Magnetic field-tuned Aharonov-Bohm oscillations and evidence for non-Abelian anyons at v=5/2

4 MB·English

by  R. L. Willett

Magneticfield-tunedAharonov–Bohmoscillationsandevidencefornon-Abeliananyonsatν = 5/2
R. L. Willett,1 C. Nayak,2,3 K. Shtengel,2,4 L. N. Pfeiffer,5 and K. W. West5
1Bell Laboratories, Alcatel-Lucent, Murray Hill, New Jersey 07974, USA
2MicrosoftResearch,StationQ,ElingsHall,UniversityofCalifornia,SantaBarbara,California93106,USA
3Department of Physics, University of California, Santa Barbara, California 93106, USA
4DepartmentofPhysicsandAstronomy, UniversityofCalifornia, Riverside, California92521, USA
5DepartmentofElectricalEngineering, PrincetonUniversity, Princeton, NewJersey08544, USA
(Dated:January15,2013)
We show that the resistance of the ν = 5/2 quantum Hall state, confined to an interferometer, oscillates
withmagneticfieldconsistentwithanIsing-typenon-Abelianstate. InthreequantumHallinterferometersof
3
different sizes, resistance oscillations at ν = 7/3 and integer filling factors have the magnetic field period
1
expectedifthenumberofquasiparticlescontainedwithintheinterferometerchangessoastokeeptheareaand
0
thetotalchargewithintheinterferometerconstant.Undertheseconditions,anAbelianstatesuchasthe(3,3,1)
2
statewouldshowoscillationswiththesameperiodasatanintegerquantumHallstate. However,inanIsing-

n typenon-Abelianstatetherewouldbearapidoscillationassociatedwiththe“even-oddeffect”andaslowerone
a associatedwiththeaccumulatedAbelianphaseduetoboththeAharonov-BohmeffectandtheAbelianpartof
J
thequasiparticlebraidingstatistics.Ourmeasurementsatν =5/2areconsistentwiththelatter.

2
1
Introduction. The origin of the fractional quantum Hall erostructures.A40nmSiNlayerisappliedtotheheterostruc-

]
l effect [1] at filling factor ν = 5/2 [2–4] has been a long- ture. The size and shape of the 2D electron channel, which
l
a standing open issue, which is important because it has been is 200 nm below the surface, is controlled by 100 nm thick
h conjectured that this state of matter supports non-Abelian Al top gates that are deposited on the SiN layer, as shown
-
anyons [5–9].Twopoint-contactFabry–Pe´rotinterferometers in Fig. 1. Prior to charging the top gates, the samples can
s
e havebeenproposedtoobservetheAharonov–Bohm(AB)ef- bebrieflyilluminatedtoenhancemobilityandtoprovidedif-
m
fectandtheanyonicbraidingstatisticsofquasiparticles[10]. ferentsamplepreparationssincetheilluminationchangesthe
. Inanon-Abelianstate,notonlythephasebutthealsotheam- distribution of localized charges in the device, as do differ-
t
a plitudeoftheobservedoscillationsisindicativeofthebraid- entcool-downs[15,16,21]. Furtherdescriptionofthedevice
m
ing statistics [11–14]. Specifically, if the ν = 5/2 state is preparationsandmeasurementdetailsispresentedinSupple-
- indeednon-Abelian, thequasiparticleparitywithintheinter- mental Material. Two interfering edge currents result from
d
ferometerdictateswhethertheresistanceofaninterferometer quasiparticle tunneling across constrictions defined by gate
n
o oscillates with enclosed area (controlled by a side gate) with sets1and3. ThelongitudinalresistanceRLismeasuredwith
c a period associated with charge e/4 quasiparticles. Such os- contacts labeled a through d in the electron-micrograph in
[
cillations should only be seen when the parity is even – the Fig.1(a)bythevoltagedropfromcontactatod,withcurrent

1 “even-oddeffect”[13,14]. Previousexperiments[15,16]are driven from b to c, using standard lock-in techniques. The
v broadly consistent with these predictions [17–20], although twostandardtopgatedesignsshowninelectronmicrographs
9 somepuzzlesremain,aswediscussbelow. in Fig. 1(a) are labeled with device dimension parameters x
3
and y adjusted to produce three separate samples with ratios
6 Inthispaper,weexaminethemagneticfielddependenceof
ofareasofroughly3:2:1. Inaddition,thefunctionalareasin
2 theresistanceofaseriesofinterferometerswithalargerange
. of active areas. Two of them are shown in Fig. 1. We for- thesedevicesaredefinedbyapplyingthegatevoltages,result-
1
inginarangeofareasfrom0.1to0.6µm2. Thetemperature
0 mulateamodelbasedontheassumptionthatthetotalcharge
3 in the interferometer and the enclosed area both remain con- ofthemeasurementsis 20mKinalldatapresentedhere.
