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Aharonov–Bohm effect inthenon-Abelian quantum Hallfluid Lachezar S. Georgiev1 and Michael R. Geller2 1InstituteforNuclearResearchandNuclearEnergy, 72TsarigradskoChaussee, 1784Sofia, Bulgaria 2Department of Physicsand Astronomy, University of Georgia, Athens, Georgia 30602-2451 (Dated:November8,2005) 6 0 The ν = 5/2 fractional quantum Hall effect state has attracted great interest recently, both as an arena 0 toexplorethephysicsof non-Abelianquasiparticleexcitations, and asapossiblearchitecturefortopological 2 quantuminformationprocessing.HereweusetheconformalfieldtheoreticdescriptionoftheMoore–Readstate toprovidecleartunnelingsignaturesofthisstateinanAharonov–Bohmgeometry. Whilenotprobingstatistics n directly, the measurements proposed here would provide a first, experimentally tractable step towards a full a J characterizationofthe5/2state. 7 PACSnumbers:71.10.Pm,73.43.–f,03.67.Lx 2 ] ThequantumHallfluidhasbecomeaparadigmofstrongly ThemostspectacularpredictionofMooreandReadisthat l l a correlated quantum systems [1, 2]. A combination of two- thequasiparticleexcitationsofthe5/2statearenon-Abelian: h dimensional confinement and strong magnetic field leads to An excited state of 2n quasiholeshas degeneracy2n 1, and − - rich phenomena driven by electron-electron interaction and the braiding of their worldlines generates elements from the s e disorder.Assuch,theuseoftraditionaltheoreticaltechniques orthogonalgroupSO(2n)actingonthedegeneratemultiplet. m such as many-bodyperturbationtheory has had only limited This property follows from the correlation function (3), and . success, and, in an approachpioneered in 1983 by Laughlin alsofromanalternativepictureofthestate(1)asaBCScon- t a [3], some of the most important advances have been made densate of l = 1 pairs of composite fermions [16], which m − by correctly guessing the many-particlewave function. The supportsexoticnon-Abelianvortexexcitations[17,18]. - statesdescribedbyLaughlin,andtheirgeneralizations[1,2], d In addition to its intrinsic interest as a system to explore haveHallconductancesσ given(inunitsofe2/h)byfrac- n xy non-Abelianquantummechanics,DasSarmaetal. [19]have o tions with odd denominatorsonly. The charged excitations, proposed to use a pair of antidots at ν = 5/2 to construct c which have fractional charge [4, 5, 6] and statistics [7], are a topological quantum NOT gate, building on an intriguing [ abelian anyons. In 1987, however, evidence for an even- idea by Kitaev to use the transformationsgenerated by non- 2 denominatorquantizedHallstateatν = 5/2wasdiscovered Abelian anyon braiding for fault-tolerantquantum computa- v inthefirstexcitedLandaulevel[8],andthestateisnowrou- tion [20]. Unfortunately, the braiding matrices generated by 6 tinelyobservedinultrahigh-mobilitysystems[9,10,11].Mo- (3)arenotcomputationallyuniversal.Butbygeneralizingthe 3 tivatedinpartbythissurprisingresult,MooreandRead(MR) 2 pairing present in ΨMR to clusters of k > 2 particles, Read introducedtheground-statetrialwavefunction[12] 1 andRezayi[21]haveproposedahierarchyofgroundandex- 1 citedstates—CFT correlatorsofparafermioncurrentsreduc- 1 5 ΨMR(z1,z2,...,zN)=Pf (zi zj)2 (1) ingto(2)and(3)whenk = 2. Theseparafermionstatesare 