Composite fermion wave functions as conformal field theory correlators
Composite fermion wave functions as conformal field theory correlators
T.H. Hansson,1 C,-C. Chang,2 J.K. Jain,2 and S. Viefers3
1Department of Physics, Stockholm University, AlbaNova University Center, SE - 106 91 Stockholm, Sweden
2Physics Department, 104 Davey Lab, The Pennsylvania State University, University Park, Pennsylvania 16802
3 Department of Physics, University of Oslo, P.O. Box 1048 Blindern, 0316 Oslo, Norway
(Dated: October 25, 2018)
It is known that a subset of fractional quantum Hall wave functions has been expressed as confor-
mal field theory (CFT) correlators, notably the Laughlin wave function at filling factor ν = 1/m (m
odd) and its quasiholes, and the Pfaffian wave function at ν = 1/2 and its quasiholes. We develop a
general scheme for constructing composite-fermion (CF) wave functions from conformal field theory.
Quasiparticles at ν = 1/m are created by inserting anyonic vertex operators, P 1
(z), 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 ν = s/(2sp+ 1)
as the CFT correlators of a new type of fermionic vertex operators, Vp,n(z), constructed from n free
compactified bosons; these operators provide the CFT representation of composite fermions carry-
ing 2p flux quanta in the nth 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 K-
matrices and l- and t-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 numbers: 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 loops1. Witten’s subsequent demonstration
that the braiding of the Wilson loops are reflected in the correlation functions of certain CFTs2 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 1 + 1 dimensional CFT3. 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
CFT4,5.
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 state6, and the Haldane-Rezayi spin singlet pairing state7, 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 ν = 1/2 state, the so-called Pfaffian
wave function, which is tentatively assigned to the observed ν = 5/2 FQHE. The quasiholes in this state have charge
q = 1/4 rather than q = 1/2 expected from the filling fraction, and exhibit non-Abelian fractional statistics. To
establish the latter it was essential to use CFT technology.44
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 progress10, this is the case
for the ground state wave functions of the prominent FQHE series ν = s/(2sp ± 1), and their related quasihole or
quasiparticle excitations. (Expressions for the states in the Haldane-Halperin hierarchy were given in Ref. 4, 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 ν = 1/m and the Pfaffian wave function at ν = 1/2.45
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) formalism12,13. Here the exper-
imentally prominent Jain states at ν = s/(2sp + 1) are formed from s filled Landau levels of “composite fermions,”
which are electrons carrying 2p flux quanta. Other CF states, as e.g. the Pfaffian, which is the preferred candidate
for the observed ν = 5/2 state, can be formed by various BCS type pairing mechanisms4,8. In the CF description, a
quasihole is obtained simply by removing a composite fermion from an incompressible FQHE state, and a quasipar-
ticle is a composite fermion in a higher, otherwise empty CF Landau level (LL). (CF Landau levels are also called Λ
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 fields13.
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 statistics13,14,15,16. 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-e operator ψ†α(~x), 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 Marchetti17. 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 ν = 1/m, for which the CF wave function differs from that proposed earlier by Laughlin18. The
calculation of the Berry phase associated with two-CF quasiparticle exchange, originally performed by Kjønsberg and
Leinaas15 and subsequently by Jeon and collaborators16, 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 angle19.
In this paper we establish a firm connection between CF wave functions and CFT correlators. Specifically:
1. We construct the quasiparticles of ν = 1/m (m odd) using a new kind of anyonic vertex operators P 1
. For a
single quasiparticle, the resulting wave function is identical to that obtained using the CF theory. A generaliza-
tion to two or more quasiparticles produces wave functions that are very similar to the CF wave functions but
not identical. For two quasiparticles at ν = 1/3, 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 ν = n/(2np+ 1) are exactly given by sums of
CFT correlators of a set of vertex operators, Vnp, which in the CF language correspond to creating composite
fermions in higher CF Landau levels.
3. We generalize the construction of the quasiparticle operator P 1
, as well as of the quasihole operators, to higher
levels in the Jain sequence; at level n, there are n independent hole operators and one quasiparticle operator.
The vertex operator Vn,p at level n is closely related to the quasiparticle operator at level n− 1.
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 Wen3.
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 ν = 1/m 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 previously20.
II. ONE AND TWO QUASIPARTICLES IN THE LAUGHLIN STATE
A. 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 ν = 1/m, where m is an odd
integer. Following Moore and Read4, we introduce the normal-ordered vertex operators,
V1(z) = : e
mϕ1(z) : (1)
(η) = : e
ϕ1(η) : , (2)
where the normal ordering symbol : :, will be suppressed in the following. The free massless boson field, ϕ1, 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)
∼ (z − w)αβei(α+β)ϕ1(w) (4)
where the last line expresses the operator product expansion (OPE) in the limit z → w. From (4) follows V1(z)V1(w)+
V1(w)V1(z) = 0, and H 1
(z)H 1
(w) − eiπ/mH 1
(w)H 1
(z) = 0. The first of these reflects that the electrons are
fermions, while the second is appropriate for fractional statistics as discussed in reference [4].
We normalize the (holomorphic) U(1) charge density operator as
J(z) =
∂zϕ1(z) (5)
so the corresponding charge is given by
dz ∂zϕ1(z), (6)
where the contour encircles the whole system. The U(1) charges, Q = 1 of the electron and Q = 1/m of the quasihole,
can be read directly from the commutators [Q, V1(z)] = V1(z) and [Q, H 1
(η)] = 1
(η). It is noted that Q 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 U(1) charge Q is given by
∆n = δn−Q, (7)
where the integer δn is the number of electrons added by the operator. If the argument of the operator is an electron
coordinate, zi, one electron is added, while no electron is added if the argument is a quasihole coordinate ηi. (The idea
of binding of an electron and m vortices was implicit in Laughlin’s original work, and was made explicit by Halperin6,
Girvin and MacDonald21 and Read22.)
The total electric charge of a particle is given by Qel = −e∆n = e(Q− δn). 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) ν = 1/m Laughlin wave function can now be written as (for notational convenience, we write
Ψ(zi) instead of Ψ({zi})):
ΨL(zi) = 〈0|R{V1(z1)V1(z2) . . . V1(zN−1)V1(zN )e−i
d2z ϕ1(z)}|0〉 (8)
≡ 〈V1(z1)V1(z2) . . . V1(zN−1)V1(zN )〉1/m
(zi − zj)me−
|zi|2/4`2 ,
where R denotes radial ordering. The second line defines the average 〈. . . 〉1/m, and the third follows for the ordering
|z1| ≥ |z2| ≥ . . . |zN |, which will be assumed below unless indicated otherwise. In the following, we shall suppress the
subscript 1/m whenever it is clear to what ground state we are referring. The exponential operator in (8) corresponds
to a constant background particle density, ρm = −ρ0/m, where ρ0 = 1/2π`2 is the density of a filled Landau level.
This is necessary since the U(1) charge neutrality condition, known from the Coulomb gas formulation, in the CFT
ensures that the correlator vanishes unless N = ρm
d2z = ρmA, which defines the area, A, of the system. As
explained in reference [4], the background charge will produce the correct gaussian factor e−
|zi|2/4`2 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) = 〈H 1
(η1)H 1
(η2) . . . H 1
(ηn)V1(z1)V1(z2) . . . V1(zN−1)V1(zN )〉. (9)
In this case the charge neutrality condition reads N + n/m = ρmA′, indicating an expansion of the droplet. From
the general relation (4) we get H 1
(z)V1(w) + V1(w)H 1
(z) = 0 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, ϕi, compactified on circles of radius Ri =
where mi are odd integers. The primary fields are given by the chiral vertex operators V (z) = e
ϕi(z) where
the integers qi 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
mi; 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. 23.
B. One quasiparticle
The most immediate guess4 for a quasiparticle operator would be to simply change the sign in the exponent in the
quasihole operator of (2), i.e. to use e−
ϕ1(η). That, however, introduces unacceptable singular terms ∼
i(zi−η)
in the electronic wave function. Inspired by the CF wave functions, we instead define a quasiparticle operator, P 1
which has a U(1) charge (1 − 1/m), and that will replace one of the the original electron operators V1(z). We can
thus think of P (z) 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 V1 and P 1
∆Qel = e((1− 1/m)− 1) = −e/m, as appropriate for a quasiparticle at ν = 1/m. The modified electron operator is
given by
(z) = ∂ei(
m− 1√
)ϕ1(z), (10)
and the wave function for a single quasiparticle with angular momentum l is written as
Ψ(l)1qp(zi) = A{z
−|z1|2/4m`2〈P 1
(z1)V1(z2) . . . . . . V1(zN )〉} (11)
(−1)i+1zli e
−|zi|2/4m`2〈P 1
j 6=i
V1(zj)〉
(−1)izli
(zj − zk)m∂i
l 6=i
(zl − zi)m−1,
where A 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 P 1
(z)V1(w)−
V1(w)P 1
(z) = 0, 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 V1(z)V1(w) + V1(w)V1(z) = 0. The
charge neutrality condition now reads N − 1 + (1− 1
) = ρmA′′, 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 (11) can be shown to be identically zero. Technically, P 1
(z) is a descendant of the
primary field, ei(
m− 1√
)ϕ(z), a construction that naturally generalizes to more complicated QH states24. Note that
the derivative in (11) acts only on the holomorphic part of the wave function.46
The quasiparticle wave function of (11) 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 f1(zi) = zlie
−|zi|2/4m`2 .
