sin θ s ) k ) k. (D1)1This gapless mode corresponds to the rotation of N´eel vector within the xy plane. Another spin wave mode acquiresa gap : ω (CAF) = 2 (cid:113) ( (cid:15) Z cos θ s + ( ρ n sin θ s + ρ s cos θ s ) k )(2 | u ⊥ | + ρ s k ) . (D2)The Zeeman field also modifies the dispersion of the sublattice/ π modes : ω , (CAF) = 2 (cid:113) ( u z + u ⊥ + (cid:15) Z cos θ s + ( ρ ⊥ sin θ s + ρ π cos θ s ) k )( u z − u ⊥ + ρ π k ) , (D3)which remain gapped in the CAF phase and become gapless at the CAF/KD phase boundary u z + u ⊥ + (cid:15) Z cos θ s = 0 .At the SO(5) point u z + u ⊥ = 0, the gapped spin wave mode ω (CAF) and sublattice/ π modes ω , (CAF) becomedegenerate. The degeneracy is due to the unbroken part of the SO(5) symmetry in the presence of Zeeman field. † Electronic address: [email protected] ‡ Electronic address: [email protected] Y. Zhang, Z. Jiang, J. P. Small, M. S. Purewal, Y.-W.Tan, M. Fazlollahi, J. D. Chudow, J. A. Jaszczak, H. L.Stormer, and P. Kim, Phys. Rev. Lett. , 136806 (2006). A. F. Young, C. R. Dean, L. Wang, H. Ren, P. Cadden-Zimansky, K. Watanabe, T. Taniguchi, J. Hone, K. L.Shepard, and P. Kim, Nature Physics , 550-556 (2012). X. Du, I. Skachko, F. Duerr, A. Luican, and E. Y. Andrei,Nature , 192-195 (2009). K. I. Bolotin, F. Ghahari, M. D. Shulman, H. L. Stormer,and P. Kim, Nature , 196-199 (2009). C. R. Dean, A. F. Young, P. Cadden-Zimansky, L. Wang,H. Ren, K. Watanabe, T. Taniguchi, P. Kim, J. Hone, andK. L. Shepard, Nature Physics , 693-696 (2011). K. Nomura and A. H. MacDonald, Phys. Rev. Lett. ,256602 (2006). K. Yang, S. Das Sarma, and A. H. MacDonald, Phys. Rev.B , 075423 (2006). B. E. Feldman, B. Krauss, J. H. Smet, and A. Yacoby,Science , 1196 (2012). B. E. Feldman, A. J. Levin, B. Krauss, D. A. Abanin, B.I. Halperin, J. H. Smet, and A. Yacoby, Phys. Rev. Lett. , 076802 (2013). A. F. Young, J. D. Sanchez-Yamagishi, B. Hunt, S. H.Choi, K. Watanabe, T. Taniguchi, R. C. Ashoori, and P.Jarillo-Herrero, Nature , 528-532 (2014). D. A. Abanin, B. E. Feldman, A. Yacoby, and B. I.Halperin, Phys. Rev. B , 115407 (2013). I. Sodemann and A. H. MacDonald, Phys. Rev. Lett. ,126804 (2014). D. M. Basko and I. L. Aleiner, Phys. Rev. B , 041409(R)(2008). J. Alicea and M. P. A. Fisher, Phys. Rev. B , 075422(2006). I. F. Herbut, Phys. Rev. Lett. , 146401 (2006). J.-N. Fuchs and P. Lederer, Phys. Rev. Lett. , 016803(2007). J. Jung and A. H. MacDonald, Phys. Rev. B , 235417(2009). K. Nomura, S. Ryu, and D.-H. Lee, Phys. Rev. Lett. ,216801 (2009). C.-Y. Hou, C. Chamon, and C. Mudry, Phys. Rev. B , 075427 (2010). M. Kharitonov, Phys. Rev. B , 155439 (2012). E. Demler, W. Hanke, and S.-C. Zhang, Rev. Mod. Phys. , 909 (2004). I. L. Aleiner, D. E. Kharzeev, and A. M. Tsvelik, Phys.Rev. B , 195415 (2007). S. Ryu, C. Mudry, C.-Y. Hou, and C. Chamon, Phys. Rev.B , 205319 (2009) I. F. Herbut, Phys. Rev. B , 085304 (2012) M. Kharitonov, Phys. Rev. B , 075450 (2012). M. Kharitonov, Phys. Rev. Lett. , 046803 (2012). M. Kharitonov, Phys. Rev. B , 195435 (2012). M. Hamermesh,
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