The fractional quantum Hall effect is the paradigmatic example of topologically ordered phases. One of its most fascinating aspects is the large variety of different topological orders that may be realized, in particular nonabelian ones. Here we analyze a class of nonabelian fractional quantum Hall model states which are generalizations of the abelian Haldane-Halperin hierarchy. We derive their topological properties and show that the quasiparticles obey nonabelian fusion rules of type su(q) k . For a subset of these states we are able to derive the conformal field theory description that makes the topological properties -in particular braiding -of the state manifest. The model states we study provide explicit wave functions for a large variety of interesting topological orders, which may be relevant for certain fractional quantum Hall states observed in the first excited Landau level.
We propose a Ginzburg-Landau theory for a large and important part of the abelian quantum Hall hierarchy, including the prominently observed Jain sequences. By a generalized ``flux attachment" construction we extend the Ginzburg-Landau-Chern-Simons composite boson theory to states obtained by both quasielectron and quasihole condensation, and express the corresponding wave functions as correlators in conformal field theories. This yields a precise identification of the relativistic scalar fields entering these correlators in terms of the original electron field.
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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