Non-Abelian quantum Hall states and their quasiparticles: From the pattern of zeros to vertex algebra

Transcription

1 Non-Abelian quantum Hall states and their quasipartiles: From the pattern of zeros to vertex algebra Yuan-Ming Lu, 1 Xiao-Gang Wen,, Zhenghan Wang, 4 and Ziqiang Wang 1 1 Department of Physis, Boston College, Chestnut Hill, Massahusetts 0467, USA Department of Physis, Massahusetts Institute of Tehnology, Cambridge, Massahusetts 019, USA Perimeter Institute for Theoretial Physis, 1 Caroline Street North, Waterloo, Ontario, Canada NJ Y5 4 Mirosoft Station Q, CNSI Bldg. Rm 7, University of California, Santa Barbara, California 9106, USA Reeived 5 Otober 009; revised manuript reeived 1 January 010; published 17 Marh 010 In the pattern-of-zeros approah to quantum Hall states, a set of data n;m;s a a=1,...,n;n,m,s a N alled the pattern of zeros is introdued to haraterize a quantum Hall wave funtion. In this paper we find suffiient onditions on the pattern of zeros so that the data orrespond to a valid wave funtion. Some times, a set of data n;m;s a orresponds to a unique quantum Hall state, while other times, a set of data orresponds to several different quantum Hall states. So in the latter ases, the pattern of zeros alone does not ompletely haraterize the quantum Hall states. In this paper, we find that the following expanded set of data n;m;s a ; a=1,...,n;n,m,s a N; R provides a more omplete haraterization of quantum Hall states. Eah expanded set of data ompletely haraterizes a unique quantum Hall state, at least for the examples diussed in this paper. The result is obtained by ombining the pattern of zeros and Z n simple-urrent vertex algebra whih deribes a large lass of Abelian and non-abelian quantum Hall states Zn. The more omplete haraterization in terms of n;m;s a ; allows us to obtain more topologial properties of those states, whih inlude the entral harge of edge states, the aling dimensions and the statistis of quasipartile exitations. DOI: /PhysRevB PACS number s : 7.4. f, 0.65.Fd I. INTRODUCTION Materials an have many different forms, whih is partially due to the very rih ways in whih atoms and eletrons an organize. The different organizations orrespond to different phases of matter or states of matter. It is very important for physiists to understand these different states of matter and the phase transitions between them. At zero temperature, the phases are deribed by the ground-state wave funtions, whih are omplex wave funtions r 1,r,...,r N with N variables. So mathematially, to deribe zero-temperature phases, we need to haraterize and lassify the ground-state wave funtions with variables, whih is a very hallenging mathematial problem. For a long time we believe that all states of matter and all phase transitions between them are haraterized by their broken symmetries and the assoiated order parameters. 1 A general theory for phases and phase transitions is developed based on this symmetry breaking piture. So within the paradigm of symmetry breaking, a many-body wave funtion is haraterized by its symmetry properties. Landau s symmetry breaking theory is a very suessful theory and has dominated the theory of phases and phase transitions until the diovery of frational quantum Hall FQH effet., FQH states annot be deribed by symmetry breaking sine different FQH states have exatly the same symmetry. So different FQH states must ontain a new kind of order. The new order is alled topologial order 4 6 and the assoiated phase alled topologial phase beause their harateristi universal properties suh as the ground states degeneray on a torus 4 are invariant under any small perturbations of the system. Unlike symmetry-breaking phases deribed by loal order parameters, a topologial phase is haraterized by a pattern of long-range quantum entanglement. 7 9 In Ref. 10, the non-abelian Berry phases for the degenerate ground states are introdued to systematially haraterize and lassify topologial orders in FQH states as well as other topologially ordered states. In this paper, we further develop another systemati haraterization of the topologial orders in FQH states based on the pattern of zeros approah. 11,1 In the strong magneti field limit, a FQH wave funtion with filling fator 1 is an antisymmetri holomorphi polynomial of omplex oordinates z i =x i +iy i exept for a ommon fator that depends on geometry: say, a Gaussian z i fator exp i 4 for a planar geometry. After fatoring out an antisymmetri fator of i j z i z j, we an deribe a quantum Hall state by a symmetri polynomial z 1,...,z N in the N limit. 11 So the haraterization and lassifiation of long-range quantum entanglements in FQH states beome a problem of haraterizing and lassifying symmetri polynomials with infinite variables. In a reent series of work, 11 1 the pattern of zeros is introdued to haraterize and lassify symmetri polynomials of infinite variables. The pattern of zeros is deribed by a sequene of integers S a a=1,,..., where S a is the lowest order of zeros of the symmetri polynomial when we fuse a different variables together. The data S a a=1,,... an be further ompatified into a finite set n;m;s a a =1,,...,n;n,m N for n-luster quantum Hall states. Here N= 0,1,,... is the set of non-negative integers. It has been shown 11,1 that all known one-omponent Abelian and non-abelian quantum Hall states an be partially haraterized by pattern of zeros. It is also shown 11,1 that, for any given pattern of zeros S a, we an onstrut an ideal loal Hamiltonian H Sa suh that the FQH state with the pattern of zeros is a zero energy ground state of the Hamiltonian /010/81 11 / The Amerian Physial Soiety