1 stant as the magnetic field is varied. We test it at ν = 7/3
PreviousResults. OscillationsinR havepreviouslybeen
: L
v and integer filling factors and show that it is consistent with observedasafunctionofsidegatevoltage,V ,whichchanges
s
i theexperimentaldata–intheν = 7/3case, itpredictsare-
X the area of the interferometer [15, 16]. Interpreted as due to
sistance oscillation with the somewhat surprising flux period
theAharonov-Bohmeffect,theexpectedperiodofoscillation
r
a Φ0/2. We thereby determine the effective area of the inter- is∆V ∝∆A=(e/e∗)Φ /B(Φ =hc/eisthefundamental
s 0 0
ference loop in each device (and each ‘preparation’ of each fluxquantumande∗ isthechargeoftheinterferingquasipar-
device,whichwedescribeinthenextparagraph). Themodel ticles), from which the quasiparticle charge e∗ could be ob-
alsopredictsthattheresistanceintheν = 5/2statewillos-
tained if the proportionality constant between ∆V and ∆A
s
cillateastheproductoftwooscillations,onewithfluxperiod
were known. Assuming its independence of magnetic field,
Φ /5 and the other with flux period Φ , as we explain and
0 0 thisconstantcouldbedeterminedfromtheperiodofR oscil-
L
comparetoourexperimentaldatabelow. lationsatintegerfillingfactors,wheree∗ = e,oratν = 7/3,
Interferometers. The interferometers used in this paper wheree∗ =e/3isexpected.Bothfillingfractionsgivesimilar
are fabricated from high-mobility (28×106cm2/V·s), high- proportionality constants between ∆V and ∆A, which sup-
s
density (4.2×1011cm−2) GaAs/AlGaAs quantum well het- portstheideathatthisconstantisapproximatelyindependent
2
other words, if the Coulomb energy dominates. In such a
case, as the magnetic field is varied, one of two possibilities
will occur. Quasiparticles will be created in the bulk or else
thequantumHalldropletwillshrinkorexpand;intheformer
case, the area of the interference loop will remain constant.
Weexpectthisscenariotoholdiftherearelocalizedstatesin
the bulk that have very low energy as a result of disorder so
thatitisenergeticallyfavorabletocreatequasiparticlesthere,
ratherthantochangethechargedensityattheedge.Whenthis
scenarioholds,increasingthefluxthroughtheinterferometer
byΦcausesthenumberofchargee∗quasiparticlestochange
byN =(νΦ/Φ )/(e∗/e).
e∗ 0
Meanwhile, changing the flux by Φ and the number of
charge e∗ quasiparticles by N causes a change ∆γ in the
e∗
phaseacquiredbyaquasiparticletakingonepatharoundthe
interferometerrelativetothephaseacquiredbyaquasiparticle
goingaroundtheother:
∆γ = 2π(Φ/Φ )(e∗/e)−2θ N
0 e∗ e∗
= (Φ/Φ )[2π(e∗/e)−2θ (νe/e∗)] (1)
0 e∗
Figure1:(top)Electronmicrographsoftwoofthethreeinterferom-
Thefirsttermontheright-hand-sideisthe(ordinaryelectro-
etersusedinthemeasurementsreportedinthispaper. Thecontacts
areindicatedschematicallyintheinterferometerontheleftbya-d. magnetic) AB phase seen by a charge e∗ quasiparticle encir-
Currentinjectedatcontactbcanbebackscatteredatthetwoquan- clingfluxΦ. Thesecondtermisthestatisticalphaseseenby
tumpointcontactsshown, therebydefininganinterferenceloopof achargee∗quasiparticlewhenitencirclesN suchquasipar-
e∗
area A. (bottom) The longitudinal resistance RL for the two sam- ticles; the phase acquired when a single charge e∗ quasipar-
plesshowsminimacorrespondingtofractionalquantumHallstates
ticleencirclesanotheris2θ ,assumingthattheparticlesare
e∗
atν =7/3,5/2,and8/3.
