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W~algebras in the quantum Hall effect Dimitra Karabali l,2 Physics Department, Syracuse University, Syracuse, NY 13244, USA Received 26 May 1994; revised 19 July 1994; accepted 15 August 1994

Abstract We show that a large class of incompressible quantum Hall states correspond to different representations of the W~algebra by explicit construction of the second-quantized generators of the algebra in terms of fermion and vortex operators. These are parametrized by a set of integers

which are related to the filling fraction. The class of states we consider includes multilayer Hall states and the states proposed by Jam to explain the hierarchical filling fractions. The corresponding second-quantized order parameters are also given.

1. Introduclion The quantum Hall effect [1,21 (QHE) appears in two-dimensional systems of electrons in the presence of a strong perpendicular uniform magnetic field B. It is characterized by the existence of a series of plateaux where the Hall conductivity is quantized and the longitudinal conductivity vanishes. The Hall conductivity is proportional to the filling fraction i.’, the ratio between the number of electrons and the degeneracy of the Landau levels. This generic feature, which appears in both the integer (IQHE) and the fractional (FQHE) quantum Hall effect, is attributed to the existence of a gap, which gives rise to an incompressible ground state. These correspond to incompressible droplet configurations of uniform density p = vB/2ir. Recently, the notion of incompressibility has been related to the existence of an infinite-dimensional algebraic structure, the W~algebra, which emerges quite naturally at least in the case of the IQHE [3,4]. In this case (v = n) the energy gap is the cyclotron 2

E-mail address: [email protected] Address after September 1, 1994: The Institute for Advanced Study, School of Natural Sciences, Princeton,

NJ 08540, USA. Elsevier Science By. SSDIO55O-3213(94)00361 -0

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D. Karabali/Nuclear Physics B 428 IFS] (1994) 531—544

energy separating adjacent Landau levels and the phenomenon can be understood in terms of noninteracting fermions. The underlying W~algebra emerges as the algebra of unitary transformations which preserve the particle number at each Landau level and it plays the role of a spectrum-generating algebra. The corresponding ground state satisfies highest-weight conditions [4]. The noninteracting picture is nonapplicable in the case of the FQHE, where the repulsive Coulomb interactions among electrons become important in producing an energy gap. Much of our understanding of the FQHE relies on successful trial wavefunctions. For example, in an attempt to explain the FQHE for z = 1/rn, where m is an odd integer, Laughlin proposed [5} a set of wavefunctions which turn out to be quite close to the exact numerical solutions for a large class of repulsive potentials [6]. Based on Laughlin wavefunctions, a hierarchy scheme [6,71 has been developed to explain the other rational values of v. A somewhat different approach has been developed by Jam [81. In this approach the FQHE wavefunction is constructed by attaching an even number of magnetic fluxes to electrons occupying an integer number of Landau levels. This way the incompressibility of the IQHE wavefunctions is carried over to the FQHE wavefunctions. Following the example of the IQHE, attempts have been made to extend the connection between the incompressibility of the ground state and the W~algebra in the case of the FQHE [4,9—11].In Ref. [101, using the relation between the ~ = 1 and ii = 1/rn ground state, we found that there exists a second-quantized expression of a W~algebra which plays the role of a spectrum-generating algebra for Laughlin wavefunctions such that the Laughlin ground state satisfies a highest-weight condition. This provides a specific one-parameter family of W<~,representations, the parameter being related to the filling fraction. In this paper we extend these ideas to more general filling fractions v. We shall derive explicit representations, in terms of second-quantized fermion and vortex operators, of the W~algebra generators for the cases of multilayer incompressible states, the states proposed by Jam [81 to explain the hierarchical filling fractions and the ones related to these by particle—hole conjugation. All these ground states satisfy highest-weight conditions. We shall also derive second-quantized expressions for the corresponding order parameters. More generally incompressible states can be thought of as highest-weight states of the W~algebra [11]. Expressing the W~generators in terms of Fock operators is a useful technique for constructing representations and of course the highest-weight state. It is thus possible to go beyond the particular examples we have considered here and

obtain other candidate states for describing the quantum Hall effect.

