Table Of ContentSymmetry characterization of the collective modes of the phase diagram of the ν = 0
quantum Hall state in graphene: mean-field and spontaneously broken symmetries
J. R. M. de Nova1,2 and I. Zapata2
1Department of Physics, Technion-Israel Institute of Technology, Technion City, Haifa 32000, Israel
2Departamento de F´ısica de Materiales, Universidad Complutense de Madrid, E-28040 Madrid, Spain
(Dated: January 13, 2017)
Wedevotethisworktothestudyofthemean-fieldphasediagramoftheν =0quantumHallstate
7
in bilayer graphene and the computation of the corresponding neutral collective modes, extending
1
0 the results of recent works in the literature. Specifically, we provide a detailed classification of
2 the complete orbital-valley-spin structure of the collective modes and show that phase transitions
are characterized by singlet modes in orbital pseudospin, which are independent of the Coulomb
n
strength and sufferstrong many-bodycorrections from short-range interactions at low momentum.
a
We describe the symmetry breaking mechanism for phase transitions in terms of the valley-spin
J
structureoftheGoldstonemodes. Fortheremainingphaseboundaries,weprovethattheassociated
2
exact SO(5) symmetry existing at zero Zeeman energy and interlayer voltage survives as a weaker
1
mean-field symmetry of the Hartree-Fock equations. We extend the previous results for bilayer
graphenetothemonolayerscenario. Finally,weshowthattakingintoaccountLandaulevelmixing
]
l through screening does not modify the physicalpicture explained above.
l
a
h PACSnumbers: 11.30.Qc71.10.-w73.22.Pr73.43.-f73.43.Lp73.43.Nq75.30.Ds
-
s
e
I. INTRODUCTION [7, 8, 13, 24, 26, 27, 29, 30]. For instance, the spectrum
m
of inter-LL collective excitations was obtained in Refs.
t. Since the discovery of the integer [1, 2] and fractional [8, 13, 30]. On the other hand, the spectrum of intra-
a
LL excitations within the zero-energy Landau level was
m quantumHall(QH)effects[3–5],alargenumberofworks
computed for the ν = 0 QH state [29] and the rest of
have been devoted to the study of two-dimensional(2D)
- integer QH states [24] allowing only for long range and
d systemsinthepresenceofstrongperpendicularmagnetic
interlayerCoulombinteractions. InRef. [26],the disper-
n fields. Due to its chiral character, rich valley-spin struc-
o tureandrelativelylargecyclotronfrequency,grapheneis sionrelationofthemodesfortheKekul´edistortion(KD)
c a particulary interesting scenario to test QH features. andCAFphasesoftheν =0QHstatewascomputedfor
[ MLG using an effective low-energy model. Also within
In this way, the study of QH states in graphene has
the ν = 0 QH state in the monolayer scenario, the bulk
1 been a hot topic of research in the last years [6–28].
v Among all the possible QH states, the ν = 0 QH state, andedgecollectivemodesoftheCAFandFphaseswere
8 obtained in Ref. [27].
correspondingtothechargeneutralitypoint,hasreceived
7
specialattentiondue to its intriguingstronglyinsulating We devote this work to the computation of the mean-
2
behavior in both bilayer (BLG) and monolayer (MLG) field energies and the intra-LL collective modes of the
3
0 graphene at zero in-plane magnetic field. In particu- whole phase diagram of the ν = 0 QH state of bilayer
. lar, BLG provides a more interesting scenario because graphene,aspresentedinRef. [21]. Specifically,wecom-
1
of the richer structure of its zero-energy Landau level pute the dispersion relation within the time-dependent
0
7 (LL) and the possibility of introducing an energy bias Hartree-Fock approximation (TDHFA), considering the
1 between the two valleys with the help of a perpendic- complete Hamiltonian of short-range interactions [21]
: ular electric field. In fact, Ref. [21] presented a com- and also including explicitly the interaction of the elec-
v
pletecharacterizationofthemean-fieldphasediagramof tronswiththefilledDiracsea[19],devotingspecialatten-
i
X the ν = 0 QH state in bilayer graphene, taking into ac- tiontothecharacterizationoftherichorbital-valley-spin
r count the most general spin symmetric interactions (in- structure of the modes, induced by the short-range val-
a cluding short-range valley/sublattice asymmetric inter- ley asymmetric interactions, and to the study of phase
actions) and the introduction of a voltage between the transitions in terms of these modes. In this way, the
two layers. The identified phases are ferromagnetic (F), work here presented provides a natural continuation to
canted anti-ferromagnetic (CAF), fully layer-polarized the recent literature on the field.
(FLP) and partially layer-polarized (PLP) [21]. The re- In particular, we show that only the singlet modes in
sulting phase diagram is analog to that of monolayer orbital pseudospin are strongly modified by many-body
graphene (MLG) [20] due to the identical structure of effects arisingfromshort-rangeinteractionswhile the re-
the Hamiltonian governing short-range valley/sublattice mainingmodesarealmostunaffectedbythemduetothe
asymmetric interactions. dominant character of long range Coulomb interactions.
Complementarily, the computation of the spectrum of Indeed, as these orbital singlet modes are independent
the collective excitations in QH integer states has been of the Coulomb interaction strength at low momentum,
also the subject of study in an important number works they also represent the lowest-energyneutral excitations
2
ofthesystem,playingacrucialroleinunderstandingthe II. EFFECTIVE HAMILTONIAN FOR THE ν =0
stability of the different phases. Regarding the valley- QUANTUM HALL STATE IN BILAYER
spin structure of the modes, we classify them according GRAPHENE
to the conserved valley and spin quantum numbers of
the total Hamiltonian. We find that, while the FLP- A. Low-energy Hamiltonian
PLP transition is governed by a spin singlet mode, the
F-CAFtransitionisgovernedbyavalleytripletone;this Inthefirstplace,webrieflypresenttheeffectivemodel
contrast arises due to the different nature of the spon- usedinthisworkforbilayergraphene,followingRefs. [6,
taneous symmetry breaking mechanism of the CAF and 20–22, 30], where the reader is referred for more details.