Abelian. Fornon-Abelianparticles,morecareisrequired,as
wewillseebelow. Therelativeminussigncanbeunderstood
usingtheargumentin[10],whereitisexplainedwhytheAB
ofmagneticfield. Atν = 5/2,R oscillationsinsomeinter-
L andstatisticalphasesshouldcancelundercertainconditions.
valsofVs appearconsistentwithABoscillationscorrespond- In an integer quantum Hall state, e∗ = e and θ = π, so
e
ingtoe/4chargeswhileinotherintervalstheyseemconsis-
∆γ = 2π(Φ/Φ ) and R will oscillate with magnetic field
0 L
tent with e/2 charges [15, 16]. These results have been in- period∆B ·A=Φ ≈41Gµm2.Nowconsidertheν =7/3
0 0
terpretedasamanifestationofthe“even-oddeffect”[13,14]:
state. If it is in the same universality class as the ν = 1/3
charge e/4 oscillations should be observed only when there Laughlinstate,thene∗ =e/3and2θ =2π/3.Then∆γ =
e/3
is an even number of charge e/4 quasiparticles in the inter-
−4π(Φ/Φ ). Consequently, R will show oscillations with
0 L
ferometer;chargee/2oscillationsshouldalwaysbeobserved. period ∆B · A = Φ /2 ≈ 20Gµm2 – i.e half that in an
1 0
Whenthereisanoddnumberofchargee/4quasiparticlesin
integerquantumHallstate.
theinterferometer, thisshouldbetheonlytypeofoscillation
Now consider the case of ν = 5/2. If the system is in an
visible [18]. In Refs. 15, 16 only e/2 oscillations are visible
Ising-typetopologicalphasesuchastheMoore–Readstate[5]
incertainside-gatevoltageintervals(when,accordingtothis
ortheanti-Pfaffianstate[22,23], thenwhenthereisaneven
interpretation, an odd number of e/4 quasiparticles is in the
numberofchargee/4quasiparticlesintheinterferenceloop,
interferometer), but it is not clear whether both e/4 and e/2
the Ising topological charge will be 1 or ψ, but when there
oscillations–oronlye/4–arepresentintheotherintervals.
isanoddnumberintheinterferenceloop,theIsingtopologi-
By contrast, in this letter we focus on RL measurements calchargewillbeσ. Asaresult,ifoneparticulartopological
during magnetic field sweeps. At integer filling, a B-field charge is energetically favorable for even quasiparticle num-
sweepproducesABoscillationsofRLwithperiod∆B·A= ber–letussuppose,forthesakeofconcreteness,thatitis1–
Φ0 ≈ 41Gµm2,whereAisthecurrent-encircledareaofthe thenthenon-AbelianIsingtopologicalchargehasaperiodic-
interferometer;wetherebydeterminetheactiveareaforeach ityoftwoquasiparticlesor,takingN =(νeΦ/Φ )/(e/4),
e/4 0
ofthedifferentdevicesandsamplepreparations. a flux period Φ = 2Φ (e/4e)/ν = Φ /5. Hence, R oscil-
0 0 L
Model. The key assumption in our interpretation of the lateswithmagneticfieldperiod∆B ·A≈8Gµm2.However,
2
experimental data is that the charge contained within the in- thereisalsoanAbelianphase(1)whichcanhaveadifferent
terference loop and the area of the loop remain constant as periodicity. The Abelian phase acquired when a charge e/4
the magnetic field is varied. It is natural to assume that the quasiparticleencirclesan2N quasiparticleswithIsingcharge
chargecontainedwithintheloopremainsconstantifitispri- 1 (or, equivalently, N charge e/2 quasiparticles) is θ = π/4
marily determined by the local electrostatic potential or, in andN =(νeΦ/Φ )/(e/2). Hence,Eq.(1)nowreads: ∆γ =
0
3
Figure2: Oscillationswithmagneticfieldattheintegerstateν = 4
andalsoatν =7/3(insets),withFFTsoftheseoscillationsshownin
thelefthandpanels.Theratiobetweenthesetwooscillationperiods
isthesameineightdevicepreparationsofvaryingsize.
−2π(Φ/Φ ). Therefore, there is also a slower oscillation in
0
R withmagneticfieldperiod∆B ·A=Φ ≈41Gµm2.
L 0 0
If, however, theIsingchargeisnotfixedto1foranyeven
numberofquasiparticles,butmayberandomlyeither1orψ,
thentheslower,periodΦ ,oscillationwillbeafflictedbyran-
0
domπ phaseshiftsthatcouldwashitout. Ifthesystemwere Figure3: OscillationsinRL asafunctionofmagneticfieldandthe
associatedFouriertransformsat(a)ν =4,(b)ν =7/3,and(c)ν =
in an Abelian (3,3,1) state, then similar considerations lead
5/2. Theverticalbluelinesmark5x,2x,andtheintegerfrequency.
toaperiodΦ oscillationbutnorapidperiodΦ /5oscillation.
