Neutral Fermion Excitations in the Moore-Read state at ν=5/2
NNeutral Fermion Excitations in the Moore-Read state at ν = 5 / Gunnar M¨oller, Arkadiusz W´ojs, , and Nigel R. Cooper TCM Group, Cavendish Laboratory, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom Institute of Physics, Wroclaw University of Technology, 50-370 Wroclaw, Poland (Dated: September 24, 2010)We present evidence supporting the weakly paired Moore-Read phase in the half-filled secondLandau level, focusing on some of the qualitative features of its excitations. Based on numericalstudies, we show that systems with odd particle number at the flux N φ = 2 N − ψ . Previous studies of the ν = 5 / ν = 5 / qualitative properties of the ν = 5 / NF of the order of the charge gap ∆ c [18]. Second, westudy the energetics of a NF in the presence of chargedquasiparticles (QPs): positive quasiholes (QHs) or neg-ative quasielectrons (QEs). In this case, our thermody-namic extrapolations of the energy are consistent with agapless NF. This confirms one of the core features of thenon-Abelian statistics of the MR state: the topologicaldegeneracy of two possible fusion channels 1 and ψ of apair of two distant QPs, corresponding to the absenceor presence of an additional fermion. Furthermore, we determine the NF dispersion and propose an experimentto probe it directly. We also give the first evidence thatthe QHs and QEs of the ν = 5 / ψ channel.For our studies, we perform exact diagonalizations ofmodel Hamiltonians for N ≤
20 spin-polarized, quasitwo-dimensional (2D) electrons on a sphere of radius R pierced by N φ = 2 N − σ magnetic flux quanta. The MRPfaffian state is at the shift of σ = 3. We consider threemodel Hamiltonians: First, the Coulomb interaction H C in the second LL, as defined by the pseudopotential co-efficients either for a 2D layer of effective width w = 0or w = 3 λ ( λ being the magnetic length). Second, amodified Coulomb interaction H , with the short-rangepseudopotential V (for pairs with relative angular mo-mentum m = 1) increased by δV = 0 . e /λ . Thisincrease in V is known to yield maximum overlap withthe MR Pfaffian [9, 19], and was found to mimic LL mix-ing [20] in the perturbative analysis of Bishara-Nayak[14, 21]. Third, and finally, the “Pfaffian (Pf) model”given by the projector on triplets of minimal relative an-gular momentum ( m = 3), H Pf = (cid:80) i,j,k P (3) ijk . Thoughwe focus on the Pfaffian state, the particle-hole symmetryof the two-body interactions H C , H makes our conclu-sions equally valid for its conjugate, the “anti-Pfaffian”.We now analyze the excitation spectra of these Hamil-tonians. A typical set of raw data is shown in Fig. 1 for N = 15 particles. Here we first focus on Figs. 1(d-f), inwhich we confirm the existence of a dispersive mode asso-ciated with a single fermionic QP for finite systems withan odd N [3]. All these three spectra feature a groundstate at nonzero angular momentum within a low-lyingband of collective excitations. Empirically, we find thatthis well-separated band extends up to the angular mo-mentum L = N/
2, as seen most clearly for H Pf . Theminima of the dispersion are far below the continuum.Eigenstates within the band are spaced by ∆ L = 1, asexpected for a single mobile NF in the background of anunderlying quantum liquid.In Fig. 2 we collected data from energy spectra fordifferent N ≤
19 to estimate the NF dispersion, which a r X i v : . [ c ond - m a t . m e s - h a ll ] J u l is compared to the magnetoroton mode in Fig. 2(c). Toreduce finite-size effects, for each N we used polynomialinterpolation of our data to locate the wave vector k and the energy E corresponding to the minimum of thedispersion. We find that k ( N ) is essentially constant,and the minimum E is well described by E ( N ) (cid:39) ∆ NF + β/N , converging to a finite NF gap ∆ NF measured withrespect to the MR ground state energy [22]. Subtractingthe finite-size scaling, our data reduce to one well-definedcurve (most accurately for H Pf ).The shape of the NF mode varies significantly betweenour model Hamiltonians. A general feature emerging forall spectra is a NF dispersion with two minima [these arebest seen in panel (c)]: a deeper “NF ” near k λ (cid:39) ” near kλ (cid:39)
