Relating Jack wavefunctions to \textrm{WA}_{k-1} theories

Estienne, Benoît; Santachiara, Raoul

doi:10.1088/1751-8113/42/44/445209

Cited by 34 publications

(55 citation statements)

References 33 publications

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“…Since Laughlin's contribution, several types of trial wavefunctions have been proposed for the FQHE at various filling fractions. These include for example hierarchical states [3], composite fermion wavefunctions [4], or the important family of states given by conformal blocks introduced by Moore and Read (MR) [5] (states corresponding to Jack polynomials have also been proposed later [6], they can actually be included in the latter family [7]). In this paper we focus on these MR trial wavefunctions and their quantum entanglement properties.…”

Section: A Motivationmentioning

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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Section: A Motivationmentioning

confidence: 99%

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

2012

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“…In particular, for the fields Φ (1|2) and Φ (2|1) , this gives an order 2 differential equation which can be related to the Calogero-Sutherland Hamiltonian [19,20,21,22]. Consider the most generic conformal block containing the field Φ (1|2) , namely…”

Section: Degenerate Fields and Differential Equationsmentioning

confidence: 99%

“…For k = 2 these models coincide with the Virasoro models from the section 4, with g = (2+r)/3. When r is integer, the WA k−1 (k +1, k +r) theories correspond to Z (r) k parafermions considered in [21,22,41,42,44,47] . The results in [22] generalize straightforwardly to any value of r, not necessarily integer, in the same manner the results for the Ising CFT were extended to generic Virasoro models in the previous section.…”

Section: Wa K−1 Theoriesmentioning

confidence: 99%

“…These findings were extended to the conformal blocks of N primaries fields of a general WA k−1 theory [21]. The main motivation in [21] was the study of a class of trial many-body wavefunctions for fractional quantum Hall effect (FQHE) at bosonic filling fraction ν = k/r [21,22].…”

Section: Introductionmentioning

confidence: 97%

“…The main motivation in [21] was the study of a class of trial many-body wavefunctions for fractional quantum Hall effect (FQHE) at bosonic filling fraction ν = k/r [21,22]. These states can be constructed from the conformal blocks of a series of minimal models of WA k−1 algebras, the WA k−1 (k + 1, k + r) theories.…”

Section: Introductionmentioning

confidence: 99%

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Conformal blocks in Virasoro and W theories: Duality and the Calogero–Sutherland model

et al. 2012

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We study the properties of the conformal blocks of the conformal field theories with Virasoro or W-extended symmetry. When the conformal blocks contain only second-order degenerate fields, the conformal blocks obey second order differential equations and they can be interpreted as ground-state wave functions of a trigonometric Calogero-Sutherland Hamiltonian with nontrivial braiding properties. A generalized duality property relates the two types of second order degenerate fields. By studying this duality we found that the excited states of the CalogeroSutherland Hamiltonian are characterized by two partitions, or in the case of WA k−1 theories by k partitions. By extending the conformal field theories under consideration by a u(1) field, we find that we can put in correspondence the states in the Hilbert state of the extended CFT with the excited non-polynomial eigenstates of the Calogero-Sutherland Hamiltonian. When the action of the Calogero-Sutherland integrals of motion is translated on the Hilbert space, they become identical to the integrals of motion recently discovered by Alba, Fateev, Litvinov and Tarnopolsky in Liouville theory in the context of the AGT conjecture. Upon bosonisation, these integrals of motion can be expressed as a sum of two, or in general k, bosonic Calogero-Sutherland Hamiltonian coupled by an interaction term with a triangular structure. For special values of the coupling constant, the conformal blocks can be expressed in terms of Jack polynomials with pairing properties, and they give electron wave functions for special Fractional Quantum Hall states.

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