2. W~algebras for IQHE The single-body hamiltonian for spin polarized (scalar) fermions confined in a twodimensional plane, in the presence of a uniform magnetic field B perpendicular to the planeis (in units h=c=e=l)

D. Karabali/Nuclear Physics B 428 [ES] (1994) 531—544

H=

533

(2.1)

.±..(Jfl2

where H1 =p~ At(x), i = x,y and V x A = —B. We define the following two sets of independent raising and lowering operators: —

a=~=(H~—iIP), b

=

~/~(X

+ iY),

at=~=(II5+iIV),

bt

=

~(X

—

iY),

[a,at]=l,

[b, btl

=

1,

(2.2)

where X, Y are the guiding center coordinates X=x—~fP,

Y=y+~[Ix,

(2.3)

[X,Y]=-~.

The two sets of operators a, at and b, bt are mutually commuting: [a, b] The hamiltonian can be written as H=w(ata+~),

—

Any state can be now decomposed in terms of the basis tensor product of Fock states defined by

btbjl)

=

ill),

[a, bt I

=

0.

(2.4)

w=B/M

and the angular momentum operator is given by J = btb

a~aIn)=nln),

=

ata.

In) ® 1) an

In, 1), which is a

(n’In) ~

(i’ll)

=

(2.5)

~

Let us also introduce the coherent basis representation for the b-oscillators bI~)=~C), where ~)__exp(bt)

(~I~’)=exp(~C’),

(lI~)=~=

(2.6)

0)andfd2~exp(—j~I2)IC)(~I=1, d2~=dRe~dIm~/ir.

The fermionic operator can be similarly decomposed as ~(x

t)

=

~

Cf(t)~~(x)

(27)

n,1=0

where (1, nI an (01C7 and ‘It’~(x)

w(n +

~) and angular momentum 1

—

(x~n,1) is the one-body wavefunction of energy n. The subset of states with fixed n form the nth

Landau level which is obviously infinitely degenerate with respect to the angular momentum. In a second-quantized language the operators Cf~satisfy the usual anticommutation relations

{C~’~,Cr’} = ~n,n’&,1’

(2.8)

which, as it is obvious from Eq. (2.7) corresponds to the standard quantization condition {~P(x,t),~P’t(x’, t)} = S(x — x’).

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D. Karabali/Nuclear Physics B 428 [ES] (1994) 531—544

So far we have not made a gauge choice. From now on we choose to work in the symmetric gauge A

(xIn,~)=

=

exp(—~IzI2+ ~),

~

where In,~) In) 0 now of the form =

~B(y,—x). Using the relation [12]

I~)and z

=

~/~(x

(2.9)

+ iy), we find that the fermion operator is

/~exp(_~IzI2)>i~Cf(t) (z —8~

=~Jexp(_~IzI2)~~(z,Z,t).

(2.10)

The quantization condition for the fermion operator constrained to be in the nth Landau level is ~

(2.11)

The r.h.s. of (2.11) is essentially the projection of the ö-function on the nth Landau

level. Let us now consider the case of noninteracting fermions that occupy the first n Landau levels. In the absence of an external potential the system is symmetric under independent unitary transformations in the space of C’s acting at each Landau level, written as:

C[(t)

=

UlkCk(t)

=

(lIuIk)Ck’(t),

I

=

0, 1

n

—

1.

