Composite fermion wave functions as conformal field theory correlators

# Composite fermion wave functions as conformal field theory correlators

## Abstract

It is known that a subset of fractional quantum Hall wave functions has been expressed as conformal field theory (CFT) correlators, notably the Laughlin wave function at filling factor ( odd) and its quasiholes, and the Pfaffian wave function at and its quasiholes. We develop a general scheme for constructing composite-fermion (CF) wave functions from conformal field theory. Quasiparticles at are created by inserting anyonic vertex operators, , that replace a subset of the electron operators in the correlator. The one-quasiparticle wave function is identical to the corresponding CF wave function, and the two-quasiparticle wave function has correct fractional charge and statistics and is numerically almost identical to the corresponding CF wave function. We further show how to exactly represent the CF wavefunctions in the Jain series as the CFT correlators of a new type of fermionic vertex operators, , constructed from free compactified bosons; these operators provide the CFT representation of composite fermions carrying flux quanta in the CF Landau level. We also construct the corresponding quasiparticle- and quasihole operators and argue that they have the expected fractional charge and statistics. For filling fractions 2/5 and 3/7 we show that the chiral CFTs that describe the bulk wave functions are identical to those given by Wen’s general classification of quantum Hall states in terms of -matrices and - and -vectors, and we propose that to be generally true. Our results suggest a general procedure for constructing quasiparticle wave functions for other fractional Hall states, as well as for constructing ground states at filling fractions not contained in the principal Jain series.

###### pacs:
73.43.-f, 11.25.Hf

## I Introduction

The evidence for an intriguing connection between conformal field theory (CFT) and the fractional quantum Hall effect (FQHE) was accumulating in the 1980s. It was realized that the effective low-energy theory of the FQHE is a topological field theory of the Chern-Simons type, where the exchange phases of the anyonic quasiparticles and quasiholes are coded in the braiding properties of the corresponding Wilson loopsCStheory (). Witten’s subsequent demonstration that the braiding of the Wilson loops are reflected in the correlation functions of certain CFTswitten () suggested a CFT-FQHE relationship, which was further strengthened by Wen, who proposed that the gapless chiral edge modes of a FQH-droplet are described by a chiral dimensional CFTwen (). It was also noticed that the holomorphic part of the Laughlin wave function takes the form of a correlator of bosonic exponents, or vertex operators, in a two dimensional CFTMR (); Fubini ().

The 1991 paper by Moore and Read was particularly important since it synthesized many of these ideas and made an explicit conjecture about the CFT description of quantum Hall (QH) states containing two parts: 1. “Representative” electronic wave functions for the ground state and its quasiparticle and quasihole excitations are correlation functions, or, more precisely, conformal blocks, in a rational conformal field theory (RCFT) where the various particles correspond to different primary fields. 2. The very same RCFT describes the edge excitations of the corresponding FQH droplet. In their paper Moore and Read gave some striking circumstantial arguments to support their conjecture, and they also showed that many FQH states, namely the Laughlin state, the states in the Halperin-Haldane hierarchy, their quasihole excitations, the Halperin spin singlet statehalperin (), and the Haldane-Rezayi spin singlet pairing staterezhal (), may be represented in terms of conformal blocks. All this might have been criticized for being just a reformulation of old results, but Moore and Read also used the CFT formalism to propose a new state, the so-called Pfaffian wave function, which is tentatively assigned to the observed FQHE. The quasiholes in this state have charge rather than expected from the filling fraction, and exhibit non-Abelian fractional statistics. To establish the latter it was essential to use CFT technology.1

Despite this advance one and a half decades ago, the program of establishing a one-to-one correspondence between QH states and conformal field theory has remained incomplete. No explicit conformal field theory expressions have so far been established for many important FQHE states; in particular, despite interesting progressflohr (), this is the case for the ground state wave functions of the prominent FQHE series , and their related quasihole or quasiparticle excitations. (Expressions for the states in the Haldane-Halperin hierarchy were given in Ref. MR, , but these are indirect, involving multiple integrals over auxillary quasihole coordinates.) Surprisingly, a proper conformal field theory representation does not exist even for the quasiparticles – as opposed to quasiholes – of the FQHE state at and the Pfaffian wave function at . 2