0 0 Theoscillationsatν =7/3areobservedtohavetwicethefrequency
Comparisonwithexperiment. TheoverallB-sweeptrace
of those at ν = 4, which is consistent with the theoretical model
between filling factors 2 to 3 of RL across two of the in- explained in the text. The oscillations at ν = 5/2 show beating
terferometers is shown in the bottom two panels of Fig. 1; betweenafastoscillationwithaperiodthatis1/5thatatν =4and
they clearly demonstrate fractional quantum Hall states at aslowonewiththesameperiodasatν =4
ν =7/3,8/3,and5/2. Thisoveralltraceisaveragedlocallyto .
defineabackgroundwhichwesubtractfromtherawR mea-
L
surement to make the oscillations clearer. Measurement of
these oscillation sets was repeated for multiple interferomet- ingν =7/3andintegerfillingfactorswerecarriedoutonthe
ricareas. The∆B periodshouldchangewithareaaccording threedifferentdevicesanddifferentpreparationsofthisstudy,
0
to our model. From the three devices used and the multiple assummarizedintherightpanelofFig.2.TheRLoscillations
preparations and gate values employed, the measured ∆B atν = 7/3consistentlyoccurattwicethefrequencyoftheir
0
periodsshowactiveareasrangingfrom0.1µm2to∼0.6µm2. respectiveintegerfillingfactoroscillationsoverthefullrange
Toputourpicturetotest,wefirstconsiderν = 4and7/3. ofdeviceareasstudied.Weconcludethattheassumptionsand
OscillationsofR withBareshownintheupperleftofFig.2. analysisoutlinedabovearevalid.
L
Theperiod∆B ofoscillationsisfoundtobesimilarnearin- Fig.3presentsthecomparisonbetweeninterferenceoscil-
0
tegerfilling2,3and4foreachdevice,consistentwiththisbe- lations of ∆R at ν = 5/2, 7/3, and ν = 4 observed in the
L
inganABoscillationandnotCoulombeffects[24]. Fromthe samesample/prepapration. (TheoverallB-sweeptraceofR
L
periodicity ∆B , we determine the active area of this prepa- for this interferometer is shown in the bottom right panel of
0
ration. (For instance, for the preparation displayed in Fig. 4, Fig. 1.) Sets of oscillations are shown with their respective
∆B ≈110G,fromwhichwededuceA≈0.36µm2.) Fouriertransforms. Onceagain,theν = 7/3oscillationsare
0
We now turn to ν = 7/3. The results for B-sweeps are observedathalfthemagneticfieldperiod ofthoseatν = 3.
showninFig.2. Oscillationsat7/3andintegerfillingfactors Interestingly,theoscillationsat5/2containahigherfrequency
are shown in Fig. 2 insets with their corresponding Fourier component,andtheFFTspectrumdemonstratesthepredom-
transforms, which show peaks. The 7/3 peak frequency is inant frequency is 5 times that of the integer oscillation fre-
twicetheν =4peakfrequency(orhalftheperiod),consistent quency. Thisvalueisconsistentwiththeexpectedoscillation
withtheanalysisabove. ThesameR measurementscompar- frequencyforexpression/suppressionofnon-Abeliane/4in-
L
4
in a series of samples with different interferometer sizes
and different sample preparations is further demonstrated in
Figs. 4, 5. Fig. 4, top panel, shows the B-sweep results for
one of the sample preparations in device area 2, focusing on
thesmallperiodresistiveoscillationscorrespondingtomulti-
ple parity changes in the enclosed e/4 quasiparticle number
near 5/2. The set of oscillations is measured with no adjust-
mentstothevoltagesofthequantumpointcontactsorcentral
top gates. If our model is correct, the five periods of oscil-
lation shown here represent ten parity changes. The B-field
rangenear5/2wherethisdataistakenismarkedintheover-
all R trace. The distinct resistive oscillations (black trace;
L
the blue trace is a coarse smoothing of the data) near 5/2 in
this preparation have a period ∆B ≈ 22G, compared to a
2
period at integer fillings of ∆B ≈ 110G. This is precisely
0
the same fivefold ratio shown in Fig. 3, once again in agree-
mentwithourmodel.Noteherethatsplittinginthe5/2peakis
notresolved,whichmaybeanindicationthatthefermionpar-
ityintheinterferometerisnotconstant,apossibilitydiscussed
above.Or,alternatively,sweepsthroughawiderB-fieldinter-
Figure4:Oscillationsatν =2(rightinset),ν =3(leftinset)andat valmaybenecessarytoobservethisslowoscillationinsome
ν =5/2(toppanel). FromtheFouriertransformsofRLvs. B(left samples/preparations. Wider sweeps may also reveal oscil-
panel),weseethattheoscillationperiodatν = 5/2is1/5aslarge lations due to transport by charge-e/2 quasiparticles, which
asatν = integer. Sometimesitshowsbeatingwithamorerapid
shouldhaveaperiod∆B·A=Φ /2(byessentiallythesame
oscillationwiththesameperiodasatν =2. 0
argumentasfor7/3). Theyarenotapparentinthemagnetic
fieldsweepsinFigs.3,4,eventhoughtheyareseeninside-
terferenceduetothechangingnumberofquasiparticleswith gatevoltagesweeps[15,16].