2 LU et al. We would like to point out that, stritly speaking, a FQH state must be a state with a finite energy gap. But in this paper, we will use the term more loosely. We will all one state a FQH state if it an be an zero energy state of an ideal Hamiltonian whose interation potential is a sum of funtions and derivatives of funtions. Suh a FQH state is alled an ideal FQH state. So our ideal FQH states may not be gapped. The ideal FQH states has a nie property that they an be haraterized by pattern of zeros or CFT. In addition to those ideal FQH wave funtions, one an also onstrut FQH wave funtion through omposite fermion approah. 19 Those omposite fermion wave funtions an be very low energies for Coulomb interation, for example, for the =/5 state. However, the omposite fermion wave funtions are in general not the ideal FQH wave funtion defined above. We still do not know how to extrat topologial properties from the omposite fermion wave funtions. On the other hand, the ideal FQH wave funtions introdued above are muh easier to handle and we an indeed extrat topologial properties from the ideal FQH wave funtions provided that they are gapped. Due to the length of this paper, in the following, we are going to summarize the issues that we are going to diuss in this paper. We will also summarize the main results that we obtain on those issues. A. Suffiient onditions on pattern of zeros Within the pattern-of-zero approah, two questions naturally arise: 1 Does any pattern of zeros, i.e., an arbitrary integer sequene n;m;s a orresponds to a symmetri polynomial z 1,...,z N? Are there any illegal patterns of zeros that do not orrespond to any symmetri polynomial? Given a legal pattern of zeros, an we onstrut a orresponding FQH many-body wave funtion? Is the FQH many-body wave funtion uniquely determined by the pattern of zeros? For question 1, it turns out that the pattern of zeros must satisfy some onsistent onditions 11,1 in order to deribe an existing symmetri polynomial. In other words, some sequenes n;m;s a do not orrespond to any symmetri polynomials. However, Refs. 11 and 1 only obtain some neessary onditions on the pattern of zeros n;m;s a. We still do not have a set of suffiient onditions on pattern of zeros that guarantee a pattern of zeros to orrespond to an existing symmetri polynomial. For question, right now, we do not have an effiient way to obtain orresponding FQH many-body wave funtion from a legal pattern of zeros. Further more, while some patterns of zeros an uniquely determine the FQH wave funtion, it is known that some other patterns of zeros annot uniquely determine the FQH wave funtion: i.e., in those ases, two different FQH wave funtions an have the same pattern of zeros. 11,0 This means that some patterns of zeros do not provide omplete information to fully haraterize FQH states. In this ase it is important to expand the data of pattern of zeros to obtain a more omplete haraterization of FQH states. We see that the above two questions are atually losely related. In this paper, we will try to address those questions. Motivated by the onformal field theory CFT onstrution of FQH wave funtions, 1 5 we will try to use the patterns of zeros to define and onstrut vertex algebras whih are CFTs. Sine the orrelation funtion of the eletron operator in the onstruted vertex algebra gives us the FQH wave funtion, one the vertex algebra is obtained from a pattern of zeros, we effetively find the orresponding FQH wave funtion for the pattern of zeros. In this way, we establish the onnetion between the pattern of zeros and the FQH wave funtion through the vertex algebra. In order for the orrelation of eletron operators in the vertex algebra to produe a single-valued eletron wave funtion with respet to eletron variables z 1,...,z N, eletron operators need to satisfy a so-alled simple-urrent property see Eqs. 4 and 60. Also the vertex algebra need to satisfy the generalized Jaobi identity GJI whih guarantees the assoiativity of the orresponding vertex algebra. 6 We find that only a ertain set of patterns of zeros an give rise to simple-urrent vertex algebras that satisfy the GJI. So the GJIs in simple-urrent vertex algebras give us a set of suffiient onditions on a pattern of zeros so that this pattern of zeros does orrespond to an existing symmetri polynomial. In this paper, we first try to use the pattern of zeros n;m;s a to define a Z n vertex algebra. From some of the GJI of the Z n vertex algebra, we obtain more neessary onditions on the pattern of zeros n;m;s a than those obtained in Refs. 11 and 1 see Se. III. It is not lear if those onditions are atually suffiient or not. Then, we try to use the pattern of zeros n;m;s a to define a Z n simple-urrent vertex algebra. From the omplete GJI of the Z n simple-urrent vertex algebra, we obtain suffiient onditions on the pattern of zeros n;m;s a see Se. V. B. How to expand the pattern-of-zeros data to ompletely haraterize the topologial order If a pattern of zeros n;m;s a an uniquely deribe the topologial order in a quantum Hall ground state, then from suh a quantitative deription, we should be able to alulate the topologial properties from the data n;m;s a. Indeed, this an be done. First different types of quasipartiles an also be quantitatively deribed and labeled by a set of sequenes S ;a that an be determined from the pattern-ofzeros data n;m;s a. 1 Those quantitative haraterizations of the quantum Hall ground state and quasipartiles allow us to alulate the number of different quasipartile types, quasipartile harges, fusion algebra between the quasipartiles, and topologial ground-state degeneray on a Riemann surfae of any genus. 1,1 However, from the pattern-of-zeros data, n;m;s a and S ;a, we still do not know how to alulate the quasipartile statistis and aling dimensions, as well as the entral harge of the edge states. This diffiulty is related to the fat that some patterns of zeros do not uniquely haraterize a FQH state. Thus one annot expet to alulate the topo

3 NON-ABELIAN QUANTUM HALL STATES AND THEIR logial properties of FQH state from the pattern-of-zeros data alone in those ases. We would like to point out that for a partiular pattern of zeros for the Z parafermion Moore- Read state, there is a suessful alulation of the quasipartile statistis from the pattern of zeros and thin ylinder limit. 7 But we do not know how to apply suh an approah to more general pattern of zeros. In this paper, we will try to solve this problem using a very different approah and for generi patterns of zeros. We first introdue a more omplete haraterization for FQH states in terms of a expanded data set: n;m;s a ;. Then, we use the data set n;m;s a ; to define a so alled Z n simpleurrent vertex algebra. The Z n simple-urrent vertex algebra ontain a subalgebra, Virasoro algebra, generated by the energy-momentum tensor T and is the entral harge of the Virasoro algebra. It ontains only n primary fields a, a =0,1,...,n 1 of the Virasoro algebra, with a Z n fusion rule a b a+b mod n, n = 0. Those a are alled simple urrents. The extra data is the one of the struture onstants of the Z n simple-urrent vertex algebra. One may want to inlude all the struture onstants C ab in the data set to have a omplete haraterization. But for the examples diussed in this paper, we find that data set n;m;s a ; already provides a omplete haraterization. So in this paper, we will use n;m;s a ; to haraterize FQH states. If later we find that n;m;s a ; is not suffiient, we