IASSNS-HEP-94/93

W∞

ALGEBRAS AND INCOMPRESSIBILITY IN THE QUANTUM HALL EFFECT *

arXiv:hep-th/9411082v1 11 Nov 1994

DIMITRA KARABALI † Institute for Advanced Study Princeton, NJ 08540, USA ABSTRACT We discuss how a large class of incompressible quantum Hall states can be characterized as highest weight states of different representations of the W∞ algebra. Second quantized expressions of the W∞ generators are explicitly derived in the cases of multilayer Hall states, the states proposed by Jain to explain the hierarchical filling fractions and the ones related by particle-hole conjugation.

The study of planar, charged nonrelativistic fermions in a strong magnetic field finds important applications in condensed matter problems, such as the quantum Hall effect (QHE)1,2. Such systems have a further, less obvious connection to (1+1)dimensional problems, such as the c = 1 string model3,4 . In this talk I will outline the emergence of an infinite dimensional algebraic structure, the W∞ algebra, and its role in the integer and fractional QHE (IQHE and FQHE), based mainly on work presented in refs. 6 and 7. 1. W∞ algebras for IQHE As is well known, the spectrum of planar, nonrelativistic fermions in the presence of a transverse, uniform magnetic field B , consists of infinitely degenerate levels, the so-called Landau levels. The energy gap between adjacent Landau levels is ω = B/M , where M is the fermionic mass (¯h = c = e = 1). For large B we can consider the fermions restricted to the lowest Landau level (LLL). We further assume that B is sufficiently strong to align all electronic spins, so we can neglect the spin degree of ~ x) = B (y, −x), the LLL condition can be written freedom. In the symmetric gauge A(~ 2 as (∂z + 21 z¯) Ψ(~x) = 0

where z =

q

B 2 (x

+ iy), z¯ =

q

B 2 (x

− iy).

(1.1)

The LLL wavefunctions are of the form

Ψ(~x) = f (¯ z ) exp − 12 |z|2

(1.2)

* Talk presented in Mt. Sorak Summer School, S. Korea, June 27-July 2, 1994 and XXth Int. Colloquium on Group Theoretical Methods in Physics, Osaka, Japan, July 4- July 9, 1994, to appear in proceedings. † This work was supported in part by the DOE grants DE-FG02-85ER40231 and DE-FG02-90ER40542

where f (¯z ) is a polynomial in z¯. Upto an exponential factor which can be absorbed in the definition of the measure, the LLL wavefunctions depend only on the antiholomorphic variables z¯, while the holomorphic ones become essentially the canonical momenta (z → ∂z¯ after taking into account the appropriate ordering)8 . This reflects the fact that the original coordinate space of electrons constrained in the LLL becomes the phase space of a one-dimensional system3 . In a second quantized language the LLL condition can be promoted to an operator equation and the corresponding LLL fermion operator has the form Ψ(~x, t) =

r

∞

z¯l B − 1 |z|2 X Cl (t) √ ≡ e 2 2π l! l=0

r

B − 1 |z|2 z , t) e 2 ψ(¯ 2π

(1.3)

where Cl ’s are operators which annihilate fermions of angular momentum l and satisfy the usual anticommutation relations {Cl† , Cl′ } = δl,l′ . In the absence of an external potential and interactions there is an infinite degeneracy with respect to angular momentum, so the system is symmetric under independent unitary transformations in the space of C ’s: Cl (t) = ulk Ck (t) = hl|u|kiCk (t)

(1.4)

The corresponding infinitesimal transformation for the LLL fermion operator is z δΨI (~x, t) = i‡ξ(∂z¯ + , z¯)‡ Ψ(~x, t) 2

(1.5)

where ξ(z, z¯) is a real function and ‡ ‡ indicates that the operators ∂z¯ + z2 act from the left. RThese transformations preserve the LLL condition and the particle number, i.e., d~xδρ(~x, t) = 0, where ρ(~x, t) = Ψ† (~x, t)Ψ(~x, t) is the LLL fermion density. The corresponding generators are given by4−7 ρ[ξ] ≡

where d2 z ≡

B 2π dxdy

Z

2

d2 ze−|z| ψ † (z) ‡ξ(∂z¯, z¯)‡ ψ(¯ z)

(1.6)

, and they satisfy an infinite dimensional algebra given by i ρ[{{ξ1 , ξ2 }}] B ∞ X (−)n n {{ξ1 , ξ2 }} = iB (∂z ξ1 ∂z¯n ξ2 − ∂z¯n ξ1 ∂zn ξ2 ) n! n=1

[ ρ[ ξ1 ], ρ[ ξ2 ] ] =

(1.7)

is the so-called Moyal bracket. This is the W∞ algebra9 . In the absence of an external potential and interactions, the W∞ algebra corresponds to a symmetry of the problem. In realistic situations the electrons are confined in a finite region. In the infinite plane geometry this can be achieved by introducing an external confining potential V (~x), for example a central harmonic oscillator potential. Such a term spoils the infinite degeneracy with respect to angular momentum by assigning higher energy to higher angular momentum states, R but the resulting Hamiltonian, H = d2 xV (~x)ρ(~x, t), is a member of the W∞ algebra4,5. {{}}

In this case the W∞ algebra does not correspond to a symmetry anymore; instead it provides a spectrum generating algebra. In order to illuminate this role of the W∞ algebra we consider the action of the W∞ generators on the many-body ground state of electrons filling up the first Landau level, i.e. ν = 1 (where ν is the filling fraction, the ratio between the number of electrons and the degeneracy of the Landau level). The ν = 1 ground state is † |Ψν=1 i0 = C0† ...CN −1 |0i

(1.8)

for N electrons. This forms an incompressible, circular droplet of radius ∼ N/B and uniform density ρ = B/2π. Compression corresponds to lowering the angular momentum, but since all available states in the LLL are occupied (ν = 1), that would require an electron to jump to a higher Landau level. However, for a large magnetic field, this is not energetically allowed due to the big energy gap. On the other hand, deformations that would result in transitions to states with higher angular momentum are allowed and cost some energy due to the confining potential. These excitations can be generated by the action of W∞ generators on the ground state. Inspection of the W∞ generators in the basis ξ(z, z¯) = z l z¯k shows that the operators ρlk decrease the angular momentum for l > k and increase the angular momentum R 2 for l < k, where ρlk ≡ d2 ze−|z| ψ† (z)(∂z¯)l (¯z )k ψ(¯z ). Thus we find that5−7 p

ρlk |Ψν=1 i0 = 0

ρlk |Ψν=1 i0 = |Ψ >

if l > k if l ≤ k

(1.9)

where |Ψi corresponds to excitations of higher angular momentum. The first line in Eq. (1.9) provides an algebraic statement for the incompressibility of the ground state, by characterizing the ground state as the highest weight state of the W∞ algebra5 . In fact we shall show that even for more general filling fractions the incompressibility of the corresponding ground states can be algebraically expressed as the ground state being the highest weight state of a W∞ algebra. Before generalizing to other filling fractions I would like to briefly mention the relation between the W∞ algebra and the algebra of area preserving diffeomorphisms. This can be easily understood in the case of LLL fermions. W∞ transformations were earlier introduced as unitary transformations. Their classical analogue, therefore, are canonical transformations which preserve the area element of the phase space. Given that the LLL phase space corresponds to the original two-dimensional coordinate space of the system3 , as mentioned earlier, these canonical transformations are the area preserving diffeomorphisms. In terms of excitations one can understand the relation between the two algebraic structures by considering the edge excitations10, i.e., k − l ∼ O(1). These low energy excitations correspond to boundary fluctuations of the ground state droplet and can be described by one-dimensional chiral boson (fermion) fields. In terms of these, and upon restriction to the edge excitations, one can show that the original W∞ algebra reduces to the algebra of area-preserving diffeomorphisms4,11 .