varying magnetic field. Moreover, the peak centered around
fivetimestheintegerfrequencyissplit,withthesplittingbe- Fig. 5 summarizes the principal result of this study. The
ing roughly twice the frequency observed at integer plateau. oscillation period in units of flux (i.e., ∆B2 ·A) at ν = 5/2
This corresponds to beats which are further consistent with measuredfordifferentsamples/preparationsisapproximately
theabovepredictionfortheinterplaybetweentheABandsta- independent of the device area derived from ∆B0. The ob-
tisticalcontributionsforanon-Abelianν = 5/2state. Other servedvaluesof∆B2·Aisshowntobeinreasonableagree-
suchdatasetsarepresentedinsupplementalmaterials. ment with the expected value of 8Gµm2 which corresponds
tothechangeintheparityofenclosedquasiparticles.
To conclude, this experiment provides the necessary com-
plementtopriormeasurements[15,16]whereABoscillations
wereexaminedasafunctionoftheactiveinterferometerarea
A controlled by the gate voltage. By sweeping the B -field
in these multiple area devices instead, the previous experi-
mental limitation coming from slow gate charging has been
avoided. Theresistanceoscillationsobservednearfillingfac-
torν =5/2inmultipledevisesshowaperiodconsistentwith
the additional magnetic field needed to add one quasihole to
theirrespective(different)activeareas(therebychangingthe
quasiparticleparity). Westressthatthepresenceofsuchape-
riodisindicativeofanon-Abeliannatureoftheν =5/2state.
Whilethisinterpretationisbasedonseveralassumptionsdis-
cussedearlier,usingtheν = 7/3FQHstateforcontrolmea-
surementssignificantlystrengthensourcase.
Figure 5: The oscillation period in units of flux is independent of
thedeviceareaandisapproximately8Gµm2 =Φ /5.Equivalently
0
(inset) the oscillation period in units of magnetic field is inversely RLW,CNandKSacknowledgethehospitalityofKITPsup-
proportionaltothearea. portedinpartbytheNSFundergrantPHY11-25915. CNand
KSaresupportedinpartbytheDARPA-QuESTprogram.KS
The observation of a small oscillation period at ν = 5/2 issupportedinpartbytheNSFundergrantDMR-0748925.
5
[1] D.C.Tsui,H.L.Stormer,andA.C.Gossard, Phys.Rev.Lett.
48,1559(1982).
[2] R.Willett,J.P.Eisenstein,H.L.Stormer,D.C.Tsui,A.C.Gos-
sard,andJ.H.English,Phys.Rev.Lett.59,1776(1987).
[3] W.Pan,J.-S.Xia,V.Shvarts,D.E.Adams,H.L.Stormer,D.C.
Tsui,L.N.Pfeiffer,K.W.Baldwin,andK.W.West,Phys.Rev.
Lett.83,3530(1999),cond-mat/9907356.
[4] J. P. Eisenstein, K. B. Cooper, L. N. Pfeiffer, and K. W. West,
Phys.Rev.Lett.88,076801(2002),cond-mat/0110477.
[5] G.MooreandN.Read,Nucl.Phys.B360,362(1991).
[6] M.Greiter,X.G.Wen,andF.Wilczek,Nucl.Phys.B374,567
(1992).
[7] C.NayakandF.Wilczek,Nucl.Phys.B479,529(1996),cond-
mat/9605145.