an always add additional data, suh as C ab. Every Z n simple-urrent vertex algebra uniquely defines a FQH state, and the data n;m;s a ; that define a Z n simple-urrent vertex algebra also ompletely haraterizes a FQH state. We would like to remark that although the data n;m;s a ; and the orresponding Z n simple-urrent vertex algebras deribe a large lass of FQH states, they do not deribe all FQH states. For example let Ai be the FQH wave funtion deribed by a Z ni simple-urrent vertex algebra A i,,. Then, in general, the FQH state deribed by the produt wave funtion = A1 A annot be deribed by a simple-urrent vertex algebra. Suh a produt state is deribed by the produt vertex algebra A 1 A, whih is in general no longer a simple-urrent vertex algebra. So a more general FQH state should have the form = Ai. 1 i The study in Refs reveal that many FQH states deribed by pattern of zeros have the following form: z i = k Znaa z i, a k where k Znaa z i is the wave funtion deribed by Z a na parafermion vertex algebra. 1 The Z n simple-urrent vertex algebra mentioned above is a natural generalization of the k Z a na parafermion vertex algebra, and Eq. 1 naturally generalizes Eq.. Note that there are many Z n simple-urrent vertex algebras even for a fixed n, so there are many different Z n simple-urrent states. For the sublass of FQH states deribed by Z n simpleurrent vertex algebra whih inludes Virasoro algebra as an essential part, the quasipartile statistis and aling dimensions, as well as the entral harge of the edge states an be alulated from the data n;m;s a ;. Certainly, we an also alulate the number of different quasipartile types, quasipartile harges, fusion algebra between the quasipartiles, and topologial ground-state degeneray on a Riemann surfae of any genus. Obviously, not every olletion n;m;s a ; orresponds to a Z n simple-urrent vertex algebra and a FQH state. GJIs of the Z n simple-urrent vertex algebra generate the onsistent onditions on the data set n;m;s a ;. Only those data sets n;m;s a ; that satisfy the GJIs an deribe a Z n simple-urrent vertex algebra and FQH states. C. Organization of the paper This paper is organized as follows. In Se. II, we review and extend the pattern-of-zeros approah to quantum Hall states. In Se. III we use the pattern of zeros to define Z n vertex algebra, and then use assoiativity onditions i.e., the GJIs of the vertex algebra to obtain extra onditions on the pattern of zeros that deribe generi FQH states. In Se. IV we list some numerial solutions of the pattern of zeros for the generi FQH states that also satisfy those extra onsistent onditions found in Se. III. In Se. V we define and onstrut the Z n simple-urrent vertex algebra from the pattern of zeros. We list the onsistent onditions obtained from GJIs of Z n simple-urrent vertex algebra. The detailed derivations of those onsistent onditions are diussed in Appendixes D F. The onsistent onditions on the patterns of zeros that deribe a Z n simple-urrent vertex algebra are more restritive than those for a generi Z n vertex algebra. Some of the solutions of the Z n simple-urrent pattern of zeros are listed in Se. VII. In Se. VI, we diuss how to represent quasipartiles in the Z n simple-urrent vertex algebra, and to alulate the topologial properties of quasipartiles from the Z n simple-urrent pattern of zeros. In Se. VII, we apply the vertex-algebra approah developed here to study some simple but nontrivial examples of FQH states, whih inlude Z n parafermion states the Read-Rezayi states 17, the Z n simple-urrent FQH states of Z n Z n type, a Z 4 simple-urrent FQH state of Z 4 Z type, et. II. PATTERN-OF-ZEROS APPROACH TO GENERIC FQH STATES In this setion, we will review how to use the pattern of zeros to haraterize and lassify different FQH states that have one omponent A diussion on two-omponent FQH states an be find in Ref. 8. A. FQH wave funtions and symmetri polynomials Generally speaking, to lassify generi omplex wave funtions r 1,...,r N is not even a well-defined problem. Fortunately, under a strong magneti field, eletrons are spin polarized in the lowest Landau level LLL when the eletron filling fration e is less than 1. The wave funtion of a single eletron in LLL we set magneti length l B = /eb to be unity hereafter is m z =z m e z /4 in a planar geometry

4 LU et al. m is the angular momentum of this single partile state. Thus the many-body wave funtion of spin-polarized eletrons in the LLL should be N zi e z 1,...,z N = e z 1,...,z N exp 4 where e z i is an antisymmetri holomorphi polynomial of eletron oordinates z i =x i +iy i. The eletron filling fration e is defined as N N e = lim = lim, 4 N N N N p where N is the total number of flux quanta piering through the sample, and N p is the total degree of polynomial e z i. For FQH states e 1, we an extrat a Jastraw fator i j z i z j and the remaining part z 1,...,z N = e z 1,...,z N i j z i z j would be a symmetri polynomial of z i. We will onentrate on this symmetri polynomial to haraterize and lassify FQH states. For the symmetri polynomial z i we an also define a filling fration in the same way as in Eq. 4, only N p replaed by the total degree of bosoni polynomial z i. The eletron filling fration e has the following relation with this bosoni filling fration : 1 e = B. Fusion of a variables: The pattern of zeros The pattern of zeros 11,1 is introdued to deribe symmetri polynomials z i through ertain loal properties, i.e., fusion of a different variables z 1,...,z a. More speifially, we bring these a variables together, viewing z a+1,...,z N as fixed oordinates. By writing the a variables in the following manner z i = i +z a,,...,a, where z a = z 1+ +z a and a a =0, we an bring these a variables together by letting tend to zero. Then we an expand the polynomial z i in powers of, lim 1 + z a,..., a + z a ;z a+1,...,z N 0 + = S ap Sa z a, 1,..., a ;z a+1,...,z N + O S a In other words, S a is the lowest order of zeros when we fuse a variables together. The pattern of zeros, by definition, is this sequene of integers S a. In this paper, we will only onsider the polynomials that satisfy a unique fusion ondition: the fusion of a variables is unique, i.e., P Sa in Eq. 7 has the same form exept for an overall fator no matter how i are hosen. a 5 There are other equivalent deriptions of the pattern of zeros. One of them is the orbital deription, l a = S a S a 1, a =1,,..., 8 where l a labels the orbital angular momentum of the singlepartile state oupied by the ath partile. Another is the oupation deription in terms of a sequene of integers n l, 9 denoting the number of partiles oupying the orbital with angular momentum l. C. Consistent onditions for the pattern of zeros In this setion, we will review and summarize the onsistent onditions on S a derived in Refs. 11 and Translational invariane A translational invariant wave funtion z 1,...,z N = z 1 z,...,z N z satisfies 0,z,...,z N 0. As a result we have S 1 =0.. Symmetry ondition After we fuse a variables together to form an a-partile luster a-luster, it is natural to ask: what happens when we fuse an a-luster and another b-luster together? Let D a,b be the order of zeros obtained by fusing an a-luster and another b-luster together. It satisfies D a,b =D b,a 0. Sine the final state is the same as fusing a+b variables together, we find an one-to-one relation between the two sets of data D a,b and S a, 11 Da,b = Sa+b Sa Sb, a 1 S a = D i,1. 