SU(4) spin waves in the ν=\pm1 quantum Hall ferromagnet in graphene
SSU(4) spin waves in the ν = ± quantum Hall ferromagnet in graphene Jonathan Atteia ∗ and Mark Oliver Goerbig † Laboratoire de Physique des Solides, Universit´e Paris Saclay,CNRS UMR 8502, F-91405 Orsay Cedex, France (Dated: February 25, 2021)We study generalized spin waves in graphene under a strong magnetic field when the Landau-level filling factor is ν = ±
1. In this case, the ground state is a particular SU(4) quantum Hallferromagnet, in which not only the physical spin is fully polarized but also the pseudo-spin associatedwith the valley degree of freedom. The nature of the ground state and the spin-valley polarizationdepend on explicit symmetry breaking terms that are also reflected in the generalised spin-wavespectrum. In addition to pure spin waves, one encounters valley-pseudo-spin waves as well asmore exotic entanglement waves that have a mixed spin-valley character. Most saliently, the SU(4)symmetry-breaking terms do not only yield gaps in the spectra, but under certain circumstances,namely in the case of residual ground-state symmetries, render the originally quadratic (in the wavevector) spin-wave dispersion linear.
I. INTRODUCTION
Graphene, a one-atom-thick layer of carbon atomsarranged in a honeycomb lattice, is the prototypeof a large class of two-dimensional materials suchas transition metal dichalchogenoids , van der Waalsheterostructures or twisted bilayers and multilayers that present striking properties such as topological, cor-related or superconducting phases. It is the paradigmof Dirac fermions in condensed matter since its disper-sion is described by the Dirac-Weyl equation in twodimensions . These fermions come in two flavours withdifferent chiralities, represented here by the valley index,which acts as an effective ”pseudo-spin”.Upon the application of a magnetic field B perpendic-ular to the graphene plane, the relativistic character ofthe Dirac fermions is at the origin of an anomalous quan-tum Hall effect. While the effect is still a consequence ofthe quantization of the electrons’ energy into highly de-generate Landau levels (LLs), the latter inherit from the B = 0 system a twofold valley degeneracy, in additionto the spin degeneracy, such that the low-energy Hamil-tonian is invariant under SU(4) spin-valley transforma-tions. This SU(4) symmetry is furthermore respected toleading order by the Coulomb interaction between theelectrons, which constitutes the dominant energy scalein partially filled LLs due to the flatness of the latter. Ifonly some spin-valley branches of a specific LL are filled,all the electrons inside this LL choose to spontaneouslybreak the SU(4) symmetry and to be polarized in a cer-tain spin and pseudo-spin state. This marks the onset of SU(4) quantum Hall ferromagnetism .The physics inside a LL is thus dominated by theCoulomb interaction E C = e /εl B = 625 (cid:112) B [ T ]K /ε ,where ε is the dielectric constant of the environment thegraphene sheet is embedded into, and l B = (cid:112) (cid:126) /eB isthe magnetic length. However, at much smaller energies,explicit symmetry breaking terms become relevant, suchas the Zeeman term, short-range electron-electron inter-actions, electron-phonon interactions or coupling to the substrate . These symmetry-breaking terms, whichhappen to be all on the same order of magnitude, de-termine thus the spin-valley polarization of the groundstate. At half-filling of the n = 0 LL ( ν = 0) severalphases have been proposed such as a ferromagnetic (F),charge density wave (CDW), Kekul´e distortion (KD) andcanted anti-ferromagnetic (CAF) phase as a function ofthe symmetry breaking terms . Notice that there isexperimental evidence for three of these phases , indi-cating that the nature of the SU(4) ferromagnetic groundstate may be sample and/or substrate dependent. Atquarter filling ν = ± et al. using the same symmetry breaking termsas Kharitonov , and one obtains similar phases as in the ν = 0 case.Spin waves are the lowest energy excitations in aferromagnet. They have been observed in a widevariety of materials and are promising platformsfor spintronics . In a two-dimensional electron gas(2DEG) in GaAs/AlGaAs heterostructures at filling ν =1, the first example of a quantum Hall ferromagnet, theground state consists of all spins pointing in the direc-tion of the magnetic field, and the spin waves correspondsimply to the precession of the spins around their groundstate position. Generalized spin waves have also been ex-tensively studied and observed in bilayer 2DEGs wherethe layer index plays the role of the pseudo-spin. Whenthe distance d is on the order of the magnetic length l B ,quantum Hall ferromagnetism of the layer pseudo-spinis observed and manifests itself in the form of a globalphase coherence between electrons in the two layers .At ν = 1 (quarter filling of the n = 0 LL), the groundstate is an interlayer coherent state where each electronis in a superposition of the two layers, and the physicalspin is fully polarized. This ground state can be viewedas a condensate of electron-hole pairs which then pos-sesses a gapless, linearly dispersing superfluid mode .This mode was observed experimentally using tunnel-ing spectroscopy. Put differently, this superfluid modeis associated with a U(1) symmetry of the ground state a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b that corresponds to the phase of the electron-hole super-position. At ν = 2 (half-filling of the n = 0 LL), one isconfronted with a frustrated situation: a complete spinpolarization excludes a full pseudo-spin polarization, and vice versa . Depending on the relative strength of the Zee-man and interlayer tunneling term, the ground state canthus be a spin ferromagnet, a spin-singlet or an inter-mediate phase with CAF order . The dispersion ofthe modes at ν = 2 are presented in Ref. [41]. Thepeculiarity of the CAF phase is that it possesses a U(1)symmetry associated with the invariance under spin ro-tation around the z axis. Such a symmetry implies alsoa gapless linearly dispersing mode which was observedexperimentally by inelastic light scattering and nuclearmagnetic resonance .In graphene, due the SU(4) spin-valley symmetry,one can have valley pseudo-spin waves in addition tospin waves, and what we call “entanglement” waves ofmixed spin-valley character. Recent experiments have managed to electrically emit and detect spin waves using local gates. This is a highly promising result in theprospective of probing and controlling the spin degree offreedom in quantum-Hall systems. So far, the observedthreshold for the emission of a spin wave is equal to thesize of the Zeeman gap, a strong indication of the emis-sion a pure spin wave. However, Ref. [50] has suggesteda setup susceptible to generate valley waves at the edgelocated at the interface between two regions with fillingfactors ( ν , ν ) = (+1 , − ν = 0 has been studiedin Refs. [51] and [52], while the low-energy dispersionand gaps of the KD and CAF state spin-waves was ob-tained using a non-linear sigma model in Ref. [53], whichshowed the presence of gapless linearly dispersing modesin these two phases. Ref. [50] has studied the trans-mission of spin waves at a junction between regions withdifferent filling factors.Motivated by these recent experiments considering in-terfaces between regions at ν = 1, 0 and −
1, we present inthis paper a classification of the dispersion relations andthe associated gaps in the graphene quantum Hall ferro-magnet at ν = 1( −
