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Interaction and Particle–Hole Symmetry of Laughlin Quasiparticles Arkadiusz Wójs Department of Physics, University of Tennessee, Knoxville, Tennessee 37996, and Institute of Physics, Wroclaw University of Technology, 50-370 Wroclaw, Poland 1 The pseudopotentials describing interaction of Laughlin quasielectrons (QE) and quasiholes (QH) 0 in an infinite fractional quantum Hall system are studied. The QE and QH pseudopotentials are 0 similarwhichsuggeststhe(approximate)particle–holesymmetryrecoveredinthethermodynamical 2 limit. Theproblemofthehypotheticalsymmetry-breakingQEhard-corerepulsionisresolvedbythe n estimatethatthe“forbidden”QEpairstatehastoohighenergyandisunstable. Strongoscillations a oftheQEandQHpseudopotentialspersistinaninfinitesystem,andtheanalogousQEandQHpair J stateswithsmallrelativeangularmomentumandnearlyvanishinginteractionenergyarepredicted. 0 1 73.43.Lp, 71.10.Pm ] l An important element in our understanding of the TheconceptofLaughlingroundstatesformedbyLaugh- l a incompressibile-fluid ground states1–3 formed in a two- linQP’sgaverisetoHaldane’shierarchy9 ofincompress- h dimensionalelectrongas (2DEG) in high magnetic fields ible“daughter”statesatν =(2p +1)−1,inaddition - QP QP s has been the identification of Laughlin correlations1 in tothe“parent”statesatν =(2p+1)−1. Forexample,the e a partially filled lowest Landau level (LL). These cor- incompressibleν = 2 statecanbeviewedastheν = 1 m 7 QH 3 relations can be defined4,5 as a tendency to avoid pair state of QH’s in the parent ν = 1 state of electrons. t. eigenstates with the largest repulsion (smallest relative The criterion for the “shor3t range” of the two- a pairangularmomentum )inthelow-energymany-body body repulsion that causes Laughlin correlations is ex- m R states. The incompressibility results at a series of filling pressedintermsoftheinteractionpseudopotentialV( ), - factors (number of particles divided by the number of defined4,10 asthe pairinteractionenergyV asafunctRion d n states) ν =(2p+1)−1 at which the p leading pair states of . Therefore, the knowledge of V( ) is necessary to R R o at = 1, 3, ..., 2p 1 are completely avoided in the predictthetypeofcorrelations(andpossibleincompress- R − c non-degenerate (Laughlin) ground state, but not in any ibility) in a given many-body system. [ of the excited states. In this note we continue our earlier study11 of inter- 1 Each Laughlin-correlated state can be understood in actions between Laughlin QP’s. The QE and QH inter- v terms of two types of quasiparticles (QP): quasielec- action pseudopotentials are calculated for the Laughlin 4 trons (QE) and quasiholes (QH), moving in an under- ν = 1 and 1 states of up to 8 and 12 electrons on a 3 5 3 lying Laughlin ground state (“reference” or “vacuum” Haldanesphere,9 respectively,andextrapolatedtoanin- 1 state). The QP’s are the elementary excitations of the finite planar system. Our results lead to the following 1 Laughlin fluid and correspond to an excessive (QH) or two main conclusions: 0 1 missing (QE) single-particle state, compared to an ex- (i) Opposite to what seemed to follow from finite-size 0 act ν = (2p + 1)−1 filling. They have finite size and calculations,11,12 the sign and magnitude of the pseu- / (fractional) electric chargeof (2p+1)−1e, and thus (in dopotential coefficients calculated for an infinite plane t ± a analogy to LL’s of electrons) the single-QP spectrum in agree with the expectation that, being charge excita- m a magnetic field is degenerate at a finite energy denoted tions, the QP’s of the same type must repel and not - as εQP. For the QP’s at a complex coordinate z = 0, attractone another. However,the oscillations in the QP d their wavefunctions are obtained by applying the pref- charge density cause oscillations in V( ), and the QP con wacatvoerfsuQnckti∂on/∂Φz2kp+(Q1E=)Qani<djQ(zik−zkz(jQ)2Hp+)1t.o the Laughlin