It is worth reminding ourselves what we can hope to accomplish using CFT techniques: We cannot “derive” the FQHE wave functions, since the CFT does not contain any information about the actual interelectron interaction. It is true that the short distance behavior of the electronic wave functions is reflected in the operator product expansion of the pertinent CFT vertex operators, but only in the simplest cases can this be directly related to a potential of the Haldane-Kivelson-Trugman type. Thus we can only hope to get ”representative wave functions” in the sense of Moore and Read, and any new candidate wave function suggested by the CFT approach must be tested and confirmed against exact solutions of the Schrödinger equation known for small systems. The crucial question is if the CFT wave functions are sufficiently natural and simple to give new insight into the physics of the problem, facilitate computations of quantities like local charge and braiding statistics, and most importantly, inspire new generalizations. Finally, we should point out that we know of no general microscopic principle that requires that the correlated quantum mechanical wave functions of interacting electrons in the lowest Landau level should be expressible as simple correlation functions of certain vertex operators in a two dimensional Euclidean rational conformal field theory.

An insight into the general FQHE states comes from the composite fermion (CF) formalismJain (); review (). Here the experimentally prominent Jain states at are formed from filled Landau levels of “composite fermions,” which are electrons carrying flux quanta. Other CF states, as e.g. the Pfaffian, which is the preferred candidate for the observed state, can be formed by various BCS type pairing mechanismsMR (); pwave (). In the CF description, a quasihole is obtained simply by removing a composite fermion from an incompressible FQHE state, and a quasiparticle is a composite fermion in a higher, otherwise empty CF Landau level (LL). (CF Landau levels are also called levels.) Explicit wave functions are constructed for all ground states and their quasiparticle and quasihole excitations. (The asymmetry between quasiparticles and quasiholes occurs since they reside in different CF Landau levels.) The CF approach is very successful, both in comparison with experiments and with numerical studies of two-dimensional electron gases in strong magnetic fieldsreview ().

The issue of fractional charge and fractional statistics of the composite fermions is a subtle one. The quasiparticles and quasiholes are composite fermions added to or removed from a CF Landau level. From one perspective, they have unit charge and fermionic statistics. Indeed, the addition of one composite fermion increases the number of electrons, and hence the net charge, by one unit, and the fermionic statistics of composite fermions has been confirmed by numerous experiments (e.g. the observation of their Fermi sea). On the other hand, the CF quasiparticles and quasiholes have a fractional “local charge” (where the local charge is the charge measured relative to the background FQHE state) and a fractional braiding statistics Leinaas (); kjons (); jain2 (); review (). These properties capture the physics that adding or removing a composite fermion causes nonlocal changes in the state, because the vortex, a constituent of the composite fermion, is a nonlocal object. This should be contrasted with the analogous process in the integral QHE, which is essentially local (the Landau level projection destroys locality only on the scale of the magnetic length ), and can be described by a local, charge- operator , where the subscript denotes the Landau level index. No such local operator can be constructed for the creation of a composite fermion, since the local charge of the quasiparticle differs from that of the electron. The fractional statistics of the quasiparticles also implies that they cannot be described by local operators, as emphasized by Fröhlich and Marchettifroh (). Even though fractional charge and fractional statistics cannot be read off directly from the CF wave functions, they nonetheless contain that information, not surprising in view of the fact that the CF construction provides a good description of all the low energy states. We mention here the quasiparticles at , for which the CF wave function differs from that proposed earlier by Laughlinlaughlin83 (). The calculation of the Berry phase associated with two-CF quasiparticle exchange, originally performed by Kjønsberg and Leinaaskjons () and subsequently by Jeon and collaboratorsjain2 (), shows that the braiding statistics for the CF quasiparticles has a sharply defined fractional value; for the Laughlin quasiparticles, in contrast, numerical calculations do not produce a convergent result for the statistical anglekjons2 ().

In this paper we establish a firm connection between CF wave functions and CFT correlators. Specifically:

1. We construct the quasiparticles of ( odd) using a new kind of anyonic vertex operators . For a single quasiparticle, the resulting wave function is identical to that obtained using the CF theory. A generalization to two or more quasiparticles produces wave functions that are very similar to the CF wave functions but not identical. For two quasiparticles at , the overlap between the two wave functions is typically 99.99% for as many as 40 electrons.