1) when one sub-LL is empty (filled).We consider the spin waves in the four phases introducedin Ref. [22] with the addition of a “valley Zeeman” term.However, since this term does not modify substantiallythe phases but rather the location of their phase tran-sitions, we consider only the dispersion in the phases ofRef. [22]. At ν = −
1, there are three Goldstone modecorresponding to flipping one electron from the filled sub-LL to each one of the three empty sub-LLs. In the simplephases such as KD or CDW, the three modes correspondto a pure spin wave, a pseudo-spin wave and an entan-glement wave. We derive a non-linear sigma model validat long wave lengths generalized to the CP coset spacecorresponding to the space of broken symmetries. In theabsence of explicit symmetry-breaking terms at low en-ergies, all the dispersions are gapless and quadratic inthe wave vector, corresponding thus to true Goldstone modes. In the presence of the symmetry breaking terms,some modes acquire a gap, while others remain gaplessbut acquire a linear dispersion relation until a certainmomentum at which they recover their quadratic disper-sion at higher momentum. We find that this behaviororiginates from a residual symmetry of the ground state.We also find that at several high-symmetry points in thephase diagram, some originally gapped modes becomegapless.The paper is organized as follows. In Sec. II, wepresent the phase diagram originally introduced in Ref.[22] using a different labelling for the phases and also dis-cuss the introduction of a valley Zeeman term. In Sec.III, we present our non-linear sigma model using a La-grangian formalism, while in Sec. IV, we present our re-sults for the dispersion relation in the different regions ofthe phase diagram. In the conclusion section, we presenta summary of the various spin waves one encounters ineach phase, in view of their dispersion, i.e. whether theyare quadratic and gapped or linear and gapless. II. QHFM GROUND STATE
In a single particle picture, flat Landau levels (LLs)are formed in graphene under a magnetic field with en-ergies E λn = λ (cid:126) ω c √ n where λ = ± is the band index, n is the LL index, ω c = √ v/l B is the cyclotron energy,and v is the Fermi velocity of graphene. For a sufficientlystrong magnetic field, the low-energy physics of a quan-tum Hall ferromagnet in the n = 0 LL is dominated bythe Coulomb interactionˆ V C = 12 (cid:88) q (cid:54) =0 v ( q )¯ ρ ( q )¯ ρ ( − q ) , (1)in terms of the Coulomb potential multiplied by the low-est Landau level (LLL) form factor, v ( q ) = 1 A πe ε | q | |F ( q ) | , (2)where A is the area of the sample and F ( q ) is the formfactor of the LLL (see eg. Ref. [9]). Furthermore, ¯ ρ ( q )represents the density operator in momentum space pro-jected into the LLL. This Hamiltonian is approximatelySU(4) invariant under spin-valley rotations. The ex-change terms favors a completely antisymmetric orbitalwavefunction to minimize the Coulomb replusion, whichthen favors a completely symmetric spin-valley spinor.At filling ν = −
1, there is thus one electron per orbitalsite and the uniform ground state is described by theSlater determinant | ψ (cid:105) = (cid:89) m (cid:32)(cid:88) µ F µ c † m,µ (cid:33) | (cid:105) (3)where µ = { σ, ξ } runs over the spin ( σ ∈ {↑ , ↓} ) and val-ley ( ξ ∈ { K, K (cid:48) } ) indices, m is the Landau site index and F is a normalized four-component spinor which describesthe QHFM ground state. A. Parametrization of the spinor
The Coulomb Hamiltonian is SU(4) symmetric, whilethe broken symmetry ground state is invariant underSU(3) ⊗ U(1) rotations corresponding to rotations be-tween the three empty sub-LL and the relative phasebetween the empty and filled sub-LL. The coset spaceis thus CP = U (4) /U (3) ⊗ U (1) which has 6 realdimensions . A general spinor describing the brokensymmetry ground state is thus parametrized by 6 an-gles. In order to describe the spinor F , we express it as aSchmidt decomposition in the basis {| K ↑(cid:105) , | K ↓(cid:105) , | K (cid:48) ↑(cid:105) , | K (cid:48) ↓(cid:105)} as | F (cid:105) = cos α | n (cid:105)| s (cid:105) + e iβ sin α | − n (cid:105)| − s (cid:105) , (4)where | n (cid:105)| s (cid:105) = | n (cid:105) ⊗ | s (cid:105) is the tensor product of thespinors | n (cid:105) = (cid:18) cos θ P sin θ P e iϕ P (cid:19) , (5) | s (cid:105) = (cid:18) cos θ S sin θ S e iϕ S (cid:19) , (6)acting in valley and spin spaces respectively. We have σ · s | ± s (cid:105) = ±| ± s (cid:105) and τ · n | ± n (cid:105) = ±| ± n (cid:105) , where s , n = sin θ S,P cos ϕ S,P sin θ S,P sin ϕ S,P cos θ S,P (7)are the unit vectors on the spin and pseudo-spin Blochspheres, respectively, with θ S , θ P ∈ [0 , π ] and ϕ S , ϕ P ∈ [0 , π ]. The angles α ∈ [0 , π ] and β ∈ [0 , π ] are the an-gles of the ”entanglement” Bloch sphere of the particle .The spinors | − s (cid:105) and | − n (cid:105) are obtained from | s (cid:105) and | n (cid:105) by the replacement θ → π − θ and ϕ → ϕ + π suchthat we have (cid:104) s | − s (cid:105) = (cid:104) n | − n (cid:105) = 0.When θ P = 0( π ), the vector n lies at the north (south)pole of the pseudo-spin Bloch sphere corresponding to apolarization in valley K ( K (cid:48) ). Analogously, for θ S = 0( π ),the vector n lies at the north (south) pole of the spinBloch sphere corresponding to spin up (down) polariza-tion. Finally, this parametrization includes the possibil-ity of “entanglement” between the spin and the pseudo-spin. In fact, this decomposition of the spinors does notcorrespond to real entanglement between two particlesbecause here it is the spin and pseudo-spin of the same particle which is “entangled”, and the Schmidt decompo-sition can be viewed as a decomposition of SU(4) spinorsin the basis of SU(2) ⊗ SU(2) spinors. Because of this rem-iniscence and the relevance of the spin and pseudospinmagnetizations in experimental measurements, we willrefer loosely to the angle α as entanglement angle forsimplicity. B. Symmetry breaking terms
Inspired by earlier works that focus on short-range electron-electron and electron-phonon interac-tions at the lattice scale, we consider the local anisotropicHamiltonian H A = 12 (cid:90) d r (cid:8) U ⊥ [ P x ( r ) + P y ( r )] + U z P z ( r ) (cid:9) − (cid:90) d r { ∆ Z S z ( r ) + ∆ P P z ( r ) } , (8)where P ( r ) = Ψ † ( r )( σ ⊗ τ )Ψ( r ) , (9) S ( r ) = Ψ † ( r )( σ ⊗ τ )Ψ( r ) (10)are the local spin and pseudo-spin densities, respectively,in terms of the vectors σ and τ of Pauli matrices vectorsacting in spin and pseudo-spin spaces, respectively, while σ and τ are the identity matrices. In the following,we neglect the identity and consider σ ≡ σ ⊗ τ and τ ≡ σ ⊗ τ . The potentials U ⊥ and U z correspond to localinteractions that act when two electrons are at the sameposition, and they act only in valley space thus favoringin-plane or out-of-plane pseudo-spin polarizations. Therelative values of ∆ Z , ∆ P , U z and U ⊥ determine thus thespin or pseudo-spin polarization of the ground state.The first term in Eq. (8) represents the electrons’ inter-action with ”frozen” in-plane phonons and is estimatedto be of the order of U ⊥ ∼ . B [( T )] K . This term cre-ates a Kekul´e-like distortion. The term U z originatesfrom short-range Hubbard type interactions and inter-valley scattering which originate from the SU(4) symme-try breaking the in Coulomb interaction . Out-of-planephonons also contribute to U z and is estimated to beof the order of ∼ . B [( T )] K . The Zeeman coupling∆ Z = gµ B B is of the order of ∼ . B [( T )] K . Finally,∆ P corresponds to a staggered potential on the A andB sublattice which generates a mass term in the Diracequation and can be generated by the interaction with asubstrate, eg hexagonal Boron-Nitride (hBN) . Dueto the locking of the sublattice and