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Probing Non-Abelian Statistics in ν = 12/5 Quantum Hall State K.T. Law Department of Physics, Brown University, Providence, Rhode Island 02912, USA ThetunnelingcurrentandshotnoiseofthecurrentbetweentwoFractionalQuantumHall(FQH) edges in the ν = 12/5 FQH state in electronic Mach-Zehnder interferometer are studied. It is shown that the tunneling current and shot noise can be used to probe the existence of k = 3 parafermion statistics in the ν =12/5 FQH state. More specifically, thedependenceof the current on the Aharonov-Bohm flux in the Read-Rezayi state is asymmetric under the change of the sign 8 0 of the applied voltage. This property is absent in the Abelian Laughlin states. Moreover the Fano 0 factor can exceed 12.7 electron charges in the ν =12/5 FQH state . This number well exceeds the 2 maximum possible Fano factor in all Laughlin states and the ν =5/2 Moore-Read state which was shown previously to be e and 3.2e respectively. n a J I. INTRODUCTION from this limitation, as long as the fluctuations are suf- 5 ficiently slow compared with the quasiparticle tunneling 1 rate. WebelievethatasimilardeviceintheFQHregime Particles other than bosons and fermions can exist in can be realized with higher magnetic fields and will pro- ] twodimensions. Onepossibilityisthatwhenoneparticle l videapracticalwaytoprobetheexistenceofnon-Abelian l makes a circle around another particle, the total many- a anyons. h particlewavefunctionacquiresanon-trivialphasefactor - eiφ where φ can be a real number, not constrainedto be In this paper, we suggest that the tunneling current s 0 or 2π. Those particles are called Abelian anyons.1,2 A and shot noise of the current between two FQH edges in e m more exotic situation can happen when the state of the MZIgeometrywithtwoquantumpointcontacts(QPCs) systemisdescribedbyamulti-componentstatevectoras canbeusedtoprobetheexistenceofnon-Abeliananyons . t the positions of the particles are specified. The actions in ν =12/5 FQH state. We show that: i) The tunneling a m of braiding a particle around another are represented by current can be reduced to a sinusoidal form I(Γ1,Γ2) = unitary matricesacting onthe state vector. If the braid- I (Γ ,Γ )+I (Γ ,Γ )cos(2πΦ +const)whenΓ Γ , - 0 1 2 Φ 1 2 a 2 ≪ 1 d ing matrices do not commute with eachother, the parti- where Γ and Γ are the tunneling amplitudes at the 1 2 n cles under study are called non-Abelian anyons.3,4 QPCsshownin Fig. 1 andΦ denotes the magnetic flux a o BothAbelianandnon-Abeliananyonsareproposedto enclosed by the two FQH edges. There is a scaling rela- c [ existinFractionalQuantumHall(FQH)systems. Arovas tion between I0 and IΦ. The scaling exponent b, defined et al.5 show that the fractionally charged quasiparticles as IΦ(Γ1,Γ2)∼[I0(Γ1,Γ2)−I0(Γ1,0)]b, equals to 5/2 in 3 inLaughlinstatesobeyfractionalstatisticswithφ=2πν the ν = 12/5 Read-Rezayi state. This value is the same v where ν is the filling factor. In the paper by Moore and as the one in ν = 1/5 Laughlin state but different from 5 Read,6 itwasarguedthatquasiparticlesinFQHsystems b=2 in the ν =5/2 Moore-Read state.24,26 ii) The flux 9 9 with filling factor ν = 5/2 possibly obey non-Abelian dependence of the tunneling current is asymmetric un- 3 statistics. Later, Read and Rezayi suggested that quasi- derthechangeofthesignoftheappliedvoltage. iii)The . particles in the more recently observed ν = 12/5 FQH Fanofactor,whichisdefinedastheratiobetweentheshot 7 0 state7,8,9 may also obey non-Abelian statistics but with noise and the tunneling current, can be as large as 12.7 7 an even richer structure,10 which can support universal in units of one electron charge in the ν =12/5 state. As 0 topological quantum computing.11 it was previously shown in Ref. 27, the maximum Fano : factor is one electron charge and 3.2 electron charge for v Severaltheoreticalproposalshavebeenmadeto probe the Laughlin states and the ν = 5/2 Moore-Read state i Abelianandnon-AbeliananyonsinFQHsystems. Using X respectively. The last two properties, the asymmetric aneleganttwopointcontactinterferometerwithananti- r I-V curve and larger than one electron charge Fano fac- a dotinthemiddleofaquantumhallbartoprobeAbelian tor,aredirectconsequencesofthenon-trivialfusionrules anyons in Laughlin states was initially proposed in Ref. and braiding rules of the quasiholes. Their observations 12. Later, the same idea was extended to probing non- wouldprovideexperimentalevidence for the existence of Abelian quasiparticles.13,14,15,16,17,18,19 Other proposals non-Abelian statistics in the ν =12/5 FQH state. are also available,20,21,22,23 but there are no experimental realizations of those ideas so far. Recently an elec- Calculating the tunneling current and shot noise in tronicMach-Zehnderinterferometer(MZI)intheinteger the MZI geometry is more complicated than in the sim- quantumHallregime,whoseschematicdiagramshownin ple Fabry-Perot geometry.12 This is due to the