(2.12)

(We mostly use the notation of Ref. [31.) An infinitesimal unitary transformation is

generated by a hermitian operator which we write as ~( h, bt ) ~ with the antinormal ordering symbol, where ~ is a real function when b and bt are replaced by z and Z, respectively. Using Eq. (2.12) we obtain the following infinitesimal transformation for the Ith Landau level fermionic operator: ô!P’(x, t) =i(xI

(b, bt)~II,k)C,~(t)

=if ~ =i(a~+~z,~z—o~)1!P’(x,t), 1=0,1 where ~ ~ indicates that the operators 8~+

f

(2.13)

act from the left. The transformations

(2.13) preserve the particle number at each Landau level dxôp’(x,t)=O, 1=0,1 n—i, where p’(x,t)

n— 1,

(2.14)

= ~P’t(x,t)~P’(x,t) is the Ith Landau level fermion density. The generators of the transformations (2.13) are given by

D. Karabali/Nuclear Physics B 428 IFS] (1994) 531—544

535

~

(2.15)

where d2z an dx dy B/2ir. These operators satisfy the commutation rules 1=0,1

n—i,

fi,~

2}=iB~~—~-(o~1a~2 —8~19~2),

(2.16)

where .j[} is the so-called Moyal bracket.The algebra (2.16) is the direct sum ofwen 1z~ copies of mutually commuting W~algebras [13,i4]. By choosing ~(z, Z) = z obtain min(s,l)

[p~~~p~I

,~

n1.

(1— n)!(s

—

n)!’~~

—

(s

k,r

l))~ (2.17)

where ~

(2.i8)

This can be written in terms of C’s as —

P1k—

‘c~ L. 1

219

(n+k)! Ct~’ C’ ~/n!(nH-k—l)! n+k-l

(

~

.

)

n=max(O,1—k)

Let us consider the action of these operators on the ground state

I!I’~,~~)o, where (2.20)

for N’ = nN electrons. It is clear that since the operators p[k decrease the angular momentum for / > k and increase the angular momentum for I < k they satisfy [41

p!kI!P~P=n)o= 0, P1kI’~’v’n)O

!P~~=~j),

if 1> k, if 1 ~ k,

I = 0,1 n 1, I =0,1,...,n— 1, —

(2.21)

where !I’~~,) correspond to higher angular-momentum excitations at the Ith level. The first line in (2.21) can be considered as the algebraic statement of incompressibility of the ii = n ground state. As far as excitations are concerned, if k 1 0(1), they correspond to edge excitations (for a review on edge excitations see Ref. [151). We expect that in the semiclassical limit the W~algebra in Eq. (2.16), reduces to the algebra of area-preserving diffeomorphisms [3,4,161 (see Eq. (2.14)). Upon restriction to the low-energy edge excitations it gives rise to a (U( 1))” Kac—Moody algebra describing n independent chiral bosons [15,17—191,one for each Landau level. —

D. Karabali/Nuclear Physics B 428 [ES] (1994) 531—544

536

3. W~algebras in the lowest Landau level In this section we consider cases where the electrons are confined in the lowest Landau level. First, we briefly review the derivation of W~algebras for v = i/rn Laughlin states and their relation to ii = 1 W~algebras, as given in Ref. [101, which will be crucial in constructing similar algebraic structures for quantum Hail fluids of general filling fraction. The main point in this derivation is the simple observation that the ii = 1/rn Laughlin ground-state wavefunction is related to the r’ = 1 wavefunction by attaching 2p (where rn = 2p + i) flux quanta to each electron (3.1)

~v=i/mJ(Zij)”~1~vrri~

i

Based on this we found in Ref. [101that the W~algebra generators for the v = 1/rn case can be written in terms of second-quantized fermion operators as (in this section we have dropped the superscript “0” used to denote the lowest Landau level fermion operators) W2~[fl=fd2z exp(_pzI2)~t(z)exp(2pa(Z))~(ô~,Z)~ x exp(—2pa(Z))ifr(Z), where a(Z)

=

f

d~z’exp(—Iz’12) ln(Z

—

Z’)~t(z’)~(~’),

(3.2)

(3.3)

exp(a ( Z)) is the quasihole operator since exP(a(z))I~)=f d2zi

d2zN exp

...

(_~Iz~I2)

ZN)[J(Z—Zi)Izl...zN),

(3.4)

where I~)=fd2zl...d2zNexP(_~IziI2)F(~i

ZN)IZi...ZN)

and

Izl..

