Hall Viscosity Revealed via Density Response

0.09 MB·English

by  Biao Huang

HallViscosityRevealedviaDensity Response
Biao Huang
Department of Physics, The Ohio State University, Columbus, OH 43210, USA
(Dated:June4,2015)
QuantumHallsystemsarerecentlyshowntopossessaquantitysensitivetothespatialgeometryandtopology
ofthesystem,dubbedtheHallviscosityη . Despitetheextensivetheoreticaldiscussionsonitsproperties,the
H
question of how to measure η still poses a challenge. In this paper, we present a general relation between
H
HallviscosityandsusceptibilityforsystemswithGalileaninvariance. Thus,itallowsfordeterminationofη
H
through density response signatures. The relations are verified in the integer quantum Hall example, and is
furtherillustratedinaneffectivehydrodynamicanalysis. SincethederivationisbasedonKuboformulaeand
5
1 assumesnomorethanconservationlawsandtranslationalsymmetry,theresultsareapplicabletoawiderange
0 ofsystems.
2

n
I. INTRODUCTION II. KUBOFORMULAEMETHOD
u
J
Wefirstsketchtheideaofrelatingviscosityandconductiv-
3 In recentyears, there has been heated discussionson Hall
viscosityη (or“Lorentzshearmodulus”)1,thethirdviscos- ity to susceptibility before the strict derivation. In a viscous
H
] ity coefficient unique in 2-dimensional isotropic, parity odd fluid,thestresstensorwouldrespondtothevelocitygradient
el systems. It can be regarded as the Berry curvature in the vαβ = 21(∂αvβ+∂βvα)through
-
r parameterspace of metric tensor1, and is proportionalto the δT =η v , (1)
st Wen-Zeeshift2,aquantitycharacterizingthesystem’sspatial µν µναβ αβ
t. topology3. whereηµνρσaretheviscositycoefficients.Inthreedimensions,
a
m Besides the topologicalsignificance, the influence of Hall rotation symmetry reduces the number of ηµνρσ to 2, corre-
viscosity on varioussystemsshowsup in multipleways. As spondingtotheusualshearandbulkviscosity
d- discovered recently, for integer quantum Hall systems4 and 2
n Laughlinwavefunctions5, theHallviscosityisrelatedtothe ηS(δµαδνβ+δµβδνα δµνδαβ), ζδµνδαβ, (2)
− d
o densityresponsetothevariationofthesample’sscalarcurva-
c ture. It acts as an anomolous force on vortices and changes wheredisspatialdimension.Butin2D,thein-planeisotropy
[ their streamlines in an eulerian vortex fluid6. More gener- allowsforanadditionalonecalledHallviscosity

2 ally,theHallviscosityappearsinthelow-momentumexpan-
v sionofHallconductivityσ (q)forGalileaninvariantsystems ηH(δναǫµβ δµβǫαν). (3)
H −
0 subject to inhomogeneouselectric field7,8. Clearly, studying
4 Meanwhile,thedensityandcurrentofasystemwouldrespond
thesephysicalconsequencesnotonlybroadensourperspetive
2 topotentialandelectricfieldperturbations
onHallviscosity,butalsofacilitatesdesigningexperimentsto
5
measureit. δn(q,ω)=χ(q,ω)ϕ(q,ω), (4)
0
. Density response measurement is a powerful and widely δj (q,ω)=σ (q,ω)E (q,ω), (5)
1 α αβ β
0 applicable tool to probe various systems. In recent works,
where χ(q,ω),σ (q,ω) are the (charge-)susceptibility and
5 manyauthorshaverelatedHallviscosity to densityresponse αβ
conductivity respectively. The key idea is that since in lin-
1 functions,i.e. thestatic structurefactor4,9, usingcertaintrial
: wave functions. Itis intriguingto see whetherthere are uni- ear regime, the different transport processes probe the same
v equilibriumpropertyofthe system, thetransportcoefficients
i versalrelationsbetweendensityresponsefunctionsandHall
X are all related to each other. Technically, the essential in-
viscosity. In this paper, we relate the Hall viscosity η to
H gredientsin the Kubo formulae for the transport coefficients
r theq4terminthesusceptibilityχ(q,ω)forageneralGalilean
a are the correlators χ [n,n] , σ [j , j ] , and
invariantsystem,thusallowingformeasurementsofHallvis- ∼ h i0 αβ ∼ h α β i0
η [T ,T ] . ForGalileaninvariantsystems,thecur-
cositythroughdensityresponseexperiments. We firstderive reµnναtβis∼rehlateµdν toαmβ oi0mentumbyj= ep. Thus,theparticleand
the relation in the microscopic level using Kubo formulae, m
momentumconservationforthesystematequilibrium,
and verify the results in the integer quantum Hall example.