fact that Fig. 1, has beenfabricatedatthe WeizmannInstitute.25 the number of quasiparticles trapped inside the interfer- Unlike the Fabry-Perot interferometer proposed in Ref. ometer changes when a quasiparticle tunnels from one 12 in which the interference pattern can be destroyed edge to the other. As a result, the statistical phase due by fluctuations of the number of quasiparticles trapped to the quasiparticles enclosed by the interference paths inside the interferometer, the MZI interferometer is free changes. Hence,thetunnelingprobabilitiesofthetunnel- 2 ingquasiparticleschangesaccordinglyaftereverytunnel- ing event.24,26,27 However,we show that in the ν =12/5 FQH state, quasiparticles trapped inside the interferometer can fuse together to form only ten non-equivalent classes of states (or superselection sectors) which are characterized by their electric and topological charges. Tunneling of a quasihole from one edge to another edge changes one state into another. The transition rates be- tweenthetennon-equivalentclassesdependonthefusion rules and braiding rules of the quasiparticles as well as FIG. 1: Schematic picture of the Mach-Zehnderinterferome- other experimental parameters. We will calculated the ter. S and D denotesources and drains. Arrows indicate the tunneling current and the corresponding shot noise in propagation direction of the chiral edge modes. Quasiparti- sections V and VI. For the purpose of illustration, some cles tunnelbetween edges 1 and 2. oftherelevantresults24,26,27concerningtheν =5/2FQH state are reproduced throughout this paper. This paper is organized as follows. In section II, the flux enclosed by the two tunneling paths. For small Γ1 structure of electronic MZI is explained. In section III, and Γ2, the transition rate can be written as24 we work out the fusion rules and braiding rules for the quasiparticles in the ν = 5/2 and ν = 12/5 FQH states. p=r0{(|Γ1|2+|Γ2|2)+2u|Γ∗1Γ2|cos[2πΦa/Φ0+δ]}, (2) In section IV, we calculate the transition rates between where r and u are functions of temperature T, the ap- the non-equivalent classes in the MZI. In section V and 0 pliedvoltageV andtheinterferometersizeLwhoseexact VI, the tunneling current and shot noise is calculated. form can be calculated by using the Hamiltonian in Eq. Section VII presents the conclusion. (1), δ = arg(Γ Γ ). The tunneling current in the inte- ∗1 2 ger Quantum Hall regime is simply I = ep.24 The zero I frequency shot noise can be written as S = eI .27 The II. ELECTRONIC MACH-ZEHNDER I I FanofactorS /I isindependentoftheappliedmagnetic INTERFEROMETER I I flux and equals 1 in the units of an electron charge. In the FQH regime, the tunneling quasiparticles are In this section, the structure of an electronic MZI will anyons with fractional electric charge qe, where q is a be explained. We willalsoseehow the non-trivialstatis- fractional number. In addition to the Aharonov-Bohm tical phase φ can affect the tunneling current and noise s phase φ =2πqΦ /Φ , a tunneling quasiparticle expe- AB a 0 dramatically. rience a statistical phase φ originatedfrom other quasi- s A schematic diagram of an electronic MZI is depicted particles enclosed by the tunneling paths. in Fig. 1. S1, S2, D1, D2 denote the sources and drains IntheLaughlinstateswithν =1/m,φ =n2πν where s of the correspondingFQH edges 1 and 2. The arrowson n is the number of quasiparticles inside the interferome- theedgesindicatetheedgemodepropagationdirections. ter. The transition rate can be written as: A and B are two points on the edges. Quasiparticles oonthtehrethedroguegshartewaollQowPeCdstodetnuontneedlbfryomQPoCne1eadngdeQtoPtCh2e pn =r0{(|Γ1|2+|Γ2|2)+2u|Γ∗1Γ2|cos[2πqΦa/Φ0+φs+δ(]3}). respectively. As the bulk excitations are gaped, the low It is important to note that the transition rate depends energy physics of the MZI is determined by the edges. on n mode m. At zero temperature, there is a sim- Hence, the Hamiltonian can be written as ple way to calculate the tunneling current. Let us assume that there are n = km quasiparticles inside the Hˆ =Hˆ +[(Γ Tˆ +Γ Tˆ )+H.c.], (1) edge 1 1 2 2 interferometer initially. Then, the transition rate is p 0 and the average time to transfer a quasiparticle from where Hêdge denotes the Hamiltonian for the two edges S1 to D2 is t0 = 1/p0. After one quasiparticle tunnel- and Tˆ and Tˆ are tunneling operators which transfer ing, n is increased by one, the transition rate becomes 1 2 a quasihole from edge 1 to edge 2 at QPC1 and QPC2 p and t = 1/p . After m tunneling events, the tran- 1 1 1 respectively. sition rate returns to the initial value. The total time m 1 A voltage difference V between S1 and S2 will result needed to transfer m quasiparticles is t¯= − t . Hence, i in a tunneling current from one edge to the other, say, i=0 from edge 1 to edge 2. For a quasiparticle to arrive at in terms of the transition rates, the tunneliPng current in D2 from S1, there are two possible tunneling paths S1- the ν = 1/m Laughlin state is I = e/(n=m−11/p ). QPC1-A-QPC2-D2 and S1-QPC1-B-QPC2-D2. In the 1/m n n=0 integer Quantum Hall regime when the tunneling quasi- ThiszerotemperatureresulttogetherwiththePfinitetem- particlesareelectrons,thetransitionratedependsonthe perature ones can be derived rigorously using the chiral tunnelingamplitudesΓ andΓ aswellastheAharonov- Luttinger liquid theory of the edge states.24 Moreover, 1 2 Bohm phase φ =2πΦ /Φ , where Φ is the magnetic one can also show that the zero frequency shot noise is AB a 0 a 3 S =e2(n=m−11/p2)/(n=m−11/p )3.27 The Fano fac- and the operator product expansions for the conformal 1/m n n fieldsaregiveninRefs. 