.

ZN)

=

(zl)

..

.

(ZN)

0).

The appropriate insertion of the vortex (quasihole) operators in (3.2) reflects an underlying similarity transformation between the W~generators for the v = i and the = 1/rn case [10]. Using the fact that

D. Karabali/Nuclear Physics B 428 [ES] (1994) 531—544

~‘(Z’) exp(na(Z))

=

(~

—

Z’~exp(na(z))~1i(Z’),

~

(3.5)

where n is an integer and [a(Z), a(Z’)] satisfy a

537

=

0. We can show that the operators W2,,

W~algebra

strong

(3.6)

[W2~[~1],W2~[t2J] =W2~[{~1,~2}],

where (} is the Moyal defined earlier. states, They further 3 in bracket the space of Laughlin since play the role of a spectrumgenerating algebra (W2p)lkI!Pvl/m)O 0, if I > k, 2p 1k

v=1/m 0

v=l/m

=

1

‘

~

where I~v=i/m)is a higher angular-momentum state of the form

f

I~v=i~m) =

d~zi

d2ZN

...

exp

(—

Iz~I~)

~

XIJ(~f_~J)mp(~1,...,ZN)IZ1...ZN)

(3.8)

i

and P(~i ZN) is a homogeneous symmetric polynomial. The operators W 2,,, p = 0, i form a one-parameter family of representations for the W~algebra and the corresponding Laughlin ground state is the highest-weight state. This provides an algebraic statement of incompressibility for the Laughlin ground states. We would like now to extend these ideas to include other incompressible states corresponding to filling fractions ii ~ 1/rn. We still consider the case of lowest Landau level fermions. First we identify the W~algebra structure corresponding to the state with filling fraction v = i — i/rn. Using the idea of particle—hole conjugation [20], we can write the v = 1 i/rn ground state, in the thermodynamic limit and up to normalization factors, as —

~f

I~v=1_i/m)O

&z1

. ..

x fJ(z1 — i

f

exp

(—

Zj)~’(Zi)

. .

&ZM

~

IziI~)

.V’(~M)I!P~l)o.

(3.9)

3(Z)

Let us now introduce the operator / ~(z) = d~’ exp(—IZ’12) ln(z

—

z’)~(z1)~t(Z’).

(3.10)

The operator exp(/3(z)) satisfies equations similar to (3.5): We have used the infinite-plane geometry where the electrons are confined by the existence of an external confining potential [4,101, for example a central harmonic-oscillator potential. Such a potential can be thought of controlling the maximum single-particle angular momentum L available within a Landau level, which also determines the size of the droplet corresponding to the ground-state configuration. In the case of the v = 1/rn ground state, N and L are related by L = rn(N — 1).

538

D. Karabali/Nuclear Physics B 428 [ES](1994) 531—544

exp(n/3(z))çl’(Z’)=(Z

cf,t(Z’)exp(n/3(Z))=(z

—r9~’)~1’(Z’)exp(nf3(Z)),

Based on these relations and the fact that W

(3.11)

_Z’)flexp(n~(z))~,t(z/). ~

(z~)I ~Pv=i)o = 0, we find that the operators

2)~(Z)exp(2pfl(Z))*~(8~,z)~ exp(—IzI xexp(_2p,8(z))clrt(z)

2p[fl=fd2z

(3.12)

satisfy a strong W~algebra and the state (3.9) satisfies a highest-weight condition (W2p)1kI~yll/m)O

=

1

0,

>

k.

(3.13)

The operator W 2~is essentially the charge-conjugated version of W2,,. In order to obtain more general filling fractions, one may consider systems where several distinct species of electrons are involved (cases where the electrons have different spin or they are in separate layers). Trial ground-state wavefunctions of the form IPK(xl)

[J[J(2J V’)’~’~ fJfJ(~’ Z~’)”~’exp(_~ ~ I~I~) —

—

!