Thenwepresentanindependentderivationbyapplyingeffec- ∂n(x,t)+∂ p (x,t)=0, (6)
t α α
tive hydrodynamics. The latter method has been adopted11 ∂ p (x,t)+∂ T (x,t)= f (x,t) (7)
to analyze collective modes of fractional quantum Hall sys- t α β αβ α
tematfillingν=1/3,whichgivesqualitativeagreementwith naturally provides a bridge among the transport coefficients
experiment.12Thetwoderivationsgivethesameresultinthe of differenttypes. Roughly speaking, by writing the conser-
longwave-lengthexpansion.Sincebothofthederivationsare vationlawsinFourierspace,weseetheviscosityisrelatedto
independentof specific wave functions, type of interactions, theq2 andq4 termsinconductivityandsusceptibilityrespec-
filling factors, and etc., the results (13), (14) are valid for a tively. Analogous ideas have been adopted recently to dis-
generalclassofGalileaninvariantsystems. cusstheunitaryFermigases10 andtheconductivity-viscosity
2
relations8, while in this paper we focus on the density re- we see that even if one only considers the susceptibility at
sponsesignaturesfor2Dsystemssubjecttouniformmagnetic zero-frequency, it reflects the linear-in-frequencypart of the
field. conductivity. As weshallsee later, itisessentialto incorpo-
The Hamiltonian describing density coupling to external ratetheω-dependencein(10)beforetakingthelimit. Explic-
potential is H = d2xnˆ(x)ϕ(x,t) = d2q nˆ(q)ϕ( q,t), itly,(10)and(12)showsthesusceptibility-viscosityrelation
1 (2π)2 −
Fvwaohrueiarrineetrnˆst(yrxas)tne=smfoP,rtmhNj=e1nˆsδ(uqR(sx)c−e=pxˆtPijb)Nij=liis1tyeth1−3eiqi·dsxˆejg.nisvFieotynrRboaypterarantsolar,tiwonitahltihne- χ(4)(q,ω)= m2(ω2q−4 ω2c)2ωhκi−n1t −2ωcωηH2(+ω)ω2
+ζ(ω) +ηS(ω) c . (13)
χ(q,ω)= i ∞dteiω+t d2xe iqx [nˆ(x,t),nˆ(0,0)] . i iω #
−~Z0 Z − · h i0 Andatzerofrequencyω=0,thestaticsusceptibilityis
(8)
Heqeurielihb.r.iu.im0smysetaenms,eannsdemωb+le=aωve+raiǫgewiinthtǫhe fu0lleynisnutreirnagcttihnagt χ(4)(q,ω=0)= mq2ω44 κi−n1t −2ωcηH(ω=0)
the perturbationadiabaticallysets in. On th→e otherhand, for c h
ηS(ω)
currentresponsetoelectricfield, theKuboformulagivesthe +ω2lim ilim(ωζ(ω)) . (14)
conductivity14 cω 0 iω !− ω 0 (cid:21)
→ →
σαβ(q,ω)= mieω2n¯+δαβ+ ~ω1+ Z ∞dteiω+t sWuemspttrieosnsothfaGttahlielseeanreilnavtiaorniasnacree.oTbhtauisn,etdheunHdaelrltvhiescoonsliytyasis-
0 guaranteedtoshowupintheq4 terminthedensityresponse
d2xe iqx [ˆj (x,t),ˆj (0,0)] , (9) functionforawiderangeofsystems.
