Quantum Hall ferroelectrics and nematics in multivalley systems

1.2 MB·

by  Inti Sodemann

QuantumHallferroelectricsandnematicsinmultivalleysystems
Inti Sodemann,1 Zheng Zhu,1 and Liang Fu1
1DepartmentofPhysics,MassachusettsInstituteofTechnology,Cambridge,Massachusetts02139,USA
(Dated:January30,2017)
WestudybrokensymmetrystatesatintegerLandaulevelfillingsinmultivalleyquantumHallsystemswhose
lowenergydispersionsareanisotropic. WhentheFermisurfaceofindividualpocketslackstwofoldrotational
symmetry,likeinBismuth(111)surfaces[1],interactionstendtodrivetheformationofquantumHallferroelec-
tricstates. WedemonstratethatthedipolemomentinthesestateshasanintimaterelationtotheFermisurface
geometryoftheparentmetal. InquantumHallnematicstates, likethosearisinginAlAsquantumwells, we
demonstratetheexistenceofunusuallyrobustskyrmionquasiparticles.
7
1 Introduction—Quantum Hall liquids with nearly degener- anisotropy, the long-range Coulomb interaction generically
0 ateinternaldegreesoffreedomhavelongbeenthesourceofa favors states with full valley polarization. We compute the
2
rich variety of phenomena. Aside from multilayer systems, electric polarization of quantum Hall ferroelectrics using a

n there exist today an array of multi-valley two-dimensional quantummechanicalapproachbasedonBerryphaseandalso
a electron systems exhibiting quantum Hall effect, including usingasemiclassicalapproachthatdirectlyrelatesthedipole
J
AlAsheterostructures[2],monolayerandbilayergraphene[4, momenttotheunderlyingFermisurfacegeometry. Forquan-

6 5], Si surfaces [6], PbTe (111) quantum wells [7], surface tumHallnematicstates,westudytheenergeticsofSkyrmion-
2
statesofthetopologicalcrystallineinsulatorSn Pb Se[8], typechargedexcitations[13]usingthedensity-matrixrenor-
1−x x
and more recently the (111) surface of Bismuth [1]. These malizationgroupmethod,andfindtheyaresuprisinglyrobust
]
l systemsallhavemultiplevalleyscenteredatdifferentregions againstmassanisotropyofthevalleys,sheddingnewinsights
e
- in momentum space, which are related to each other by the ontheexperimentsonAlAsquantumwells[14].
tr rotationalsymmetryofthecrystal. Generalsetup—Westartfromsymmetryconsiderationson
s
Among these multivalley systems, AlAs heterostructures, two-dimensional(2D)multivalleysystems. Weconsidersys-
.
t
a Si surfaces, PbTe (111) quantum wells and Bi(111) surfaces temswiththree, four, orsixfoldrotationalsymmetry, which
m all have highly anisotropic pockets, whose orientations are rulesoutanyprincipalaxiswithinthe2Dplane. Inorderfor
valley-dependent. The shape of Fermi surface, which is de- eachvalleytohaveananisotropicdispersion,werequirethat
-
d termined by the electronic band structure of the host mate- the symmetry group that leaves each valley invariant, or the
n rial, is crucial in determining the pattern of valley symmetry “littlegroup”,isonlyasubgroupofC ,orequivalently,con-
2v
o
breaking. For AlAs, and PbTe, the pockets are centered at tains no more than a twofold rotation (x,y) → (−x,−y) and
c
time-reversal-invariant momenta, and therefore each pocket at most two mirror planes that are orthogonal to each other,
[
hasanellipticalshapewithtwo-foldsymmetry. Thistwofold (x,y)→(−x,y)and(x,y)→(x,−y).

1 Fermi surface anisotropy favors a valley-polarized quantum There are two types of systems with multiple anisotropic
v
Hall state where a subset of valley degenerate Landau levels valleys: type-I are those whose little group contains at most
6
3 are fully occupied [2, 10]. This state spontaneously breaks a single mirror plane; type-II are those with larger symme-
8 thelargercrystalrotationalsymmetryintoatwofoldoneand try. For systems of the first type, each valley is of such low
7 isthusanematicquantumHallstate. symmetrythattheelectrondispersionatzerofieldhasnoin-
0 However, the recently studied Bi(111) surface and versioncenter, i.e., (cid:15)(k) (cid:44) (cid:15)(−k)wherekisthe“small”mo-
.
1 Sn1−xPbxSe (001) surface, brings in a novel ingredient into mentum within the valley. As we will show, in the quantum
0 this problem. As shown in (Fig. 1), Bi (111) surface has six Hall regime and at odd-integer fillings, Coulomb interaction
7 tadpole-shaped hole pockets [11]; Sn Pb Se (001) surface induces a valley-polarized state which breaks all rotational
1−x x
1
hasfourcrescent-shapedpockets[12]. Thesepocketscomein symmetry of the crystal and is therefore a ferroelectric. For
:
v time-reversed pairs located at opposite momenta away from systems of the second type, the electron dispersions at each
i time-reversal-invariant points in the Brillouin zone. Impor- valley have twofold symmetry (cid:15)(k) = (cid:15)(−k). In this case,
X
tantly, Fermi surface of each pocket does not have a twofold valley-polarized quantum Hall states exhibit nematic instead
r
rotational symmetry. Therefore, we expect Landau orbitals offerroelectricorders. TheBi(111)surface[1]isarepresen-
a
associated with each valley generically to carry an in-plane tativeoftype-IsystemsandtheAlAs(001)quantumwell[2]
electric dipole moment. This implies that a valley-polarized arepresentativeoftype-II.
state can possess an electric polarization, rendering the re- Tostudyinteractingelectronsinmulti-valleyquantumHall
sulting quantum Hall state a ferroelectric. The giant fermi systemsweassumethatthemagneticfieldislargeenoughso
surface anisotropy of Bi(111) and Sn1−xPbxSe (001) surface that the Hamiltonian can be projected into a set of M nearly
statesmakesthempromisingcandidatesystemstostudythis degenerateLandaulevelslevelsassociatedwithdifferentval-
novelferroelectricityinquantumHallstates(Fig.1). leys. The system is at some partial integer filling ν ∈ Z
In this letter we provide a unified description of these fer- (ν ≤ M). The interaction between an electron with position
roelectric and nematic states in multi-valley integer quan- r from valley i and an electron with position r from valley
1 2
tum Hall systems. We establish that due to Fermi surface j is dominated by the long-range part of the Coulomb inter-
2
singlevalley. Theproofisprovidedinthesupplementaryma-
a) b) E
terial[34].Tocheckthatformfactorsarelinearlyindependent
onesimplyneedstoshowthatdet(X )(cid:44)0.
ij
Form factors are generically linearly independent for val-
k
y leyswithanisotropicFermisurfacesthataredistinct. Weuse
the word distinct here in a strong sense. For example, two
�¯ elliptical Fermi surfaces that are rotated with respect to each
k other, as in AlAs, are distinct. Similarly, all the Fermi sur-
x
k facesattheBi(111)surfacearedistinct(Fig.1).Therefore,we
k y
x
expect generically that the ground states of type-I and type-
II systems is a fully valley polarized ferroelectric or nematic
stateatν=1.