1=1 i

(3.14)

ii

have been suggested [21,22] as candidates for describing incompressible Hall states for an r-layer system, where K,, are odd integers so that the wavefunctions are antisymmetric under exchange of identical fermions. The corresponding filling fraction is given by z’ = ~ ~ ),j, where K is an r x r symmetric matrix. In order to identify the quantum W~algebra structure associated with (3.14) we introduce r independent lowest Landau level fermion operators such that {(Z,t),~1,tJ(Z’,t)}=&’1’exp(Zz’),

1i

r,

{~sJ(i,t),~(e’(z’,t)}={~rt’(Z,t),c1rtJ(Z’,t)}=0,

(3.i5)

and the corresponding quasihole operators exp(a’ (Z)), where ln(Z 2) — Z’)~t’(Z’)~’(~’) (3.16) = d~z’exp(—~z’~ A relation similar to (3.i) applies between (3.i4) and the u = I state. The main difference now is that vortices from different species are involved. We find that the corresponding W~generators have the form

f

W~[fl=fd2zexp(_IZI2)~t1(Z)exp (~~iia~(Z))~(8~,Z)~

xexp

—~R,ja~(Z)

ifr’(z),

(3.i7)

where I = i r and ~ = K I (I is the identity matrix). Using Eqs. (3.5) one can show that the operators W~give rise to r commuting copies of W~algebras. They —

D. Karabali/Nuclear Physics B 428 [ES](1994) 531—544

539

provide a family of representations of W~algebra parametrized by the symmetric matrix K. Further the corresponding ground state (3.14) satisfies highest-weight condition, namely

=0,

5

(W[~)1kI1I1’

if 1> k,

i

=

1,.. .,r.

(3.18)

Using the expressions (3.16) for the quasihole operators we can also construct secondquantized operators for the order parameters [23,24] of the r-layer system qlt

=

f

d2~exp(_IzI2)~lt(z)exp (~K!LaL(z-)).

(3.i9)

This provides a generalization of the r’ = i/rn order parameter operator [i 0]. Using Eqs. (3.5) and the commutativity of a”s we can show that qltqft

(3.20)

= _(_)KIJqJtqlt.

Thus for odd K

1,, the operators q/t are bosonic, as they should be in order to describe order parameters. For even K,~they are fermionic and so we consider the redefined operators

2)~lt(Z)~~(Z), (3.21) Q’f_q’texp(~i~s),

S=~A,fd2Z

where A, are integers such that K,,, + ~A,

—

exp(_IzI

A~= odd. S is a “cocycle” factor [24]

appropriately chosen to impose commutativity conditions for Q’t, namely [Qlt, QJt I

=

0,

I, J = 1

r.

(3.22)

The many-body ground state (3.14) is now created out of the vacuum by the action of these bosonic operators

I~) = ~

(~~~Q1t~1)

0),

(3.23)

As we noticed in Ref. [iO]the W~generators, Eq. (3.17), can be further written as W~[fl

=f

d2z exp(_IZI2)~tl(z)

x~ (a~

—

d2z’

~

~

~‘(z).

(3.24)

The term d2z~

plays the role of a gauge potential and it is similar to the one induced by the Chern— Simons interaction [25].Analogous first-quantized expressions were also derived in Ref. [9].

D. Karabali/Nuclear Physics B 428 [ES] (1994) 531—544

540

Further by applying particle—hole conjugation as in (3.9)—(3.13) we can derive second-quantized expressions for the W~generators for 1 = 1 i’ by charge-conjugating WI’?. The wavefunctions (3.i4) are also associated to the standard hierarchy scheme [6,7,22]. They are written in terms of electron coordinates and the coordinates of the quasiparticles formed at each level of the hierarchy. According to the hierarchy scenario, the quasiparticles at each level are the condensates of the quasiparticles of the previous level. Although it is clear from Eq. (3.24) that at a first-quantized level there is a W~structure [9], the precise second-quantized representation of W~generators is not clear. The main difficulty in this case is the assignment of creation and annihilation operators to the quasiparticles at each level of the hierarchy. Given this difficulty we are going to use in the next section, the alternative description of the FQHE, proposed by Jan, where everything is formulated in terms of electrons. Such a framework will be very convenient in displaying the W~algebra structure corresponding to the FQHE states, as described by Jam wavefunctions. —

