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

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 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.


Introduction
The discovery of the fractional quantum Hall effect [1] has led to the prediction of fractionally charged quasiparticle excitations [2], quasiholes and quasielectrons, obeying fractional statistics [3,4]. For most quantum Hall states the quasiparticle statistics is expected to be Abelian, i.e. the many-quasiparticle wave function picks up a fractional phase under the exchange of the quasiparticle coordinates. However, certain states are thought to host non-Abelian excitations [5], in which case the many-quasiparticle wave function has multiple components which transform according to a unitary braid matrix U ij when quasiparticles at positions w i and w j are exchanged.
One way in which the theoretical understanding of the fractional quantum Hall effect has progressed is by proposing trial wave functions for ground states and excited states, with the goal of capturing topological properties such as the fractional charges and braiding statistics Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. of the quasiparticle excitations. Examples of Abelian trial wave functions are the Laughlin wave function [2], the hierarchy wave functions [6,7] and the composite fermion (CF) wave functions [8]; examples of non-Abelian wave functions include the Moore-Read [5] and, more generally, the Read-Rezayi [9] series. The Moore-Read state, or rather its particle-hole conjugate, the 'anti-pfaffian' [10,11], are leading candidates for describing the plateau at ν = 5 2 based on numerical studies, see for instance [12]. A powerful tool in proposing and analyzing such model wave functions has been conformal field theory (CFT). Various trial wave functions can be expressed as conformal blocks and it was conjectured that this description makes the topological properties, in particular the braiding properties, of the wave function manifest [5]. The braiding statistics of quasiholes is represented by the Berry holonomy which has contributions from Berry phase 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 [13,28,29] 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 [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 the rank-level duality between their CFT descriptions.
The paired spin-singlet states are also closely related to a set of non-Abelian hierarchy wave functions proposed in [36] based on a picture of successive condensation of non-Abelian quasiparticles. This set of trial wave functions, which we refer to as Hermanns hierarchy wave functions, can be thought of as bilayer composite fermion wave functions where one performs a symmetrization (or antisymmetrization) over the layer index. 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.
The paper is organized as follows. In section 2, we briefly review the connection between trial wave functions in the fractional quantum Hall effect and conformal field theory. In section 3 we discuss the paired spin-singlet states in detail and introduce 'master formulas' that relate two representations of the paired spin-singlet states, which allows us to find explicit wave functions for four quasiholes. In section 4, we present the calculation of the braiding properties in the paired spin-singlet state Ψ (4,2) after which we present the calculation for a general paired spin-singlet state in section 5. Finally, in section 6, we comment on the relation between the paired spin-singlet states and the Hermanns hierarchy states. In the appendices, we provide details on the WZW CFTs, the associated parafermion CFTs and the consequences of rank-level duality for the braid matrices studied in this paper.