0 (cid:18)zi−zj(cid:19)i<j − alreadycomputationallyuniversalatk =3. / Y t a for N (even) electrons in a partially occupied Landau level Computation with non-Abelian quasiparticles will be in- m withcomplexcoordinatesz ,wherethefirsttermisthePfaf- credibly challengingexperimentally. Demonstratingthat the i - fian and the standard Gaussian factor is suppressed. Exact actual 5/2 state is in the universality class of (1), and that d diagonalizationstudies[13,14]indicatethattheexactground thequasiparticlesareindeednon-Abelian,arenecessaryfirst n o stateatν =5/2iscloseto(1). steps. Direct interferometric and thermodynamic probes c MooreandReadalsoconstructedexcited-statewavefunc- of the non-Abelian statistics have been proposed recently : tions. By identifying(1) with a two-dimensionalchiralcon- [22, 23]. Here we propose an even simpler test of the MR v i formalfieldtheory(CFT)correlationfunction state, in a similar antidot geometry. Although the tunnel- X ingmeasurementproposedbelowdoesnotdirectlyprobethe r ΨMR(z1,z2,...,zN)= ψ(z1)ψ(z2) ψ(zN) (2) non-Abelian nature of the excitations, it can distinguish be- a h ··· i tweenthetunnelingofordinaryelectronsandthatofabosonic ofcharge1fermionfieldsψ :ei√2φ:ψ ,whereφisau(1) ≡ M charge1excitationwecallκ,whichisallowedbytheCFTand boson[15]andψ aneutralMajoranafermion,MRproposed M whichhaslowerscalingdimension.Andtheexistenceoftwo CFT-basedexcitedstatesoftheform inequivalentcharge1 excitationsis itself a fingerprintof the ψqh(η1) ψqh(η2n)ψ(z1) ψ(zN) . (3) non-Abeliannatureofthefundamentalquasiholeψqh. Exper- h ··· ··· i imentalobservationoftheelectronandκtunnelingchannels Hereψ :e(i/√8)φ:σ isthefundamentalcharge1/4quasi- wouldgiveconfidenceintheMRstate,thepowerfulCFTap- qh ≡ holefieldoftheCFT,withσthechiralspinfieldofthecritical proach, and, by extension, the k > 2 parafermionhierarchy Isingmodel. necessaryforuniversaltopologicalquantumcomputation. 2 needtoconsider.Weemphasizethatκisanallowedexcitation of the Pfaffian state satisfying the parity rule [29, 30]. The x I 1 I scalingdimensionofκis1,andifittunnelsitwilldominate L R electrontunnelinginthelow-temperaturelimit,leavingaclear (cid:80)L (cid:81)(cid:32)5/2 (cid:77) (cid:81)(cid:32)5/2 (cid:80)R experimentalsignature. Weturnnowtoacalculationofthesource-draincurrent x µ 2 I(V,T,ϕ)=I (V,T)+I (V,T) cos +ϕ 0 AB ∆ǫ (cid:16) (cid:17) asafunctionofvoltageV andtemperatureT,whichaccord- ingtoouranalysiscanbedecomposedintoflux-independent FIG.1:(coloronline).Antidotinsideaν =5/2Hallbarwithweak and period-one oscillatory Aharonov–Bohm (AB) compo- tunnelingpointsatx1andx2,threadedbyanABfluxϕ. nents. Here µ is the mean electrochemical potential of the contacts,∆ǫ 2πv/Listhenoninteractinglevelspacingon ≡ theantidotwithedgevelocityv andcircumferenceL, andϕ The geometry we consider is illustrated in Fig. 1. The istheABfluxinunitsofh/e. ν = 1/3realizationofthissystemwasconsideredbyoneof TheHamiltonianinthestrong-antidot-couplingregimeis uspreviously[24],andadualconfiguration,thedoublepoint- contactinterferometer,wasstudiedpreviouslybyChamonet al. [25]. In the strong-antidot-couplingregime pictured, the H =HL+HR+δH, with δH = ΓiBi+Γ∗iBi† . edgestates,indicatedbythearrows,arestronglyreflectedby iX=1,2(cid:16) (cid:17) (5) theantidot.Thisregimecanberealizedexperimentallyintwo The Hamiltonians for the uncoupled right- and left-moving physically distinct ways: (i) The Hall fluid can be pinched edgestatesoflengthL sys offnearthe x , leaving largetunnelingbarriersor “vacuum” i regions. Onlyelectronscantunnelthroughthesevacuumre- H = 2πv L c and H = 2πv L¯ c R 0 L 0 gions.