We determine the exclusion statistics properties of the fundamental edge
quasi-particles over a specific $\nu=\half$ non-abelian quantum Hall state
known as the pfaffian. The fundamental excitations are the edge electrons of
charge $-e$ and the edge quasi-holes of charge $+{e \over 4}$. We explicitly
determine thermodynamic distribution functions and establish a duality which
generalizes the duality for fractional exclusion statistics in the sense of
Haldane.Comment: LaTeX, 4 pages, no figures; results for 1/q pfaffian adde

Schoutens, K.

English

arXiv

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UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl)UvA-DARE (Digital Academic Repository)Exclusion statistics for non-abelian quantum Hall statesSchoutens, K.DOI10.1103/PhysRevLett.81.1929Publication date1998Published inPhysical Review LettersLink to publicationCitation for published version (APA):Schoutens, K. (1998). Exclusion statistics for non-abelian quantum Hall states. PhysicalReview Letters, 81, 1929-1933. https://doi.org/10.1103/PhysRevLett.81.1929General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s)and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an opencontent license (like Creative Commons).Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, pleaselet the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the materialinaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letterto: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. Youwill be contacted as soon as possible.Download date:10 Mar 2023VOLUME 81, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 31 AUGUST 1998ecificonalExclusion Statistics for Non-Abelian Quantum Hall StatesKareljan SchoutensInstitute for Theoretical Physics, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands(Received 13 March 1998)We determine the exclusion statistics properties of the fundamental edge quasiparticles over a spn ­ 12 non-Abelian quantum Hall state known as theq ­ 2 Pfaffian state. The fundamental excitationsare the edge electrons of charge2 and the edge quasiholes of charge1 y4. We explicitly determinethermodynamic distribution functions and establish a duality which generalizes the duality for fractiexclusion statistics as described by F. D. M. Haldane. [S0031-9007(98)06847-1]PACS numbers: 73.40.Hm, 05.30.–dfordisloutngnssndenge:icsfterotrenedSoon after the discovery of the fractional quantuHall effect, Laughlin proposed the many body wavfunctions that have ever since served as the theoretbasis for the understanding of the phenomenon. It wquickly established that the fundamental quasiparticover the Laughlin states are unusual in a number of wain particular, they have fractional charge and fractionbraiding statistics.The prediction of fractionally charged excitations harecently been confirmed in a number of independeexperiments based on shot noise [1] and on the physof the Coulomb blockade [2]. The fractional braidinstatistics of Laughlin quasiparticles are harder to detin a direct way. Braiding particles in a two-dimensionasystem is a theoretical notion that does not have a direxperimental counterpart.To explore experimental consequences of fractionbraiding statistics, one may as a first step try to translathis notion into a related notion of fractionalexclusionstatistics, i.e., into some generalization of the Pauexclusion principle [3,4]. The exclusion statistics leada generalization of the Fermi-Dirac distribution, and frothis one derives predictions for observable quantities.In recent work [5,6], we have carried out part of thabove program for the quasiparticles over then ­ 1mLaughlin state. In particular, we have linked the braidinstatistics of these excitations to the exclusion statistof the edge quasiparticles in the same system. Tconnection is established with the help of a conformfield theory (CFT) whose conformal blocks reproducthe wave functions for the quasiparticle excitations awhose finite size spectrum coincides with the spectruof edge excitations. The braiding statistics of excitatioover the Laughlin state are Abelian with statistics angu ­pm for the charge6em excitations andu ­ mp forthe excitations of charge6e [7]. In [5,6] we establishedthat the exclusion statistics of edge electrons (of cha2e) and edge quasiholes (of charge1 em ) are given byfractional exclusion statistics in the sense of Haldane [with statistics parameterg ­ m andg ­ 1m , respectively,and no mutual statistics between the two. This result w0031-9007y98y81(9)y1929(4)$15.00meicalaslesys:alsnticsgectlectaltelitomegicshealendmnslerge3],asestablished by a method based on recursion relationstruncated conformal spectra [5].For a quantum Hall state with fundamental chargeedge excitations of chargeq (measured in units ofe) anddistribution functionnsed (generalizing the Fermi-Diracdistribution), the zero temperature Hall conductancequickly shown to besH ­ q2nmaxe2h, (1)with nmax the maximum (reached fore ! 