[8] N. Read and D. Green, Phys. Rev. B 61, 10267 (2000), cond-
mat/9906453.
[9] P.Bonderson,V.Gurarie,andC.Nayak,Phys.Rev.B83,075303
(2011),arXiv:1008.5194.
[10] C.deC.Chamon,D.E.Freed,S.A.Kivelson,S.L.Sondhi,and
X.G.Wen,Phys.Rev.B55,2331(1997),cond-mat/9607195.
[11] E.Fradkin,C.Nayak,A.Tsvelik,andF.Wilczek,Nucl.Phys.
B516,704(1998),cond-mat/9711087.
[12] S.DasSarma,M.Freedman,andC.Nayak,Phys.Rev.Lett.94,
166802(2005),cond-mat/0412343.
[13] A.SternandB.I.Halperin,Phys.Rev.Lett.96,016802(2006),
cond-mat/0508447.
[14] P.Bonderson,A.Kitaev,andK.Shtengel,Phys.Rev.Lett.96,
016803(2006),cond-mat/0508616.
[15] R.L.Willett,L.N.Pfeiffer,andK.W.West,PNAS106,8853
(2009).
[16] R.L.Willett,L.N.Pfeiffer,andK.W.West,Phys.Rev.B82,
205301(2010),arXiv:0911.0345.
[17] X.Wan,Z.-X.Hu,E.H.Rezayi,andK.Yang,Phys.Rev.B77,
165316(2008),arXiv:0712.2095.
[18] W. Bishara, P. Bonderson, C. Nayak, K. Shtengel, and J. K.
Slingerland,Phys.Rev.B80,155303(2009),arXiv:0903.3108.
[19] W. Bishara and C. Nayak, Phys. Rev. B 80, 155304 (2009),
arXiv:0906.0327.
[20] B.Rosenow, B.I.Halperin, S.H.Simon, andA.Stern, Phys.
Rev.B80,155305(2009),arXiv:0906.0310.
[21] R. L. Willett, M. J. Manfra, L. N. Pfeiffer, and K. W. West,
Appl.Phys.Lett.91,052105(2007),cond-mat/0703197.
[22] M.Levin,B.I.Halperin,andB.Rosenow,Phys.Rev.Lett.99,
236806(2007),arXiv:0707.0483.
[23] S.-S. Lee, S. Ryu, C. Nayak, and M. P. A. Fisher, Phys. Rev.
Lett.p.236807(2007),arXiv:0707.0478.
[24] R. L. Willett, M. J. Manfra, L. N. Pfeiffer, and K. W. West
(2013),unpublished.
6
SUPPLEMENTALMATERIALS coherencelengthismuchlonger,whichinturnenhancestheir
contribution to the coherent interference signal [S2, S4]. It
TheoreticalresultsforB-sweepand(3,3,1)state therefore remains a puzzle that period Φ0/2 oscillations are
notconvincinglyseeninourdata, particularlyinviewofthe
Inthissectionwecomparetheexpectedoscillationperiods factthate/2oscillationswereaprominentfeatureseeninthe
ofmagneticfieldsweepsforAbelianandnon-Abeliancandi- side-gate voltage sweeps [S5, S6]. One possible explanation
date states at ν = 5/2. As explained in the main text, the is the relatively narrow magnetic field window available for
netphaseaccumulationinaquantumHallinterferometercon- B-sweepsatν = 5/2, whichinturnmakesitdifficulttosee
sistsoftwocontributions: theABphaseduetothemagnetic longer-periodoscillations. Thiswindowislimitedbythenar-
fluxchangebyΦandthestatisticalphaseduetointroducing row width of the ν = 5/2 plateau immediately flanked by
N additionalquasiparticlesofchargee∗ totheinterferome- thereentrantcompressibleintegerstates[S7,S8]–seemore
e∗
terarea: on this in the next section. It is also worth mentioning that
the data presented here was measured at T ≈ 20mK, while
∆γ =2π(Φ/Φ )(e∗/e)−2θ N (S1)
0 e∗ e∗ previouslyreportedside-gatevoltagesweepswereperformed
Ifweassumethattheactiveareaoftheinterferometerremains at T ≈ 25mK and higher; lowering the temperature should
constant throughout a magnetic field sweep, these two con- leadtothesuppressionofthee/2contributiontooscillations.
tributions are not independent; N = (νΦ/Φ )/(e∗/e) and TestingthisargumentbymeasuringB-sweepsathighertem-
e∗ 0
hence peraturesandfordifferentdevicesizesisanimportantfuture
direction.