FIG.1:(coloronline)MeasuredARPESfermisurfacesofa)Bi(111)
Itispossible,however,thattwoFermisurfacesfordistinct
surfacefromRef11andb)Sn Pb Se(001)surfacefromRef12.
1−x x valleys are, in a small momentum approximation, identical
(as in the case of graphene within the Dirac approximation),
action, which depends only on their distance |r − r |, i.e., hence the form factors are linearly dependent. These system
v(r1 −r2) = 1/A(cid:80)qvqeiq·(r1−r2) [35]. Notethatv1isin2depen- in the presence of long-range Coulomb interaction have an
emergent SU(2) symmetry. Also, an important condition for
dent of the relative orientation of the two electrons, because
theabovetheoremonmulticomponentquantumHallstatesis
systems considered here all have isotropic dielectric proper-
that all electrons reside in the same layer so that the interac-
ties. DespitethebareCoulombinteractionbeingvalleyinde-
tionsbeforeprojectingtotheLandaulevelareidentical. Oth-
pendent,itbecomesvalleydependentafterprojectionintothe
erwise,non-trivialHartreetermsmayfavorsuperpositionsof
Landaulevels:
components in different layers, as it happens for the exciton
condensateinquantumHallbilayers[32].
P0v(r1−r2)P0 = A1 (cid:88)vqFi(q)Fj(−q)eiq·(R1−R2), (1) spaOcuerDHMarRtrGee[-1F6o–c2k0a]nsaimlyusilsatiisocnosnofinrtmheedtobryusoguerommoemtreyn.tTuhmis-
q
isimplementedbymappingthesingleparticleorbitalsinthe
whereR≡r−l2zˆ×pistheintra-Landaulevelguidingcenter Landau gauge into a one dimensional chain. Details can be
operator, p = ∇/i − eA is the mechanical momentum, and foundinSupplementaryMaterial [34].
F(q)≡(cid:104)i,n|e−il2zˆ·q×p|i,n(cid:105)istheformfactordeterminedbythe Quantum Hall ferroelectrics—In this section we consider
i
wavefunction,|i,n(cid:105),oftheLandaulevelofinterestassociated multi-valley systems of the type-I, where anisotropic valleys
withvalleyi((cid:126)≡1,l2 ≡1/eB).Theseformfactorsarecrucial are located at non-time-reversal-invariant momenta and each
fortheenergeticsofvalleysymmetrybreakingquantumHall valleylackstwofoldsymmetry. Thistypeofmulti-valleysys-
statesandaredescribedindetailfortype-Iandtype-IImulti- temsisrealizedonthe(111)surfaceofbismuth[11],andon
valleysystemsintheSupplementarymaterial[34]. the (001) surface of topological crystalline insulators SnTe,
We study the Hamiltonian (S10) using both Hartree- SnxPb1−xSe and SnxPb1−xTe [21]. In both materials, surface
Fock approximation and DMRG numerical method. Within stateLandaulevelshavebeenobservedbyscanningtunneling
Hartree-Fock,weconsidertranslationallyinvarianttrialSlater microscopy[1,8].
determinantstatesthatarearbitrarycoherentsuperpositionsof Type-I systems exhibit a fully valley polarized state at to-
the M valleys. Theirenergy(includingtheneutralizingback- tal filling ν = 1 [34] that fully breaks rotational symmetry.
ground)canbefoundtobe: Because the quantum Hall state is insulating in the bulk, we
expect a spontaneous electric polarization to develop, whose
magnitude can be computed by adapting the modern theory
(cid:88)
E[P]=−N X (cid:104)i|P|j(cid:105)(cid:104)j|P|i(cid:105), ofpolarizationbasedonBerryphase[5]. Wefindthedipole
φ ij
i,j momentperelectronalongtheydirectionisgivenby:
(cid:88)ν 1(cid:90) d2q (2)
P= |χ (cid:105)(cid:104)χ |, X = v F(q)F∗(q).
a=1 a a ij 2 (2π)2 q i j D =−i|e|l2 (cid:90) Lx/l2dk (cid:104)u |∂ |u (cid:105). (3)
y L y ky ky ky
where i labels the valleys and |χ (cid:105) are ν orthonormal vectors x 0
a
describingtheoccupiedcoherentcombinationsofthevalleys. where|u (cid:105)aresingleparticleorbitalsinagaugethatistrans-
ky
TrialstatesarefoundbyminimizingE[P]asafunctionofP. lationally invariant along y for the Landau level of inter-
Atν = 1thestatethatminimizestheHartree-Fockenergy est[34]. Ananalogousexpressionholdsforthedipolealong
is maximally polarized into a single valley under very gen- the x axis. Alternatively, the dipole can also be obtained by
eral conditions. In fact we prove the following theorem: if computing the displacement of the average position of each
thesetofformfactorsF(q)withi=1,...,νarelinearlyinde- occupied single particle state. Both calculations agree, as
i
pendent functions of q and the interaction is strictly positive demonstratedin[34].
v >0,theminimumofE[P]atν=1forasystemwithmul- We now introduce a semiclassical approach to establish a
q
tipleanisotropicvalleysisastatemaximallypolarizedintoa direct connection of the dipole moment with the underlying
3
Fermi surface geometry at zero field. Consider a valley de- a) b)
scribedbyasingle-bandHamiltonianH(p)ofarbitraryform.
BychoosingtheLandaugauge A = 0and A = Bx,wecan
x y
view the eigenvalue problem as effectively one-dimensional.