pisahiirngstaintetesrwacitthionsmeanlelrRgy(asrmeapllrerdadicituesd)R.anTdhenevaarnlyishvainng- : Partially filled lowest LL is not the only many-body of repulsion in these states rules out incompressibility of v system with Laughlin correlations, which generally oc- suchhypothetical11 groundstatesinHaldane’shierarchy i X cur when the single-particle Hilbert space is degenerate as ν = 6 or 6 , and limits the family of valid hierarchy 17 19 r and the two-body interaction is repulsive and has short statestothe(experimentallyobserved)Jain’ssequence13 a range.4,5 AmongotherLaughlin-correlatedsystemsarea at ν =n(2pn 1)−1. This vanishing is also essential for two-componentsystemofelectronsandchargedexcitons the stability o±f fractionally charged excitons14 hQE (n − n (X ,twoelectronsboundtoavalencehole)formedinan QE’s bound to a valence hole) observed15 in photolumi- electron–hole plasma in a magnetic field,6,7 or a system nescence of the 2DEG. of (bosonic) electron pairs formed near the half-filling of (ii) From the similarity of QE and QH pseudopoten- the first excited LL5 (Moore–Read8 state at ν = 5). 2 tialsweconcludethat,inagreementwithHaldane’sintu- Due to their LL-like macroscopic degeneracy and the itivepicture ofQP’splaced“between”the electrons,9 no Coulomb nature of their interaction, Laughlin correla- asymmetrybetweenQEandQHHilbertspaces(angular tions canalso be expected in a systemofLaughlinQP’s. 1 momenta) exists that should require introduction of a 11.5 phenomenological hard-core QE–QE repulsion16 to pre- 13.2 continuum dict the correct number of many-QE states in numerical 0.02 ) energyspectra. Instead,therepulsionenergyofthe“for- l2E (e/ (c) N=11, 2S=28 (d) N=12, 2S=31 0.01 bidden” QE pair state is finite but higher than that of a correspondingQHpairstate. Itevenexceedsthe Laugh- 11.4 lin gap ∆ = ε +ε to create an additional QE–QH 0.01 13.1 QE QH pairandmakesthisQEpairstateunstable(andpushesit ) 0 4 L 8 12 0 4 L 8 12 0.00 l2/ into the 3QE+QHcontinuum). This instability explains e why Jain’s composite fermion (CF) picture13 correctly V ( N= 8 N= 9 predicts the lowest-energy bands of states, despite the 0.00 N=10 asymmetry of QE and QH LL’s introduced by an (un- N=11 -0.01 N=12 physical) effective magnetic field. Also, the similarity of the QE andQHpair states and energiesprecludes quali- tativelydifferentresponseofaLaughlin-correlated2DEG -0.01 to a positively and negatively chargedperturbation.14 -0.02 The knowledge of pseudopotentials defining interac- (a) V (b) V QH QE tionsofLaughlinQP’sisessentialinHaldane’shierarchy9 of the fractional quantum Hall effect,1–3 in which they 1 3 5 7 9 11 13 1 3 5 7 9 11 13 determine those of Laughlin fillings at which the QP’s FIG.1. Thecomparisonofquasihole(a)andquasielectron form (daughter) Laughlin incompressible states of their (b)pseudopotentialsV(R)calculated at ν = 13 inN-electron own. Although they are to a large extent equivalent, systems on a Haldane sphere. Insets: The comparison of Haldane’s hierarchydiffers from Jain’sCF picture in the 11-electron (c) and 12-electron (d) energy spectra in which “symmetric” description of the two types of QP’s. Hal- the lowest-energy band contains two quasielectrons. In (c), dane’s elegant argument9 that both QE and QH exci- opencirclesshowthe(shiftedinenergy)11-electronspectrum of two quasiholes. tations are bosons placed “between” the N (effectively one-dimensional) electrons yields equal numbers of possible QE and QH states, g˜ = g˜ = N + 1 (tildes QE QH hard-core in the QE–QE repulsion (and its absence in mean bosons), which on a sphere correspond to equal ∗ V ) or the resulting asymmetry between l and l . single-particle angular momenta, ˜lQE = ˜lQH = 21N (be- QTHhis asymmetry is inherent in Jain’s CFQpHicture,1Q3Ein cause g˜ = 2˜l+1; the lowest LL on a Haldane sphere is which QE’s and QH’s are converted into particles and an angular momentum shell of l = S, half the strength vacances in different CF LL’s whose (different) angular of Dirac’s magnetic monopole