(ii)Thesystemcanbeginintheweak-antidot-coupling Lsys − 24 Lsys − 24 (cid:16) (cid:17) (cid:16) (cid:17) regime, where current from the upper edge tunnels through are given by the zero modes of the CFT stress tensors T(z) the antidot (acting as a macroscopic impurity) to the lower andT¯(z¯), one,andthetemperatureisthenlowered.Theweaktunneling regime, which permits quasiparticletunneling, is unstable at dz dz¯ L zT(z) and L¯ z¯T¯(z¯) low temperatures, as in the ν = 1/3 quantum point contact 0 ≡ 2πi 0 ≡ 2πi I I [26], and the system then flows to the stable strong-antidot- whichsatisfytheVirasoroalgebrawithcentralchargec=3/2 coupling fixed point, which can be described by an effective [29,30,31]. Then weak-tunnelingtheory[27]. Intheν = 1/3case, thiseffec- tive theory contains electron tunneling only, and is identical 1 ∞ ∞ tcoomtheastofrfocmastehe(i)e.xTachteBmeothsetdAranmsaatzticsoillluutsiotrnattihoenochfitrhailsLfauct-t L0 = 2J02+n=1J−nJn+n=1(n− 21)ψ−Mn+21ψnM−21, X X tinger liquid model for a ν = 1/3 point contact containing and similarly for L¯ . The Laurent mode expansion of the 0 only charge 1/3 quasiparticles, which nonetheless describes Majorana field with antiperiodicboundaryconditionson the electron-liketunnelingfaroffresonance[28]. cylinder(z =e2πix/Lsys)is In principle, the effective weak-tunneling theory for case dz (ii)canbedifferentthanthatof(i),andwewillconsiderthis ψ (z)= ψM z n with ψM = zn 1ψ (z). possibilityhere. Because thereis noexactsolutionavailable M n Z n−12 − n−21 I 2πi − M fortheν = 5/2quasiparticletunnelingmodel,wewillguess X∈ the form of the effective theory. Guided by the reasonable Theu(1)currentJ(z) i∂zφhasmodeexpansion ≡ condition that any charge 1 excitation of the CFT preserv- dz ing the stability of the fixed pointcan potentially tunnel(the J(z)= Jnz−n−1, with Jn = znJ(z). 2πi charge requirement introduced to recover the ν = 1/3 re- n Z I X∈ sult),weneedtoconsideraclusterofatleastfourfundamental HereJ /√2andJ¯/√2aretheusualg = 1/2Luttingerliq- 0 0 quasiholesψ . Accordingtothefusionruleσ σ =1+ψ qh × M uidnumbercurrentsNRandNL. Theoperators [29,30],theproductψ ψ ψ ψ =ψ+κyieldstwo qh qh qh qh × × × distincttunnelingobjects,theelectron/holeψ,andacharge1 Bi ≡ψL(xi)ψR†(xi), i=1,2, boson enteringEq.(5)aretunnelingoperatorsactingatpointsx in i κ :ei√2φ: (4) Fig. 1. The fields ψ and ψ appearingin B dependon the L R ≡ tunnelingobject. Thetunnelingamplitudesatx andx are 1 2 whosequantumnumbersaresummarizedinTableI.Thereare alsoexcitationslessrelevantthantheelectronthatwedonot Γ =Γeiπ(µ/∆ǫ+ϕ) and Γ =Γe iπ(µ/∆ǫ+ϕ), 1 2 − 3 istakenwithrespecttotheHamiltonian TABLE I: Some chiral conformal fields and their quantum num- bers. ∆c and ∆0 are the scaling dimensions in the charged and neutral sectors, ∆ ≡ ∆c + ∆0 is the total CFT dimension, and H H +H µ N µ N . θ≡2π∆(mod2π)isthestatisticsangle. 