− · α β 0
×Z h i
wheren¯ = N/V istheaverageparticlenumberdensity. Here
thecurrentdensity ˆj (x)= e N πˆi,δ(x xˆi) ,where ... III. EXAMPLE:INTEGERQUANTUMHALLSYSTEMS
α 2m i=1{ α − } { }
isanti-commutator,andπˆi = pˆPi eAˆi(xˆi).
HerewefollowtheKubαoformαu−laecαonstructionin8andob- Weapplytheaboveresultstodiscusstheparadigmexample
tain the frequency-dependent conductivity-viscosity relation ofintegerquantumHalleffect.TheHamiltonianinvolvesonly
inlowmomentumexpansionuptoq2term,validforGalilean thekineticpartHˆ0 = iπˆiµπˆiµ/2m=~ωc i(aˆ†iaˆi+1/2),where
invariantsystems, theladderoperatorfoPrLandaulevelsisaˆP=(πˆi+iπˆi)/√2~eB,
i x y
σ(2)(q,ω)= e2 with the commutationrelation [aˆi,aˆ†j] = δij. Workingin the
µν m2(ω2 ω2)2 sphericalgaugeAˆi = B( yˆi,xˆi,0),anddefineRˆi = pˆi+eAˆi,
− c 2 −
×[(ζ(ω)− κii−ωn1t) ω2qµqν+iωωcq2ǫµν+ω2cǫµβǫνδqβqδ wspeecciafineisntthroedduecgeebnˆie=rac(yRˆixw−itihRˆiiyn)/ea√c2h~LeBan,dwaiuthle[vbˆei,lbˆi†nj]t=ermδisj.oIft
(cid:16) (cid:17) angularmomentumLˆz =~ i(bˆ†ibˆi−aˆ†iaˆi).Consideraground
+ηS(ω)q2 (ω2+ω2c)δµν+2iωωcǫµν state with the lowest ν LanPdau levels fully filled; each level
−ηH(ω) iω(cid:16)ωc(qµqν+q2δµν)−(ω2+(cid:17) ω2c)q2ǫµν ], (10) hleaasddineggetenremrasciynVs/u2scπel2Bp.tibUilsiitnygdeirqeucatltyion(8) we calculate the
(cid:16) (cid:17)
whereω =eB/misthecyclotronfrequency,andthe“internal n¯ω
c χ(2)(q,ω)=(ql )2 c , (15)
compressiblity”is B ~((ω+)2 ω2)
− c
ε 1 1
κi−n1t =−V ∂∂PVint!N,ν = B2 ∂∂B2ε2!ν, (11) χ(4)(q,ω)=(qlB)4~2 (ω+)2−4ω2c − (ω+)2−ω2c!.(16)
Here the energydensity is ε = (eBν)2/4πm, and the average
where V means area. Here the “internal pressure” P =
int densityisn¯ =ν/2πl2,wherethemagneticlengthl2 =~/eB.
P BM/V excludesthepressurecontributionfromthemag- B B
− Next we use (12) (13) to calculate the same terms in sus-
netization M duetoedgecurrent8,16. Notethatthederivative
ceptibilitysoastoverifythesusceptibility-viscosityrelations.
istakenatfixedνinsteadoffixed B. Thus,theinternalcom-
Thezerothorderterminconductivityisdirectlyobtainedfrom
pressibilityhasafinitevalue.
inI(n10p)rewviaosunsolittegriavteunr,esa7s,8,fothretheexpdliisccitufsrseiqounetnhceyredeitpewnadsenncoet wthheiKchubroedfuocrmesultao(9th)e,σf(µa0νm)(ωili)a=r omienω2en¯+(σω+(ω0)2)2−(ωω2cδ=µν+0)m(n¯ω=e2c2−ωωce22)νǫµ/νh,
µν
important. But from the relation between susceptibility and at zero frequency. Using (12) we obtain (15). Further8,
conductivity18,obtainedfromKuboformulae(8)(9)andpar- κ 1 = 2ε,ζ(ω) = 0,ηS(ω) = iω+ε ,η (ω) = 2ωcε .