valley indices in the n = 0 LL, this term is analogous to a Zeeman term actingin pseudo-spin space, we thereby dub it ”valley Zeeman”term. This terms favors a polarization in one valley andthus on one sublattice. The energies U ⊥ and U z are pro-portional to the perpendicular magnetic field while ∆ z is proportional to the total magnetic field. Moreover, ∆ P is an intrinsic effect and thus independent of the magneticfield. Notice that these energy scales are all on the sameorder of magnitude and are likely to be strongly sample-dependent. We thus consider them, here, as tunable pa-rameters that determine the phase diagram of the QHFMground states as well as that of the skyrmions formed ontop of these states.Applying the Hartree-Fock approximation, the energyof the anisotropic energy E A = (cid:104) F | H A | F (cid:105) can be ex-pressed as E A [ F ] = N φ (cid:104) u ⊥ (cid:16) M P x + M P y (cid:17) + u z M P z (cid:105) (11) − N φ [∆ Z M S z + ∆ P M P Z ] , (12)where N φ = A/ (2 πl B ) is the number of flux quantathreading the area A of the sample and M P = (cid:104) F | τ | F (cid:105) = n cos α (13) M S = (cid:104) F | σ | F (cid:105) = s cos α (14)are the spin and pseudo-spin magnetization respectively.The parameters u ⊥ ,z are obtained as u ⊥ ,z = V H ⊥ ,z − V F ⊥ ,z (15)where V H ⊥ ,z and V F ⊥ ,z are the Hartree and Fock potentials,respectively, associated with the potentials U ⊥ ,z . For a δ ( r ) interaction, at ν = ±
1, the Hartree and Fock poten-tials are identical and thus cancel each other . We thuspostulate a slightly non-local interaction.As a function of the angles, we obtain the expression E A [ F ] = N φ (cid:20)
12 cos α ( u ⊥ sin θ P + u z cos θ P ) − ∆ P cos α cos θ S − ∆ Z cos α cos θ S ] . (16)The phase diagram is obtained by minimizing Eq. (16).We first consider the phase diagram without the valleyZeeman term ∆ P in Sec. II C, while we show its effect inSec. II D. C. Phase Diagram without valley Zeeman term
The phase diagram of the QHFM at ν = ± et al .Here, we briefly review the different phases in order todiscuss the spin waves associated with each ground state.There is a Z redundancy in the parametrization of thespinors (see appendix of Ref. [22]) such that without lossof generality we can assume α ∈ [0 , π/ θ S = 1 everywhere.Minimizing Eq. (16), we find the four phases shown inFig. (1) which can be separated in two types : for u ⊥ >u z , an easy-axis pseudo-spin polarization is favored,which is the case of the charge density wave (CDW) andanti-ferrimagnetic (AFI) phases, while for u z > u ⊥ , aneasy-plane polarization is favored, namely, the Kekul´edistortion (KD) and canted anti-ferromagnetic (CAF)phase. In addition to that, the phases can present entan-glement ( α (cid:54) = 0) or not ( α = { , π } ). The CDW and KDphases are not entangled and they have maximal spin andpseudo-spin magnetizations, they are thereby ferromag-netic phases. The AFI and CAF phases are entangled,such that their spin and pseudo-spin magnetizations are CDWKD AFICAF ( a )- - - - - - u ⟂ / Δ Z u z / Δ Z FIG. 1. (a) Phase diagram of the QHFM ground state com-posed of four phase : charge density wave (CDW), Kekul´edistortion (KD), anti-ferrimagnetic (AFI) and canted anti-ferromagnetic (CAF). (b)-(e) Spin magnetization on the Aand B sublattices of the different phase. (b) CDW (c) KD,(d) AFI and (e) CAF. reduced. These phases are realized in the regions of pos-itive u ⊥ and u z because entanglement allows to reducethe pseudo-spin magnetization thus making a compro-mise between the spin and pseudo-spin magnetizations.In the limit of vanishing Zeeman term (compared to u ⊥ and u z ), these two phases are maximally entangled be-come both anti-ferromagnetic. We mention that, as op-posed to the ν = 0 case, at ν = ±
1, the spin and pseudo-spin can be maximal at the same time. Thus the CDWand KD phases are pseudo-spin polarized and spin ferro-magnetic, whereas at ν = 0, the phases can be either bespin polarized and pseudo-spin unpolarized (F), pseudo-spin polarized and spin unpolarized (KD and CDW) orentangled (CAF). Notice that in Ref. [22], these phaseswere named after their valley pseudo-spin magnetization :the CDW (AFI) phases are associated with an unentan-gled (entangled) easy-axis pseudo-spin order, while theKD (CAF) comes along with an unentangled (entangled)easy-plane pseudo-spin magnetization.In order to characterize the different phases, we fo-cus on experimentally measurable quantities such as thespin magnetization and electronic density on the A andB sublattices ρ A,B = 12 (cid:104) F | ( τ ± τ z ) | F (cid:105) , (17) M S A,B = 12 (cid:104) F | σ ( τ ± τ z ) | F (cid:105) , (18)respectively.The spinor of the CDW phase is | F (cid:105) = | n z (cid:105)| s z (cid:105) , (19)where n z = (1 , T and s z = (1 , T correspond to aspin and pseudo-spin both polarized at the north of theirrespective Bloch spheres, such that the electrons havespin-up and are polarized in valley K or K (cid:48) correspond-ing thus to a ferromagnetic phase restricted to a singlesublattice. The sublattice polarization is given by ρ A = 1and ρ B = 0 or ρ A = 0 and ρ B = 1 and there is thus aspontaneous Z sublattice symmetry breaking. The spinmagnetizations on sublattices A and B are M S A = s z and M S B = 0.The spinor of the KD phase is given by | F (cid:105) = | n ⊥ (cid:105)| s z (cid:105) , (20)where | n ⊥ (cid:105) = √ (1 , e iϕ ) T points to a position at theequator of the pseudo-spin Bloch sphere and correspondsthus to a superposition of the two valleys. The angle ϕ corresponds to the orientation of the pseudo-spin mag-netization in the xy plane. There is thus a residual U (1)symmetry corresponding to the angle ϕ . Both sublat-tices are equally populated such that ρ A = ρ B = 1 / M S A = M S B = s z .The spinor of the AFI phase has the expression | F (cid:105) = cos α | n z (cid:105)| s z (cid:105) + e iβ sin α | − n z (cid:105)| − s z (cid:105) , (21)with cos α = ∆ Z u z . (22)This phase corresponds thus to an entangled phase whichin turn reduces the amplitude of the spin magnetizationin order to minimize the anisotropic energy. The spinmagnetization on the A and B sublattices are M S A = (1 + cos α ) s z and M S B = ( − α ) s z such thatthe spin magnetization on each sublattice points alongthe z direction but there is an imbalance between thespin magnetization in sublattices A and B. For u z = ∆ Z ( α = 0), namely at the CDW-AFI transition, we recoverthe CDW phase, while for u z (cid:29) ∆ Z ( α → π/ M S A = − M S B = s z which is anti-ferromagnetic, as we would expect in thelimit of a vanishing Zeeman effect.The spinor of the CAF phase has the expression | F (cid:105) = cos α | n ⊥ (cid:105)| s z (cid:105) + e iβ sin α | − n ⊥ (cid:105)| − s z (cid:105) , (23)with cos α = ∆ Z u ⊥ . (24)This phase has its pseudo-spin polarized in the xy plane of the Bloch sphere and presents entangle-ment analogously to the AFI phase. Both sublatticesare populated equally ρ A = ρ B = 1 /
2. The spinmagnetization on the A and B sublattices forms acanted anti-ferromagnetic pattern with M S A , B =( ± sin α cos( β − ϕ ) , ± sin α sin( β − ϕ ) , cos α ) suchthat the z component of the magnetization is iden-tical on both sublattices, but there is a canting ofthe spin in the xy plane with opposite orientation onthe sublattices. At the transition with the KD phase(∆ Z = u ⊥ → α = 0), we recover a ferromagnetic phasewith equal weight on the K and K’ valleys, while in thefully entangled limit ( u ⊥ (cid:29) ∆ z → α = π/ xy plane. D. Phase diagram with valley Zeeman