2`) of nsed.For Haldaneg-ons one hasnmaxg ­1g and this gives thefamiliar result sH ­1me2h , both for the edge electronsand the edge quasiholes over then ­ 1m Laughlin state.In this quasiparticle formalism, the fact that the Halconductance does not depend on temperature comes abas a consequence of the duality between the statisticsgand 1g distribution functions [8].In the present Letter, we extend the above reasonito the case of a quantum Hall state (the so-calledq ­ 2Pfaffian state) with excitations that obey non-Abeliabraiding statistics. In this context, “non-Abelian” meanthat the wave function for a set of quasiparticle excitationat given positions has more than one component, athat the action of braiding (exchanging) two of thosexcitations is represented by a nondiagonal matrix action the multi-component wave function.The issue addressed in this Letter is a fundamental onwe are exploring the consequences for thermodynamand transport of the non-Abelian braiding statistics oa set of fundamental excitations in a condensed matsystem. To our knowledge, a similar problem has nbeen addressed until now [9]. As a characteristic featuof “non-Abelian exclusion statistics,” we shall identify anontrivial prefactor in the high-energy Boltzmann tail of ageneralized distribution function.After the experimental observation [10] of a quantumHall effect at the even denominator filling fractionn ­52 ,a number of candidate ground states with filling fraction ­ 12 have been proposed. One of these is the so-call© 1998 The American Physical Society 1929VOLUME 81, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 31 AUGUST 1998rtheisoleicsloyeasisreutedtesn-ledrnendaner-q ­ 2 Pfaffian state [7,11] with wave functionCPf ­ Pf√1zi 2 zj! Yi.jszi 2 zjd2Yie2sjzi j2y4,2d (2)with Pf s 1zi2zj d ­ As1z12z21z32z4 . . .d denoting the anti-symmetrized product over pairs of electrons and w, ­qh̄ceB the magnetic length. Following [7], one mainterpret this state as a conformal block ofn chiral vertexoperators in ac ­ 32 CFT. In the same manner, the wavfunctions for electron and quasihole excitations over tPfaffian state are obtained from conformal blocks with tappropriate insertions of electron and quasihole operatoTo write these operators, we represent the CFT asproduct of ac ­ 12 (Ising) CFT with Majorana fermionc and spin operators and a c ­ 1 Gaussian theorywith scalar fieldw. In terms of these fields, the electrooperatorGszd (of charge2e) and the quasihole operatofszd (of charge1 e4 ) take the following form:Gszd ; sce2ip2wd szd, fszd ; sseis1y2p2dwd szd .(3)The electron and quasihole operators play a dourole in the theory. Via their conformal blocks they givbulk wave functions for excited states. At the same timthey act as spectrum generating fields in the edge CFIn terms of statistics, the implications of the underlyinCFT are again twofold. On the one hand, the braidistatistics in the CFT directly translate into braidinstatistics of electron and quasihole (bulk) excitations. Othe other hand, the exclusion statistics properties ofelectron and quasihole operators (see below) dictatethermodynamic (and, to some extent, transport) properof the low-energy edge theories. For the Pfaffian stathe (non-Abelian) braiding statistics were studied in gredetail in [7,12]. The study of the exclusion statisticaspects is the subject of this Letter.The statistics of quasiparticles over then ­ 1m Laugh-lin states are Abelian and can be inferred from the propties of two-particle states. As such, they can be studiedinspecting the short distance operator product expansi(OPE) of the electron and quasihole fields. The OPElead to Abelian braiding statistics, with anglesmp and pmfor the electrons and quasiholes, respectively. The cresponding exclusion statistics are of the type proposby Haldane, with statistics parametersg ­ m, g ­ 1m , re-spectively [6]. The OPE’s for the electron and quasihoedge excitations over the the Pfaffian state have a mcomplicated structure, and it is no longer possible to drive the statistics from here. Indeed, the non-Abelian nture of the braiding statistics of the quasihole excitatiobecomes apparent only at the four-particle level.In [12], it was argued that the wave-function specifyina collection of 2n quasiholes (at given locations) ovethe Pfaffian state has2n21 independent components. Inthe thermodynamic limit, this degeneracy amounts1930ithyehehers.anrbleee,T.gnggnthethetieste,atser-byons’sor-edleoree-a-nsgrtoa degeneracy ofp2 per quasihole. This nonintegedegeneracy factor, which ends up as a prefactor inBoltzmann tail of the generalized distribution function,a manifestation of the non-Abelian nature of the quasihexclusion statistics.To derive quantitative results for the exclusion statistof the edge excitations over the Pfaffian state, we empthe method of truncated conformal spectra [5]. The idof this method is to establish an explicit quasiparticle baof the CFT chiral finite size spectrum, and from thederive the statistics. In [6], this procedure was carried oin great detail for the Laughlin states and for the chargector for more general hierarchical quantum Hall sta(Jain series).The complication that is new for the case of noAbelian quantum Hall states is the occurrence of so-calfusion channels in the specification of a multi-f state.These have their origin in the following fusion rule fothe spin field of the Ising CFT:fsg ? fsg ­ f1g 1 fcg . (4)Each time we apply af mode to a state containing aodd number of quanta, we have to specify a choicfor either the first or the second term on the right-haside of (4). For a state containingN quasiholes, the totalfusion path is encoded in a Bratteli diagram, which isarrowpath on the following grid:-. . . . -.