2. The dispersionof H C shows strong finite-size effects, which we interpretas a consequence of the proximity to a phase transitioninto a charge-density wave phase [19]. Indeed, H whichis known to yield a state well inside the weakly pairedphase also produces cleanly defined dispersions, particu-larly for the NF. (The magnetoroton dispersion for H C or H – not shown – has stronger finite-size effects than theNF, as it involves two interacting QPs instead of one.)Comparing different panels, it is remarkable that as soon . . E ( e / λ ) . . E H C (LL ) H ( δ V =0.04 e / λ ) H Pf . . E ( e / λ ) . . E NF+2QHs (a) (b) (c)(d) (e) (f) s h i ft ed s h i ft ed NF+2QEs L . . E ( e / λ ) L L . . E (g) (h) (i) s h i ft ed NF FIG. 1. (color online) Energy spectra (bare interaction en-ergy E versus angular momentum L ) of N = 15 electronsin a half-filled second LL, interpreted in terms of the neutralfermion (NF) excitation in the Pfaffian (Pf) ground state.Different values of the magnetic flux are: N φ = 26 (top), 27(center), and 28 (bottom), corresponding to the NF with ad-ditional pair of charged quasielectrons (QEs), NF alone, andNF with additional pair of charged quasiholes (QHs), respec-tively. Different interactions are: H C = pure Coulomb (left), H = Coulomb with an additional enhancement of the m = 1pair pseudopotential (center), and H Pf = three-body PfaffianHamiltonian with the only triplet pseudopotential at m = 3(right). Labels indicate squared overlaps |(cid:104)H C/ |H Pf (cid:105)| forthe low-lying bands. as the two minima of the NF dispersion actually form(which for H seems to require δV (cid:38) . e /λ ), they re-main located at virtually unchanged wave vectors, whilethe bandwidth depends significantly on the particularmodel (e.g., on δV ). The NF lies slightly below theFermi surface of CFs, i.e. k (cid:39) k F (for a half-filled LLof spinless fermions, k F = 1 /λ ). This confirms the ex-pectation of Bogoliubov theory, that in a weakly pairedphase of CFs, and for weak coupling, the minimum isclose to k F [4]. The presence of the second minimumNF is more surprising. It could arise as a superpositionof a NF with additional magnetoroton excitations. How-ever, the combined energy for a NF and magnetorotonis found to be larger than NF . Tentatively, this featurecould be related to a bound state of these objects. Inany case, we conclude that NF cannot decay into a NFand a magnetoroton, so it is a genuine feature of the NFdispersion describing a long-lived excitation, and can betested in experiment.Direct observation of the NF requires a probe changingthe fermion number of the second LL. One such probe isphotoluminescence (PL), in which an electron in this LLrecombines with a photoexcited valence band hole (the‘1,0’ or ‘1,1’ PL lines in Ref. [23]). The PL spectrumdepends on the nature of the state into which the holerelaxes prior to recombination. Often for fractional quan-tum Hall systems, this is an “excitonic” state in which thehole binds a charge e to form a neutral exciton moving inthe background incompressible liquid [24]. Interestingly,for the ν = 5 / N e [26]. Which of these two states has the lower energyis difficult to predict: this amounts to determining thefusion channel of four QEs in the presence of the hole.However, if such fusion can be viewed as pairwise, thenthe two pairs will fuse either both to 1 or both to ψ , giv-ing an overall even N e . PL recombination removes oneelectron, so it leaves a final state with opposite parity tothe initial state. For an odd initial N e , recombination canoccur to the MR ground state, yielding a sharp PL line k λ . . E ( e / λ ) k λ k λ . . E H C (LL ) H ( δ V =0.04 e / λ ) H Pf (a) (b) (c)neutralfermion N=12 1416 18N=11 13 1517 19 δ V =0.02 e / λ magneto-roton FIG. 2. (color online) Dispersions (energy E versus wavevector k ) of the neutral fermion (NF) collective modes of thehalf-filled second LL, estimated from the systems of N ≤ N φ = 2 N −