Experimentally, graphene is generally placed on topof a substrate. In the case of hBN, a potential differ-ence is generated between the A and B sites of grapheneand yields a valley-dependent potential due to the valley-sublattice equivalence in the LLL of graphene. Such aterm favors a polarization on one sublattice and thusin one valley, analougously to a Zeeman term in valleyspace. The evolution of the phase diagram in the pres-ence of the valley Zeeman term is shown in Fig. 2. Thephases CDW and AFI are not modified by the valley Zee-man term because their pseudo-spin is already polarizedin one valley. However, the presence of the valley Zee-man breaks the Z symmetry between the two valleys byfavoring one valley corresponding to the sublattice withsmallest on-site potential. However, the KD and CAFphases are modified such that their pseudo-spin polariza-tion is now canted towards the north pole of the Blochsphere (or the south pole if the staggered potential is re-versed). The KD phase becomes a canted KD phase withspinor | F (cid:105) = | n (cid:105)| s z (cid:105) , (25)with cos θ P = ∆ P ( u z − u ⊥ ) . (26)There is thus a continuous phase transition between theCDW and CKD phase transition located at u z − u ⊥ = CDWCKD AFICAF' - - - - - - u ⟂ / Δ Z u z / Δ Z FIG. 2. (a)Phase diagram of the QHFM ground state withthe valley Zeeman term ∆ P such that ∆ P = ∆ Z . The KDand CAF phases are modified compared to the case withoutthe valley Zeeman term and are turned to a canted KD phase(CDW) and a different CAF phase (CAF’). ∆ P , where the pseudo-spin is progressively canted rela-tive to the z direction. For u z − u ⊥ (cid:29) ∆ P , we recoverthe KD phase. The CDW occupied thus a larger portionof the phase diagram compared to the ∆ P = 0 case (seeFig. 1).The transition between the CDW and AFI phase is alsomodified because the cost to entangle the easy-axis phaseimplies a non-zero weight on the valley K (cid:48) . Thereby, thetransition occurs at u z = (∆ P + ∆ Z ) and the entangle-ment angle in the AFI phase α is now given bycos α = ∆ Z + ∆ P u z . (27)Finally, the CAF phase is also modified into a differentCAF phase such that the spinor reads | F (cid:105) = cos α | n (cid:105)| s z (cid:105) + e iβ sin α | − n (cid:105)| − s z (cid:105) , (28)wherecos α = ∆ Z u ⊥ and cos θ P = ∆ Z ∆ P u ⊥ ( u z − u ⊥ ) . (29)Once again, the AFI phase is favored in a larger part ofthe phase diagram and the transition between the AFIand CAF phases is located at u z = u ⊥ (∆ P / ∆ Z + 1).The four phase transitions meet at the point ( u ⊥ , u z ) =(∆ Z , ∆ Z + ∆ P ). | � � 〉 | � � 〉 | � � 〉 | � 〉 � � � FIG. 3. Four sub-LLs of the n = 0 LL and the three associatedspin wave modes corresponding to the mixing of the filled sub-LL described by the spinor | F (cid:105) with each of the three emptysub-LLs described by the spinors | C i (cid:105) . III. NON-LINEAR SIGMA MODEL
In order to find the dispersion relations of the Gold-stone modes, we derive an effective Lagrangian which de-scribes the low-energy (long-wavelength) excitations ofthe ground state. In the SU(4) invariant limit (in theabsence of symmetry breaking terms), this Lagrangianconsists of a non-linear sigma model describing the fieldsassociated with the broken symmetries. The collectivemodes of this Lagrangian are the different Goldstonemodes. In the presence of the symmetry breaking terms,the Goldstone modes acquire a mass gap.
A. Broken symmetries and their generators
At filling factor ν = ±
1, the spontaneous symmetrybreaking mechanism corresponds to filling one sub-LLout of the four with any SU(4) spin-valley orientation(in the absence of symmetry breaking term). Explicitely,this symmetry breaking mechanism corresponds to SU (4) → SU (3) ⊗ U (1) , (30)where SU(4) is the original symmetry of the Hamiltonianwhich in composed of 15 generators and SU(3) ⊗ U(1) isthe residual symmetry the ground state which is invari-ant under tranformations that mixes the 3 empty sub-levels corresponding to 8 generators times the relativeU(1) phase between the empty and the occupied sub-LLs. According to Refs. [60] and [54], there are thus15 − − CP = U (4) / [ U (3) ⊗ U (1)] which has six dimensions .In order to find an explicit expression for the brokengenerators, we consider for simplicity the CDW groundstate | F (cid:105) = | n z (cid:105)| s z (cid:105) = | K ↑(cid:105) to be the filled sub-LL in thebasis A = {| F (cid:105) , | C (cid:105) , | C (cid:105) , | C (cid:105)} = {| K ↑(cid:105) , | K ↓(cid:105) , | K (cid:48) ↑(cid:105) , | K (cid:48) ↓(cid:105)} as shown in Fig. 3. The spinors | C i (cid:105) define theempty sub-LLs of the basis A . In this basis, we are ableto define the six broken generatorsΓ x = 12 σ x P + n z Γ x = 12 τ x P + s z Γ x = 14 ( σ x τ x − σ y τ y ) Γ y = 12 σ y P + n z Γ y = 12 τ y P + s z Γ y = 14 ( σ x τ y + σ y τ x ) , (31)where P + s z = (1 + σ z ) and P + n z = (1 + τ z ) are theprojectors over the spin up and valley K , respectively.Here, the matrices σ and τ are the usual Pauli matricesacting in the spin and pseudo-spin spaces, respectively.Explicitely, the Γ x operators areΓ x = Γ x = Γ x = . (32)The matrices Γ x,y mix | F (cid:105) and | C (cid:105) , the matrices Γ x,y mix | F (cid:105) and | C (cid:105) while the matrices Γ x,y mix | F (cid:105) and | C (cid:105) . We have thus three sets of canonically conjugatematrices such that for each mode a [Γ aµ , Γ aν ] = iε µνλ Γ aλ (33) { Γ aµ , Γ aν } = i δ µν , (34)where µ, ν, λ ∈ { x, y, z } , a ∈ { , , } , ε µνλ is the three-dimensional Levi-Civita tensor, δ µν is the identity matrixand we have introduced the additional matricesΓ z = 12 σ z P + n z , Γ z = 12 τ z P + s z , Γ z = 14 ( σ z + τ z ) , (35)to complete the algebra. To study the spin waves foranother phase, we simply rotate the spinors and the gen-erators by a SU(4) unitary transformation U | ˜ F (cid:105) = U | F (cid:105) , (36a) | ˜ C i (cid:105) = U | C i (cid:105) , (36b)˜Γ aµ = U Γ aµ U † . (36c)An important object that characterizes the spin wavesin a (anti-)ferromagnet is the matrix of the commutatorsof the broken generators over the ground state M abµν = (cid:104) F | [Γ aµ , Γ bν ] | F (cid:105) , (37)with µ, ν ∈ { x, y } . We find that it is independent ofthe basis and defines the number and dispersion of theGoldstone modes associated with the number of bro-ken symmetry (in the absence of explicit symmetry breaking terms). We find that this matrix has the ex-pression for any phase (cid:104) F | [Γ aµ , Γ bν ] | F (cid:105) = i ε µν δ ab . (38)where ε µν is the two-dimensional Levi-Civita tensor for µ, ν ∈ { x, y } . According to the general theory of Refs.[62] and [63], the number of quadratic spin waves is equalto Rank[ M ] / B. Lagrangian
The effective low-energy Lagrangian is obtained anal-ogously to Ref. [33] by constructing a coherent state | ψ [ π ] (cid:105) = e i (cid:80) r i O ( r i ,t ) | ψ (cid:105) , (39)where | ψ (cid:105) is the second quantized QHFM ground state(3) and O ( r i , t ) = π aµ ( r i , t )Γ aµ ( r i ) , (40)where π aµ ( r i , t ) are six real fields associated with the bro-ken generators Γ aµ ( r i ) acting at the Landau site r i andwe have assumed summation over repeated indices. Theycorrespond to generalized local spin-valley rotations andthus describe the quantum state | ψ [ π ] (cid:105) with spin-valleytextures.The total Lagrangian L is the sum of the kinetic term L K , the Coulomb term L C and the symmetry breaking L SB terms L = L K − L C − L SB , (41) L K = (cid:104) ψ [ π ] | i∂ t | ψ [ π ] (cid:105) , (42) L C = (cid:104) ψ [ π ] | H C | ψ [ π ] (cid:105) , (43) L SB = (cid:104) ψ [ π ] | H A | ψ [ π ] (cid:105) − (cid:104) ψ | H A | ψ (cid:105) . (44)In order to derive the effective non-linear sigma model atlow-energy, we follow closely Refs. [60], [54] and [15].