Model wave functions and conformal field theory
We consider model wave functions for fractional quantum Hall states of the form where M 0 and the magnetic length l B has been set to 1. In equation (1), Φ is a symmetric holomorphic function of the particle coordinates z j = x j + iy j , obeying certain vanishing conditions. In this paper we only consider paired states, for which Φ (z 1 , . . . , z N ) zi=zj = 0, but Φ (z 1 , . . . , z N ) zi=zj=z k = 0 for any distinct z i , z j , z k . The wave function Ψ M=0 is bosonic and has the same pairing property, while the simplest fermionic wave function corresponds to is the polynomial of lowest degree with the pairing property as described above. To simplify the discussion we set M = 0 from here on, denoting Ψ M=0 by Ψ. We will consider the general wave functions with M > 0 at a later stage. We also suppress the Gaussian factors. In the following we make extensive use of the connection between CFT and the fractional quantum Hall effect [5,39], by means of which trial wave functions are expressed as (chiral) conformal blocks in a certain CFT. In particular, the wave function is represented by a vacuum expectation value of (radially ordered) operators in the CFT which describe the constituent (quasi)particles. The trial wave function for the ground state reads Here the operator V represents an electron 2 and the operator O bg is a background charge operator which is needed to ensure a nonzero result: it can chosen in such a way that the Gaussian factors are reproduced [5]. By a simple change of the operators V, the wave function Ψ M for general M can also be represented in this way. Similarly, model wave functions for quasiholes can be obtained by including appropriate operators H at positions w = w x + iw y . Not all CFTs give appropriate trial wave functions: there are certain conditions to be satisfied [39,40], most notably the existence of appropriate operators to represent the electrons and quasiholes. For the quasihole operators H a requirement is that of mutual locality with respect to the electrons, which means that the braiding of quasiholes and electrons is trivial. 2 Although the particles described by V are bosons for M = 0, we refer to them as electrons.
This requirement implies that the operator product expansion (OPE) of the fields H and V is of the form where is a non-negative integer and H denotes the field resulting from the fusion of H with V. This condition places a constraint on the possible types of quasiholes. A well-known example of a model wave function-that is, a trial wave function with a known parent Hamiltonian-is the Moore-Read wave function [5], which we denote by Ψ (2,2) . Here, the notation Ψ (n+1,k) refers to a wave function with n internal degrees of freedom and a k-clustering property which we refer to as a pairing property for k = 2. For M = 0, Ψ (n+1,k) has an su (n + 1) k symmetry, while for M > 0, this is broken down to su (n) k . The relevant CFT for the Moore-Read model wave function is the product of the Ising CFT and the u (1) chiral boson CFT, where the correlator of the boson field φ is given by φ (z) φ (w) = − log (z − w). The electron and quasihole operators read where the Majorana fermion ψ and the 'spin field' σ are the primary fields of the Ising CFT and the vertex operator e iαφ is a primary field of the free boson CFT. Writing {z} for the collection z 1 , . . . , z N , the model wave function for the ground state is Because of the fusion rule σ × σ = 1 + ψ of the spin field σ, the many-quasihole wave 'function' has different components labeled by a fusion channel index p , i.e. the specific way in which the spin fields fuse to the identity. The wave function with 2m quasiholes has 2 m−1 components [32], or (chiral) conformal blocks, given by 1 4 ij .

(6)
Here w ij = w i − w j , the w i are assumed to be radially ordered, i.e. |w 1 | < ... < |w 2m | and we have adopted the notation with X denoting a string of Majorana fermions ψ. The explicit wave functions involving arbitrarily many quasiholes and electrons for the Moore-Read wave function were found in [16,41]. The implementation of the (four-) quasi-hole states as topological protected q-bits was studied [42,43].
The conformal blocks Ψ ( p) (2,2) transform non-trivially amongst themselves when the quasiparticle coordinates are exchanged. That is, exchanging w i and w j and analytically continuing the wave function, Ψ ( p) a unitary braid matrix. The collection of braid matrices, which were found in [32], forms a unitary representation of the braid group on 2m strands.