4. W~algebras for Jam states One of the theories proposed to explain the observed fractions for the FQHE is the one suggested by Jan [81, in which there is a strong connection between IQHE and FQHE. As far as the hierarchical filling fractions t’ = n/ (2pn + 1) are concerned, the essential idea of this scenario is that the corresponding incompressible FQHE groundstate wavefunction ~ is related to the IQHE wavefunction ~ as =

[I(~

—

(4.1)

2”W°

Z)

i

This relation is a straightforward generalization of (3.1) and we can therefore use previous ideas in order to write down the second-quantized W~generators which would express the incompressibility of (4.1). The main difference now is that fermions in higher Landau levels are involved and some of the previously used operator relations

might change. In order to write down the corresponding Jam state we first notice that the I~~’~)ø state defined in Eq. (2.20) can be also written in terms of fermion operators as (up to normalization factors)

~

exp(_IZjhI2)Ll1(x~

x~t’(Z~,i[)~

x~,)

(4.2)

where zl’(x 1 XN) is the properly antisymmetrized wavefunction with 2) removed and l/’ (Z, Z) N-body is the fermion operator at the the exponential factorsEq. exp( — I zJ I Ith Landau level, (2.10). Antisymmetrization is done independently at each Landau

541

D. Karabali/Nuclear Physics B 428 [ES](1994) 531—544

level. In particular, given the single-body wavefunctions at the Ith Landau level (see Eqs. (2.7) and (2.iO)), 1—1 (x) ~P,

=

V ~—exp(—~IzI I

2 )i .1

(z

~—

8)’ -~Z~

2 anV___exp(_~IZI )x~”~

(4.3)

4

the term ~i’(x1

XN) is defined as N—I—I (x 1)

x,

XN)

=det

N—I—I

x,

(x2)

...

x1

1

I—I Xi

N—I—I

1

Xi (X2) Xi’(’2)

(Xl)

X,’(’I)

(XN)

... ...

x,~(‘N)

(4.4)

X,’(’N)

Given the fact that the Jan wavefunction for v = n/ (2pn + i) is related to the !i’~°.~ wavefunction as in Eq. (4.i), it is rather trivial to show that the corresponding Jan state can be written as (for N’ = nN electrons) ~

[(ft

1=0 I~~)o~f[J

2z~’exP(_IziuI2))

i=I

xli [ii’(x~

d

[J(z’

Zi)2P]

HH(z’ 1

i

x~,,)~,tI(Z(,~jl)~

z~)2P

i,,j

10).

(4.5)

In Section 2 we expressed the incompressibility property of the IQHE ground states in terms of the first line of Eq. (2.21). This equation was easily derived by using the expressions (2.i9) and (2.20) in terms of C’s. We can instead use the alternative expressions (2.18) and (4.2) in terms of the fermion operators. We then find that Eq. (2.21) is satisfied if zl’(x~ obeys the differential equation

4,)

—

8~,)~’(xi

XN)

=

0,

1> k,

VI.

(4.6)

Given the definition of ~l’,Eqs. (4.3) and (4.4), we can explicitly check that the above equation is satisfied. Eq. (4.6) is simply re-expressing the incompressibility condition of Eq. (2.21) in terms of many-body wavefunctions and is equivalent to it. Let us now consider the operators a’(fl, I = 0,1 n i, —

=

f

d2z’exp(—~z’~2) ln(Z

—

Z’)~t’(z’,~’)~’(z’,~’).