(cid:112) B
Writing the wavefunction as ψky(x,y) = uky(x)eikyy/ Ly, we /l
have: y
(cid:32)(cid:126) d (cid:33)
H ,k −eBx u (x)=(cid:15) u (x). (4)
i dx y ky n ky
x/l x/l
Wesetk = 0(othersolutionsareobtainedbyaglobaltrans- B B
y
lation). The WKB method leads to an approximate wave-
functionoftheform:
FIG. 2: (color online) Probability amplitude contours for coherent
statesinthefirstLandaulevelofaDiracconewithanisotropicve-
u0(x)≈(cid:88) (cid:113) 1 exp(cid:32)(cid:126)i (cid:90) xdx(cid:48)pcxl,s(x(cid:48),(cid:15)n)(cid:33), (5) Tlohceitiwesavvexf/uvnyc≈tio2n.2ca(rar)i,esanaddwipiothleamniosmoteronptypearnpdentidlitcδuvlaxr≈to0t.h3e(tbi)lt.
s vcxl,s(x,(cid:15)n) oftheDiraccone.
where pcl (x(cid:48),(cid:15) )isthe srootoftheclassicalenergyrelation:
x,s n cones[7,22]. Weconsiderthen = 1Landaulevel[36]. The
H(pcl ,−eBx)=(cid:15) ,andvcl (x,(cid:15) )=∂H(p ,−eBx)/∂p . Typ-
x,s n x,s n x x wavefunctionsassociatedwitheachvalleybreakrotationand
ically we have two roots s = +,− and two turning points
inversionsymmetryasillustratedinFig.2. TheHartree-Fock
x+,− that separate the classically allowed region from the energy is (up to a global constant) E/A = −βn2, where n =
classically forbidden region. Matching boundary conditions z z
tr(σ P) and σ is a Pauli matrix in the valley indices, and β
z z
between classically allowed and forbidden regions leads to
ispositive[34], inagreementwithourtheoremoffullvalley
the Onsager quantization condition: l2A((cid:15)n) = 2π(n+1/2), polarization. Therefore,thegroundstateisofIsingtype(n =
z
where A((cid:15) ) is the area in momentum space inside the iso-
n ±1)withcoexistingferroelectricandnematicproperties. The
energetic contour H(k) = (cid:15) and n is the Landau level in-
n dipolemoment,toleadingorderintheDiracconetilt,canbe
dex. From Eq. (S46) the dipole can be written as D =
(cid:82) y foundtobe:
−el2 dxIm(u∗(x)∂ u (x)). Inthesemiclassicallimitoflarge
0 x 0
nthisexpressionreducestoasimpleform[34]andonefinds (cid:32) (cid:33)
δv 1
thatdipolemomentisgivenbytimeaveragedpositionofthe D ≈∓|el|sign(B)√ x 1− √ , D =0. (8)
electronoverthesemiclassicalorbit: y vxvy 2 2 x
As expected the dipole of a given valley is orthogonal to the
(cid:72)
dtr(t,(cid:15) ) directionoftheDiracconetilt,andreverseswiththedirection
n
D≈−|e|(cid:104)r(t,(cid:15)n)(cid:105)t ≡−|e| (cid:72) , (6) of the perpendicular field. This is allowed because the mag-
dt
neticfieldbreaksthemirrorsymmetrypresentatzerofield.
We now discuss potential experimental manifestations of
where the integral is perfomed over the semiclassical cy-
clotron orbit r(t) = l2k(t) × zˆ, with k(t) tracing a constant the quantum Hall ferroelectric states on the Bi(111) sur-
energycontour H(k(t)) = (cid:15) ataspeedv = |∂(cid:15)/∂k|[6]. This face[1],whereweexpectferroelectricstatestoappearatodd
n
integer fillings. One potential signature is the appearance of
picture predicts intuitively and generically that the dipole is
additional ferroelectric domain walls in addition to those as-
orthogonaltothedirectionofthedistortionoftheFermisur-
sociated with the nematicity seen at even fillings. Another
facesincetherealspaceorbitisarotatedversionoftheFermi
interesting possibility is the existence of gapless conducting
surface[34].
edge states at the ferroelectric domain walls. If one neglects
TheabovesemiclassicalapproachisapplicabletohighLan-
inter-valleyscattering,thevalleypseudospinisconservedand
daulevelsn (cid:29) 1suchasthoseobservedinBi(111)surface.
oneexpectsgaplesschargecarryingcounter-propagatingedge
Tostudyferroelectricityinn∼1Landaulevels,weconsidera
modelwithtwovalleys(M = 2)ofDiracfermionslocatedat modes that arise at the domain walls [37]. Due to their fer-
oppositemomenta,atfillingν = 1. Thismodeldescribesthe roelectricnature,thesedomainsmaybemanipulatedbySTM
biasvoltageorin-planeexternalelectricfield.Finally,wenote
low-energy dispersion of the [001] surface of SnTe and Pb-
thatBi(111)holepocketslocatedatoppositemomentacarry
SnTetopologicalcrystallineinsulatorsinthelowtemperature
opposite in-plane spin-polarizations, which cancel when the
phase[22]. Here,theDiracconesaregenerically“tilted”[7]
two are equally occupied. Therefore, valley-polarized quan-
anddescribedbythefollowingHamiltonian:
tum Hall states at odd-integer filling also carry an in-plane
H =±v σ p +v σ p ±δv p , (7) spinpolarization,andthuscanbemanipulatedbyanin-plane
0 x x x y y x x x
magneticfield.
The only discrete symmetry that leaves each valley invariant Valley skyrmions in quantum Hall nematics— In this sec-
isasinglemirrorplanealongthelinethatconnectstheDirac tion we consider multi-valley systems of type-II that de-
4
5 1 .0 wenowturntostudyitschargedexcitations. IntheSU(2)in-
S k y r m io n Q u a s ip a r tic le variantlimit,i.e. whenthemasstensorsarethesameforboth
valleys, we expect the lowest energy excitation to be infinite
4
sizedskyrmions[13]. Whenthemasstensorsareslightlydif-
ferent,weexpectthattheskyrmionswillhaveafinitesizedic-
0 .9
3 tated by the competition between the Ising anisotropy which
e
rg wantstoshrinkthemandtheCoulombenergywhichwantsto
a
fl h expand them to smear the charge over large distances [38].
c
N2 D To be able to study accurately the properties of skyrmions
0 .8
we resort to DMRG [39]. The skyrmion quasi-electron can
1 be obtained as the ground state of the Hamiltonian when
N = N +1[40]. InFig.3weshowthatnon-trivialskyrmions
φ
(those involving at least one spin flip) survive up to large
0 0 .7
2 4 6 8 mass ratio mx/my ≈ 3.8 for the case of two valleys rotated
m / m by π/2 (as in AlAs). AlAs has a mass anisotropy of about
x y m /m ≈ 5 [2]. The experiment of Ref. [14] found a non-
x y
lineardependenceofthechargegaponstrain,whichwasin-
terpretedasevidenceforskyrmions.Ourfindingssuggestthat
FIG.3:(coloronline)Numberofelectronsintheminorityvalley(left
thisisscenarioisnotunlikelysincethecriticalmassratioto
verticalaxis)foraquasiparticlesasafunctionmassanisotropyration
inatwovalleysystem(AlAs)andthechargegap(rightverticalaxis) observeskyrmionsisultimatelydependentonthedetailsofin-
computedwithDMRG. teractionsandcaneasilychangebyeffectsbeyondourmodel
(e.g. finitewellwidthsandLandaulevelmixing).