Model wave functions
The Moore-Read wave function Ψ (2,2) is the simplest example of a paired 'spin-singlet' state, denoted Ψ (n+1,2) , which are studied in this paper. The pairing property of Ψ (2,2) may be verified by inspection of equation (5), or by considering the OPE between the electron operators, using ψ (z) ψ (z ) ∼ 1/(z − z ). In fact, the Moore-Read wave function is the unique, densest zero-energy eigenstate of a certain three-body Hamiltonian [44,45]. Consequently, the Moore-Read wave function may also be obtained by symmetrizing two bosonic Laughlin wave functions as observed by Cappelli et al [46]. Denoting the Laughlin wave functions by Ψ (2,1) , as equation (9) obeys the same vanishing properties and has the same degree. Here, the coordinates {z} are partitioned into two 'layers' S 1 , S 2 of equal size 3 , and the sum is over all inequivalent partitions. We consider two partitions to be equivalent if they are related by a layer permutation S 1 ↔ S 2 . The paired spin-singlet state Ψ (n+1,2) can be viewed as a generalization of the Moore-Read wave function to particles having n internal quantum numbers. These wave functions have an underlying su (n + 1) 2 symmetry. Additionally, they are also non-zero when two particles are at the same position, and vanish when three particles are brought together (quadratically when the three particles are identical, linearly otherwise).
Generalizing equation (4) there are n electron operators V α which factor into a 'parafermion' [47] ψ α generalizing the Majorana fermion ψ, and a vertex operator of n independent chiral bosons φ = (φ 1 , . . . , φ n ): Here α = 1, . . . , n and v α is a vector: to avoid clutter in the notation, we do not write vector-superscripts. The factor √ 2 in the vertex operator is included so that the vectors v α are simple in terms of the roots of su (n + 1), see appendix B.3. In particular, they should obey v α · v β = 1 + δ αβ , so that the OPE of two electron operators reads in accordance with the pairing property of Ψ (n+1,2) . The paired spin-singlet states are the unique densest, zero-energy eigenstates of the same three-body Hamiltonian that has the Moore-Read state as its ground state (it is understood that the Hamiltonian treats all particle types equally). Generalizing equation (9), Ψ (n+1,2) can be obtained by symmetrizing the following generalized Halperin wave functions Here N α denotes the number of particles with index α, with coordinates z α i . Hence, the model wave functions Ψ (n+1,2) can be expressed as In the symmetrized representation, each layer S a = {S 1 a , S 2 a , . . . , S n a } with a = 1, 2 contains half the coordinates with a given index α.
The relevant CFT that describes the paired spin-singlet states is the su (n + 1) 2 WZW CFT (see [48] for an introduction). These CFTs can be written as a product of a parafermion theory su (n + 1) 2 /u (1) n and n free boson CFTs [47], which leads to the expression equations (10) and (4) for n = 1 in which case the parafermion CFT is the Ising CFT. The more general parafermion CFTs are described in appendix B. The electron operators V α are currents of the su (n + 1) 2 WZW model, as described in appendix A. The fundamental quasiholes are represented by primary fields H µ of the WZW model, where µ = 0, 1, . . . , n labels the different types: a quasihole with a pseudospin index (µ = 1, . . . , n) or a 'spinless' quasihole (µ = 0). These operators read where σ µ is a spin field of the parafermion theory. In order that the operators H µ have the correct OPEs with the electron operators, equation (3) with = 0, the inner products have to satisfy The quasihole wave function can be expressed as a correlator of operators H µ and V α , or in terms of two copies of Ψ (n+1,1) with quasiholes inserted in the layers S 1 , S 2 . In particular, the operator H µ is equivalent to the insertion a quasihole in one of the layers, which becomes a non-Abelian quasihole after the symmetrization procedure. We are mainly interested in four-quasihole wave functions. For the simplest case, where all quasiholes carry the index µ = 1, the two conformal blocks ( p = 0, 1) read Here, X denotes a string of parafermions X = α,i ψ α (z α i ). To express the conformal blocks in a symmetrized representation, we first define the wave functions where S 1 a denotes the coordinates with index µ = 1 in layer a. Only two of the three possible symmetrized wave functions Ψ 12;34 , Ψ 13;24 , and Ψ 14;23 are linearly independent, as was seen in [32] for the case n = 1. In particular, the wave functions Ψ ab;cd are related by (18) in terms of the following anharmonic ratio We note that the convention for the anharmonic ratio used here differs from the one used in for instance [33,34], but agrees with the convention in [32]. The reason for picking the current convention is that the w i are properly radially ordered, namely, after an appropriate conformal transformation, we have The Ψ ab;cd obey the same vanishing properties as the conformal blocks equation (16), when either electrons or electrons and quasiholes are taken to the same point. As a result, and by virtue of equation (18), each conformal block may be expanded in the basis Ψ 12;34 , Ψ 13;24 as (20) where the expansion coefficients A ( p) , B ( p) depend only on the w i and ensure the correct behavior when quasiholes are brought to the same position.