∆γ =(Φ/Φ )[2π(e∗/e)−2θ (νe/e∗)]. (S2)
0 e∗
At ν = 5/2, the smallest charge excitation is an e/4 quasi-
hole/quasiparticeirrespectiveoftheexactnatureofthestate. Additionalinteger,7/3,and5/2datasets;5/2splitting
Since the smallest charge carriers are expected to dominate
tunneling at the low temperature, weak tunneling limit [S1,
Additional sets of data showing Fourier transforms of os-
S2](or,inthecaseoftheanti-Pfaffianstate,tovarywithtem-
cillationsatinteger,7/3,and5/2fillingfactorsfromdifferent
perature in the same way as transport due to e/2 quasiparti-
sample preparations are shown in Fig. S1. Note that the 5/2
cles), we will first focus on these e/4 excitations. As shown
peakissplitandiscenteredat5Φ forallthedatasetsinthis
0
inthemaintext,iftheν =5/2stateisnon-Abelian(whether
figure.
itisMoore–Readoranti-Pfaffian),thisshouldmanifestitself
Inroughlyhalfthedatasetsexamined,wewereabletore-
via a Φ /5 oscillation period corresponding to the even-odd
0 solvethepredictedsplittingofthe5/2FFTpeakatfivetimes
effect. An additional Φ period of purely Abelian nature is
0 the fundamental integer oscillation frequency (1/Φ ). This
0
alsoexpected,althoughitmaybewashedoutbyfermionpar-
splitting at 5/Φ to 5/Φ ±1/Φ , due to modulation of the
0 0 0
ityfluctuations.
non-AbelianoscillationbythelowfrequencyAB/anyonicos-
If, on the other hand, the ν = 5/2 state is an Abelian
cillations at ν = 5/2, can be seen only when we take the
(3,3,1)state[S3],theΦ /5oscillationperiodshouldnotbe
0 FouriertransformtheresistanceoverasubstantialrangeinB-
observed–thereisnoeven-oddeffectinthiscase.AnAbelian
field: suchdataisdisplayedinFig.3ofthemaintext,andin
phasecanbecalculatedusingEq.(2)withaslightcaveat.The
Fig.S1.
statisticalangleθ canbeeither3π/8or−π/8,dependingon
Some B-field sweeps in this study ran over only a small
whether the spin of a quasiparticle going around the inter-
range of B near 5/2 filling to focus on measuring the small
ferometer and that of a quasiparticle inside the loop are the
period (Φ /5) oscillations attributed to non-Abelian e/4 ex-
sameoropposite. Sincethe(3,3,1)stateisspin-unpolarized, 0
pression/suppression: theseresultsareshowninFig.4ofthe
excitations of both spins may carry charge around the inter-
maintext,andhereinFig.S2. Theothermeasurements,cov-
ference loop. Hence we can use the average value of the
eringalargerrangeinB,notedabove,facilitateresolutionof
statistical angle, i.e. π/8, per quasiparticle (or, more pre-
the splitting at 5/2. However, the range of B-field of either
cisely, π/4 per pair with opposite spins). This yields ∆γ =
of these oscillations is limited around 5/2 by the presence of
2π(Φ/Φ )[(1/4)−(1/8)×10]=−2π(Φ/Φ ).Theresult-
0 0 reentrantintegerstates[S7,S8]adjacentto5/2fillingonboth
ingoscillationperiodisΦ .
0 thehigh-andlow-fieldsides. Thisisamoreprominenteffect
So far in our discussion we have neglected other types of
inthelargerareadevices.
charged excitations, particularly the charge e/2 excitations
that are also expected at ν = 5/2. These excitations are
always Abelian and their statistical angle is θ = π/2, ir-
respective of the nature of the state. Eq. (2) then yields Gatesweepexamplesformultipledevicesizes
∆γ = −4π(Φ/Φ ) implying a Φ /2 periodicity. While
0 0
backscattering of these quasiparticles at quantum point con- When side gate sweeps are applied rather than B-field
tacts should be suppressed by comparison to e/4 quasiparti- sweeps,themultipledevicesofdifferentareasexaminedhere
cles (except in the anti-Pfaffian state), it is likely that their displaythesameABpropertiesat5/2asobservedinprevious
7
studies[S5,S6],namelyalternationofe/4ande/2periodos-
cillations. See Fig. S3. This alternation is attributed to the
sidegateexcursionchangingnotonlytheenclosedmagnetic
fluxnumberbutalsothelocalizednon-Abeliane/4quasiparti-
clenumber. ThereareoscillationscorrespondingtoABinter-
ference of e/4 particles when an even number of e/4 quasi-
particles are enclosed, and the e/2 oscillations are apparent
when that number is odd. According to theoretical predic-
tions[S2],thee/2oscillationsarepervasivebutaremoreeas-
ily observed when the larger amplitude e/4 oscillations are
suppressed. The data of the Figure show this alternation for
allthreerudimentarydevicesizes. Furtherdemonstrationand
detailsofthesemeasurementscanbefoundinreference[S9].