in the center4,9,17). momenta are equal to l∗ and l , respectively. How- QE QH In a system of n QP’s, a mean-field Chern–Simons ever, the effective magnetic field leading to the correct transformation (MFCST)18–20 can further be used to values of g∗ and g in the CF picture does not phys- QE QH convert such bosonic QP’s to more convenient fermions ically exist. While for the QH states the effective field withg =g˜+(n−1)yieldingl =˜l+12(n−1). However,for is one of possible physical realizations of the MFCST QE’s this value of l seemed to predict incorrect number describing Laughlin correlations (the avoidance of most of low-energy states in the numerical energy spectra un- strongly repulsive pair states) in the underlying electron less the pair state at the maximum angular momentum system,4 no explanation for g∗ being smaller than g Lmax =2l 1 was forbidden.16 Ona sphere, the relation is possible within the CF modQeEl itself. QH between L−= l1 +l2 and is L = 2l , and thus To resolve this puzzle we have examined the QE and | | R −R the exlusion of the pair state at Lmax is equivalent to a QH pseudopotentials calculated for the systems of N hypothetical hard-core repulsion, VQE(1)=∞. 12 electrons at ν = 31 and 51. In Fig. 1 we compare VQ≤E Suchinteractionhard-corecanbeformallyremovedby (a) and V (b) obtained at ν = 1 for different values an appropriate redefinition of the single-particle Hilbert QH 3 of N. In both frames, = 2l L, with l = l = QE QH space. This is accomplished by a fermion-to-fermion 1(N +1). To obtain theRvalues−of V, the energies of the MFCST4,11,21,22 which replaces g by g∗ = g 2(n 1), 2 LaughlingroundstateandofthetwoQP’saresubtracted and l = 12N + 12(n − 1) by l∗ = 12N − 21(n−− 1).−By from the energies of the appropriate QP pair states11,12 “elimination” we mean that the angular momenta L nQE (such as the QE pair states for N =11 and 12 shown in of states containing n QE’s can be obtained by simple the insets). The energy is measuredin the units ofe2/λ, andunrestrictedadditionofnindividualangularmomen- and λ is the magnetic length. tum vectors l∗QE followed by antisymmetrization (QE’s In the limit of N , the sphere radius R √N are treated as indistinguishable fermions), just as LnQH diverges and the num→eri∞cal values of V( ) conve∼rge to could be obtained by antisymmetric combination of n those describing an infinite 2DEG on aRplane. In this vectors lQH. Although they seem to agree with the “nu- (planar) geometry, is the usual relative pair angu- merical experiments,”4,11,16 no explanation exists for a lar momentum. ReRmarkably, when is defined as QE R 2 complicated objects than electrons, and the oscillations (c) QH, n =1/5 (d) QE, n =1/5 in V ( ) reflect the oscillations in their more compli- 0.06 0.04 QP R 0.02 0.02 cated charge density profile (similar oscillations occur in the electron pseudopotentials in higher LL’s). The con- 0.00 0.00 sequences of this fact are even more important. 0.04 0.02 First, from a general criterion4,5 for Laughlin correlations at ν (2p+1)−1 (defined as the avoiding of pair 0.0 0.1 0.2 0.0 0.1 0.2 ≈ states with < 2p + 1 in the low-energy many-body ) R l2e/0.02 0.00 states) in a system interacting through a pseudopoten- V ( tial V( ) we find that the QP’s of the parent Laughlin R state of electrons form Laughlin states of their own only at ν = 1. These states and their ν = 1 daugh- QP 3 QE 3 0.00 -0.02 tersexhaustJain’sν =n(2pn 1)−1 sequence. No other ± incompressibledaughterstatesoccurinthehierarchy,in- =1 cluding the (ruled out earlier11) ν = 4 or 4 states or =3 11 13 -0.02 =5 -0.04 the hypothetical11 ν = 6 or 6 states. Despite all the 17 19 (a) QH, n =1/3 =7 (b) QE, n =1/3 differences between Haldane’s hierarchy and Jain’s CF model, our conclusion makes their predictions of the in- 0.0 0.1 0.2 0.0 0.1 0.2 1/N 1/N compressibility at a given ν completely equivalent. FIG.2. Theleadingquasihole(ac)andquasielectron(bd) Second, the (near) vanishing of V (3) explains the