Insummary,wehaveshownthatmultivalleysystemswith
velop nematicity but which carry no dipole moment. We fo- anisotropicdispersionsleadgenericallytoferroelectricorne-
cus on the simplest non-trivial case with two valleys whose maticquantumHallstatesatoddintegerLandaulevelfillings.
anisotropic mass tensors are rotated by a π/2 angle relative TheferroelectricstatesarisewhentheparentFermisurfaceof
to each other. This is the scenario realized in AlAs sys- a single valley lacks an inversion center, such as the case of
tems [2, 2]. Other experimentally relevant scenarios along Bi(111)[1]. Wehaveshownthattheresultingdipolemoment
withusefulformulaefortheirformfactorsarediscussedinthe hasanintimaterelationwiththeunderlyingfermisurfacege-
supplementary material [34]. Our main objective is to study ometry of the parent metal. We also demonstrated the exis-
non-trivialchargeexcitationsonthesesystems,butwebriefly tence of non-trivial skyrmion-type charged excitations in the
reviewgroundstateproperties. Atfillingν = 1onefindsthat nematicstateswithanunexpectedlylargestabilitytotheIsing
the energy is (up to a global constant) E/A = −αn2, where symmetrybreakingterms,sheddinglightintothequestionof
z
n = tr(σ P), and α > 0 [34]. The system has Ising charac- thepresenceoftheseexcitationsinAlAs[14].
z z
terinagreementwithourtheoremoffullvalleypolarization. We would like to thank Vidya Madhavan, Benjamin Feld-
The Ising nematic state is an exact eigenstate of the Hamil- man,MalikaRanderia,AliYazdani,DonnaSheng,HeunMo
tonianprojectedintothevalleydegenerateLandaulevel. We Yoo, Joonho Jang and Ray Ashoori for stimulating discus-
confirmedthatitisthegroundstateviaDMRGapproachem- sions. I.S.issupportedbythePappalardoFellowship. Z.Z.is
ploying a torus with a square aspect ratio (1 : 1) [34] for as supportedbytheDavidandLucilePackardfoundation.L.F.is
many as 24 electrons and for mass anisotropies as large as supportedbytheDOEOfficeofBasicEnergySciences,Divi-
m /m =16[34]. sionofMaterialsSciencesandEngineeringunderAwardNo.
x y
Havingestablishedthatthegroundstateisanematicstate, DE-SC0010526.
[1] B. E. Feldman, M. T. Randeria, A. Gyenis, F. Wu, H. Ji, R. [6] K.Eng,R.N.McFarland,andB.E.Kane,Phys.Rev.Lett.99,
J. Cava, A. H. MacDonald, and A. Yazdani, Science 354, 316 016801(2007).
(2016). [7] V.A.Chitta,W.Desrat,D.K.Maude,B.A.Piot,N.F.Oliveira
[2] M.Shayegan,E.P.DePoortere,O.Gunawan,Y.P.Shkolnikov, Jr., P. H. O. Rappl, A. Y. Ueta, E. Abramof, Physica E: Low-
E.Tutuc,K.Vakili,Phys.Stat.Sol.b243,3629(2006). dimensionalSystemsandNanostructures34,124(2006).
[3] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. [8] Yoshinori Okada, Maksym Serbyn, Hsin Lin, Daniel Walkup,
Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, Wenwen Zhou, Chetan Dhital, Madhab Neupane, Suyang Xu,
Nature438,197(2005). Yung Jui Wang, R. Sankar, Fangcheng Chou, Arun Bansil, M.
[4] Y.Zhang,Y-W.Tan,H.L.Stormer,andP.Kim,Nature438,201 Zahid Hasan, Stephen D. Wilson, Liang Fu, Vidya Madhavan,
(2005). Science341,6153(2013)
[5] K. S. Novoselov, E. McCann, S. V. Morozov, V. I. Fal’ko, M. [9] D. A. Abanin, S. A. Parameswaran, S. A. Kivelson, and S.
I.Katsnelson,U.Zeitler,D.Jiang,F.Schedin,andA.K.Geim, L. Sondhi, Phys. Rev. B 82, 035428 (2010); A. Kumar, S. A.
NaturePhysics2,177(2006). Parameswaran,andS.L.SondhiPhys.Rev.B88,045133(2013).
5
[10] X. Li, F. Zhang, A. H. MacDonald, Phys. Rev. Lett. 116, ofthistypeofestimatesforsmallskyrmions[3].
026803(2016). [40] Quasi-holeandquasi-electronexcitationsarerelatedbyparticle
[11] Y.Ohtsubo, J.Mauchain, J.Faure, E.Papalazarou, M.Marsi, holesymmetry.
P.LeFvre,F.Bertran,A.Taleb-Ibrahimi,andL.Perfetti,Phys.
Rev.Lett.109,226404(2012).
[12] P. Dziawa, B. J. Kowalski, K. Dybko, R. Buczko, A. Szczer-
bakow,M.Szot,E.Lusakowska,T.Balasubramanian,B.M.Wo-
jek,M.H.Berntsen,O.TjernbergandT.Story,NatureMaterials
11,1023(2012).
[13] S.L.Sondhi, A.Karlhede, S.A.Kivelson, andE.H.Rezayi,
Phys.Rev.B47,16419(1993).
[14] Y.P.Shkolnikov,S.Misra,N.C.Bishop,E.P.DePoortere,and
M.Shayegan,Phys.Rev.Lett.95,066809(2005).
[15] K.Moon,H.Mori,K.Yang,S.M.Girvin,A.H.MacDonald,
L.Zheng,D.Yoshioka,andS-C.Zhang,Phys.Rev.B51,5138
(1995).
[16] N.ShibataandD.Yoshioka,Phys.Rev.Lett.86,5755(2001).
[17] A.E.Feiguin, E.Rezayi, C.Nayak, andS.DasSarma, Phys.
Rev.Lett.100,166803(2008).