[S1] P. Fendley, M. P. A. Fisher, and C. Nayak, Phys. Rev. B 75,
045317(2007),cond-mat/0607431.
[S2] W. Bishara, P. Bonderson, C. Nayak, K. Shtengel, and J. K.
Slingerland,Phys.Rev.B80,155303(2009),arXiv:0903.3108.
[S3] B.I.Halperin,Helv.Phys.Acta56,75(1983).
[S4] X.Wan,Z.-X.Hu,E.H.Rezayi,andK.Yang,Phys.Rev.B77,
165316(2008),arXiv:0712.2095.
[S5] R.L.Willett,L.N.Pfeiffer,andK.W.West,PNAS106,8853
(2009).
[S6] R.L.Willett,L.N.Pfeiffer,andK.W.West,Phys.Rev.B82,
205301(2010),arXiv:0911.0345.
[S7] J.P.Eisenstein,K.B.Cooper,L.N.Pfeiffer,andK.W.West,
Phys.Rev.Lett.88,076801(2002),cond-mat/0110477.
[S8] J. S. Xia, W. Pan, C. L. Vicente, E. D. Adams, N. S. Sulli-
van,H.L.Stormer,D.C.Tsui,L.N.Pfeiffer,K.W.Baldwin,
and K. W. West, Phys. Rev. Lett. 93, 176809 (2004), cond-
mat/0406724.
[S9] R. L. Willett, M. J. Manfra, L. N. Pfeiffer, and K. W. West
(2013),unpublished.
8
FigureS1: Fourseparatesamplepreparationsshowingtransportthroughthedevice(toppanels),andRLoscillations(leftpanels)atintegral,
7/3,and5/2fillingfactors. TherighthandpanelsshowrespectiveFFTsforthosefillingfactors. Thethreeverticalbluelinesmarktheinteger
frequency1/Φ ,2/Φ ,and5/Φ .The7/3oscillationfrequencyisconsistentlyattwicethatoftheintegerfilling,andthe5/2peakcomplexis
0 0 0
centerednear5timestheintegerfrequency.DataaretakenatT ≈20mK.
9
FigureS2: ABoscillationsatintegerfillingusedtodeterminethearea(leftcolumn)andthecorrespondingtransportnearν = 5/2(center)
showingsmallperiodoscillationsconsistentwithexpression/suppressionofnon-Abeliane/4interference,alongwithFFTsofbothspectra
(right)foraseriesofinterferometerdevicesofdifferentareas. Inourmodel,themagneticfield∆B necessarytoaddtwoe/4quasiparticles
2
isdeterminedfrom∆B /∆B =0.2,or∆B A≈8Gµm2.Theblacklineisanaverageofseveral(typically,eight)B-fieldsweeps,andthe
2 0 2
bluelineshowsthesamedatasmoothedbylocalaveraging. IntheFFTpanelthetwoverticalbluelinesdifferbyafactorof5infrequency.
DataaretakenatT ≈20mK.
10
FigureS3:Magneto-transportandgate-voltagesweeposcillationsatν =5/2.Alldevicesusedherearefabricatedfromthesameheterostruc-
turewafer,withrepresentativebulktransportshownintheleftpanel.Thecentralcolumnshowsrepresentativetransportthrougheachdevice;
noteprominenceoftheν =5/2minimumandthepresenceofFQHEstateatν =7/3intheselongitudinalresistancetraces. Therighthand
columnshowslongitudinalresistancechangewithsidegate(2)sweepatν = 5/2;eachdevicedemonstratesoscillationsconsistentwiththe
ABeffectatperiodscorrespondingtochargee/4.Themarkedverticallinesoftheseperiodsarederivedfromsimilarmeasurementsatintegers
and7/3fillings,definingtheperiodcorrespondingtoe/4charge. Ineachdevicethepreviouslyobserved[S5,S6]alternationofe/4ande/2
periodsisaffirmedinlargesidegatevoltageexcursions.TemperatureinalldataisT ≈20mK.