ticleconservation(6), Ti−hntenfrom(13)wehave(16),as(eωx+)p2e−c4ωte2cd.H 4ω2c−(ω+)2
q q We can further obtain the dynamic structure fac-
χ(q,ω)= ie2αωβ+σαβ(q,ω), (12) tor S(q,ω)13 using the fluctuation-dissipation theorem
3
Imχ(q,ω) = π (S(q,ω) S( q, ω)), and the identity Then applying the alternative expression for compressibility
x±1iǫ =P1x ∓πiδ−(x~)V, − − − dκiennts=ityn1re∂∂Psnpinto,nwsehitcohpgoitvenestia∇lPpienrttu=rbκai−tni1to(n∇δδnn)=/n¯,χϕw,ewhhaevreetthhee
N
S(2)(q,ω)=(ql )2 δ(ω ω ), (17) susceptibility(inmomentumspace)reads
B c
2 −
Nν n¯2q2
S(4)(q,ω)=(ql )4 [δ(ω 2ω ) 2δ(ω ω )]. (18) χ(q,ω)= .
B 8 − c − − c mn¯ω2 κi−n1t ηS ζ iωq2+ (mn¯ωc−ηHq2)2
The static structure factor13 S(q) = (1/N) ∞dωS(q,ω) = −iω − −  ηSq2/iω mn¯
nˆ(q)nˆ( q) isthen R0   − (24)
h − i Inthelowmomentumexpansion,theq2termisexactlygiven
S(q)= (qlB)2 ν(ql )4+ ((ql )6). (19) bythatforintegerquantumHall effect(15). Thisisbecause
2 − 8 B O B the intra-Landau level excitations start from the q4 term15.
Forν=1,itreducestothefirsttwotermsintheexpansionof Thus, the q2 contribution must entirely come from inter-
Landaulevelexcitations,whosecharactersarecapturedbythe
fthuellyw-fiellll-ekdnloowwnesrtesLualntdSa(uql)ev=el115−. e−(qlB)2/2 fora systemwith integerquantumHalleffect. Theq4termisgivenbythesame
equation(13),asbeingderivedfromtheKuboformula.Thus,
inthelowmomentumregime,theclassicalhydrodynamicsre-
producesthe strict susceptibility-viscosity relations obtained
IV. HYDRODYNAMICANALYSIS
fromKuboformulae.
(Ingeneral, theviscosity coefficientsζ,ηS,η wouldhave
H
Having performedthe strict microscopic analysis, it is in- momentum dependence also4,5. In the expansion of (24)
teresting to see whether there is a macroscopicderivationof discussed above, we have taken the viscosity coefficients
the susceptibility-viscosity relations (13) (14), which would ζ,ηS,η to be constant (only depends on frequency). This
H
provideasimplerandmoreintuitivewaytounderstandthem. is suffficient when we expandχ(q,ω) up to q4 terms, as can
To this end, we next show that (13) (14) can be reproduces
be seen from (24). But when applying(24) to higher orders
bypureclassicalhydrodynamicequations,despitetheunder- inq,oneneedstofirstexpandζ,ηS,η tohigherpowersinq
H
lyinghighlyquantumstructureoftheelectronicliquid. Such (i.e. q0 andq2termsforζ,ηS,η )beforeexpanding(24)(i.e.
H
amethodhasbeenadoptedtodiscussthefractionalquantum toq6termforχ(q,ω))).
HallliquidsandBose-Einsteincondensatesrecently11,17,19.