2. We show that the ground state wave functions in the Jain series are exactly given by sums of CFT correlators of a set of vertex operators, , which in the CF language correspond to creating composite fermions in higher CF Landau levels.

3. We generalize the construction of the quasiparticle operator , as well as of the quasihole operators, to higher levels in the Jain sequence; at level , there are independent hole operators and one quasiparticle operator. The vertex operator at level is closely related to the quasiparticle operator at level .

4. We demonstrate that the very CFT that yields the CF wave functions also directly defines an edge theory for the Jain states that is precisely the one expected from the general arguments given by Wenwen ().

Our CFT construction has many advantages. (i) At the technical level, it produces accurate wave functions directly in the lowest Landau level with no need for projection, and the charge and statistics of the quasiparticles are revealed in the algebraic properties of the corresponding operators, just as in the case of the quasiholes of the states. (ii) Although the effective edge theory for the Jain states was known from general principles, we provide a direct derivation from a CFT where the conformal blocks yield microscopically accurate bulk wave functions. (iii) It gives a new insight and suggests new extensions; a generalization of this work produces natural ansätze for quasiparticle wave functions for more complicated CF states such as the Moore-Read Pfaffian state, as well as for ground states at fractions (e.g., 4/11), which do not belong to the principal Jain series.

The paper is organized as follows. In the next section we explain the basic ideas behind our construction and give explicit wave functions for one- and two-quasiparticles, as well as that for a quasiparticle-quasihole pair. The general structure of the CFT description of the states in the Jain series is discussed in section III, while the detailed technical proof for the equivalence between the CF and the CFT wave functions is left for Appendix B. In section IV we explain the construction of the edge theory, and in section V we construct localized quasiparticle states and show how to extract charge and statistics from the relevant Berry phases; the latter can be ascertained analytically if we make a random phase assumption. Some details of the calculations are found in Appendix C. Section V presents numerical calculations supporting our claims in sections II and V and, finally, a summary is found in section VII. A short report on parts of this work has been published previouslyHanssonI ().

## Ii One and two quasiparticles in the Laughlin state

### ii.1 The ground state and the quasihole states

We first review some of the basic formalism of the CFT construction of QHE wave functions, in particular the construction of the ground state and quasihole wave functions at the Laughlin fractions , where is an odd integer. Following Moore and ReadMR (), we introduce the normal-ordered vertex operators,

 V1(z) = :ei√mφ1(z): (1) H1m(η) = :ei√mφ1(η):, (2)

where the normal ordering symbol , will be suppressed in the following. The free massless boson field, , is normalized so as to have the (holomorphic) two point function

 ⟨φ1(z)φ1(w)⟩=−ln(z−w), (3)

so that the the vertex operators obey the relations

 eiαφ1(z)eiβφ1(w) = eiπαβeiβφ1(w)eiαφ1(z)=(z−w)αβeiαφ1(z)+iβφ1(w) (4) ∼ (z−w)αβei(α+β)φ1(w)

where the last line expresses the operator product expansion (OPE) in the limit . From (4) follows , and . The first of these reflects that the electrons are fermions, while the second is appropriate for fractional statistics as discussed in reference [MR, ].

We normalize the (holomorphic) charge density operator as

 J(z)=i√m∂zφ1(z) (5)

so the corresponding charge is given by

 Q=1√m12π∮dz∂zφ1(z), (6)

where the contour encircles the whole system. The charges, of the electron and of the quasihole, can be read directly from the commutators and . It is noted that does not give the electric charge; rather it has the interpretation of vorticity as seen from (4). Introducing a positive vorticity in a homogenous state corresponds to a local depletion of the electron liquid, while a negative vorticity amounts to a local increase in density. Thus the excess electron number compared with the ground state created by an operator with charge is given by

 Δn=δn−Q, (7)

where the integer is the number of electrons added by the operator. If the argument of the operator is an electron coordinate, , one electron is added, while no electron is added if the argument is a quasihole coordinate . (The idea of binding of an electron and vortices was implicit in Laughlin’s original work, and was made explicit by Halperin halperin (), Girvin and MacDonald GM () and Read read ().)

The total electric charge of a particle is given by . Note that the excess charge associated with the addition of an electron is zero, as expected, because this expands the droplet without creating any local charge variation.