0 ≡ R L− R R− L L field charge ∆c ∆0 ∆ θ/π The finite-temperature correlation function X (t) is com- quasiholeψqh 14 116 116 81 14 putedinthreesteps: (i)First,itissplitintoprodiujctsoffinite- κparticle 1 1 0 1 0 temperature correlation functions (with chirality ) of the Majoranaψ 0 0 1 1 1 ± electronψM 1 1 122 223 1 fthoermrmhaψlc±†o(rxre,lta)tψio±n(fxu′n,ct′t)ioiβn;s(airie)oTbhteanintehdeaseszoenreo--dteimmepnesraiotunrael correlation functions (after Wick rotation to imaginary time τ)onacylinderwithcircumferenceL v/(k T);(iii)Fi- T B where, with no loss of generality, Γ can be taken to be real. nally,wemapthecylindertothecomplex≡planebytheconfor- ThebareamplitudeΓdependsonmicroscopicdetailsandtype mal transformationz = e2π(ivτ±x)/LT where, for primary oftunnelingobject. fieldswithscalingdim±ension∆, ThetunnelingcurrentI(V,T,ϕ)canbecalculatedbylinear responsetheoryalongthelinesofRefs.[24]and[25],leading to ψ†(z)ψ (z′) =(z z′)−2∆ for z > z′ . (6) h ± ± i − | | | | I (V,T) = 4eΓ2ImX (ω =eV) and 0 11 Undertheconformalmapachiralprimaryfieldtransformsas I (V,T) = 4eΓ2ImX (ω =eV), AB 12 ψ (x,τ) (2π) 1/2[i dz ]∆ψ (z),andbygoingback to±realtim→eweob−tain d(ivτ±x) ± where X (ω) is the Fouriertransformof the responsefunc- ij tion ( iπ/L )4∆ Xij(t)≡−iθ(t)h[Bi(t),Bj†(0)]iβ, hψ±†(x,t)ψ±(0)iβ = 2πsh2∆[±π(x Tvt iε)/LT], (7) ∓ ± withBi(t) eiH0tBie−iH0tandthethermalaverage ≡ whereεisapositiveinfinitesimalrequiredby(6). Finally,the A = trAe−βH0 desiredresponsefunctionis h iβ tre−βH0 4∆ 2∆ θ(t) π − X (t)= Im sh[π(x x +vt+iε)/L ] sh[π(x x vt iε)/L ] . (8) ij −2π2 L i− j T i− j − − T (cid:18) T(cid:19) (cid:26) (cid:27) When2∆isaninteger,whichwillbethecasehere,thetrans- Weexpectthatboththeelectronsandκparticleswillcon- formX (ω)hasaninfinitenumberofpolesoforder2∆and tributetotheobservedtunnelingcurrent,aswillthefilledLan- ij canbeobtainedbyresiduesummation.Notethatthelocalre- daulevel. Ifonlyelectronstunnel,theywillcontribute sponsefunctionwhichdeterminesthedirectcurrentI could 0 beobtainedasthelimit X (ω)= lim X (ω). 11 12 |x1−x2|→0 e γ2∆ǫ T 4 eV T 2 eV 3 eV 5 I(el) = el 64 2π +20 2π + 2π , (9) 0 h240π2 T ∆ǫ T ∆ǫ ∆ǫ ( (cid:18) 0(cid:19) (cid:18) (cid:19) (cid:18) 0(cid:19) (cid:18) (cid:19) (cid:18) (cid:19) ) and e2γ2∆ǫ (T/T )3 πeV 2 T 2 T πeV πeV T T πeV I(el) = el 0 2 +2 1 3cth2 sin +6 cth cos , AB h π sh3(T/T0)(cid:26)(cid:20) (cid:18) ∆ǫ (cid:19) (cid:18)T0(cid:19) (cid:18) − (cid:18)T0(cid:19)(cid:19)(cid:21) (cid:18) ∆ǫ (cid:19) (cid:18) ∆ǫ (cid:19)T0 (cid:18)T0(cid:19) (cid:18) ∆ǫ (cid:19)(cid:27) (10) 4 whichisidenticaltothatoftheg =1/3chiralLuttingerliquid I(FL) =(e2/h)2γ2 V and 0 FL [24,25]. Here e ∆ǫ (T/T ) πeV T0 ≡~v/πkBL, (11) IA(FBL) = h2γF2L π sh(T/T00)sin(cid:18) ∆ǫ (cid:19). (12) isatemperaturescaleassociatedwiththelevelspacingofthe Thecontributionfromtheκchannelaloneis antidotedgestate ofvelocityv andcircumferenceL. Above T0, the thermal length LT becomes smaller than L, and the I(κ) = e24γκ2 T 2V 1+ 1 eV 2 , (13) AB oscillations become washed out. γ Γ/(vL2∆ 1) is 0 h 3 T 4 πk T ≡ − (cid:18) 0(cid:19) " (cid:18) B (cid:19) # a dimensionless tunneling amplitude. Each of the two filled Landau levels also contributes a Fermi liquid (FL) current and e2γ2∆ǫ (T/T )2 T T eV eV eV I(κ) = κ 0 2 cth sin π 2π cos π . (14) AB h π sh2(T/T0)(cid:26) (cid:18)T0(cid:19) (cid:18)T0(cid:19) (cid:18) ∆ǫ(cid:19)− ∆ǫ (cid:18) ∆ǫ(cid:19)(cid:27) TheoscillationsoftheABcurrentsasfunctionsofthevoltage h] for the three channels at temperature T = T0 are shown in 22 |e/ 1.0 1/2 Fermi liquid) FigA.n2.experiment will observe these two or three transport nits [2| 00..89 ==13 / 2 ((e lbeoctsroonn)) channels in parallel, the contribution of each determined by u n thebareamplitudesγκ,γel,andγFL.However,theκchannel 2 i 0.7 will always dominatethe electron channelin the low-energy =5/ 0.6 limit. T) for 0.5 (B 0.4 A F∆IG=.21:(κ(ctoulnonreolinnlgin)ea)sTahfeunActhiaornoonfovth–eBaophpmliecduvrroelntatgIeAVB(cVo,mTp)arfeodr ce G 0.3 n to∆ = 3/2(electrontunneling)reducedbyafactorof100andthe cta 0.2 chiralFermiliquidcurrent(∆=1/2)multipliedbyafactorof10at u d T =T0. on 0.1 C 0.0 0 1 2 3 4 5 Temperature T [T0] 80 T/T0=1 FIG.3: (coloronline). Aharonov–Bohm conductances forelectron 60 andκtunnelingatν = 5/2. TheFermiliquidconductance isalso ] / shown. 40 2 | 2| h) 20 e/ V) [( 0 nt I(AB-20 Curre -40 wrehspereectGiv0elayn.dFoGrAthBeaκrechthanendeilrethcetyanredatdhe AB conductances, -60 =1/2 (Fermi Liquid) =1 ( boson) -80 =3/2 (electron) 0 2 4 6 8 10 12 14 16 18 20 Voltage V in units of V0= /e e2 4γ2 T 2 ThelinearconductanceG (dI/dV) forthesechan- G(κ)(T) = κ , nelstakestheform ≡ V→0 0 (cid:18)h(cid:19) 3 (cid:18)T0(cid:19) e2 (T/T )2 T T µ G(κ)(T) = 4γ2 0 cth 1 , G(φ,T)=G0(T)+cos 2π ∆ǫ +φ GAB(T), AB (cid:18)h(cid:19) κsh2(T/T0)(cid:20)(cid:18)T0(cid:19) (cid:18)T0(cid:19)− (cid:21) h (cid:16) (cid:17)i 5 ContractNo.F-1406. MRGwassupportedbytheNSFunder TABLE II: Asymptotic tunneling conductance. T0 is a crossover grantsDMR-0093217andCMS-040403. temperaturedefinedinEq.(11). eV ≪kBT ≪kBT0 eV ≪kBT0 ≪kBT Fermiliquid G0 ∼const G0 ∼const [1] R.E.PrangeandS.M.Girvin,eds.,TheQuantumHallEffect ∆= 12 GAB ∼const GAB ∼Te−T/T0 (Springer-Verlag,Berlin,1990),2nded. κparticle G0 ∼T2 G0 ∼T2 [2] S. 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Finally,wealsonotethatthelimitofasingle [24] M.R.GellerandD.Loss,Phys.Rev.B56,9692(1997). ν = 5/2quantumpointcontact(inthestable,strongtunnel- [25] C.Chamon,D.E.Freed,S.A.Kivelson,S.L.Sondhi,andX.G. ingregime)followsfromourresultsbylettingT ;the Wen,Phys.Rev.B55,2331(1997). 0 → ∞ [26] K.Moon,H.Yi,C.L.Kane,S.M.Girvin,andM.P.A.Fisher, electroncontributioninthiscaseagreeswiththatfortunneling Phys.Rev.Lett.71,4381(1993). betweenaFLandν =5/2edgestate[32]. [27] Thisassumes,ofcourse,thatthereisastablestrong-tunneling Inconclusion,wehaveusedtheCFTpictureofMooreand fixedpointhere,andthattherearenootherstablefixedpoints. Read [12] to calculate the AB tunneling spectrum in a ν = [28] P.Fendley, A.W.W.Ludwig, andH.Saleur, Phys.Rev.Lett. 5/2antidotgeometry. Observingtheelectronandpossiblyκ 74,3005(1995). transport channels will give evidence in support of the non- [29] A.Cappelli,L.S.Georgiev,andI.T.Todorov,Commun.Math. Abeliannatureofthe5/2state. Phys.205,657(1999). [30] L.S.Georgiev,Nuc.Phys.B651,331(2003). WethankAdySternforusefuldiscussions. LSGhasbeen [31] M.MilovanovicandN.Read,Phys.Rev.B53,13559(1996). partiallysupportedbytheFP5-EUCLIDNetworkProgramof [32] N.ReadandE.Rezayi,Phys.Rev.B54,16864(1996). the EC under Contract No. HPRN-CT-2002-00325 and by theBulgarianNationalCouncilforScientificResearchunder