29and30andcanbe writtenas: n=0 n=0 tor is flux dePpendent sincePthe tunneling rates pn depend on the applied magnetic flux. Finally, we have min(l+l′,2k−l−l′) n=m 1 n=m 1 Φl Φl′ = Φn . (5) S /eI =( − 1/p2)/( − 1/p )2 1.27 m× m′ m+m′ 1/m 1/m n=0 n n=0 n ≤ n=X|l−l′| For the non-AbePlianstates, thPe situationis morecom- plicated. A quasiparticle in the non-Abelian states is characterized by its electric charge and its topological Φlm(z)Φlm′′(0)= Cmll′mn′z∆hΦnm+m′(0) (6) charge. The statistical phase φ depends on the number n s X of quasiparticles inside the interferometer as well as the wherethe expansioncoefficientsCll′n areconstantsand topological charges of the tunneling quasiparticle and of mm ′ the quasiparticles inside the interferometer. The statis- ∆h = hnm+m′ − hlm − hlm′′. The exponent ∆h in Eq. tical phase φ in the cases of the ν = 5/2 Moore-Read (6) is important as it gives the statistical phase, 2π∆h, s state and the ν = 12/5 Read-Rezayi state will be cal- acquired when a conformal field Φlm makes a full circle culated in the section III. Moreover, the result of fusing around another conformal field Φl′ when the result of twotopologicalchargestogethermaynotbe unique. For fusing these two fields is Φn . m′ m+m ′ instance, the topologicalchargeof aquasihole inthe 5/2 On the other hand, a free chiral bosonic field is gov- stateis σ. Accordingtothe fusionrules,twoσ fields can erned by the action fuse together and the resulting field can be I or ψ with 1 equal probability 1/2. The factor 1/2 modifies the tran- S = dxdt[∂ φ ∂ φ +(∂ φ )2]. (7) t c x c x c sition rates. In the ν =12/5 FQH state, the situation is −4π Z slightly more complicated. The transition rates will be studied in detail in section IV. The vertex operator Va =: eiaφc : has conformal dimension a2.31 Theoperatorproductexpansionsbetweenver- 2 tex operators have the form III. STATISTICAL PHASE V (z)V (0)=C zabV (0) (8) a b ab a+b Generalizing the idea of Moore and Read,6 Read and so that the fusion rule between vertex operators is Rezayipointedoutthattheν =5/2andν =12/5states can be described by the k = 2 and k = 3 parafermion V V =V . (9) a b a+b conformal field theories respectively.10 More specifically, × a quasiparticle operator can be written as Ψ =Φl V q.p m α where the parafermion field Φl describes the topologi- B. ν =5/2 Moore-Read state m cal charge of the quasiparticle and the vertex operator Va =: eiaφc : describes its electric charge. The braiding The ν = 5/2 Moore-Read state can be described by properties between quasiparticles depend on both of the the k = 2 parafermion theory. According to IIIA, we quantum numbers. In this section, we review the fusion can easily see that Φ0, Φ0 and Φ1 are the only three 0 2 1 rules and braiding rules of the parafermion theories and independentfields inthetheory. Followingthe notations thechiralbosontheory. Thestatisticalphaseφsacquired in Ref. 10, we label the fields as I, ψ and σ respectively. when a quasiparticle makes a full circle around another The conformaldimension of the three fields are 0, 1 and 2 in the non-Abelian states is calculated. 1 respectively according to Eq. (4). Following Eq. (5), 16 the fusion rules are: ψ ψ = 1, ψ σ =σ and σ σ = × × × 1+ψ. A. Parafermion and chiral boson theories One ofthe importantobservationsofMoore andRead isthatthegroundstatetrialwavefunctionoftheν =5/2 Theparafermionconformalfieldtheory29,30hascentral FQH state can be expressed as the correlation function charge c = 2kk+−22 in the Virasoro algebra. The primary of operators of the form ψ :e√2φc :, fields in the theory arelabeled as Φl andhave conformal gdeimneernastieosnahsle=ries2lk(o(kkf−+fil2)e)ldwshΦelmrewli=th0cl,o1n,f.o..r,mka−ld1i.mEeancshioΦnsll <Ψ5ψ/2(z=1)P,ψf((zz2i)−1·z·j·)ψQ(zi>Nj)(ezii√−2φz(zj1))2ei=√2φ(z2)···ei√2φ(zN)Φbg >, (10) hlm =hl+ (l−m4)k(l+m), for −l≤m<l, (4) where Pf is the Pfaffian and Φbg = e−i d2z√2ρ0φ(z) hlm =hl+ (m−l)(24kk−l−m), forl≤m≤2k−l. icshathrgeebdaecnksgirtyo.unEdvicdheanrtgley,otpheeraetloecrtrwointhopρ0erRdateonrotceasntbhee Theconformalfieldsaresubjecttotheconstraintsl+m identified as Ψ = ψ : e√2φc :. Moreover, one can 5/2,el. 0 (mod2) and Φlm = Φkk+−ml = Φlm+2k. The fusion rule≡s show that the states with quasiholes can be obtained by 4 inserting operators σ :e2√12φc : into the correlatorin Eq. TABLE I: Statistical phase originating from topological (10). As a result, the quasihole operator can be written charges in theν =12/5 state as Ψ5/2,q.h. = σ : e2√12φc :. In other words, a quasihole carries electric charge 4e and topological charge σ. With φβσ1α φψσ12ǫ φσσ11ǫ φǫσ1ψ1 φσσ21ψ2 φψσ11σ1 φσσ21σ1 φIσ1σ2 φǫσ1σ2 the identification of the quasiparticle operators with the Phase 2π1 2π3 2π2 2π1 2π 8 2π14 2π13 2π 4 5 5 3 3 15 15 15 15 conformal fields, we may calculate the statistical phases φ acquired when a quasihole makes a full circle around s another excitation with electric charge ne and topolog- 4 ical charge α. In the ν = 5/2 state, α can take three values: I, ψ and σ. The statisticalphase canbe written as the sum oftwo contributions: π φ =n +φβ . (11) s 4 σα FIG.2: Thenumbersfrom1to6labelsthesixnon-equivalent The first term on the right hand side of Eq. (11) orig- classes of thecomposite particleenclosed bytheinterferome- inates from the vertex operators. It can be calculated ter in the ν =5/2 Moore-Read state. The quantum number from substituting a = 1 , b = n and z = e2π 2√2 2√2 inside the bracket denote the electric charge and the topo- into Eq. (8). The second term φβ originates from the logical charge of the state respectively. The arrows indicate σα parafermionfieldswhereαdenotesthetopologicalcharge the transition direction at zero temperature. pl denote the ofthe other excitationandβ denotes the fusionresultof transition rates and the factors of 1/2 are due to the fusion probabilities. σ and α. According to Eqs. (5) and (6), if α = I , φσ = 0; if α = ψ, φσ = π; however, if α = σ, there σI σψ are two possibilities, φI = π and φψ = 3π. These σσ −4 σσ 4 IV. TRANSITION RATES results are consistent with the ones derived by using the algebraic theory of anyons in Ref. 26. In MZI geometry, the quasiholes inside the area en- closedbyQPC1-A-QPC2-B-QPC1canbeseenasacom- C. ν =12/5 Read-Rezayi state posite particle with electric charge nqe, and topological charge α from the point of view of a tunneling quasihole Read and Rezayi proposed that the ν = 12/5 FQH ontheFQHedges. Intheν =5/2state,fornquasiholes, state can be described by the k =3 parafermion theory. the electric charge is ne/4 and the topological charge α According to IIIA, one can find six nonequivalent con- may take value I, ψ or σ, as a result of fusing n σ fields. formal fields in this theory. Following the notations in When n is odd, the topological charge is σ. When n Refs. 10and19, wedefine the sixnonequivalentfields as is even, the topological charge can be I or ψ. We can Φ0 = I, Φ0 = ψ , Φl = σ and Φ2 = ǫ, where l = 1,2. obtain six non-equivalent classes of states by fusing n Th0e fusion2lruleslwhilch arel relevan0t to our calculations quasiholes. The six classesare: 1(−4e,σ), 2(0,I), 3(0,ψ), can be written as: 4(e,σ), 5(e,I) and 6(e,ψ). The numbers in the bracket 4 2 2 representthe electriccharge(mode)andthe topological I σ =σ , ψ σ =ǫ, × l l l× l charge respectively. The six classes are represented as σψ3−σl×=σlψ=+σσ3−l,, ǫσ×σl =σψ3=−lI++σǫl., (12) vertexes in Fig. 2. State (34e,σ) is identified with state l× l l 3−l 3−l× l (−4e,σ) since they differ from each other by an electron. The ground state trial wave function of the FQH state The arrows represent the transition directions between can be obtained by the correlation function of electron states at zero temperature. The transition rates are la- operators with the form Ψ12/5,el. = ψ1 : ei√53φc : and beled as pk where thequasiholeoperatorcanbewrittenasΨ =σ : ei√115φc :. Evidently,aquasiholecarries 5e e1l2e/c5t,qr.ihc.char1ge pk =r0{(|Γ1|2+|Γ2|2)+2u|Γ∗1Γ2|cos[2πΦa/4Φ0+kπ/2(1+3δ)]}. and topological charge σ . Similar with the case of the 1 TheseresultscanbeobtainedfromEq. (3)withq = 1 ν =5/2state,thestatisticalphaseofaquasiholemaking 4 andtheresultsforφ insectionIIIB.Thetransitionrates a full circle around an excitation with electric chargene s 5 from 1 2, 1 3, 4 5 and 4 6 are modified andtopologicalchargeαcanbewrittenasthesumoftwo → → → → by a factor of 1/2 which is called the fusion probability. contributions φ = n2π +φβ . The first contribution s 15 σ1,α Heuristically, if the quasiparticle inside the interferom- can be obtained by substituting a = 1 , b=n 2π and √15 √15 eter has topological charge σ, the fusion result of the z = ei2π into Eq. (8). The non-Abelian contribution quasiparticlewith a quasihole is not unique accordingto φβ = 2π(h h h ) can be calculated from the the fusionruleσ σ =1+ψ. There areequalprobabili- σ1,α β − α − σ1 × conformaldimensions ofthe fields. The results arelisted tiesthattheresultingfieldisI orψ. Onemayfollowthe in Table I. arguments in Refs. 27,32 to calculate the fusion proba- 5 FIG. 3: Two pairs of particles a¯a and ¯bb are created out of the vacuum and particles a(a¯) and b(¯b) are fused together to result in particle c(c¯). FIG. 4: The numbers from 1 to 10 labels the ten non- equivalent classes of the composite particle enclosed by the interferometer in the ν = 12/5 FQH state . The quantum bilities pIσσ and pψσσ. numberinside the bracket denotetheelectric charge and the In the context of algebraic theory of anyons, suppose topological charge of thestate respectively. The arrows indi- two particle-antiparticle pairs aa¯ and b¯b are created out cate the transition direction at zero temperature. pβb P+ σαa a→b of the vacuum, if a and b are fused together, what is denotethe forward transition rates. the probability pc that the resulting particle is c? This ab processisdepictedinFig. 