The (un-normalized) Laughlin wave function can now be written as (for notational convenience, we write instead of ):

 ΨL(zi) = ⟨0|R{V1(z1)V1(z2)…V1(zN−1)V1(zN)e−i√mρm∫d2zφ1(z)}|0⟩ ≡ ⟨V1(z1)V1(z2)…V1(zN−1)V1(zN)⟩1/m = ∏i

where denotes radial ordering. The second line defines the average , and the third follows for the ordering , which will be assumed below unless indicated otherwise. In the following, we shall suppress the subscript whenever it is clear to what ground state we are referring. The exponential operator in (II.1) corresponds to a constant background particle density, , where is the density of a filled Landau level. This is necessary since the charge neutrality condition, known from the Coulomb gas formulation, in the CFT ensures that the correlator vanishes unless , which defines the area, , of the system. As explained in reference [MR, ], the background charge will produce the correct gaussian factor characteristic of the lowest Landau level wave function. For a more detailed discussion of this background charge prescription, see Appendix A.

The wave function for a collection of Laughlin quasiholes is also easily written:

 ΨL(η1…ηn;zi)=⟨H1m(η1)H1m(η2)…H1m(ηn)V1(z1)V1(z2)…V1(zN−1)V1(zN)⟩. (9)

In this case the charge neutrality condition reads , indicating an expansion of the droplet. From the general relation (4) we get which guarantees that (9) is uniquely defined and analytic in the electron coordinates.

Very little of the rather sophisticated mathematics of CFT will be used in this paper, but a few formal comments are in order. A CFT is in general not defined by a Lagrangian, but by an operator product algebra, or set of fusion rules, together with a specification of the field content defined by the so-called primary fields. The CFTs of interest here are defined by a Lagrangian describing a collection of free bosons, , compactified on circles of radius where are odd integers. The primary fields are given by the chiral vertex operators where the integers define the charge lattice describing the possible “electric” charges in the Coulomb gas formulation of the CFT. The vertex operators satisfy an extended chiral algebra that, together with the charge lattice, defines the relevant CFT, which in this case is called a “rational torus” with radii ; this is an example of a rational CFT. Acting on the primary fields with the generators of the conformal group gives families of “descendant fields”, which can be expressed using derivatives of the parent primary fields. Such descendant fields will be important in the construction of quasiparticle operators presented in the next section. The full CFT contains fields of both chiralities and has correlation functions that can be written as (in general a sum over) products of holomorphic and anti-holomorphic factors, so-called conformal blocks. The holomorphic blocks are precisely the correlation functions of chiral vertex operators that we have identified with the electronic wave functions. In general, these blocks also depend parametrically on quasiparticle and quasihole coordinates, and acquire nontrivial phase factors, called monodromies, when these coordinates are transported along closed loops. It is these monodromies that reproduce the braiding phases that also can be calculated from the expectation values of Wilson loops in a Chern-Simons theory. A detailed discussion of the conditions that a CFT has to fulfill in order to describe a QH state can be found in Ref. frohlich, .

### ii.2 One quasiparticle

The most immediate guessMR () for a quasiparticle operator would be to simply change the sign in the exponent in the quasihole operator of (2), i.e. to use . That, however, introduces unacceptable singular terms in the electronic wave function. Inspired by the CF wave functions, we instead define a quasiparticle operator, , which has a charge , and that will replace one of the the original electron operators . We can thus think of as a modified electron operator, but with a different amount of vorticity. The excess electric charge associated with such a modification is the difference between the charges of the operators and i.e. , as appropriate for a quasiparticle at . The modified electron operator is given by

 P1m(z)=∂ei(√m−1√m)φ1(z), (10)

and the wave function for a single quasiparticle with angular momentum is written as

 Ψ(l)1qp(zi) = A{zl1e−|z1|2/4mℓ2⟨P1m(z1)V1(z2)……V1(zN)⟩} = ∑i(−1)i+1zlie−|zi|2/4mℓ2⟨P1m(zi)∏j≠iV1(zj)⟩ = ∑i(−1)izli(i)∏j

where denotes anti-symmetrization of the coordinates. The second line follows by noting that the anti-symmetrized product has the form of a Slater determinant which is then expanded by the first row. From (4) we get , so the radial reordering of the quasiparticle operator does not give rise to any sign change. The anti-symmetrization with respect to the remaining coordinates is trivial since . The charge neutrality condition now reads , so the droplet has undergone a small contraction, as expected for a quasiparticle.