PLP phases. Interestingly, we prove that the remaining Theeffectivedynamicsatlowenergiescanbedescribed
phase boundaries, F-FLP and CAF-PLP, present a gap- by a two-band model [6], in which the field operator for
less mode arising from a mean-field symmetry inherited electrons has 8 components and reads:
fromthefullexactSO(5)symmetryexistingatthesame
boundaries for zero Zeeman energy and zero interlayer ψˆ (x)
ψˆ(x)= + (1)
voltage [26]. Moreover,we show that the CAF and PLP ψˆ (x)
phases are able to present dynamical instabilities as a (cid:20) − (cid:21)
result of their spontaneously broken symmetries. ψˆKAσ(x) ψˆKA¯σ(x)
bilTayheersatnrdonmgoannoalaloygeyrgbreatpwheeennetahlelowνs=us0toQHstrsatiagthetsfoirn- ψˆσ(x)=ψψψˆˆˆKKK′′BBA˜˜σσσ(((xxx)))≡ ψψψˆˆˆKKK′′BBA¯¯¯σσσ(((xxx))) , σ =±
wardly translate most of these results to the monolayer
with A,B˜ the most far apart sublattices, K,K the two
scenario, recovering essentially the same results of Ref. ′
valleys and σ the spin polarization. We note that the
[27]. We alsostudy the effects ofLandaulevelmixing by
two sublattices are interchanged in the K valley so, as
consideringthescreeningeffectofCoulombinteractionin ′
usuallydone,wewillrefertothecorrespondingsubspace
the large-N approximation[20, 31–34], showing that the
as A¯B¯ in order to avoid confusions. The 8 components
described orbital-valley-spin structure still holds, quan-
of the field operator then correspond to the total space
titatively (but not qualitatively)changingthe dispersion
KK A¯B¯ s, s being the spin space.
relation of the modes. Finally, we relate the results pre- ′⊗ ⊗
ThecorrespondingeffectiveHamiltonianofthesystem
sented in this work with experimental scenarios, includ-
isHˆ =Hˆ +Hˆ +Hˆ . Afterneglectingtrigonalwarping
ing a computation of the mean-field transport gaps and 0 C sr
effects and other smallcorrections [6, 22, 25, 30, 37], the
a discussion on the collective modes detection.
single-particle Hamiltonian Hˆ reads:
0
We remark that the collective modes calculated in Hˆ = d2x ψˆ (x)[H +ǫ T ǫ σ ]ψˆ(x), (2)
the present work are neutral and, therefore, topologi- 0 ˆ † B V zz − Z z
cally trivial [35, 36]. The charged, topologically non-
trivialexcitations(skyrmions)havebeenstudiedinboth whereTij =τiKK′⊗τjA¯B¯⊗1ˆs andi,j =0,x,y,z withτi,
monolayer [7] and bilayer [11] graphene, and the effect i=x,y,z,theusualPaulimatricesandτ =1ˆwhileσ is
0 z
of short-rangeinteractions [26] and screening [20] on the the correspondingPauli matrix in the spin space. In the
non-linearsigma-modelstiffnesscoefficientshasbeenad- following,the Paulimatricesinvalleyorsublatticespace
dressedforMLG.Hereweprovidethecorrectionsdue to are denoted using the letter τ and the Pauli matrices in
shortrangeinteractionstothestiffnesscoefficientsinthe spin space are denoted using the letter σ.
bilayer case as well. The first term between square brackets in Eq. (2),
H , is a 2 2 matrix acting in the A¯B¯ subspace and
B
×
corresponds to the kinetic energy:
The article is arranged as follows: we first introduce
the effective projected Hamiltonian considered in this
0 a2
work in Sec. II. We re-derive in Sec. III the phase dia- H =~ω B (3)
gram of Ref. [21] using a Hartree-Fock (HF) mean-field B B(cid:20) (a†B)2 0 (cid:21)
scheme and compute the corresponding mean-field ener- ωB = eB⊥ =1.76 1011meB [T] Hz
gies and transport gaps. The dispersion relation of the m × m ⊥
differentcollectivemodes,computedwithintheTDHFA, =6.28 1012B [T] Hz ,
is shown in Sec. IV. We translate the same calculations × ⊥
to monolayer graphene in Sec. V. Effects of LL mixing m being the effective mass for which we take the exper-
arestudiedinSec. VI.Adiscussiononexperimentalfea- imental value m = 0.028me [38], with me the electron
tures is presented in Sec. VII. Finally, the conclusions mass and aB the magnetic annihilation operator
are drawn in Sec. VIII. Technical details about the di-
agonalization of the HF equations and the TDHFA are lB πy +iπx ~ 25.7
a = , l = = nm, (4)
given in Appendices A-C. B ~ √2 B eB B [T]
r
⊥ ⊥
p
3
with π = i~∂ +eA , i = 1,2, the components of the With respect to the interacting part of the Hamilto-
i i i
momentum−operator after the Peierls substitution. The nian, Hˆ representsthe longrangeCoulombinteraction:
C
magnitude l is referred as the magnetic length.
B 1
The second term, ǫVTzz, arises from a voltage differ- HˆC = 2ˆ d2xd2x′ :[ψˆ†(x)ψˆ(x)]V0(x−x′)[ψˆ†(x′)ψˆ(x′)]:
ence between the two layers, ǫ = Ea /2, with E the
V z (8)
perpendicularelectricfieldanda 0.35nmthesepara-
z ≈ Here, : denotes normal ordering of the field operators
tion between the layers, while the third term takes into
andV (x)=e2/κx istheCoulombpotential,withe2
account the Zeeman effect, ǫZ = µBB,B = B2+B2. e2/4π0ǫ andκcthe|d|ielectricconstantoftheenvironmecn≡t.
0
k ⊥
Here,weassumethatthemagneticfieldisnotqnecessarily Finally,fortheshort-rangeinteractionHamiltonian,Hˆ ,
sr
in the z direction, i.e., it can present a parallel compo- weconsiderthe mostgeneralexpressioncompatible with
nent to the graphene plane. The polarizations σ = all the symmetries of the problem [20]
±
correspond to the spin components that are antiparallel
1
(parallel) to the total magnetic field B, respectively. Hˆ = ′ g d2x[ψˆ (x)T ψˆ(x)]2 : (9)
Taking into account that only the perpendicular mag- sr 2 ijˆ † ij
i,j
X
netic field affects the orbital motion and using the Lan-
dau gauge, we can write the potential vector as A(x) = The ′ in this sum denotes that we exclude the symmet-
[0,B x,0]. In this particular gauge, the eigenstates of ric term i = j = 0, already accounted by the long-range
H ⊥are characterized by the following quantum num- Coulomb interaction. These interactions are asymmet-
B
bers: the magnetic index n, which is an integer number ric in the valley and sublattice spaces. The origin of
and characterizes the energy of the corresponding Lan- these short-range interactions are the Coulomb interac-
dau level, ǫ ; the momentum in the y-direction, k, and tion between sublattice/valley spaces and the electron-
n
the polarization in the valley-spin (KK s) space, α. phononinteractions,which we alsotreatas short-ranged
′
Specifically, they are given by Ψ0 (x)⊗= Ψ0 (x)χ , [20]. For shortness, we refer in the following to the long
n,k,α n,k α
where χ is an arbitrary 4-component spinor in valley- range Coulomb interactions as simply Coulomb interac-
α
spin space while the orbital wave function with compo- tions. Except for the kinetic energy term, this Hamilto-
nents in the space A¯B¯ is nian is formally similar to that of monolayer graphene,
see Sec. V.