The previous analysis can be easily extended to the case where the fermions fill up the first n Landau levels, ν = n. A simple analysis of the Hamiltonian and the corresponding single-body energy and angular momentum wavefunctions shows that the fermion operator can be now expanded as Ψ(~x, t) =

r

n

∞

B − 1 |z|2 X X I I (z − ∂z¯)I z¯l √ ≡ i Cl (t) √ e 2 2π I! l! I=0 l=0

r

n

B − 1 |z|2 X I ψ (z, z¯, t) e 2 2π

(1.10)

I=0

where I indicates the Landau level and is related to the energy and l − I measures ′ the angular momentum. The operators ClI satisfy {Cl†I , ClI′ } = δI,I ′ δl,l′ . There are now n mutually commuting W∞ generators corresponding to independent unitary transformations acting at each Landau level. They are of the form I

ρ [ξ] ≡

Z

2

d2 ze−|z| ψ I† (z, z¯) ‡ξ(∂z¯, z¯ − ∂z )‡ ψ I (z, z¯)

I = 0, 1, .., n

(1.11)

Their action on the ν = n ground state |Ψν=n i0 , where (for N ′ = nN electrons) |Ψν=n i0 =

n−1 Y

†I (C0†I ...CN −1 )|0i,

(1.12)

I=0

is quite similar to Eq. (1.9). We find that5,7 ρIlk |ΨIν=n i0 = 0

ρIlk |ΨIν=n i0

= |ΨI >

if l > k if l ≤ k

I = 0, 1, ..., n − 1

I = 0, 1, ..., n − 1

(1.13)

2

where ρIlk ≡ d2 ze−|z| ψI† (z, z¯)(∂z¯)l (¯z − ∂z )k ψ(z, z¯) and |ΨI i corresponds to excitations of higher angular momentum at the I -th level. As in the ν = 1 case, the incompressibility of the ν = n ground state is algebraically expressed as the ground state being the highest weight state of a W∞ algebra. In the next section we shall seek the generalization of this to the fractional quantum Hall states. R

2. W∞ algebras for FQHE The main experimental feature of both the IQHE and FQHE, namely the appearance of a series of plateaux where the Hall conductivity is quantized and proportional to the filling fraction ν , while the longitudinal conductivity vanishes, is attributed to the existence of a gap, which gives rise to an incompressible ground state. For IQHE, the essential physics can be well understood in terms of noninteracting fermions. The energy gap is the cyclotron energy separating adjacent Landau levels and for a large magnetic field the Coulomb interaction can be neglected. The noninteracting picture is nonapplicable in the case of the FQHE, where the Coulomb interaction among electrons is important in producing an energy gap. Much of our understanding of the FQHE relies on successful trial wavefunctions, such as the Laughlin wavefunctions12,13 and the ones proposed more recently by Jain14 . In both cases they correspond to incompressible configurations of uniform density ρ = νB/2π.

Earlier, in the case of the IQHE, we have seen that the incompressibility of the ground state is closely related to the existence of the W∞ algebra structure. This relation can be extended to the FQHE6,7,14,15 . I shall first describe the derivation of W∞ algebras for ν = 1/m Laughlin states and their relation to ν = 1 W∞ algebras which will be crucial in constructing similar algebraic structures for quantum Hall fluids of general filling fraction. Here we consider the electrons confined in the lowest Landau level. The main point in this derivation is the simple observation that the ν = 1/m Laughlin ground state wavefunction is related to the ν = 1 wavefunction by attaching 2p (where m = 2p + 1) flux quanta to each electron Ψ0ν=1/m =

Y (¯ zi − z¯j )2p Ψ0ν=1

(2.1)

i k

(2.5)

This provides an algebraic statement of incompressibility for the Laughlin ground states. The operators W2p form a one-parameter family of W∞ representations. These ideas can be now extended7 to include other incompressible states corresponding to filling fractions ν 6= 1/m. In general all these states can be characterized as the highest weight states of different realizations of a W∞ algebra. I shall briefly present three such cases and explicitly write down the second quantized expressions of the corresponding W∞ generators 1) ν = 1 − 1/m states 2) multilayer systems and 3) Jain states. 2.1. ν = 1 − 1/m states Using the idea of particle-hole conjugation17 , we can write the ν = 1 − m1 ground state, in the thermodynamic limit and up to normalization factors, as |Ψν=1−1/m i0 ∼

Z

d2 z1 ...d2 zM e−

P

|zi |2

Y i

W∞

ALGEBRAS AND INCOMPRESSIBILITY IN THE QUANTUM HALL EFFECT *

arXiv:hep-th/9411082v1 11 Nov 1994

DIMITRA KARABALI † Institute for Advanced Study Princeton, NJ 08540, USA ABSTRACT We discuss how a large class of incompressible quantum Hall states can be characterized as highest weight states of different representations of the W∞ algebra. Second quantized expressions of the W∞ generators are explicitly derived in the cases of multilayer Hall states, the states proposed by Jain to explain the hierarchical filling fractions and the ones related by particle-hole conjugation.