3,theamplitudeoftheprocess the resulting state can be 3(e/5,Ψ ) or 4(e/5,σ ) with is shown to be ddacdb, where dα denotes the quantum probability τ12 and τ1 respectively. 2 1 dimensionofapqarticlewithtopologicalchargeα. Hence, At zero temperature, when the voltage difference V the fusion probability is27,28,32 betweenS1andD2ispositive,onlytransitionsindicated by the arrows in Fig. 4 can happen. For convenience, d d pc = c =Nc c , (14) we denote this forward transition (the transition along ab d d abd d a b a b the directions of the arrows) rate from state a to state i X b as P+ . The total forward transition rate, defined as where Nacb is called the fusion multiplicity which gives P+ ma→odbifiedbythecorrespondingfusionprobability,is a b tthoerensuumltbienrco.f ways that a and b can be fused together tdheen→ottoepdoalosgRica+a→l bch=arpgβσeb1αoafPtah+→ebs,twatheerbeefαoareanadndβbafdteenrotthees In the ν = 5/2 state, d = d = 1 and d = √2. Nc I ψ σ ab fusion with a quasihole. canbezeroor1accordingtothefusionrules. Asaresult, Atfinitetemperature,quasiholescantunnelfromedge we have pI =pψ =1/2. σσ σσ 2 to edge 1 such that transitions in Fig. 4 can occur in In the ν = 12/5 FQH state, the fusion results of the directionsbothaloneandagainstthedirectionsofthear- quasiparticles inside the interferometer can be classified rows. Abackwardtunnelingeventcanberegardedasthe into ten non-equivalent classes. The ten classes are rep- followingphysicalprocess: aquasihole-quasiparticlepair resented by the vertexes in Fig. 4, from state 1 to state iscreatedoutofthevacuumnearthetunnelingpointcon- 10. The quantum numbers inside the bracketdenote the tact, the quasihole tunnels to edge 1 while the quasipar- electric charge and the topological charge of the state. ticle withcharge e/5andtopologicalchargeσ tunnels 2 State 1(2) and state 1(2) are in the same class because − ′ ′ intoedge2andfuseswiththequasiholesenclosedbythe they differ from each other by an electron which can be interference paths. As a result, the backward transition slatabteelesdaraesd(e−noet,eψd1)b.y pTlhwehterraensition rates between the risatoebtRaa−in→ebd=fropmβσb2αthaPea−d→etbawilehderbeaPlaa−n→ceb c=onPdb+i→tiaoen−.eV/5kBT p =r (Γ 2+Γ 2)+2uΓ Γ cos[2π(Φ /Φ +l)/5+δ] . l 0{ | 1| | 2| | ∗1 2| a 0 } (15) V. ZERO TEMPERATURE CURRENT Similarwiththecaseintheν =5/2FQHstate,p canbe l obtained from Eq. (3) and the results of the statistical phase φ = n2π +φβ from section III. The transition After classifying the states and obtaining the transi- s 15 σ1,α rates are modified by the corresponding fusion proba- tion rates between them in section IV, we are ready to bility. Quasiparticles with topological charge I,ψ and calculate the tunneling current between the two FQH 1 ψ have quantum dimension 1 and quasiparticles with edges with the help of Figs. 2 and 4. In this section, we 2 topologicalcharge σ ,σ and ǫ have quantum dimension assume that the condition k T qeV is satisfied such 1 2 B ≪ τ = 2cos(π/5). Hence, from Eq. (14), the fusion proba- that quasiparticle tunneling can happen only from one bilities can take three values 1, 1 and 1 . For example, edge to the other. τ τ2 when a quasihole with electric charge e/5 and topologi- Intheν =5/2FQHstate,therearefourpossibleways cal charge σ fuses with a quasiparticle in state 1(0,I), totransferoneelectronfromS1toD2. Thefourpossible 1 the resulting state has electric charge e/5 and topolog- ways are denoted as four different paths in Fig. 2. If ical charge σ with probability 1. On the other hand, wetakestate 1(0,I)asthe startingpoint, the fourpaths 1 if a quasihole fuse with a quasiparticle in state 2(0,ǫ), are 1-2-4-5-1’,1-2-4-6-1’,1-3-4-5-1’and 1-3-4-6-1’respec- 6 tively. The average time it takes to transfer an electron rates Ri− j equal zero. In this case, the steady state by path 1-2-4-5-1’ is t = 1 + 1 + 1 + 1 . The tun- solutions→f for the kinetic equations can be found easily 1 p3 p0 p0 p1 i neling probabilities p are given in Eq. (13). The prob- in the following way: k acabnilidtyentootetatkheeapvaetrhag1eistimq1e=takpe3p+n3pt1opt0pr+0apn2s.feSrimanilealrelcyt,rwone First, we define fk′ = fkj1=01Rk+→j, R˜i−1→i = through path i by ti and the corresponding probability R+i 1 i and R˜ P= R+i 1 i+1 , by qi. The average time to transfer an electron from S1 R+i 1 i+−R→+i 1 i+1 i−1→i+1 R+i 1 i−+R→+i 1 i+1 4 wh−er→e i =− →1,3,5,7,9 and we have used− →the n−ot→ation to D2 is t¯5/2 = tiqi