The factor zli 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 P 1
(zi) gives rise to an
exponential factor exp(−|zi|2(1− 1/m)/4`2), and the compensating prefactor ensures that the overall gaussian factor
is exp{−
j |zj |
2/(4`2)}. 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 f1 precisely constitute the
angular momentum l wave function ψl(z) = zle−|z|
2/4m`2 for a charge e/m 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 P 1
was the observation that the wave function (11) is identical to the corresponding CF wave function (Eq.
5 of ref. 25), 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 z̄:s by derivatives in the resulting polynomial. Technically, of course, when deriving the
one-quasiparticle wave function, the derivatives in (11) enter in the exact same places as those due to projection in
the CF construction – but no projection is needed in the present formalism26. We return to this point in section
IV, where we construct the ground states of the Jain sequences at ν = n/(2np + 1).47 Second, in spite of the close
relation to composite fermions, the operator P 1
(z) is not fermionic, as can be seen from the commutation relation
(z)P 1
(w)− eiπ(m−2+1/m)P 1
(w)P 1
(z) = 0 or the OPE P 1
(z)P 1
(w) ∼ (z −w)m−4+
2(m−1)√
ϕ1(w), that follow
from (4). The precise connection to composite fermions will be discussed in the section on the ν = 2/5 state below.
Although the fractional exponent 1/m 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.
C. Two or more quasiparticles
Based on the experience with the single quasiparticle case, we expect the wave function for M quasiparticles to be
of the form
Ψ(l)Mqp(zi) = A{fM (z1 . . . zM )〈P 1m (z1) . . . P 1m (zM )V1(zM+1) . . . . . . V1(zN )〉}. (12)
The form of fM is determined by the condition that the final electronic wave function be analytic and antisymmetric,
with limiting behavior ∼ (zp−zq)m−1+lpq , with the relative angular momenta lpq ≥ 1 and odd. Because the correlator
gives non-analytic factors of the type ∂p∂q(zp − zq)m−2+1/m from all contractions among quasiparticle operators, we
choose
fM (z1 . . . zM ) = g(Z)
(zp − zq)1+lpq−1/me−
i |zi|
2/4m`2 ,
where Z = 1
i=1 zi is the center of mass coordinate. Again, the exponential factors are included to give the correct
gaussian factor exp[−
j |zj |
2/(4`2)] in the N -electron wave function. As anticipated in the case of one quasiparticle,
fM is just the LLL wave function of M anyons with fractional charge e/m.
To cast (12) in a form suitable for computation, we will use the following formula, which generalizes the expansion
by a row used in (11) above:
(zp − zq)1−1/m+lpqP 1
(z1) . . . P 1
(zM )V1(zM+1) . . . . . . V1(zN )} (13)
p=1 ipR{
(zip − ziq )
1−1/m+lpqP 1
(zi1) . . . P 1
(ziM )V1(zīM+1) . . . V1(zīN )},
where the sum is over all subsets {i1 . . . iM} of M of the N integers, and {̄i1 . . . īM} is the conjugate subset of N −M
integers. The proof is found in Appendix B 1.
Using this result, the wave functions for two quasiparticles with total angular momentum L and relative angular
momentum l can be written as
Ψ2qp(zi) = =
(−1)i+jZLij(zi − zj)
1+l− 1
(|zi|2+|zj |2)〈P 1
(zi)P 1
k 6=i,j
V1(zk)〉, (14)
where Zij = (zi + zj)/2. Evaluating the correlator we obtain the following explicit form for the wave function for two
quasiparticles with relative angular momentum l and center of mass angular momentum L,
Ψl,L2qp(zi) =
(−1)i+jZLij(zi − zj)
1+l− 1
m ∂zi∂zj (zi − zj)
m−2+ 1
m (15)
(ij)(zk − zi)m−1
(ij)(zl − zj)m−1
(ij)∏
(zm − zn)m,
where the derivatives act on the whole expression to their right, and
(ij) =
k 6=i,j
∏(kl)
i<j =
i,j 6=k,l
The corresponding wave function in the CF approach is given by25
Ψ̃l,L2qp(zi) =
(−1)i+jZLij(zi − zj)
l∂zi∂zj (zi − zj)
m−1 (16)
(ij)(zk − zi)m−1
(ij)(zl − zj)m−1
(ij)∏
(zm − zn)m .
The two wave functions differ by terms wherein the derivatives in (16) act on the factor (zi − zj)1−
m . It is known25
that the CF wave function in (16) 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 (16) act on a function which is a polynomial of order N in both zi and zj , this will generate O(N2)
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 (zi − zj) between the quasiparticles, and thus unlikely to affect
qualitative properties.
D. 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 η, given by
Ψqp−qh(zi, η) = A{e−|z1|
2/4m`2〈P 1
(z1)V1(z2) . . . . . . V1(zN )H 1
(η)〉} (17)
(−1)i+1 e−|zi|
2/4m`2〈P 1
j 6=i
V1(zj)H 1
(−1)i
(zj − zk)m
j 6=i
(zj − η) ∂i
l 6=i
(zl − zi)m−1 (zi − η)1−
where the antisymmetrization acts on the electron coordinates zi only. More generally, a quasiparticle localized at
some position η′ away from the origin may be constructed as a coherent superposition of the angular momentum
states given in (11).
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
A. The ν = 2/5 composite fermion ground state
In the composite fermion picture, the ground state wave functions at fillings ν = n/(2np+ 1) are constructed as n
filled Landau levels of composite fermions with 2p flux quanta attached. In particular, the ν = 2/5 state corresponds
to filling the lowest two CF Landau levels. This state may thus be viewed as a “compact” state of N/2 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 (15) of the
1/m state to the M -quasiparticle case, with M = N/2, and consider a maximum density circular droplet obtained by
putting all the quasiparticle pairs in their lowest allowed relative angular momentum (` = 1), and with zero angular
momentum for the center of mass (L = 0). For simplicity we shall also take m = 3 (and suppress the subscript m on
the operators) since the generalization to arbitrary odd m is obvious. Using (13) and evaluating the correlators, the
wave function for M quasiparticles reads
ΨMqp(zi) =
i1<i2<···<iM
(zik − zil)
3 ∂zi1∂zi2 . . . ∂ziM
k′<l′
− zi′
3 (18)
(i2,i3...iM )(zk1 − zi1)
(i1,i3...iM )(zk2 − zi2)
2 . . .
(i1,i2...iM )(zkM − ziM )
(i1,i2...iM )∏
(zm − zn)3 .
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 ϕ2(z). In fact, by defining
Ṽ (z) = ei
3ϕ2(z)∂e
ϕ1(z) , (19)
we find that (18) may be written in the following compact form
ΨMqp(zi) = A{〈
Ṽ (zi)
j=M+1
V1(zj)〉} (20)
i.e. as a sum of correlators of M Ṽ :s and (N −M) V1: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 (18). 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) = ∂e
ϕ1(z)ei
3ϕ2(z) , (21)
where the derivative now acts on both the exponentials, and consider the case N = 2M . We then find that the
following sum of correlators of M V2:s and M V1:s:
ΨCF2/5(zi) = A{〈
V2(zi)
j=M+1
V1(zj)〉} (22)
i1<i2...iM
ī1<ī2...̄iM
k ik〈V2(zi1) . . . V2(ziM )V1(zī1) . . . V1(zīM )〉
exactly reproduces the (N = 2M)-electron CF wavefunction for ν = 2/5.
The operators V2(zi), as opposed to the P (zi):s, are real fermionic operators in that they anticommute among
themselves, but commute with V1(zi):s, just as the P (zi):s. Note that the form of V2 was determined entirely from
the form of the maximum density M -quasiparticle wave function, so its fermionic nature was not an input. If we want
to interpret V2 as a composite electron operator, it should have the same charge as V1. This is ensured if we redefine
the charge density operator as
J(z) =
∂ϕ1(z) +
∂ϕ2(z). (23)
This construction may seem ad hoc in the sense that we fix the coefficient of ϕ2 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 ν = 2/5 state.
Fulfillment of the charge neutrality condition for the vertex operators V2 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
〈· · · 〉2/5 ≡ 〈0| . . . e−i
15ρ̃3
d2z ϕ2(z)e−i
d2z ϕ1(z)|0〉, (24)
where ρ̃3 = (1/15)ρ0, so the total background electron density is (1/3 + 1/15)ρ0 = (2/5)ρ0. We stress that this value
is not an input, but follows from demanding that V2 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 N/2 = M V1:s and the M V2:s have the same area. Charge neutrality gives the following
conditions on the areas A and Ã integrated over in (24),
3ρ3A (25)√
15ρ̃3Ã ,
which implies A = Ã 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 (18), and more generally the ”compact” CF states13, 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” ν = 1/3, so there is no natural way
to obtain non-zero correlators even if we were to introduce the field ϕ2(z). As we see below, this would also be in
conflict with the known properties of the charge 1/3 quasiholes.
B. The quasihole operators
To create quasiholes in the 2/5 state, the operator H 1
(η) of (2) is no longer appropriate since it does not give
holomorphic electron wave functions, as is seen from, e.g. , 〈V2(z)H1/3(η)〉 ∼ (z − η)2/3. Instead, it is necessary
to include the second Bose field, ϕ2, and construct quasihole operators of the form Hpq(η) = e
ϕ1(η)+i
ϕ2(η).