While the exponent of (10) follows naturally from the above charge requirement (and may be viewed as a combination of an electron operator and an “inverse” quasihole operator), the derivative has been put in “by hand”. Without the derivative, the wave function (II.2) can be shown to be identically zero. Technically, is a descendant of the primary field, , a construction that naturally generalizes to more complicated QH states HHV (). Note that the derivative in (II.2) acts only on the holomorphic part of the wave function. 3

The quasiparticle wave function of (II.2) has a different character than those written above for the ground and the quasihole states, in that it is a sum over correlators, and that it involves prefactors . The factor sets the angular momentum, while the exponential factor is chosen to give the correct lowest Landau level (LLL) electronic wave function: Due to its modified charge, the quasiparticle operator gives rise to an exponential factor , and the compensating prefactor ensures that the overall gaussian factor is . Here and in the following, we suppress exponential factors of the correlators whenever convenient, but fully display all prefactors for clarity. It is suggestive that the prefactors precisely constitute the angular momentum wave function for a charge particle in the LLL. Although we have no formal derivation of this, we find below a similar interpretation in the case of several quasiparticles, where their anyonic nature is also manifest.

As pointed out previously, the quasiparticle wave function above is obtained by modifying one of the electron operators, rather than inserting a new operator. This is very suggestive of the CF picture of a quasiparticle as an excitation of a composite fermion to a higher CF Landau level. In fact, what originally led us to construct the operator was the observation that the wave function (II.2) is identical to the corresponding CF wave function (Eq. 5 of ref. jain3, ), which is known to have a good variational energy and the correct fractional charge. In spite of this identity, however, there are two differences between the present derivation and the CF construction that deserve to be noted: First, the present formalism is entirely within the lowest Landau level. The CF construction of wave functions, on the other hand, involves placing composite fermions in higher CF Landau levels and subsequently projecting onto the LLL by replacing all :s by derivatives in the resulting polynomial. Technically, of course, when deriving the one-quasiparticle wave function, the derivatives in (II.2) enter in the exact same places as those due to projection in the CF construction – but no projection is needed in the present formalismhax (). We return to this point in section IV, where we construct the ground states of the Jain sequences at . 4 Second, in spite of the close relation to composite fermions, the operator is not fermionic, as can be seen from the commutation relation or the OPE , that follow from (4). The precise connection to composite fermions will be discussed in the section on the state below. Although the fractional exponent suggests fractional statistics, one cannot directly read the statistical angle from the two-point function. This issue is discussed in more detail in section V.

### ii.3 Two or more quasiparticles

Based on the experience with the single quasiparticle case, we expect the wave function for quasiparticles to be of the form

 Ψ(l)Mqp(zi) = A{fM(z1…zM)⟨P1m(z1)…P1m(zM)V1(zM+1)……V1(zN)⟩}. (12)

The form of is determined by the condition that the final electronic wave function be analytic and antisymmetric, with limiting behavior , with the relative angular momenta and odd. Because the correlator gives non-analytic factors of the type from all contractions among quasiparticle operators, we choose

 fM(z1…zM)=g(Z)M∏p

where is the center of mass coordinate. Again, the exponential factors are included to give the correct gaussian factor in the -electron wave function. As anticipated in the case of one quasiparticle, is just the LLL wave function of anyons with fractional charge .

To cast (12) in a form suitable for computation, we will use the following formula, which generalizes the expansion by a row used in (II.2) above:

 A {M∏p

where the sum is over all subsets of of the integers, and is the conjugate subset of integers. The proof is found in Appendix B.1.

Using this result, the wave functions for two quasiparticles with total angular momentum and relative angular momentum can be written as

 Ψ2qp(zi)= = ∑i

where . Evaluating the correlator we obtain the following explicit form for the wave function for two quasiparticles with relative angular momentum and center of mass angular momentum ,

 Ψl,L2qp(zi) = ∑i

where the derivatives act on the whole expression to their right, and and .

