Experimental constraints and a possible quantum Hall state at ν=5/2
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J u l Experimental constraints and a possible quantum Hall state at ν = 5 / Guang Yang and D. E. Feldman
Department of Physics, Brown University, Providence, Rhode Island 02912, USA (Dated: June 7, 2018)Several topological orders have been proposed to explain the quantum Hall plateau at ν = 5 / Fractional quantum Hall effect (QHE) exhibits remark-ably rich phenomenology. More than 70 filling factorshave been discovered. Some of them are well understoodbut many are not. In particular, the nature of the fragilestates in the second Landau level remains a puzzle.The quantum Hall plateaus at the filling factors [1] ν = 5 / ν = 7 / ν = 5 / i.e. , the 5 / ν = 5 / SU (2) andanti- SU (2) states have attracted much interest as pos-sible candidates to explain the QHE plateau at ν = 5 / K = 8 and anti-331 states [9, 10].Most above-mentioned states were invented before ex-perimental information beyond the existence of the 5 / / ν = 5 /
2. We argue that all previ-ously proposed ground state wave functions are excludedby those constraints. To explain the 5 / / T g − , where the exponent g depends on the topological order and is universal inthe absence of long-range interactions [2]. Theory pre-dicts g = 1 / SU (2) states and g = 3 / / /
8. In the earliest experiment [18], the best fit for g at the fixed charge e ∗ = e/ g = 0 .
45. This was interpreted initially as asignature of the anti-Pfaffian or SU (2) state. Subse-quent experiments [8, 19] in other sample geometries pro-duced g between 0 .
37 and 0 .
42 as the best fits at fixed e ∗ = e/
4. This supports the case for the 331 state. Itwas argued that the measured exponents are affected bylong-range electrostatic forces [10]. Their effect dependson the sample geometry and in all cases increases the ob-served g . Thus, all tunneling data are compatible withthe 331 state [10].At the same time, the 331 state is incompatible withthe observation [12] of an upstream neutral mode. Thismeans that no proposed ground state wave function fitsall existing data. Below we identify a different groundstate that agrees with the existing experiments.We propose that the 5 / / nnm wave function isΨ usual = Π k 2. In addition to the second Landau level at ν = 1 / 2, the first Landau level is filled in the 5 / K -matrix [2] K = (cid:18) (cid:19) (3)and the charge vector q = (1 , l = (1 , 0) and l = (0 , e ∗ = e q K − l T , = e/ θ = 2 π l K − l T = 3 π/ . (5)The statistics of two identical particles is given by thephase, accumulated when they exchange their positions: θ = θ = π l K − l T = − π/ . (6)The simplest interpretation of the two quasiparticleflavors implies a spin-unpolarized state, where excitationswith two different spin projections are allowed. The 113order is also possible in a spin-polarized system. Forexample, one can rewrite Eq. (3) as K = W T K ′ W ; K ′ = (cid:18) − (cid:19) ; W = (cid:18) (cid:19) . (7)The matrix K ′ describes the same 113 topological orderand can be interpreted within the hierarchical construc-tion for spin-polarized electrons: a condensate of charge-2 e quasiholes forms on top of the integer QHE state.We now turn to the edge physics [2]. The edge actionis L = − ¯ h π Z dxdt { X i,j =1 , K ij ∂ t φ i ∂ x φ j + X j = I ,I ∂ t φ j ∂ x φ j + X i,j =1 , ,I ,I V ij ∂ x φ i ∂ x φ j } , (8)where the fields φ , φ describe the fractional QHE edgemodes and φ I , φ I describe the two integer edge chan-nels. By introducing the charged mode φ ρ = φ + φ andthe neutral mode φ n = φ − φ we can rewrite the actionin the form L = − ¯ h π Z dtdx [2 ∂ t φ ρ ∂ x φ ρ − ∂ t φ n ∂ x φ n +2 v ρ ( ∂ x φ ρ ) + v n ( ∂ x φ n ) + 2 v ρn ∂ x φ ρ ∂ x φ n ] − ¯ h π Z dtdx [ X i = I ,I ( ∂ t φ i ∂ x φ i + v i ( ∂ x φ i ) )+2 u ∂ x φ I ∂ x φ I + 2 X i = ρ,n ; j = I ,I w ij ∂ x φ i ∂ x φ j ] , (9)where the charge density in the fractional edge channels ρ F = e∂ x φ ρ / (2 π ) and the charge density in the integerchannels is ρ I = e ( ∂ x φ I + ∂ x φ I ) / (2 π ). In what followswe will ignore the integer edge channels and concentrateon the first two lines in the action (9). The integer chan-nels have little effect on our results as discussed in Sup-plemental material [26].The minus sign in front of ∂ t φ n ∂ x φ n signifies the exis-tence of an upstream neutral mode in agreement with theexperiment. We now turn to the quasiparticle tunneling.The most relevant quasiparticle operators create excita-tions of charge e/ O , = exp( iφ , ) =exp( i [ φ ρ ± φ n ] / φ → φ , φ → φ . In such case v ρn = 0 in Eq. (9) and the scalingdimensions of the operators O , are identical and equal∆ = 3 / 16. Weak quasiparticle tunneling between twoedges at the point x = 0 is described by the contributionto the action [2] L T = Z dt X i =1 , Γ i O ( t ) i ( x = 0) O ( b ) † i ( x = 0) + h . c ., (10)where the indices t and b refer to the top and bottomedges. The low-temperature tunneling conductance G ( T )can be estimated [2] by performing the renormalizationgroup procedure up to the energy scale E ∼ T and setting G ( T ) ∼ Γ ( T ). Thus, G ( T ) ∼ T g − , where g = 3 / v ρn changes the above result. Thescaling dimensions of the operators O , become differentand correspond to two contributions to the conductance G , ( T ) ∼ T g ± − with g ± = 1 √ − c (cid:18) ± c √ (cid:19) , (11)where c = √ v ρn / ( v ρ + v n ). This might suggest that thetunneling conductance is nonuniversal and no meaningfulcomparison with the experiment is possible. However, weshow below that a theory, based on the 113 order, doespredict the scaling of the conductance with g , close to3 / 8, even without the flavor symmetry. In contrast tothe famous spin-polarized 2 / G = νe /h in a bar geometry in a system with upstreammodes, if the edges are long enough. At the same time,the tunneling conductance only depends on what hap-pens within a thermal length from the tunneling contact.Hence, the low-temperature tunneling conductance is af-fected by disorder only if disorder is relevant in the renor-malization group sense [27]. In our problem, disorder is responsible for electron tunneling between the two frac-tional modes φ and φ . The corresponding contributionto the action L D = R dtdx { ζ ( x ) exp[2 iφ n ( x )] + h . c . } ,where ζ ( x ) is a random complex number. One can checkthat L D is always irrelevant. Thus, disorder does not leadto a universal conductance scaling and Eq. (11) applies.The explanation for the observed g ≈ / c ≪ v n , v ρ and v ρn (9) in the tunnel-ing experiments [8, 18, 19]. In all those experiments, theedges are defined by top gates. As observed in Ref. 28,the charge density profile in such situation is almost thesame as in the absence of the magnetic field. The edgeconsists of several compressible strips, separated by nar-row incompressible strips of fixed charge density (Fig. 1).The widths of the compressible strips and their distancesfrom the gates depend on the filling factor and other de-tails and are estimated to be between hundreds nm anda few µ m [28]. The widths of the gates and their dis-tances from the electron gas are within the same range.This gives us an estimate of the distance between theedge states and the gates. In the simplest picture, thewidths of various edge channels can be estimated fromthe widths of the compressible strips. Quantum localiza-tion modifies such picture [28]. We expect that an edgechannel is located within a compressible region betweentwo incompressible strips with filling factors ν < ν .The part of the