The study of planar, charged nonrelativistic fermions in a strong magnetic field finds important applications in condensed matter problems, such as the quantum Hall effect (QHE)1,2. Such systems have a further, less obvious connection to (1+1)dimensional problems, such as the c = 1 string model3,4 . In this talk I will outline the emergence of an infinite dimensional algebraic structure, the W∞ algebra, and its role in the integer and fractional QHE (IQHE and FQHE), based mainly on work presented in refs. 6 and 7. 1. W∞ algebras for IQHE As is well known, the spectrum of planar, nonrelativistic fermions in the presence of a transverse, uniform magnetic field B , consists of infinitely degenerate levels, the so-called Landau levels. The energy gap between adjacent Landau levels is ω = B/M , where M is the fermionic mass (¯h = c = e = 1). For large B we can consider the fermions restricted to the lowest Landau level (LLL). We further assume that B is sufficiently strong to align all electronic spins, so we can neglect the spin degree of ~ x) = B (y, −x), the LLL condition can be written freedom. In the symmetric gauge A(~ 2 as (∂z + 21 z¯) Ψ(~x) = 0

where z =

q

B 2 (x

+ iy), z¯ =

q

B 2 (x

− iy).

(1.1)

The LLL wavefunctions are of the form

Ψ(~x) = f (¯ z ) exp − 12 |z|2

(1.2)

* Talk presented in Mt. Sorak Summer School, S. Korea, June 27-July 2, 1994 and XXth Int. Colloquium on Group Theoretical Methods in Physics, Osaka, Japan, July 4- July 9, 1994, to appear in proceedings. † This work was supported in part by the DOE grants DE-FG02-85ER40231 and DE-FG02-90ER40542

where f (¯z ) is a polynomial in z¯. Upto an exponential factor which can be absorbed in the definition of the measure, the LLL wavefunctions depend only on the antiholomorphic variables z¯, while the holomorphic ones become essentially the canonical momenta (z → ∂z¯ after taking into account the appropriate ordering)8 . This reflects the fact that the original coordinate space of electrons constrained in the LLL becomes the phase space of a one-dimensional system3 . In a second quantized language the LLL condition can be promoted to an operator equation and the corresponding LLL fermion operator has the form Ψ(~x, t) =

r

∞

z¯l B − 1 |z|2 X Cl (t) √ ≡ e 2 2π l! l=0

r

B − 1 |z|2 z , t) e 2 ψ(¯ 2π

(1.3)

where Cl ’s are operators which annihilate fermions of angular momentum l and satisfy the usual anticommutation relations {Cl† , Cl′ } = δl,l′ . In the absence of an external potential and interactions there is an infinite degeneracy with respect to angular momentum, so the system is symmetric under independent unitary transformations in the space of C ’s: Cl (t) = ulk Ck (t) = hl|u|kiCk (t)

(1.4)

The corresponding infinitesimal transformation for the LLL fermion operator is z δΨI (~x, t) = i‡ξ(∂z¯ + , z¯)‡ Ψ(~x, t) 2

(1.5)

where ξ(z, z¯) is a real function and ‡ ‡ indicates that the operators ∂z¯ + z2 act from the left. RThese transformations preserve the LLL condition and the particle number, i.e., d~xδρ(~x, t) = 0, where ρ(~x, t) = Ψ† (~x, t)Ψ(~x, t) is the LLL fermion density. The corresponding generators are given by4−7 ρ[ξ] ≡

where d2 z ≡

B 2π dxdy

Z

2

d2 ze−|z| ψ † (z) ‡ξ(∂z¯, z¯)‡ ψ(¯ z)