[18] T.Xiang,Phys.Rev.B53,R10445(1996).
[19] S.R.White,Phys.Rev.Lett.69,2863(1992).
[20] J.Zhao,D.N.Sheng,andF.D.M.Haldane,Phys.Rev.B83,
195135(2011).
[21] Junwei Liu, Wenhui Duan, and Liang Fu, Phys. Rev. B 88,
241303(2013).
[22] M.SerbynandL.Fu,Phys.Rev.B90,035402(2014).
[23] I.SodemannandL.Fu,Phys.Rev.Lett.115,216806(2015).
[24] D.J.Thouless,Phys.Rev.B27,6083(1983).
[25] R.D.King-SmithandD.Vanderbilt,Phys.Rev.B47,1651(R)
(1993).
[26] L. A. Falkovskii, J. Exptl. Theoret. Phys. (U.S.S.R.) 49, 609
(1965); A. Yu. Ozerin and L. A. Falkovsky, Phys. Rev. B 85,
205143(2012).
[27] C. Kittel, Introduction to solid state, John Wiley & Sons,
(1966).
[28] L.Onsager,Phil.Mag.43,1006(1952).
[29] D-W. Wang, E. Demler, and S. Das Sarma, Phys. Rev. B 68,
165303(2003).
[30] Y.Barlas,R.Ct,K.Nomura,andA.H.MacDonald,Phys.Rev.
Lett.101,097601(2008).
[31] B. Yang, Z. Papic´, E. H. Rezayi, R. N. Bhatt, and F. D. M.
Haldane,Phys.Rev.B85,165318(2012).
[32] K.Moon,H.Mori,KunYang,S.M.Girvin,A.H.MacDonald,
L. Zheng, D. Yoshioka, and Shou-Cheng Zhang, Phys. Rev. B
51,5138(1995).
[33] M.Abolfath,J.J.Palacios,H.A.Fertig,S.M.Girvin,andA.
H.MacDonald,Phys.Rev.B56,6795(1997).
[34] Supplementarymaterial.
[35] Weneglectshortdistancevalleydependentcorrections.
[36] ThezerothLandauleveldoesnotcarrydipoleeveninthepres-
ence of the tilt because the Hamiltonian retains an intra-valley
particle-hole times inversion symmetry which forbids a dipole
moment for the zeroth level. However, all other Landau levels
willcarryadipole.
[37] WewishtonotethatothertypesofquantumHallferroelectrics
arisinginwidequantumwells[29]andinbilayergraphene[30]
havebeenpreviouslyproposed,buttoourknowledgethesepro-
posalshavenotbeenexperimentallyrealizedsofar.
[38] Thestiffnesscostisscaleinvariantandhenceindifferenttothe
sizeoftheskyrmion.
[39] The energetics can be captured by a non-linear sigma
model [13, 32], but using using long-wavelength estimates for
non-linearsigmamodelparametersleadstoasubstantialunder-
estimationthesizeoftheskrymions[34].Thisisaknownpitfall
6
QUANTUMHALLFERROELECTRICSAND with elliptic Fermi surfaces that are related by discrete ro-
NEMATICSINMULTIVALLEYSYSTEMS: tations. Sections VI and VII discuss the proof of the the-
SUPPLEMENTARYMATERIAL orem of full valley polarization unifying the specific exam-
ples that are discussed in the main text and in Sec. V. Sec-
tion VIII derives the Berry phase formula for the dipole us-
ingthemoderntheoryofpolarizationandalsoshowsthatthis
Contents formula is equivalent to the calculation based on the shift of
the position of the electron measured from the guiding cen-
I. Introduction 6 ter. Section IX describes the WKB calculation of Landau
level wavefunctions and their corresponding dipole moment
II. Landaulevelsanddensityformfactorsfor inthesemiclassicallimit. SectionXconsidersthesemiclassi-
anisotropicparabolicdispersions 6 calpictureofthedipoleforthecaseofthetiltedDiraccone.
Section XI discussed the details of the DMRG implementa-
III. Landaulevelsanddensityformfactorsfor tionandtheextrapolationofthegapstothermodynamiclimit
anisotropicDiraccones 7 along with the estimate of the skyrmion quasiparticle stabil-
ityasafunctionofmassanisotropy. Comparisonbetweenthe
IV. Landaulevelsanddensityformfactorsintilted
Hartree-Fock and DMRG charge gaps are discussed in Sec-
Diraccones 7
tion XII illustrating the agreement of both approaches in the
caseofconventionalquasiparticlesandtheirsignificantdevi-
V. Hartree-Focktheoryformultiplevalleys 8
ation for skyrmions. Section XIII discusses the failure of a
A. Twocomponentsatν=1 8
naive estimate for the stability of the skyrmions using bare
B. three-foldsymmetricthree-componentsatν=1 9
long-wavelength parameters and a non-linear sigma model.
C. four-foldsymmetricfour-componentsatν=1 9
AndfinallySec.XIVshowsasimplifiedwaytocomputethe
chargegapinthetwocomponentcaseofellipticpocketsus-
VI. Proofoffullvalleypolarizationfortwo
ing particle-hole symmetry and the knowledge of the energy
anisotropicvalleysatν=1withinHartree-Fock 9
ofthequasi-electron.
VII. Proofoffullvalleypolarizationformultiple
valleysatν=1withinHartree-Fock 10
II. LANDAULEVELSANDDENSITYFORMFACTORS
FORANISOTROPICPARABOLICDISPERSIONS
VIII. Berryphaseandpositionshiftformulaeofthe
electricDipolemomentinaLandaulevel 11
ConsidertheproblemofLandaulevelswithananisotropic
IX. WKBexpressionfortheDipolemoment 12 masstensor:
1
X. Semiclassicalpictureofthecyclotronmotionin H = p g p . (S1)
2m∗ a ab b
tiltedDiraccones 13
Wherep=∇/i−eA,andgisatensorthatcanbediagonalized
XI. DMRGimplementation 13 as g = QTS2Q, where Q ∈ SO(2) and S is diagonal with
positive eigenvalues and detS = 1. We can define rescaled
XII. ComparisonbetweenHartree-FockandDMRG momenta along the principal axes of the tensor to be: π =
a
quasiparticlegap 14
(SQ) p ,thesesatisfythesamecommutationrelationsasthe
ab b
originalones:
XIII. Isingskyrmionsfromnon-linearsigmamodel 15
XIV. Simplifiedcalculationofchargegapfrom [π ,π ]=il−2(cid:15) . (S2)
a b ab
particle-holesymmetry 16
Which allows to solve the LL problem by defining the LL
References 17 raisingoperators:
l
I. INTRODUCTION a≡ √ (π +iπ ), (S3)
x y
2
This supplementary material contains details of calcula-
satisfying[a,a†] = 1. Theguidingcenteroperatorsareintra-
tionspresentedinthemaintextbutalsoextensionsintoafew
Landauleveloperatorsdefinedas:
other cases of interest. In Sections II to IV we derive wave-
functions and form factors for a variety of anisotropic sys-
tems and for different Landau levels. Section V explains the R ≡r +l2(cid:15) π . (S4)
a a ab b
Hartree-FocktheoryandpresentsthesolutionsoftheHartree-
Fock minimization explicitly for the nematic case of valleys Theysatisfy:
7
wherenistheLandaulevelofinterestandL aretheLaguerre
n
polynomials.ThereforetheinteractionprojectedtothenLan-
[R ,R ]=−il2(cid:15) ,
a b ab daulevelis:
(S5)
[R ,p ]=[R ,π ]=0.
