We propose a new method for investigating the exclusion statistics of
quasi-particles in Conformal Field Theory (CFT) spectra. The method leads to
one-particle distribution functions, which generalize the Fermi-Dirac
distribution. For the simplest $su(n)$ invariant CFTs we find a generalization
of Gentile parafermions, and we obtain new distributions for the simplest
$Z_N$-invariant CFTs. In special examples, our approach reproduces
distributions based on `fractional exclusion statistics' in the sense of
Haldane. We comment on applications to fractional quantum Hall effect edge
theories.Comment: 4 pages, 1 figure, LaTeX (uses revtex

A. B. Zamolodchikov

C. Nayak

D. Bernard

E. Frenkel

F. D. M. Haldane

G. Gentile

H. W. J. Blöte

I. Affleck

K. Schoutens

Kareljan Schoutens

P. Bouwknegt

S. B. Isakov

S. Iso

V. Bazhanov

Y. Kato

Y. S. Wu

Y.-S. Wu

English

arXiv

Schoutens, K.

International Migration, Integration and Social Cohesion online publications

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 in conformal field theory spectraSchoutens, K.DOI10.1103/PhysRevLett.79.2608Publication date1997Published inPhysical Review LettersLink to publicationCitation for published version (APA):Schoutens, K. (1997). Exclusion statistics in conformal field theory spectra. Physical ReviewLetters, 79, 2608-2611. https://doi.org/10.1103/PhysRevLett.79.2608General 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:11 Feb 2023VOLUME 79, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R S 6 OCTOBER 1997rmallizes” inries.2608Exclusion Statistics in Conformal Field Theory SpectraKareljan SchoutensInstitute for Theoretical Physics, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands(Received 17 June 1997)We propose a new method for investigating the exclusion statistics of quasiparticles in confofield theory (CFT) spectra. The method leads to one-particle distribution functions, which generathe Fermi-Dirac distribution. For the simplest SUsnd invariant CFTs we find a generalization ofGentile parafermions, and we obtain new distributions for the simplestZN -invariant CFTs. Inspecial examples, our approach reproduces distributions based on “fractional exclusion statisticthe sense of Haldane. We comment on applications to fractional quantum Hall effect edge theo[S0031-9007(97)04183-5]PACS numbers: 05.30.–d, 05.70.Ce, 11.25.Hfnsnisehlor-tcefnTre-ofsl-ee-s-tetg.,-mthx-frofini-trreConformal field theory (CFT) in two dimensions is ainvaluable tool in the analysis of (among other thingthe low-temperature properties of a variety of condensmatter systems. In the literature on CFT (which is vasthere is a certain dichotomy between, on the one hadescriptions based onbosonizationand, on the other,descriptions which give a central role toquasiparticles.In the standard approach to rational CFT, the spectruis described in terms of representations of a bosonic crent algebra called thechiral algebra. Examples are affineKac-Moody (KM) algebras and higher spin extension(called W algebras) of the Virasoro algebra. In applcations to condensed matter systems, a similar descriptis often used. Examples are the Luttinger liquids for 1interacting electrons, which have a Us1d affine KM sym-metry and are usually treated in bosonized form. Othexamples are the low-temperature theories for the muchannel Kondo effect, which have been analyzed on tbasis of their SUs2dk affine KM symmetry.When dealing with the CFT for noninteracting electrons, one clearly does not need bosonization, but ufree fermions (satisfying canonical anticommutation rlations) instead. While interacting electrons give risemore general CFTs, it is entirely natural to look for descriptions that mimic the treatment of free electrons. Tidea is to identify fundamental excitations (quasiparticleover the many-body ground state and to study their propties. For integrable models (analyzed using Bethe ansand factorizable scattering) such an approach is by nstandard.Until now, a general approach to “CFT quasiparticleshas been lacking. In special cases, progress has bmade by viewing specific CFTs as massless limits of itegrable particle theories, leading to “masslessS matricesfor CFT quasiparticles” [1]. Related to this is a new “integrable” approach to CFT; see [2]. Other special exampare CFTs which can be viewed as continuum limitsintegrable models of lattice electrons. Examples are tSUsnd1 CFTs, which can be cast in a spinon formulatioanalogous to that of the SUsnd Haldane-Shastry (HS) spinchains [3].0031-9007y97y79(14)y2608(4)$10.00)edt),d,mur-s-ionDerlti-he-es-to-es)er-atzow”eenn--esfhenIn this Letter, we propose an approach to CFT quasipaticles which is intrinsic to the CFT, i.e., which does nomake reference to associated integrable particle or lattimodels. The starting point is the finite size spectrum oa CFT defined on a cylinder. In particular, we focus othe chiral Hilbert spaces, which together build up a CFpartition function. In a quasiparticle formulation, a chi-ral Hilbert space is viewed as a collection of multiparticle(quasiparticle) states (we shall write “particle” for “quasi-particle” where no confusion can arise). Forfermionicquasiparticles, the systematics of multiparticle states asimply given by the Pauli principle, resulting in the Fermi-Dirac distribution function. For more general quasiparticles one may try to understand the spectrum in termsmore general distributions that correspond to various formof exclusion statistics.The notion of exclusion statistics was introduced by Hadane [4], in the context of integrable theories with inverssquare interactions. The main idea to study the way onparticle levels in the spectrum are filled to form allowedmany-particle states. The simplest scenario [4] is to asume that the act of filling an available one-particle staof type i reduces the dimension of the available Hilberspace for particles of typej by an amountgij. In theabsence of other interactions, the statistics matrixG ­sgijd completely determines the thermodynamics (see, e.[5,6]). Concrete examples of this type of exclusion statisics are the Calogero-Sutherland (CS) models of quantumechanics with inverse square two-body interactions (wiadjustableg).We here propose a new method for studying the eclusion statistics of CFT quasiparticles. At the heart oour method is what can be called a “transfer matrix fotruncated chiral spectra.” We shall present a numberexamples where CFT spectra are completely encodedone-particle distribution functions (generalizing the FermDirac distribution). In special examples, the statistics thawe find are of the type proposed by Haldane, while in othecases we find more general results.One check on the distributions that we propose heis the coefficientg in the low-temperature specific heat,© 1997 The American Physical SocietyVOLUME 79, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R S 6 OCTOBER 1997ndfordtov--dwhich is known to be related to the central chargecCFT ofthe CFT according to [7]CL­ gk2Br0T , g ­p6cCFT , (1)where r0 ­ sh̄yFd21 is the density of states per unilength.To introduce our new method we focus on the simpleSUs2d invariant CFT, which is the SUs2d1 Wess-Zumino-Witten model. For this CFT, a quasiparticle formulatiohas been proposed in [8] and worked out in great detin [9]. The formulation uses operatorsf62s which createquasiparticles called spinons. The spinons form a doubunder SUs2d and carry (dimensionless) energyL0 ­ s.The chiral spectrum of the SUs2d1 CFT may be built inthe following manner. One starts by writing the followingpolarizedN-spinon states:f12fs2N21dy4g2nN · · · f12s3y4d2n2 f12s1y4d2n1 j0l , (2)with nN $ · · · $ n2 $ n1 $ 0. One then uses the Yan-gian symmetry algebra to construct multispinon statwith mixed 1 and2 indices. The collection of all thesestates forms a basis of the full chiral spectrum. The SUs2dcontent of the Yangian multiplet labeled byn1, . . . , nNfollows from the generalized commutation relations satified by the spinon modes, or, equivalently, from the reresentation theory of the Yangian [8,9].Comparing the allowed spinon modesf62s with freefermion modesc2s1y2d2l , we observe that the fermionmodel 1 12 has “split” into an odd modes ­ l 114 andan even modes ­ l 1 34 . To maintain a level spacingof one unit, we view these two spinon modes as formina single “one-particle level” in the spectrum. What wwould like to do is to factorize the full chiral partitionsum of the CFT into a product over these one-particlevels, so that the free energy becomes a sum of oparticle contributions.While we cannot straightforwardly extract the contrbution of thelth level, we may proceed as follows. Wetstnailletess-p-gelene-i-truncate the chiral spectrum by only allowing the spinomodes of the firstl levels. We denote byPlsq, x1, x2d,Qlsq, x1, x2d the partition functions of the firstl levels,where in Pl the highest occupied mode should be odwhile in Ql it should be even. In formulaPlsq, x1, x2d ­ traces#ldodd sqL0 xN11 xN22 d (3)and similar forQlsq, x1, x2d, where we introduce chemi-cal potentialsm6, write x6 ­ ebm6 , q ­ e2bs2pyLds1yr0d,and denote byN6 the number of6 spinons in a state.Restricting to fully polarized states (puttingx2 ­ 0), weobtain the following recursion relation√PlQl!