The coefficients p and q 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 V1(z) or V2(z) 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 p and q, and lead to the following two
fundamental quasihole operators for the ν = 2/5 state:
H01 = e
ϕ2(η) (26)
H10 = e
ϕ1(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 V2 to have unit charge. All
other allowed vertex operators can be constructed as products of H01(η) and H10(η); 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 (22). Not surprisingly, a direct correspondence with the composite fermion picture is again revealed:
Inserting the operator H10(η) (with η = 0 for simplicity) into the ν = 2/5 ground state (22) exactly gives the wave
function of a quasihole in the center of the lowest CF Landau level, while H01 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 η = 0) two quasiholes at the origin, one in each
CF-Landau level.13 Section V clarifies the relation between these quasihole operators and Wen’s effective bulk and
edge theories for the ν = 2/5 quantum Hall state.
If we would attempt to use the operators V1 and V2 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 V2: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.
C. The quasiparticle operator
The quasiparticle operator of the ν = 2/5 state is constructed in the same spirit as P 1
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 (H01 and H10 in (26)) and
two electron operators (V1 and V2), it superficially looks as if as there are four quasiparticle candidates. However, it
can be shown24 that three of these are excluded as they do not produce non-zero wave functions, and one is left with
P2/5(z) = ∂
ϕ1(z)+
ϕ2(z) (27)
which corresponds to combining H01 (a quasihole in the second CF Landau level) with V2 (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)
V2(zi)
2M+1∏
j=M+2
V1(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.
D. The ν = 3/7 state and the Jain series
As a final explicit example, let us construct the ground state and quasiholes of the ν = 3/7 state, i.e. the third
level of the ν = s/(2s+ 1) 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 V1:s, V2:s and the new operator V3:
V3(z) = P2/5(z)e
ϕ3(z) = ∂2ei[
ϕ1(z)+
ϕ2(z)+
ϕ3(z)] (29)
and again, the result is precisely the ν = 3/7 CF wave function (see appendix B 3). The relevant charge density
operator, which ensures unit charge of V3, is given by
J(z) =
∂ϕ1(z) +
∂ϕ2(z) +
∂ϕ3(z). (30)
It is easy to check that V3(z) is fermionic, but commutes with both V1 and V2, and that the wave function written in
analogy with (22) has filling fraction ν = 3/7. 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) = e
ϕ3(η)
H010(η) = e
ϕ2(η)− 2√35ϕ3(η)
H100(η) = e
ϕ1(η)− 2√15ϕ2(η)−
ϕ3(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 (31) span the charge lattice. In direct
generalization of the ν = 2/5 case, the ν = 3/7 quasiparticle operator is given by a combination of the inverse hole
operator in the highest occupied CF Landau level, i.e. H001, and V3, with one additional derivative,
P3/7(z) = ∂
i[ 2√
ϕ1(z)+
ϕ2(z)+
ϕ3(z)] . (32)
The pattern for construction of higher level operators in the ν = s/(2s + 1) series should now be obvious, and in
Appendix B 3 we give the general expressions for the operators Vpn describing the electrons at the nth level in the
n/(2np+ 1) series, as well as the corresponding current density operator. The proof that the CF wave functions for n
filled CF Landau levels are reproduced by sums of correlators with an equal number of Vpn:s (for fixed p) 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 Vpn(zi) and Vpn(zj) at the same
level give a factor (zi − zj)2p+1 in the correlation function, while two operators Vp,n1(zi) and Vp,n2(zj) at different
levels produce a factor (zi − zj)2p (see appendix B 3). This gives an alternative way to calculate the filling fraction,
and also demonstrates that the limiting value for n→∞ is ν = 1/2p.
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 aIµ with a Chern-Simons action3,
L = −
KII′aIµ∂νaI′λ ε
µνλ −
Aµ∂νtIaIλε
µνλ , (33)
where the matrix K and the “charge vector” tT = (t1 . . . tp) have integer elements. The filling fraction is given by ν =
tTK−1t. A generic quasiparticle carries integral charges of the aIµ field, and is thus labeled by p integers constituting
the vector l = (l1 . . . lp). The electric charge and the statistics of the quasiparticle are given by q = −etTK−1l and
θ = πlTK−1l, respectively. This description is not unique; as explained in reference [3], an equivalent description is
given by (K ′, t′, l′) = (WKWT ,W t,W l) where W is an element of SL(p, Z), i.e. an integer valued p× p matrix with
unit determinant.
As an example of the above, the ν = 2/5 state is described by the K matrix and t vector,
K2/5 =
tT = (1, 1) . (34)
This is an example of what Wen refers to as the symmetric basis, where in general tT = (1, 1, . . . , 1). By an SL(2, Z)
transformation, we can represent the same state in the “hierarchy basis” (which naturally occurs when constructing
states in the Halperin-Haldane hierarchy) characterized by tT = (1, 0, . . . , 0).
K ′2/5 = WKW
; t′T = tTWT = (1, 0) ; W =
, (35)
Starting from the Chern-Simons theory (33) defined on a finite two dimensional domain, one can derive a dynamical
theory for the edge excitations. The details can be found in [3] and references therein, and the resulting theory is
Sed =
dtdx [KIJ∂tφI∂xφJ − VIJ∂xφI∂xφJ + 2eAµ�µν∂νtIφI ], (36)
where K and t, as well as the quasiparticle vector l, are the same as in the effective bulk theory (33). This is a
multicomponent chiral boson theory with the current operator given by
Jµ = −
�µνtI∂νφI . (37)
The quasiparticle operators (including the electron operator) take the generic form
Ψ ∼ ei
q lqφq , (38)
familiar from abelian bosonization of one-dimensional fermion systems. The numbers VIJ are the non-univeral edge
velocities, which depend on the details of the confining potential.
In their original paper on the connection between QH liquids and conformal field theories, Moore and Read made
two basic claims. The first, which we already have discussed, is that the electronic wave functions can be expressed as
conformal blocks of certain CFT:s. The second is that this very same CFT is the one dimensional theory describing
the dynamical edge excitations. This last claim should not be taken literally since it is known that the edge dynamics
is non-universal. Not only the edge velocities, but also the character, and even the number of edge modes can depend
on details of the edge potential. Examples are the polarization edge modes related to edge spin texture27 and the
counter-propagating modes resulting from edge reconstruction as first discussed by Shamon and Wen28. Thus we can
only hope that the CFT will provide a “minimal” edge theory consistent with the topological properties of the bulk,
i.e. that it supports excitations with the same charges. In spite of these limitations, the Moore-Read conjecture about
the edge theory has been very fruitful, especially in the search for effective field theories for the non-abelian Pfaffian
state29.
We shall now demonstrate the connection between the CFT construction of the Jain states and Wen’s K-matrix
formulation by explicitly working out the case of ν = 2/5. Led by the Moore-Read conjecture, we will start from
our CFT bulk theory, read off the K-matrix and the charge vector, and show that in the basis where (26) are the
fundamental quasihole operators, one exactly recovers Wen’s K-matrix and t-vector in the symmetric basis. This is
consistent with Read’s earlier result30 that the symmetric basis naturally describes the Jain states. Alternatively, we
may choose a basis consisting of either of the charge 1/5 quasiholes in (26), along with the charge 2/5 vortex (i.e. the
product of the two 1/5-hole operators); as we shall see, this instead corresponds to the hierarchical basis.
The conformal field theory contains the two uncoupled bosonic fields ϕ1 and ϕ2, compacitfied on radii R2 = 3 and
15, respectively. The corresponding action, S =
d2xLcft for the full scalar fields φi(x, t) = ϕi(z) + ϕ̄i(z̄) is obtained
from the Lagrangian,
Lcft =
(φ̄1, φ̄2)
µ �µν∂νφI ≡
KIJφI∂µ∂
µφJ −AµJµ . (39)
where the information about the compactification radii is contained in the charge vector t̃T = (1/
3, 1/
15). The
Lagrangian (39) contains both right and left moving fields, but these decouple, and it is known that the dynamics
of a single chiral component, such as ϕi(z), is described by the first order Lagrangian (36) with the same K matrix
and t vector31. In order to directly compare with Wen’s formalism, we rescale the Bose fields such as to obtain an
integer charge vector, tT = (1, 1): (ϕ′1, ϕ
2) ≡ (ϕ1/
3, ϕ2/
15). Naively, the corresponding K matrix would then be
diag(3, 15). It is however important to remember that a CFT is not defined only by the Lagrangian of the fields ϕi,
which gives the operator product expansions, or fusion rules, of the primary fields (i.e. the vertex operators), but
also by primary field content, i.e. the allowed vertex operators. In the case of the ν = 2/5 state these allowed fields
define a charge lattice with the basis vectors given by the quasihole operators (26). Thus, we will change to a basis
(χ1, χ2) where the fundamental quasihole operators spanning the charge lattice are given by Hi = eiχi . As can be
seen from (26), this is achieved by the field redefinition,
ϕ2 = 3ϕ
2 (40)
ϕ2 = ϕ
1 − 2ϕ
Inverting this transformation and inserting into (39), it is now easy to verify that the resulting K matrix and t vector
are precisely the K and t in (34). Alternatively (and equivalently), if we start from a basis of one of the 1/5 quasihole
operators, say H10, together with the charge 2/5 “vortex” H11 ≡ H10H01, corresponding to the change of basis
χ1 = ϕ
1 + ϕ
2 (41)
χ2 = ϕ
1 − 2ϕ
we find that the corresponding K-matrix and t-vector are the ones given in (35), i.e. the hierarchical basis. This
equivalence, at the effective Chern-Simons theory level, of the Jain states and the hierarchy scheme, has been previously
pointed out by several authors30,32. These authors arrive at this result by a general argument, based on similarity
between the Jain states and filled Landau levels, that ignores the projection on the lowest Landau level. It is reassuring
that the above demonstration, based on explicitly holomorphic wave functions, leads to the same result.