Master formulas and braiding
Following [33], relations like equation (20) which relate the conformal blocks to symmetrized wave functions open up the possibility of finding explicit expressions for two-quasihole and four-quasihole wave functions. In turn, this allows us to study the braiding properties of the quasiholes by finding the monodromies of the four-quasihole wave functions, i.e. the transformation properties of the conformal blocks under exchanges of quasihole positions. Such equations are therefore referred to as 'master formulas'.
In the following, we obtain various master formulas for different types of quasiholes. By taking limits of the master formulas, letting the electron positions coincide with each other or with the quasihole positions, the expansion coefficients A ( p) , B ( p) are determined [33]. In particular, we employ operator product expansions of the parafermions ψ α and spin fields σ µ , found in appendix B, to reduce the correlator to a four-point function of spin fields. The latter can be determined using the results obtained in [34], where closely related four point functions of primary fields in the su (n + 1) 2 WZW CFT were found explicitly by solving the Knizhnik-Zamolodchikov equation. The spin field four point functions are presented in appendix C.
Using the solutions of the coefficients A ( p) , B ( p) in terms of the w i we find the manifest transformation of the conformal block Ψ Here, U (n+1,2) ij p p is the 2 × 2 braid matrix corresponding to the given transformation. In particular, we determine the matrices corresponding to the transformations in terms of the anharmonic ratio x defined in equation (19). The braid matrices for the more general wave functions Ψ M (see equation (1)) are obtained afterwards and differ from the bosonic (M = 0) braid matrices by a global phase only. This analysis hinges on the explicit form of the four point functions of spin fields. Unfortunately the explicit form of correlators involving more than four spin fields is much harder to obtain. Therefore, although the conformal blocks and symmetrized wave functions can be written down, the expansion coefficients A ( p) , B ( p) , . . . can not be determined easily in the same way. Additionally we assume that the braiding statistics is determined by the manifest transformation of the wave function alone (holonomy = monodromy), i.e. that there is no additional contribution to the statistics coming from the Berry phase.

The paired su (4) 2 spin-singlet state
The (bosonic) model wave function Ψ (4,2) has the two equivalent representations An explicit representation of the vectors v α and q µ that satisfy the correct inner products in this case are given in equation (46). The number of electrons of each pseudospin type must be even-this ensures the parafermions fuse to the identity, or that the sets of coordinates can be partitioned into two equal sized sets in the symmetrized representation. The prefactor N may be fixed by taking pairs of parafermions to the same point, i.e. letting z α 2j → z α 2j−1 for j = 1, . . . , N α /2 and α = 1, 2, 3. Using the OPEs of the parafermions (see appendix B.3) and taking the aforementioned limit of equation (22), one finds N = 2 1 2 (N1+N2+N3)−1 . There are four quasihole operators: a spinless quasihole H 0 , as well spinful quasiholes H 1 , H 2 , H 3 . The simplest two-quasihole wave function is obtained by inserting two identical quasiholes H µ ; the wave function reads Here S µ a denotes all coordinates with pseudospin index µ, where we adopt the convention S 0 a = S a . Indeed, the spinless quasiholes 'couple' to all types of electrons. In equation (24), the additional factor A depending on w 1 , w 2 is fixed by requiring that both sides are equal in the limit w 2 → w 1 . Using the OPEs of the spin fields, this yields A = w 1 8 12 . In the following sections, we present the relevant master formulas for the four-quasihole wave functions. We insert two pairs of identical quasiholes for simplicity, which leaves the cases where we consider two orderings of quasiholes corresponding to the insertions σ µ σ µ σ µ σ µ X and σ µ σ µ σ µ σ µ X .
For the bosonic wave function the different quasihole types are related by a symmetry so that their braid matrices are identical. In particular, it is enough to consider the above cases for (I) µ = 1 and (II) µ = 1, µ = 2. The symmetry relating quasihole types is broken for M > 0, and the braid matrices differ by an overall (global) phase from the bosonic braid matrices. The braid matrices for the wave functions with M > 0 are presented in section 4.6.