9 Sine z i is a symmetri polynomial, it deribes a state of bosoni partiles seated at oordinates z i. Thus the a a-luster seated at z i an also be regarded as a bosoni partile. The derived polynomial see P Sa in Eq. 7 as an example should be symmetri with respet to interhange of two idential bosons seated at z a 1 and z a. When we fuse suh two idential bosoni lusters z a 1 and z a together, we have P z a 1,z a,... = z a 1 z a a,ap D z 1 a a + z,... lim z 1 a z b + O z a 1 z a D a,a This leads to the symmetry ondition D a,a = even S a = even.. Conave onditions 11 The first onave ondition is the non-negativity of D a,b, D a,b 0 S a+b S a + S b. It omes naturally from the fusion of two lusters

5 NON-ABELIAN QUANTUM HALL STATES AND THEIR When we fuse three lusters together, we find the total order of off-partile zeros to be a,b, = D a,b+ D a,b D a, 0. This gives the nd onave ondition: 1 a,b, = S a+b+ + S a + S b + S S a+b S a+ S b n-luster ondition The above onditions, Eqs. 11, 1, and 14, have many solutions S a. Many of those solutions has a periodi struture that the whole sequene S a an be determined from first a few terms, k k 1 S a+kn = S a + ks n + mn + kma, 15 where m D n,1. 16 We will all suh a pattern of zeros the one that satisfies an n-luster ondition. We see that, for an n-luster sequene, only the first n terms, S,..., S n+1, are independent, and the whole sequene is determined by the first n terms. To understand the physial meaning of the n-luster ondition, we note that Eq. 15 is equivalent to the following ondition: kn,b, = 0 for any k. 17 This means that a symmetri polynomial that satisfies the n-luster ondition has the following defining property: as a funtion of the n-luster oordinate z n, the derived polynomial P z n,z 1 a, has no off-partile zeros. Under the n-luster ondition, we see that D n,n = nm = even. We also note that the filling fration is given by 18 = n 19 m sine S a 1 m n a as a. We like to mention that the luster ondition plays a very important role in the Jak polynomial approah to FQH. 1 However, in the pattern of zeros approah, the n-luster ondition only play a role of grouping and tabulating solutions of the onsistent onditions. The solutions with larger n orrespond to more omplex wave funtions whih usually orrespond to less stable FQH states. Later, we will diuss the relation between the pattern of zeros and CFT. We find that the solutions that do not satisfy the n-luster ondition i.e., with n= orrespond to irrational CFT, whih may always orrespond to gapless FQH states. The Jak polynomial approah and the pattern-of-zeros approah have some lose relations. The Jak polynomials are speial ases of the polynomials haraterized by pattern of zeros. 5. Summary To summarize, we see that the pattern of zeros for an n-luster polynomial is deribed by a set of positive integers n;m;s,...,s n. Introduing S 1 =0 and k k 1 S a+kn = S a + ks n + mn + kma, 0 whih define S n+1, S n+,..., we find that the data n;m;s,...,s n must satisfy D a,b = S a+b S a S b 0 D a,a = even, 1 a,b, = S a+b+ + S a + S b + S S a+b S a+ S b+ 0 for all a,b,=1,,,... Conditions 1 and are neessary onditions for a pattern of zeros to represent a symmetri polynomial. Although Eqs. 1 and are very simple, they are quite restritive and are quite lose to be suffiient onditions. In fat if we add an additional ondition a,b, = even, the three onditions, 1, may even beome suffiient onditions for a pattern of zeros to represent a symmetri polynomial. 11,1 However, this ondition is too strong to inlude many valid symmetri polynomials suh as Gaffnian, 4 a nontrivial Z 4 state diussed in detail in Se. VII. We will obtain some additional onditions in Se. III C, whih ombined with Eqs. 1 and form a set of neessary and potentially suffiient onditions for a valid pattern of zeros. D. Label the pattern of zeros by h a In this setion, we will introdue a new labeling heme of the pattern of zeros. We an label the pattern of zeros in terms of h a = S a as n n + am a m n. 4 This labeling heme is intimately onneted to the vertex algebra approah that we will diuss later. The n-luster ondition 0 of S a implies that h a is periodi h 0 =0, h a = h a+n 5 The two sets of data n;m;s,...,s n and n;m;h 1,...,h n 1 has a one-to-one orrespondene sine S a = h a ah a a 1 m n We an translate the onditions on m;s a to the equivalent onditions on m;h a. First, we have ns a =nh a nah 1 + a a 1 m = 0 mod n,

6 LU et al. ns a = nh a nah 1 + a a 1 m = 0 mod n, m 0, mn = even, 7 ns n =0 mod n in Eq. 7 leads to nh 1 +m=0 mod, from whih we see that nh 1 is an integer. From nh a a nh 1 +a a 1 m=even integer, we see that nh a are always integers. Also nh a are always even integers, and nh a+1 are either all even or all odd. Sine h n =0, thus when n=odd, nh a are all even. Only when n=even an nh a+1 either be all even or all odd. When m=even, nh a+1 are all even. When m=odd, nh a+1 are all odd. The two onave onditions beome h a+b h a h b + abm n = D ab = integer 0, 8 h a+b+ h a+b h b+ h a+ + h a + h b + h = a,b, = integer 0. 9 The valid data n;m;h 1,...,h n 1 an be obtained by solving Eqs. 5 and 7 9. Choosing 1 a,b a+b n in Eq. 9, we have 0 a,b,n a b = h n a b h a+b h n a h a h n b h b = n a,n b,a + b 0 whih implies the following refletion ondition on h a : h a h n a = a h 1 h n 1 =0. 0 From Eq. 0 we see that partially solving onditions 9 redues the number of independent variables haraterizing a pattern of zeros from n 1 in S,...,S n to n in h 1,...,h n/. However, being a sequene of frations rather than integers, h a labeling heme imposes some diffiulty in numerially solving onditions 5 and 7 9. In Appendixes A 1 and A we will further use onsistent onditions 9 to introdue two hemes labeling the pattern of zeros with a sequene of non-negative integers or halfintegers. They turn out to be quite effiient for numerial studies sine onsistent onditions 5 and 7 9 an be redued to a muh smaller set after introduing a new labeling heme M k ; p;m as in Appendix A. In partiular, this M k ; p;m labeling heme is the same one as adopted in the literature of parafermion vertex algebra. 