Research Diary

The description of the ground states given above is in terms of a coherent state rep-resentation for matrices. To make connections with the more traditional form of the wavefunctions, we need to find a map between the creation operatorsZ^†and the posi-tion space representaposi-tion as discussed in PartIII. We first briefly review the key points of the Abelian case, and then provide the generalization to theSU(p)matrix model.

p = 1 and the Laughlin Wavefunctions

Recall that at the formal level, there was a clear similarity between the ground state for p= 1theories,

|groundik =

^a1...aN(Z⁰ϕ)^†_a1(Zϕ)^†_a2. . .(Z^N−1ϕ)^†_aNk

|0i (12.10) and the Laughlin wavefunctions at filling fractionν= 1/m

ψ_m^Laughlin(z_a) = Y

a<b

(z_a−z_b)^me^−BP|za|2/4

=

^a1...aNz_a⁰₁z¹_a2. . . z_a^N−1

N

m

e^−BP|za|2/4.

In Part III we described two strategies which essentially integrate out ϕ and the off-diagonal elements of Z to transform to a wavefunction representation. Fork = 0, the wavefunctions coincided with the Slater determinant for a fully-filled Landau level

hz_a|groundi^k=0 =Y

a<b

(z_a−z_b)e^−BP|za|2/4. (12.11) The exponential factor is the usual factor arising form the normalization of coherent states. The single factor of the Vandermonde determinant, which is not obvious in (12.10) when k = 0, is a Jacobian that arises in the transformation from matrix-valued objects to their eigenvalues.

Meanwhile, fork ≥1, neither representation of the wavefunction coincided with the Laughlin state. Nonetheless, both had the property that

hza|groundi^k → ψ_k+1^Laughlin(za) as |za−zb| → ∞.

In other words, the wavefunctions that arise from the matrix model coincide with the Laughlin wavefunctions only at large distances. This ensures that the matrix model ground state has filling fraction

ν = 1 k+ 1 .

However, the matrix model state differs from the Laughlin wavefunction as particles approach to within a magnetic length.

We reiterate that although the matrix model and Laughlin states differ in detail, this is not a matter of concern. There is nothing privileged about the Laughlin wavefunc-tion: it is merely a representative of a universality class of states, characterized by their topological order – and moreover, it may be that there are other more mathematically convenient representatives. Indeed, in Part III, we explicitly saw two things: firstly, that the coherent state representation (12.10) reproduces the key aspects of Laughlin physics, and secondly, that for certain calculations it offers greater analytic control.

Wavefunctions for p ≥ 2

For the casep = 1 described above, all physical states have the same dependence on ϕ^†excitations; they differ only in theirZ^†excitations. This is the reason that noϕ vari-ables were needed when writing the wavefunctions. In contrast, whenp≥2, different physical states can have a different structure ofϕ^†_i excitations. These capture the way the state transforms under theSU(p)symmetry.

We repeat the procedure described above, moving from coherent state representation to wavefunction. Fork = 0the wavefunction knows nothing about the spin degrees of freedom. This means that thek = 0wavefunction is again given by (12.11), describing a fully-filled Landau level withν = 1.

However, forp ≥ 2and k > 1, we have a new ingredient. Apart from the Vander-monde determinant (12.11), each time that a power of a particle coordinatez_aappears in the wavefunction, it is accompanied by a spin degree of freedom,

σ_a ∈ {1, . . . , p}

wherea= 1, . . . , N labels the particle. Concretely, a term in the wavefunction contain-ing an operatorϕ^†_{i a}gives particleaa spin degree of freedom pointing in theidirection.

For example, whenk = 1, the a^th particle has a single spin degree of freedomσ_a. This reflects the fact that, as we saw earlier, each particle transforms in the fundamental representation ofSU(p). More generally, the internal state of each particle is determined byk independent spin labelsσ_a. As we will explain in some detail in Chapter13, this is to be interpreted as specifying thek^th symmetric representation underSU(p). (The symmetry simply arises from the fact thatϕ^†_{i a}ϕ^†_{j a}=ϕ^†_{j a}ϕ^†_{i a}.)

When N is divisible byp, the ground state wavefunction (12.8) is a an SU(p) spin-singlet. The states have filling fraction

ν= p

k+p (12.12)

and have the property that

hza|groundi^k → ψBW(za) as |za−zb| → ∞

whereψ_BW(z_a)are a class of non-Abelian wavefunctions constructed some time ago by Blok and Wen [183]. Like many non-Abelian quantum Hall states, the explicit descrip-tion of the wavefuncdescrip-tionsψBW(za)is straightforward, but somewhat fiddly. We devote the next chapter to a more detailed description of these quantum Hall states and their properties.

13 The Blok-Wen States

In this chapter, we describe the Blok-Wen wavefunctions in some detail. The original construction of [183] was in terms of conformal blocks of aSU(p)WZW model and we will revisit this approach in Chapter15. Here we provide an alternative, more down-to-earth construction of the states. We start with wavefunctions carrying spin under SU(2), moving on to the more generalSU(p)case in Section13.2.