pseudopotential coefficients V(R) calculated at ν = 31 (ab) stability of the hQE complex14 in tQhEe 2DEG interact- and ν = 1 (cd) in N-electron systems on a Haldane sphere, 2 5 ing with an (optically injected) valence hole. Being the plottedasafunction ofN−1. Thindottedlinesshowextrap- most strongly bound and the only radiative state of all olation to N →∞. “fractionally chargedexcitons” hQE , the hQE is most n 2 likely the complex observed15 in the PL spectra of the 2l L rather than 2l∗ L, the QE and QH pseu- 2DEG at ν > 1. QE − QE − 3 dopotentials become quite similar. The main difference Third,sinceVQE(5)isabouttwicelargerthanVQH(5), isthe obviouslackofthe QE =1stateandstrongeros- itisalsoplausiblethatVQE(1)couldbemuchlargerthan R cillationsintheVQE(R),butthemaximumatR=5and VQH(1), sothat the RQE =1 state would fallin the con- the minima at = 3 and 7 are common for both V tinuumandcouldnotbeidentifiedintheenergyspectra. QE R and V . The same structure occurs also for the QP’as InFig.1(c),ontopofthe11-electronspectrumatDirac’s QH in the ν = 1 state.11 Most unexpected in Fig. 1 are the monopole strength (the number of magnetic flux quanta 5 negativesignsofV . Theonlypositivepseudopotential piercing the Haldane sphere9,4,17) 2S =28, marked with QP coefficient is V (1), which might indicate that, despite full dots, in which the lowest-energy states contain two QH QP’sbeingchargeexcitations,bothQE–QEandQH–QH QE’s at ν = 1, with open circles we have marked an- 3 interactions are generally attractive. other spectrum calculated for the same N = 11 but at In Fig. 2 we plot a few leading pseudopotential coeffi- 2S = 32, whose lowest-energy band contains two QH’s. cients (those at the smallest values of ) V and V The second spectrum is vertically shifted so that the en- QH QE at ν = 1 and 1 as a function of N−1. CRlearly,the corre- ergies of the QE and QH pair states coincide at L = 1 3 5 sponding coefficients of all four pseudopotentials behave (i.e. at = 11) at which VQE and VQH are both negli- similarly which confirms the correct use of l rather gible), bRut the energy units (e2/λ) are the same. Since QE thanl∗ inthe definitionof . Itisalsoclearthatall the Laughlin gap ∆ to the continuum of states with ad- QE RQE coefficients V increase with increasing N (although at a ditional QE–QH pairs involves the sum of QE and QH differentrate for QE’sandQH’s)and itseems that none energies, it is roughly the same in both spectra. How- of them will remain negative in the N limit. In ever, the minima and maxima in VQE( ) are stronger attempt to estimate the magnitude of V →in t∞his limit we thanthoseinVQH( ),andthedifferenceRVQE VQH in- R | − | have drawn straight linest that approximately extrapo- creasesat larger L. While it is hardly possible to rescale late our data for some of the coefficients. Most notewor- VQH so as to reproduce VQE at L 9 and convincingly ≤ tthhyreevatliumeessalraer:geVrQtHh,aνn=1V/3Q(H1,)ν=≈1/50(.013)ea2s/λexbpeeicntgedafbrooumt pthraedticVtQEit(s1v1a)luiseiantdeLed=la1r1ge(rRtQhEan=∆1,1)w,hitichseewmosulldikeexly- the comparison of interacting charges (1e and 1e, re- plain the absence of the QE = 11 state below the con- 3 5 tinuum. An example ofRsuch “rescaling” procedure is spectively), the V(3) coefficients (seemingly) vanishing in all four plots, and V (5) 0.005e2/λ being shown in Fig. 1(c) with the line obtained by stretching QH,ν=1/3 ≈ V sothatitcrossesV (5)andV (3)atL=7and9, about twice smaller than V (5). QH QE QE QE,ν=1/3 respectively. Similar lines are shown in Fig. 1(d) for the The predicted small value of V(3) and of some other 12-electron spectrum corresponding to two QE’s in the leadingcoefficientsisbyitselfquiteinteresting,although lowest band (2S = 31). Certainly, this procedure, based it can be understood from the fact that QP’s are more 3 ontheassumptionthatVQE(3)andVQH(3)aresmalland 2D.C. Tsui, H.L. Störmer, and A. C. Gossard, Phys. Rev. that V(1) is proportional to V(5), is not