This construction straightforwardly carries over to the other fractions in the Jain sequence; for example, in the case
of ν = 3/7, one may pick the three charge-1/7 quasihole operators of (31) as basis of the charge lattice, corresponding
to the field redefinition
ϕ3 (42)
Again, this brings us to the symmetric basis, with tT = (1, 1, 1) and the K-matrix given by Kij = 2+δij . Alternatively,
we may construct a basis consisting of quasihole operators with charge 1/7, 2/7 and 3/7, respectively, by appropriate
combinations of the charge-1/7 quasiholes in (31). As before, this corresponds to the hierarchical basis, with tT =
(1, 0, 0) and the same K-matrix as that given by Wen3,
Kh3/7 =
 3 −1 0−1 2 −1
0 −1 2
 . (43)
V. LOCALIZED QUASIPARTICLES AND FRACTIONAL CHARGE AND STATISTICS
The present formulation already gives a strong hint for fractional charge and fractional statistics of the CF quasi-
particles: We have seen from (7) that the operator P 1
(z) corresponds to a localized charge at z, and the presence
of the factor (zi − zj)
m is suggestive of fractional statistics with angle π
. This is not a proof, however. The usual
argument for fractional charge and statistics proceeds via the Berry phases produced by adiabatic braidings of local-
ized quasiparticles. In this section we construct the wave functions for localized states of one and two quasiparticles,
and use these to calculate the Berry phases relevant for charge and statistics within what we call a “random phase
assumption”.
A localized quasiparticle state is constructed as a coherent superposition of a the angular momentum states given
in (11) and (15). For a single quasiparticle at location η̄ we have (putting ` = 1)
Ψ1qp(η, η̄; zi) = Ñ1(η̄η)e−
4m |η̄|
(2m)ll!
Ψl1qp(zi) (44)
= Ñ1(η̄η)
(−1)ie−
4m (|zi|
2+|η̄|2−2η̄zi)〈P (zi)
(i)V (zj)〉 .
Notice that the normalization constant N1(η̄η) = Ñ1(η̄η)e−
4m |η̄|
only depends on the combination η̄η. Likewise, we
construct the wave function for two quasiparticles at positions η̄± = N̄ ± η̄/2 as
Ψ2qp(N, N̄, η, η̄; zi) = Ñ2(N, N̄, η, η̄)e−
8m |η̄|
2− 12m |N̄ |
l=1,3,...
L=0,1,...
η)l−1
Ψl,L2qp(zi) (45)
= Ñ2
(−1)i+je−
8m (|η̄|
2+|zij |2) sinh
η̄zij
2m (|N̄ |
2+|Zij |2−2N̄Z)z
ij 〈P 1m (zi)P 1m (zj)
(ij)V1(zk)〉 ,
where zij = zi − zj . For η̄ = 0 and N̄ = η̄ = 0, respectively, these expressions reduce to Ψ01qp and Ψ
2qp, the wave
functions with minimum angular momentum. The explicit wave functions obtained by evaluating the correlators in
(44) and (45), are very similar, but not identical, to the corresponding CF wave functions. One source of difference is
the slight deviation between the angular momentum eigenstates given by (15) and (16), pointed out in section II C,
and shown to be numerically insignificant in section VI. The other source of difference can be seen already for the one
quasiparticle state. The CF wave function reads
ΨCF =
(−1)i e−
(zj − zk)3
(zi − zn)2
zi − zn
 exp
|zk|2
, (46)
but the term proportional to (1/m−1)η̄ is missing in the corresponding CFT wave function. This difference, however,
is a finite size effect. This term contributes to the wave function only when the exponential factor exp(η̄zi/2m) is
expanded to the N th power, and thus amounts to a (nonuniversal) boundary term.
Before proceeding to the calculation of the charge and statistics of the quasiparticles, it is helpful to recall, as a
background, the corresponding calculation for the quasiholes of ν = 1/m. Consider the normalized wave function for
one quasihole, given by (9):
Ψ(η, η̄; zi) = N ′e−
|η|2ΨL(η; zi) (47)
where ΨL(η; zi) =
i(zi − η1)
k<l(zk − zl)
|zi|2 is the Laughlin unnormalized wave function for a single
quasihole. The plasma analogy shows that N ′ is independent of η; the normalization integral of (47) is the partition
function of a Coloumb plasma with a charged impurity, which is independent of the position of the impurity as
long as it is farther than a screening length (`) from the edge. The Berry phase associated with a circular loop
η = Reiθ; θ ∈ {0, 2π}, is given by (with ~ = c = 1)
dθ 〈Ψ(η, η̄)|i∂θ|Ψ(η, η̄)〉 (48)
dθ 〈Ψ(η, η̄)|(η̄∂η̄ − η∂η)|Ψ(η̄, η)〉
where A is the area enclosed by the loop, and the last line employs integration by parts and the observation that the
only η̄ dependence in the wave function is from the gaussian factor. The result, as expected, is the Aharanov-Bohm
phase for a particle of charge e/m.
Turning to two quasiholes and again using (9) we have
Ψ(η̄a, ηa; zi) = N ′′(η1 − η2)
(|η1|2+|η2|2)ΨL(ηa; zi) (49)
where ΨL(ηa; zi) =
i(zi − η1)
j(zj − η2)
k<l(zk − zl)
|zi|2 is the unnormalized Laughlin wave function
for two quasiholes. Naively we would read the fractional statistics parameter as θ = π/m directly form the factor
(η1−η2)
m in (49) but that gives the correct result only if there is no extra Berry phase (other than the usual Aharonov-
Bohm phase) associated with the exchange path (here integrating θ from 0 to π)33. The absence of additional phases
can be confirmed by an explicit calculation using the plasma analogy for a system with two impurities separated
farther than the screening distance, `0. Taking η1 = −η2 = η, we get γexB = −
BπR2, which is nothing but the
AB-phase expected from the exchange of two charge e
particles through a circular path with radius R. That allows
us to read the exchange statistics phase directly from the factor (η1 − η2)
m in (49). Wilczek and Nayak33 suggest
that it is no coincidence that (9) yields a wave function with no Berry contribution to the statistics angle; they make
a conjecture, supported by arguments, that QH wave functions given directly as correlators, or conformal blocks, of
a CFT have vanishing Berry phase (forgetting the Aharanov-Bohm contribution) so the exchange statistics can be
obtained from the so-called monodromy, which in this simple case is just the phase ei2π/m produced by the factor
(η1 − η2)
m when the quasiholes braid around each other along closed paths defined via an analytic continuation of
the original wave function. In the more general case of non-Abelian fractional statistics, several conformal blocks
correspond to the same configuration of quasihole coordinates, and the monodromies are now matrices that encode
how these conformal blocks transform into each other under braidings of the coordinates.
Rather than using (48), we follow Kjønsberg and Leinaas who showed that for a (normalized) wave function of the
form, Ψ(η, η̄; zi) = N (η̄η)Ψh(η, η̄; zi), where N is a real function only of η̄η, the Berry phase is given by34
dθ 〈Ψ(η, η̄)|i∂θ|Ψ(η)〉 = ΘR2
lnN (η̄η)2 . (50)
The Laughlin wave function for a single quasihole at η or two quasiholes at ±η has this form. The upper limit is
taken to be Θ = 2π for a single quasihole, and Θ = π for an exchange of two quasiholes. From (47) and (49), the
normalization constants are given by (up to an η independent factor)
N1(η̄η) = e−
|η|2 (51)
N2(η̄iηi) = |η1 − η2|
(|η1|2+|η2|2),
so the formula (50) can be applied. (We have assumed sufficiently far separated quasiholes.) For a single quasihole loop
of radius R, (50) gives the Berry phase −2πAB(e/m). For two quasiholes at ±η, the Berry phase is (π/m)−AB(e/m)
(with Θ = π). The difference, π/m, gives the contribution from fractional statistics. In this case, there is no
monodromy, and the the full statistical phase appears as a Berry phase.48
We now turn to the case of quasiparticles, where we need to calculate the relevant normalization constants from
the wave functions (44) and (45). Because the sum in (45) extends only over even powers of η̄, the holomorphic part
is single valued under η̄ → −η̄, implying an absence of monodromy contribution to the statistical angle, and thus
both the charge and statistics can be extracted directly from N1 and N2, provided that they can be chosen as real
functions of η̄η only. For quasiparticles, the normalized wave function has the form N (η̄η)Ψ(η̄), and the Berry phase
is given by
γB = −ΘR2
lnN (η̄η)2 . (52)
Unfortunately, the calculation of the normalization factors is more difficult than in the case of the quasiholes, because
the electronic wave functions (44) and (45) involve sums over correlators; e.g. for two quasiparticles the relevant
integral is ∼
ijkl(−1)
i+j+k+l〈P 1
(zi)P 1
(ij)V1(zp)〉〈P 1
(z̄k)P 1
(z̄l)
(lm)V1(zj)〉. If, however, we keep only
the diagonal terms in the sums, which amounts to a kind of random phase assumption discussed in the next section,
then the normalization factors can be calculated as shown in Appendix C. The calculation of the normalization
constants, outlined in Appendix C, gives the result:
N 21 = Ñ
|η|2 ∼ (R2)−1e−
R2 ; η = Reiθ (53)
N 22 = Ñ
|η|2 ∼ (R2)
R2 ; η =
eiθ (54)
Using (52) we can extract the fractional charge from the Berry phase corresponding to a single quasiparticle,
Bπ(R2 + 2m`2) =
BA+ 2π , (55)
so the leading term is precisely the expected AB phase for a charge e/m object. The O(`2) term is a quantum correction
to the “classical” R2. Such corrections have been discovered earlier both in the context of CF quasiparticles16 and
noncommutative matrix models35. The statistical angle is obtained by subtracting this result from the Berry phase
extracted from the exchange path. Remembering that the effective area enclosed by the exchange path is πR2/4, we
θ = γexcB −
γB = −
. (56)
This reproduces the result obtained from a direct numerical evaluation of the Berry phases from the composite fermion
wave functions (44) and (45)15,16,19. The statistics of the quasiholes and quasiparticles obtained above differ in sign,
contrary to the expectation from general considerations based on effective Chern-Simons or the CF theory, and we
want to comment upon this. Both normalization constants in (53) are of the form N ∼ (R2)ae−bR
, and from the
derivation in the Appendix one sees that the constant a is unambiguously determined, and the same holds for the
fractional part of b, while the integer part of b could in principle be shifted by using a different prescription for the
ordering of the derivatives. However, this does not necessarily mean that the frational part of θ is well determined
since it is sensitive to a cancellation of a large number of terms proportional to R2 in the Berry phases. The above
result is based on the assumption that we have correctly identified |η| as the distance between the two quasiparticles.