In hydrodynamics, the microscopic details are averaged
over,givingafeweffectivemacroscopicvariables,
V. MEASURINGHALLVISCOSITY
n(r,t)=n¯ +δn(r,t), u(r,t)=u¯ +v(r,t). (20)
The relations (13), (14) connects the Hall viscosity with
Here n,u are the macroscopically averaged number density the susceptibility, which can be measured by scattering ex-
andthevelocityrespectively,includingtheequilibriumvalue periments. Since many recent works only concern the zero
n¯,u¯, and the small deviation δn(r,t),v(r,t) caused by exter- frequency value of Hall viscosity η (ω = 0), we focus on
H
nalperturbation.ThedynamicsisgivenbytheNavier-Stokes equation(14)here. Notethatinmostcasesthebulkviscosity
equation (momentum conservation) (7), where the momen- ζ(ω) doesnotdiverge8 atzerofrequency. Thenthelast term
tump=mnu. TheforcecomesfromtheLorentzforceofthe inequation(14)vanishesandtheformulacanberewrittenas
uniformmagneticfieldandtheexternalpotentialperturbation
f =−enBez×uT−n=∇ϕP(r,δt). T+hmenstureusstenδsTor,is (21) ηH = 2κωi−n1tc + ω2c ωli→m0 ηiωS!− m22ω3c χ(4q)4(q). (25)
µν int µν µ ν µν
−
In order to extract η from the measurement of χ(4)(q), one
where the viscous part δT is given by (1)-(3), and P is H
µν int has to determine the first two terms in (25). The inverse in-
theinternalpressureintroducedearlier. WestressthatP ex-
int ternal compressibility κ 1 = B2(∂2ε(B)/∂B2) can be deter-
cludesthe contributiontopressurefromLorentzforceexert- i−nt ν
minedbytheauxiliarymeasurementofmagneticsusceptibil-
ingonedgecurrent,andthereforeissuitabletobeusedhere,
ityχ = µ ∂M = µ ∂2ε 20 atconstantfillingfraction,
astheNaivier-Stokesequationonlyconcernsbulkproperties M 0 ∂B ν − 0 ∂B2 ν
andassumesnoboundaryeffect. Considerthelinearresponse oralocalcur(cid:16)ren(cid:17)tmeasure(cid:16)men(cid:17)tinresponsetotheinhomoge-
neousmagneticfield7δj= δM= ε (B)e δB.Inpar-
regime where δn(r,t),v(r,t) are kept up to linear order, and ′′ z
∇× − ×∇
use the equilibrium value n¯ = constant,u¯ = 0, the Navier- ticular, in high magnetic field where Landau level mixing is
negligible,theinteractionenergycanbeneglectedcompared
Stokesequation(7)becomes
withthekineticenergy.Thenwehavethefreeparticleresults
mn¯∂tv=(en¯B+ηH∆)(v ez)+ (ζ v n¯ϕ Pint)+ηS∆v, (22) ε = (eBν)2 and ε = (eB)2ν for integer and fractional quantum
× ∇ ∇· − − 4πm 4πm
Hallsystemsrespectively,givingκ 1 =2ε.
On the other hand, the particle conservation, also kept up to i−nt
Further,lim (ηS/iω)generallyyieldsafinitevalue,(See
linearorder,gives ω 0
theintegerquan→tumHallsystemforexample),andneedscare-
∂δn+n¯ v=0, (23) ful evaluations. Here we invoke the Kubo formulafor shear
t
∇·
4
viscosity8, ηS = ~ωV+ 0∞eiω+th[Jˆ12(t),Jˆ12(0)]i0, to compute Thequalitativebehaviorofχ(q)insmall-qlimitcanbeob-
thisquantity. Jˆ istheRoff-diagonalelementofthestraingen- served as follows. Using compressibility sum rule χ(q
eratorJˆµν. Inth1e2spectralrepresentation, 0) = −NκT−1 andthe incompressiblefeatureofquantumHa→ll
states,weseetheconstantterminχ(q)alwaysvanishes.Here
ηiωS = 2V~ ωn(0ω|h0+|)J2ˆ12(0ω)2|ni|2, (26) κMT−o1r=eo−vVer(,∂thPe/∂loVw)N-l,yBiinsgtheexcuistautsiaolnissowthitehrimnathlecolomwpersetsLsiabnilditayu.