­√1 ql2s3y4d x1ql2s1y4d x1 1 1 q2l21 x21! √Pl21Ql21!, (4)with P0 ­ 0 and Q0 ­ 1. We now make the impor-tant step of approximating these exact expressionsPlsq, x1d andQlsq, x1d by the product fromj ­ 1 to l ofthe largest eigenvaluels jd1 sq, xd of the jth recursion ma-trix. Clearly, this brings the partition sums in the desirefactorized form and reduces thermodynamic quantitiessums of independent single-level contributions. The aerage occupation of thelth level is found to bensldsq, xd ­ x≠x ln lsld1 sq, xd ­2p1 1 4 q122lx22­2p1 1 4 e22bsm12el d, (5)with el ­ sl 112 d2pL1r0. Note that the maximal occupation of a given level is 2. Interestingly, this distributionfunction for polarized SUs2d spinons is identical to that ofparticles withg ­ 12 (semionic) exclusion statistics [5].If we now include the multispinon states with mixeindices, we find that thelth recursion matrix becomes√1 2 q2l22x2 ql2s3y4dxsz 1 1z dql2s1y4dxsz 1 1z d s1 2 q2l22x2d 1 1 q2l21x2sz2 1 1 1 1z2 d!, (6)enatwhere we putx6 ­ x z61. Following the same logic, wederive the distribution functionsnsldsq, x, zd. Interestingspecial cases arensldsq, xd ­ x≠x ln lsld1 sq, x, z ­ 1d ­21 1 qs1y2d2lx21(7)for the expected total number of spinons in levell, andQsldsq, zd ­e2z≠z ln lsld1 sq, x ­ 1, zd­e ql2s1y2dsz 2 z21dpq2l21sz 1 z21d2 1 4 s1 2 q2l21d(8)for the expected charge at levell (we assume charges6 e2for the 6 spinons). The distributions (7) and (8) agrewith the distributions obtained from fractional exclusiostatistics withG ­√12121212!.Note that (7) implies that forz ­ 1 (zero voltage) thethermodynamics of the spinon system is identical to thof two free fermions and the central charge iscCFT ­2 3 12 ­ 1. Note also that, withz ­ es1y2dbeV , theintegrated charge1LP`l­1 Qsldsq, zd ­ 14p e2Vro , which2609VOLUME 79, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R S 6 OCTOBER 1997odc-kls-is half of the value obtained for two charge6e freefermions.The correspondence with Haldane statistics is satisfyisince the spinons of the associated SUs2d HS spin chainsatisfy these same statistics [4]. This confirms the validof our new approach, which in no way relied on the exasolution of the HS chain.We now present some special examples.(1) TheSUsnd1 CFTs.—The first generalization of theSUs2d1 results concerns the SUsnd1 spinons. The Yangiansymmetry of the SUsnd1 CFT was established in [10]while the spinon formulation was presented in [3]. Thearen fundamental spinon speciesfi , transforming in therepresentation of SUsnd. Repeating the analysis shownabove, finding explicitly the SUsnd analog of the recursionmatrices (4) and (6) (see [11]), we find that (i) a singspinon speciesfi0 (in absence of any others) satisfieHaldane statistics withg ­ n21n , (ii) when exciting alln spinon species symmetrically (choosing allxi equal tox ­ ebm), the expected total occupation of thelth level isgiven bynsldsq, xd ­ x≠x lnf1 1 qlx 1 . . . 1 sqlxdn21gn. (9)Comparing these results, one finds that forn . 2 there isnegative mutual exclusion among different spinons.Interestingly, the statistics going with the distributio(9) were proposed by Gentile as early as 1940 [12]. Ofinds that a single “Gentile parafermion” contributes thamount n21n to the central charge, so that the full resubecomescCFT ­ nn21n ­ n 2 1, as expected.Our results are consistent with [13], where, by differemethods, the link between SUsnd HS spin chains andGentile parafermions has also been established.(2) ZN parafermions.—Within the context of CFT, thesimplest generalization of the Majorana fermion is the scalled ZN parafermion. It features in a CFT of centrachargecN ­2sN21dN12 as a primary field of dimensionhN ­N21N and ZN charge 1 [14]. By applying the methodoutlined above, we obtained a distribution function for thZN parafermion and established that the full CFT spectruis reproduced by a gas of noninteracting quasiparticlesthis type. For the purpose of explaining these results,focus on the caseN ­ 3.It is well known that the chiral spectrum of theZ3parafermion CFT can be interpreted in terms of twparafermionsc6 of oppositeZ3 charge. However, byexploiting the generalized commutation relations obeyby the modes ofc6 [14], one easily shows that themodes ofc1 alone can generate the full spectrum, anthat thec2 quasiparticle can be viewed as a composiof two c1 quasiparticles. Having understood how thc1 modes alone build the chiral spectrum (see [11one may define truncated partition sums. We founthe following recursion matrix between thelth andthe sl 2 1dth truncated sums (which each have thre2610ngityctrelesnneeltnto-lemofweoeddtee]),decomponents)0B@ s1 2 y3d y2 yys1 2 y3d 1 2y22y2s1 2 y3d ys1 1 y3d 1 1 2y31CA , (10)with y ­ x ql. The Z3 parafermion distribution functionis expressed in terms of the largest eigenvaluel1 of thismatrixnsldsq, xd ­ s y≠y ln l1d s y ­ xqld . (11)From the asymptoticsl1s yd ~ y3, one finds that the maxi-mum occupation per level equals 3. Using the Cardanformula one may writensld in closed form. We herepresent a plot (Fig. 1), which displays the functionn as afunction of energy. As a check we (numerically) evaluatethe coefficientg of the specific heat, reproducing theexpected valueg ­ p645 .For generalN, the distribution forZN parafermionsallows a maximum of12 NsN 2 1d particles per level.(3) Quantum Hall effect edge theories.—As a furtherapplication we briefly discuss edge theories for the frational quantum Hall effect (FQHE). We shall come bacto this topic in a separate publication [15]. For then ­ 1mFQHE (withm an odd integer), the edge theory is a chiraFIG. 1. Distribution functions forZ3 parafermions (solidline), for ordinary fermions (dashed line), and for particles satifying g ­ 13 exclusion statistics (dotted line). All distributionsare atm ­ 0; the energy is given in unitsb21.VOLUME 79, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R S 6 OCTOBER 1997eicdisopis,-s.d,s.PcCFT ­ 1 CFT at compactification radiusR2 ­ m. Thenatural quasiparticles to consider are the edge electroncharge2e) and the edge quasihole (of chargeem ). Writ-ing the spectrum in terms of these quasiparticles, and aplying the above procedure, we find that the fundamenquasiparticles are independent, and obey Haldane excsion statistics withg ­ m and g ­ 1ym, respectively.This is consistent with the result of bosonization appliedg-ons [16].The (known) results for the specific heat (cCFT ­ 1)and the response to voltage (QL ­1m12p e2Vr0) are easilyreproduced by exploiting the duality betweeng ­ m andg ­ 1ym statistics [15]. The central charge arises assum cm 1 c1ym. For m ­ 2 we find c2 ­25 , c1y2 ­35 ,while for m ­ 3, c3 ­ 0.343 . . . , c1y3 ­ 0.655 . . . .These quantum Hall results can be appreciated onbasis of the analogy with CS quantum mechanics wiinverse square interactions (see, e.g., [17]). We strehowever, that our derivation does not rely on this analogWhen applied to composite edges for the FQHE in thJain series, at filling fraction ­ n2np11 , the combinedresults of this paper lead to a formulation in terms of (i)single charged mode, satisfying Haldane statistics wg ­ n, and (ii) a set ofn spinons for SUsnd1, satisfyingthe generalized Gentile statistics described above. Tnew quasiparticle formulation forms a suitable startinpoint for studying finite-T features (including tunnelingcharacteristics) of these edges.The potential applications of our new approach to CFspectra are manifold, especially when the extensionboundary CFTs is considered. We mention edge state sctering, state counting, and a variety of finite-T characteris-tics of (non-Abelian) FQHE edges, and non-Fermi liquifeatures in quantum impurity problems such as the mulchannel Kondo effect.The method presented here can successfully be appto many CFTs other than those mentioned here [11]. Ainteresting example is thecCFT ­ 2225 CFT for the Yang-Lee edge singularity where the Virasoro generators taon the role ofg ­ 2 quasiparticles, giving the correcteffective central chargẽcCFT ­25 [18].To conclude, we should stress that the CFT quasipticles considered in this Letter should not be confused wi(ofp-tallu-toathethss,y.eaithhisgTtoat-dti-liednkear-ththe asymptotic states of the “masslessS matrix” approachto CFT [1]. The two are very different, and they give risto different distribution functions in the thermodynamlimit.Many thanks to A. W. W. Ludwig, P. Bouwknegt, anR. van Elburg for helpful discussions. Part of thwork was done at the ITP Santa Barbara Workshon “Quantum Field Theory in Low Dimensions: fromCondensed Matter to Particle Physics.” The authorsupported in part by the foundation FOM.[1] A. B. Zamolodchikov and Al. B. Zamolodchikov, Nucl.Phys.B379, 602 (1992).[2] V. Bazhanov, S. Lukyanov, and A. B. ZamolodchikovCommun. Math. Phys.11, 381 (1996); hep-th/9604044.[3] P. Bouwknegt and K. Schoutens, Nucl. Phys.B482, 345(1996); Report No. q-alg/9703021.[4] F. D. M. Haldane, Phys. Rev. Lett.67, 937 (1991).[5] Y. S. Wu, Phys. Rev. Lett.73, 922 (1994); C. Nayak andF. Wilczek, Phys. Rev. Lett.73, 2740 (1994); S. B. Isakov,Mod. Phys. Lett. B8, 319 (1994).[6] S. B. Isakov, D. P. Arovas, J. Myrheim, and A. P. Polychronakos, Phys. Lett. A212, 299 (1996).[7] H. W. J. Blöte, J. L. Cardy, and M. P. Nightingale, PhyRev. Lett.56, 742 (1986); I. Affleck, Phys. Rev. Lett.56,742 (1986).[8] F. D. M. Haldane, Phys. Rev. Lett.66, 1529 (1991);F. D. M. Haldane, Z. N. C. Ha, J. C. Talstra, D. Bernarand V. Pasquier, Phys. Rev. Lett.69, 2021 (1992).[9] D. Bernard, V. Pasquier, and D. Serban, Nucl. PhyB428, 612 (1994); P. Bouwknegt, A. W. W. Ludwig, andK. Schoutens, Phys. Lett. B338, 448 (1994).[10] K. Schoutens, Phys. Lett. B331, 335 (1994).[11] K. Schoutens (to be published).[12] G. Gentile, Nuovo Cimento17, 493 (1940).[13] Y. Kato and Y. Kuramoto, J. Phys. Soc. Jpn.64, 4518(1995).[14] A. B. Zamolodchikov and V. A. Fateev, Sov. Phys. JET62, 215 (1985).[15] R. van Elburg and K. Schoutens (to be published).[16] Y.-S. Wu and Y. Yu, Phys. Rev. Lett.75, 890 (1995).[17] S. Iso, Nucl. Phys.B443, 581 (1995).[18] E. Frenkel and A. Szenes, Duke Math. J.69, 53 (1993).2611