This question is discussed in some detail in in Ref. 16 where it is shown that the distance between two quasiparticles
is slightly different from |η|; correcting for the distance produces the same statistics as for quasiholes (modulo an
integer). Because our wave function is closely related to the CF wave function, similar considerations should apply
here as well.
VI. NUMERICAL TESTS
This section concerns quantitative tests of the CFT quasiparticle wave functions. The numerical tests consist of
two parts. In the first part, we compare, at filling factor ν = 1/3, the two-quasiparticle wave functions (15) obtained
from the conformal field theory with the standard wave functions from the CF theory by calculating their Coulomb
interaction energies and overlap. We will find that the two are practically identical. In the second part, the random
phase approximation used in section V is examined.
A. Two-quasiparticle wave function
The N -composite fermion wave function for two quasiparticles at filling factor ν = 1/3 is constructed by compactly
filling the lowest CF-LL by N − 2 composite fermions, and placing the remaining two composite fermions in the
second CF-LL. We consider below the state in which the two “excited” composite fermions are in angular momentum
orbitals, and occupy the smallest angular momenta. The wave function for this state is written as
ΨCF = PLLL
∣∣∣∣∣∣∣∣∣∣∣∣
η1,−1(z1) η1,−1(z2) . . . η1,−1(zN )
η1,0(z1) η1,0(z2) . . . η1,0(zN )
η0,0(z1) η0,0(z2) . . . η0,0(zN )
η0,1(z1) η0,1(z2) . . . η0,1(zN )
η0,N−3(z1) η0,N−3(z2) . . . η0,N−3(zN )
∣∣∣∣∣∣∣∣∣∣∣∣
(zi − zj)2 exp
|zk|2
, (57)
where PLLL denotes projection into the lowest Laudau level (LLL), and ηn,m(z) is the single particle eigenstate in
symmetric gauge:
ηn,m(z, z̄) = Nn,m e
−|z|2/4
(−1)k
z̄kzk+m, (58)
in which n = 0, 1, 2, . . . labels the Landau level index, m = −n,−n+ 1,−n+ 2, . . . is the angular momentum quantum
number, k0 = max(0,−m), and
Nn,m =
2π2m(n+m)!
is the normalization constant. As usual, the magnetic length has been set to unity. The wave function can be shown
to be equal to
ΨCF = PLLL
∣∣∣∣∣∣∣∣∣∣∣∣∣
z̄1z1 z̄2z2 . . . z̄NzN
z̄1 z̄2 . . . z̄N
1 1 . . . 1
z1 z2 . . . zN
zN31 z
2 . . . z
∣∣∣∣∣∣∣∣∣∣∣∣∣
(zi − zj)2 exp
|zk|2
. (60)
Following the standard procedure, the projection procedure is accomplished by expanding the determinant, moving
all z̄’s to the left, and replacing z̄ by 2∂/∂z (with the convention that the derivatives do not act on the Gaussian
part). A technique developed in Ref. [36] has made it possible to perform the projection in a more convenient manner,
which is the one we use below. (This method gives projected states very close to those obtained by “brute force”
projection.) The projected wave function is written as
ΨCF =
(−1)i+j(zi − zj)3
(ij)∏
(zi − zk)2
(ij)∏
(zj − zl)2
(ij)∏
(zm − zn)3
 −2(zi − zj)2 +
(ij)∑
(zi − zk)(zj − zl)
(ij)∑
(zk − zi)(zk − zj)
 exp
|zk|2
. (61)
The symbol (ij) denotes that indices i and j are excluded in the summation or the product. The total angular
momentum of (61) is
N + 2. (62)
Numerical simulations for the wave function in (61) have shown that it produces better variational energy than the
two quasiparticles wave function obtained by generalizing Laughlin’s single quasiparticle state25.
The CFT wave function for two quasiparticles located at the origin is given by (14). To make contact with the
above CF wave function, we set the center of mass angular momentum to zero and put the two quasiparticles in
the smallest relative angular momentum channel l = 1; that produces a wave function that has the total angular
momentum given in (62) . For these parameters, the CFT ansatz for the two quasiparticles wave function ΨCFT at
ν = 1/3 reduces to
ΨCFT =
(−1)i+j(zi − zj)3
(ij)∏
(zk − zi)2
(ij)∏
(zl − zj)2
(ij)∏
(zm − zn)3 (63)
−49 1(zi − zj)2 + 83
(ij)∑
(zk − zi)(zk − zj)
(i,j)∑
(zk − zi)(zl − zj)
 exp
|zk|2
To compare the two-quasiparticle wave functions in Eqs. (63) and (61), we compare their Coulomb interaction
energies and also calculate their overlaps. The Coulomb energy (in units of e2/�`0) is defined as
∣∣∣∣∣∣ 12
i 6=j
|ri − rj |
∣∣∣∣∣∣Ψ
〈Ψ|Ψ〉
, (64)
We do not include here electron-background and background-background contributions; these are not necessary for
the present purpose, as they are identical for the two wave functions. The overlap is defined as
|〈ΨCFT|ΨCF〉|√
〈ΨCFT|ΨCFT〉〈ΨCF|ΨCF〉
. (65)
Both quantities are evaluated using Metropolis Monte Carlo integration. A single data point is obtained by averaging
100 independent Monte Carlo runs, with ∼ 1.2 × 105 iterations in each run. For N = 50 electrons, the total
N ECFT ECF O
10 7.76619(62) 7.76600(62) 0.9999301(2)
20 24.1403(19) 24.1402(19) 0.9999274(4)
30 46.3258(18) 46.3257(18) 0.9999274(3)
40 73.2339(17) 73.2339(17) 0.9999266(2)
50 102.0588(16) 102.0585(16) 0.999613(5)
TABLE I: The Coulomb energy ECFT (ECF), quoted in units of e
2/�`, for the CFT (composite fermion) two-quasiparticle wave
function ΨCFT (ΨCF). O is the properly normalized overlap between the two candidate wave functions for a number of system
sizes N . In ΨCFT, we set L = 0 and l = 1. The definitions of energy and overlap are given in the text. The Monte Carlo
statistical uncertainty is shown in brackets.
 0  2  4  6  8  10  12  14  16  18
excess charge
 0  5  10  15  20
excess charge
FIG. 1: Density profiles of ΨCFT and ΨCF for N = 40 (left panel), and 50 (right panel) particles. “GS” denotes the ν = 1/3
background obtained from Laughlin’s wave function. The horizontal dashed line indicates 2/3 units of charge. In both cases, the
excess integrated charge for ΨCFT, denoted by thick dashed line, shows the correct quantized value. The statistical undertainty
in Monte Carlo is smaller than the widths of the lines.
computational time is approximately 100 hours on a single node of a Beowulf-type PC cluster consisting of dual 3.06
GHz Intel Xeon Processors. The results are summarized in Table I. The excellent agreement demonstrates that the
two two-quasiparticle wave functions are essentially identical. Figure 1 depicts the density profiles of the two wave
functions for N = 40 and 50 electrons. The excess integrated charge (i.e. the charge measured relative to the ν = 1/3
background) for the CFT wave function is also shown. The excess charge has a well-defined plateau at 2/3 before
the edge distortion. (The creation of two quasiparticles near the center of the droplet induces two quasiholes of equal
charge at the edge.)
B. Random Phase Approximation
The composite fermion quasiparticles have been shown to possess well-defined fractional braiding statistics16. The
comparisons in the previous subsection imply that this property, in principle, carries over to the CFT quasiparticles.
However, one of the strengths of the CFT description of the FQHE states is that it reveals the braiding properties in
a transparent manner.