a b a b
v ≡ P v(r −r )P =
ThesingleparticleHilbertspacecanbedecomposedintoa 12 n 1 2 n
(cid:32) (cid:33) (cid:32) (cid:33)
toennswohripcrhodau,cat†|na(cid:105)c⊗t a|mnd(cid:105),|wmh(cid:105)earree|ni(cid:105)ntarrae-LthaendLaaundleavuellevinedliicnedsicoens A1 (cid:88)vqLn l2|q21|2 Ln l2|q22|2 e−l2|q1|2+4|q2|2eiq·(R1−R2). (S12)
whichR acts. NowtheprojectedinteractionintotheLandau q
a
levelofinterestisobtainedbyimaginingwehavetwoflavors
of particles with different mass tensors (g = QTS2Q and
1 1 1 1 III. LANDAULEVELSANDDENSITYFORMFACTORS
g = QTS2Q ) which interact via a potential that depends
2 2 2 2 FORANISOTROPICDIRACCONES
onlyoninterparticledistance:
ThederivationisverysimilartothatofGalileanelectrons.
1 (cid:88) WestartfromananisotropicDiracHamiltonian:
V(r1−r2)= A Vqeiq·(r1−r2). (S6)
q
H =σ g p =vσ(cid:48)π . (S13)
wherer andr aboveareunderstoodtobeoperators. Using a ab b a a
1 2
Eq.(S4)wedecomposethepositionofeachparticleas: wheregisa3×2tensortowhichweapplyasingularvalue
decompositionoftheformg = RTSQ,whereR ∈ SO(3)de-
scribes a rotation of Pauli matrices σ in pseudospin space,
r1 ≡R1−l2(cid:15)p1 =R1−l2(cid:15)QT1S1−1π1, (S7) Q ∈ SO(2)describesthetransformatioanoftheprincipalaxes
in real space, and S is a 3 × 2 matrix whose upper 2 × 2
where we are using matrix notation for the two component
block is diagonal and characterizes the anisotropy of veloci-
levi-civitasymbol(cid:15). Wehaveasimilarexpressionforr2. Us- tiesS =diag{(v /v )1/2,(v /v )1/2},andwhoselowerrowhas
ingthisweget: zero entries, andx vy= (v vy)1/x2. Using the same definition of
x y
ladderoperatorsasinEq.(S3),theHamiltonianandthespec-
eiq·(r1−r2) =eiq·(R1−R2)e−il2(q1·π1−q2·π2), trumare:
(S8)
q =−S−1Q(cid:15)q. √
i i i (cid:32) (cid:33) (cid:32) (cid:33)
2v 0 a† 1 s|n−1,k(cid:105)
H = , |n,k,s(cid:105) = √ , forn>0
In the above expression the terms containing the operator π1 l a 0 D 2 |n,k(cid:105)
produce inter-Landau level mixing and we proceed by pro- (cid:32) (cid:33) √
0 2nv
jecting them into the zeroth landau level which is defined as |0,k(cid:105) = , E = s
a|0(cid:105) = 0. By using Eq. (S3) in combination with the BCH D |0,k(cid:105) nks l
formulaonecanshowthat: (S14)
where s = ±labelsnegativeandpositiveenergyLandaulev-
eil2q·π =e−l2|4q|2e√il2qa†e√il2q∗a, (S9) elsandkistheguidingcenterintra-Landaulevelindex. From
theseexpressionswecanobtaintheformfactorsoftheinter-
whereq=q +iq .ThenoneobtainsthattheprojectedHamil- actionprojectedtotheDiracLandaulevels.Onecanshowthat
x y
thezeroLandaulevelisidenticaltotheGalileancase,andfor
tonianis:
the excited Landau levels one gets the same expressions dis-
cussedintheprevioussectionbutwiththemodification:
V12 ≡ P0V(r1−r2)P0 = A1 (cid:88)Vqe−l2|q1|2+4|q2|2eiq·(R1−R2). (S10) (cid:32) (cid:33) (cid:32) (cid:32) (cid:33) (cid:32) (cid:33)(cid:33)
q l2|q|2 1 l2|q|2 l2|q|2
L → L +L (S15)
n n n−1
2 2 2 2
Noticethatq arelinearfunctionsofqdescribedinEq.(S8).