(1.6)

, and they satisfy an infinite dimensional algebra given by i ρ[{{ξ1 , ξ2 }}] B ∞ X (−)n n {{ξ1 , ξ2 }} = iB (∂z ξ1 ∂z¯n ξ2 − ∂z¯n ξ1 ∂zn ξ2 ) n! n=1

[ ρ[ ξ1 ], ρ[ ξ2 ] ] =

(1.7)

is the so-called Moyal bracket. This is the W∞ algebra9 . In the absence of an external potential and interactions, the W∞ algebra corresponds to a symmetry of the problem. In realistic situations the electrons are confined in a finite region. In the infinite plane geometry this can be achieved by introducing an external confining potential V (~x), for example a central harmonic oscillator potential. Such a term spoils the infinite degeneracy with respect to angular momentum by assigning higher energy to higher angular momentum states, R but the resulting Hamiltonian, H = d2 xV (~x)ρ(~x, t), is a member of the W∞ algebra4,5. {{}}

In this case the W∞ algebra does not correspond to a symmetry anymore; instead it provides a spectrum generating algebra. In order to illuminate this role of the W∞ algebra we consider the action of the W∞ generators on the many-body ground state of electrons filling up the first Landau level, i.e. ν = 1 (where ν is the filling fraction, the ratio between the number of electrons and the degeneracy of the Landau level). The ν = 1 ground state is † |Ψν=1 i0 = C0† ...CN −1 |0i

(1.8)

for N electrons. This forms an incompressible, circular droplet of radius ∼ N/B and uniform density ρ = B/2π. Compression corresponds to lowering the angular momentum, but since all available states in the LLL are occupied (ν = 1), that would require an electron to jump to a higher Landau level. However, for a large magnetic field, this is not energetically allowed due to the big energy gap. On the other hand, deformations that would result in transitions to states with higher angular momentum are allowed and cost some energy due to the confining potential. These excitations can be generated by the action of W∞ generators on the ground state. Inspection of the W∞ generators in the basis ξ(z, z¯) = z l z¯k shows that the operators ρlk decrease the angular momentum for l > k and increase the angular momentum R 2 for l < k, where ρlk ≡ d2 ze−|z| ψ† (z)(∂z¯)l (¯z )k ψ(¯z ). Thus we find that5−7 p

ρlk |Ψν=1 i0 = 0

ρlk |Ψν=1 i0 = |Ψ >

if l > k if l ≤ k

(1.9)

where |Ψi corresponds to excitations of higher angular momentum. The first line in Eq. (1.9) provides an algebraic statement for the incompressibility of the ground state, by characterizing the ground state as the highest weight state of the W∞ algebra5 . In fact we shall show that even for more general filling fractions the incompressibility of the corresponding ground states can be algebraically expressed as the ground state being the highest weight state of a W∞ algebra. Before generalizing to other filling fractions I would like to briefly mention the relation between the W∞ algebra and the algebra of area preserving diffeomorphisms. This can be easily understood in the case of LLL fermions. W∞ transformations were earlier introduced as unitary transformations. Their classical analogue, therefore, are canonical transformations which preserve the area element of the phase space. Given that the LLL phase space corresponds to the original two-dimensional coordinate space of the system3 , as mentioned earlier, these canonical transformations are the area preserving diffeomorphisms. In terms of excitations one can understand the relation between the two algebraic structures by considering the edge excitations10, i.e., k − l ∼ O(1). These low energy excitations correspond to boundary fluctuations of the ground state droplet and can be described by one-dimensional chiral boson (fermion) fields. In terms of these, and upon restriction to the edge excitations, one can show that the original W∞ algebra reduces to the algebra of area-preserving diffeomorphisms4,11 .