1. Kinetic term
In the continuum limit, the kinetic term can be ex-pressed as L K = ρ (cid:90) d rZ † ( r , t ) i∂ t Z ( r , t ) , (45)in terms of the spinor field Z ( r , t ) = e iO ( r ,t ) | F (cid:105) , (46)where | F (cid:105) is the ground state spinor corresponding toEq. (3). Expanding O ( r , t ) up to second order in the π fields, with the help of Eq. (38), we obtain L K = ρ (cid:90) d rε µν π aµ ∂ t π aν (47)= ρ (cid:90) d r A a [ π ] · ∂ t π a , (48)where ρ = (2 πl B ) − is the electron density, and A a [ π ] =( − π ay , π ax ,
0) is the Berry connection associated with themode a .
2. Gradient term
To lowest order in the spatial derivatives, the energyassociated with the Coulomb Hamiltonian gives rises toa gradient term L C = ρ s (cid:90) d r Tr [ ∇ P ∇ P ] (49)= 2 ρ s (cid:90) d r∂ j Z † (1 − ZZ † ) ∂ j Z, (50)where P ( r , t ) = ZZ † (51)is the (space-time dependent) order parameter of the fer-romagnet and ρ s = 116 √ π e εl B (52)is the spin stiffness. This gradient term corresponds tothe cost in exchange energy associated with the misalign-ment of neighboring spins.The matrix P is a projector that obeys P = P , P † = P and Tr[ P ] = 1. Up to second order in the π -fields, the gradient term is given by L C = ρ s (cid:90) d r ( ∇ π aµ ) , (53)where we have used the property that (cid:104) F | Γ aµ Γ bν | F (cid:105) = δ ab ( δ µν + iε µν ). We recover thus the usual non-linearsigma model term extended to the six fields in the CP space.
3. Anisotropic terms
Finally, the symmetry breaking terms correspond tothe anisotropic energy E A [ Z ] of the slowly varying field Z minus the anisotropic energy of the ground state such that we consider only the excess energy corresponding tothe spin wave L A = E A [ Z ] − E A [ F ] , (54)where E A [ F ] is given by Eq. (12) and E A [ Z ] = ρ (cid:90) d r (cid:110) (cid:88) i u i M P i [ Z ] − ∆ Z M S z [ Z ] (cid:111) , (55)with i ∈ { x, y, z } , u x = u y = u ⊥ , and M P [ Z ] = (cid:104) Z | τ | Z (cid:105) (56) M S [ Z ] = (cid:104) Z | σ | Z (cid:105) (57)are the spin and pseudo-spin magnetizations analogousto (14) generalized to the field Z . We can express theanisotropic Lagrangian in a more compact way L A = ρ (cid:90) d r (cid:88) i u i t i − ∆ Z s z , (58)with t i = (cid:104) Z | τ i | Z (cid:105) − (cid:104) F | τ i | F (cid:105) (59) s z = (cid:104) Z | σ z | Z (cid:105) − (cid:104) F | σ z | F (cid:105) . (60)We now expand the pseudo-spin magnetization up to sec-ond order in the π -fields (cid:104) Z | τ i | Z (cid:105) = (cid:104) F | e − iO τ i e iO | F (cid:105) = (cid:104) F | τ i | F (cid:105) − iπ aµ (cid:104) F | [Γ aµ , τ i ] | F (cid:105)− π aµ (cid:104) F | [Γ aµ , [Γ bν , τ i ]] | F (cid:105) π bν , (61)and we have a similar expression for the spin magnetiza-tion. Upon squaring, the pseudo-spin anisotropy has alinear and a quadratic term in the π -fields t i = R aµ π aµ + π aµ R abi,µν π bν , (62)with R aiµ = − i (cid:104) F | τ i | F (cid:105)(cid:104) F | [Γ aµ , τ i ] | F (cid:105) (63) R abi,µν = − (cid:104) F | [Γ aµ , τ i ] | F (cid:105)(cid:104) F | [Γ bν , τ i ] | F (cid:105)− (cid:104) F | τ i | F (cid:105)(cid:104) F | [Γ aµ , [Γ bν , τ i ]] | F (cid:105) . (64)The Zeeman term is linear in the spin magnetization suchthat we have s z = R aZµ π aµ + π aµ R abZ,µν π bν , (65)where R aZµ = − i (cid:104) F | [Γ aµ , σ z ] | F (cid:105) (66) R abZ,µν = − (cid:104) F | [Γ aµ , [Γ bν , σ z ]] | F (cid:105) (67)For every state | F (cid:105) , the linear terms cancel each other (cid:88) i u i R aiµ − ∆ Z R aZµ = 0 (68)for all µ and a . The anisotropic Lagrangian can thus bewritten as L A = (cid:90) d r π T R π (69)where π = ( π aµ ) is the six-component vector made of the π -fields and R abµν = (cid:88) i u i R abiµν − ∆ Z R abZµν (70)is a 6 × { µ, a } that we call theanisotropy matrix.We now consider the effective action S = (cid:82) dt L andFourier transform the kinetic and gradient Lagrangians(47) and (53) in space and time S = (cid:90) dωd k π T ( k , ω ) M π ( − k , − ω ) , (71)with M abµν = (cid:16) ρ iωε µν − ρ s k δ µν (cid:17) δ ab − ρ R abµν . (72)The dispersion relations of the collective mode are ob-tained by minimizing the action, δ S /δπ ( k , ω ) = 0, whichgives the equation M ( k , ω ) π ( k , ω ) = 0 . (73)Because the matrix M ( k , ω ) is hermitian, the frequenciesalways come in pairs ± ω ( k ). However, we only considerthe three positive eigenfrequencies ω α ( k ), which corre-spond to the physically relevant modes, and discard thenegative-energy solutions. The corresponding fields π areobtained by finding the null space of M . The resultingspinor is thus given by | Z α (cid:105) = (cid:18) + iπ aµ,α Γ aµ − π aµ,α π bν,α Γ aµ Γ bν (cid:19) | F (cid:105) , (74)where π aµα is the eigenstate corresponding to the fre-quency ω α . When the matrix is block-diagonal M abµν ∝ δ ab , the different modes are decoupled and the eigenstatelabels are identical to the mode label α = a . This is thecase for the CDW and KD phases. C. Change of ground state
The general analysis of the previous sections has beenperformed by considering the ground state spinor | F (cid:105) = | n z (cid:105)| s z (cid:105) . To consider a different ground state, we perform the unitary rotation given by Eqs. (36). The spinor Z isthus transformed as˜ Z = U Z = e i ˜ π aµ ˜Γ aµ | ˜ F (cid:105) , (75)where we have introduced the the fields ˜ π aµ which cor-respond now to the modes a associated with the brokengenerators ˜Γ aµ . However, for simplicity, we will keep thenotation π aµ in every basis and assume that the π -fieldscorrespond to the modes in the corresponding basis.The kinetic and gradient terms are independent ofthe basis because the SU(4) transformation matrix U isglobal L K [ ˜ Z ] = L K [ Z ] and L C [ ˜ Z ] = L C [ Z ]. However,the symmetry breaking terms are basis dependent. Thespin and pseudo-spin magnetization read (cid:104) ˜ Z | τ | ˜ Z (cid:105) = (cid:104) Z | P | Z (cid:105) (76) (cid:104) ˜ Z | σ z | ˜ Z (cid:105) = (cid:104) Z | S z | Z (cid:105) , (77)such that instead of computing the commutators in Eq.(61) using the transformed matrices ˜Γ aµ , we simply re-place the matrices τ and σ z by P = U † τ U (78) S z = U † σ z U, (79)such that the pseudo-spin magnetization reads (cid:104) ˜ Z | τ i | ˜ Z (cid:105) = (cid:104) F | P i | F (cid:105) − iπ aµ (cid:104) F | [Γ aµ , P i ] | F (cid:105)− π aµ (cid:104) F | [Γ aµ , [Γ bν , P i ]] | F (cid:105) π bν , (80)where | F (cid:105) = | n z (cid:105)| s z (cid:105) and the matrices Γ aµ are given byEqs. (31). We have a similar expression for the spinmagnetization in the transformed basis. Thus instead ofcomputing the transformed matrices and spinors in thenew basis, we simply express the matrices τ and σ z inthe basis ˜ A . Thus the anisotropic Lagrangian reads L A [ ˜ Z ] = (cid:90) d r π ˜ R π , (81)where ˜ R abµν = (cid:88) i u i ˜ R abiµν − ∆ Z ˜ R abZµν , (82)and the matrices ˜ R abiµν and ˜ R abZµν are obtained from Eqs.(64) and (67) by the replacements τ i → P i and σ z → S z . IV. DISPERSION RELATIONS
Using the formalism developped in the previous sec-tion, we now diagonalize the matrix (72) to find the dis-persion relations of the three different modes and theirassociated gaps. We only consider the four phases of Sec.II C without the valley Zeeman term since they are notsubstantially modified upon its introduction.0