A calculation of the braiding statistics requires wave functions for spatially localized quasiparticles, which are
constructed in the previous section. A key observation is that the braiding statistics can be obtained from the wave
function normalization factor, which depends only on the coordinates of localized quasiparticles (c.f. (45) ). However
it is difficult to obtain an explicit analytical expression for the normalization factor for localized quasiparticle states,
because the localized quasiparticle wave functions are sums over many correlators; for example, the relevant term in
the integral for the normalization factor of a single quasiparticle is∑
〈Pm(zi)
k 6=i
Vm(zk)〉〈Pm(z̄j)
l 6=j
Vm(z̄l)〉. (66)
The analytic form of the normalization factor can be obtained by assuming that only the “diagonal” elements in (66)
are relevant, which we have referred to as the random-phase assumption; the braiding properties of the quasiparticles
can be derived under this approximation and are in agreement with the known results. In this section we test the
random phase assumption for a single localized quasiparticle.
The wave function for a single quasiparticle at ν = 1/m localized at η̄ is written as
Ψ1qp(η̄) =
(−1)i e−
4m (|η̄|
2+2η̄zi)〈Pm(zi)
j 6=i
Vm(zj)〉, (67)
with the correlator 〈Pm(zi)
j 6=i Vm(zj)〉 given by
〈Pm(zi)
j 6=i
Vm(zj)〉 =
(zj − zk)m
j 6=i
(zi − zj)m−1
j 6=i
zi − zj
· exp
|zl|2
. (68)
For simplicity, we define the notation
F (zi) ≡ (−1)i e−
2m (η̄zi)〈Pm(zi)
j 6=i
Vm(zj)〉. (69)
Then the wave function Ψ1qp(η̄) can be expressed as
Ψ1qp(η̄) = e
− 14m |η̄|
F (zi). (70)
The square of the normalization factor, N1(η, η̄), is given by the integral
N 21 (η, η̄) =
dzk Ψ
1qp(η̄)Ψ1qp(η̄)
2m |η̄|
|F (zi)|2 +
[F ∗(zi)F (zj) + F (zi)F
∗(zj)]
≡ Mdiag +Moff−diag. (71)
Mdiag andMoff−diag are the “diagonal” and “off-diagonal” contributions to the full normalization factor, respectively.
We calculate the ratio Mdiag/N1(η, η̄)2 for several quasiparticle locations for ν = 1/3. The result is shown in Fig. 2.
The principal conclusion is that the contribution of the single diagonal term is of the same order as that of a large
number of off-diagonal terms. Although not conclusive, this suggests that the diagonal term used to calculate the
quasiparticle charge and statistics in section V will be dominant, thus providing a partial justification for the neglect
of the off-diagonal terms.
VII. SUMMARY AND OUTLOOK
In this paper we have extended the class of QH wave functions that can be expressed as correlators in a conformal field
theory to include the quasiparticle states at the Laughlin fractions, as well as all the ground states in the positive Jain
sequence and their quasiparticle and quasihole excitations. The connection between the CFT operators and composite
fermions was explicitly demonstrated by constructing n fermionic vertex operators Vp,n, built from n compactified
scalar fields, corresponding to the n filled CF Landau levels at ν = n/(2np+ 1). For these states we also constructed
the fractionally charged excitations: At ν = n/(2np + 1), there are n independent hole operators (corresponding to
removing a composite fermion from any one of the n filled CF Landau levels) and one unique quasiparticle operator
(corresponding to putting a CF in the empty (n + 1)st Landau level). The fermionic vertex operator Vp,n at level n
is closely related to the quasiparticle operator of the Jain state with n − 1 filled CF LLs; in this sense, the ground
state at the fraction n/(2pn+ 1) of the Jain sequence may be viewed as a condensate of quasiparticles of the state at
(n − 1)/[2p(n − 1) + 1]. It should be noted, however, that these quasiparticles obey well defined fractional statistics
only in the dilute limit. We also showed that the conformal field theories used to obtain the CF wave functions give
precisely the chiral edge theories that are expected from general considerations within the effective field theory scheme
developed by Wen, thus giving microscopic support to that approach.
 0  0.05  0.1  0.15  0.2  0.25  0.3  0.35
!=(0.0,0.0)
!=(2.0,0.0)
!=(4.0,0.0)
!=(6.0,0.0)
FIG. 2: The ratio Mdiag/N1(η, η̄)2 for several quasiparticle location η. The coordinates are in units of magnetic length. The
results for η = (6.0, 0.0) are not plotted for N < 10 because of significant edge effects.
An attractive aspect of the methods developed in this paper is that they can be extended and applied to other
quantum Hall states. For example, in a recent paper, a straightforward generalization of our vertex operators was
employed38 to describe the states observed recently by Pan et.al.39, which do not belong to the principal series
ν = n/(2pn + 1) but have been modeled as the FQHE of composite fermions39,40. Generalizing our methods to
include the negative Jain sequence, ν = n/(2np − 1), or more generally to states obtained from condensing holes, is
a more challenging problem. Another interesting challenge is to find a CFT operator that directly creates a localized
quasiparticle, rather than having to construct it as a coherent superposition of angular momentum states as was done
in this paper; work on this question is in progress24.
Acknowledgements: We would like to thank Jürgen Fuchs, Maria Hermanns, Anders Karlhede and Jon Magne
Leinaas for helpful discussions. We thank Prof. B. Janko for hospitality at the ITS, a joint institute of ANL and
the University of Notre Dame, funded through DOE contract W-31-109-ENG-38 and Notre Dame Office of Research.
This work was supported in part by the Swedish Research Council, by Nordforsk, and by NSF under grant no.
DMR-0240458
APPENDIX A: THE BACKGROUND CHARGE
Definition of correlators of vertex operators such as (8) requires introduction of a compensating charge to satisfy the
neutrality condition implied by the conserved U(1) charge37. This can be done in different ways. The simplest is to
put a compensating charge Vbg(z∞) = e−i
Nmϕ(z∞) at the position z∞, taken to infinity, and define the correlator by
a limiting procedure:41
ΨL(zi) = lim
〈V1(z1)V1(z2) . . . V1(zN−1)V1(zN )〉 =
(zi − zj)m . (A1)
This prescription does not produce the exponential factor e−
|zi|2/4`2 , characteristic of a lowest Landau level wave
function.
In this paper we use the prescription given in Ref. 4, which corresponds to a smeared background charge given by
the operator
d2z ϕ(z) , (A2)
where ρm = ρ0/m with ρ0 = eB/2π the density of filled Landau level. The difficulty with this prescription is that a
direct evaluation of the correlator gives a contribution
e−mρm
d2z ln(z−zi) , (A3)
where the presence of the logarithm makes the imaginary part of the integral undefined. The aim of this appendix is
to give a regularized version of the smeared background charge that: i) is well defined; ii) reproduces the pertinent
gaussian factor in (8); and iii) differs from (8) only through a well defined (although singular) gauge transformation.
The idea behind our regularization is to replace the continuous background field by a lattice of singular flux tubes
of strength
, (A4)
which defines the integer n.
First consider a single electron in the presence of a single (fractional) flux tube at the origin. The wave function
close to the flux tube behaves as
ψ(z) ∼ z−
2π , (A5)
so for a single electron in a flux tube lattice, it is natural to consider a wave function of the type
ψ(z; {z~n}) = f(z)
(z − z~n)−
2π , (A6)
which has the correct singular behaviour at the lattice points z~n and satisfies the Laplace equation,
∇2ψ(z; {z~n}) = 4∂z∂z̄ψ(z; {z~n}) = 0 ; z /∈ {z~n}. (A7)
This is not a sufficient condition for an acceptable electron wave function – we must also require that ψ(z; {z~n}) is
normalizable. As we now show, this will determine the allowed analytic functions f(z). Without any loss of generality,
we specialize to a regular lattice of K points z~n = (nx + iny)a with spacing a that covers a total area A - this is our
regularized version of a uniform flux Φ = Kδφ corresponding to a uniform field of strength B = Φ/A inside the area
A. We will ignore edge effects.
Since we have limz→∞ ψ(z; {z~n}) ∼ f(z)z−K
2π , and normalizability requires49 limz→∞ ψ(z; {z~n}) ∼ 1/z, we de-
mand f(z) ∼ zk where,
= k −
≤ −1, (A8)
where we choose A so that Φ/2π is an integer. This can be rewritten as
k + 1 ≤
= Aρ0 = N0, (A9)
where N0 is the number of states in the lowest Landau level. This result makes it plausible that, in the limit of large
n, the functions (A6) with
f(z) =
N0−1∑
will give a good description of the lowest Landau level at a magnetic field of strength B. To show this, we rewrite ψ
ψ(z; {z~n}) = f(z)
z − z~n
z̄ − z̄~n
)− δφ4π ∏
|z − z~n|−
2π , (A10)
and approximate
|z − z~n|−
2π = exp
ln |z − z~n|
≈ exp
d2r′ ln |~r − ~r′|
. (A11)
The last integral was calculated as∫
d2r′ ln |~r − ~r′| =
d2r′ ln |~r − ~r′|∇2r′r
d2r′ δ2(~r − ~r′)r′2 =
, (A12)
where we integrated by parts and neglected boundary terms. (The justification is that there is an understood density
function ρ(~r′) that rapidly falls to zero outside the area A but is essentially constant inside. This still leaves an edge
correction due to the derivatives acting on the profile that we ignore.) Finally we note that δφ
= Kδφ 1
= Φ 1
, where ` is the magnetic length corresponding to the field strength B, so the approximate wave functions
are of the form
ψ(z; {z~n}) =
z − z~n
z̄ − z̄~n
)− δφ4π
f(z)e−
|z|2 (A13)
and are expected to be valid in the limit of a/`→ 0, and z well inside the lattice. We have thus recovered the standard
lowest Landau level wave functions, albeit in an unconventional gauge defined by the (arbitrarily chosen) flux lattice.