-cs.- . . . .-.- 1 ,N . . . . . 4 3 2 1(5)with the vertical direction indicating the Ising sectors1,, andc and the horizontal indexi ­ 1, . . . , N labelingthe f modes. The starting point of an arrow path is thCFT vacuumj0l. The multi-f states for a given fusionpath are [withfszd ­Ps f2szs21y8]f21y82DN2nN . . . f21y82D22n2f21y82D12n1j0l , (6)with integersnN $ . . . n2 $ n1 $ 0 and with incrementsDi determined by the Bratteli diagram according toDi11 ­ Di 112 for--i 1 1 iand..i 1 1 iDi11 ­ Di 114 for - .i 1 1 iand. -i 1 1 iDi11 ­ Di for . -i 1 1 iand- .i 1 1 iwith D1 ­ 0. For example, the lowest energy fouquasihole states for the two possible fusion paths are.-.- f23y8f23y8f21y8f21y8j0l, E ­ 1 ,.-. -f211y8f27y8f25y8f21y8j0l, E ­ 3 .(7)VOLUME 81, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 31 AUGUST 1998anal-le-cs.r-enewedofoveIn the construction of multielectron states the issuefusion rules does not explicitly arise, but the details othe rules are different from what we had for the Laughlistates [6]. On the vacuumj0l, we allow the followingmultielectron states [withGszd ­Pt G2tzt23y2]:G2s1y2d2mM . . . G2s1y2d2m2 G2 122m1 j0l (8)with integersmi satisfyingmi $ maxsmi21 1 1, mi22 1 4d , (9)with m0 ­ 22, m21 ­ 23. For example, the lowestenergy multi-G states overj0l are of the form. . . G213y2G211y2G25y2G23y2j0l , (10)with increments alternating between1 and 3. It isquickly seen that the number of states allowed by therules is larger than for the case withg ­ 2, where theconstruction rule ismi $ mi21 1 2.We are now ready to specify a complete set of indepedent multiparticle states. We start by constructing a towof states overj0l (which we shall write asj0, 1l from nowon) by acting independently withf andG modes accord-ing to the rules that we specified. We then introduce fivadditional reference states for which we repeat the prcedure. The lowestf andG modes that are allowed oneach of the reference states are specified in Table I.The central claim of this Letter is that the union of all thquasiparticle states in all six sectors precisely forms a baof the chiral Hilbert space of the CFT, and thereby (uto certain projections, see [6]) of physical edge theorieThis is a mathematical result which, however, has a clephysical interpretation in terms of exclusion statistics (sebelow). The chiral spectrum of the CFT for the Pfaffiastate is usually understood in terms of representations oUs1d superaffine (Kac-Moody) current algebra. The nepoint of view that we advocate here is very different andirectly refers to the charge carrying excitations in thsystem. UsingMAPLE, we have checked that the characteidentities that are implied by our claim are satisfied to thfirst 25 nontrivial orders inq.The absence of any interference between the constrtion rules for thef and theG quanta means that there isTABLE I. Reference states for the quasiparticle constructioof the chiral Hilbert space over theq ­ 2 Pfaffian state. Thecreation operatorsf2s and G2t act on these states accordingto the exclusion rules specified in the text. The symbols1, c,and s denote Ising sectors and the quantitiesD1 and m0, m21determine the lowestf andG modes that are allowed.State Charge Energy D1 m0, m21j0, 1l 0 0 0 22, 23j0, cl 0 1212 21, 24j 2e4 , sl 2e4183421, 23j 2e2 , 1l 2e2143421, 22j 2e2 , cl 2e23414 0, 23j 23e4 , sl 23e45812 0, 22offnsen-ereo-esisps.arenf awdereuc-nno mutual exclusion statistics between the two. This isfirst sign of a (generalized) particle-hole duality betweethe two types of quasiparticles. We shall make this duity precise in what follows.Having specified the quasiparticle basis we can impment the procedure of [5] and derive the thermodynamiFor thef modes, we define truncated and projected patition sumsXlsx, qd andYlsx, qd, l ­ 18 ,38 , . . . , as expres-sions of the formtrace#lsxNf qEd , (11)with Nf the number of modes,E the dimensionlessenergy, x ­ ebmqh and q ­ e2bs2pyLr0d with r0 thedensity of states per unit length. The subscript#lindicates that the highest occupied modef2s is suchthat l 2 s is a nonnegative integer.Xl (Yl) is furtherrestricted by requiring that the total charge be an ev(odd) multiple of e4 . The rules specified above lead to threcursion relationsXl ­ Xl21 1 xqlsYl 1 Yl21y2d ,Yl ­ Yl21 1 xqlXl21y4 .