3, for the differentHamiltonians of Fig. 1. Gray dashed lines in (b) show theevolution of ∆ NF with δV . For comparison, (c) also showsthe magnetoroton mode for H Pf . (symmetrically broadened by disorder). For even initial N e , recombination leaves an odd final N e , and so mustinvolve the creation of a NF . Since the excitonic stateis typically easily localized by disorder [24], the result-ing PL spectrum will probe the density of states of theNF band. The minima in the NF band will appear astwo asymmetrically broadened peaks [27]. Observationof this double-peak structure in PL would allow directmeasurements of the minima of the NF band.The evolution of the dispersion minimum NF as afunction of δV , sketched as gray lines in Fig. 2(b), givessome insight into the nature of phase transitions. Atsmall δV , the NF gap ∆ NF remains nonzero, so we ex-pect the pairing nature of the phase to survive up to thetransition into a charge-density wave [19]. The collectiveNF mode flattens at small δV ; however, the spectrum isdominated by finite-size effects below V ≈ .
02. At large δV (approaching interactions resembling the lowest LL)a smooth decay of ∆ NF indicates a continuous weakeningof pairing (see also [9]). The minimum becomes steeper,and the gap collapses near k ≈ k F = 1 /λ , consistent witha crossover into the CF Fermi liquid state.We now turn to investigate the physics of a neutralfermion in the presence of two QEs or QHs. For a pairof QHs in the MR state (even N ), H Pf features a bandof zero-energy states spaced by ∆ L = 2 and terminatingat L = N/ H Pf at even N ,again with ∆ L = 2 but terminating at L = N/ −
2. For2QE+NF configurations, a band with the same angularmomenta is obtained by removing one particle and twoflux from the system, so the lowest energy NF states canbe thought of as holelike. Figures 1 (a-c) show example2QE+NF spectra.In the presence of QPs the low-lying excitations are notas well separated in the spectrum as for the NF alone, andmore significant finite-size effects are expected. Thus, weproceed carefully in analyzing the energetics. The pres-ence of the NF can affect the angular momentum, relativepositions, and shape of the QPs, changing their interac-tion energy in an unknown way. Since we are unableto subtract these effects systematically, instead we av-erage the energy over all states in the low-energy bandassociated with the two QP(+NF) excitations. Thus,we evaluate the (properly normalized) average (cid:104) E α (cid:105) ∼ (cid:80) L (2 L + 1) E α ( L ). The energy of each eigenstate E α ( L )is measured with respect to the ground state energy [22](where α indicates a set of QP numbers, α =2QE, 2QH,2QE+NF, and 2QH+NF). For Coulomb Hamiltonians,we apply standard corrections to the energies E α ( L ), in-cluding using rescaled magnetic length, applying an elec-trostatic charging correction of the energies, and correct-ing for the Coulomb interactions of QPs, δV QP , in the . . 