Exclusion statistics in conformal field theory spectra

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Exclusion statistics in conformal field theory spectra
Schoutens, K.
Published in:
Physical Review Letters
DOI:
10.1103/PhysRevLett.79.2608
Link to publication
Citation for published version (APA):
Schoutens, K. (1997). Exclusion statistics in conformal field theory spectra. Physical Review Letters, 79, 2608-
2611. https://doi.org/10.1103/PhysRevLett.79.2608
General rights
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Download date: 04 Feb 2021
VOLUME 79, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R S 6 OCTOBER 1997
rmal
lize
s” in
ries.
2608Exclusion Statistics in Conformal Field Theory Spectra
Kareljan Schoutens
Institute for Theoretical Physics, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands
(Received 17 June 1997)
We propose a new method for investigating the exclusion statistics of quasiparticles in confo
field theory (CFT) spectra. The method leads to one-particle distribution functions, which genera
the Fermi-Dirac distribution. For the simplest SUsnd invariant CFTs we find a generalization of
Gentile parafermions, and we obtain new distributions for the simplestZN -invariant CFTs. In
special examples, our approach reproduces distributions based on “fractional exclusion statistic
the sense of Haldane. We comment on applications to fractional quantum Hall effect edge theo
[S0031-9007(97)04183-5]
PACS numbers: 05.30.–d, 05.70.Ce, 11.25.Hfn
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reConformal field theory (CFT) in two dimensions is a
invaluable tool in the analysis of (among other thing
the low-temperature properties of a variety of condens
matter systems. In the literature on CFT (which is vas
there is a certain dichotomy between, on the one ha
descriptions based onbosonizationand, on the other,
descriptions which give a central role toquasiparticles.
In the standard approach to rational CFT, the spectru
is described in terms of representations of a bosonic c
rent algebra called thechiral algebra. Examples are affine
Kac-Moody (KM) algebras and higher spin extension
(called W algebras) of the Virasoro algebra. In appl
cations to condensed matter systems, a similar descript
is often used. Examples are the Luttinger liquids for 1
interacting electrons, which have a Us1d affine KM sym-
metry and are usually treated in bosonized form. Oth
examples are the low-temperature theories for the mu
channel Kondo effect, which have been analyzed on t
basis of their SUs2dk affine KM symmetry.
When dealing with the CFT for noninteracting elec
trons, one clearly does not need bosonization, but u
free fermions (satisfying canonical anticommutation r
lations) instead. While interacting electrons give rise
more general CFTs, it is entirely natural to look for de
scriptions that mimic the treatment of free electrons. T
idea is to identify fundamental excitations (quasiparticle
over the many-body ground state and to study their prop
ties. For integrable models (analyzed using Bethe ans
and factorizable scattering) such an approach is by n
standard.
Until now, a general approach to “CFT quasiparticles
has been lacking. In special cases, progress has b
made by viewing specific CFTs as massless limits of i
tegrable particle theories, leading to “masslessS matrices
for CFT quasiparticles” [1]. Related to this is a new “inte
grable” approach to CFT; see [2]. Other special examp
are CFTs which can be viewed as continuum limits
integrable models of lattice electrons. Examples are t
SUsnd1 CFTs, which can be cast in a spinon formulatio
analogous to that of the SUsnd Haldane-Shastry (HS) spin
chains [3].0031-9007y97y79(14)y2608(4)$10.00)
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-
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n
In this Letter, we propose an approach to CFT quasipa
ticles which is intrinsic to the CFT, i.e., which does no
make reference to associated integrable particle or latti
models. The starting point is the finite size spectrum o
a CFT defined on a cylinder. In particular, we focus o
the chiral Hilbert spaces, which together build up a CF
partition function. In a quasiparticle formulation, a chi-
ral Hilbert space is viewed as a collection of multiparticle
(quasiparticle) states (we shall write “particle” for “quasi-
particle” where no confusion can arise). Forfermionic
quasiparticles, the systematics of multiparticle states a
simply given by the Pauli principle, resulting in the Fermi-
Dirac distribution function. For more general quasipar
ticles one may try to understand the spectrum in terms
more general distributions that correspond to various form
of exclusion statistics.
The notion of exclusion statistics was introduced by Ha
dane [4], in the context of integrable theories with invers
square interactions. The main idea to study the way on
particle levels in the spectrum are filled to form allowed
many-particle states. The simplest scenario [4] is to a
sume that the act of filling an available one-particle sta
of type i reduces the dimension of the available Hilber
space for particles of typej by an amountgij. In the
absence of other interactions, the statistics matrixG ­
sgijd completely determines the thermodynamics (see, e.
[5,6]). Concrete examples of this type of exclusion statis
ics are the Calogero-Sutherland (CS) models of quantu
mechanics with inverse square two-body interactions (wi
adjustableg).
We here propose a new method for studying the e
clusion statistics of CFT quasiparticles. At the heart o
our method is what can be called a “transfer matrix fo
truncated chiral spectra.” We shall present a number
examples where CFT spectra are completely encoded
one-particle distribution functions (generalizing the Ferm
Dirac distribution). In special examples, the statistics tha
we find are of the type proposed by Haldane, while in othe
cases we find more general results.
One check on the distributions that we propose he
is the coefficientg in the low-temperature specific heat,© 1997 The American Physical Society
VOLUME 79, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R S 6 OCTOBER 1997
n
d
for
d
to
v-
-
d
which is known to be related to the central chargecCFT of
the CFT according to [7]
C
L
­ gk2Br0T , g ­
p
6
cCFT , (1)
where r0 ­ sh̄yFd21 is the density of states per uni
length.
To introduce our new method we focus on the simple
SUs2d invariant CFT, which is the SUs2d1 Wess-Zumino-
Witten model. For this CFT, a quasiparticle formulatio
has been proposed in [8] and worked out in great det
in [9]. The formulation uses operatorsf62s which create
quasiparticles called spinons. The spinons form a doub
under SUs2d and carry (dimensionless) energyL0 ­ s.
The chiral spectrum of the SUs2d1 CFT may be built in
the following manner. One starts by writing the following
polarizedN-spinon states:
f
1
2fs2N21dy4g2nN · · · f
1
2s3y4d2n2 f
1
2s1y4d2n1 j0l , (2)
with nN $ · · · $ n2 $ n1 $ 0. One then uses the Yan-
gian symmetry algebra to construct multispinon stat
with mixed 1 and2 indices. The collection of all these
states forms a basis of the full chiral spectrum. The SUs2d
content of the Yangian multiplet labeled byn1, . . . , nN
follows from the generalized commutation relations sati
fied by the spinon modes, or, equivalently, from the re
resentation theory of the Yangian [8,9].
Comparing the allowed spinon modesf62s with free
fermion modesc2s1y2d2l , we observe that the fermion
model 1 12 has “split” into an odd modes ­ l 1
1
4 and
an even modes ­ l 1 34 . To maintain a level spacing
of one unit, we view these two spinon modes as formin
a single “one-particle level” in the spectrum. What w
would like to do is to factorize the full chiral partition
sum of the CFT into a product over these one-partic
levels, so that the free energy becomes a sum of o
particle contributions.
While we cannot straightforwardly extract the contr
bution of thelth level, we may proceed as follows. Wet
st
n
ail
let
es
s-
p-
g
e
le
ne-
i-
truncate the chiral spectrum by only allowing the spino
modes of the firstl levels. We denote byPlsq, x1, x2d,
Qlsq, x1, x2d the partition functions of the firstl levels,
where in Pl the highest occupied mode should be od
while in Ql it should be even. In formula
Plsq, x1, x2d ­ trace
s#ld
odd sqL0 x
N1
1 x
N2
2 d (3)
and similar forQlsq, x1, x2d, where we introduce chemi-
cal potentialsm6, write x6 ­ ebm6 , q ­ e2bs2pyLds1yr0d,
and denote byN6 the number of6 spinons in a state.
Restricting to fully polarized states (puttingx2 ­ 0), we
obtain the following recursion relation√
Pl
Ql
!
­
√
1 ql2s3y4d x1
ql2s1y4d x1 1 1 q2l21 x21
! √
Pl21
Ql21
!
, (4)
with P0 ­ 0 and Q0 ­ 1. We now make the impor-
tant step of approximating these exact expressions
Plsq, x1d andQlsq, x1d by the product fromj ­ 1 to l of
the largest eigenvaluel
s jd
1 sq, xd of the jth recursion ma-
trix. Clearly, this brings the partition sums in the desire
factorized form and reduces thermodynamic quantities
sums of independent single-level contributions. The a
erage occupation of thelth level is found to be
nsldsq, xd ­ x≠x ln l
sld
1 sq, xd ­
2p
1 1 4 q122lx22
­
2
p
1 1 4 e22bsm12el d
, (5)
with el ­ sl 1
1
2 d
2p
L
1
r0
. Note that the maximal occupa
tion of a given level is 2. Interestingly, this distribution
function for polarized SUs2d spinons is identical to that of
particles withg ­ 12 (semionic) exclusion statistics [5].
If we now include the multispinon states with mixe
indices, we find that thelth recursion matrix becomes√