(12)The initial conditions, which are sector dependent, follofrom the information in Table I. The solution of (12) canbe approximated asXl ,Qli­1 l1sqixd, with l1szqhd thelargest real solution of the equationsl 2 1d sl1y2 2 1d ­ z2qhl5y4. (13)This leads to a distribution functionnqhsed ­ fzqh≠zqh logsl1dg szqh ­ ebsmqh2edd (14)for the quasihole excitations.For theG quanta, the above rules are easily convertinto a recursion relation for the quantityVls y, qd, whichis the trace ofyNG qE, with NG the number ofG quantaand y ­ ebme , over all multi-G states with modes#l 2 12 . We findVl ­ Vl21 1 yql2s1y2dVl22 1 y2q2l22Vl242 y3q3l2s13y2dVl26 . (15)The corresponding distribution function isnesed ­ fze≠ze logsm1dg sze ­ ebsme2edd , (16)with m1szed the largest real solution ofsm4 2 z2e d sm2 2 zed ­ m5. (17)For temperatureT ­ 0, the distribution functionsnqhsed andnesed are step-down functions with maximumnmaxqh ­ 8 and nmaxe ­12 . At finite T , the asymptoticbehavior fore ! ` is nqh ,p2 e2be and ne , e2be.We stress that the prefactorp2 is a direct manifestationof the non-Abelian nature of the exclusion statisticsthe quasiholes. As an easy consequence of the abformulas, we establish the duality relation2nesed ­ 1 218nqhµ2e4∂, (18)1931VOLUME 81, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 31 AUGUST 1998yor.st.sprovidedmqh ­ 214 me. This result generalizes the duality for Haldane’s fractional exclusion statistics [8]. Thduality allows us to interpret the spectrum in differenways. One picture has quasiholes with energies rangfrom 2` to `, with the CFT vacuum and other referencstates represented as filled seas of quasiholes. An anative is a picture based on edge electrons, and then this the “mixed” picture that we started from, withf andGquanta of positive energies only.In the mixed picture, the total central chargec ­ 32 ofthe edge CFT is established as the sum of a contributc1qh from the quasiholes and a piece1e from the electrons.Using MAPLE, we evaluatedc1qh ­ 1.059 . . . , c1e ­ 0.440 . . . . (19)To compute the Hall conductance, we insertqqh ­14andnmaxqh ­ 8 or qe ­ 21 andnmaxe ­12 into (1), givingsH ­12e2h as expected. The fact that this resultindependent of temperature is guaranteed by the duarelation (18). We refer to [6] for the details of this way ocomputing and for further applications of the quasiparticformalism.In conclusion, we have established an appropriate geralization of the Pauli principle for a situation wherelectrons and quasiholes obey non-Abelian braiding stistics. We have also demonstrated a particle-hole duain this non-Abelian theory. In a future publication, wshall give a more extensive treatment of exclusion statics in CFT’s with non-Abelian braiding and give morexamples. These will include the so-called parafermionquantum Hall states, higher level Wess-Zumino-Wittetheories, and a spinon representation for the SOs5d1 CFT,which is relevant for SOs5d invariant ladders models forcorrelated electrons.The author thanks P. Bouwknegt, N. Read, and R. Avan Elburg for discussions. Part of this work wadone at the 1997 ITP Santa Barbara WorkshopLow-dimensional Field Theory: from Particle Physics tCondensed Matter. This research was supported in pby the foundation FOM of the Netherlands.1932-etingelter-ereionislityfleen-eta-lityetis-eicn. J.sonoartNote added.—The results presented here are easilgeneralized to the more general Pfaffian states with factszi 2 zjdq in (2). Equations (13) and (17) becomesl 2 1d sl1y2 2 1d ­ z2qhls3q21dys2qdsm2q 2 z2e d smq 2 zed ­ m3q21.(20)and the duality relation readsqnesed ­ 1 214qnqhµ2e2q∂. (21)After submitting this manuscript, we learned that S. BIsakov (private communication; to be published) hadetermined the exclusion statistics forbulk quasiholesover the generalq Pfaffian state, and found results thaare identical to those presented here foredgequasiholes.[1] R. de-Piciottoet al., Nature (London)389, 162 (1997);L. Saminadayaret al., Phys. Rev. Lett.79, 2526 (1997).[2] See, e.g., V. J. Goldman, Physica (Amsterdam)1E, 15(1997).[3] F. D. M. Haldane, Phys. Rev. Lett.67, 937 (1991).[4] S. B. Isakov, Mod. Phys. Lett. B8, 319 (1994);A. Dasnières de Veigy and S. Ouvry, Phys. Rev. Lett72,600 (1994); Mod. Phys. Lett. B9, 271 (1995); Y.-S. Wu,Phys. Rev. Lett.73, 922 (1994).[5] K. Schoutens, Phys. Rev. Lett.79, 2608 (1997).[6] R. A. J. van Elburg and K. Schoutens, cond-mat/9801272[7] G. Moore and N. Read, Nucl. Phys. B360, 362 (1991).[8] C. Nayak and F. Wilczek, Phys. Rev. Lett.73, 2740(1994); A. K. Rajagopal, Phys. Rev. Lett.74, 1048 (1995).[9] The analysis of T. Fukui, N. Kawakami, and S.-K. Yang[ J. Phys. Soc. Jpn.65, 1617 (1996)], which uses theterminology “non-Abelian exclusion statistics” is verydifferent from ours, as it is not based on the statisticof the fundamental spinons in the problem.[10] R. Willett et al., Phys. Rev. Lett.59, 1776 (1987).[11] M. Greiter, X. G. Wen, and F. Wilczek, Nucl. Phys. B374, 567 (1992).[12] C. Nayak and F. Wilczek, Nucl. Phys. B479, 529 (1996).