1,2
For higher Landau levels we need to modify the density
form factors. Using Eq. (S9) and the algebra of raising and
IV. LANDAULEVELSANDDENSITYFORMFACTORSIN
loweringoperatorsonecanshowthefollowingidentities:
TILTEDDIRACCONES
(cid:104)n|eil2q·π|n(cid:105)= TotheHamiltonian, H, appearinginEqs.(S13),(S14)we
=e−l2|4q|2 (cid:88)n (cid:32)−l2|q|2(cid:33)m n(n−1)...(n−m+1), athdedxa-dpierertcutriobnat:iondescribingthetiltoftheDiracconealong
2 (m!)2 (S11)
m=0
(cid:32) (cid:33) (cid:114)
=e−l2|4q|2Ln l2|2q|2 , H1 =δvxπx =δvx vvyx(a+a†), (S16)
8
whereaisdefinedinEq.(S3).The(un-normalized)perturbed V. HARTREE-FOCKTHEORYFORMULTIPLEVALLEYS
zeroenergyLandauleveltofirstorderinδv isfoundtobe:
x
We assume the magnetic field is large enough so that the
(cid:32) (cid:33)
|0(cid:105) Hamiltoniancanbeprojectedintoasetof M nearlydegener-
|0(cid:105)1 = τ|0(cid:105) , (S17) ateLandaulevelslevelsassociatedwithdifferentvalleys. The
√ systemisatsomepartialintegerfillingν∈Z(ν≤ M). Were-
where τ = δv /( 2v ). This verifies that the zero Landau stricttotranslationallyinvarianttrialSlaterdeterminantstates
x y
level does not carry dipole as expected since particle-hole that are otherwise arbitrary coherent combinations of the M
times inversion is a symmetry. Generically the n-th Landau valleys:
level will break inversion symmetry however. The valence
andconductionfirstLandaulevelshaveaperturbedform:
|±1(cid:105) = √1 (cid:32)|1(cid:105)±τ(√12|2(cid:105)−|0(cid:105))(cid:33). (S18) |Ψ(cid:105)=(cid:89)a=ν1(cid:89)kN=φ1(cid:88)i=M1(cid:104)χa|i(cid:105)c†ki|O(cid:105), (S24)
1 2 ±|0(cid:105)+τ|1(cid:105)
where|O(cid:105)isthereferencevacuuminwhichtheLandaulevels
As described in the main text we imagine now two Dirac ofinterestareempty,kisanintraLandaulevellabel,ilabels
valleysatoppositemomentaandwithtiltsδv ofoppositesign the M valleys, |χ (cid:105)are N orthonormalvectorsdescribingthe
x a
and equal magnitude. Let us assume we are filling only one occupied coherent combinations of the valleys. The expec-
ofthetwofirstLandaulevels. Theformfactorobtainedfrom tation value of the energy (including the neutralizing back-
theaboveperturbativeexpressionis: ground)is:
Fτ(q)≡1(cid:104)1|eil2q·π(cid:48)|1(cid:105)1 = e−lZ2|q|2/4(cid:20)(1+τ2)(1− q42)+ E[P]=−Nφ(cid:88)Xij(cid:104)i|P|j(cid:105)(cid:104)j|P|i(cid:105),
τ (S19) i,j
τ42 f22(q)+ 2√τ2(f12(q)+ f21(q))(cid:21). P=(cid:88)ν |χa(cid:105)(cid:104)χa|, Xij = 21(cid:90) (d2π2q)2vqFi(q)Fj(−q). (S25)
a=1
where f (q) ≡ e−l2|q|2/4(cid:104)n|eil2q·π(cid:48)|m(cid:105), andZ isthenormaliza-
nm τ
ThesolutiontotheHartree-Fockproblemcorrespondstofind-
tion factor of the perturbed states in Eq. (S18). To leading
ing the minimum of E with respect to variations of the pro-
orderinτwecanwritetheformfactoras:
jector P. One case in which the integrals can be performed
analyticallyisthecaseinwhichtheinteractioncorrespondto
F (q)≈e−l2|q|2/4(1− q2)(1+ iτ√qx). (S20) delta-functions in real space, so that vq = g. In this case we
τ
4 2 have:
Onevalleywillhaveaformfactorcorrespondingtoτand
tshigenos.thWerecloabrreelstphoenmdinbgyαto=−{τ+s,i−n}c.eIfthoenetilptserhfoarvmesotphpeocsaitle- Xij = 2π(cid:113)Det(1+S QgQTS−2Q QTS ). (S26)
culationoftheexchangeenergywiththeseformfactorsthen i i j j j i i
onefindsthattheexchangeintegralsinEq.(S25):
Thebehavioroftheexchangeintegralsasafunctionofval-
leyanisotropyandrelativeorientationisdepictedinFigs.S1-
(cid:90)
1 d2q S10. ThebehaviorofhigherLandaulevelsandtheDiraccase
X = v F (q¯)F (q¯), q¯ =−S−1Q(cid:15)q. (S21)
αβ 2 (2π)2 q α β isqualitativelydifferentbutqualitativelysimilartothen = 0
LandaulevelasdepictedinFigs.S1-S10.
Theexchangeenergycanthenbeshowntobe:
(1−n2) A. Twocomponentsatν=1
E/N = z X +const. (S22)
φ τ
2
with: In this case the projector is P = |χ(cid:105)(cid:104)χ|, and the exchange
energyis:
Xτ =X+++X−−−2 X+−+X−+ E =−N (cid:88)X p p , p =|(cid:104)i|χ(cid:105)|2. (S27)
=τ2 (cid:90) d2q v e−l2|q¯|2/4(cid:32)1− l2|q¯|2(cid:33)2l2q¯2 (S23) φ ij ij i j i
2 (2π)2 q 4 x Theminimizationisperformedundertheconstraints(cid:80) p =
i i
Clearly for a repulsive interaction with v > 0 we get that 1, p ∈ [0,1]. We restrict to the case when the two mass
q i
X isstrictlypositive. ThereforethegroundstateistheIsing tensors have the same eigenvalues but different orientations.
τ
nematicferroelectricstate: n =±1. ThentheHartree-Fockenergybecomes:
z
9
GalileanDeltan=0(blue),n=1(yellow), mx/my=1/2
X12
E =−2(X −X )(p2−p )−X , (S28) X0
Nφ 11 12 1 1 11 1.0
In Figs. S1- S10 we verify explicitly that in all cases X ≥ 0.8
11
X ≥0.Thereforetheenergyhasminimaatp ={0,1}under
12 1
the constraints in question. Namely the ground state sponta- 0.6
neouslychoosesoneflavorandnocoherentcombinationsand
itisspontaneouslynematic. 0.4
0.2
B. three-foldsymmetricthree-componentsatν=1
θ
0.5 1.0 1.5 2.0 2.5 3.0
Consider now the case of three-components with
anisotropic mass tensors oriented in a symmetric three-
FIG. S1: Comparison between the zero and first Landau level of
fold fashion with relative angles 2π/3. In this case the Galileancasefordeltafunctioninteractions.
Hartree-Fockenergyis:
NEφ =−pTXp, X = XXX111122 XXX111212 XXX111221 . (S29) p1 = √12 010 , p2 = √12−101, λ1 =λ2 =0,
−1 0
XhaXst=he(Xfollo+w2inXge)ipgepnTva+lu(eXdec−omXpo)(sIit−ionp:pT), p3 = 12−−1111, λ3 =2(X11−X12), (S32)
11 12 0 0 11 12 0 0
   
p0 = √13111. (S30) p4 = 21111, λ3 =2(X11+X12).