1 2 q2l22x2 ql2s3y4dxsz 1 1z d
ql2s1y4dxsz 1 1z d s1 2 q
2l22x2d 1 1 q2l21x2sz2 1 1 1 1z2 d
!
, (6)e
n
atwhere we putx6 ­ x z61. Following the same logic, we
derive the distribution functionsnsldsq, x, zd. Interesting
special cases are
nsldsq, xd ­ x≠x ln l
sld
1 sq, x, z ­ 1d ­
2
1 1 qs1y2d2lx21
(7)
for the expected total number of spinons in levell, and
Q
sldsq, zd ­
e
2
z≠z ln l
sld
1 sq, x ­ 1, zd
­
e ql2s1y2dsz 2 z21dp
q2l21sz 1 z21d2 1 4 s1 2 q2l21d
(8)for the expected charge at levell (we assume charges6 e2
for the 6 spinons). The distributions (7) and (8) agre
with the distributions obtained from fractional exclusio
statistics with
G ­
√
1
2
1
2
1
2
1
2
!
.
Note that (7) implies that forz ­ 1 (zero voltage) the
thermodynamics of the spinon system is identical to th
of two free fermions and the central charge iscCFT ­
2 3 12 ­ 1. Note also that, withz ­ e
s1y2dbeV , the
integrated charge1L
P
`
l­1 Q
sldsq, zd ­ 14p e
2Vro , which2609
VOLUME 79, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R S 6 OCTOBER 1997
o
d
c-
k
l
s-is half of the value obtained for two charge6e free
fermions.
The correspondence with Haldane statistics is satisfyi
since the spinons of the associated SUs2d HS spin chain
satisfy these same statistics [4]. This confirms the valid
of our new approach, which in no way relied on the exa
solution of the HS chain.
We now present some special examples.
(1) TheSUsnd1 CFTs.—The first generalization of the
SUs2d1 results concerns the SUsnd1 spinons. The Yangian
symmetry of the SUsnd1 CFT was established in [10]
while the spinon formulation was presented in [3]. The
aren fundamental spinon speciesfi , transforming in the
representation of SUsnd. Repeating the analysis shown
above, finding explicitly the SUsnd analog of the recursion
matrices (4) and (6) (see [11]), we find that (i) a sing
spinon speciesfi0 (in absence of any others) satisfie
Haldane statistics withg ­ n21n , (ii) when exciting all
n spinon species symmetrically (choosing allxi equal to
x ­ ebm), the expected total occupation of thelth level is
given by
nsldsq, xd ­ x≠x lnf1 1 qlx 1 . . . 1 sqlxdn21gn. (9)
Comparing these results, one finds that forn . 2 there is
negative mutual exclusion among different spinons.
Interestingly, the statistics going with the distributio
(9) were proposed by Gentile as early as 1940 [12]. O
finds that a single “Gentile parafermion” contributes th
amount n21n to the central charge, so that the full resu
becomescCFT ­ n
n21
n ­ n 2 1, as expected.
Our results are consistent with [13], where, by differe
methods, the link between SUsnd HS spin chains and
Gentile parafermions has also been established.
(2) ZN parafermions.—Within the context of CFT, the
simplest generalization of the Majorana fermion is the s
called ZN parafermion. It features in a CFT of centra
chargecN ­
2sN21d
N12 as a primary field of dimensionhN ­
N21
N and ZN charge 1 [14]. By applying the method
outlined above, we obtained a distribution function for th
ZN parafermion and established that the full CFT spectru
is reproduced by a gas of noninteracting quasiparticles
this type. For the purpose of explaining these results,
focus on the caseN ­ 3.
It is well known that the chiral spectrum of theZ3
parafermion CFT can be interpreted in terms of tw
parafermionsc6 of oppositeZ3 charge. However, by
exploiting the generalized commutation relations obey
by the modes ofc6 [14], one easily shows that the
modes ofc1 alone can generate the full spectrum, an
that thec2 quasiparticle can be viewed as a composi
of two c1 quasiparticles. Having understood how th
c1 modes alone build the chiral spectrum (see [11
one may define truncated partition sums. We foun
the following recursion matrix between thelth and
the sl 2 1dth truncated sums (which each have thre2610ng
ity
ct
re
le
s
n
ne
e
lt
nt
o-
l
e
m
of
we
o
ed
d
te
e
]),
d
e
components)0B@ s1 2 y3d y2 yys1 2 y3d 1 2y2
2y2s1 2 y3d ys1 1 y3d 1 1 2y3
1CA , (10)
with y ­ x ql. The Z3 parafermion distribution function
is expressed in terms of the largest eigenvaluel1 of this
matrix
nsldsq, xd ­ s y≠y ln l1d s y ­ xqld . (11)
From the asymptoticsl1s yd ~ y3, one finds that the maxi-
mum occupation per level equals 3. Using the Cardan
formula one may writensld in closed form. We here
present a plot (Fig. 1), which displays the functionn as a
function of energy. As a check we (numerically) evaluate
the coefficientg of the specific heat, reproducing the
expected valueg ­ p6
4
5 .
For generalN, the distribution forZN parafermions
allows a maximum of12 NsN 2 1d particles per level.
(3) Quantum Hall effect edge theories.—As a further
application we briefly discuss edge theories for the fra
tional quantum Hall effect (FQHE). We shall come bac
to this topic in a separate publication [15]. For then ­ 1m
FQHE (withm an odd integer), the edge theory is a chira
FIG. 1. Distribution functions forZ3 parafermions (solid
line), for ordinary fermions (dashed line), and for particles sati
fying g ­ 13 exclusion statistics (dotted line). All distributions
are atm ­ 0; the energy is given in unitsb21.
VOLUME 79, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R S 6 OCTOBER 1997
e
ic
d
is
op
is
,
-
s.
d,
s.
P
cCFT ­ 1 CFT at compactification radiusR2 ­ m. The
natural quasiparticles to consider are the edge electron
charge2e) and the edge quasihole (of chargeem ). Writ-
ing the spectrum in terms of these quasiparticles, and a
plying the above procedure, we find that the fundamen
quasiparticles are independent, and obey Haldane exc
sion statistics withg ­ m and g ­ 1ym, respectively.
This is consistent with the result of bosonization applied
g-ons [16].
The (known) results for the specific heat (cCFT ­ 1)
and the response to voltage (QL ­
1
m
1
2p e
2Vr0) are easily
reproduced by exploiting the duality betweeng ­ m and
g ­ 1ym statistics [15]. The central charge arises as
sum cm 1 c1ym. For m ­ 2 we find c2 ­
2
5 , c1y2 ­
3
5 ,
while for m ­ 3, c3 ­ 0.343 . . . , c1y3 ­ 0.655 . . . .
These quantum Hall results can be appreciated on
basis of the analogy with CS quantum mechanics wi
inverse square interactions (see, e.g., [17]). We stre
however, that our derivation does not rely on this analog
When applied to composite edges for the FQHE in th
Jain series, at filling fraction ­ n2np11 , the combined
results of this paper lead to a formulation in terms of (i)
single charged mode, satisfying Haldane statistics w
g ­ n, and (ii) a set ofn spinons for SUsnd1, satisfying
the generalized Gentile statistics described above. T
new quasiparticle formulation forms a suitable startin
point for studying finite-T features (including tunneling
characteristics) of these edges.
The potential applications of our new approach to CF
spectra are manifold, especially when the extension
boundary CFTs is considered. We mention edge state sc
tering, state counting, and a variety of finite-T characteris-
tics of (non-Abelian) FQHE edges, and non-Fermi liqui
features in quantum impurity problems such as the mul
channel Kondo effect.
The method presented here can successfully be app
to many CFTs other than those mentioned here [11]. A
interesting example is thecCFT ­ 2
22
5 CFT for the Yang-
Lee edge singularity where the Virasoro generators ta
on the role ofg ­ 2 quasiparticles, giving the correct
effective central chargẽcCFT ­
2
5 [18].
To conclude, we should stress that the CFT quasip
ticles considered in this Letter should not be confused wi(of
p-
tal
lu-
to
a
the
th
ss,
y.
e
a
ith
his
g
T
to
at-
d
ti-
lied
n
ke
ar-
th
the asymptotic states of the “masslessS matrix” approach
to CFT [1]. The two are very different, and they give ris
to different distribution functions in the thermodynam
limit.
Many thanks to A. W. W. Ludwig, P. Bouwknegt, an
R. van Elburg for helpful discussions. Part of th
work was done at the ITP Santa Barbara Worksh
on “Quantum Field Theory in Low Dimensions: from
Condensed Matter to Particle Physics.” The author
supported in part by the foundation FOM.
[1] A. B. Zamolodchikov and Al. B. Zamolodchikov, Nucl.
Phys.B379, 602 (1992).
[2] V. Bazhanov, S. Lukyanov, and A. B. Zamolodchikov
Commun. Math. Phys.11, 381 (1996); hep-th/9604044.
[3] P. Bouwknegt and K. Schoutens, Nucl. Phys.B482, 345
(1996); Report No. q-alg/9703021.
[4] F. D. M. Haldane, Phys. Rev. Lett.67, 937 (1991).
[5] Y. S. Wu, Phys. Rev. Lett.73, 922 (1994); C. Nayak and
F. Wilczek, Phys. Rev. Lett.73, 2740 (1994); S. B. Isakov,
Mod. Phys. Lett. B8, 319 (1994).
[6] S. B. Isakov, D. P. Arovas, J. Myrheim, and A. P. Poly
chronakos, Phys. Lett. A212, 299 (1996).
[7] H. W. J. Blöte, J. L. Cardy, and M. P. Nightingale, Phy
Rev. Lett.56, 742 (1986); I. Affleck, Phys. Rev. Lett.56,
742 (1986).
[8] F. D. M. Haldane, Phys. Rev. Lett.66, 1529 (1991);
F. D. M. Haldane, Z. N. C. Ha, J. C. Talstra, D. Bernar
and V. Pasquier, Phys. Rev. Lett.69, 2021 (1992).
[9] D. Bernard, V. Pasquier, and D. Serban, Nucl. Phy
B428, 612 (1994); P. Bouwknegt, A. W. W. Ludwig, and
K. Schoutens, Phys. Lett. B338, 448 (1994).
[10] K. Schoutens, Phys. Lett. B331, 335 (1994).
[11] K. Schoutens (to be published).
[12] G. Gentile, Nuovo Cimento17, 493 (1940).
[13] Y. Kato and Y. Kuramoto, J. Phys. Soc. Jpn.64, 4518
(1995).
[14] A. B. Zamolodchikov and V. A. Fateev, Sov. Phys. JET
62, 215 (1985).
[15] R. van Elburg and K. Schoutens (to be published).
[16] Y.-S. Wu and Y. Yu, Phys. Rev. Lett.75, 890 (1995).
[17] S. Iso, Nucl. Phys.B443, 581 (1995).
[18] E. Frenkel and A. Szenes, Duke Math. J.69, 53 (1993).2611