Exclusion statistics for non-abelian quantum Hall states

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UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl)UvA-DARE (Digital Academic Repository)Exclusion statistics for non-abelian quantum Hall statesSchoutens, K.DOI10.1103/PhysRevLett.81.1929Publication date1998Published inPhysical Review LettersLink to publicationCitation for published version (APA):Schoutens, K. (1998). Exclusion statistics for non-abelian quantum Hall states. PhysicalReview Letters, 81, 1929-1933. https://doi.org/10.1103/PhysRevLett.81.1929General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s)and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an opencontent license (like Creative Commons).Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, pleaselet the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the materialinaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letterto: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. Youwill be contacted as soon as possible.Download date:25 Jul 2022VOLUME 81, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 31 AUGUST 1998ecificonalExclusion Statistics for Non-Abelian Quantum Hall StatesKareljan SchoutensInstitute for Theoretical Physics, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands(Received 13 March 1998)We determine the exclusion statistics properties of the fundamental edge quasiparticles over a spn ­ 12 non-Abelian quantum Hall state known as theq ­ 2 Pfaffian state. The fundamental excitationsare the edge electrons of charge2 and the edge quasiholes of charge1 y4. We explicitly determinethermodynamic distribution functions and establish a duality which generalizes the duality for fractiexclusion statistics as described by F. D. M. Haldane. [S0031-9007(98)06847-1]PACS numbers: 73.40.Hm, 05.30.–dfordisloutngnssndenge:icsfterotrenedSoon after the discovery of the fractional quantuHall effect, Laughlin proposed the many body wavfunctions that have ever since served as the theoretbasis for the understanding of the phenomenon. It wquickly established that the fundamental quasiparticover the Laughlin states are unusual in a number of wain particular, they have fractional charge and fractionbraiding statistics.The prediction of fractionally charged excitations harecently been confirmed in a number of independeexperiments based on shot noise [1] and on the physof the Coulomb blockade [2]. The fractional braidinstatistics of Laughlin quasiparticles are harder to detin a direct way. Braiding particles in a two-dimensionasystem is a theoretical notion that does not have a direxperimental counterpart.To explore experimental consequences of fractionbraiding statistics, one may as a first step try to translathis notion into a related notion of fractionalexclusionstatistics, i.e., into some generalization of the Pauexclusion principle [3,4]. The exclusion statistics leada generalization of the Fermi-Dirac distribution, and frothis one derives predictions for observable quantities.In recent work [5,6], we have carried out part of thabove program for the quasiparticles over then ­ 1mLaughlin state. In particular, we have linked the braidinstatistics of these excitations to the exclusion statistof the edge quasiparticles in the same system. Tconnection is established with the help of a conformfield theory (CFT) whose conformal blocks reproducthe wave functions for the quasiparticle excitations awhose finite size spectrum coincides with the spectruof edge excitations. The braiding statistics of excitatioover the Laughlin state are Abelian with statistics angu ­pm for the charge6em excitations andu ­ mp forthe excitations of charge6e [7]. In [5,6] we establishedthat the exclusion statistics of edge electrons (of cha2e) and edge quasiholes (of charge1 em ) are given byfractional exclusion statistics in the sense of Haldane [with statistics parameterg ­ m andg ­ 1m , respectively,and no mutual statistics between the two. This result w0031-9007y98y81(9)y1929(4)$15.00meicalaslesys:alsnticsgectlectaltelitomegicshealendmnslerge3],asestablished by a method based on recursion relationstruncated conformal spectra [5].For a quantum Hall state with fundamental chargeedge excitations of chargeq (measured in units ofe) anddistribution functionnsed (generalizing the Fermi-Diracdistribution), the zero temperature Hall conductancequickly shown to besH ­ q2nmaxe2h, (1)with nmax the maximum (reached fore ! 