and the current is I5/2 = e/t¯5/2. n = n+10. With the steady state condition dfi(t) = 0, 1 dt See Ref. 27 forPa full derivation. The explicit expres- we can rewrite the kinetic equations into a set of matrix sion of the tunneling current in terms of the tunneling equations: probabilities is:27 f 0 R˜ f I5/2(V)= 1 (2+ p3 + p1)+e 1 (2+ p0 + p2). fi′+i′1 != 1 R˜ii−11→i+i1 ! fii′′−21 !. (20) p1+p3 p0 p2 p0+p2 p1 p3 − → − (16) FromEq. (20), wecanseethatf +f =C whereC is It is important to note that when we change the sign i′ i′+1 of the applied voltage, the tunneling current can be cal- independent of i because R˜i 1 i+R˜i 1 i+1 = 1. As a result,thetunnelingcurrent−ca→nbewr−itt→enasI =5e C. culated in the same way as above. However, the cor- ∗ responding diagram in Fig. 2 is modified in two ways. Denoting the 2 by 2 matrix in Eq. (20) by Mi 1, we obtain a self-consistent matrix equation for f a−nd f First, the directions of the arrows are reversed. More 1′ 2′ importantly, the fusion probabilities are modified from which reads: 1/2(1) to 1(1/2). Hence, the resulting current is differ- f f ent from the original one besides a change in the sign of 1′ =M M M M M 1′ . (21) 10 8 6 4 2 the current. In terms of the tunneling probabilities, we f2′ ! f2′ ! have: I5/2(−V)= 1 (2+ p1 + p3)+e 1 (2+ p2 + p0). R˜2Th4eR˜s4olu6Rt˜io6n8of+ER˜q2. (23R1˜)6is8f)1R′˜=8 N9]R[˜11−0 (1R˜=2→N4R˜f41′′→a5n+d This observpa1t+iopn3 is sipg0nificpa2nt bep0c+aup2se it ips1a rpe(3s1u7l)t fN2′f→2=′′ wNh→[e1re−N(→1i−s a(Rñ4o→→r5m+al→iRzã4t→io6nR→˜6fa→c8t)oRr˜→.8→f91′)′R˜(1f02′→′)1]ca=n of the non-unity fusion probability and this property is be interpretedasthe transitionprobabilitytostate 1(2) absent in the Laughlin states. The experimental obser- fromstate2(1)after5tunnelingevents. UsingEq. (20), vation of the asymmetric I-V curve provides evidence we can generate fi′ = Nfi′′ from f1′ and f2′. From the for the existence of the non-Abelian excitations in the solutions of Eq. (21), one caneasily show that f1′+f2′ = ν = 5/2 FQH state.26 As shown below, the I-V curve in fi′ +fi′+1 = N(1 +R˜2 3R˜4 5R˜6 7R˜8 9R˜10 1) = C. → → → → → the ν =12/5 state is also asymmetric. 10 Withthenormalizationcondition f =1,weconclude The diagram shown in Fig. 4 for the ν = 12/5 FQH l l=1 state is considerablymore complicated than the ones for that P the Laughlinstatesandν =5/2state. Inthe restofthis 1 section, we will calculate the zero temperature current N = . with the kinetic equation approach. 4 [f 1 +f 1 ] The transitions between the 10 non-equivalent states 2′′l+1R+ 2′′l+2R+ +R+ l=0 2l+1 2l+4 2l+2 2l+3 2l+2 2l+4 → → → labeled in Fig. 4 are governed by the kinetic equations P (22) for the system:24,27 As a result, in terms of the forward transition rates, the tunneling current can be expressed formally as: 10 df (t) dit =Xj=1[−fi(t)(Ri+→j +Ri−→j)+fj(t)(Rj+→i+Rj−→i)], I12/5 = 4 5e∗(1+R˜2→3R˜4→5R˜6→7R˜8→9R˜10→1) . (18) [f 1 +f 1 ] where fi(t) is the probability of the composite particle l=0 2′′l+1R+2l+1→2l+4 2′′l+2R+2l+2→2l+3+R+2l+2→2l+4 inside the interferometerto be found in statei attime t. P (23) Inthisnotation,thetunnelingcurrenthastheexpression: The form of the functional dependence of the current onthetransitionratesinEq. (5)doesnotdependonthe 10 10 specificvaluesofthetransitionrates. However,exploring I =e∗ fi (Ri+→j −Ri−→j), (19) the symmetry in the transitions rates in Fig. 4, we have i=1 j=1 X X wherefi arethesteadystatesolutionsofEq. (18)ande∗ f1′+2k(Φa) = f1′(Φa+2k′Φ0) (24) equalse/5. Atzerotemperature,thebackwardtransition f2′+2k(Φa) ! f2′(Φa+2k′Φ0)! 7 where 2k = 2k mod 5 and k runs from 0 to 4. As a ′ result, the expression of the tunneling can be expressed in a more manageable form. 5e (1+R˜ R˜ R˜ R˜ R˜ ) ∗ 2 3 4 5 6 7 8 9 10 1 I12/5 = 5 → → → → → [f 1 (Φ +2k Φ )+f 1 (Φ +2k Φ )] 1′′R+ a ′ 0 2′′R+ +R+ a ′ 0 k=0 1 4 2 3 2 4 → → → P (25) Eq. (25) can be further simplified as r (Γ 2+ Γ 2)[α+βcos(2πΦ +5δ)] 0 1 2 a I12/5 =5e∗γ+|κc|os(2π|Φ|+5δ)+ξsin(2πΦ +5δ) a a (26) FIG.5: ShowsthefluxdependentofI(Γ1,Γ2). Γ1 =1forall where α, β, γ, κ, ξ are functions of R = u|Γ∗1Γ2| as thefourcurves. Forthethreesolid curves,fromtheonewith well as the applied voltage but independen|Γt1o|2f+t|Γh2e|2ap- tΓh2eelaqrugaelssttaom1p,l0it.u9daentdot0h.8eorensepwecitthivetlhye.smThaleledstasahmedplictuurdvee, plied magnetic flux. Their exact algebraic expressions depicts the magnitude of the current when V is changed to are lengthy and will not be shown here. The important −V with Γ2 =1. point is that α,γ 1+... and β,κ,ξ R5+... where ... ∼ ∼ denotes the higher order terms in R. When R 1, for example,whenΓ Γ ,thetunnelingcurrentca≪nbere- solid curves from the one with the largest amplitude to 2 1 ducedtoasinusoid≪alformI =I0+IΦcos(2πΦa+const). theonewiththesmallestamplitude,Γ2 equals1,0.9and In this limit, one can show that 0.8respectively. Thedashedcurvedepictsthemagnitude of the current when V is changed to V with Γ = 1. 