compressible strip on one side of the edgechannel should be understood as an incompressible QHEliquid with the filling factor ν and localized quasiholes.The part of the compressible strip on the other side ofthe channel should be understood as an incompressibleQHE liquid of the filling factor ν with localized quasi-particles. The width a of the edge channel depends onthe localization length and is less than the total width ofthe compressible strip. We expect a > l B . A localizationlength < l B would mean that disorder is too strong forQHE correlations to exist.Let us now estimate v ρ . If the distance from the edgeto the gate is comparable to a then the energy cost of theaverage linear charge density ρ = e∂ x φ ρ / (2 π ) in a regionof size a × a is δE ∼ ( aρ ) / ( ǫa ) , (12) n ( y ) y FIG. 1: Density profile of a 2D electron gas in a magneticfield. The density is constant in narrow incompressible strips. where ǫ is the dielectric constant. This energy cost entersthe action (9) as a ¯ hv ρ ( ∂ x φ ρ ) / (2 π ). Hence, v ρ ∼ e / (¯ hǫ ).The velocity v ρ increases by a factor of ln( d/a ), if theedge width a is much smaller than the distance d ≫ l B from the gate [29]. This comes from the energy cost ofthe interaction between the sections of the edge at thedistances l , d > l > a .We expect that the neutral mode runs at the sameplace as the charged mode and the excitations of the neu-tral mode redistribute the two electron flavors withoutchanging the overall charge density beyond the magneticlength l B . Thus, the neutral mode only participates inshort-range interaction of radius l B . We find v n by es-timating the energy cost of the disbalance ρ n = ρ − ρ between the charge densities of the two flavors. We getan estimate, similar to Eq. (12), but with an additionalfactor l B /a to account for the short-range character ofthe interaction [30]. This is similar to the calculation ofthe charge velocity in the presence of the top gate [31],where the interaction radius is set by the distance to thegate. Thus, v n ∼ v ρn ∼ l B a [1+ln( d/a )] v ρ ≪ v ρ . Hence, c ≪ g ≈ / 8. The latter conclusion is not affectedby the integer edge channels [26]. Indeed, due to spacialseparation, the interaction of the neutral mode with theinteger channels is weaker than its interaction with thefractional charged mode.Our physical picture differs from the simplest pictureof the charged and neutral channels in a very clean ν = 2system. There, two spin channels correspond to two widecompressible strips, separated by an incompressible re-gion. Nevertheless, even at ν = 2 one expects the chargedmode to be much faster than the neutral mode [32]. Thisis the only thing that matters for our estimate of c . Be-sides, with two contra-propagating channels, the gener-alization of the ν = 2 picture for ν = 5 / < ν < ν max and the other with 5 / < ν < ν max , and an incompress-ible strip with ν = ν max > / ν > / s ≫ l B differs significantly from the chargedistribution in the absence of the magnetic field and isunlikely. QPC QPC FIG. 2: Aharonov-Bohm interferometer. Quasiparticles movealong the edges and tunnel between the edges at the quantumpoint contacts QPC1 and QPC2. Several quasiparticles arelocalized between the edges. What are the signatures of the 113 state in anAharonov-Bohm interferometer [1], Fig. 2? We con-sider two possibilities: 1) only one quasiparticle flavorcan tunnel through the tunneling contacts in Fig. 2; 2)the flavor-symmetric situation: tunneling amplitudes areidentical for both flavors. The realistic situation is likelyin between. In the first case the current through the in-terferometer changes periodically as a function of its areawith the period, corresponding to the additional mag-netic flux Φ / = hc/e ∗ = 4 hc/e through the device. Inthe