eiky 1 sgn n φ (x+kl2) The two-band model is expected to work quite well in
Ψ0n,k(x)= Ly √2(cid:20) φ|n||(nx|−+2 klB2) B (cid:21) aciawlliydeforratnhgeeZoLfLm[a3g7n].etFicorfileolwdser1mTag.neBti⊥cfi.eld3s0tTri,gospnea-l
ǫn =spgn n n(n 1)~ωB (5) warping effects become important and for larger mag-
| | | |−
neticfieldsonehastousethecompletefour-bandmodel,
p
for n =0,1 and as the overlap of the wave function of the n = 1 level
| |6
withtheignoredsublatticesisnotnegligible. Othersmall
Ψ0 (x)= eiky 0 , ǫ =0 (6) correctionstothetwo-bandHamiltonianhereconsidered
n,k Ly (cid:20)φ|n|(x+klB2) (cid:21) n breakthedegeneracybetweenthesingle-particleenergies
of the n=0 and n=1 levels [24] but they are negligible
p
forthedegeneratelevels n =0,1withzeroenergy(note comparedtotheanalogLambshift,discussedinthenext
that n = 1 are indeed|th|e same state). Hereafter, we section.
±
refer to this manifold of states as the zero Landau level
(ZLL). In the previous equations, L is the length of the
y
systemintheydirectionandφ (x)istheusualharmonic B. Projection onto the zero Landau Level
n
oscillator wave function [see Eq. (A2)]. We see that the
kinetic energy is degenerate in y-momentum and valley- We now address the study of the ν = 0 QH state,
spin polarization; in particular, the degeneracy in k for which corresponds to a half-filling of the ZLL and com-
each magnetic level n is N = S/2πl2, S being the to- plete filling of all LLs with n 2. For that purpose,
B B ≤ −
tal area of the system. Interestingly, the wave functions we make an estimation of the order of magnitude of the
in the ZLL only have non-vanishing components in the differenttermsintheHamiltonianandwecomparethem
subspace KK B¯ s. withthetypicalenergydifferencebetweenLLs,~ω . For
′ B
⊗ ⊗
Usingthepreviouseigenfunctions,thefieldoperatoris the Zeeman term, we find
decomposed as
ǫ B
Z
=0.014 1 (10)
~ω B ≪
B
ψˆ(x)= ∞ ′ Ψ0 (x)cˆ (7) ⊥
n,k,α n,k,α for realistic values of the ratio B/B while for the
n=X−∞Xk,α Coulomb interaction one has ⊥
where ′ means that n takes every integer value except FC = 13.58 , F e2c (11)
n= 1. ~ωB κ B [T] C ≡ κlB
− ⊥
p
4
Usual values for the perpendicular magnetic field are As aresultofthe aboveanalysis,the only termthatis
B & 1 T and the highest available continuous mag- notsmallcomparedtotheseparationbetweenLLsisthat
ne⊥tic field in the laboratory is B 80 T [39], which relatedwithCoulombinteractions. Webeginbytreating
meansthatthestrengthoftheCou⊥lo≃mbinteractionveri- Coulomb interactions as weak, F ~ω ; for instance,
C B
fies F &~ω whenthe environmentis vacuum (κ=1). by supposing a typical value κ = 5≪and a magnetic field
C B
The interlayer voltage can in principle vary in a wide B 20 T, F /~ω 0.5, which can be regarded as
C B
∼ ∼
range of values [24, 37] but we keep its value sufficiently sufficiently small to treat it perturbatively. In Sec. VI,
small here, ǫ ~ω . Finally, a dimensional analysis we address the usual situation F & ~ω and explain
V B C B
≪
of the short-rangeterms gives an estimation for the cou- how to deal with it.
pling constants g e2d/κ with d 0.1 nm the typical
ij ∼ c ∼ Takingintoaccountthepreviousconsiderationsandin
length scale of the lattice. Then, the energy scale asso-
order to study the lowest energy excitations, we neglect
e2d
ciated to the short-range interactions is c , which is LLmixingandrestrictourselvestotheZLLbyprojecting
small compared to the LL separation as ∼ κl2B the full Hamiltonian into that subspace, as previously
done in Refs. [20, 21, 24, 29]. We remark [see Eq. (6)
e2d F d 0.1 andensuingdiscussion]thatthestatesintheZLLbelong
κlB2c~ωB ∼ ~ωCB lB ∼ κ ≪1 (12) to the KK′ ⊗B¯ ⊗s space, which means that they are
localized, for each valley, in one sublattice or the other
Wenotethattheenergyassociatedtotheshort-rangein- and correspondingly, in one layer or the other. Thus,
teractionsscaleslinearlywiththemagneticfieldalthough withintheZLL,thesublatticedegreeoffreedombecomes
forlowvaluesofthemagneticfieldthecouplingconstants equivalentto the valley degreeoffreedom. The resulting
themselvescanberenormalized[20,21],seealsoSec. VI. effective Hamiltonian for the ZLL is:
1
Hˆ(0) = d2x ψˆ (x)[ ǫ τ ǫ σ ]ψˆ(x)+ d2x d2x :[ψˆ (x)ψˆ(x)]V (x x)[ψˆ (x)ψˆ(x)]:
ˆ † − V z − Z z 2ˆ ′ † 0 − ′ † ′ ′
1
+ d2x g :[ψˆ (x)τ ψˆ(x)]2 :+ d2x d2xψˆ (x)V (x,x)ψˆ(x) (13)
2ˆ i † i ˆ ′ † DS ′ ′
i
X
where τi ≡τiKK′ and gi =gi0+giz,i=1,2,3. As we re- III. HARTREE-FOCK EQUATIONS AND
strict to the ZLL, the kinetic energy term is suppressed. MEAN-FIELD PHASE DIAGRAM
We have neglected the symmetric short-range interac-
tion, arising from the coupling g , due to its smallness
0z
compared to the symmetric Coulomb interaction [20].