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Exclusion statistics in conformal field theory spectra
Schoutens, K.
Published in:
Physical Review Letters
DOI:
10.1103/PhysRevLett.79.2608
Link to publication
Citation for published version (APA):
Schoutens, K. (1997). Exclusion statistics in conformal field theory spectra. Physical Review Letters, 79, 2608-
2611. https://doi.org/10.1103/PhysRevLett.79.2608
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Download date: 08 Feb 2021
VOLUME 79, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R S 6 OCTOBER 1997
rmal
lize
s” in
ries.
2608Exclusion Statistics in Conformal Field Theory Spectra
Kareljan Schoutens
Institute for Theoretical Physics, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands
(Received 17 June 1997)
We propose a new method for investigating the exclusion statistics of quasiparticles in confo
field theory (CFT) spectra. The method leads to one-particle distribution functions, which genera
the Fermi-Dirac distribution. For the simplest SUsnd invariant CFTs we find a generalization of
Gentile parafermions, and we obtain new distributions for the simplestZN -invariant CFTs. In
special examples, our approach reproduces distributions based on “fractional exclusion statistic
the sense of Haldane. We comment on applications to fractional quantum Hall effect edge theo
[S0031-9007(97)04183-5]
PACS numbers: 05.30.–d, 05.70.Ce, 11.25.Hfn
s
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i
s
e
h
l
o
r-
t
ce
f
n
T
re
-
of
s
l-
e
e-
s-
te
t
g.,
-
m
th
x-
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of
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i-
t
r
reConformal field theory (CFT) in two dimensions is a
invaluable tool in the analysis of (among other thing
the low-temperature properties of a variety of condens
matter systems. In the literature on CFT (which is vas
there is a certain dichotomy between, on the one ha
descriptions based onbosonizationand, on the other,
descriptions which give a central role toquasiparticles.
In the standard approach to rational CFT, the spectru
is described in terms of representations of a bosonic c
rent algebra called thechiral algebra. Examples are affine
Kac-Moody (KM) algebras and higher spin extension
(called W algebras) of the Virasoro algebra. In appl
cations to condensed matter systems, a similar descript
is often used. Examples are the Luttinger liquids for 1
interacting electrons, which have a Us1d affine KM sym-
metry and are usually treated in bosonized form. Oth
examples are the low-temperature theories for the mu
channel Kondo effect, which have been analyzed on t
basis of their SUs2dk affine KM symmetry.
When dealing with the CFT for noninteracting elec
trons, one clearly does not need bosonization, but u
free fermions (satisfying canonical anticommutation r
lations) instead. While interacting electrons give rise
more general CFTs, it is entirely natural to look for de
scriptions that mimic the treatment of free electrons. T
idea is to identify fundamental excitations (quasiparticle
over the many-body ground state and to study their prop
ties. For integrable models (analyzed using Bethe ans
and factorizable scattering) such an approach is by n
standard.
Until now, a general approach to “CFT quasiparticles
has been lacking. In special cases, progress has b
made by viewing specific CFTs as massless limits of i
tegrable particle theories, leading to “masslessS matrices
for CFT quasiparticles” [1]. Related to this is a new “inte
grable” approach to CFT; see [2]. Other special examp
are CFTs which can be viewed as continuum limits
integrable models of lattice electrons. Examples are t
SUsnd1 CFTs, which can be cast in a spinon formulatio
analogous to that of the SUsnd Haldane-Shastry (HS) spin
chains [3].0031-9007y97y79(14)y2608(4)$10.00)
ed
t),
d,
m
ur-
s
-
ion
D
er
lti-
he
-
es
-
to
-
e
s)
er-
atz
ow
”
een
n-
-
es
f
he
n
In this Letter, we propose an approach to CFT quasipa
ticles which is intrinsic to the CFT, i.e., which does no
make reference to associated integrable particle or latti
models. The starting point is the finite size spectrum o
a CFT defined on a cylinder. In particular, we focus o
the chiral Hilbert spaces, which together build up a CF
partition function. In a quasiparticle formulation, a chi-
ral Hilbert space is viewed as a collection of multiparticle
(quasiparticle) states (we shall write “particle” for “quasi-
particle” where no confusion can arise). Forfermionic
quasiparticles, the systematics of multiparticle states a
simply given by the Pauli principle, resulting in the Fermi-
Dirac distribution function. For more general quasipar
ticles one may try to understand the spectrum in terms
more general distributions that correspond to various form
of exclusion statistics.
The notion of exclusion statistics was introduced by Ha
dane [4], in the context of integrable theories with invers
square interactions. The main idea to study the way on
particle levels in the spectrum are filled to form allowed
many-particle states. The simplest scenario [4] is to a
sume that the act of filling an available one-particle sta
of type i reduces the dimension of the available Hilber
space for particles of typej by an amountgij. In the
absence of other interactions, the statistics matrixG ­
sgijd completely determines the thermodynamics (see, e.
[5,6]). Concrete examples of this type of exclusion statis
ics are the Calogero-Sutherland (CS) models of quantu
mechanics with inverse square two-body interactions (wi
adjustableg).
We here propose a new method for studying the e
clusion statistics of CFT quasiparticles. At the heart o
our method is what can be called a “transfer matrix fo
truncated chiral spectra.” We shall present a number
examples where CFT spectra are completely encoded
one-particle distribution functions (generalizing the Ferm
Dirac distribution). In special examples, the statistics tha
we find are of the type proposed by Haldane, while in othe
cases we find more general results.
One check on the distributions that we propose he
is the coefficientg in the low-temperature specific heat,© 1997 The American Physical Society
VOLUME 79, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R S 6 OCTOBER 1997
n
d
for
d
to
v-
-
d
which is known to be related to the central chargecCFT of
the CFT according to [7]
C
L
­ gk2Br0T , g ­
p
6
cCFT , (1)
where r0 ­ sh̄yFd21 is the density of states per uni
length.
To introduce our new method we focus on the simple
SUs2d invariant CFT, which is the SUs2d1 Wess-Zumino-
Witten model. For this CFT, a quasiparticle formulatio
has been proposed in [8] and worked out in great det
in [9]. The formulation uses operatorsf62s which create
quasiparticles called spinons. The spinons form a doub
under SUs2d and carry (dimensionless) energyL0 ­ s.
The chiral spectrum of the SUs2d1 CFT may be built in
the following manner. One starts by writing the following
polarizedN-spinon states:
f
1
2fs2N21dy4g2nN · · · f
1
2s3y4d2n2 f
1
2s1y4d2n1 j0l , (2)
with nN $ · · · $ n2 $ n1 $ 0. One then uses the Yan-
gian symmetry algebra to construct multispinon stat
with mixed 1 and2 indices. The collection of all these
states forms a basis of the full chiral spectrum. The SUs2d
content of the Yangian multiplet labeled byn1, . . . , nN
follows from the generalized commutation relations sati
fied by the spinon modes, or, equivalently, from the re
resentation theory of the Yangian [8,9].
Comparing the allowed spinon modesf62s with free
fermion modesc2s1y2d2l , we observe that the fermion
model 1 12 has “split” into an odd modes ­ l 1
1
4 and
an even modes ­ l 1 34 . To maintain a level spacing
of one unit, we view these two spinon modes as formin
a single “one-particle level” in the spectrum. What w
would like to do is to factorize the full chiral partition
sum of the CFT into a product over these one-partic
levels, so that the free energy becomes a sum of o
particle contributions.
While we cannot straightforwardly extract the contr
bution of thelth level, we may proceed as follows. Wet
st
n
ail
let
es
s-
p-
g
e
le
ne-
i-
truncate the chiral spectrum by only allowing the spino
modes of the firstl levels. We denote byPlsq, x1, x2d,
Qlsq, x1, x2d the partition functions of the firstl levels,
where in Pl the highest occupied mode should be od
while in Ql it should be even. In formula
Plsq, x1, x2d ­ trace
s#ld
odd sqL0 x
N1
1 x
N2
2 d (3)
and similar forQlsq, x1, x2d, where we introduce chemi-
cal potentialsm6, write x6 ­ ebm6 , q ­ e2bs2pyLds1yr0d,
and denote byN6 the number of6 spinons in a state.
Restricting to fully polarized states (puttingx2 ­ 0), we
obtain the following recursion relation√
Pl
Ql
!
­
√
1 ql2s3y4d x1
ql2s1y4d x1 1 1 q2l21 x21
! √
Pl21
Ql21
!
, (4)
with P0 ­ 0 and Q0 ­ 1. We now make the impor-
tant step of approximating these exact expressions
Plsq, x1d andQlsq, x1d by the product fromj ­ 1 to l of
the largest eigenvaluel
s jd
1 sq, xd of the jth recursion ma-
trix. Clearly, this brings the partition sums in the desire
factorized form and reduces thermodynamic quantities
sums of independent single-level contributions. The a
erage occupation of thelth level is found to be
nsldsq, xd ­ x≠x ln l
sld
1 sq, xd ­
2p
1 1 4 q122lx22
­
2
p
1 1 4 e22bsm12el d
, (5)
with el ­ sl 1
1
2 d
2p
L
1
r0
. Note that the maximal occupa
tion of a given level is 2. Interestingly, this distribution
function for polarized SUs2d spinons is identical to that of
particles withg ­ 12 (semionic) exclusion statistics [5].
If we now include the multispinon states with mixe