5 III. CONSTRUCTING FQH WAVE FUNCTIONS FROM Z N VERTEX ALGEBRAS If we use n;m;h a to haraterize n-luster symmetri polynomial z i, onditions 7 9 are required by the single-valueness of the symmetri polynomial. Or more preisely, Eqs. 7 9 ome from a simple requirement that the zeros in z i all have integer orders. However, onditions 7 9 are inomplete in the sense that some patterns of zeros n;m;h a an satisfy those onditions but still do not orrespond to any valid polynomial. A. FQH wave funtion as a orrelation funtion in Z n vertex algebra To find more onsistent onditions, in the rest of this paper, we will introdue a new requirement for the symmetri polynomial. We require that the symmetri polynomial an be expressed as a orrelation funtion in a vertex algebra. More speifially, we have 1 z i = lim z N h z N V z V e z i, 1 where V e z is an eletron operator and V represents a positive bakground to guarantee the harge neutral ondition. This new requirement, or more preisely, the assoiativity of the vertex algebra, leads to new onditions on h a. The eletron operator has the following form: V e z = z :e i z / :, where: e i z / : stands for normal ordering, whih is impliitly understood hereafter is a vertex operator in a Gaussian model. It has a aling dimension of 1 and the following operator produt expansion OPE Ref. 6 : e ia z e ib w = z w ab e i a+b w + O z w ab+1. The operator is a primary field of Virasoro algebra obeys an quasi-abelian fusion rule, a b a+b +, a a, 4 where represent other primary fields of Virasoro algebra whose aling dimensions are higher than that of a+b by some integer values. We believe that the integral differene of the aling dimensions is neessary to produe a singlevalued orrelation funtion see Eq. 1. Let h a be the aling dimension of the simple urrent a. Therefore the a-luster operator has a aling dimension of V a V e a = a z e ia z / 5 h a = h a + a. 6 The vertex algebra is defined through the following OPE of the a-luster operators: V V a z V b w = C a,b V a,b w z w h a +h b h + O z w h a+b h a h b +1, a+b 7 V where C a,b are the struture onstants. However, the above OPE is not quite enough. To fully define the vertex algebra, we also need to define the relation between V a z V b w and V b w V a z. The orrelation funtions is alulated through the expetation value of radial-ordered operator produt. 6,6,7 The radial-ordered operator produt is defined through

7 NON-ABELIAN QUANTUM HALL STATES AND THEIR z w V a V br V a z V b w = z w V a V bv a z V b w, ab w z V a V bv b w V a z, where Va V b = h a + h b h a+b. z 8 z w 9 Note that the extra omplex fator ab is introdued in the above definition of radial order. In the ase of standard onformal algebras, where Va V b Z, we hoose ab = e i V a V b if both V a and V b are fermioni and ab =e i V a V b if at least one of them is bosoni. But in general, the ommutation fator an be different from 1 and an be hosen more arbitrarily. To gain an intuitive understanding of the above definition of radial order, let us onsider the Gaussian model and hoose V a =e ia and V b =e ib. The aling dimensions of V a and V b are h a = a and h b= b. V a V b =h a +h b h a+b. We see that Va V b Z if a,b Z and suh a Gaussian model is an example of standard onformal algebras. If both a and b are odd, then h a and h b are half-integers and V a and V b are fermioni operators. In this ase Va V b = ab=odd. So under the standard hoie ab = e i V a V b, we have ab =1. If one of a and b is even, Va V b = ab=even and one of V a and V b is bosoni operators. Under the standard hoie ab =e i V a V b,we have again ab =1. Even when a and b are not integers, in the Gaussian model, the radial order of V a =e ia and V b =e ib is still defined with a hoie ab =1. This is a part of the definition of the Gaussian model. In this paper, we will hoose a more general definition of radial order where ab are assumed to be generi omplex phases ab =1. The vertex algebra generated by have a form a z b w = C a,b where a+b w z w h a +h b h a+b C a,b 0. + O z w h a+b h a h b +1, When ombined with the U 1 Gaussian model, the above vertex algebra an produe the wave funtion for a FQH state see Eq. 1. We will also limit ourselves to the vertex algebra that satisfies the n-luster ondition, n =1, 4 where 1 stands for the identity operator defined in Appendix B. Those vertex algebras are in some sense finite and orrespond to rational onformal field theory. We will all suh vertex algebra Z n vertex algebra. We see that in general, a FQH state an be deribed by the diret produt of a U 1 Gaussian model and a Z n vertex algebra. Some exmaples of Z n vertex algebra are studied in Refs. 8 and 9. Note that the Z n vertex algebras are different from the Z n simple-urrent vertex algebras that will be defined in Se. V. The Z n simple-urrent vertex algebras are speial ases of the Z n vertex algebras. In this and the next setions, we will onsider Z n vertex algebras. We will further limit ourselves to Z n simple-urrent vertex algebras in Se. V and later. As a result h a = h a+n, h n =0, ab = a+n,b = a,b+n, n,a = a,n =1, C a,b = C a+n,b = C a,b+n, C n,a = C a,n =1, C a,b = a,b C b,a. 4 By hoosing proper normalizations for the operators a,we an have = 1, a mod n n/ C a, a a, a, a mod n n/, C a,b =1 ifa or b = 0 mod n. 44 To summarize, we see that the Z n vertex algebras whose orrelation funtions give rise to eletron wave funtions are haraterized by the following set of data n;m;h a ;C a,b,... a,b=1,...,n, where m=n/. Here represent other struture onstants in the subleading terms. The ommutation fators ab are not inluded in the above data beause they an be expressed in terms of h a and are not independent see Eq. E8. Sine the Z n vertex algebra enodes the many-body wave funtion of eletrons, we an say that the data n;m;h a ;C a,b,... a,b=1,...,n also haraterize the eletron wave funtion. We an study all the properties of eletron wave funtions by studying the data n;m;h a ;C a,b,... a,b=1,...,n. In the pattern-of-zero approah, we use data n;m;h a to haraterize the wave funtions. We will see that the n;m;h a ;C a,b,... a,b=1,...,n haraterization is more omplete, whih allows us to obtain some new results. B. Relation between h a and h a What is the relation between the two haraterizations: n;m;h a a=1,...,n and n;m;h a ;C ab a,b=1,...,n? The single valueness of the orrelation funtion z i requires that the zeros in z i all have integer orders. In this setion, we derive onditions on the aling dimension h a, just from this integral-zero ondition. This allows us to find a simple relation between n;m;h a a=1,...,n and n;m;h a ;C ab a,b=1,...,n. From the definition of D ab and the OPE 7, we see that D a,b S a+b S a S b = h a+b h a h b = h a+b h a h b + ab = D b,a

