Clustering properties, Jack polynomials and unitary conformal field theories

Estienne, Benoît; Regnault, Nicolas; Santachiara, Raoul

doi:10.1016/j.nuclphysb.2009.09.002

Cited by 21 publications

(23 citation statements)

References 46 publications

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“…k0 r−1 k]. The Jack wave functions can be related to WA k−1 conformal field theories and can be classified in terms of symmetric polynomial categories [19][20][21][22][23][24][25] . Moreover, the quasiparticle excitations of the trial state systems can also be written as coherent state superpositions of Jacks 26,27 .…”

Section: Bosonic Statesmentioning

confidence: 99%

Decomposition of fractional quantum Hall model states: Product rule symmetries and approximations

et al. 2011

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We provide a detailed description of a new product rule structure of the monomial (Slater) expansion coefficients of bosonic (fermionic) fractional quantum Hall (FQH) states derived recently, which we now extend to spin-singlet states. We show that the Haldane-Rezayi spin-singlet state can be obtained without exact diagonalization through a differential equation method that we conjecture to be generic to other FQH model states. The product rule symmetries allow us to build approximations of FQH states that exhibit increasing overlap with the exact state (as a function of system size) even though our approximation omits more than half of the Hilbert space. We show that the product rule is valid for any FQH state which can be written as an expectation value of parafermionic operators.

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Section: Bosonic Statesmentioning

confidence: 99%

Decomposition of fractional quantum Hall model states: Product rule symmetries and approximations

et al. 2011

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“…However, this program has been observed recently to break down for non-unitary CFTs. While large sets of bulk trial wavefunctions can be written as correlation functions in a non-unitary CFT [5][6][7][8][9], the bulk and edge CFT can no longer match. Indeed the edge CFT is a low-energy effective theory describing the physical edge states, and as any proper quantum field theory it has to be unitary [10].…”

mentioning

confidence: 99%

Correlation Lengths and Topological Entanglement Entropies of Unitary and Nonunitary Fractional Quantum Hall Wave Functions

2015

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Using the newly developed Matrix Product State (MPS) formalism for non-abelian Fractional Quantum Hall (FQH) states, we address the question of whether a FQH trial wave function written as a correlation function in a non-unitary Conformal Field Theory (CFT) can describe the bulk of a gapped FQH phase. We show that the non-unitary Gaffnian state exhibits clear signatures of a pathological behavior. As a benchmark we compute the correlation length of Moore-Read state and find it to be finite in the thermodynamic limit. By contrast, the Gaffnian state has infinite correlation length in (at least) the non-Abelian sector, and is therefore gapless. We also compute the topological entanglement entropy of several non-abelian states with and without quasiholes. For the first time in FQH the results are in excellent agreement in all topological sectors with the CFT prediction for unitary states. For the non-unitary Gaffnian state in finite size systems, the topological entanglement entropy seems to behave like that of the Composite Fermion Jain state at equal filling.PACS numbers: 03.67. Mn, 05.30.Pr, Our understanding of the Fractional Quantum Hall (FQH) effect has benefited substantially from the use of model wavefunctions [1][2][3][4]. These wavefunctions, although not ground-states of realistic hamiltonians, are nonetheless supposed to capture the universal behavior of the state such as quasiparticle charge, statistics, braiding in the gapped bulk, as well as electron and quasihole exponents on the gapless edge. In a seminal paper [4] Moore and Read proposed to use conformal blocks, i.e. correlation functions in a Conformal Field Theory (CFT), as a building block to write down bulk model wavefunctions for the ground state and its quasihole excitations. This construction relies on a number of conjectures, the most important being that such a model bulk wavefunction describes a gapped topological state. Another assumption is that the universality class of the fractional quantum Hall state -most notably the braiding and fusion properties of the excitations -can be read off directly from the bulk CFT. Finally the bulk-edge correspondence is usually assumed. It states that (most) properties of the physical gapless edge states should be described by the same CFT that was used to build the bulk wavefunctions. Despite the nontrivial nature of these conjectures, there is a large body of (mostly exact diagonalization) evidence that supports the Moore-Read construction.However, this program has been observed recently to break down for non-unitary CFTs. While large sets of bulk trial wavefunctions can be written as correlation functions in a non-unitary CFT [5][6][7][8][9], the bulk and edge CFT can no longer match. Indeed the edge CFT is a low-energy effective theory describing the physical edge states, and as any proper quantum field theory it has to be unitary [10]. In that case one of the aforementioned hypothesis has to break down : either the edge CFT is different from the one used to write the bulk statewhich was shown to ...

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“…We can wonder what would happen if we started from some Laughlin ν = 1/(2n) state (even denominator fractions are imposed by the bosonic nature of the particles). For two decoupled layers, the symmetrization of two Laughlin ν = 1/4 is related to the Hafffnian state 24 while the symmetrization of two Laughlin ν = 1/6 is connected to the N = 1 superconformal field theories 60 . For this latest case, a local model Hamiltonian that reproduces both the ground state and the quasihole excitations is generally not known 61 (Note that the symmetrization of three Laughlin ν = 1/4 states has a connection to the S 3 conformal field theories 62 , but it also suffers from the absence of a local model Hamiltonian for the quasihole excitations).…”

Section: Discussionmentioning

confidence: 99%

Projective construction of theZkRead-Rezayi fractional quantum Hall states and their excitations on the torus geometry

et al. 2015

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Multilayer fractional quantum Hall wave functions can be used to construct the non-Abelian states of the Z k Read-Rezayi series upon symmetrization over the layer index. Unfortunately, this construction does not yield the complete set of Z k ground states on the torus. We develop an alternative projective construction of Z k Read-Rezayi states that complements the existing one. On the multi-layer torus geometry, our construction consists of introducing twisted boundary conditions connecting the layers before performing the symmetrization. We give a comprehensive account of this construction for bosonic states, and numerically show that the full ground state and quasihole manifolds are recovered for all computationally accessible system sizes. Furthermore, we analyze the neutral excitation modes above the Moore-Read on the torus through an extensive exact diagonalization study. We show numerically that our construction can be used to obtain excellent approximations to these modes. Finally, we extend the new symmetrization scheme to the plane and sphere geometries.

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Clustering properties, Jack polynomials and unitary conformal field theories

Cited by 21 publications

References 46 publications

Decomposition of fractional quantum Hall model states: Product rule symmetries and approximations

Decomposition of fractional quantum Hall model states: Product rule symmetries and approximations

Correlation Lengths and Topological Entanglement Entropies of Unitary and Nonunitary Fractional Quantum Hall Wave Functions

Projective construction of theZkRead-Rezayi fractional quantum Hall states and their excitations on the torus geometry

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