indices, we find that thelth recursion matrix becomes√
1 2 q2l22x2 ql2s3y4dxsz 1 1z d
ql2s1y4dxsz 1 1z d s1 2 q
2l22x2d 1 1 q2l21x2sz2 1 1 1 1z2 d
!
, (6)e
n
atwhere we putx6 ­ x z61. Following the same logic, we
derive the distribution functionsnsldsq, x, zd. Interesting
special cases are
nsldsq, xd ­ x≠x ln l
sld
1 sq, x, z ­ 1d ­
2
1 1 qs1y2d2lx21
(7)
for the expected total number of spinons in levell, and
Q
sldsq, zd ­
e
2
z≠z ln l
sld
1 sq, x ­ 1, zd
­
e ql2s1y2dsz 2 z21dp
q2l21sz 1 z21d2 1 4 s1 2 q2l21d
(8)for the expected charge at levell (we assume charges6 e2
for the 6 spinons). The distributions (7) and (8) agre
with the distributions obtained from fractional exclusio
statistics with
G ­
√
1
2
1
2
1
2
1
2
!
.
Note that (7) implies that forz ­ 1 (zero voltage) the
thermodynamics of the spinon system is identical to th
of two free fermions and the central charge iscCFT ­
2 3 12 ­ 1. Note also that, withz ­ e
s1y2dbeV , the
integrated charge1L
P
`
l­1 Q
sldsq, zd ­ 14p e
2Vro , which2609
VOLUME 79, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R S 6 OCTOBER 1997
o
d
c-
k
l
s-is half of the value obtained for two charge6e free
fermions.
The correspondence with Haldane statistics is satisfyi
since the spinons of the associated SUs2d HS spin chain
satisfy these same statistics [4]. This confirms the valid
of our new approach, which in no way relied on the exa
solution of the HS chain.
We now present some special examples.
(1) TheSUsnd1 CFTs.—The first generalization of the
SUs2d1 results concerns the SUsnd1 spinons. The Yangian
symmetry of the SUsnd1 CFT was established in [10]
while the spinon formulation was presented in [3]. The
aren fundamental spinon speciesfi , transforming in the
representation of SUsnd. Repeating the analysis shown
above, finding explicitly the SUsnd analog of the recursion
matrices (4) and (6) (see [11]), we find that (i) a sing
spinon speciesfi0 (in absence of any others) satisfie
Haldane statistics withg ­ n21n , (ii) when exciting all
n spinon species symmetrically (choosing allxi equal to
x ­ ebm), the expected total occupation of thelth level is
given by
nsldsq, xd ­ x≠x lnf1 1 qlx 1 . . . 1 sqlxdn21gn. (9)
Comparing these results, one finds that forn . 2 there is
negative mutual exclusion among different spinons.
Interestingly, the statistics going with the distributio
(9) were proposed by Gentile as early as 1940 [12]. O
finds that a single “Gentile parafermion” contributes th
amount n21n to the central charge, so that the full resu
becomescCFT ­ n
n21
n ­ n 2 1, as expected.
Our results are consistent with [13], where, by differe
methods, the link between SUsnd HS spin chains and
Gentile parafermions has also been established.
(2) ZN parafermions.—Within the context of CFT, the
simplest generalization of the Majorana fermion is the s
called ZN parafermion. It features in a CFT of centra
chargecN ­
2sN21d
N12 as a primary field of dimensionhN ­
N21
N and ZN charge 1 [14]. By applying the method
outlined above, we obtained a distribution function for th
ZN parafermion and established that the full CFT spectru
is reproduced by a gas of noninteracting quasiparticles
this type. For the purpose of explaining these results,
focus on the caseN ­ 3.
It is well known that the chiral spectrum of theZ3
parafermion CFT can be interpreted in terms of tw
parafermionsc6 of oppositeZ3 charge. However, by
exploiting the generalized commutation relations obey
by the modes ofc6 [14], one easily shows that the
modes ofc1 alone can generate the full spectrum, an
that thec2 quasiparticle can be viewed as a composi
of two c1 quasiparticles. Having understood how th
c1 modes alone build the chiral spectrum (see [11
one may define truncated partition sums. We foun
the following recursion matrix between thelth and
the sl 2 1dth truncated sums (which each have thre2610ng
ity
ct
re
le
s
n
ne
e
lt
nt
o-
l
e
m
of
we
o
ed
d
te
e
]),
d
e
components)0B@ s1 2 y3d y2 yys1 2 y3d 1 2y2
2y2s1 2 y3d ys1 1 y3d 1 1 2y3
1CA , (10)
with y ­ x ql. The Z3 parafermion distribution function
is expressed in terms of the largest eigenvaluel1 of this
matrix
nsldsq, xd ­ s y≠y ln l1d s y ­ xqld . (11)
From the asymptoticsl1s yd ~ y3, one finds that the maxi-
mum occupation per level equals 3. Using the Cardan
formula one may writensld in closed form. We here
present a plot (Fig. 1), which displays the functionn as a
function of energy. As a check we (numerically) evaluate
the coefficientg of the specific heat, reproducing the
expected valueg ­ p6
4
5 .
For generalN, the distribution forZN parafermions
allows a maximum of12 NsN 2 1d particles per level.
(3) Quantum Hall effect edge theories.—As a further
application we briefly discuss edge theories for the fra
tional quantum Hall effect (FQHE). We shall come bac
to this topic in a separate publication [15]. For then ­ 1m
FQHE (withm an odd integer), the edge theory is a chira
FIG. 1. Distribution functions forZ3 parafermions (solid
line), for ordinary fermions (dashed line), and for particles sati
fying g ­ 13 exclusion statistics (dotted line). All distributions
are atm ­ 0; the energy is given in unitsb21.
VOLUME 79, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R S 6 OCTOBER 1997
e
ic
d
is
op
is
,
-
s.
d,
s.
P
cCFT ­ 1 CFT at compactification radiusR2 ­ m. The
natural quasiparticles to consider are the edge electron
charge2e) and the edge quasihole (of chargeem ). Writ-
ing the spectrum in terms of these quasiparticles, and a
plying the above procedure, we find that the fundamen
quasiparticles are independent, and obey Haldane exc
sion statistics withg ­ m and g ­ 1ym, respectively.
This is consistent with the result of bosonization applied
g-ons [16].
The (known) results for the specific heat (cCFT ­ 1)
and the response to voltage (QL ­
1
m
1
2p e
2Vr0) are easily
reproduced by exploiting the duality betweeng ­ m and
g ­ 1ym statistics [15]. The central charge arises as
sum cm 1 c1ym. For m ­ 2 we find c2 ­
2
5 , c1y2 ­
3
5 ,
while for m ­ 3, c3 ­ 0.343 . . . , c1y3 ­ 0.655 . . . .
These quantum Hall results can be appreciated on
basis of the analogy with CS quantum mechanics wi
inverse square interactions (see, e.g., [17]). We stre
however, that our derivation does not rely on this analog
When applied to composite edges for the FQHE in th
Jain series, at filling fraction ­ n2np11 , the combined
results of this paper lead to a formulation in terms of (i)
single charged mode, satisfying Haldane statistics w
g ­ n, and (ii) a set ofn spinons for SUsnd1, satisfying
the generalized Gentile statistics described above. T
new quasiparticle formulation forms a suitable startin
point for studying finite-T features (including tunneling
characteristics) of these edges.
The potential applications of our new approach to CF
spectra are manifold, especially when the extension
boundary CFTs is considered. We mention edge state sc
tering, state counting, and a variety of finite-T characteris-
tics of (non-Abelian) FQHE edges, and non-Fermi liqui
features in quantum impurity problems such as the mul
channel Kondo effect.
The method presented here can successfully be app
to many CFTs other than those mentioned here [11]. A
interesting example is thecCFT ­ 2
22
5 CFT for the Yang-
Lee edge singularity where the Virasoro generators ta
on the role ofg ­ 2 quasiparticles, giving the correct
effective central chargẽcCFT ­
2
5 [18].
To conclude, we should stress that the CFT quasip
ticles considered in this Letter should not be confused wi(of
p-
tal
lu-
to
a
the
th
ss,
y.
e
a
ith
his
g
T
to
at-
d
ti-
lied
n
ke
ar-
th
the asymptotic states of the “masslessS matrix” approach
to CFT [1]. The two are very different, and they give ris
to different distribution functions in the thermodynam
limit.
Many thanks to A. W. W. Ludwig, P. Bouwknegt, an
R. van Elburg for helpful discussions. Part of th
work was done at the ITP Santa Barbara Worksh
on “Quantum Field Theory in Low Dimensions: from
Condensed Matter to Particle Physics.” The author
supported in part by the foundation FOM.
[1] A. B. Zamolodchikov and Al. B. Zamolodchikov, Nucl.
Phys.B379, 602 (1992).
[2] V. Bazhanov, S. Lukyanov, and A. B. Zamolodchikov
Commun. Math. Phys.11, 381 (1996); hep-th/9604044.
[3] P. Bouwknegt and K. Schoutens, Nucl. Phys.B482, 345
(1996); Report No. q-alg/9703021.
[4] F. D. M. Haldane, Phys. Rev. Lett.67, 937 (1991).
[5] Y. S. Wu, Phys. Rev. Lett.73, 922 (1994); C. Nayak and
F. Wilczek, Phys. Rev. Lett.73, 2740 (1994); S. B. Isakov,
Mod. Phys. Lett. B8, 319 (1994).
[6] S. B. Isakov, D. P. Arovas, J. Myrheim, and A. P. Poly
chronakos, Phys. Lett. A212, 299 (1996).
[7] H. W. J. Blöte, J. L. Cardy, and M. P. Nightingale, Phy
Rev. Lett.56, 742 (1986); I. Affleck, Phys. Rev. Lett.56,
742 (1986).
[8] F. D. M. Haldane, Phys. Rev. Lett.66, 1529 (1991);
F. D. M. Haldane, Z. N. C. Ha, J. C. Talstra, D. Bernar
and V. Pasquier, Phys. Rev. Lett.69, 2021 (1992).
[9] D. Bernard, V. Pasquier, and D. Serban, Nucl. Phy
B428, 612 (1994); P. Bouwknegt, A. W. W. Ludwig, and
K. Schoutens, Phys. Lett. B338, 448 (1994).
[10] K. Schoutens, Phys. Lett. B331, 335 (1994).
[11] K. Schoutens (to be published).
[12] G. Gentile, Nuovo Cimento17, 493 (1940).
[13] Y. Kato and Y. Kuramoto, J. Phys. Soc. Jpn.64, 4518
(1995).
[14] A. B. Zamolodchikov and V. A. Fateev, Sov. Phys. JET
62, 215 (1985).
[15] R. van Elburg and K. Schoutens (to be published).
[16] Y.-S. Wu and Y. Yu, Phys. Rev. Lett.75, 890 (1995).
[17] S. Iso, Nucl. Phys.B443, 581 (1995).
[18] E. Frenkel and A. Szenes, Duke Math. J.69, 53 (1993).2611