8 LU et al. We see that D 1,n = n.so n is an positive integer whih is alled m. From Eq. 45, we an show that 11,1 and a 1 S a = D i,1 = h a ah 1 = h a ah 1 a a 1 + h a = S a as n n + am a m n Therefore, the h a introdued before is nothing but the aling dimensions h a of the simple urrents a see Eq. 4. In the following, we will use h a to deribe the aling dimensions of a. Thus the data n;m;h a ;C a,b a,b=1,...,n an be rewritten as n;m;h a ;C a,b a,b=1,...,n. Those h a satisfy Eqs As emphasized in Ref. 11 and 1, the onditions 7 9, although neessary, are not suffiient. In the following, we will try to find more onditions from the vertex algebra. C. Conditions on h a and C a,b from the assoiativity of vertex algebra The multipoint orrelation of a Z n vertex algebra an be obtained by fusing operators together, thus reduing the original problem to alulating a orrelation of fewer points. 5 It is the assoiativity of this vertex algebra that guarantees any different ways of fusing operators would yield the same orrelation in the end 6 so that the eletron wave funtion would be single valued. The assoiativity of a Z n vertex algebra requires h a and C a,b to satisfy many onsistent onditions. Those are the extra onsistent onditions we are looking for. The onsistent onditions ome from two soures. The first soure is the onsistent onditions on the ommutation fators a,b as diussed in Appendix B. When applied to our vertex algebra 40, we find that some onsistent onditions on a,b allow us to express a,b in terms of h a. Then other onsistent onditions on a,b will beome onsistent onditions on h a see Appendix E 1. The seond soure is GJI for the vertex algebra 40 as diussed in Appendix E. We like to stress that the diussions so far are very general. The onsistent onditions that we have obtained for generi Z n vertex algebra are neessary onditions for any FQH states. A detailed derivation of those onditions on h a and C a,b is given in Appendix E. Here we just summarize the new and old onditions in a ompat form. The onsistent onditions an be divided in two lasses. The first type of onsistent onditions at only on the pattern of zeros n;m;h a see Eqs. 7 9, E9, E10, E1, E14, and E1, nh a nah 1 + a a 1 m = 0 mod n, m 0, mn = even, h a+b+ where h a =h a+n h a+b h a+b h a h b + abm N, n h b+ h a+ + h a + h b + h N, n 1,1 = even, a 1,1 a,a = even, a =1,,...,n 1, n, n, n =4h n/ 1 if n = even, 48 and a,b =h a +h b h a+b. The seond type of onsistent onditions at on the struture onstants see Eqs. E1, E, E7, and E8. For any a, b, and, C a,b C a+b, = C b, C a,b+ = a,b C a, C b,a+ if a,b, =0, C a,b C a+b, = C b, C a,b+ + a,b C a, C b,a+ if a,b, =1, 49 where a,b is a funtion of the pattern of zeros h a, For any a n/ ij = 1 ij 1,1 i,j = 1. C a, a = C a,a C a, a =1 if a,a, a =0, C a, a = C a,a C a, a if a,a, a =1. 50 Here C a,b satisfies the normalization ondition 44. There may be additional onditions when a,b, 0,1. But we do not know how to derive those onditions systematially at this time. IV. EXAMPLES OF GENERIC FQH STATES DESCRIBED BY THE Z N VERTEX ALGEBRA To obtain the examples of generi FQH states, we have numerial solved onditions 48. We do not require Eq. E16 to be satisfied in order to inlude some valid interesting solutions, like Gaffnian whih violates Eq. E16. In this setion, we list some of those solutions in terms of n;m;h a a=1,...,n 1. First we note that, for two n-luster symmetri polynomial 1 and deribed by n;m 1 ;h 1,a and n;m ;h,a, the produt = 1 is also an n-luster symmetri polynomial. is deribed by the pattern of zeros n;m;h a = n;m 1 + m ;h 1,a + h,a. 51 Most of the solutions an be deomposed aording to Eq. 51. We will all the solutions that annot be deomposed primitive solutions. We will only list those primitive solutions. We only searhed solutions with a filling fration 1/4. We an see that most solutions shown also satisfy ondition E16, whih means they obey OPE 68 and orrespond to speial Z n vertex algebras. However, some solu

9 NON-ABELIAN QUANTUM HALL STATES AND THEIR tions suh as a four-luster state alled Gaffnian, expliitly violates ondition E16 and their OPE s take the more general form E18 and E19. They are deribed by generi Z n vertex algebras. A. n=1 ase There is only one n=1 primitive solution, n =1: =0, m;h 1 h n 1 = ;, n =: =4/5 Z parafermion state, m;h 1 h n 1 = ;, p;m 1 M n 1 = ;0 0, n 0 n m 1 = 0. It is the Z parafermion state Z. 55 p;m 1 M n 1 = 0;, n 0 n m 1 = It is =1/ Laughlin state. Note that h a =0, indiating that the simple-urrent part of vertex algebra is trivial and has a zero entral harge =0. The vertex algebra ontains only the U 1 Gaussian part. B. n= ase We note that the n=1 primitive solution also appears as a n= primitive solution. We find only one new n= primitive solution, n =: =1/ Z parafermion state, m;h 1 h n 1 = ;, 1 p;m 1 M n 1 = 1;0, n 0 n m 1 = 0. 5 It is the =1 Pfaffian state Z. The simple urrent part of the vertex algebra is a Z parafermion CFT. If we only use onditions 7 9 obtained in Refs. 11 and 1, then n =, m;h 1 h n 1 = ; 4, 1 p;m 1 M n 1 = 1 ;0, n 0 n m 1 = will be a solution. Suh a solution does not orrespond to any symmetri polynomial, indiating that onditions 7 9 are inomplete. An extra ondition E1 from ommutation fators remove suh an inorret solution. C. n= ase Apart from the n=1 primitive solution, we find only one new n= primitive solution, D. n=4 ase Apart from the n=1 primitive solutions, we find only two new n=4 primitive solutions using onditions 7 9, E10, E1, and E14, n =4: =1 Z 4 parafermion state, m;h 1 h n 1 = ; 4 1 4, p;m 1 M n 1 = ;000, n 0 n m 1 = 4 0. whih is the Z 4 parafermion state Z4, and n =4: m;h 1 h n 1 = ; , 1 p;m 1 M n 1 = 1; 1 1 1, 56 n 0 n m 1 = We like to point out that a nonprimitive solution m;h 1,...,h n 1 = ; 1 4,0, 1 4 = 4; 1,0, 1 is the Z parafermion state the Pfaffian state. Consistent onditions from a study of useful GJI s show that it has entral harge =1/ the same as Z Pfaffian state and,a =1, =0, indiating that =1 is the identity operator here. In other words, this Z 4 simple-urrent vertex algebra is generated by a Z simple urrent. Another nonprimitive solution m;h 1,...,h n 1 = ; 1 4,0, 1 4 = 6; 4,0, 4 is the Gaffnian state.4 Gaffnian vertex algebra is a Z 4 simple-urrent vertex algebra with 1, = 1, =, = 1 and =0. In omparison with Z 4 Pfaffian, this Z 4 Gaffnian vertex algebra annot be generated by any Z simple urrent. This example will be analyzed in detail in Se. VII. E. Inluding onditions (49) and (50) In the above, we only onsidered onditions 48. Those patterns of zeros that satisfy Eq. 48 may not satisfy ondi