The case m 1 = 4
We consider the quasihole wave function with m 1 = 4 and take N 1 = 2 and N 2 = N 3 = 6. We label the coordinates z 1 , z 2 , z 3 , . . . , z 14 , omitting the pseudospin index. Then, the master formula reads: We then take the following three limits of equation (25): To obtain expressions for A ( p) and B ( p) , only two limits are strictly necessary. The third limit is taken to fix the phases of the four-point function of spin fields: this is explained in more detail in appendix C. These limits reduce the full correlators to four-point functions, namely Taking the limits of the symmetrized wave functions Ψ 12;34 and Ψ 13;24 as well, one finds the equations  34 . (30) By using the four point functions of spin fields, which are determined in appendix C, we find In equation (31), We consider the four quasihole wave function with m 1 = m 2 = 2, taking N 1 = N 2 = 2 and N 3 = 4. As in the previous section, we label the coordinates z 1 , z 2 , . . . , z 8 , omitting the pseudospin indices. We consider two possible orderings of the four operators, corresponding to Note that in this case, there are only two natural ways of dividing the quasiholes over the two layers in the Cappelli representation. For the first case, the symmetrized wave functions are Ψ 13;24 and Ψ 14;23 . For the second case, they are Ψ 12;34 and Ψ 14;23 .

First case.
For the first case, we write We then consider the limits which give the equations As in the previous section, we use the four point functions of the spin fields to find In this case, we take the limits which give the solutions

The case m 0 = 4
We take m 0 = 4 and N 1 = N 2 = N 3 = 2. The conformal blocks have the expressions We take the limits which reduce the four point functions to . This yields the same equations as in equation (31), i.e. the braid matrices for spinless quasiholes in the bosonic case are be the same as the braid matrices for the spinful quasiholes. This result was to be expected: it follows from the su (4) symmetry which is unbroken in the case M = 0.

Braiding transformations
By keeping track of how the coefficients A ( p) , B ( p) and the symmetrized quasihole wave functions Ψ ab;cd transform, we find the manifest transformation of the conformal blocks Ψ ( p) (4,2) . The transformations of the Ψ ab;cd are obtained straightforwardly, using equation (18). The transformations of the coefficients A ( p) , B ( p) follow from the transformations of the anharmonic ratios and the transformations of the functions F p i which are presented in appendix D. The combined transformation yields where (U (4,2) ij ) p p is the 2 × 2 braid matrix corresponding to the transformation w i w j . We now present the matrices corresponding to the transformations w 1 w 2 , w 1 w 3 and All of the matrices are found for the case m 1 = 4, while the case m 1 = m 2 = 2 yields only the matrices U (4,2) 12 and U (4,2) 13 , one for each ordering. As was mentioned before, the symmetry between the quasiholes in the bosonic case means these are the correct braid matrices in the general cases m µ = 4 and m µ = m µ = 2 as well.
The matrices U ij constitute a two-dimensional representation of the braid group, as they are unitary and satisfy the 'Yang-Baxter' relation U 13 = U 12 U 23 U 12 = U 23 U 12 U 23 . Moreover, the braid matrices are closely related to the braid matrices associated with quasiholes in the k = 4 Read-Rezayi wave function [9], which we write Ψ (2,4) . The braid matrices for the Read-Rezayi states are given in [33,49]. The close relation is due to the rank-level duality between the WZW CFTs su (4) 2 for the paired spin-singlet and su (2) 4 for the Read-Rezayi state [50].
Denoting the braid matrices for Ψ (n+1,k) by U (n+1,k) ij , the matrices satisfy U (4,2) 12 where the overline indicates that the rows and columns (i.e. the order of the fusion channels) of the matrix are swapped. This is explained in more detail in appendix E.