〈 E α 〉 ( e / λ ' ) 〈 E α 〉 H C (LL ) H ( δ V =0.04 e / λ ) H Pf (a) (b) (c) /N . . E p r o x ( e / λ ' ) /N /N E p r o x small symbols: w=3 λ (d) (e) (f) FIG. 3. (color online) Comparison of the total energies of apair of QEs or QHs, with and without an additional NF (seetext for the precise definition), for systems of different size N and for the different Hamiltonians of Fig. 1. Top: ener-gies (cid:104) E α (cid:105) averaged over all 2QE/2QH (+NF) states; bottom:energies E prox taken for the smallest average QE–QE or QH–QH distance. While (cid:104) E α (cid:105) extrapolate to similar values withor without NF, the differences are significant for E prox . excited configurations [29–31].The values (cid:104) E α (cid:105) obtained after applying the chargingcorrections are shown in Figs. 3(a-c) and reflect the totalenergy of a system with two QEs or QHs, with or withoutan additional NF [31]. Importantly, for odd N it containsthe energy cost for adding a NF in the presence of a pairof QEs (∆ − NF ) or QHs (∆ +NF ), that includes the interac-tion of the NF with these QPs. Since we average over allpositions of the QPs, in finite-size systems one expects anonzero splitting ∆ ± NF between the ψ and 1 channels fromconfigurations with overlapping QPs. Estimates fromtrial states of QHs at close separation [32] suggest that forCoulomb interactions the splitting is (cid:39) . e /λ ≡ ∆ max .When averaged over all possible QP positions, the con-tribution would be significantly smaller, due in part toits oscillatory behaviour. For finite systems, we find the(average) splitting of fusion channels, including its finite-size effects, satisfies ∆ ± NF (cid:46) ∆ max . (For H Pf in Fig. 3(c),the splitting ∆ +NF vanishes by construction [28].)If the QPs are non-Abelian Ising anyons, as in theMoore-Read phase, then the energy splitting ∆ ± NF shouldscale to zero in the thermodynamic limit. The extrapola-tions in Fig. 3(a-c) are consistent with the vanishing ofboth splittings ∆ ± NF (in each case, the extrapolated valueis considerably smaller than its standard deviation). Al-though we cannot prove that the splittings vanish exactly,we emphasize that their best estimates are at least an or-der of magnitude smaller than the charge gap ∆ c or ∆ NF .It is highly nontrivial to find a near degeneracy on thisscale. We have examined the behaviour as a functionof δV in our model Hamiltonian H . We find that thesplitting remains similarly small over the same range ofinteractions for which the L = 0 ground state has a largegap and a high overlap with the Moore-Read Pfaffian [11,12], i.e. , extending upwards in δV from about H C to-wards the point of collapse of the NF gap. We take theseresults as evidence that, over this range of δV , the QEsand the QHs in these realistic systems have non-Abelianexchange statistics of the form of the Moore-Read phase.To investigate which fusion channel is preferred atshort distance, we have estimated the splitting ∆ ± NF forQPs at near-coincident points. In Fig. 3(d-f), we show E prox = E α ( L max ) using the low-lying 2QP / 2QP+NFstates with the largest angular momentum (and closestseparation of the QPs) [33]. In this case the odd–evensplitting opens for each of the considered Hamiltonians.The splitting also remains in the thermodynamic limit,revealing a slightly negative ∆ ± NF , signaling a preferencefor the ψ -channel [e.g., ∆ +NF = − . e /λ and∆ − NF = − . e /λ for H , and ∆ +NF = − . H Pf ]. Previously, the splitting had been known onlyfor QHs, and was based on variational wave functions[32]. Here, we report the splittings for both QE and QHbased on exact calculations in finite systems.In conclusion, we have analyzed the neutral fermionexcitations of the ν = 5 / ν = 5 / [1] R. Willett, J. P. Eisenstein, H. L. Stormer, D. C. Tsui,A. C. Gossard, and J. H. English, Phys. Rev. Lett., , 1776 (1987); J. P. Eisenstein, R. Willett, H. L.Stormer, D. C. Tsui, A. C. Gossard, and J. H. En-glish, ibid ., , 997 (1988); W. Pan, J. Xia, V. Shvarts,D. Adams, H. Stormer, D. Tsui, L. Pfeiffer, K. Baldwin,and K. West, ibid ., , 3530 (1999).[2] G. Moore and N. Read, Nucl. Phys., B360 , 362 (1991).[3] M. 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