Using symmetry considerations, it can be proven that
g = g g [20, 33], so there are only two indepen-
x y
dentcoup≡ling⊥constantsg ,g . The potential V (x,x)
z DS ′
representsthemean-field⊥interactionoftheZLLwiththe
(inert) Dirac sea compound by all the occupied states Inordertoobtainthe mean-fieldphasediagramofthe
with n 2 [19, 25, 40]. As shown in Appendix A2, ν =0QHstateatzerotemperature,weuse the Hartree-
≤ −
this potential is diagonalwithin the ZLL: its explicit ex- Fock (HF) approximation for the self-consistent single-
pression is given by Eq. (A39). The projection onto the particle wave functions, which we denote as Ψ . The
n,k,α
ZLL leads to a field operator of the form corresponding HF equations for the Hamiltonian (13)
read(we refer the reader to Appendix A for allthe tech-
ψˆ(x)= Ψ0 (x)cˆ (14) nical details)
n,k,α n,k,α
n=0,1k,α
X X
ǫn,αΨn,k,α(x)=ˆ d2x′ VDS(x,x′)Ψn,k,α(x′)− νm,βˆ d2x′ V0(x−x′)Ψm,p,β(x)Ψ†m,p,β(x′)Ψn,k,α(x′) (15)
m,p,β
X
+ νm,βgi [Ψ†m,p,β(x)τiΨm,p,β(x)]τiΨn,k,α(x)−τiΨm,p,β(x)Ψ†m,p,β(x)τiΨn,k,α(x)
Xi mX,p,β (cid:16) (cid:17)
ǫ τ Ψ (x) ǫ σ Ψ (x)
V z n,k,α Z z n,k,α
− −
where the indices n,m = 0,1 label the magnetic levels, sent the polarization in the valley-spin space. We have
k,p are the momenta in the y-direction and α,β repre-
5
made explicit that we are dealing with an integer QH magnetic level is through the Coulomb and Lamb-shift
state, so every orbital p is filled in the same way and terms, while the other contributions to the energy only
then, the occupation number ν of each state solely depend on the spinor χ . Note that, although the inter-
n,p,α α
depends on n and α, ν =ν . In particular, for the action with the Dirac sea favors the filling of the mag-
n,p,α n,α
ν = 0 QH state, only half of the ZLL is filled. Thus, netic level n = 1, it is still more energetically favorable
for each value of the y-momentum k, only four states of to occupy in the same way the levels n=0,1 due to the
the eightfold space, formed by the 0,1 magnetic states exchange interaction. Indeed, the mean-field state (16)
and the valley-spin degrees of freedom, are occupied. As is an exact eigenvalue of the effective Hamiltonian (13)
usual, the direct (Hartree) term for the Coulomb inter- whenshort-rangeinteractionsareneglected. AsCoulomb
action is suppressed by the positive charge background. interactions dominate over short-range interactions, this
Animportantresultisthattheorbitalpartofthe self- mean-fieldsolutionisexpectedtoprovideaverygoodap-
consistent HF wave functions are equal to that of the proximation to the actual ground state [7, 20], showing
non-interactingwavefunctions,giveninEq. (6); seeAp- the robustness of the formalism here considered.
pendix A for the proof. Hence, the only remaining task As a resultof the previousdiscussion,the HF energies
is to specify the spinors χ . In order to minimize the can be written as:
α
dominant Coulomb interaction, the electrons occupy in
F F
the same way the valley-spin space for the two magnetic ǫ = n +ǫ , ǫ = n +ǫ (20)
levels, i.e., ν = ν = ν = ν = 1, with χ two n,(a,b) − 2 (a,b) n,(c,d) 2 (c,d)
0,a 0,b 1,a 1,b a,b
orthogonalspinors[11,21]sothemean-fieldgroundstate
withǫ dependingonlyonthepolarizationα. Theenergy
α
is of the form
of the total state (per wave vector state) is:
|Ψ0i= cˆ†0,p,acˆ†1,p,acˆ†0,p,bcˆ†1,p,b|DSi, (16) (F0+F1)
p EHF = +2E(P)
Y − 2
1
with DS the Dirac sea formed by all occupied Landau E(P)= u [tr(Pτ )]2 tr(Pτ Pτ )
| i i i i i
levels with n 2. 2 −
The four re≤m−aining unoccupied states of the ZLL are Xi (cid:8) (cid:9)
ǫ tr(Pτ ) ǫ tr(Pσ ) (21)
V z Z z
characterized by the spinors χ . In this way, the occu- − −
c,d
pation number only depends on the valley-spin polariza- The contribution from Coulomb interaction to the to-
tion α, νn,α = να, taking the values α = a,b,c,d that tal energy turns to be degenerate and does not depend
correspond to an orthonormal basis of the valley-spin on the specific form of the occupied spinors χ . The
a,b
space. These spinors are computed after projecting the actualground state of the system is determined by com-
HFequationsintothe orbitalpartofthe wavefunctions, paring the energies corresponding to all possible solu-
obtainingclosedalgebraicequationsinvalley-spinspace: tions to the HF equations and selecting that with low-
est energy E(P), in the same fashion of Refs. [20, 21].
F
n
ǫn,αχα = χα FnPχα ǫVτzχα ǫZσzχα Hence, the corresponding mean-field phase diagram for
2 − − −
the ν = 0 QH state is the same of those references and
+ ui([tr(Pτi)]τi τiPτi)χα (17) is represented in Fig. 1. The different possible phases
−
i areferromagnetic(F), cantedanti-ferromagnetic(CAF),
X
where u =g /πl2 and F =F +F , with: fully layer-polarized (FLP) and partially layer-polarized
i i B n n0 n1
(PLP).Theexpectedphaseforthe ν =0QHstateofbi-
π 1 3 layergrapheneforǫ =0andzeroin-planecomponentof
V
F = F , F =F = F , F = F (18)
00 2 C 01 10 2 00 11 4 00 themagneticfieldisprobablytheCAFphase[21,22,41],
r
which implies that u > u > 0. The exact values of
z
so F00 >F01 >F11 and then F0 >F1 as these short-rangeenergies−rem⊥ain unknown but their or-
3 5 der of magnitude is uz, u 0.1~ωB [23, 42]. In the
F0 = F00, F1 = F00 (19) following, we treat them|a⊥s|ph∼enomenological inputs for
2 4
the theory. All phase boundaries intersect at the critical
The values of the factors F are obtained by inserting
nm point
the Coulomb potential in Eq. (A24).