This conclusion is also strongly suggested by the numerical calculations of Pryor, who shows that already for n = 8,
at least four flat bands are identifiable in the electron spectrum, corresponding to the four lowest Landau levels42. A
generalization of the analytic argument presented above to include higher Landau levels would be interesting.
Returning to our original goal of regularizing the operator insertion (A2), we see that
d2z ϕ(z) →
Vb(z~n) =
m2π`2
φ(z~n) (A14)
will do the job in the limit a/`2 → 0, because the total U(1) charge of the K vertex operators Vb equals
−Ka2eB/2πm = −Aρm = −N , where N is the total number of electrons. Making the replacement (A14) in (8)
(A14) in (8) 〈
V1(z1)V1(z2) . . . V1(zN )
m2π`2
φ(z~n)
, (A15)
and using the same approximation as in (A11), we regain the correct exponential factor (The contraction of V1(zj)
~n Vb(z~n) gives the factor
~n |zj − z~n|
−δφ/2π which, according to (A11), gives the Gaussian factor for zj), an
unimportant constant from the contractions between the different Vb’s, and also a singular and rapidly changing phase
factor just as in (A13). This is the regularized version of the statement in reference 4 that (8) “is trying to give us
the answer in a gauge where the vector potential is zero, which means it differs by an everywhere-singular gauge
transformation from the usual symmetric gauge vector potential for the uniform background magnetic field.” Here
we should also mention that in correlation functions involving the full scalar field, φ(z, z̄), the singular phases will
cancel out, so these are well behaved functions even in the limit of vanishing lattice spacing. Finally, we note that the
regularization procedure outlined above suffers from a formal difficulty - it introduces vertex operators with charges
that do not belong to the charge lattice of the CFT under consideration. Putting a compensating charge at infinity
would not suffer from this problem, but would also not correspond to a homogeneous system.
APPENDIX B: EQUIVALENCE BETWEEN CFT AND CF WAVE FUNCTIONS
In this appendix we provide derivations for some of the formulae in the main text, and prove that the CFT wave
functions for the states in the Jain series indeed reproduce the wave functions from the CF framework using a direct
projection on the LLL.
1. An identity
We begin by deriving the central relation (13). The basic idea is to express the antisymmetrization as a Slater
determinant and then use the Laplace expansion of an (N +M)× (N +M) determinant,
detA =
�PidetBidetCi, (B1)
where the sum is over the
ways in which a N ×N matrix Bi can be formed from the first N rows of A,
Ci is the complementary M ×M matrix and �Pi is the sign of the permutation needed to bring the N columns of Bi
followed by the M columns of Ci into the original order.43 This formula generalizes the expansion by a row used to
derive (11) for the one quasiparticle case.
We have,
(zp − zq)1/m+lpqP 1
(z1) . . . P 1
(zM )V1(zM+1) . . . . . . V1(zN )} (B2)
(zrp − zrq )
1/m+lpqP 1
(zr1) . . . P 1
(zrM )V1(zrM+1) . . . . . . V1(zrN )
�{in}
sgnPs(zsp − zsq )
1/m+lpqP 1
(zs1) . . . P 1
(zsM )
sgnPtV1(ztM+1) . . . V1(ztN )
p=1 ipR{
(zip − ziq )
1/m+lpqP 1
(zi1) . . . P 1
(ziM )V1(zīM+1) . . . V1(zīN )} ,
where {r1 . . . rN} is a permutation, Pr, of {1 . . . N}; {s1 . . . sM} is a permutation, Ps, of a subset {i1 . . . iM} of the
N integers; {tM+1 . . . tN} a permutation, Pt, of the conjugate set {̄i1 . . . īM} of N −M integers; and �Pr etc. the
corresponding sign factors. The symbol lpq denotes the relative angular momentum, and R is the radial ordering
operator. The sum
{in} is over all the N !/M !(N −M)! ways of doing this partition, and �{in} is the sign of the
permutation needed to order the M rows of the first partition to the left in the Slater determinant. In the above
expression, the second line is the definition, the third row follows from the Laplace expansion, and the last by noting
that the prefactor makes the expression explicitly antisymmetric under exchange of coordinates in the subset {in}.
Antisymmetry under exchange in the second subset is already guaranteed by the anti-commutation relations for the
V1:s. As in the single quasiparticle case, no extra signs are obtained by the final radial ordering because P and V1
commute.
2. Equivalence between the ν = 2/5 CF and CFT wave functions.
We now prove that the CFT wave function (22) for ν = 2/5 is identical to that of composite fermions. Recalling
that (22) differs from (18) only in that all the derivatives are on the left, we have the following explicit expression
(recall that N = 2M so there are M V1:s and M V2:s in the correlator),
ΨCFT2/5 (zi) =
i1<i2...iM
ik∂zi1∂zi2 . . . ∂ziM
(zik − zil)
3 (B3)
(i2,i3...iM )(zk1 − zi1)
(i1,i3...iM )(zk2 − zi2)
2 . . .
(i1,i2...iM )(zkN − zin)
(i1,i2...iM )∏
(zm − zn)3 .
To write this in the CF form, we factor out a full Jastrow factor:
ΨCFT2/5 (zi) =
i1<i2...iM
ik∂zi1∂zi2 . . . ∂ziM
(zik − zil)
(i1,i2...iM )∏
(zm − zn)1
N=2M∏
(zp − zq)2 (B4)
The first two Jastow factors are nothing but the Vandermonde determinants of the subset I = {zi1 . . . ziM } and the
conjugate subset J = {zī1 . . . zīM }. Also useful is the operator identity
∂z1∂z2 . . . ∂zM
∣∣∣∣∣∣∣∣∣∣∣
1 1 . . 1
z1 z2 . . zM
z21 z
2 . . z
. . . . .
zM−11 . . . z
∣∣∣∣∣∣∣∣∣∣∣
∣∣∣∣∣∣∣∣∣∣∣
∂z1 ∂z2 . . ∂zM
∂z1z1 ∂z2z2 . . ∂zM zM
1 ∂z2z
2 . . ∂zM z
. . . . .
1 . . . ∂zM z
∣∣∣∣∣∣∣∣∣∣∣
which follows because each coordinate, as well as the corresponding derivative, appears once and only once in every
term when the determinant is expanded. We can now use the Laplace formula, (B1), in the opposite direction to
write
ΨCFT2/5 (zi) =
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
∂z1 ∂z2 . . ∂zN
∂z1z1 ∂z2z2 . . ∂zN zN
1 ∂z2z
2 . . ∂zN z
. . . . .
1 . . . ∂zN z
1 1 . . 1
z1 z2 . . zN
z21 z
2 . . z
. . . . .
zM−11 . . . z
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
(zp − zq)2 =
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
∂z1 ∂z2 . . ∂zN
z1∂z1 z2∂z2 . . zN∂zN
z21∂z1 z
2∂z2 . . z
. . . . .
zM−11 ∂z1 . . . z
N ∂zN
1 1 . . 1
z1 z2 . . zN
z21 z
2 . . z
. . . . .
zM−11 . . . z
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
(zp − zq)2, (B6)
where we omitted an unimportant sign. The last expression is, up to an overall normalization factor, the CF wave
function as given in reference36.
The last identity in (B6) follows because when the derivatives act on the factors in the determinant they give a row
that is already present in the lower part of the determinant. For this to be true it is necessary that all the angular
momentum states are present and that there are at least as many rows without derivatives as those with derivatives.
These conditions correspond to having a maximum density droplet of electrons in the second CF Landau level which
is no larger then the droplet formed by the electrons in the lowest CF Landau level. This completes the proof of the
statement in the main text.
3. The general CF operators and the Jain series
We now extend the previous analysis to a general state in the Jain series. First we give the explicit expressions for
the operators Vp,n discussed in section III D:
Vp,1(z) = e
2p+1ϕ1(z)
Vp,2(z) = ∂e
ϕ1(z)e
2p+1ϕ2(z)
Vp,3(z) = ∂
ϕ1(z)e
(2p+1)(4p+1)
ϕ2(z)
4p+1ϕ3(z) (B7)
. . .
Vp,n(z) = ∂
ϕ1(z)e
(2p+1)(4p+1)
ϕ2(z)
. . . e
[2p(n−2)+1][2p(n−1)+1]
ϕn−1(z)
2np+1
2(n−1)p+1ϕn(z).
