Conformal field theory construction for non-Abelian hierarchy wave functions

Tournois, Yoran; Hermanns, Maria

doi:10.1103/physrevb.96.245107

Cited by 2 publications

(7 citation statements)

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“…In Ref. [38], the Hermanns hierarchy states were given a CFT description by using their close relation to the paired spin-singlet states. Referring to Refs.…”

Section: Application To Hermanns Hierarchy Statesmentioning

confidence: 99%

“…Referring to Refs. [36,38] for more details, the (bosonic) Hermanns hierarchy wave functions read Ψ Her;n ({z}) = 1 N S1,S2 Ψ CF;n (S 1 ) Ψ CF;n (S 2 ) ,…”

Section: Application To Hermanns Hierarchy Statesmentioning

confidence: 99%

“…To obtain the braid matrices for the M > 0 wave functions Ψ M (4,2) , we modify the electron and quasihole operators for the M = 0 case. First, we adopt the following representation [38] of the vectors v and q…”

Section: Wave Functions For General Mmentioning

confidence: 99%

“…These have been studied numerically in [37], showing that they are promising candidates for the second Landau level. The simplest (non-trivial) Hermanns hierarchy state was shown to be closely related to the non-Abelian spin-singlet state [36]; in [38] it was shown that the other Hermanns hierarchy states are similarly related to the paired spin-singlet states. Using this relation, we argue that the braiding properties of quasiholes in the Hermanns hierarchy states should be the same as those in the paired spin-singlet states.…”

Section: Introductionmentioning

confidence: 99%

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Braiding properties of paired spin-singlet and non-Abelian hierarchy states

Tournois

Ardonne

2020

J. Phys. A: Math. Theor.

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We study explicit model wave functions describing the fundamental quasiholes in a class of non-abelian fractional quantum Hall states. This class is a family of paired spin-singlet states with n ≥ 1 internal degrees of freedom. We determine the braid statistics of the quasiholes by determining the monodromy of the explicit quasihole wave functions, that is how they transform under exchanges of quasihole coordinates. The statistics is shown to be the same as that of the quasiholes in the Read-Rezayi states, up to a phase. We also discuss the application of this result to a class of non-abelian hierarchy wave functions. accumulated during the exchange as well as the explicit transformation -the monodromy -of the wave function [13]. For the Laughlin [14,15] as well as the Moore-Read case [16] (among other "Ising type" states, see also [17,18]) it was shown that the CFT description is one in which the statistics is given by the monodromy, with a trivial Berry phase. This was verified numerically in the Laughlin case [19], and in the Moore-Read and Z 3 Read-Rezayi [20] cases using the matrix product state formulation of [21]. In those cases, therefore, the braid statistics of quasiholes can be inferred from the manifest transformation of the quasihole wave function.In this paper, we study the braiding properties of quasiholes in a one-parameter family of non-abelian model wave functions denoted Ψ (n+1,2) , with n ≥ 1. Referred to as paired spinsinglet states, this family is a generalization of the spin polarized Moore-Read wave function (n = 1) and the non-abelian spin-singlet (NASS) [22] wave function (n = 2), to particles carrying n quantum numbers determining the charge and (pseudo-) spin. Such model wave functions have been considered in the context of rotating spin-1 bosons for n = 3 [23, 24], graphene [25], as well as fractional Chern insulators [26,27] with Chern number C > 1. Related wave functions were studied in [28,13,29] using a parton construction. Recently, progress was made on the Landau-Ginzburg theories describing these states [30].According to the 'Moore-Read conjecture' [5] (see [31] for a review) the CFT representation of the paired spin-singlet states should make the braiding properties manifest in the monodromy. By finding explicit quasihole wave functions, the braid matrices for the Moore-Read wave functions were found in [32], and those for the Read-Rezayi and NASS cases were determined in [33]. We study the manifest transformation properties of the paired spin-singlet states by obtaining explicit expressions for four-quasihole wave functions using conformal field theory techniques. This calculation relies on explicit four-point functions in certain Wess-Zumino-Witten (WZW) models which were obtained in Ref. [34], as well as the properties of the closely related parafermion CFTs [35] which are presented in Appendix B. We show that the braiding properties of the quasiholes for Ψ (n+1,2) are, up to a phase, the same as those of the quasiholes in the Z n+1 Read-Rezayi states [9], which reflects th...

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“…In Ref. [38], the Hermanns hierarchy states were given a CFT description by using their close relation to the paired spin-singlet states. Referring to Refs.…”

Section: Application To Hermanns Hierarchy Statesmentioning

confidence: 99%

“…Referring to Refs. [36,38] for more details, the (bosonic) Hermanns hierarchy wave functions read Ψ Her;n ({z}) = 1 N S1,S2 Ψ CF;n (S 1 ) Ψ CF;n (S 2 ) ,…”

Section: Application To Hermanns Hierarchy Statesmentioning

confidence: 99%

Section: Wave Functions For General Mmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Braiding properties of paired spin-singlet and non-Abelian hierarchy states

Tournois

Ardonne

2020

J. Phys. A: Math. Theor.

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“…The connection between CFT and quantum impurity models was greatly strengthened because the ground state wave functions correspond to CFT correlators. This idea was largely motivated by the practice of using CFT to construct fractional quantum Hall states [50][51][52][53][54][55]. From the perspective of tensor networks, the CFT approach generates infinite-dimensional matrix product state [56][57][58][59] for quantum impurity models.…”

Section: Contentsmentioning

confidence: 99%

Quantum impurity models from conformal field theory

2019

J. High Energ. Phys.

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The coupling between localized magnetic moments and itinerant electrons presents a plethora of interesting physics. The low-energy physics of some quantum impurity systems can be described using conformal field theory (CFT). In this paper, the connection between quantum impurity models and CFT is further strengthened as we construct a class of exactly solvable models with ground states given by CFT correlators. The method developed here is completely analytical and can be applied to fermions with an arbitrary number of colors and multiple impurities. Numerical calculations are performed to characterize certain aspects of our models for which we do not have analytical results.

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Conformal field theory construction for non-Abelian hierarchy wave functions

Cited by 2 publications

References 58 publications

Braiding properties of paired spin-singlet and non-Abelian hierarchy states

Braiding properties of paired spin-singlet and non-Abelian hierarchy states

Quantum impurity models from conformal field theory

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