SpinPseudo - SpinEntanglement kl B ω / Δ Z FIG. 4. Dispersion relation of the three modes in the KDphase for u z = − ∆ Z and u ⊥ = 2∆ Z . The three modes aregapped and quadratically dispersing. A. Charge density wave phase
In the charge density wave, the ground state spinorand the empty sub-LL | C a (cid:105) defining the three mode a have the expression | F (cid:105) = | n z (cid:105)| s z (cid:105) = (1 , , , T (83a) | C (cid:105) = | n z (cid:105)| − s z (cid:105) = (0 , , , T (83b) | C (cid:105) = | − n z (cid:105)| s z (cid:105) = (0 , , , T (83c) | C (cid:105) = | − n z (cid:105)| − s z (cid:105) = (0 , , , T (83d)in the basis {| K ↑(cid:105) , | K ↓(cid:105) , | K (cid:48) ↑(cid:105) , | K (cid:48) ↓(cid:105)} . We havechosen here a ground state polarized in valley K , butone can also choose a polarization in valley K (cid:48) by thereplacement n z → − n z . The mode a = 1 which mixes | F (cid:105) and | C (cid:105) corresponds to a pure spin wave such thatthe pseudo-spin remains unaffected. The mode a = 2mixes | F (cid:105) and | C (cid:105) and corresponds to a pseudo-spinwave where the spin remains unaffected. The mode a = 3corresponds to an entanglement wave in which inversesboth the spin and pseudo-spin such that the spinor Z isin a superposition of | n z (cid:105)| s z (cid:105) and | − n z (cid:105)| − s z (cid:105) .The anisotropy matrix R is block diagonal R abµν ∝ δ ab such that the three modes are decoupled. We find the dis-persion relations ω a ( k ) corresponding to the three modes a = 1 , , ω ( k ) = 2 πρ s ( k l B ) + ∆ Z (84) ω ( k ) = 2 πρ s ( k l B ) + u ⊥ − u z (85) ω ( k ) = 2 πρ s ( k l B ) + ∆ Z − u z . (86)The three modes have a quadratic dispersion and a massterm proportional to the anisotropic energy terms. TheCDW region is defined by u ⊥ > u z and u z < ∆ Z suchthat the three modes have a positive gap in the region.The three eigenmodes have the same expression for eachmode such that the spinor with wavevector k correspond- | ↑ ⟩ | ↓ ⟩ | K ⟩ | K ′ ⟩ | K ↑ ⟩ | K ′ ↓ ⟩ ( a ) ( b ) ( c ) FIG. 5. Bloch spheres corresponding to each modes in theCDW phase. (a) Spin Bloch sphere of the pure spin mode1. (b) Pseudo-spin Bloch sphere of the pseudo-spin mode 2.(c) Entanglement Bloch sphere corresponding to the entan-glement mode 3. The black arrow indicates the ground statepolarization, while the red arrow corresponds to the magne-tization at a point in space in the presence of a spin wave.The red arrow rotates periodically around the ground statepolarization according to Eq. (87). ing to mode a reads Z k a ( r , t ) = (cid:18) − π (cid:19) | F (cid:105) + i π e i ( k · r − ω a t ) | C a (cid:105) (87)where π (cid:28) | C a (cid:105) with the phaseoscillating at frequency ω a and wavevector k .The first mode corresponds to a pure spin wave, its gapis unaffected by the anisotropic term and depends onlyon the Zeeman term. Because the pseudo-spin remainsunaffected by the spin wave and is polarized in one valley,the spins live only on one sublattice (we choose sublatticeA here for illustration) and spin the magnetization is M S A = π cos( k · r − ω t ) π sin( k · r − ω t )1 − π , M S B = 0 . (88)The spin wave consist thus of the spins of sublattice Aprecessing around the axis z at frequency ω as shown inFig. 6.(a).The second mode corresponds to a pseudo-spin wavefor which the gap depend only on the pseudo-spinanisotropic terms and not on the Zeeman term. In theCDW region, we have chosen for simplicity a polariza-tion in the valley K (a similar treatment can be doneif the polarization is in valley K (cid:48) ) and thus the pseudo-spin points towards the north pole of the Bloch sphere.Because the pseudo-spin magnetization points along the z direction, the anisotropic energy of the ground statedepends only on u z . The presence of a pseudo-spin waveintroduces a pseudo-spin magnetization in the xy planeof the Bloch sphere, such that the magnetization out-of-plane anisotropic energy u z is reduced, while there is acost in in-plane anisotropic energy u ⊥ , hence the gap isproportional to ( u ⊥ − u z ). The pseudo-spin magnetiza-1 (a)(b) (c)FIG. 6. Three modes of the CDW phase. (a) ”Snapshot”of the pure spin wave mode a = 1 seen from the top withwavevector k along the axis y . We observe the precession ofthe spins around the z axis of the spins in the A sublattice. (b)Sublattice polarization of the pseudo-spin wave mode a = 2.We observe a small electronic density on sublattice B. Thedynamic part of the field is encoded in the relative phase ofthe superposition between valley K and K (cid:48) . The spin mag-netization is proportional to the sublattice density and pointsalong s z . (c) Spin magnetization on the A and B sublatticesof the entanglement mode a = 3, there is a small spin mag-netization on the B sublattice with opposite direction as onsublattice A. tion is given by M P = π cos( k · r − ω t ) π sin( k · r − ω t )1 − π . (89)This expression for the pseudo-spin is analogous to thespin magnetization (88) of the pure spin wave. It isnow the pseudo-spin that precesses around the z axis,such that it corresponds to a superposition of the val-ley K and K (cid:48) with a relative phase oscillating at fre-quency ω . However, the electronic density imbalance ofthe sublattice, which corresponds to the z component ofthe pseudo-spin magnetization ( M P z = ρ A − ρ B ) remainsuniform ρ A = 1 − π ρ B = π B . Because the spinors | F (cid:105) and | C (cid:105) both have spins pointing along the z direction,the spin magnetization on sublattices A and B is sim-ply proportional to the electronic density, M S A = ρ A s z and M S B = ρ B s z . The total spin magnetization is thus M S = s z . b) c) Δ a / Δ Z a) SpinPseudo - Spin Entanglement - - - - - u ⟂ / Δ Z u z / Δ Z FIG. 7. Size of the gap of a) the pseudo-spin and b) theentanglement waves as a function of u ⊥ and u z in the CDWregion. We observe that the pseudo-spin gap ∆ vanishes atthe boundary with the KD phase, and the entanglement gap∆ vanishes at the boundary with the AFI entangled phase.c) ”Phase diagram” of the spin mode with the lowest gap. Wecan see that the pseudo-spin and entanglement modes havethe lowest energy near the phase boundaries, whereas the spinmode dominates elsewhere. The spinors of the third mode cannot be expressed asa tensor product of a spin and a valley spinors. Thereby,this mode is an entanglement mode which mixes the sub-LLs | K ↑(cid:105) and | K (cid:48) ↓(cid:105) . It corresponds to the electron be-ing mainly polarized on sublatice A with spin up with asmall polarization on sublatice B with spin down with therelative phase oscillating at frequency ω . Analogouslyto the pseudo-spin wave, the pseudo-spin magnetizationalong the z direction is reduced ( M P z = 1 − π /