accurate. Nev- Lett. 48, 1559 (1982). ertheless,havinginmindthesimilaritiesofV andV 3The Quantum Hall Effect, edited by R. E. Prange and S. QE QH in Figs. 1 and 2, and in the absence of any physical rea- M. Girvin, Springer-Verlag, New York 1987. son why the = 1 state might not exist while the 4A.Wójs and J. J. Quinn,Phil. Mag. B 80, 1405 (2000). QE = 1 statRe does, we believe that it is more reason- 5A.Wójs, Phys. Rev.B (to appear, cond-mat/0009046). QH RabletoassumethatV (1)isfinite,althoughlargerthan 6A. Wójs, P. Hawrylak, and J. J. Quinn, Phys. Rev. B 60 QE ∆. The fact that the = 1 state is pushed into the 11661 (1999). RQE 7A. Wójs, I. Szlufarska, K.-S. Yi, and J. J. Quinn, Phys. 3QE+QHcontinuumsimplymeansthatitisunstableto- Rev.B 60 11273 (1999). ward spontaneous creation of a low-energy QE–QH pair 8G. Moore and N. Read, Nucl.Phys. B 360, 362 (1991). with finite angular momentum (magneto-roton). 9F. D.M. Haldane, Phys.Rev.Lett. 51, 605 (1983). The assumption that ∆ < V (1) < restores the QE ∞ 10F. D. M. Haldane and E. H. Rezayi, Phys. Rev. Lett. 60, elegant symmetry of Haldane’s picture of QP’s “placed” 956 (1988). betweenelectrons.9 Itreplacestheproblemofexplaining 11A.Wójs and J. J. Quinn,Phys. Rev.B 61 2846 (2000). the QE hard-core16 by a question of why VQE is larger 12S. N. Yi, X. M. Chen, and J. J. Quinn, Phys. Rev. B 53, than V at short distance (e.g., at = 1 and 5; see QH R 9599 (1996). Fig. 1). But the fact that VQE and VQH are not equal 13J. Jain, Phys.Rev. Lett. 63, 199 (1989). at shortdistance is by no means surprising since the QE 14A.WójsandJ.J.Quinn,Phys.Rev.B63,045303 (2001); and QH have different wavefunctions. ibid.63, 045304 (2001). In conclusion, we have calculated the pseudopoten- 15D. Heiman, B. B. Goldberg, A. Pinczuk, C. W. Tu, A. tials VQP,ν( ) describing interaction of QE’s and QH’s C. Gossard, and J. H. English, Phys. Rev. Lett. 61, 605 in LaughlinRν = (2p+1)−1 states of an infinite 2DEG. (1988). Thesepseudopotentialsareallsimilar,showingstrongre- 16S. He, X.-C. Xie, and F.-C. Zhang, Phys. Rev. Lett. 68, pulsion at = 1 and 5, and virtually no interaction at 3460 (1992). =3. TheRunexpected QE–QE and QH–QH attraction 17X.M.ChenandJ.J.Quinn,SolidStateCommun.92,865 R which results in few-electron calculations disappears in (1996). the limit of an infinite system. Because the QP charge 18G. S. Canright, S. M. Girvin, and A. Brass, Phys. Rev. atν =(2p+1)−1 decreaseswithincreasingp,theQPin- Lett. 63, 2291 (1989). teraction at ν = 1 is stronger than at ν = 1. Because of 19F.Wilczek, Fractional Statistics and AnyonConductivity, 3 5 different QE and QH wavefunctions, V is larger than World Scientific, 1990. QE 20J. J. Quinn, A. Wójs, J. J. Quinn, and A. T. Benjamin, V at small (short distance). The coefficient V (1) QH QE R Physica E (toappear, cond-mat/0007481). exceeds the Laughlingap∆ to create anadditional QE– 21A.Lopez and E. Fradkin,Phys. Rev.B 44, 5246 (1991). QHpair,whichmakestheQEpairstateat =1unsta- R 22B. I. Halperin, P. A. Lee, and N. Read, Phys. Rev. B 47, ble. This instability, rather than a mysterious QE hard- 7312 (1993). core or an inherent asymmetry between the QE and QH angular momenta, is the reason for the overcounting of few-QEstateswhen,followingHaldane,QE’saretreated as bosonswith ˜l = 1N. In particular,itexplains the ab- 2 sence of the L = N multiplet in the low-energy band of states in the N-electron numerical spectra at the values of 2S =(2p+1)(N 1) 2, corresponding to two QE’s in the Laughlinν =−(2p+−1)−1 state. The (near)vanish- ingofV (3)isthereasonwhynohierarchystatesother QP than those from Jain’s ν = n(2pn 1)−1 sequence are ± stable. It is also the reasonfor the strong binding of the fractionally charged exciton hQE . 2 The gratefully acknowledges discussions with John J. Quinn,PawelHawrylak,LucjanJacak,andIzabelaSzlu- farska,andsupportfromthe PolishState Committee for Scientific Research (KBN) Grant No. 2P03B11118. 1R.Laughlin, Phys.Rev.Lett. 50, 1395 (1983). 4