1
Because X ≥ X ≥ 0alltheeigenvaluesarepositive. This
11 12 thestructureofeigenvaluesimpliesthatminimumwillcorre-
implies that the minimum of the energy in Eq. (S29) corre- spondtomaking|p +p −p −p |aslargeaspossiblewhile
spondstothespontaneouslychosennematics,namely p1 =1, satisfying the const1raint3s (cid:80) 2p =4 1, p ∈ [0,1]. The solu-
and p2 = p3 = 0, and symmetry related copies. The case of tionsare p + p =1,and pα+αp =0,aαndsymmetryrelated
ν = 2 is equivalent after performing a particle-hole transfor- copies. No1ticet3hatCDWty2peso4lutions, p = p = 1/2,are
mation. 1 3
degenerate with the pure nematic solution in this case, and
there is indeed an SU(2) degeneracy remaining that would
needtobesplitbyinteractionsbeyondCoulomb.
C. four-foldsymmetricfour-componentsatν=1
VI. PROOFOFFULLVALLEYPOLARIZATIONFOR
Thiscaseisspecialasthegroundstateisnotnecessarilyne- TWOANISOTROPICVALLEYSATν=1WITHIN
maticbutindeedthereisadegeneratemanifold. Weconsider HARTREE-FOCK
four-components with mass tensors oriented in a symmetric
four-fold fashion with relative angles π/2 between adjacent
Consider the situation of two anisotropic valleys labeled
pockets. InthiscasetheHartree-Fockenergyis:
1,2. Assumethevalleysarerelatedbyadiscreterotationby
some angle α. This implies that the Landau level have the
same energies in both valleys and the eigenstates are related
 
NEφ =−pTXp, X = XXX111121 XXX111212 XXX111121 XXX111212 . (S31) biannytdraan-dLisiasancdrleaatbueellsafybomerlmthfoeetrLrytahne|2d,ganuu,ildkei(cid:105)vneg=lcoRefnαint|1eter,rnde,eskgt(cid:105).r,eAewssshuoemfreferkeweiedsoaamrne
X12 X11 X12 X11 atpartialfillingν=1ofanyofthedoublydegenerateLandau
levelsofinterests. FromourgeneralHartree-Fockexpression
ThespectrumofXis: inEq.(S25)weobtainthatthevariationalenergyis:
10
GalileanCoulombn=0(blue),n=1(yellow), mx/my=1/2 thatvq onlydependsonthenorm|q|,impliesthat X11 = X22,
X12 X12 = X21 andthattheseexchangeintegralsarerealnumbers.
Usingthis,wecanrewritetheenergyas:
1.0
E X +X
0.8 =−2(X −X )(p −1/2)2− 11 12. (S36)
N 11 12 1 2
φ
0.6
The above expression implies that the ground state either
0.4 spontaneously polarizes into a single valley (Ising type) for
X > X ,orintoanequalsuperpositionofbothvalleys(XY
11 12
0.2 type) for X < X . The marginal case for X = X will
11 12 11 12
haveeffectivelySU(2)symmetryandcanbeanarbitraryco-
θ
0.5 1.0 1.5 2.0 2.5 3.0 herentsuperpositionofthetwovalleys.
The form factor satisfies F(q) = F(−q)∗. Interestingly,
i i
thisimpliesthattheexchangeintegralcanbeseenasaninner
FIG. S2: Comparison between the zero and first Landau level of productinthespaceofformfactors:
GalileancaseforCoulombinteractions.
CoulombGalileann=1(blue),Diracn=1(yellow), 1(cid:90) d2q
(F,F )≡ X = v F(q)F (q)∗. (S37)
vx/vy= mx/my=1/2 i j ij 2 (2π)2 q i j
X12
It is easy to verify that (F,F ) satisfies all the axioms of an
0.8 i j
innerproductprovidedv isstrictlypositive(asitisthecase
q
for the Coulomb potential) and it is finite because the form
0.6 factorsalwayshaveexponentialdecayas|q| → ∞. Usingthe
Cauchy-Schwarzinequalitywecanestablishthat:
0.4
(cid:112)
(F ,F )≤ (F ,F )(F ,F )→ X ≤ X (S38)
1 2 1 1 2 2 12 11
0.2
Therefore we establish that the system generically is of the
θ Ising type and polarizes into a single valley. The inequality
0.5 1.0 1.5 2.0 2.5 3.0 is saturated for the marginal case in which X = X . This
12 11
happens if and only if the form factor for each valley is sep-
aratelyinvariantundertheR rotation,sothat F = F . This
FIG.S3: ComparisonofDiracandGalileancaseinthesecondLan- α 1 2
canonlyhappeniftheHamiltoniandescribingeachvalleyis
daulevelforCoulombinteractions.
invariantundertherotationR thatrelatesbothvalleysupto
α
acoordinateindependentunitarytransformation.
Noticethattheaboveargumentimpliesthatifwehavetwo
valleysatkand−kthatarerelatedbyaπrotation,buteachof
E
=−(X p2+X p2+p p (X +X )), p =|(cid:104)χ|i(cid:105)|2, (S33) thethefermisurfacesofagivenvalleyisnotinvariantunder
N 11 1 22 2 1 2 12 21 i
φ aπrotation,thenthesystemwillpolarizeintoasinglevalley
where |χ(cid:105) is the vector describing the coherent combination implyingthegroundstatebreaksinversionsymmetryanditis
of the two valleys in the valley Bloch sphere and |i(cid:105) are the aferroelectric.
vectors corresponding to the two poles of the Bloch sphere.
The probabilities are normalized p + p = 1. The explicit
1 2
formoftheexchangeintegralsis: VII. PROOFOFFULLVALLEYPOLARIZATIONFOR
MULTIPLEVALLEYSATν=1WITHINHARTREE-FOCK
(cid:90)
1 d2q
X = v F(q)F (−q) (S34) Consider the general case in which we have n valleys lo-
ij 2 (2π)2 q i j catedatdifferentregionsofmomentumspace. Wedonotas-
sume that the valleys are related by symmetry, but only that
Andtheexplicitformoftheformfactorsis:
theformfactorsarelinearlyindependentfunctionsofq,inthe
senseimpliedbytheinnerproductdefinedbytheexchangein-
Fi(q)≡(cid:104)i,n|e−il2qa(cid:15)abpb|i,n(cid:105). (S35) tegralinEq.(S38).Becausetheformfactorsarefunctionsofa
two-dimensionalwavevectorq,weexpectthatcasesinwhich
BecausethevalleysarerelatedbyarotationR byangleα,we the form factors are linearly dependent are either fine tuned
α
havethatthat F (q) = F (R−1q). This,togetherwiththefact accidentalcasesorcasesinwhichasymmetrythatrelatestwo
2 1 α