2) suchthat there is a gain in anisotropic energy u z . However,there is a cost in Zeeman energy, and the gap is propor-tional to ∆ Z − u z . The sublattice polarizarion is identicalto the pseudo-spin wave but the spin magnetization is M S A = (cid:18) − π (cid:19) s z M S B = − π s z (91)such that the total spin is reduced similarly to the spinwave.Figs. 7.(a) and (b) show the size of the gaps ∆ a of thepseudo-spin and entanglement modes in units of ∆ Z . Wecan see that the size of the gap decrease as we get closer tothe boundaries and eventually vanish at the boundaries.The gap of the pseudo-spin wave vanishes at theboundary with the KD phase defined by u ⊥ = u z . At2this line, as one can see from Eq. (8), the SU(2) pseudo-spin symmetry is restored and there is thus no pre-ferred orientation of the pseudo-spin. There is no costin anisotropic energy for the creation of a pseudo-spinwave. The pseudo-spin wave becomes thus a true Gold-stone mode where the spontaneously broken symmetry isthe SU(2) pseudo-spin rotation symmetry.The gap of the entanglement wave vanishes at theboundary u z = ∆ Z with the anti-ferrimagnetic phasewhich is an entangled phase. This comes from the factthat the spin and pseudo-spin magnetizations along z ofthe wave are identical M S z = M P z = (1 − π /
2) becausewe have a small imbalance over the state | K (cid:48) ↓(cid:105) with op-posite spin and pseudo-spin that of | K ↑(cid:105) . In addition,there is no spin and pseudo-spin magnetization in the xy plane M S x ,P x = M S y ,P y = 0. Thus, at the transitionline u z = ∆ Z , up to second order in π , the anisotropicenergy term E A [ Z ] = u z M P Z − ∆ Z M S z = u z − ∆ Z = E A [ F ] , (92)which is independent of the amplitude π . Thereby, forsmall amplitudes, the spin and pseudo-spin magnetiza-tions cancel each other at the transition line. This sym-metry between the spin and pseudo-spin magnetizationwill be explored further in Sec. IV C. B. Kekul´e distortion phase
In the KD phase, we apply the unitary transformation U KD = e i π n · τ , (93)with n = (sin ϕ, − cos ϕ,
0) to the spinors (83) of theCDW phase such that we have the spinors in the KDphase | ˜ F (cid:105) = | n ⊥ (cid:105)| s z (cid:105) = 1 √ , , e iϕ , T (94a) | ˜ C (cid:105) = | n ⊥ (cid:105)| − s z (cid:105) = 1 √ , , , e iϕ ) T (94b) | ˜ C (cid:105) = | − n ⊥ (cid:105)| s z (cid:105) = 1 √ − e − iϕ , , , T (94c) | ˜ C (cid:105) = | − n ⊥ (cid:105)| − s z (cid:105) = 1 √ , − e − iϕ , , T , (94d)where we have a U(1) pseudo-spin symmetry in the xy plane of the Bloch sphere. Similarly to the analysis forthe CDW phase, the mode 1 is a pure spin wave wherethe pseudo-spin is unaffected, the mode 2 is a pseudo-spinwave, while the mode 3 is an entanglement mode.The anisotropy matrix R is again block diagonal R abµν ∝ δ ab such that the three modes are decoupled. Wefind the dispersion relations ω a ( k ) corresponding to the | s z ⟩ | − s z ⟩ | n ⊥ ⟩ | s z ⟩ | − n ⊥ ⟩ | − s z ⟩ ( a ) ( b ) ( c ) | n z ⟩ | − n z ⟩ | n ⊥ ⟩ FIG. 8. Bloch spheres corresponding to each modes in theKD phase in the same way as Fig. 5. (a) Spin Bloch sphereof the spin mode 1. (b) Pseudo-spin Bloch sphere of thepseudo-spin mode 2. The ground state has a U(1) symme-try for rotations around the z axis. (c) Entanglement Blochsphere corresponding to the entanglement mode 3. For thespin and entanglement modes, the red arrow rotates period-ically around the ground state polarization according to Eq.(87). At low-energy, the pseudo-spin mode is restricted tothe equator of the pseudo-spin Bloch sphere, which costs noanisotropic energy, while at higher energy, it acquires an ele-ment along the z direction. SpinPseudo - SpinEntanglement kl B ω / Δ Z FIG. 9. Dispersion relation of the three modes in the KDphase for u z = 2∆ Z and u ⊥ = − Z . We observe that thepseudo-spin mode is gapless, linear at low momentum | k (cid:28) k and becomes quadratic at higher momentum. the two othermodes are gapped and quadratic. three modes a = 1 , , ω ( k ) = 2 πρ s ( k l B ) + ∆ Z (95) ω ( k ) = | k | l B (cid:112) πρ s (cid:112) πρ s ( k l B ) + u z − u ⊥ (96) ω ( k ) = 2 πρ s ( k l B ) + ∆ Z − u ⊥ . (97)The dispersion of the three modes is shown in Fig. 9.Analogously to the CDW case, the mode 1 correspondsto a spin mode, the mode 2 to a pseudo-spin mode andthe mode 3 to an entanglement mode.The spin and entanglement modes are quite similarto the modes observed for the CDW phase, they arequadratic gapped modes with a gap proportional to theZeeman coupling for the spin wave and a gap equal to3∆ Z − u ⊥ for the entanglement wave, which correspondsto flipping both the spin and the pseudo-spin. This gap isalways positive since in the KD phase, we have u ⊥ < ∆ Z .The space-time dependent spinor corresponding to thesetwo mode has the same expression as (87) with the basisspinors given by Eqs. (94). The pure spin wave has thespins of each sublattice oscillating at frequency ω withequal weight on both sublattices M S A = M S B = 12 π cos( k · r − ω t ) π sin( k · r − ω t )1 − π . (98)Finally, the second mode looks different, it has a gap-less linear dispersion at low-momentum k (cid:28) u z − u ⊥ while we recover a quadratic dispersion relation at highmomentum. The transition between these two regimesoccurs at a momentum of k = √ u z − u ⊥ . Similarly tothe pseudo-spin mode in the CDW phase (85), the energy u z − u ⊥ corresponds to the energy necessary to bring onepseudo-spin out of the plane, namely there is a cost inout-of-plane anisotropic energy u z but a gain in in-planeanisotropic energy u ⊥ . This energy is always positive inthe KD region since u z > u ⊥ . Thereby, at low momen-tum, there is not enough energy to bring one pseudo-spinout of the plane. The model corresponds thus to an XY model where the pseudo-spin is restricted to the equatorof the Bloch sphere and this mode is analogous to thelinearly dispersing superfluid mode in Helium and in bi-layer 2DEGs . Its gaplessness originates from theU(1) symmetry of the ground state : there is no cost inanisotropic energy cost for rotating a pseudo-spin in the xy plane. When the energy is larger than u z − u ⊥ , thereis now enough energy to bring the pseudo-spin out of theplane and we recover the usual quadratic dispersion rela-tion associated with the fact that the two generators arenow canonically conjugate. C. Anti-ferrimagnetic phase
The unitary matrix that tranforms the CDW spinors(83) into the entangled spinors of the AFI phase is givenby U AFI = e i α σ x m · τ , (99)where m = (sin β, − cos β,