NARCIS 

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 in conformal field theory spectraSchoutens, K.DOI10.1103/PhysRevLett.79.2608Publication date1997Published inPhysical Review LettersLink to publicationCitation for published version (APA):Schoutens, K. (1997). Exclusion statistics in conformal field theory spectra. Physical ReviewLetters, 79, 2608-2611. https://doi.org/10.1103/PhysRevLett.79.2608General 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 79, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R S 6 OCTOBER 1997rmallizes” inries.2608Exclusion Statistics in Conformal Field Theory SpectraKareljan SchoutensInstitute for Theoretical Physics, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands(Received 17 June 1997)We propose a new method for investigating the exclusion statistics of quasiparticles in confofield theory (CFT) spectra. The method leads to one-particle distribution functions, which generathe Fermi-Dirac distribution. For the simplest SUsnd invariant CFTs we find a generalization ofGentile parafermions, and we obtain new distributions for the simplestZN -invariant CFTs. Inspecial examples, our approach reproduces distributions based on “fractional exclusion statisticthe sense of Haldane. We comment on applications to fractional quantum Hall effect edge theo[S0031-9007(97)04183-5]PACS numbers: 05.30.–d, 05.70.Ce, 11.25.Hfnsnisehlor-tcefnTre-ofsl-ee-s-tetg.,-mthx-frofini-trreConformal field theory (CFT) in two dimensions is ainvaluable tool in the analysis of (among other thingthe low-temperature properties of a variety of condensmatter systems. In the literature on CFT (which is vasthere is a certain dichotomy between, on the one hadescriptions based onbosonizationand, on the other,descriptions which give a central role toquasiparticles.In the standard approach to rational CFT, the spectruis described in terms of representations of a bosonic crent algebra called thechiral algebra. Examples are affineKac-Moody (KM) algebras and higher spin extension(called W algebras) of the Virasoro algebra. In applcations to condensed matter systems, a similar descriptis often used. Examples are the Luttinger liquids for 1interacting electrons, which have a Us1d affine KM sym-metry and are usually treated in bosonized form. Othexamples are the low-temperature theories for the muchannel Kondo effect, which have been analyzed on tbasis of their SUs2dk affine KM symmetry.When dealing with the CFT for noninteracting electrons, one clearly does not need bosonization, but ufree fermions (satisfying canonical anticommutation rlations) instead. While interacting electrons give risemore general CFTs, it is entirely natural to look for descriptions that mimic the treatment of free electrons. Tidea is to identify fundamental excitations (quasiparticleover the many-body ground state and to study their propties. For integrable models (analyzed using Bethe ansand factorizable scattering) such an approach is by nstandard.Until now, a general approach to “CFT quasiparticleshas been lacking. In special cases, progress has bmade by viewing specific CFTs as massless limits of itegrable particle theories, leading to “masslessS matricesfor CFT quasiparticles” [1]. Related to this is a new “integrable” approach to CFT; see [2]. Other special exampare CFTs which can be viewed as continuum limitsintegrable models of lattice electrons. Examples are tSUsnd1 CFTs, which can be cast in a spinon formulatioanalogous to that of the SUsnd Haldane-Shastry (HS) spinchains [3].0031-9007y97y79(14)y2608(4)$10.00)edt),d,mur-s-ionDerlti-he-es-to-es)er-atzow”eenn--esfhenIn this Letter, we propose an approach to CFT quasipaticles which is intrinsic to the CFT, i.e., which does nomake reference to associated integrable particle or lattimodels. The starting point is the finite size spectrum oa CFT defined on a cylinder. In particular, we focus othe chiral Hilbert spaces, which together build up a CFpartition function. In a quasiparticle formulation, a chi-ral Hilbert space is viewed as a collection of multiparticle(quasiparticle) states (we shall write “particle” for “quasi-particle” where no confusion can arise). Forfermionicquasiparticles, the systematics of multiparticle states asimply given by the Pauli principle, resulting in the Fermi-Dirac distribution function. For more general quasiparticles one may try to understand the spectrum in termsmore general distributions that correspond to various formof exclusion statistics.The notion of exclusion statistics was introduced by Hadane [4], in the context of integrable theories with inverssquare interactions. The main idea to study the way onparticle levels in the spectrum are filled to form allowedmany-particle states. The simplest scenario [4] is to asume that the act of filling an available one-particle staof type i reduces the dimension of the available Hilberspace for particles of typej by an amountgij. In theabsence of other interactions, the statistics matrixG ­sgijd completely determines the thermodynamics (see, e.[5,6]). Concrete examples of this type of exclusion statisics are the Calogero-Sutherland (CS) models of quantumechanics with inverse square two-body interactions (wiadjustableg).We here propose a new method for studying the eclusion statistics of CFT quasiparticles. At the heart oour method is what can be called a “transfer matrix fotruncated chiral spectra.” We shall present a numberexamples where CFT spectra are completely encodedone-particle distribution functions (generalizing the FermDirac distribution). In special examples, the statistics thawe find are of the type proposed by Haldane, while in othecases we find more general results.One check on the distributions that we propose heis the coefficientg in the low-temperature specific heat,© 1997 The American Physical SocietyVOLUME 79, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R S 6 OCTOBER 1997ndfordtov--dwhich is known to be related to the central chargecCFT ofthe CFT according to [7]CL­ gk2Br0T , g ­p6cCFT , (1)where r0 ­ sh̄yFd21 is the density of states per unilength.To introduce our new method we focus on the simpleSUs2d invariant CFT, which is the SUs2d1 Wess-Zumino-Witten model. For this CFT, a quasiparticle formulatiohas been proposed in [8] and worked out in great detin [9]. The formulation uses operatorsf62s which createquasiparticles called spinons. The spinons form a doubunder SUs2d and carry (dimensionless) energyL0 ­ s.The chiral spectrum of the SUs2d1 CFT may be built inthe following manner. One starts by writing the followingpolarizedN-spinon states:f12fs2N21dy4g2nN · · · f12s3y4d2n2 f12s1y4d2n1 j0l , (2)with nN $ · · · $ n2 $ n1 $ 0. One then uses the Yan-gian symmetry algebra to construct multispinon statwith mixed 1 and2 indices. The collection of all thesestates forms a basis of the full chiral spectrum. The SUs2dcontent of the Yangian multiplet labeled byn1, . . . , nNfollows from the generalized commutation relations satified by the spinon modes, or, equivalently, from the reresentation theory of the Yangian [8,9].Comparing the allowed spinon modesf62s with freefermion modesc2s1y2d2l , we observe that the fermionmodel 1 12 has “split” into an odd modes ­ l 114 andan even modes ­ l 1 34 . To maintain a level spacingof one unit, we view these two spinon modes as formina single “one-particle level” in the spectrum. What wwould like to do is to factorize the full chiral partitionsum of the CFT into a product over these one-particlevels, so that the free energy becomes a sum of oparticle contributions.While we cannot straightforwardly extract the contrbution of thelth level, we may proceed as follows. Wetstnailletess-p-gelene-i-truncate the chiral spectrum by only allowing the spinomodes of the firstl levels. We denote byPlsq, x1, x2d,Qlsq, x1, x2d the partition functions of the firstl levels,where in Pl the highest occupied mode should be odwhile in Ql it should be even. In formulaPlsq, x1, x2d ­ traces#ldodd sqL0 xN11 xN22 d (3)and similar forQlsq, x1, x2d, where we introduce chemi-cal potentialsm6, write x6 ­ ebm6 , q ­ e2bs2pyLds1yr0d,and denote byN6 the number of6 spinons in a state.Restricting to fully polarized states (puttingx2 ­ 0), weobtain the following recursion relation√PlQl!