2`) of nsed.For Haldaneg-ons one hasnmaxg ­1g and this gives thefamiliar result sH ­1me2h , both for the edge electronsand the edge quasiholes over then ­ 1m Laughlin state.In this quasiparticle formalism, the fact that the Halconductance does not depend on temperature comes abas a consequence of the duality between the statisticsgand 1g distribution functions [8].In the present Letter, we extend the above reasonito the case of a quantum Hall state (the so-calledq ­ 2Pfaffian state) with excitations that obey non-Abeliabraiding statistics. In this context, “non-Abelian” meanthat the wave function for a set of quasiparticle excitationat given positions has more than one component, athat the action of braiding (exchanging) two of thosexcitations is represented by a nondiagonal matrix action the multi-component wave function.The issue addressed in this Letter is a fundamental onwe are exploring the consequences for thermodynamand transport of the non-Abelian braiding statistics oa set of fundamental excitations in a condensed matsystem. To our knowledge, a similar problem has nbeen addressed until now [9]. As a characteristic featuof “non-Abelian exclusion statistics,” we shall identify anontrivial prefactor in the high-energy Boltzmann tail of ageneralized distribution function.After the experimental observation [10] of a quantumHall effect at the even denominator filling fractionn ­52 ,a number of candidate ground states with filling fraction ­ 12 have been proposed. One of these is the so-call© 1998 The American Physical Society 1929VOLUME 81, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 31 AUGUST 1998rtheisoleicsloyeasisreutedtesn-ledrnendaner-q ­ 2 Pfaffian state [7,11] with wave functionCPf ­ Pf√1zi 2 zj! Yi.jszi 2 zjd2Yie2sjzi j2y4,2d (2)with Pf s 1zi2zj d ­ As1z12z21z32z4 . . .d denoting the anti-symmetrized product over pairs of electrons and w, ­qh̄ceB the magnetic length. Following [7], one mainterpret this state as a conformal block ofn chiral vertexoperators in ac ­ 32 CFT. In the same manner, the wavfunctions for electron and quasihole excitations over tPfaffian state are obtained from conformal blocks with tappropriate insertions of electron and quasihole operatoTo write these operators, we represent the CFT asproduct of ac ­ 12 (Ising) CFT with Majorana fermionc and spin operators and a c ­ 1 Gaussian theorywith scalar fieldw. In terms of these fields, the electrooperatorGszd (of charge2e) and the quasihole operatofszd (of charge1 e4 ) take the following form:Gszd ; sce2ip2wd szd, fszd ; sseis1y2p2dwd szd .(3)The electron and quasihole operators play a dourole in the theory. Via their conformal blocks they givbulk wave functions for excited states. At the same timthey act as spectrum generating fields in the edge CFIn terms of statistics, the implications of the underlyinCFT are again twofold. On the one hand, the braidistatistics in the CFT directly translate into braidinstatistics of electron and quasihole (bulk) excitations. Othe other hand, the exclusion statistics properties ofelectron and quasihole operators (see below) dictatethermodynamic (and, to some extent, transport) properof the low-energy edge theories. For the Pfaffian stathe (non-Abelian) braiding statistics were studied in gredetail in [7,12]. The study of the exclusion statisticaspects is the subject of this Letter.The statistics of quasiparticles over then ­ 1m Laugh-lin states are Abelian and can be inferred from the propties of two-particle states. As such, they can be studiedinspecting the short distance operator product expansi(OPE) of the electron and quasihole fields. The OPElead to Abelian braiding statistics, with anglesmp and pmfor the electrons and quasiholes, respectively. The cresponding exclusion statistics are of the type proposby Haldane, with statistics parametersg ­ m, g ­ 1m , re-spectively [6]. The OPE’s for the electron and quasihoedge excitations over the the Pfaffian state have a mcomplicated structure, and it is no longer possible to drive the statistics from here. Indeed, the non-Abelian nture of the braiding statistics of the quasihole excitatiobecomes apparent only at the four-particle level.In [12], it was argued that the wave-function specifyina collection of 2n quasiholes (at given locations) ovethe Pfaffian state has2n21 independent components. Inthe thermodynamic limit, this degeneracy amounts1930ithyehehers.anrbleee,T.gnggnthethetieste,atser-byons’sor-edleoree-a-nsgrtoa degeneracy ofp2 per quasihole. This nonintegedegeneracy factor, which ends up as a prefactor inBoltzmann tail of the generalized distribution function,a manifestation of the non-Abelian nature of the quasihexclusion statistics.To derive quantitative results for the exclusion statistof the edge excitations over the Pfaffian state, we empthe method of truncated conformal spectra [5]. The idof this method is to establish an explicit quasiparticle baof the CFT chiral finite size spectrum, and from thederive the statistics. In [6], this procedure was carried oin great detail for the Laughlin states and for the chargector for more general hierarchical quantum Hall sta(Jain series).The complication that is new for the case of noAbelian quantum Hall states is the occurrence of so-calfusion channels in the specification of a multi-f state.These have their origin in the following fusion rule fothe spin field of the Ising CFT:fsg ? fsg ­ f1g 1 fcg . (4)Each time we apply af mode to a state containing aodd number of quanta, we have to specify a choicfor either the first or the second term on the right-haside of (4). For a state containingN quasiholes, the totalfusion path is encoded in a Bratteli diagram, which isarrowpath on the following grid:-. . . . -.