Xn − n0 levelcorrespondtoχ(q)startingfromq4 term21. Thatmeans
whereω =ω ω ,withω thefrequencyofenergyeigen- if the inter-Landau level excitation is supressed in the large
n0 n 0 n
− magnetic field limit, (or we can cancel such contribution by
state n . (InthepresenceofLandaulevelmixing, n alsoin-
clude|shiigherLandauleveleigenstates).Therefore,|ifiω =0 subtracting the χ(2) given by equation (15)), χ(4) is the lead-
n0
ing orderin χ(q). In sum, the algebraicrelation (29) clearly
or the matrix element vanishes, the corresponding term is
meansthat it will be sufficientto extract the q4 term in χ(q)
zero;otherwise,wehavethefiniteresult
bymeasuringonlythelongwavelengthpartofS(q,ω),asis
ηS 2~ 0Jˆ (0)n 2 usuallythecaseincurrentexperiments.
lim = ′ |h | 12 | i| , (27)
ω→0 iω! −V Xn ωn0
where meanssummingoverenergyeigenstateswhichare
′
not degenerate with the ground state. Explicitly, the off-
P
diagonal element of the strain generator assumes the form
VI. CONCLUSIONANDOUTLOOK
Jˆ (0) = xˆi,πˆi eBxˆi2 /2~, and can be written con-
12 i −{ y}−
venientlyPint(cid:16)hesphericalgauge(cid:17)Aˆi = 1B rˆias
2 × Therelationbetweensusceptibilityandviscosity(13)(14)
xˆipˆi 1 is presentedanddiscussed, showingtherole ofviscosity co-
Jˆ12(0)=− ~y =−4 [(aˆi−aˆ†i)2−(bˆi−bˆ†i)2]. (28) efficientsintheq4termofsusceptibilityforageneralclassof
Xi Xi Galileaninvariantsystem. Itsuggeststhepossibilityofmea-
suringtheHallviscosityintermsofdensityresponseexperi-
whereaˆi,bˆi areinter-andintra-Landaulevelladderoperators mentsforawiderangeofquantumHallsystems.
defined previously. Consider, for instance, integer quantum
Hall effect. Then the only non-zero matrix element in (27) In addition, it is worth mentioning the connection of this
is 0aˆ22 , where 2 meansthe state with oneelectron from workwithcoldatomsystems,wherequantumHallstatescan
h | i| i | i besimulatedusingrotatingatomicgases22.ThequantumHall
either of the top two filled Landaulevels in the groundstate
stateisapproachedwhentherotationfrequencyofthetrapped
beingexcited2levelsupwards,correspondingtoanexcitation
energyω = 2ω . Thus, lim (ηS/iω) = ε/4ω2, in con- gas approaches the critical frequency, where the centrifugal
sistency w20ith thecpreviousresωu→lt.0 The evalua−tionfocr various force almost cancels the trapping force, and the Galilean in-
variance is approximately satisfied. Since in cold atom sys-
fractionalquantumHallsystemsisleftforfuturework,which
tems, thedensitymeasurementandthepotentialengineering
generally requiresa numericalevaluation of (27) (28) in the
arethemoststandardexperimentaltools,andcanbemadeto
presence or absence of Landau level mixings depending on
highaccuracy,ouranalysispavestheway forfurtherdiscus-
specificexperimentalsituations.
Thus, with κ 1 andlim (ηS/iω) being measuredor cal- sionsofHallviscositysignaturesincoldatomexperiments.
i−nt ω 0
culated,onecanuseequati→on(25) todeterminetheHallvis-
cosity η through the long wavelength part of static sus-
H
ceptibility χ(4)(q,ω = 0), which is related to scattering
experiments13. Explicitly,thequantitybeingmeasuredisthe
dynamic structure factor S(q,ω) = S (q)δ(ω ω (q)),
n n − n VII. ACKNOWLEDGEMENT
where ω (q)’s are excitation modes. Using Kramers-Kronig
n P
relation Reχ(q,ω) = (1/π) ∞ dz (Imχ(q,z))/(z ω) and
P −
fluctuation dissipation theoreRm−∞, we have the static suscepti- IthankTin-LunHo,Yuan-MingLu,andAlexanderAbanov
bilityχ(q)=χ(q,ω=0)as21
fordiscussions,andespeciallyT.-L.Hoforhelpfulcomments
andsuggestionsaboutthemanuscript.Thisworkissupported
2 S (q)
χ(q)= n . (29) byNSFGrantDMR-1309615,andtheNASAGrant1501430.