13.1 Particles with SU (2) Spin

The simplest examples of wavefunctions describing particles with spin are due to Hal-perin [187]. We takeN particles, withN even, and split them into two groups ofN/2 particles, with positionsz_aandw_awhere each index now runs overa= 1, . . . , N/2. The (m, m, n)wavefunctions are

ψ(z, w) =

N/2

Y

a<b

(z_a−z_b)^m

N/2

Y

c<d

(w_c−w_d)^mY

a,d

(z_a−w_d)ⁿ (13.1)

where, as throughout this chapter, we will omit the overall exponential factor common to all wavefunctions. Counting the angular momentum of particles shows that these states have filling fraction

ν = 2 m+n.

The (m, m, n) states (13.1) are really shorthand for wavefunctions with spin. As we review below, they should be dressed with explicit spin wavefunctions. This will result in the Blok-Wen states. These are actually a slightly more general class of states than those that emerge from the matrix model. We will see that the matrix model gives states withn= 1andm=k+ 1.

Usually one thinks of the Halperin states as describing spin-¹₂particles, withz_aandw_a labelling the positions of those which are spin-up and spin-down respectively. With this interpretation the(n+1, n+1, n)states are spin-singlets. However, we will show that we can also view (13.1) as describing particles with spins > ¹₂. This is perhaps surprising as these particles have2s+ 1spin states and it is not obvious how to decompose these into two groups. We will see that, with this interpretation, the(m, m, n)states are spin-singlets whens = (m −n)/2. Matching to the matrix model parameters, this means s=k/2.

Spin

¹₂

The standard interpretation of (13.1) is as a wavefunction for spin-¹₂ particles. To dress the wavefunction with these spin states, it’s useful to change notation slightly and label the positions of allN particles asz_a. Each particle carries a further internal spin degree of freedom σ_a which takes values | ↑i or | ↓i. The (m, m, n) state for m > n is then written as

ψ(z, σ) =A

" _N Y

a<b

(z_a−z_b)ⁿ Y

a<bodd

(z_a−z_b)^m−n Y

c<deven

(z_c−z_d)^m−n | ↑↓↑↓. . .↑↓i

#

(13.2)

where A stands for antisymmetrization over all particles, exchanging both positions and spins. This wavefunction describes fermions formodd and bosons formeven.

It is well known that only the states withm−n = 1 are spin-singlets [188]. In this case, the wavefunction factorizes as

ψ_n+1,n+1,n(z, σ) =

N

Y

a<b

(z_a−z_b)ⁿΦ(z, σ).

This describes fermions for n even and bosons for n odd. Here the first factor takes the familiar Laughlin-Jastrow form, while the second factor is the Slater determinant of two fully filled Landau levels, one for the up spins and one for the down spins. The resulting wavefunction can be written as

Φ(z, σ) = Ah Y

a<bodd

(z_a−z_b) Y

c<deven

(z_c−z_d)| ↑¹i| ↓²i| ↑³i. . .| ↓^Nii

or, equivalently, as

Φ(z, σ) = _a1...aN(z_a1z_a2)⁰(z_a3z_a4)¹. . .(z_aN−1z_aN)^N/2−1

×h

| ↑^a1i| ↓^a2i| ↑^a3i| ↓^a4i. . .| ↑^aN−1i| ↓^aNii

. (13.3)

In particular, this latter expression makes it clear that the spins are paired in singlet states of the form| ↑a1i| ↓a2i − | ↓a1i| ↑a2i.

Spin 1

So far, we have just reproduced the usual story. Suppose now thatm=n+ 2. We claim that the following is a spin-singlet wavefunction for spin 1 particles,

ψ_n+2,n+2,n(z, σ) = Y

a<b

(z_a−z_b)ⁿP

Φ²(z, σ)

. (13.4)

This is a wavefunction for fermions whennis odd and bosons whennis even.

Our first task is to explain what this means. The factorΦ² includes two spin states for each particle. The tensor product of two spin 1/2 states gives2⊗2 = 1⊕3. The operator P projects onto the symmetric 3. (In the present case, this operation is not required as it is implemented automatically by the form ofΦ². However, we include it in our expression for clarity.) This means that we can interpret (13.4) as a quantum Hall state for spin 1 particles, with the map

| ↑i| ↑i=|1i , | ↓i| ↓i=|−1i , | ↑i| ↓i=| ↓i| ↑i=|0i. (13.5) We further claim that (13.4) is a spin-singlet. We will first motivate this by looking at the kinds of terms that arise. We will subsequently provide a proof in the course of rewriting the wavefunction in a more familiar form.

Consider two particles, labelled 1 and 2, each of which carries spin¹₂. The spin-singlet state is

|12i¹₂ =|↑¹i|↓²i − |↓¹i|↑²i

where the subscript1/2is there to remind us that this is the singlet built from two spin 1/2 particles. The simplest terms that occur in (13.4) are of the form|12i¹₂|12i¹₂. Using the map (13.5), we have

|12i¹₂|12i¹₂ = |1₁i|−1₂i+|−1₁i|1₂i −2|0₁i|0₂i

which is indeed the singlet formed from two spin 1 states. To highlight this, we write the above equation as

|12i¹₂|12i¹₂ =|12i¹.