-cs.- . . . .-.- 1 ,N . . . . . 4 3 2 1(5)with the vertical direction indicating the Ising sectors1,, andc and the horizontal indexi ­ 1, . . . , N labelingthe f modes. The starting point of an arrow path is thCFT vacuumj0l. The multi-f states for a given fusionpath are [withfszd ­Ps f2szs21y8]f21y82DN2nN . . . f21y82D22n2f21y82D12n1j0l , (6)with integersnN $ . . . n2 $ n1 $ 0 and with incrementsDi determined by the Bratteli diagram according toDi11 ­ Di 112 for--i 1 1 iand..i 1 1 iDi11 ­ Di 114 for - .i 1 1 iand. -i 1 1 iDi11 ­ Di for . -i 1 1 iand- .i 1 1 iwith D1 ­ 0. For example, the lowest energy fouquasihole states for the two possible fusion paths are.-.- f23y8f23y8f21y8f21y8j0l, E ­ 1 ,.-. -f211y8f27y8f25y8f21y8j0l, E ­ 3 .(7)VOLUME 81, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 31 AUGUST 1998anal-le-cs.r-enewedofoveIn the construction of multielectron states the issuefusion rules does not explicitly arise, but the details othe rules are different from what we had for the Laughlistates [6]. On the vacuumj0l, we allow the followingmultielectron states [withGszd ­Pt G2tzt23y2]:G2s1y2d2mM . . . G2s1y2d2m2 G2 122m1 j0l (8)with integersmi satisfyingmi $ maxsmi21 1 1, mi22 1 4d , (9)with m0 ­ 22, m21 ­ 23. For example, the lowestenergy multi-G states overj0l are of the form. . . G213y2G211y2G25y2G23y2j0l , (10)with increments alternating between1 and 3. It isquickly seen that the number of states allowed by therules is larger than for the case withg ­ 2, where theconstruction rule ismi $ mi21 1 2.We are now ready to specify a complete set of indepedent multiparticle states. We start by constructing a towof states overj0l (which we shall write asj0, 1l from nowon) by acting independently withf andG modes accord-ing to the rules that we specified. We then introduce fivadditional reference states for which we repeat the prcedure. The lowestf andG modes that are allowed oneach of the reference states are specified in Table I.The central claim of this Letter is that the union of all thquasiparticle states in all six sectors precisely forms a baof the chiral Hilbert space of the CFT, and thereby (uto certain projections, see [6]) of physical edge theorieThis is a mathematical result which, however, has a clephysical interpretation in terms of exclusion statistics (sebelow). The chiral spectrum of the CFT for the Pfaffiastate is usually understood in terms of representations oUs1d superaffine (Kac-Moody) current algebra. The nepoint of view that we advocate here is very different andirectly refers to the charge carrying excitations in thsystem. UsingMAPLE, we have checked that the characteidentities that are implied by our claim are satisfied to thfirst 25 nontrivial orders inq.The absence of any interference between the constrtion rules for thef and theG quanta means that there isTABLE I. Reference states for the quasiparticle constructioof the chiral Hilbert space over theq ­ 2 Pfaffian state. Thecreation operatorsf2s and G2t act on these states accordingto the exclusion rules specified in the text. The symbols1, c,and s denote Ising sectors and the quantitiesD1 and m0, m21determine the lowestf andG modes that are allowed.State Charge Energy D1 m0, m21j0, 1l 0 0 0 22, 23j0, cl 0 1212 21, 24j 2e4 , sl 2e4183421, 23j 2e2 , 1l 2e2143421, 22j 2e2 , cl 2e23414 0, 23j 23e4 , sl 23e45812 0, 22offnsen-ereo-esisps.arenf awdereuc-nno mutual exclusion statistics between the two. This isfirst sign of a (generalized) particle-hole duality betweethe two types of quasiparticles. We shall make this duity precise in what follows.Having specified the quasiparticle basis we can impment the procedure of [5] and derive the thermodynamiFor thef modes, we define truncated and projected patition sumsXlsx, qd andYlsx, qd, l ­ 18 ,38 , . . . , as expres-sions of the formtrace#lsxNf qEd , (11)with Nf the number of modes,E the dimensionlessenergy, x ­ ebmqh and q ­ e2bs2pyLr0d with r0 thedensity of states per unit length. The subscript#lindicates that the highest occupied modef2s is suchthat l 2 s is a nonnegative integer.Xl (Yl) is furtherrestricted by requiring that the total charge be an ev(odd) multiple of e4 . The rules specified above lead to threcursion relationsXl ­ Xl21 1 xqlsYl 1 Yl21y2d ,Yl ­ Yl21 1 xqlXl21y4 .