2 − I (Γ ,Γ ) [I (Γ ,Γ ) I (Γ ,0)]b (27) We set the arbitrary phase δ =0 in the calculation. Φ 1 2 0 1 2 0 1 ∼ − From Eq. (25) we see that the current is a periodic with b=5/2. The exponent b in the k =3 Read-Rezayi function ofthe magnetic flux with periodΦ , this agrees 0 state is the same as the one in the ν = 1/5 Laughlin with the Byers-Yang theorem.33 The minimums of the states24 but different from the one in the Moore-Read current occur at Φ /Φ = 1/2 mod 1 when δ = 0. This a 0 state in which b=2.26 can be explained by the fact that whenever Φ /Φ = a 0 If wechangethe signofthe voltagedifference between 1/2 mod 1, one of the tunneling probabilities p in Eq. l S1 and S2 such that quasiholes propagate from S2 to (15) can be much smaller than 1 given that u 1. For ≈ D1, a similar diagram as the one in Fig. 4 with differ- example,ifp isverysmall,thesystemwillbe“trapped” 2 ent transition rates is obtained, resulting in a different at state 3(e,ψ ) for a long time before any tunneling 5 2 functional dependence of the tunneling current on the event can happen. This results in a minimum in the appliedmagneticflux. Morespecifically,thecoefficientξ tunneling current. of sin(2πΦ +5δ) in Eq. (26) will change its sign to ξ a − besides an overallchange in the sign of the current. In order to work out the diagram with V changed VI. ZERO TEMPERATURE SHOT NOISE to V, we can imagine the following physical process: − a quasihole-quasiparticle pair is created out of the vac- It was pointed out in Ref. 27 that measuring the uum near S2, the quasihole with electric charge e/5 and Fano factor e˜, which is defined as the ratio between the topological charge σ1 will tunnel to D1 through two shot noise and the tunneling current, can be an effective possible paths, S2-QPC1-B-QPC2-D1 and S2-QPC1-A- way to probe non-Abelian statistics. It was shown that QPC2-D1. Thequasiparticlewithcharge e/5andtopo- the maximum Fano factor in the Laughlin state and the − logicalchargeσ2 willfusewiththecompositeparticlein- Moore-Read state is e and 3.2e respectively where e is side the area enclosed by QPC1-A-QPC2-B-QPC1. Ac- anelectroncharge. Itis shownbelowthatthe maximum cordingly,wecanobtaina newdiagramsimilarto Fig. 4 Fano factor in the k = 3 Read-Rezayi state can be as andcalculatethecurrentwiththeproceduresmentioned large as 12.7e. above. The new diagram can be obtained by changing Shot noise is defined as the Fourier transform of the thedirectionofthearrowsinFig. 4withthefusionprob- current-currentcorrelation function, abilities modified from 1 (1) to 1( 1 ). τ2 τ2 WeassumethattheinterferometersizeLandtheexci- S(ω)= 1 +∞ <Iˆ(0)Iˆ(t)+Iˆ(t)Iˆ(0)>exp(iωt)dt. tation velocity satisfy the condition eVL/5hν 1 such 2 thatthefunctionuinEq. (15)canbesetto1. M≪oreover, Z−∞ (28) we absorb the function r0 into the tunneling amplitudes Inthispaper,weareinterestedinthelowfrequencylimit Γ1 andΓ2. Thefluxdependenceofthetunnelingcurrent of the shot noise. In this limit, S can be written as27 I(Γ ,Γ ) in Eq. (25) is plotted in Fig. 5 with Γ = 1 1 2 1 in the appropriate units in all four curves. For the three S =<δQ2(t)>/t (29) 8 where δQ(t) is the fluctuation of the charge Q(t) trans- t + when λ (x) is negative. From Eq. (34), we k → ∞ mitted during a period of time t when the measurement see that M(x) has diagonal elements which are nega- time t is fixed. tive, off-diagonal elements which are positive or zero, Inthis section,we use the generatingfunction method 10 and M (x = 1) = 0 . Rohrbach theorem34 tells us developed in Ref. 27 to calculate the zero temperature ij j=1 shot noise and the Fano factor. Without loss of general- that zPero is a non-degenerate eigenvalue of M(x) and all ity,we canassumethatthe fusionresultofthe quasipar- other eigenvalues are negative when x = 1. If we de- ticle inside the interferometer at time t = 0 is in state 1 note λ(x) as the unique eigenvalue which has the prop- orstate2aslabeledinFig. 4. LetusdefinePk+5n,i(t)as 10 theprobabilitythatk+5nquasiparticleshavetransferred erty λ(x = 1)= 0 then fi(x,t) can be written as the fromS1toD2 attime tandthe resultingcompositepar- i=1 sumofg(x)eλ(x)tandothPerunimportanttermswhichwill ticle is in state i. Here, we denote k =[i 1] where [y] is −2 eventuallygotozerowhenwesetx=1attheendofthe the largest integer k satisfying k y, n is an arbitrary ≤ calculations at large t. Together with the normalization integer and i runs from 1 to 10. From the definition of condition g(x=1)=1, we can show that P (t), one can show that l,i P˙l,i(t)= 10 [−Pl,i(t)(Ri+→j +Ri−→j) I = <Qt(t)> = e∗ddx(g(x)teλ(x)t)|x=1 =e∗dλd(xx)|x=(31,5) j=1 X +Pl−1,j(t)Rj+→i+Pl+1,j(t)Rj−→i]. (30) S = <δQt2(t)> =e∗2(d2dλx(2x)|x=1+ dλd(xx)|x=1), (36) By defining and the Fano factor ∞ f (x,t)= P (t)xk+5n, (31) i k+5n,i d2λ(x) dλ(x) n=X−∞ e˜=e∗(1+ dx2 |x=1/ dx |x=1). (37) we immediately see that The derivatives dλ(x) and d2λ(x) can be deter- dx |x=1 dx2 |x=1 d 10 mined by applying the first andsecond derivative opera- <Q(t)>=e∗( fi)x=1 (32) tors on the characteristicequation, det[M(x) λ(x)I]= dx | − i=1 0, and setting x=1 at the end of the calculation. X For the purpose of calculating the first and second and derivativeofλ(x), weonlyneedtoknowthelowestorder 10 terms in λ(x) of the characteristic equation because of d d <δQ2(t)>=e∗2( x fi)x=1 <Q(t)>2 . the propertyλ(x=1)=0. If the characteristicequation dx dx | − i=1 is written as: X (33) Thetimeevolutionoffi(x,t)isgovernedbythefollowing ...