Vertex Operator Algebra Arising from the Minimal Series M(3,p) and Monomial Basis

Feigin, Boris; Jimbo, Michio; Miwa, Tetsuji

doi:10.1007/978-1-4612-0087-1_8

Cited by 17 publications

(37 citation statements)

References 5 publications

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“…[22] for k = 2 is insufficient to answer the question, and contains undefined notation. Fortunately, the full results and notation were given by Feigin and coworkers [30], and using their results one can show that as N → ∞ (with N even) the counting of the (2, R) admissible partitions and of the corresponding polynomials exactly coincides with the fermionic form of the character χ R+2,3 1,1…”

Section: Some Recently-proposed Trial Statesmentioning

confidence: 93%

Conformal invariance of chiral edge theories

2009

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The low-energy effective quantum field theory of the edge excitations of a fully-gapped bulk topological phase corresponding to a local Hamiltonian must be local and unitary. Here it is shown that whenever all the edge excitations propagate in the same direction with the same velocity, it is a conformal field theory. In particular, this is the case in the quantum Hall effect for model "special Hamiltonians", for which the ground state, quasihole, and edge excitations can be found exactly as zero-energy eigenstates, provided the spectrum in the interior of the system is fully gapped. In addition, other conserved quantities in the bulk, such as particle number and spin, lead to affine Lie algebra symmetries in the edge theory. Applying the arguments to some trial wavefunctions related to non-unitary conformal field theories, it is argued that the Gaffnian state and an infinite number of others cannot describe a gapped topological phase because the numbers of edge excitations do not match any unitary conformal field theory.

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Section: Some Recently-proposed Trial Statesmentioning

confidence: 93%

Conformal invariance of chiral edge theories

2009

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“…which we use as a wavefunction when it is non-zero (these wavefunctions appeared previously in [38,39] For the Laughlin wavefunction, we have a(z) = e iϕ(z)/ √ ν . There is no statistics sector.…”

Section: B Edge Excitationsmentioning

confidence: 99%

“…The negative modes J n (n < 0) do not need to be considered, since they annihilate the out vacuum N |. In general, the positive mode J n of the U (1) current J(z) produces the corresponding sum of powers 1 √ ν z n i [38,39]. These wavefunctions are the wellknown (neutral) edge states for the Laughlin wavefunction [19].…”

Section: B Edge Excitationsmentioning

confidence: 99%

Edge-state inner products and real-space entanglement spectrum of trial quantum Hall states

2012

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We consider the trial wavefunctions for the fractional quantum Hall effect that are given by conformal blocks, and construct their associated edge excited states in full generality. The inner products between these edge states are computed in the thermodynamic limit, assuming generalized screening (i.e. short-range correlations only) inside the quantum Hall droplet, and using the language of boundary conformal field theory (boundary CFT). These inner products take universal values in this limit: they are equal to the corresponding inner products in the bulk 2d chiral CFT which underlies the trial wavefunction. This is a bulk/edge correspondence; it shows the equality between equal-time correlators along the edge and the correlators of the bulk CFT up to a Wick rotation.This approach is then used to analyze the entanglement spectrum of the ground state obtained with a bipartition A ∪ B in real-space. Starting from our universal result for inner products in the thermodynamic limit, we tackle corrections to scaling using standard field-theoretic and renormalization group arguments. We prove that generalized screening implies that the entanglement Hamiltonian HE = − log ρA is isospectral to an operator that is local along the cut between A and B. We also show that a similar analysis can be carried out for particle partition. We discuss the close analogy between the formalism of trial wavefunctions given by conformal blocks and Tensor Product States, for which results analogous to ours have appeared recently. Finally, the edge theory and entanglement spectrum of px ± ipy paired superfluids are treated in a similar fashion in the appendix.

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“…The generating function (equivalent of Eq. (1)) for the number of (2, r) partitions λ i − λ i+2 ≥ r has been obtained using the theory of jagged partitions [20] [21]. After some algebra, we find a ground-state energy r N 2…”

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confidence: 99%

Properties of Non-Abelian Fractional Quantum Hall States at Fillingν=k/r

2008

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We compute the physical properties of non-Abelian Fractional Quantum Hall (FQH) states described by Jack polynomials at general filling ν = k r . For r = 2, these states are identical to the Z k Read-Rezayi parafermions, whereas for r > 2 they represent new FQH states. The r = k + 1 states, multiplied by a Vandermonde determinant, are a non-Abelian alternative construction of states at fermionic filling 2/5, 3/7, 4/9.... We obtain the thermal Hall coefficient, the quantum dimensions, the electron scaling exponent, and show that the non-Abelian quasihole has a well-defined propagator falling off with the distance. The clustering properties of the Jack polynomials, provide a strong indication that the states with r > 2 can be obtained as correlators of fields of non-unitary conformal field theories, but the CFT-FQH connection fails when invoked to compute physical properties such as thermal Hall coefficient or, more importantly, the quasihole propagator. The quasihole wavefuntion, when written as a coherent state representation of Jack polynomials, has an identical structure for all non-Abelian states at filling ν = k r .

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Vertex Operator Algebra Arising from the Minimal Series M(3,p) and Monomial Basis

Cited by 17 publications

References 5 publications

Conformal invariance of chiral edge theories

Conformal invariance of chiral edge theories

Edge-state inner products and real-space entanglement spectrum of trial quantum Hall states

Properties of Non-Abelian Fractional Quantum Hall States at Fillingν=k/r

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