(12)The initial conditions, which are sector dependent, follofrom the information in Table I. The solution of (12) canbe approximated asXl ,Qli­1 l1sqixd, with l1szqhd thelargest real solution of the equationsl 2 1d sl1y2 2 1d ­ z2qhl5y4. (13)This leads to a distribution functionnqhsed ­ fzqh≠zqh logsl1dg szqh ­ ebsmqh2edd (14)for the quasihole excitations.For theG quanta, the above rules are easily convertinto a recursion relation for the quantityVls y, qd, whichis the trace ofyNG qE, with NG the number ofG quantaand y ­ ebme , over all multi-G states with modes#l 2 12 . We findVl ­ Vl21 1 yql2s1y2dVl22 1 y2q2l22Vl242 y3q3l2s13y2dVl26 . (15)The corresponding distribution function isnesed ­ fze≠ze logsm1dg sze ­ ebsme2edd , (16)with m1szed the largest real solution ofsm4 2 z2e d sm2 2 zed ­ m5. (17)For temperatureT ­ 0, the distribution functionsnqhsed andnesed are step-down functions with maximumnmaxqh ­ 8 and nmaxe ­12 . At finite T , the asymptoticbehavior fore ! ` is nqh ,p2 e2be and ne , e2be.We stress that the prefactorp2 is a direct manifestationof the non-Abelian nature of the exclusion statisticsthe quasiholes. As an easy consequence of the abformulas, we establish the duality relation2nesed ­ 1 218nqhµ2e4∂, (18)1931VOLUME 81, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 31 AUGUST 1998yor.st.sprovidedmqh ­ 214 me. This result generalizes the duality for Haldane’s fractional exclusion statistics [8]. Thduality allows us to interpret the spectrum in differenways. One picture has quasiholes with energies rangfrom 2` to `, with the CFT vacuum and other referencstates represented as filled seas of quasiholes. An anative is a picture based on edge electrons, and then this the “mixed” picture that we started from, withf andGquanta of positive energies only.In the mixed picture, the total central chargec ­ 32 ofthe edge CFT is established as the sum of a contributc1qh from the quasiholes and a piece1e from the electrons.Using MAPLE, we evaluatedc1qh ­ 1.059 . . . , c1e ­ 0.440 . . . . (19)To compute the Hall conductance, we insertqqh ­14andnmaxqh ­ 8 or qe ­ 21 andnmaxe ­12 into (1), givingsH ­12e2h as expected. The fact that this resultindependent of temperature is guaranteed by the duarelation (18). We refer to [6] for the details of this way ocomputing and for further applications of the quasiparticformalism.In conclusion, we have established an appropriate geralization of the Pauli principle for a situation wherelectrons and quasiholes obey non-Abelian braiding stistics. We have also demonstrated a particle-hole duain this non-Abelian theory. In a future publication, wshall give a more extensive treatment of exclusion statics in CFT’s with non-Abelian braiding and give morexamples. These will include the so-called parafermionquantum Hall states, higher level Wess-Zumino-Wittetheories, and a spinon representation for the SOs5d1 CFT,which is relevant for SOs5d invariant ladders models forcorrelated electrons.The author thanks P. Bouwknegt, N. Read, and R. Avan Elburg for discussions. Part of this work wadone at the 1997 ITP Santa Barbara WorkshopLow-dimensional Field Theory: from Particle Physics tCondensed Matter. This research was supported in pby the foundation FOM of the Netherlands.1932-etingelter-ereionislityfleen-eta-lityetis-eicn. J.sonoartNote added.—The results presented here are easilgeneralized to the more general Pfaffian states with factszi 2 zjdq in (2). Equations (13) and (17) becomesl 2 1d sl1y2 2 1d ­ z2qhls3q21dys2qdsm2q 2 z2e d smq 2 zed ­ m3q21.(20)and the duality relation readsqnesed ­ 1 214qnqhµ2e2q∂. (21)After submitting this manuscript, we learned that S. BIsakov (private communication; to be published) hadetermined the exclusion statistics forbulk quasiholesover the generalq Pfaffian state, and found results thaare identical to those presented here foredgequasiholes.[1] R. de-Piciottoet al., Nature (London)389, 162 (1997);L. Saminadayaret al., Phys. Rev. Lett.79, 2526 (1997).[2] See, e.g., V. J. Goldman, Physica (Amsterdam)1E, 15(1997).[3] F. D. M. Haldane, Phys. Rev. Lett.67, 937 (1991).[4] S. B. Isakov, Mod. Phys. Lett. B8, 319 (1994);A. Dasnières de Veigy and S. Ouvry, Phys. Rev. Lett72,600 (1994); Mod. Phys. Lett. B9, 271 (1995); Y.-S. Wu,Phys. Rev. Lett.73, 922 (1994).[5] K. Schoutens, Phys. Rev. Lett.79, 2608 (1997).[6] R. A. J. van Elburg and K. Schoutens, cond-mat/9801272[7] G. Moore and N. Read, Nucl. Phys. B360, 362 (1991).[8] C. Nayak and F. Wilczek, Phys. Rev. Lett.73, 2740(1994); A. K. Rajagopal, Phys. Rev. Lett.74, 1048 (1995).[9] The analysis of T. Fukui, N. Kawakami, and S.-K. Yang[ J. Phys. Soc. Jpn.65, 1617 (1996)], which uses theterminology “non-Abelian exclusion statistics” is verydifferent from ours, as it is not based on the statisticof the fundamental spinons in the problem.[10] R. Willett et al., Phys. Rev. Lett.59, 1776 (1987).[11] M. Greiter, X. G. Wen, and F. Wilczek, Nucl. Phys. B374, 567 (1992).[12] C. Nayak and F. Wilczek, Nucl. Phys. B479, 529 (1996).

Exclusion statistics for non-abelian quantum Hall states

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