shiTfth,eatreisrimngFfnr/o2minthEeqi.nt(e1r7a)citsiotnheofanthaleogZLofLthweitLhatmhbe V =(ǫ∗Z,ǫ∗V)=(−2u⊥,uz−u⊥). (22)
Dirac sea [19, 40]. The term FnP arises from the The complete phase diagram can be explored experi-
−
exchange Coulomb interaction, where the matrix P is mentally by manipulatingthe in-plane componentofthe
the projector onto the subspace formed by χa,b, P = magnetic field or the layer voltage. This fact can be
χaχ†a +χbχ†b and hence Pχa,b = χa,b, Pχc,d = 0. The checked by looking at right Fig. 1, where we represent
remaining terms are those related with the short-range the phase diagramas afunction ofthe Zeemanandvolt-
interactionsandthevalley-spinpartofthesingle-particle ageenergies,(ǫ ,ǫ ),forfixedvaluesofu ,u suchthat
Z V z
Hamiltonian. We see that the sole dependence on the u > u >0. ⊥
z
− ⊥
6
We now briefly describe all the phases and give the
expressions for the filled and empty valley-spin spinors,
χ = n s , χ = n s
the matrix P and the corresponding valley-spin energies a | zi⊗| ai b |− zi⊗| bi
ǫα for each phase. χc =|nzi⊗|−sai, χd =|−nzi⊗|−sbi
ǫ = u 2u cos2θ ǫ cosθ ǫ = u ǫ
a,b z S Z S V z V
− − ⊥ − ∓ − ∓
ǫ = 2u sin2θ +ǫ cosθ ǫ = 2u ǫ
c,d S Z S V V
A. Ferromagnetic phase − ⊥ ∓ − ⊥∓
I +cosθ σ +sinθ (s σ)τ
S z S z
P = k·
2
In the F phase, all the electrons have the spin aligned
E(P)= u ǫ cosθ (26)
with the magnetic field. The complete set of solutions − z− Z S
involvesthefollowing4spinorsinvalley-spinspace,with where s are spin states with polarization given by
a,b
| i
eigenvalues: the vectors s =[ sinθ cosφ , sinθ sinφ ,cosθ ],
a,b S S S S S
± ±
with tilting angle cosθ = ǫ /2u , and s =
S Z
χa = nz sz , χb = nz sz (23) [cosφ ,sinφ ,0]. We choose the−phases⊥of these sktates
| i⊗| i |− i⊗| i S S
χc = nz sz , χd = nz sz in such a way that sb =σz sa and sb = σz sa
| i⊗|− i |− i⊗|− i | i | i |− i − |− i
ǫ = 2u u ǫ ǫ , ǫ =ǫ ǫ so the CAF phase continuously matches the F phase of
a,b z Z V c,d Z V
− ⊥− − ∓ ∓ Eq. (23) for θ = 0. As the azimut φ is a free param-
I +σ S S
z
P = , E(P)= (2u +u ) 2ǫ eter, the CAF phase exhibits a U(1) symmetry. When
z Z
2 − ⊥ −
this solutions exists, it always has lower energy than the
Here, s denotes the state with spin polarizationσ = regular F phase, so the condition for the presence of the
z
, I t|h±e 4i 4 identity matrix and n = K , n = CAF phase is just
z z
± × | i | i |− i
K . The F phase is always a solution of the HF equa-
| ′i cosθS <1 ǫZ <ǫZc 2u (27)
tionsalthoughitdoesnotalwayscorrespondtotheactual ⇒ ≡− ⊥
ground-state. with ǫ the critical Zeeman field.
Zc
D. Partially layer-polarized phase
B. Full layer-polarized phase
In analogy with the relation between the FLP and F
The FLP phase is the equivalentofthe Fphase but in
phases,thePLPissimilartotheCAFphasebutinvalley
valley pseudospin, KK , which means that all the elec-
′
space:
trons are concentrated on one layer (due to the equiva-
lency of valley-sublattice-layer of the ZLL in BLG). The
corresponding spinors are now: χ = n s , χ = n s (28)
a z b z
| i⊗| i | i⊗|− i
χ = n s , χ = n s
χ = n s , χ = n s (24) c z d z
a z z b z z |− i⊗| i |− i⊗|− i
| i⊗| i | i⊗|− i ǫ =u sin2θ +u cos2θ ǫ ǫ cosθ =u ǫ
χ = n s , χ = n s a,b V z V Z V V Z
c |− zi⊗| zi d |− zi⊗|− zi ⊥ ∓ − ⊥∓
ǫa,b =uz∓ǫZ −ǫV, ǫc,d =−2u⊥−2uz∓ǫZ +ǫV ǫc,d =−2u⊥−uz−u⊥sin2θV −uzcos2θV
P = I+2τz, E(P)=uz−2ǫV +ǫIV+consθτV ∓ǫZ =−3u⊥−uz∓ǫZ
P = ·
2
In analogy to the F phase, it is always a solution to the
E(P)=u sin2θ +u cos2θ 2ǫ cosθ
HF equations but not necessarily the ground-state. By ⊥ V z V − V V
comparing the energies of the two phases, we obtain the =u ǫV cosθV
⊥−
boundary between the F and FLP phases:
with n a state with valley-polarization given by n =
| i
[sinθ cosφ ,sinθ sinφ ,cosθ ], cosθ = ǫ /(u
V V V V V V V z
−
u ). The PLP phase also presents an U(1) symmetry.
ǫV −ǫZ =u⊥+uz (25) In⊥analogy with the F-CAF phase transition, whenever
this solution exists, it has lower energy than the FLP
phase so the existence condition for the PLP phase is
C. Canted anti-ferromagnetic phase
cosθ <1 ǫ <ǫ u +u (29)
V V Vc z
⇒ ≡ ⊥
Thepreviousphaseswouldbetheonlypossiblephases with ǫ the critical voltage energy. On the other hand,
Vc
iftherewerenotshort-rangeinteractions,i.e.,u =uz = the boundary between the PLP and the CAF phases is
0. However,whentakingintoaccounttheseinte⊥ractions, placed at
the system can exhibit canted anti-ferromagnetism or
ǫ2 ǫ2
partially layer-polarization in order to minimize the in- V + Z =u +u (30)
z
teraction energy. In the CAF phase, we have that: uz u 2u ⊥
− ⊥ ⊥
7
CAF
F FLP
V
PLP
V
z V
u
ǫ
PLP
FLP F
CAF
u ǫ
⊥ Z
FIG. 1. Phase diagram of the ν = 0 QH state. Left plot: phase diagram in the parameter space u⊥,uz. The point V is the
intersection ofallphaseboundaries. Rightplot: expectedphasediagram forfixedu⊥,uz,withuz >−u⊥ >0,asafunctionof
theZeeman and voltage energies, ǫZ,ǫV.