0) and α is given by Eq. (22).The basis spinors of the AFI phase are | ˜ F (cid:105) = cos α | n z (cid:105)| s z (cid:105) + e iβ sin α | − n z (cid:105)| − s z (cid:105) (100a) | ˜ C (cid:105) = cos α | n z (cid:105)| − s z (cid:105) + e iβ sin α | − n z (cid:105)| s z (cid:105) (100b) | ˜ C (cid:105) = − sin α e − iβ | n z (cid:105)| − s z (cid:105) + cos α | − n z (cid:105)| s z (cid:105) (100c) | ˜ C (cid:105) = − sin α e − iβ | n z (cid:105)| s z (cid:105) + cos α | − n z (cid:105)| − s z (cid:105) . (100d) mode α = α = kl B ω / Δ Z FIG. 10. Dispersion relation of the three modes in the AFIphase for u z = 2∆ Z and u ⊥ = 6∆ Z . The modes a = 1 and a = 2 are coupled and form the α = 1 and α = 2 whichare quadratically dispersing while the entanglement mode islinear and gapless. We can see that the modes 1 and 2 involves the fourbasis spinors | n z (cid:105)| s z (cid:105) , |− n z (cid:105)|− s z (cid:105) , |− n z (cid:105)| s z (cid:105) and | n z (cid:105)|− s z (cid:105) and one cannot factor the spinors in order to have adefinite spin or pseudo-spin mode. We find that thesetwo modes are coupled and their dispersions are given by ω , = ± (cid:16) u ⊥ − u z (cid:17) cos α + (cid:114) πρ s ( k l B ) [2 πρ s ( k l B ) + 2 u ⊥ ] + u ⊥ α , (101)which are both positive due to the gap term inside thesquare root. The gaps ∆ α of the modes α = 1 and α = 2are ∆ = u z cos α = ∆ Z (102)∆ = ( u ⊥ − u z ) cos α . (103)For α = 0, namely at the boundary with the CDWphase, the spinors (100) simplify to the CDW spinorsand we recover the pseudo-spin mode with gap u ⊥ − u z and the spin mode with gap ∆ Z = u z .The dispersion for the entanglement mode a = 3 isgiven by ω ( k ) = (cid:112) πρ s | k | l B (cid:112) πρ s ( k l B ) + u z (1 − cos α ) . (104)We can see that for cos α = 1 (∆ Z = 2 u z ), namelyat the transition with the CDW phase, we obtain agapless quadratic dispersion. When ∆ Z < u z , wehave a linear dispersion at low momentum which trans-forms into a quadratic dispersion around momentum k = (cid:112) u z (1 − cos α ). This mode is analogous tothe pseudo-spin mode in the KD phase. The linearityat low-momentum originates from the U(1) symmetry ofthe ground state associated with the parameter β in Eqs.4 | n z ⟩ | s z ⟩ | − n z ⟩ | − s z ⟩ | F ⟩ | C ⟩ | n ⊥ ⟩ | s z ⟩ | − n ⊥ ⟩ | − s z ⟩ | F ⟩ | C ⟩ ( a ) ( b ) FIG. 11. Entanglement Bloch spheres corresponding to theentanglement mode in (a) the AFI phase and (b) the CAFphase. The spinor | F (cid:105) indicated by the black arrow corre-sponds to the ground state, while the spinor | C (cid:105) is locatedat opposite direction of the Bloch sphere. The ground statespossesses a U(1) symmetry associated with the angle β cor-responding to the latitude indicated by the circle at the tipof the black and red arrows. At low-energy, the entangle-ment wave correponds to a small deviation at equi-latitudeindicated by the red arrow. (100a) and (100d). The spinors | ˜ F (cid:105) and | C (cid:105) are both ina superposition of the states | n z (cid:105)| s z (cid:105) and | − n z (cid:105)| − s z (cid:105) as shown in Fig. 11. It costs thus no anisotropic en-ergy to move the ground state (black arrow in Fig. 11)around the parallel of the Bloch sphere at which lie boththe black and red arrows. At higher momentum, thereis enough energy to bring the entanglement mode out ofthis latitude and restore the symmetry between the xy direction and the z direction. D. Canted anti-ferromagnetic phase
The unitary matrix that tranforms the CDW spinors(83) into the entangled spinors of the canted anti-ferromagnetic phase is the product of the matrices (93)and (99) of the KD and AFI phase U CAF = e i π n · τ e i α σ x m · τ , (105)where n = (sin ϕ, − cos ϕ, m = (sin β, − cos β,
0) and α is given by Eq. (24). mode α = α = kl B ω / Δ Z FIG. 12. Dispersion relation of the three modes in the CAFregion for u z = 12∆ Z and u ⊥ = 2∆ Z . We observe two gaplessmodes : the entanglement mode and the mode α = 2 whichoriginates from the gapless mode of the KD region. The basis spinors of the AFI phase are | ˜ F (cid:105) = cos α | n ⊥ (cid:105)| s z (cid:105) + e iβ sin α | − n ⊥ (cid:105)| − s z (cid:105) (106a) | ˜ C (cid:105) = cos α | n ⊥ (cid:105)| − s z (cid:105) + e iβ sin α | − n ⊥ (cid:105)| s z (cid:105) (106b) | ˜ C (cid:105) = − sin α e − iβ | n ⊥ (cid:105)| − s z (cid:105) + cos α | − n ⊥ (cid:105)| s z (cid:105) (106c) | ˜ C (cid:105) = − sin α e − iβ | n ⊥ (cid:105)| s z (cid:105) + cos α | − n ⊥ (cid:105)| − s z (cid:105) (106d)The modes a = 1 and a = 2 are also coupled and wedon’t present their explicit expression here since it is toolengthy. We find the corresponding gaps∆ = ∆ Z (107)∆ = 0 , (108)such that one mode is gapless with a linear dispersionrelation at low-energy as can be seen in Fig. 12 andone mode has a pure Zeeman gap. We can see that themodes α = 1 and α = 2 originate from an anti-crossingaround momentum | k | l B ≈ .
03 between a linear modeand a gapped quadratic mode, which are the descendantsof the spin and the gapless pseudo-spin modes of the KDphase. The mode 1 becomes quadratic at higher energy.Once again, the mode 3 is decoupled from the others,and corresponds thus to an entanglement mode with dis-persion ω ( k ) = (cid:112) πρ s | k | l B (cid:112) πρ s ( k l B ) + u ⊥ (1 − cos α ) . (109)This mode is the analog of the entanglement mode in theAFI phase except that we are in the basis {| n ⊥ (cid:105)| s z (cid:105) , | − n ⊥ (cid:105)| − s z (cid:105)} as shown in Fig. 11. The gaplessness and lin-earity originates also from the U(1) symmetry associatedwith the angle β in Eqs. (106a) and (106d).5 CDW ω = k + Δ ω = k + Δ ω = k + Δ KD ω = k + Δ ω ∝ k ω = k + Δ AFI ω = k + Δ ω = k + Δ ω ∝ kCAF ω = k + Δ ω ∝ k ω ∝ k - - - - u ⟂ / Δ Z u z / Δ Z FIG. 13. Summary of the low-energy dispersion relation in thefour phases. The indices 1,2 and 3 refer to the spin, pseudo-spin and entanglement modes respectively, except in the CAFand AFI region where the spin and pseudo-spin modes arecoupled. In the schematic expression of the dispersion rela-tions, we have set ρ s /ρ = 2 πρ s l B ≡
1. In the CDW region,the three modes are gapped. In the KD region, there are twogapped modes and one gapless linear modes, the pseudo-spinmode. In the AFI region, the entanglement mode is gaplesswhile the two other modes are gapped. Finally, in the CAFregion, there are two gapless modes, the entanglement andthe coupled mode α = 2, the descendant of the pseudo-spinmode. V. CONCLUSION
To conclude, we have presented the dispersions of thedifferent types of spin waves, namely pure spin, valley,and entanglement waves, in graphene at filling factor ν = ±
1. We have considered the four different pos-sible ground states presented by Lian et al based on the anisotropic terms u ⊥ and u z originally introduced byKharitonov . We have introduced a non-linear sigmamodel based on a Lagrangian formalism which describesthe long wavelength space-time dependent spin-valleyrotations. The presence of small explicit symmetry-breaking terms generally opens a gap in the dispersionrelation of the different types of spin waves. However, wehave found that in each phase, except in the CDW region,there remain one or two gapless modes with a linear dis-persion relation at low momentum. The fact that thesemodes remain gapless originates from a residual symme-try of the ground state, which is present even when thesymmetry breaking terms are introduced. These modesrecover a quadratic dispersion relation at higher energieswhen the symmetry between the different directions ofoscillation is restored. The summary of our findings forthe presence or absence of a gap for the three modes ineach region is presented in Fig. 13.Our study, along with the expression for the gaps at ν = 0 for the KD and CAF phase presented in Ref. [53]opens the way to an analysis of the scattering of spinwaves at interfaces between regions with different fillingfactor taking into account the different types of spin wave(spin, pseudo-spin or entanglement). Depending on thesteepness of the scattering region, we expect a differentscattering process and emit the possibility that one wavetype in the ν = ± ν = 0. Thescattering mechanism should also depend on the phasethe region at ν = 0 is in. ACKNOWLEDGMENTS
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