+A(x)λ2+B(x)λ+C(x)=0, (38) setofequationswhichcanbederivedfromEqs. (30)and (31). where ... denotes higher order terms in λ(x), one can show 10 ddtfi(x,t)= j=1[−fi(x,t)(Ri+→j +Ri−→j) (34) dλ(x) x=1 = dC(x) x=1/dB(x) x=1 (39) dx | − dx | dx | +Pxfj(x,t)Rj+ i+ x1fj(x,t)Rj− i]. → → and Eq. (34) is reduced to Eq. (18) when x = 1 and fi(x = 1,t) = fi(t) as it is clear from their defini- d2λ(x) = −ddx22C−2ddBx dλd(xx)−2A(x)(dλd(xx))2 . (40) tions. The above set of equations can be rewritten in dx2 |x=1 B(x) |x=1 a vector form f~˙(x,t) = M(x)f~(x,t) where M(x) is a The kinetic matrix at zero temperature M(x) has T=0 real 10 10 matrix. The solution of Eq. (34) has a simple structure whichis shownin Eq. (41). The coef- × 10 the form fi(x,t) = gik(x,t)eλk(x)t, where λk(x) are ficients A(x), B(x), C(x) in Eq. (38) can be determined k=1 fromM(x)T=0 after lengthly but straightforwardcalcu- the eigenvalues of MP(x) and gik(x,t)eλk(x)t 0 for lations. → 9 P 0 0 0 0 0 0 0 0 xP4 − 0 τ2  0 −(Pτ21 + Pτ3) 0 0 0 0 0 0 xP3 xτP1  0 xP1 P 0 0 0 0 0 0 0  τ2 − 2   xP xP3 0 (P0 + P3) 0 0 0 0 0 0   0 τ − τ τ2   0 0 0 xP3 P 0 0 0 0 0   τ2 − 4  M(x)T=0 = 00 00 xP02 x0τP0 00 −(Pτ20xτP+20 Pτ2) −0P1 00 00 00  (41)  0 0 0 0 xP xP2 0 (P2 + P4) 0 0   4 τ − τ2 τ   0 0 0 0 0 0 0 xP2 P 0   0 0 0 0 0 0 xP1 xττP24 −03 −(Pτ24 + Pτ1)       The flux dependence of the Fano factor e˜ with Γ = 1 Γ = 1 is depicted in Fig. 6. We can see that when the 2 current is minimal at Φ /Φ = 1/2, e˜ 12.7e, which a 0 ≈ is much larger than the maximum Fano factor in the case of both the ν = 1/5 Laughlin state with e˜ = max 1 and the Moore-Read state with e˜ 3.2e. The max ≈ largeFanofactorcanbeunderstoodinthefollowingway: suppose we tune the applied magnetic field such that Φ /Φ 1/2 and p 1. The system can be trapped a 0 2 ≈ ≪ in state 3 for a long time before a tunneling event can happen which would drive the system to state 6. Once the system is in state 6, a series of tunneling events can happen in a relatively short period of time which would FIG. 6: A plot of the Fano factor e˜(Γ1,Γ2)/e against Φa/Φ0 drive the system to other states before getting trapped with Γ1 =Γ2 =1. at state 3 again. As a result, the Fano factor can be much larger than 1 electron charge. Since the existence FQH state. More specifically, the scaling exponent b in of those by-pass roads (by-passing state 3) is a direct therelationbetweenthefluxdependentpartandtheflux consequence of the non-trivial fusion rules and braiding independent part of the tunneling current in Eq. (27) is rules, the observation of a Fano-factor which is larger 5/2 in the ν =12/5 Read-Rezayi state, in contrast with than one electron charge would provide evidence for the b = 2 in the Moore-Read state, but is the same as the existence of non-Abelian statistics in FQH states. case in the ν = 1/5 Laughlin state. However, I-V curve All the calculations above were performed in the case in the ν = 12/5 Read-Rezayi state is asymmetric. This of k = 3 Parafermion state, which corresponds to a propertyisabsentinallLaughlinstates. Inaddition,the FQH state with ν = 13/5. As we mentioned before, Fanofactorintheν =12/5Read-Rezayistatecanexceed the ν = 12/5 state can be obtained by a particle-hole 12.7ewhenthe currentis tunedto its minimumvalueby transformation,10,19 which results in a change in a sign changing the applied magnetic flux. This number well ofstatisticalanglesaswellasasignchangeinthecharge exceeds the maximum possible Fano factor in all Laugh- carriedbythetunnelingquasiparticle. Consequently,the lin states and the ν = 5/2 Moore-Read state which was tunneling probabilities p in Eq. (15) will be changed to l shown to be e and 3.2e respectively. The two key prop- p = p . The net effect is simply the same as changing Φ′l to−lΦ and δ to δ at the end of the calculations erties, namely, the asymmetric I-V curve and the larger a − a − than one electron charge Fano factor, are direct conse- which will not affect our arguments. quences of non-Abelian statistics. Their experimental observations would provide evidence for the existence of non-Abelian statistics in the ν =12/5 FQH state. VII. CONCLUSIONS In conclusion, we have shown that in Mach-Zehnder Acknowledgments interferometer geometry, the tunneling current and the zero temperature shot noise can be used to show the ex- The author is indebted to D. Feldman for his encour- istence of k = 3 parafermion statistics in the ν = 12/5 agement and guidance, as well as inspiring discussions 10 throughout this project. 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