E. Transport gap wave function is created by the action of the operator
Mˆn†αn′α′(k) ≡ Mˆn†λσn′λ′σ′(k) on the mean-field ground
Anexperimentalmagnitudeofinterestthatcanbeob- state Ψ0 , with
| i
tained within the present mean-field computation is the
1
ttwraenesnpothrteglaopwe∆stHeFn,edrgeyfineemdpatsytshteateenearngdytdhiffeelraesntcefilbleed- Mˆn†λσn′λ′σ′(k)=rNB q e−iqkxl2Bcˆ†n,q+k2y,λσcˆn′,q−k2y,λ′σ′
state. From the above results, and within the conven- X (32)
tion chosen here for the occupied and empty levels, it is
where we have made explicit the dependence in valley
immediate to show that in all phases λ=K,K andspinσ = indicesofthetotalvalley-spin
′
±
polarization index α. As well known from the general
∆ =ǫ ǫ =F +∆bc, ∆bc ǫ ǫ , (31)
HF 1,c 1,b 1 c b theory of integer QH effect, the collective modes are ex-
− ≡ −
pressed in terms of linear combinations of magnetoexci-
in good agreement with Refs. [15, 24], where the
tons. The resulting dispersion relation is isotropic and
only asymmetric interactions considered are interlayer
only depends on k = k, as shown in Appendix C.
Coulomb interactions. | |
In order to classify these modes, we study their sym-
metries in the whole 8-dimensional space 01 KK s,
′
⊗ ⊗
formedby the tensorialproductofthe magneticlevelsof
IV. COLLECTIVE MODES theZLLandthevalley-spinspace. Forthatpurpose,fol-
lowingthe notationof Ref. [24], we define the operators:
A. Preliminary remarks
1
Sˆi = 2 cˆ†n,p,λ,σ(σi)σσ′cˆn,p,λ,σ′ (33)
We proceed to compute the neutral collective modes n,p,λσ,σ
X X′
ofthepreviousmean-fieldphasesusingtheTDHFA.The 1
general formalism of the TDHFA and its application to Lˆi = 2 cˆ†n,p,λ,σ(τi)λ,λ′cˆn,p,λ′,σ (34)
integer QH states is explained thoroughly in Appendix nX,p,σλX,λ′
BthewrheisleulAtspppreensdeinxteCdpinrotvhidisesseacltliothne. technical details on Oˆi = 12 cˆ†n,p,λ,σ(µi)nn′cˆn′,p,λ,σ (35)
p,λ,σn,n
Due to the particular form of the ν = 0 QH state, X X′
in which the unoccupied levels have different valley-spin which correspond to the components of the spin, valley
polarization with respect to the occupied levels, the ex- and orbital pseudospin, respectively. In the above ex-
citations correspond to valley-spin waves. Within the pression, µ are the corresponding Pauli matrices in the
i
TDHFA, the collective modes are obtained from the magnetic index, with µ = 1 for n=1,0.
z
eigenvalues and eigenvectors of the matrix X˜(k), given We consider the behavior±of the magnetoexcitons un-
by Eq. (C12). The wave vector k corresponds to the der transformations generated by the previous opera-
momentum of the so-called magnetoexciton [43], whose tors. Forinstance,forthespinoperator,thecommutator
8
[Sˆi,Mˆn†λσnλσ (k)] reads oftheCoulombinteractionstrength. Ontheotherhand,
′ ′ ′
the orbital triplet modes (O =1) depend on short-range
[Sˆi,Mˆn†λσn′λ′σ′(k)]= (Gi)ξξ′,σσ′Mˆn†λσn′λ′σ′(k) interactions only through the valley-spin contributions
ξξ of mean-field energies, but not through many-body cor-
X′
1 rections. Since Coulomb interactions are dominant, this
(G ) = [(σ ) δ δ (σ ) ] ,
i ξξ′,σσ′ 2 i ξσ ξ′σ′ − ξσ i σ′ξ′ structureimpliesthatthetripletmodesareshiftedabove
(36) by relatively large Coulomb energies and hence, the or-
bitalsingletmodesarethelowestenergymodesatk =0.
It is easy to prove that the matrices Gi form a represen- Specifically, the hierarchy ωµ < ωµ = ωµ < ωµ is
00 11 1 1 10
tation with spin 1/2 1/2 = 1 0 of the Lie algebra of satisfied. It is worth noting that the triplet−modes with
⊗ ⊕
SU(2). Thus, spin singlet and triplet magnetoexcitons O = 1 are degenerate due to the analog Lamb shift.
z
can be constructed according to the value of the total For±k >0,however,theorbitalsingletandtripletmag-
spin S and its z-component Sz, Mˆn†λn′λ′,SSz(k). Specifi- netoexcitons are mixed and the previous classification is
cally, the spin triplet magnetoexcitons are given by no longer valid. Then, in order to classify the orbital
structureofthedifferentmagnetoexcitonsforfixedµ,we
Mˆn†λnλ11(k)=Mˆn†λ+nλ (k) (37) introduce the discrete index N = 0,1,2,3 and denote
′ ′ ′ ′−
Mˆn†λn′λ′10(k)= √12 Mˆn†λ+n′λ′+(k)−Mˆn†λ−n′λ′−(k) ttihveelcyomrraetscphonthdeincgollferecqtiuveenmcyodaesswωNiµth(ko)rbsiotatlhpesyeurdeospspeicn-
Mˆn†λn′λ′11(k)=Mˆn†λh−n′λ′+(k) i OanOdzs=oo0n0.,1T1h,e1n−ot1a,t1i0onatiskch=os0e;nhienncseu,chω0µa(wka=y0th)a=t,ωfo0µ0r
while the spin singlet magnetoexciton reads fixed µ, ωµ(k)<ωµ (k) for N <N .
N N ′
′
As the collective modes correspond to valley-spin
1
Mˆn†λn′λ′00(k)= √2 Mˆn†λ+n′λ′+(k)+Mˆn†λ−n′λ′−(k) wωµav(eks), thωeµb(e0h)a+vioρrµokf2t,hweitmhoρdµestfhoergloewnermaloimzeedntvuamlleyis-
h (3i8) N ≃ N N N
spin stiffness [43]. However, this behavior is modified as
cDauneatlosotbhueiflodrsmimalilaanraslionggyletwaitnhdtthreipoleptemraatgonrsetLˆoie,xOˆciit,ownes ωsp0µo(nkt)a≃nevoGusklyfobrrtohkeenGsoyldmsmtoenterimeso.desofthe phaseswith
in valley and orbital pseudospin. At large momentum, kl & 1, only Coulomb inter-
B
The importance of the previous considerations arises actions are relevant since C(k) decays as k 1 while
−
fmroumtesthweitfhactthethoaptertahteoresffSˆec2t,iLˆve2,HSˆza,mLˆizlt,owniitahn (13) com- Rth(ek)mdatoreiscesitthasat∼aries−e(dkluB2e)2,thewhmeraenyC-b(ok∼d),y Rco(nkt)ribaure-
Sˆ2 = SˆSˆ, Lˆ2 = Lˆ Lˆ (39) tionsfromCoulombandshort-rangeinteractions,respec-
i i i i
tively (check Appendix C for their precise definition).
i=x,y,z i=x,y,z
X X For kl 1, the collective modes frequency approach
B
≫
soS,L,S ,L areexpectedtobegoodquantumnumbers asymptoticallythemean-fieldparticle-holeexcitationen-
z z
that characterize the magnetoexcitons whenever |Ψ0i is ergy ~ω ≃ǫn,α−ǫn′,α′.
also an eigenstate of any of these operators. Thus, the The above considerations imply that the modes N =
collective modes can be labeled by a general index µ, 1,2,3 depend very weakly on short-range interactions
which representsa set of conservedquantum numbers in since in both limits kl 1, kl 1 they solely de-
B B
≪ ≫
valley-spin space. Specifically, the F and FLP states has pend on them through the valley-spin part of the mean-
well defined quantum numbers for S,L,S ,L which, as field contributions, which only provide a constant shift,
z z
discussed in the paragraph following Eq. (B22) and in andforkl 1Coulombcontributionsarethedominant.