The corresponding wave function in the CF approach is given byjain3 ()

 ~Ψl,L2qp(zi) = ∑i

The two wave functions differ by terms wherein the derivatives in (II.3) act on the factor . It is known jain3 () that the CF wave function in (II.3) gives the correct fractional charge and statistics of the two-quasiparticle state. The first non-trivial test of our construction is therefore to check whether the CFT wave function (14) shares these good charge and statistics properties. This is indeed the case, as demonstrated by our numerical simulations, which are summarized in section V below. These results show that the two wave functions are essentially identical (for example, their overlap is 99.96% for 50 particles). This can be understood from the following heuristic arguments: First, since the derivatives in (II.3) act on a function which is a polynomial of order in both and , this will generate terms. It is unlikely that the few terms picked up by acting on the first factor will be significant. Secondly, these terms are sub-leading in the coordinate difference between the quasiparticles, and thus unlikely to affect qualitative properties.

### ii.4 Quasiparticles and quasiholes

Wave functions for pairs of quasiparticles and quasiholes can be constructed by inserting pairs of the corresponding operators into the CFT correlator for the Laughlin ground state. The simplest case is a quasiparticle at the origin together with a quasihole at position , given by

 Ψqp−qh(zi,η) = A{e−|z1|2/4mℓ2⟨P1m(z1)V1(z2)……V1(zN)H1m(η)⟩} = ∑i(−1)i+1e−|zi|2/4mℓ2⟨P1m(zi)∏j≠iV1(zj)H1m(η)⟩ = ∑i(−1)i(i)∏j

where the antisymmetrization acts on the electron coordinates only. More generally, a quasiparticle localized at some position away from the origin may be constructed as a coherent superposition of the angular momentum states given in (II.2).

For states with equally many quasiparticles and quasiholes, the background charge does not have to be changed from its ground state value. In this sense, wave functions of this type are the natural low energy bulk excitations that do not require any compensating edge charge. On a closed surface, no fractionally charged states are allowed.

## Iii Composite Fermion states in the Jain series

### iii.1 The ν=2/5 composite fermion ground state

In the composite fermion picture, the ground state wave functions at fillings are constructed as filled Landau levels of composite fermions with flux quanta attached. In particular, the state corresponds to filling the lowest two CF Landau levels. This state may thus be viewed as a “compact” state of quasiparticles, i.e. the CF:s in the second Landau level are in the lowest possible total angular momentum state.

To explore the connection to our CFT construction, we generalize the two-quasiparticle wave function (II.3) of the state to the -quasiparticle case, with , and consider a maximum density circular droplet obtained by putting all the quasiparticle pairs in their lowest allowed relative angular momentum (), and with zero angular momentum for the center of mass (). For simplicity we shall also take (and suppress the subscript on the operators) since the generalization to arbitrary odd is obvious. Using (II.3) and evaluating the correlators, the wave function for quasiparticles reads

 ΨMqp(zi) = ∑i1

Since the anyonic wave function on the first line has the form of a Jastrow factor, it is natural to introduce a second free bosonic field . In fact, by defining

 ~V(z)=ei√53φ2(z)∂ei2√3φ1(z), (19)

we find that (III.1) may be written in the following compact form

 ΨMqp(zi) = A{⟨M∏i=1~V(zi)N∏j=M+1V1(zj)⟩} (20)

i.e. as a sum of correlators of :s and :s.

Again, this expression differs from the corresponding CF wave function only in the ordering of the derivatives and the Jastrow factors in the first line of (III.1). Indeed, as demonstrated in Appendix B, the CF wave function is obtained simply by moving all the derivatives all the way to the left. Let us therefore define

 V2(z)=∂ei2√3φ1(z)ei√53φ2(z), (21)

where the derivative now acts on both the exponentials, and consider the case . We then find that the following sum of correlators of :s and :s:

 ΨCF2/5(zi) = A{⟨M∏i=1V2(zi)2M∏j=M+1V1(zj)⟩} = ∑\lx@stackreli1

exactly reproduces the -electron CF wavefunction for .