10 LU et al. tions 49 and 50, i.e., one may not be able to find C a,b that satisfy Eqs. 49 and 50. However, we do not know how to hek onditions 49 and 50 systematially. We have to hek them on a ase by ase basis. For the Z and Z parafermion states, we find that Eqs. 49 and 50 redue to trivial identities after using Eq. 44. So the nontrivial C 1,1 and C, for the Z parafermion vertex algebra annot be determined from Eqs. 49 and 50, whih means that onditions 49 and 50 an be satisfied by any hoies of C a,b that are onsistent with Eq. 44. For the state with pattern of zeros n;m;h a = 4;; , we find that by hoosing a,b, = 1,, and 1,, in Eq. 49, we an obtain the following equations: C 1, C, = C, C 1,1 = 1, 1=C, C 1, Clearly no C a,b an satisfy the above two equations. Thus the n=4 pattern of zeros m;h a = ; do not orrespond to any valid symmetri polynomial. It s interesting to note that the n=4 pattern of zeros m;h a = ; = 4; orrespond to the Z parafermion state and the n=4 pattern of zeros m;h a = ; = 6; orrespond to the Gaffnian state, both being valid symmetri polynomials. For the state with pattern of zeros n;m;h a = 4;4;1 1 1, we find that by hoosing a,b, = 1,1,1, 1,1,, and 1,,0 in Eq. 49, we an obtain the following equations: C 1,1 C,1 = C 1,1 C 1, + C 1,1 C 1,, C 1,1 = C 1, = C 1,, C 1, = C 1, = C,1, 59 whih an be redued to C 1,1 =C 1,1. We see that the only solution is C 1,1 =C 1, =C,1 =0, whih is not allowed by Eq. 41. Thus the n=4 pattern of zeros m;h a = 4;111 do not orrespond to any valid symmetri polynomial. F. Summary In Refs. 11 and 1, we have seen that onditions 7 9 are not enough sine they allow the following pattern of zeros n;m;h a = ;1; 1 4. Suh a pattern of zeros does not orrespond to any valid polynomial. Condition 48 obtained in this paper rule out the above invalid solution. So onditions 48 is more omplete than onditions 7 9. However, ondition 48 is still inomplete sine they allow the invalid patterns of zeros suh as n;m;h a = 4;; and 4;4;111. Both of them an be ruled out by onditions 49 and 50. Conditions are the onsistent onditions that we an find from some of GJI, based on the most general form of OPE 40. So those onditions are neessary but may not be suffiient. The orrespondene between the patterns of zeros n;m;h a and FQH states is not one to one. There an be many polynomials that have the same pattern of zeros. This is not surprising sine the pattern of zeros only fixes the highest-order zeros in eletron wave funtions symmetri polynomials, while different patterns of lower-order zeros ould lead to different polynomials in priniple. In other words, the leading-order OPE 40 alone might not suffie to uniquely determine the orrelation funtion of the vertex algebra. The examples studied in this setion support suh a belief. Expliit alulations for some examples suggest that the pattern of zeros together with the entral harge and simple urrent ondition would uniquely determine the FQH state. This is a reason why we introdue Z n simple-urrent vertex algebra in the next setion. V. Z n SIMPLE-CURRENT VERTEX ALGEBRA In the last setion, we diuss legal patterns of zeros that satisfy the onsistent onditions and deribe existing FQH states. If we believe that a legal pattern of zeros n;m;h a, or more preisely the data n;m;h a ;, an ompletely deribe a FQH state, then we should be able to alulate all the topologial properties of the FQH states. But so far, from the pattern of zeros n;m;h a, we an only alulate the number of different quasipartile types, quasipartile harges, and the fusion algebra between the quasipartiles. 1,1 Even with the more omplete data n;m;h a ;, we still do not know, at this time, how to alulate the quasipartile statistis and aling dimensions. One idea to alulate more topologial properties from the data n;m;h a ; is to use the data to define and onstrut the orresponding Z n vertex algebra, and then use the Z n vertex algebra to alulate the quasipartile aling dimensions and the entral harge. However, so far we do not know how to use the data n;m;h a ; to ompletely onstrut a Z n vertex algebra in a systemati manner. Starting from this setion, we will onentrate on a subset of legal patterns of zeros that orrespond to a subset of Z n vertex algebra. Suh a subset is alled Z n simple-urrent vertex algebras. The FQH states deribed by those Z n simpleurrent vertex algebras are alled Z n simple-urrent states. We will show that in many ases the quasipartile aling dimensions and the entral harge an be alulated from the data n;m;h a ; for those Z n simple-urrent states. A. OPE s of Z n simple-urrent vertex algebra The Z n simple-urrent vertex algebra is defined through an Abelian fusion rule with yli Z n symmetry for primary fields a of Virasoro algebra, 5,40 a b a+b, a a. 60 Compared to Eq. 4, here we require that a and b fuse into a single primary field of Virasoro algebra a+b. Suh operators are alled simple urrents. The Z n simple-urrent vertex algebra is defined by the following OPE of a, 5,40 a+b w a z b w = C a,b z w + O z w 1 a,b, a,b