Wave functions for general M
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 which satisfy the inner products equation (15). These vectors ensure charge neutrality in all sectors except the first, so that O bg depends on the field φ 1 only, which describes charge. To obtain the model wave function Ψ M (4,2) without quasiholes, we change the first components to which yields the appropriate modification to the wave function as in equation (1). Introducing quasiholes, the appropriate change to q 1 µ follows from the requirement of mutual locality, so that the inner products between the vectors q µ and v α is unchanged for M > 0. For the vectors q, this yields The new conformal blocks Ψ M ( p) (4,2) then differ from their bosonic M = 0 counterparts by the full Jastrow factor i<j (z i − z j ) M and similar factors of the quasihole coordinates which we denote by Ξ: The factors Ξ for the different cases are The factor Ξ leads to additional, global phases of the braid matrices, while the Jastrow factor has no effect. In particular, they break the symmetry between the braiding behavior of the different quasiholes. The resulting braid matrices for the quasiholes with spin are The updated braid matrices for the spinless quasiholes read

Braiding for the paired su (n + 1) 2 spin-singlet states
We turn to the braiding of the fundamental quasiholes in the paired su (n + 1) 2 spin-singlet state equation (52). There are n electron operators and n + 1 quasihole operators H µ , given by equation (14), in terms of parafermions ψ α and spin fields σ µ of the parafermion theory su (n + 1) 2 /u (1) n . The model wave function for the ground state reads where the OPE of the parafermion fields fixes the normalization N = 2 1 2 ( α Nα)−1 . We consider the master formula for the case m 1 = 4, equation (20), since it yields all braid matrices. The master formula is: We consider the simplest case where N 1 = 2 and N i 2 = 6, taking the three limits The four-point functions of spin fields are presented in appendix C, and yield the solutions Here 2∆ = n(n+2) (n+1)(n+3) . Further, √ h and the functions F p i are given in appendix B.3. Using the transformations of the functions F p i presented in appendix D, it is straightforward to obtain the braid matrices In these expressions, d n = 2 cos π n+3 , see appendix D. For n = 3, they reduce to the matrices in equations (42)- (44). For n = 2, these braid matrices agree with the results obtained in [33] for the NASS case 4 , while for n = 1 they agree with the braid matrices for the Moore-Read wave function [32]. Again, the matrices U ij constitute a unitary representation of the braid group, i.e. they satisfy U 13 = U 12 U 23 U 12 = U 23 U 12 U 23 . Finally, the matrices are closely related to the braid matrices (see [33]) of the Read-Rezayi Ψ (2,n+1) states, see appendix E for more detail.
Generalizing the discussion in section 4.6, the braid matrices for the wave functions for general M read for the spinless quasiholes.