second case, we need to add two periodic patterns ofperiod Φ / due to the two flavors. Their phase difference∆ θ depends on the numbers n , of the localized quasi-particles of the two flavors inside the interferometer. Onefinds ∆ θ = (2 θ − θ )( n − n ) = π ( n + n ) mod 2 π .Hence, the two interference patterns cancel, if an oddnumber of quasiparticles are localized in the device. Thesame behavior is expected for the Pfaffian state [1] andthe flavor-symmetric 331 state [21]. At the same time,a more complex Mach-Zehnder setup is known to unam-biguously distinguish the 331 and Pfaffian states [22] andmay help probe the 113 topological order.All states, reviewed in Ref. 10, exhibit universal tun-neling transport with g = integer / 8. According to Eq.(11), the 113 state is different and a very precise tunnel-ing experiment will show g = g − , where 2 / < g − < / g − ≈ g + (11) such that g − + g + = 3 / c .In contrast to our findings, numerical investigations ofsmall model systems favor the Pfaffian and anti-Pfaffianstates (see, e.g., Refs. 33–36). At the same time, ex-isting numerical results have a number of limitations.For example, Landau-level mixing [37, 38] was ignoredin many studies. 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We will alsoneglect the interaction between the fractional QHE neu-tral mode φ n and the integer charged mode φ c and willonly include the interaction of the neutral mode with thefractional QHE charged mode φ ρ and the interaction be-tween the integer and fractional charged modes. Thus,we consider the following edge action: L = − ¯ h π Z dxdt (cid:2) ∂ t φ ρ ∂ x φ ρ + 2 v ρ ( ∂ x φ ρ ) − ∂ t φ n ∂ x φ n + v n ( ∂ x φ n ) + 12 ∂ t φ c ∂ x φ c + v c ∂ x φ c ) +2 v ρn ∂ x φ n ∂ x φ ρ + 2 v ρc ∂ x φ ρ ∂ x φ c (cid:3) , (13)where we estimate the velocities of the charged modesas v ρ ∼ e ǫ ¯ h (1 + ln da ) and v c ∼ e ǫ ¯ h (1 + ln d ′ a ′ ) with a ′ and d ′ being the width of the integer edge and its distancefrom the gate. We expect v ρn ∼ v n ≪ v ρ , v n ≤ v c and v ρc ≤ v ρ , v c . We first diagonalize the action part that does not dependon φ n . This is achieved by introducing new variables θ , such that φ ρ = 1 √ θ cos α + θ sin α ); φ c = √ − θ sin α + θ cos α ) , (14)where cos α = vuuut 12 + 12 r v ρc ( v ρ − v c ) ;sin α = vuuut − r v ρc ( v ρ − v c ) . (15)The action assumes the form L = − ¯ h π Z dxdt (cid:2) − ∂ t φ n ∂ x φ n + v n ( ∂ x φ n ) + X k =1 ∂ x θ k ( ∂ t θ k + v k ∂ x θ k )+ √ v ρn ∂ x φ n (cos α∂ x θ + sin α∂ x θ ) (cid:3) , (16)where v = v ρ + v c v ρ − v c s v ρc ( v ρ − v c ) ; v = v ρ + v c − v ρ − v c s v ρc ( v ρ − v c ) . (17) One can check that v ∼ v ρ and v ∼ v c .We now find corrections to the tunneling exponent g inthe first order in v ρn . We first find the correction due tothe contribution C = √ v ρn ∂ x φ n cos α∂ x θ in the lastline of the action (16) and then the correction from thecontribution C = √ v ρn ∂ x φ n sin α∂ x θ . The strategy isthe same on both steps. In particular, during the first stepwe omit C from the action and diagonalize the remainingaction with the transformation φ n = Φ cosh γ + Θ sinh γ ; θ = Φ sinh γ + Θ cosh γ, (18)where tanh 2 γ = − √ v ρn cos αv + v n . (19)This yields the action that consists of three pieces: onedepends only on θ , the second depends only on Φ andthe third depends only on Θ. We use that action to findthe scaling dimensions of the quasiparticle operatorsˆ T = exp (cid:18) i cos α √ θ + i sin α √ θ ± i φ n (cid:19) . (20)This yields g = ∓ v ρn cos α v + v n ) . Combining the correctionsdue to both contributions to the last line of Eq. (16) onefinds g = 38 ∓ v ρn (cid:18) cos αv + v n + sin αv + v n (cid:19) . (21)By inspecting different allowed relations between the pa-rameters of the action (13), one finds that g ≈ //