(4.7)

‘ For example, in the case of the lowest Landau level ~I°(xi XN) fl~<~(si — ~ N is related to the maximal single-particle angular momentum available within a Landau level as was mentioned in the footnote (3) on p. 537. For example by counting powers of ~ we find that 2p (N’ — 1) + (N — 1) = L, where L is the maximal single-particle angular momentum in the lowest Landau level.

D. Karabali/Nuclear Physics B 428 [ES] (1994) 531—544

542

Using the commutation relations (2.11) we can show that

[aI(Z),~tI(z’,Z’)]=ln(Z_8~,)~!,tl(Z’,Z’), (4.8) Based on these relations and the fact that wavefunctions in different Landau levels are orthogonal, Eq. (4.3), we find, after some tedious algebra, that the operators w

2/~=fd2z ~

(4.9)

~

where I = 0 n 1, give rise to n commuting copies of strong W~algebras. Further using Eqs. (4.5) and (4.6) we can verify that the corresponding Jam state for v = n/ (2pn + 1) satisfies the highest-weight condition —

(Y4~)1kI~I’~)o=0,

l>k,

1=0

n—i.

(4.10)

As before we can use particle—hole conjugation to derive the charge-conjugated W~,,

operators which express the incompressibility of the states with 1

=

1

—

ii.

5. Discussion In this paper we have given an explicit second-quantized representation of the W~algebra structure characterizing the incompressibility of a large class of fractional Hall states. The corresponding generators are expressed in terms of fermion and vortex operators. In general incompressible states can be thought of as highest-weight states of W~algebra [111. Expressing the W~generators in terms of Fock operators is useful in constructing representations and of course the highest-weight state. It is thus possible to obtain new candidate states for describing quantum Hall effect. For the specific examples we considered, the multilayer incompressible states and the Jan states, we find that there are r commuting copies of W~algebras, where r is the number of layers in a multilayer system or the number of Landau levels in the Jam framework [81. This W~algebra, besides characterizing the incompressibility of the ground state, provides a spectrum-generating algebra for excitations. We expect that in the semiclassical limit the W~algebra reduces to the algebra of area-preserving diffeomorphisms, which upon restriction to the low-energy edge excitations gives rise to a ((U( 1)) r Kac—Moody algebra describing r independent chiral bosons, one for each layer. So far there is the underlying assumption that the number of electrons at each layer is conserved. If this is relaxed by considering tunneling between different layers 6. the spectrum-generating algebra is enlarged by an embedding in a nonabelian structure 6

This was pointed to me by Bunji Sakita.

D. Karabali/Nuclear Physics B 428 [ES] (1994) 531—544

543

Further in constructing the W~generators for multilayer systems and the hierarchical states based on Jan’s construction we made use of the special mapping (4.1) between integer and fractional quantum Hall states. In Jan’s scenario there are more generalized incompressible states [81, for example states which can be written as products of integer quantum Hall states. A generic feature of the corresponding incompressible fluids is the existence of nonabelian edge excitations [261, which again suggests a nonabelian W~algebra structure. The precise fermionic representation of this is under investigation. An approach somewhat similar in spirit, in linking the incompressibility of fractional quantum Hall states to the existence of a W~algebra structure, has been advocated in Ref. [ii]. Using the representation theory of a centrally extended W~algebra (in terms of the one-dimensional edge degrees of freedom) the authors of Ref. [ii] were able to classify the hierarchical (abelian) quantum Hall states. Although the relation between the two-dimensional realization of the centerless W~algebra, denoted ~ and the centrally extended algebra in terms of one-dimensional chiral fields, denoted ~ is straghtforward in the v = 1 case, it is not so for other filling fractions. In the i.’ = 1 case, the normal ordering of ~ with respect to the x’ = 1 ground state gives the centrally extended W~ algebra [19]. It is interesting to see whether the two W~structures can be similarly related in the case of other filling fractions. Acknowledgement I would like to thank T.R. Govindarajan, V.P. Nair and B. Sakita for useful discussions. This work was supported by the U.S. Department of Energy under the contract number DE-FGO2-85ER4023 1.

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