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

Abstract

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...

Non-Hermitian Skin Custers from Strong Interactions

“…When many-body interactions dominate singleparticle energetics, unexpected new physics often emerge [1][2][3][4][5][6][7][8][9][10][11][12]. A classic example is the appearance of anyonic quasiparticles in FQH systems, whose emergent statistics cannot be inferred from single-particle Chern topology alone [12][13][14][15][16][17][18].…”