The next kind of term that arises in (13.4) involves four different particles. It is the cyclic term|12i¹₂|23i¹₂|34i¹₂|41i¹₂. We can similarly expand this in terms of spin 1 states and again find that only combinations of singlet states appear:

|12i¹₂|23i¹₂|34i¹₂|41i¹₂ =|12i¹|34i¹− |13i¹|24i¹+|14i¹|23i¹.

The most general term in (13.4) has2n particles. This too can be written as the linear combinations ofnspin 1 singlet states. Rather than demonstrate this term by term, we will instead show that the wavefunction (13.4) has an alternative form written purely in terms of spin 1 singlets.

The Spin 1 Wavefunction as a Pfaffian

We will now show that the wavefunction (13.4) for spin 1 particles can be written as Φ²(z, σ) = Pf

|abi¹ za−zb

Y

a<b

(z_a−z_b) (13.6)

withPf(M_ab)the Pfaffian of the matrixM. This is a spin-singlet version of the Moore-Read state [16]. It is sensible because the spin 1 singlet |abi¹ is symmetric in the two spins, in contrast to|abi¹₂ which is antisymmetric.

It was noticed long ago [183, 190] that the(3,3,1)state is closely related to the Pfaf-fian state. In [183] the particles were spin-1 but projected onto the m = 0 spin com-ponent; in [190] the particles were taken to be spin 1/2 and the resulting state was not a spin-singlet. Our result (13.6) is clearly closely related to these earlier results, both of which are proven using the Cauchy identity. However, the proof of (13.6) requires more sophisticated machinery which appears not to have been available at the time of [183,190].

The Proof:

The projective Hilbert space associated to the two spins is a Bloch sphere CP¹. We parametrize this by the inhomogeneous coordinateζ. Formally, we then set | ↓^ai = 1 and|↑^ai=ζ_aand writeΦas the polynomial

Φ(z, ζ) = 1

2^N/2_a1...aNh

(z_a1z_a2)⁰. . .(z_aN−1z_aN)^N/2−1ih

(ζ_a1 −ζ_a2). . .(ζ_aN−1 −ζ_aN)i . This has the advantage that the right-hand-side can be viewed as the determinant of a N ×N matrix∆[z;ζ]with components given by

∆[z;ζ]_a,b=







z_a^b−1 1≤j ≤ ^N₂ ζaz_a^{b−1 N}₂ + 1 ≤j ≤2N

(13.7)

To show the result (13.6), we then need to prove the polynomial identity det²∆[z;ζ] = Pf^?

(ζ_a−ζ_b)² z_a−z_b

Y

a<b

(z_a−z_b).

In fact, this identity is a special case of a more general result proven in [191]. Theorem 2.4 of this paper shows (among other things) that two matrices∆[z;ζ]and∆[z;η], each defined by (13.7), obey the relation

det ∆[z;ζ] det ∆[z;η] = Pf

(ζ_b−ζ_a)(η_b−η_a) zb−za

det(z^b−1_a ).

Settingζa =ηayields the desired result.

Higher Spin

The generalization to higher spins is now obvious. We construct the wavefunction ψn+2s,n+2s,n(z, σ) =Y

a<b

(z_a−z_b)ⁿP

Φ^2s(z, σ)

(13.8)

wherePis there to remind us that the spin states for each particle are projected onto the fully symmetrized product. This means that this is a wavefunction for particles with spins. Once again, the final state is a spin-singlet. This follows from some trivial group theory. The infinitesimal action ofSU(2)on the tensor product of2sspin states is

T^α =

N

X

a=1

t^α_a⊗1⊗. . .⊗1+symmetric

wheret^α_a is the operator in the fundamental representation acting on thea^th particle, andα = 1,2,3labels the threesu(2) generators. BecauseP projects onto an irrep, we have

T^αP Φ²s

=P

"

X

a

t^α_aΦ⊗Φ⊗. . .⊗Φ

#

+symmetric .

But each of these terms vanishes because Φis itself a spin-singlet, which means that P

at^α_aΦ = 0. This ensures that (13.8) is indeed a spin-singlet.

Although (13.8) provides an explicit description of the state, it would be pleasing to find a simple expression purely in terms of the singlets|abi^s, analogous to the Pfaffian (13.6) fors = 1. We have not been able to do this; it may simply not be possible due to the entanglement structure between higher numbers of spins.

While the Halperin states (13.1) describe Abelian quantum Hall states, our spin-singlet states (13.6) and (13.8) with spins≥1are all non-Abelian quantum Hall states.

Indeed, it has long been known that dressing a quantum Hall state with spin degrees of freedom can change the universality class of the state. We will see in Chapter15that these states are associated toSU(2)_2sWZW models.

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