Because all ϕi’s commute, we can write
Vp,n(z) = ∂
n−1eiϕ̃n(z), (B8)
where ϕ̃1 = ϕ1 and
ϕ̃n(z) =
[2(k − 1)p+ 1](2kp+ 1)
ϕk(z) +
2np+ 1
2(n− 1)p+ 1
ϕn(z) ; n ≥ 0 (B9)
Using the sum formula:
[2p(k − 1) + 1][2pk + 1]
2pn+ 1
(B10)
and the charge density operator
J(z) =
2p+ 1
∂zϕ1(z) +
(2p+ 1)(4p+ 1)
∂ϕ2(z) · · ·+
[2p(n− 1) + 1][2pn+ 1]
∂ϕn(z) (B11)
it can be shown that the operators (B7) satisfy the properties stated in the text vis á vis charge and statistics, and
also give the filling fraction ν = n
2pn+1
. We can now construct the wave function for the general ground state in the
Jain series by a recursive procedure. For a total of N = nM electrons, it is natural to write
ΨCFp,n (zi) = A{〈
Vp,n(zi)
j=M+1
Vp,n−1(zj) · · ·
j=(n−1)M+1
Vp,1(zj)〉} (B12)
The proof that this indeed reproduces the ν = n/(2pn + 1) CF wave function, is a straightforward generalization of
that given for 2/5 in the previous section. It involves using the Laplace formula (B1) iteratively n− 1 times, breaking
the problem down into the n groups (Landau levels) of particles, in analogy with the procedure in section B 1. The
generalization of (B3) then contains n− 1 sign factors, one for each additional group of particles, and one can follow
the logic of (B4) - (B6) (with n (M ×M) subdeterminants instead of two) to derive the equivalence of the CF and
CFT wave functions.
APPENDIX C: THE NORMALIZATION FACTORS N1 AND N2
We begin with a single quasiparticle. Using the explicit form (44), and keeping only the diagonal terms in the
double sum in the normalization integral we get,
|Ñ1(η, η̄)|−2 ∼
d2zi e
− 12m |zi−η|
j 6=i
d2zj 〈P 1
V1(zi)〉 〈P 1
(z̄i)
V1(zi)〉∗ . (C1)
Here and below we use the sign ∼ to indicate that we neglect η-independent constants. We write P 1
(zi) = ∂iP̂ 1
and P 1
(z̄i) = ∂̄iP̂ 1
(z̄i) and perform the zi integral after making the approximate substitution zi → η in the
correlators. This gives
|Ñ1(η, η̄)|−2 ∼
j 6=i
∂η〈P̂ 1
j 6=i
V1(zj)〉
∂η̄〈P̂ 1
j 6=i
V1(zj)〉∗
 (C2)
Note that we first moved the derivatives in the operators P 1
(z) outside the expectation values. That this is allowed
follows either from a direct calculation, or from noting that P 1
(z) is a descendant of the primary field P̂ 1
(z) and
using standard methods to express the correlator of descendant fields as derivatives of correlators of primary fields37.
It is important that all sign factors cancel in the diagonal terms.
Next we note that the η̄ dependence of each correlator is given by 〈P̂ 1
j 6=i V1(zj)〉 ∼ exp [−(m− 1)|η|
2/(4m)].
This allows us to move the derivatives outside the full two dimensional correlation function. Reintroducing the
magnetic length, `, defining Dη = ∂η + c η̄ with c = (m− 1)/(m`2), and noting [Dη, D̄η] = 0, we get,
|Ñ1(η, η̄)|−2 ∼
DηD̄η
j 6=i
d2zj |〈P̂ 1
j 6=i
V1(zj)〉|2 (C3)
The right hand side of this equation is now in a form where plasma analogy arguments can be applied: the integral
is the free energy of an overall neutral plasma with a charged impurity at the fixed postion η. This free energy is
independent of the impurity positions because of screening, so finally, using DηD̄η1 = c2 η̄η + c, and noting that all
terms in the sum give identical contributions, we conclude that to leading order in `2/|η|2, |Ñ1(η, η̄)|−2 ∼ η̄η, which
gives (53) in the main text.
The calculation of the two quasiparticle normalization factor, |Ñ2(N, η; N̄ , η̄)| follows in an analogous manner, with
some extra complication due to the more complicated exponential factors in the expression (45). Again keeping only
the diagonal terms and completing squares in the exponents, we get,
|Ñ2(N, η; N̄ , η̄)|−2 ∼
d2zijd
2Zij e
|Zij−N |2 [e−
4m |zij−η|
4m |zij+η|
− 2 cosϑ e−
4m (|η|
2+|zij |2)]
(zijzij)
k 6=i,j
d2zk [∂i∂j〈P̂ 1
(zi)P̂ 1
k 6=i,j
V1(zk)〉][∂i∂j〈P̂ 1
(z̄i)P̂ 1
(z̄j)
k 6=i,j
V1(zk)〉], (C4)
where eiϑ = η̄zij − zijη, and overall constants are suppressed and the derivatives are moved outside the expectation
values. Because of the gaussian factors in |Zij −N |, we can approximate the integral by substituting the maximum
value Zij = N = 12 (η+ + η−). The third term in the square brackets (∼ cosϑ) is maximum at zij = 0; the integral
vanishes with this substitution because of the factor (zijzij)1−
m . This term is therefore neglected. The remaining
two terms are equal. For the first term, we have zij = η = η+ − η−. Proceeding as before, defining D+ = ∂η+ + c η̄+
etc, and using that the η̄+ dependence of each correlator is ∼ exp[−(m− 1)|η|2/(4m)], we get,
|Ñ2(N, η; N̄ , η̄)|−2 ∼
k 6=i,j
d2zk [∂i∂j〈P̂ 1
(zi)P̂ 1
k 6=i,j
V1(zk)〉][∂̄i∂̄j〈P̂ 1
(z̄i)P̂ 1
(z̄j)
k 6=i,j
V1(zk)〉] (C5)
= D+D−D̄+D̄−
k 6=i,j
d2zk |〈P̂ 1
(zi)P̂ 1
k 6=i,j
V1(zk)〉|2.
The integral is now the partition function for a neutral plasma with two impurities at positions η+ and η− and this
free energy is again independent of the impurity positions because of screening. We thus have D+D−D̄+D̄−1 =
c2[c2|η+|2|η−|2 + c|η+|2 + c|η−|2 + 1]. Finally, taking N = 0 and substituting z± = ±η/2 in the above expressions,
and noting that all terms in the sums give identical contributions, we get, to leading order in `2/|η|2, the formula (54)
quoted in the text. Retaining the leading order contribution is valid in the limit when the quasiparticles are far sepa-
rated; this suggests corrections to statistics for smaller separations, as also found in direct numerical evaluations15,16.
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24 T.H. Hansson, M. Hermanns and S. Viefers, in preparation.
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26 J.N. Ginocchio and W.C. Haxton [Phys. Rev. Lett. 77, 1568 (1996)] give an alternative way of constructing CF wave
functions directly in the LLL.
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http://arxiv.org/abs/cond-mat/0002330
http://arxiv.org/abs/hep-th/0103013
36 R. Kamilla and J.K. Jain, Int. J. Mod. Phys. B 11 2621 (1997).
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41 See e.g. E. Ardonne, PhD thesis, www.kitp.ucsb.edu/˜ ardonne/thesis/boekje.pdf, which also gives a general discussion of
the application of conformal field theory to the quantum Hall effect.
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43 M. L. Metha, Matrix Theory, Hindustan Publishing Corporation, (1989).
44 Later the non-Abelian statistics has also been understood in the context of d-wave paired superconductors8 and has also
been studied numerically9.
45 An asymmetry in the description of the quasiparticles and quasiholes at ν = 1/m has been a striking feature of other
descriptions as well. In Laughlin’s theory, a quasihole at position η is represented as a vortex
(zi−η), while a quasiparticle
is created by a complicated operator involving many derivatives. The fractional statistics of the quasiholes is easy to derive,
while the statistics of the quasiparticles eludes a precise analytical treatment. In the Ginzburg-Landau-Chern-Simons effective
theories, the quasiholes and quasiparticles are described by vortices and anti-vortices respectively, and again there is an
asymmetry in the description.11
46 A more careful evaluation of the correlators using a regularized background charge (cf. Appendix A) does give a contribution
also from the exponential. In order to have holomorphic wave function this must be cancelled by replacing ∂i with a suitable
covariant derivative. Since this in the end amounts to a mere change of notation, we simply use the rule that the derivatives
do not act on the exponential part in the correlation function.
47 As straightforward projection tends to get computationally heavy in numerical calculations with many particles and a large
number of derivatives, slightly different methods of obtaining LLL wave functions have been employed in most of the CF
literature13. These, too, are often referred to as projection. For a single quasiparticle the different prescriptions agree, while
in the general case they produce very similar but not identical wave functions. It is the brute force projection which exactly
matches with the CFT construction for the Jain sequence ground states.
48 The quantity N (η̄η) can be interpreted as the (diagonal element) of the quasihole density matrix, N (η, η̄)2 ≡ ρ(η, η̄) =
ψ(η)ψ(η̄)?, where the factor ψ(η1, η2) is the quasihole wave function.
49 This is the correct condition on a sphere (or a compactified plane). On a real plane the last mode will be marginal in the
sense that the normalization integral will have a logarithmic divergence.
http://arxiv.org/abs/cond-mat/0702516
	Introduction
	One and two quasiparticles in the Laughlin state
	The ground state and the quasihole states
	One quasiparticle
	Two or more quasiparticles
	Quasiparticles and quasiholes
	 Composite Fermion states in the Jain series
	The = 2/5  composite fermion ground state
	The quasihole operators
	The quasiparticle operator
	The = 3/7 state and the Jain series
	Connection to effective Chern-Simons theories and edge states
	Localized quasiparticles and fractional charge and statistics
	Numerical tests
	Two-quasiparticle wave function
	Random Phase Approximation
	Summary and Outlook
	The background charge
	Equivalence between CFT and CF wave functions
	An identity
	Equivalence between the = 2/5 CF and CFT wave functions. 
	The general CF operators and the Jain series
	The normalization factors N1 and N2 
	References