The previous analysis can be easily extended to the case where the fermions fill up the first n Landau levels, ν = n. A simple analysis of the Hamiltonian and the corresponding single-body energy and angular momentum wavefunctions shows that the fermion operator can be now expanded as Ψ(~x, t) =

r

n

∞

B − 1 |z|2 X X I I (z − ∂z¯)I z¯l √ ≡ i Cl (t) √ e 2 2π I! l! I=0 l=0

r

n

B − 1 |z|2 X I ψ (z, z¯, t) e 2 2π

(1.10)

I=0

where I indicates the Landau level and is related to the energy and l − I measures ′ the angular momentum. The operators ClI satisfy {Cl†I , ClI′ } = δI,I ′ δl,l′ . There are now n mutually commuting W∞ generators corresponding to independent unitary transformations acting at each Landau level. They are of the form I

ρ [ξ] ≡

Z

2

d2 ze−|z| ψ I† (z, z¯) ‡ξ(∂z¯, z¯ − ∂z )‡ ψ I (z, z¯)

I = 0, 1, .., n

(1.11)

Their action on the ν = n ground state |Ψν=n i0 , where (for N ′ = nN electrons) |Ψν=n i0 =

n−1 Y

†I (C0†I ...CN −1 )|0i,

(1.12)

I=0

is quite similar to Eq. (1.9). We find that5,7 ρIlk |ΨIν=n i0 = 0

ρIlk |ΨIν=n i0

= |ΨI >

if l > k if l ≤ k

I = 0, 1, ..., n − 1

I = 0, 1, ..., n − 1

(1.13)

2

where ρIlk ≡ d2 ze−|z| ψI† (z, z¯)(∂z¯)l (¯z − ∂z )k ψ(z, z¯) and |ΨI i corresponds to excitations of higher angular momentum at the I -th level. As in the ν = 1 case, the incompressibility of the ν = n ground state is algebraically expressed as the ground state being the highest weight state of a W∞ algebra. In the next section we shall seek the generalization of this to the fractional quantum Hall states. R

2. W∞ algebras for FQHE The main experimental feature of both the IQHE and FQHE, namely the appearance of a series of plateaux where the Hall conductivity is quantized and proportional to the filling fraction ν , while the longitudinal conductivity vanishes, is attributed to the existence of a gap, which gives rise to an incompressible ground state. For IQHE, the essential physics can be well understood in terms of noninteracting fermions. The energy gap is the cyclotron energy separating adjacent Landau levels and for a large magnetic field the Coulomb interaction can be neglected. The noninteracting picture is nonapplicable in the case of the FQHE, where the Coulomb interaction among electrons is important in producing an energy gap. Much of our understanding of the FQHE relies on successful trial wavefunctions, such as the Laughlin wavefunctions12,13 and the ones proposed more recently by Jain14 . In both cases they correspond to incompressible configurations of uniform density ρ = νB/2π.

Earlier, in the case of the IQHE, we have seen that the incompressibility of the ground state is closely related to the existence of the W∞ algebra structure. This relation can be extended to the FQHE6,7,14,15 . I shall first describe the derivation of W∞ algebras for ν = 1/m Laughlin states and their relation to ν = 1 W∞ algebras which will be crucial in constructing similar algebraic structures for quantum Hall fluids of general filling fraction. Here we consider the electrons confined in the lowest Landau level. The main point in this derivation is the simple observation that the ν = 1/m Laughlin ground state wavefunction is related to the ν = 1 wavefunction by attaching 2p (where m = 2p + 1) flux quanta to each electron Ψ0ν=1/m =

Y (¯ zi − z¯j )2p Ψ0ν=1

(2.1)

i k

(2.5)

This provides an algebraic statement of incompressibility for the Laughlin ground states. The operators W2p form a one-parameter family of W∞ representations. These ideas can be now extended7 to include other incompressible states corresponding to filling fractions ν 6= 1/m. In general all these states can be characterized as the highest weight states of different realizations of a W∞ algebra. I shall briefly present three such cases and explicitly write down the second quantized expressions of the corresponding W∞ generators 1) ν = 1 − 1/m states 2) multilayer systems and 3) Jain states. 2.1. ν = 1 − 1/m states Using the idea of particle-hole conjugation17 , we can write the ν = 1 − m1 ground state, in the thermodynamic limit and up to normalization factors, as |Ψν=1−1/m i0 ∼

Z

d2 z1 ...d2 zM e−

P

|zi |2

Y i