11 NON-ABELIAN QUANTUM HALL STATES AND THEIR a z a w = where we define 1+ h a z w T w z w h a + O z w h a, 6 a,b h a + h b h a+b, 6 a n a and a = n+a is understood due to the Z n symmetry. In the ontext the subript a of Z n simple urrents is always defined as a mod n. We like to point out here that the form of the OPE 6 is a speial property of the Z n simple-urrent vertex algebra. For a more general Z n vertex algebra that deribes a generi FQH state, the orrespond OPE has a more general form, a z a w = 1+ h a z w T w z w h + a + O z w h a, T a z w h a 64 where T a are dimension- primary fields of Virasoro algebra T a,a=1,..., n may not be linearly independent though. Also, for the Z n simple-urrent vertex algebra, the subleading terms in Eq. 61 are also determined. For more details, see Appendix F. C a,b are the struture onstants of this vertex algebra. We also have onformal symmetry T z a w = and Virasoro algebra h a z w a w + 1 z w a w + O 1, 65 / T z T w = z w 4 + T w z w + T w z w + O 1 66 where T z represents the energy-momentum tensor, whih has a aling dimension of. stands for the entral harge as usual, whih is also a struture onstant. Using the notation of generalized vertex algebra 6 see Appendix B, we have i j i,j = C i,j i+j, i + j 0 mod n, 67 i i i, i =1, i i i, i 1 =0, i i i, i = h i T, T i = h i i = i T, T i 1 = i, i T 1 = h i 1 i, TT 4 =, TT =0, TT =T, TT 1 = T, 70 with T, i = and T,T =4. We all it a speial Z n simpleurrent vertex algebra if it satisfies OPE s For example, the Z n parafermion states 17 orrespond to a series of speial Z n simple-urrent vertex algebras. The ommutation fator AB equals unity if either A or B is the energy-momentum tensor T: T, i = i,t= T,T =1. Similarly we have A,1 = 1,A =1 for the identity operator 1 and any operator A. However, i,j i, j given in Eq. E8 an be 1 in general. In deriving OPE 68 we have assumed that i, i =1, i, whih is not neessary. For example, the Z 4 Gaffnian does not satisfy i, i =1, i. So, we will adopt the more general OPE E18 and E19 instead of Eq. 68 to inlude examples like Gaffnian whih do give a FQH wave funtion. OPE 68 is for a speial Z n simple-urrent vertex algebra that satisfies i, i =1, i. For a more general Z n simple-urrent vertex algebra, they beome i i i, i = C i, i, i i i, i 1 =0, i i i, i = C i, ih i T, C i, i = 1, n i n/ mod n 71 i, i, i n/ mod so that we always have C a,b = a,b C b,a for any subripts a and b in suh an assoiative vertex algebra. OPE s 67, 71, 69, 70, 116, and 117 define the generalized Z n simple-urrent vertex algebra, or simply Z n simple-urrent vertex algebra. The Gaffnian state orresponds to a generalized Z 4 simple-urrent vertex algebra with a, a 1. When a, a =1, we have a speial Z n simpleurrent vertex algebra. What kind of pattern of zeros n;m;h a, or more preisely what kind of data n;m;h a ;,C ab, an produe a Z n simple-urrent vertex algebra? Sine the Z n simple-urrent vertex algebras are speial ases of Z n vertex algebras, the data n;m;h a ;,C ab must satisfy onditions However, the data n;m;h a ;,C ab for Z n simple-urrent vertex algebras should satisfy more onditions. Those onditions an be obtained from the GJI of Z n simple-urrent vertex algebras. In Appendix E, we derived all those extra onsistent onditions for a generi Z n vertex algebra from the useful GJI s based on OPE 40. Now based on OPE s summarized in this setion, we an similarly derive a set of extra onsistent onditions for a Z n simple-urrent vertex algebra. These onditions are summarized in Se. VB. For those valid data that satisfy all the onsistent onditions, the full properties of simple-urrent vertex algebra an be obtained. This in turn allows us to alulate the physial topologial properties of the FQH states assoiated with those valid patterns of zeros

12 LU et al. We like to point out that many examples of Z n simpleurrent vertex algebra have been studied in detail. They inlude the simplest Z n simple-urrent vertex algebra the Z n parafermion algebra. 5,40,41 More general exmaples that have been studied are the higher generations of Z n parafermion algebra 4 46 and graded parafermion algebra In those exapmles, the Z n simple-urrent algebras are studied by embedding the algebras into some known CFT, suh as oset models of Ka-Moody urrent algebras and/or Coulomb gas models. However, in this paper, we will not assume suh kind of embeding. We will try to alulate the properties of Z n simple-urrent vertex algebra diretly from the data n;m;h a,,... without assuming any embedding. B. Consistent onditions from useful GJI s In Appendix E a,we show how to obtain the onsistent onditions on the data n;m;h a ;,C ab haraterizing a generi Z n vertex algebra from a set of useful GJI s as deribed in Appendix D, requiring that OPE E1 is obeyed. Here for a Z n simple-urrent vertex algebra, requiring that OPE s 67, 71, 69, 70, 116, and 117 are obeyed, we an derive a larger set of onsistent onditions on the data n;m;h a ;,C ab. For the examples studied in this paper, we find that C ab an be uniquely determined from n;m;h a ; using those onsistent onditions. Thus, for those states, C ab are not independent and an be dropped. Sine A,B,C = A,C,B = C,A,B for an assoiative vertex algebra, we an ombine the onsistent onditions obtained from GJI s of all possible six permutations of three operators A,B,C together. In this setion we summarize the onsistent onditions obtained from useful GJI s just like in Appendix Ea and list them in a ompat manner. These extra onsistent onditions, together with onditions 48 should form a omplete set of onsistent onditions, whih allows us to obtain a valid pattern of zeros and onstrut the assoiated simple-urrent vertex algebra and FQH wave funtion. 1. {A,B,C}={ a, b, },a+b,b+,a+å0modn For a,b, =0, we have the following onsistent onditions: C a,b C a+b, = C b, C a,b+ = a,b C a, C b,a+. 7 Notie that the onsistent onditions obtained from useful GJI s of a, b, and of b, a, only differ by a fator of a,b sine C a,b = a,b C b,a. Thus they are not independent onditions. Similarly it s easy to show that other permutations yield onsistent onditions linearly dependent with the above ondition using the fat that a,b a, = a,b+ here sine a,b, =0. For a,b, =1, we have the following onsistent onditions: C a,b C a+b, = C b, C a,b+ + a,b C a, C b,a+. 7 For a,b, there are no extra onsistent onditions.. {A,B,C}={ a, b, b },a+bå0modn For a,b, b =0 we have the following independent onsistent onditions, h a h b = a, b =0, h b a 0, C a,b C a+b, b = a,b C a, b C b,a b = C b, b 74 sine we know Eq. B7 and a,0 =1. For a,b, b =1 the independent onsistent onditions are = 4h a h b a,b 1 a,b, C a,b C a+b. b = 1 a,b C b, b, a,b C a, b C b,a b = a,b C b, b. 75 For a,b, b = the independent onsistent onditions are a,b C a, b C b,a b = a,b a,b 1 C a,b C a+b, b = a,b 1 a,b + h a h b C b, b, + h a h b C b, b. 76 For a,b, b = the independent onsistent onditions are = a,b 1 a,b C a,b C a+b, b a,b C a, b C b,a b + h a h b C b, b. 77 For a,b, b 4 there are no extra onsistent onditions from useful GJI s.. {A,B,C}={ a, a, a },aånõmodn For a,a, a = a,a +h a =0 the onsistent onditions are summarized as h a = a,a =0, a 0, C a,a C a, a = C a, a = C a,a = a, a =1. 78 For a,a, a = a,a +h a =1 the orresponding onsistent onditions are a,a = 1, h a =1, =, a, a = 1, C a,a C a, a =C a, a, C a, a = C a,a. 79 For a,a, a = the independent onsistent onditions are = h a h, a C a,a C a, a =h a, C a, a = C a,a = a, a =1 80 sine we have C a, a =C a,a here. For a,a, a = a,a +h a = the extra onsistent onditions are