B
∼
Appendix C,implies thatthey cannotexhibitdynamical Indeed,itisshowninAppendix C2thatthestiffness co-
instabilities, at least within the projected model consid- efficients of the N = 1 modes depends very weakly on
eredinthis work. Onthe otherhand,the CAFandPLP u ,u while those of the N = 3 modes are completely
z
states do not satisfy this property and, as shown in the in⊥dependentofthem. Moreover,duetotheanalogLamb
next section, they are indeed able to display dynamical shift, the N = 2 modes can be computed explicitly as
instabilities in some regions of the parameter space. they correspond to a symmetric combination of the or-
Atk =0,apartfromthesevalley-spinsymmetries,the bital OO = 11,1 1 modes as given by Eqs. (C22),
z
−
z component of the orbital pseudospin, O , is also con- (C23), which show that their dependence on k only in-
z
served by the Hamiltonian as it represents the angular volves Coulomb interactions. As a result, we expect the
momentum of the magnetoexciton [8, 29, 30]. Indeed, dispersionrelationoftheN =1,2,3modestobequitein-
the total orbital pseudospin O is also a good quantum dependentoftheirvalley-spinstructureandthenrecover
number in this limit. Then, we can provide a complete essentially the same results of Refs. [24, 29], where only
classification of the modes at k = 0 using their total (intra- and inter-layer) Coulomb interactions are taken
orbital-valley-spin symmetry, denoting their frequencies into account.
as ωµ . As shownin Appendix C,the energyof the or- The situation is quite different for the N = 0 modes,
OOz
bital singlet modes OO =00 is completely independent since at k = 0 they correspond to the orbital singlet
z
9
modes which are independent of the Coulomb strength Regarding full-flip excitations, they are characterized
and the only modes that experience many-body cor- by the triplet modes L = 1. Indeed, their dispersion
z
±
rections coming from short-range interactions. Hence, relation is the same, solely shifted by the layer voltage,
their behavior at low momentum greatly depends on ωN1−1(k) = ωN11(k) + 4ǫV, so we only need to compute
the valley-spin structure of the modes. Besides, as they ω11(k)tocharacterizethem. Inparticular,fortheorbital
N
present a positive stiffness (see next section), ωµ(k) in- singlet mode,
0
creases monotonically with k. Hence, the orbital singlet
modesatk=0area)thelowestenergyexcitationsofthe ~ω0101 =2(uz+u⊥+ǫZ −ǫV) (41)
systemandb)themostsensiblemodestothevalley-spin
whichisproportionaltothedifferencebetweenthemean-
structureofinteractions. We thereforeconcludethatthe
fieldenergiesoftheFLPandFstates. Hence,inanalogy
orbitalsinglet modes are the natural candidates to char-
to the caseof spin-flipexcitations, ω11 =0 is the bound-
acterize the phase transitions. We note that the orbital 00
ary between the two phases.
singlet modes are denoted as the “even” modes in Ref.
The existence of such a gapless mode can be under-
[29].
stood from the appearance of a mean-field symmetry
rightattheboundarybetweentheFLPandtheFphases.
We define a mean-field symmetry as that whichis not of
B. Collective modes: analysis of the results
the complete Hamiltonianbut only ofthe mean-fieldHF
solutions [see discussion after Eq. (B32) for more de-
In this section, we compute the dispersion relation for
tails]. Specifically, when condition (25) is satisfied, the
the collective modes of the different ground states ob-
HF equations present a continuously degenerate mean-
tained in Sec. III, see Eqs. (23),(24),(26),(28). For each
field ground state described by the parameters t,φ and
phase, we discuss the symmetries of the excitations and
characterizedin valley-spin as
of the ground state, give the explicit expression for the
frequencies of the orbital singlet modes ω0µ0 in order to χa =|nzi⊗|szi (42)
study the stability of the corresponding phase and the χ =cost n s +eiφsint n s
b z z z z
valuesoftheassociatedstiffness, ρµ0,andplotthedisper- χ = e i|φsiin⊗t |n− i s +co|−st in⊗| is
sionrelationofthedifferentcollectivemodes,ωµ(k). We c − − | zi⊗|− zi |− zi⊗| zi
N
χ = n s
refer the reader to Appendix C2 for the specific details d z z
|− i⊗|− i
on the calculation of the collective modes. P(t,φ)= 1 I +σ cos2t+τ sin2t
z z
2
+ sin(cid:0)2t cosφ(Πx+Πy)+sinφ(Πx Πy)
x y y − x
1. Ferromagnetic phase
with energy E[P((cid:2)t,φ)] = E(P )cos2t+E(P )s(cid:3)i(cid:1)n2t,
F FLP
P being the projectors of the F, FLP phases, re-
Theferromagneticstatehasvalley-pseudospinL,L = F,FLP
z
spectively [see Eqs. (21),(23) and (24)], and the matri-
0 and spin S,S = 2N , with 4N the total number
z B B ces Πi defined as Πi 1τ σ . Thus, right at the
of electrons in the ZLL. All magnetoexcitons carry spin j j ≡ 2 i ⊗ j
phase boundary, the total mean-field energy of the state
S =1,S = 1 so we use the valley pseudospin in order
z
tocharacteri−zethecollectivemodes,ωµ=LLz(k). Theex- doesnotdependonthe parameterst,φandacontinuous
N mean-field symmetry arises from this fact. In particu-
citations of the ferromagnetic state consists of spin-flip
lar, the parameter t describes the phase transition, with
excitations and full-flip excitations. The spin-flip excita-
t=0correspondingtotheFstatewhiletheFLPstateis
tions involve transitions between two different spin po-
obtained for t = π. The associated total state, denoted