−~V ω (q)
Xn n
1 J.E.Avron,R.Seiler,andP.G.Zograf,Phys.Rev.Lett.75,697 2 N. Read, Phys. Rev. B 79, 045308 (2009); N. Read and E. H.
(1995); J. E. Avron, J. Stat.Phys. 92, 543 (1998); I. V. Tokatly Rezayi,Phys.Rev.B84,085316(2011).
andG.Vignale,J.Phys.Condens.Matter21,275603(2009). 3 X.G.WenandA.Zee,Phys.Rev.Lett.69,953(1992).
4 A.G.AbanovandA.Gromov,Phys.Rev.B90,014435(2014).
5
5 T. Can, M. Laskin, P. Wiegmann, Phys. Rev. Lett. 113, 046803 19 C.J.PethickandH.Smith,Bose-EinsteinCondensationinDilute
(2014). Gases, (Cambridge 2008); L. Pitaevskii and S. Stringari, Bose-
6 P. Wiegmann and A. G. Abanov, Phys. Rev. Lett. 113, 034501 EinsteinCondensation,(Oxford2003).
(2014). 20 Theusualconventionisχ =∂M/∂H;B=µ (1+χ )H.Butex-
M 0 M
7 C.HoyosandD.T.Son,Phys.Rev.Lett.108,066805(2012). ceptforferromagneticsystems,χ 1(QHsystemsaremostly
8 B.Bradlyn,M.Goldstein,andN.Read,Phys.Rev.B86,245309 paramagnetic).Sowehaveχ =µM∂≪M/∂B.
M 0
(2012). 21 We can compare it to the situation in single mode
9 D.X.Nguyen,D.T.SonandC.Wu,arxiv:1411.3316. approximation15. There S (q) becomes the static structure
n
10 E.TaylorandM.Randeria,Phys.Rev.A81,053610(2010). factor S(q) while ω (q) ∆(q) corresponds to the only one
n →
11 I.V.Tokatly,Phys.Rev.B73,205340(2006);I.V.Tokatly,Phys. gapped mode. But a recent experiment12 shows the existence
Rev.B74, 035333 (2006); I.V.Tokatly, G.Vignale,Phys.Rev. of two collective modes. Thus, we keep the general form (29).
Lett.98,026805(2007). But the presence of two or multiple modes does not affect the
12 C. F. Hirjibehedin, I. Dujovne, A. Pinczuk, B. S. Dennis, L. N. conclusion that intra-Landau level excitation corresponds to
Pfeiffer,andK.W.West,Phys.Rev.Lett.95,066803(2005). S(q,ω),χ(q,ω) starting from q4. This can be seen by applying
13 D.PinesandP.Nozie`res, TheTheoryofQuantum Liquids,vol.I the f-sumrule13 ∞ωS(q,ω)dω = ~Nq2/2mand(15).Thenwe
0
(Benjamin,NewYork,1966). seethesumruleisRsaturatedbytheinter-Landaulevelexcitations,
14 G.D.Mahan,Many-ParticlePhysics,3rdedition(Springer2013). meaningthattheq2termcomesentirelyfromthecyclotronmode.
15 S.M.Girvin,A.H.MacDonald,P.M.Platzman,Phys.Rev.B33, 22 IthankMarkusGreinerfordiscussingtheon-goingsmall-cluster
2481(1986). experimentinhisgroupduringthe2015INTconference.Seealso
16 N.R.Cooper, B.I.Halperin, andI.M.Ruzin, Phys.Rev.B55, A.L.Fetter,Rev.Mod.Phys.,81,647(2009)forareviewofthe
2344(1997). rotatingatomicgasesandthesimulationofquantumHallstates.
17 P.B.Wiegmann, JETP144, 617(2013); P.B.Wiegmann, Phys.
Rev.B88,241305(R)(2013).
18 X.-G. Wen, ”Quantum Field Theory of Many-Body Systems”,
OxfordUniversityPress(2004),eq(3.7.7)insection3.7.2.