Application to Hermanns hierarchy states
We apply the results obtained to a series of recently introduced trial wave functions [36] which are obtained from a hierarchy picture of successive condensation of non-Abelian quasiparticles. We refer to these wave functions as Hermanns hierarchy states. They can be thought of as symmetrized copies of composite fermion (CF) [8] wave functions and were studied numerically in [37]. In [38], the Hermanns hierarchy states were given a CFT description by using their close relation to the paired spin-singlet states. Referring to [36,38] for more details, the (bosonic) Hermanns hierarchy wave functions read where the symmetrization is similar to that in the paired spin-singlet case except one now symmetrizes over two (bosonic) CF wave functions instead of the Ψ (n+1,1) , with Here λ labels the effective Λ-levels, λ = 1, . . . , n, and ∂ m λ ≡ i ∂ m (∂z λ i ) m is a product over derivatives of coordinates in level λ. The Ψ CF;n have ν = n n+1 , and their fermionic counterparts constitute the positive Jain series with ν = n 2n+1 . Therefore, the bosonic Hermanns hierarchy wave function has ν = 2n n+1 and the corresponding fermionic wave function has filling fraction ν = 2n 3n+1 . The case n = 1 corresponds to the Moore-Read state, while for n = 2, 3, the Hermanns hierarchy wave functions are, after particle-hole conjugation, candidates for ν = 2 + 3 7 and ν = 2 + 2 5 , respectively. To see the relation between the Hermanns hierarchy wave functions and the paired spinsinglet states we recognize equation (62) in terms of the paired spin-singlet states. In equation (63), the particles have definite pseudospin indices in the paired spin-singlet state Ψ (n+1,2) and the symmetrization is a sum over the ways of assigning pseudospin to the particles. Similarly, the quasihole model wave functions in the Hermanns hierarchy are obtained by symmetrizing paired spin-singlet states with quasiholes. The latter have n + 1 distinct fundamental quasiholes, i.e. quasiholes with a definite pseudospin index µ = 1, . . . , n or the spinless quasihole with µ = 0. Although we perform a symmetrization, effectively removing internal quantum numbers, the Hermanns hierarchy wave function still has n + 1 distinct fundamental quasiholes, discernible by the short distance behavior of the many-quasihole wave function. Considering a single quasihole for simplicity 5 , a model wave function for a quasihole with the smallest charge is where w (1) denotes a quasihole with pseudospin µ = 1. The model wave functions for different choices of µ = 1, . . . , n are expected to differ slightly because of the derivatives, but to have the same topological properties. The other type of fundamental quasihole corresponds to the spinless µ = 0 quasihole in the paired spin-singlet state. It is straightforward to generalize this to several quasiholes. With the quasihole wave functions in place, we now argue that the braid properties of the quasiholes in the Hermanns hierarchy are the same as those of the paired spin-singlet states studied in this paper. In writing the Hermanns hierarchy wave functions one has to perform two symmetrizations: one over identical layers as in equation (61), and one over pseudospin as in the CF wave functions.
The symmetrization over identical layers changes the statistics of the quasiholes in the individual layers: the most famous example is the Cappelli et al construction [46] of the Moore-Read state, via the symmetrization of two Laughlin states. This symmetrization reduces the dimension of the Hilbert space of quasihole states, which effectively renders the quasiholes of the Laughlin layers non-Abelian. Likewise, this symmetrization procedure renders the quasiholes in the generalized Halperin states non-Abelian, resulting in the braiding properties of the state Ψ (n+1,2) .
Contrarily, it has been argued that the pseudospin symmetrization does not change the statistics of the quasiholes [51], in accordance with the result that one can determine the statistics for the quasiholes of the CF wave functions from the K-matrix formalism [52,53]. In symmetrizing over the pseudospin, a reduction of the dimension of the Hilbert space is expected not to occur. The difference with the layer case is that the parts of the wave functions associated with different pseudospin are not identical. Although this is not a proof, it is likely that even for the Hermanns hierarchy wave functions the symmetrization over pseudospin does not alter the statistics of the quasiholes. Assuming this argument to be correct, the fundamental quasiholes in the Hermanns hierarchy come in two types, whose braid matrices are given by equations (59) and (60) respectively.
In particular, the (non-Abelian) braid behavior of the fundamental quasiholes in the Hermanns hierarchy wave functions at ν = 2n 3n+1 is the same, up to an overall phase, as that of the quasiholes in the Z n+1 Read-Rezayi wave functions at ν = n+1 n+3 . For n = 2, the Hermanns hierarchy wave function is a trial wave function for ν = 2 + 3 7 (after particle-hole conjugation), and one expects the quasiholes to obey Z 3 statistics. For n = 3, the Hermanns hierarchy wave function has the same filling factor as the Z 3 Read-Rezayi wave function: both are wave functions for ν = 2 + 2 5 (after particle-hole conjugation). Interestingly, one expects Z 4 -type braiding in the Hermanns hierarchy case, which is non-universal for topological quantum computing [54], as opposed to the Z 3 braiding expected in the Read-Rezayi case. Additionally, this differs from the Ising (Moore-Read) statistics expected for quasiholes in the Bonderson-Slingerland [55] hierarchy state at ν = 2 + 2 5 .