The operators , as opposed to the :s, are real fermionic operators in that they anticommute among themselves, but commute with :s, just as the :s. Note that the form of was determined entirely from the form of the maximum density -quasiparticle wave function, so its fermionic nature was not an input. If we want to interpret as a composite electron operator, it should have the same charge as . This is ensured if we redefine the charge density operator as

 J(z)=i√3∂φ1(z)+i√15∂φ2(z). (23)

This construction may seem ad hoc in the sense that we fix the coefficient of by hand so as to obtain the correct charge. However, we shall see below that this choice is consistent, in that it produces the correct charge for the quasiholes in the state.

Fulfillment of the charge neutrality condition for the vertex operators requires a background charge, which for the maximum density circular droplet can be assumed to be constant. Furthermore, this density must reproduce the correct exponential factor for electrons in the LLL. The latter is achieved by redefining the expectation value as

where , so the total background electron density is . We stress that this value is not an input, but follows from demanding that describe unit charge particles in the LLL, which was what led us to the above form (23) of the charge density operator. We now show that this state is indeed homogeneous, i.e. that the droplets formed by the :s and the :s have the same area. Charge neutrality gives the following conditions on the areas and integrated over in (24),

 √3M+2√3M = √3ρ3A (25) √53M = √15~ρ3~A,

which implies and thus homogeneity. From the perspective of composite fermions, this correponds to two filled CF Landau levels, since the degeneracy is the same in all Landau levels. It would be interesting to redo the above construction on a closed manifold, where we would expect the concept of “filled CF Landau level” to emerge in a natural way from the condition that the correlators do not vanish.

Although it is possible to write general many-quasiparticle wave functions similar to the two particle wave function in (14), it is only the maximum density droplet of (III.1), and more generally the ”compact” CF statesreview (), that allow for a simple expression in terms of conformal blocks as in (20); for general relative angular momenta one still has to explicitly put in compensating (anyonic) wave functions by hand. In this general case, there is also no reason for introducing a constant background charge different from that of the “parent” , so there is no natural way to obtain non-zero correlators even if we were to introduce the field . As we see below, this would also be in conflict with the known properties of the charge 1/3 quasiholes.

### iii.2 The quasihole operators

To create quasiholes in the 2/5 state, the operator of (2) is no longer appropriate since it does not give holomorphic electron wave functions, as is seen from, e.g. , . Instead, it is necessary to include the second Bose field, , and construct quasihole operators of the form . The coefficients and are determined from the requirements that (i) the wave function of any single quasihole be holomorphic, i.e. the power of the correlator between any quasihole operator and or be a non-negative integer, and (ii) the resulting hole operator not be expressible as a combination (product) of the other quasihole or vertex operators. These conditions uniquely determine the allowed coefficients and , and lead to the following two fundamental quasihole operators for the state:

 H01 = ei3√15φ2(η) (26) H10 = ei√3φ1(η)−2i√15φ2(η) .

Using the charge operator corresponding to the charge density (23) one verifies that both these operators create quasiholes with charge 1/5. Note that this charge assignment is a prediction of our scheme, rather than an input, since the form of the charge operator (23) was determined independently from demanding to have unit charge. All other allowed vertex operators can be constructed as products of and ; the operators in (26) span the charge lattice.

It is an easy exercise to construct the explicit electron wave functions obtained by inserting the operators (26) in the correlator (III.1). Not surprisingly, a direct correspondence with the composite fermion picture is again revealed: Inserting the operator (with for simplicity) into the ground state (III.1) exactly gives the wave function of a quasihole in the center of the lowest CF Landau level, while gives a quasihole in the second CF Landau level. Taking the product of the two quasihole operators, one obtains a charge-2/5 operator which, in the CF language, reproduces the wave function of a vortex, i.e. (for ) two quasiholes at the origin, one in each CF-Landau level. review () Section V clarifies the relation between these quasihole operators and Wen’s effective bulk and edge theories for the quantum Hall state.

If we would attempt to use the operators and to describe a 1/3 state with a small number of quasiparticles (e.g. by putting a compensating charge at the edge or at infinity by hand), we would be forced to use the operators (26) for the quasiholes and thus be led either to a wrong charge assignment for the quasiholes or to redefine the charge operator as to make the :s carry fractional charge. This again stresses that the form of the charge operator as well as the various vertex operators is intimately tied to the particular ground state under consideration.