larizations, keeping the same valley polarization while 2
as Ψ(t,φ) , can be connected to the F state F by a
in full-flip excitations both valley and spin indices are | i | i
continuous unitary transformation of the form:
flipped at the same time. Specifically, the spin-flip exci-
tationsarecharacterizedbythemodeswithL=1,0and Ψ(t,φ) =eGˆ(t,φ) F
L = 0. The energy of the orbital singlet associated to | i | i
z
these modes, ~ω0L00, is: Gˆ(t,φ)= teiφcˆ†n,p,ccˆn,p,b−h.c. (43)
n,p
X
~ωL0 =2ǫ +2u ( 1)L2u (40)
00 Z ⊥− − ⊥ and, in terms of the projectors P, P(t,φ) =
The spin singlet mode ~ω00 = 2ǫ > 0 is the Larmor eG(t,φ)PFe−G(t,φ), with G(t,φ) the matrix version of the
00 Z operator Gˆ(t,φ). Note that the azimut φ simply arises
mode [27], while the spin triplet mode characterizes the
F-CAFphasetransition,~ω10 =2ǫ +4u =2(ǫ ǫ ); fromaddingatrivialrotationinvalleyand/orspinspace
precisely,ω10 =0isthe bou0n0dary[ZseeEq⊥. (27)]Zbe−twZecen to Gˆ(t,0). The generator of Eq. (43) can be further
00
thetwophases. Theappearanceofasuchagaplessmode rewritten as
correspondstothespontaneouslybrokenU(1)symmetry
Gˆ(t,φ)= it cosφ Πˆx Πˆy sinφ Πˆx+Πˆy
ofthe CAFphase. Inthe regionwherethe CAFphaseis − y − x − x y
present, the system is energetically unstable as ω00 <0. h (cid:16) (cid:17) (cid:16) (cid:17)(i44)
00
10
F F FLP FLP
1.2 1.2 1.2 1.2
1 1 1 1
B0.8 B0.8 B0.8 B0.8
ω ω ω ω
/ / / /
ω0.6 ω0.6 ω0.6 ω0.6
0.4 0.4 0.4 0.4
L=0Lz=0
0.2 L=1Lz=1 0.2 L=1Lz=0 0.2 S=1Sz=1 0.2 S=0Sz=0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
klB klB klB klB
CAF CAF PLP PLP
1.2 1.2 1.2 1.2
1 1 1 1
B0.8 B0.8 B0.8 B0.8
ω ω ω ω
/ / / /
ω0.6 ω0.6 ω0.6 ω0.6
0.4 0.4 0.4 0.4
Lz=0,+
0.2 L=1Lz=1 0.2 Lz=0,− 0.2 S=1Sz=1 0.2 S=0Sz=0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
klB klB klB klB
FIG. 2. Dispersion relation of the collective modes for the different phases of the ν = 0 QH state in bilayer graphene. We
take the values FC = 0.5~ωB, uz = 0.1~ωB and u⊥ = −0.05~ωB for the Coulomb and short-range energies, respectively. We
remarkthat,whenseveralbranchesarerepresentedinthesameplot,themodeswithN =1,2,3aresoclosetoeachotherthat
is difficult to distinguish them. Upper left corner: dispersion relations of the F phase with ǫV = 0.02~ωB and ǫZ = 0.2~ωB,
with the left plot devoted to the valley triplet mode ωN11(k) and the right plot devoted to the modes with Lz = 0, ωNL0(k),
L = 0,1 being depicted in solid (dashed) line. Upper right corner: dispersion relations of the FLP phase with ǫV = 0.2~ωB
and ǫZ =0.02~ωB. The left plot displays thespin triplet mode ωN11(k) and right plot the spin singlet mode ωN00(k). Lower left
corner: dispersionrelationsoftheCAFphasewithǫV =0.02~ωB andǫZ =0.02~ωB. Leftplotcorrespondstothevalleytriplet
mode ωN11(k) and right plot to the modes with Lz = 0, the +(−) branches being depicted in solid (dashed) line. Lower right
corner: dispersion relations of the PLP phase with ǫV =0.1~ωB and ǫZ = 0.02~ωB. Left plot corresponds to the spin triplet
mode ω11(k) and right plot thespin singlet mode ω00(k).
TheoperatorsΠˆx,Πˆy,alongwithSˆ,Lˆ ,formarepresen- (dashed) line. As the corresponding mean-field energies
i i i z
tation of SO(5) that is an exact symmetry of the total are degenerate and the modes N = 1,2,3 depend very
Hamiltonianforu +u =0andǫ =ǫ =0[26]. Thus, weaklyontheshort-rangeinteractions,thecurvesofboth
z Z V
theexoticSO(5)s⊥ymmetryexistingattheF-FLPbound- branchesareveryclosetoeachother;indeed,thisdegen-
ary for zero Zeeman and layer voltage terms survives as eracy becomes exact for the N = 2 modes as explained
a weaker mean-field version at the same phase bound- in the previous section. The characteristicnegative stiff-
ary in the realistic scenario u +u = 0. Remarkably, nessoftheN =1modescanalsobeobservedintheplots
z
this gapless mode behaves as ω11(k⊥)6 k2 for low mo- [see Eq. (C31)]. Only the frequencies ω00(k), ω10(k) are
0 ∼ 0 0
mentum in contrast to the linear dependence found for clearly distinguished due to the many-body corrections
the Goldstone modes of the phases with spontaneously arising from short-range interactions at low momentum.
broken symmetries.
The stiffness for the orbital singlet modes is given, for
dominant Coulomb interactions FC u ,uz [see Eq. When comparing the plots of spin-flip and full-flip ex-
(C30) for the exact expression], by ≫ ⊥ citations, we see that the curves for the N = 1,2,3 are
extremely similar. Again, this can be explained by in-
89 25
ρµ0 ≃ 224F00− 49uµ lB2 (45) voking the dominant character of Coulomb interactions
(cid:18) (cid:19) and the weak dependence on short-range effects. As ex-
with uµ a short-range coupling that depends on the pected, the main differences are observed once more for
valley-spin symmetry of the mode. Specifically, u00 = the lowest energy modes N = 0. Variations of ǫ ,ǫ
Z V
u 2u , u10 = u +2u and u11 =u1 1 =u . only provide a linear energy offset that depends on the
z z − z
−We−plo⊥t in the −upper left⊥corner of Fig. 2 the dis- valley-spin structure of each mode, given in Eqs. (40),
persion relation of the full-flip (left panel) and spin-flip (41). Finally, we remark that variations of the different
(right panel) excitations. In the case of spin-flip excita- parameters only change quantitatively but not qualita-
tions, the L = 0, (L = 1) branches are plotted in solid tive the plots.