­√1 ql2s3y4d x1ql2s1y4d x1 1 1 q2l21 x21! √Pl21Ql21!, (4)with P0 ­ 0 and Q0 ­ 1. We now make the impor-tant step of approximating these exact expressionsPlsq, x1d andQlsq, x1d by the product fromj ­ 1 to l ofthe largest eigenvaluels jd1 sq, xd of the jth recursion ma-trix. Clearly, this brings the partition sums in the desirefactorized form and reduces thermodynamic quantitiessums of independent single-level contributions. The aerage occupation of thelth level is found to bensldsq, xd ­ x≠x ln lsld1 sq, xd ­2p1 1 4 q122lx22­2p1 1 4 e22bsm12el d, (5)with el ­ sl 112 d2pL1r0. Note that the maximal occupation of a given level is 2. Interestingly, this distributionfunction for polarized SUs2d spinons is identical to that ofparticles withg ­ 12 (semionic) exclusion statistics [5].If we now include the multispinon states with mixeindices, we find that thelth recursion matrix becomes√1 2 q2l22x2 ql2s3y4dxsz 1 1z dql2s1y4dxsz 1 1z d s1 2 q2l22x2d 1 1 q2l21x2sz2 1 1 1 1z2 d!, (6)enatwhere we putx6 ­ x z61. Following the same logic, wederive the distribution functionsnsldsq, x, zd. Interestingspecial cases arensldsq, xd ­ x≠x ln lsld1 sq, x, z ­ 1d ­21 1 qs1y2d2lx21(7)for the expected total number of spinons in levell, andQsldsq, zd ­e2z≠z ln lsld1 sq, x ­ 1, zd­e ql2s1y2dsz 2 z21dpq2l21sz 1 z21d2 1 4 s1 2 q2l21d(8)for the expected charge at levell (we assume charges6 e2for the 6 spinons). The distributions (7) and (8) agrewith the distributions obtained from fractional exclusiostatistics withG ­√12121212!.Note that (7) implies that forz ­ 1 (zero voltage) thethermodynamics of the spinon system is identical to thof two free fermions and the central charge iscCFT ­2 3 12 ­ 1. Note also that, withz ­ es1y2dbeV , theintegrated charge1LP`l­1 Qsldsq, zd ­ 14p e2Vro , which2609VOLUME 79, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R S 6 OCTOBER 1997odc-kls-is half of the value obtained for two charge6e freefermions.The correspondence with Haldane statistics is satisfyisince the spinons of the associated SUs2d HS spin chainsatisfy these same statistics [4]. This confirms the validof our new approach, which in no way relied on the exasolution of the HS chain.We now present some special examples.(1) TheSUsnd1 CFTs.—The first generalization of theSUs2d1 results concerns the SUsnd1 spinons. The Yangiansymmetry of the SUsnd1 CFT was established in [10]while the spinon formulation was presented in [3]. Thearen fundamental spinon speciesfi , transforming in therepresentation of SUsnd. Repeating the analysis shownabove, finding explicitly the SUsnd analog of the recursionmatrices (4) and (6) (see [11]), we find that (i) a singspinon speciesfi0 (in absence of any others) satisfieHaldane statistics withg ­ n21n , (ii) when exciting alln spinon species symmetrically (choosing allxi equal tox ­ ebm), the expected total occupation of thelth level isgiven bynsldsq, xd ­ x≠x lnf1 1 qlx 1 . . . 1 sqlxdn21gn. (9)Comparing these results, one finds that forn . 2 there isnegative mutual exclusion among different spinons.Interestingly, the statistics going with the distributio(9) were proposed by Gentile as early as 1940 [12]. Ofinds that a single “Gentile parafermion” contributes thamount n21n to the central charge, so that the full resubecomescCFT ­ nn21n ­ n 2 1, as expected.Our results are consistent with [13], where, by differemethods, the link between SUsnd HS spin chains andGentile parafermions has also been established.(2) ZN parafermions.—Within the context of CFT, thesimplest generalization of the Majorana fermion is the scalled ZN parafermion. It features in a CFT of centrachargecN ­2sN21dN12 as a primary field of dimensionhN ­N21N and ZN charge 1 [14]. By applying the methodoutlined above, we obtained a distribution function for thZN parafermion and established that the full CFT spectruis reproduced by a gas of noninteracting quasiparticlesthis type. For the purpose of explaining these results,focus on the caseN ­ 3.It is well known that the chiral spectrum of theZ3parafermion CFT can be interpreted in terms of twparafermionsc6 of oppositeZ3 charge. However, byexploiting the generalized commutation relations obeyby the modes ofc6 [14], one easily shows that themodes ofc1 alone can generate the full spectrum, anthat thec2 quasiparticle can be viewed as a composiof two c1 quasiparticles. Having understood how thc1 modes alone build the chiral spectrum (see [11one may define truncated partition sums. We founthe following recursion matrix between thelth andthe sl 2 1dth truncated sums (which each have thre2610ngityctrelesnneeltnto-lemofweoeddtee]),decomponents)0B@ s1 2 y3d y2 yys1 2 y3d 1 2y22y2s1 2 y3d ys1 1 y3d 1 1 2y31CA , (10)with y ­ x ql. The Z3 parafermion distribution functionis expressed in terms of the largest eigenvaluel1 of thismatrixnsldsq, xd ­ s y≠y ln l1d s y ­ xqld . (11)From the asymptoticsl1s yd ~ y3, one finds that the maxi-mum occupation per level equals 3. Using the Cardanformula one may writensld in closed form. We herepresent a plot (Fig. 1), which displays the functionn as afunction of energy. As a check we (numerically) evaluatethe coefficientg of the specific heat, reproducing theexpected valueg ­ p645 .For generalN, the distribution forZN parafermionsallows a maximum of12 NsN 2 1d particles per level.(3) Quantum Hall effect edge theories.—As a furtherapplication we briefly discuss edge theories for the frational quantum Hall effect (FQHE). We shall come bacto this topic in a separate publication [15]. For then ­ 1mFQHE (withm an odd integer), the edge theory is a chiraFIG. 1. Distribution functions forZ3 parafermions (solidline), for ordinary fermions (dashed line), and for particles satifying g ­ 13 exclusion statistics (dotted line). All distributionsare atm ­ 0; the energy is given in unitsb21.VOLUME 79, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R S 6 OCTOBER 1997eicdisopis,-s.d,s.PcCFT ­ 1 CFT at compactification radiusR2 ­ m. Thenatural quasiparticles to consider are the edge electroncharge2e) and the edge quasihole (of chargeem ). Writ-ing the spectrum in terms of these quasiparticles, and aplying the above procedure, we find that the fundamenquasiparticles are independent, and obey Haldane excsion statistics withg ­ m and g ­ 1ym, respectively.This is consistent with the result of bosonization appliedg-ons [16].The (known) results for the specific heat (cCFT ­ 1)and the response to voltage (QL ­1m12p e2Vr0) are easilyreproduced by exploiting the duality betweeng ­ m andg ­ 1ym statistics [15]. The central charge arises assum cm 1 c1ym. For m ­ 2 we find c2 ­25 , c1y2 ­35 ,while for m ­ 3, c3 ­ 0.343 . . . , c1y3 ­ 0.655 . . . .These quantum Hall results can be appreciated onbasis of the analogy with CS quantum mechanics wiinverse square interactions (see, e.g., [17]). We strehowever, that our derivation does not rely on this analogWhen applied to composite edges for the FQHE in thJain series, at filling fraction ­ n2np11 , the combinedresults of this paper lead to a formulation in terms of (i)single charged mode, satisfying Haldane statistics wg ­ n, and (ii) a set ofn spinons for SUsnd1, satisfyingthe generalized Gentile statistics described above. Tnew quasiparticle formulation forms a suitable startinpoint for studying finite-T features (including tunnelingcharacteristics) of these edges.The potential applications of our new approach to CFspectra are manifold, especially when the extensionboundary CFTs is considered. We mention edge state sctering, state counting, and a variety of finite-T characteris-tics of (non-Abelian) FQHE edges, and non-Fermi liquifeatures in quantum impurity problems such as the mulchannel Kondo effect.The method presented here can successfully be appto many CFTs other than those mentioned here [11]. Ainteresting example is thecCFT ­ 2225 CFT for the Yang-Lee edge singularity where the Virasoro generators taon the role ofg ­ 2 quasiparticles, giving the correcteffective central chargẽcCFT ­25 [18].To conclude, we should stress that the CFT quasipticles considered in this Letter should not be confused wi(ofp-tallu-toathethss,y.eaithhisgTtoat-dti-liednkear-ththe asymptotic states of the “masslessS matrix” approachto CFT [1]. The two are very different, and they give risto different distribution functions in the thermodynamlimit.Many thanks to A. W. W. Ludwig, P. Bouwknegt, anR. van Elburg for helpful discussions. Part of thwork was done at the ITP Santa Barbara Workshon “Quantum Field Theory in Low Dimensions: fromCondensed Matter to Particle Physics.” The authorsupported in part by the foundation FOM.[1] A. B. Zamolodchikov and Al. B. Zamolodchikov, Nucl.Phys.B379, 602 (1992).[2] V. Bazhanov, S. Lukyanov, and A. B. ZamolodchikovCommun. Math. Phys.11, 381 (1996); hep-th/9604044.[3] P. Bouwknegt and K. Schoutens, Nucl. Phys.B482, 345(1996); Report No. q-alg/9703021.[4] F. D. M. Haldane, Phys. Rev. Lett.67, 937 (1991).[5] Y. S. Wu, Phys. Rev. Lett.73, 922 (1994); C. Nayak andF. Wilczek, Phys. Rev. Lett.73, 2740 (1994); S. B. Isakov,Mod. Phys. Lett. B8, 319 (1994).[6] S. B. Isakov, D. P. Arovas, J. Myrheim, and A. P. Polychronakos, Phys. Lett. A212, 299 (1996).[7] H. W. J. Blöte, J. L. Cardy, and M. P. Nightingale, PhyRev. Lett.56, 742 (1986); I. Affleck, Phys. Rev. Lett.56,742 (1986).[8] F. D. M. Haldane, Phys. Rev. Lett.66, 1529 (1991);F. D. M. Haldane, Z. N. C. Ha, J. C. Talstra, D. Bernarand V. Pasquier, Phys. Rev. Lett.69, 2021 (1992).[9] D. Bernard, V. Pasquier, and D. Serban, Nucl. PhyB428, 612 (1994); P. Bouwknegt, A. W. W. Ludwig, andK. Schoutens, Phys. Lett. B338, 448 (1994).[10] K. Schoutens, Phys. Lett. B331, 335 (1994).[11] K. Schoutens (to be published).[12] G. Gentile, Nuovo Cimento17, 493 (1940).[13] Y. 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Exclusion Statistics in Conformal Field Theory Spectra

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