### iii.3 The quasiparticle operator

The quasiparticle operator of the state is constructed in the same spirit as given in (10), i.e. as a combination of an “inverse” quasihole operator and one of the electron operators, combined with an appropriate number of derivatives. Since in the 2/5 state there are two independent hole operators ( and in (26)) and two electron operators ( and ), it superficially looks as if as there are four quasiparticle candidates. However, it can be shown HHV () that three of these are excluded as they do not produce non-zero wave functions, and one is left with

 P2/5(z)=∂2e2i√3φ1(z)+2i√15φ2(z) (27)

which corresponds to combining (a quasihole in the second CF Landau level) with (a composite fermion in the second CF Landau level). Again, the two derivatives are necessary in order to produce a non-zero wave function

 Ψ1qp(zi)=A⟨P2/5(z1)M+1∏i=2V2(zi)2M+1∏j=M+2V1(zj)⟩, (28)

and (28) is identical to the corresponding CF wave function. Note that, given the connection to composite fermions, it is very natural to have two different quasihole operators but only one quasiparticle operator: There are two filled CF LLs in which to create quasiholes, but the only way (except for higher excitations) to create a quasiparticle is to put one composite fermion in the third CF Landau level.

### iii.4 The ν=3/7 state and the Jain series

As a final explicit example, let us construct the ground state and quasiholes of the state, i.e. the third level of the Jain sequence. The generalization to the full Jain series is given in Appendix B.3 .

The 3/7 state is obtained from a correlator containing an equal number of :s, :s and the new operator :

 V3(z)=P2/5(z)ei7√35φ3(z)=∂2ei[2√3φ1(z)+2√15φ2(z)+7√35φ3(z)] (29)

and again, the result is precisely the CF wave function (see appendix B.3). The relevant charge density operator, which ensures unit charge of , is given by

 J(z)=i√3∂φ1(z)+i√15∂φ2(z)+i√35∂φ3(z). (30)

It is easy to check that is fermionic, but commutes with both and , and that the wave function written in analogy with (III.1) has filling fraction . In the language of composite fermions, this corresponds to filling up three CF Landau levels. In analogy with the 2/5 state, one finds three independent charge-1/7 quasihole operators, which exactly correspond to quasiholes in the third, second, and first CF Landau levels, respectively:

 H001(η) = ei[5√35φ3(η)] H010(η) = ei[3√15φ2(η)−2√35φ3(η)] (31) H100(η) = ei[1√3φ1(η)−2√15φ2(η)−2√35φ3(η)].

Operators for excitations with higher charge are obtained as products of these; for example, the product of all three is a charge-3/7 vortex. Again, it is straightforward to check that the operators (III.4) span the charge lattice. In direct generalization of the case, the quasiparticle operator is given by a combination of the inverse hole operator in the highest occupied CF Landau level, i.e. , and , with one additional derivative,

 P3/7(z)=∂3ei[2√3φ1(z)+2√15φ2(z)+2√35φ3(z)]. (32)

The pattern for construction of higher level operators in the series should now be obvious, and in Appendix B.3 we give the general expressions for the operators describing the electrons at the level in the series, as well as the corresponding current density operator. The proof that the CF wave functions for filled CF Landau levels are reproduced by sums of correlators with an equal number of :s (for fixed ) is outlined in Appendix B.3. The construction of the pertinent quasihole operators should be straightforward, although we have not derived the explicit formulae beyond the ones given above.

From the general expressions of the operators, it is easy to see that two operators and at the same level give a factor in the correlation function, while two operators and at different levels produce a factor (see appendix B.3). This gives an alternative way to calculate the filling fraction, and also demonstrates that the limiting value for is .

## Iv Connection to effective Chern-Simons theories and edge states

Wen has developed a general effective theory formalism for the QH liquids based on representing the currents by two dimensional gauge fields with a Chern-Simons actionwen (),

 L=−14πKII′aIμ∂νaI′λ εμνλ−e2πAμ∂νtIaIλεμνλ, (33)

where the matrix and the “charge vector” have integer elements. The filling fraction is given by . A generic quasiparticle carries integral charges of the field, and is thus labeled by integers constituting the vector . The electric charge and the statistics of the quasiparticle are given by and , respectively. This description is not unique; as explained in reference [wen, ], an equivalent description is given by where is an element of