Conclusion
In this paper we have studied the braiding properties of the fundamental quasiholes in the paired spin-singlet states by finding explicit expressions of the quasihole wave functions and obtaining their monodromies. As expected on the basis of rank-level duality, we have shown that the non-Abelian braiding properties of the quasiholes in the paired spin-singlet states are closely related to the quasiholes in the Read-Rezayi series, with the only difference an overall phase. The extension to clustered spin-singlet states Ψ (n+1,k) with k > 2 is straightforward, although additional subtleties such as fusion multiplicities will arise, and is left to future work. Additionally, we have argued that the braid behavior of quasiholes in certain (spin polarized) non-Abelian hierarchy states should agree with that of the quasiholes in the paired spin-singlet states, and have observed that if the former are the appropriate model wave functions, the expected braid properties are Z 3 -type braiding for ν = 2 + 3 7 and Z 4 -type braiding for ν = 2 + 2 5 . The latter is to be contrasted with the Z 3 -type braiding based on the Read-Rezayi wave function and Ising statistics (Z 2 -type braiding) based on the state in the Bonderson-Slingerland hierarchy.
In finding the quasihole braiding properties from the CFT wave functions, we have assumed that 'holonomy = monodromy', i.e. that no additional Berry phase contributes to the braid statistics. Additionally, we have argued that the braiding properties of the paired spin-singlet states are unchanged by a symmetrization procedure. A promising method to address these matters is the matrix product state implementation of [21], by means of which the full Berry holonomy may be calculated numerically for large system sizes.
We provide details on the WZW models that underpin the paired spin-singlet states. We discuss the current algebra as well as the WZW primary fields with respect to this current algebra. Introducing a vertex representation of the currents as well as the WZW primary fields, we explicitly identify the electron and quasihole operators used to write down the model wave functions. We refer to [34,48,56] for more information.

A.1. Current algebra
The su (n + 1) k WZW model is characterized by its current algebra, a set of OPEs of currents J a corresponding to the generators t a of su (n + 1) with f abc the structure constants, i.e. [t a , t b ] = if abc t c . As a simple example, we consider n = 1. The currents J 1 , J 2 and J 3 obey the above OPEs with f abc = abc . Alternatively one may introduce raising and lowering operators through The current algebra of the su (n + 1) k model is generated by J ± α with α = 1, . . . , n which form su (2) subalgebras with J 3 α . Following Gepner [47], the vertex representation of the su (n + 1) k current algebra is an explicit representation of the currents J a in terms of free bosons φ = (φ 1 , . . . , φ n ) and parafermions: The vectors v α obey v α · v α = 2 and v α · v β = 1 if α = β; they correspond to specific roots in the root lattice of su (n + 1). It is straightforward to show that these currents generate the su (n + 1) k current algebra, by using the OPEs as well as the OPE between vertex operators For k = 1 the parafermions ψ α are trivial, while for k = 2 they satisfy ψ † α = ψ α . The parafermions are discussed in more detail in appendix B. The connection to the paired spin-singlet states is the identification of the electron operators with the raising operators:

A.2. WZW primary fields
The primary fields in the WZW model are fields that correspond to a specific representation of the algebra su (n + 1) k . The number of irreducible representations is finite, as opposed to the algebras su (n + 1). In particular, the representations Λ of su (n + 1) k are denoted where µ Λ µ = k and the Λ µ are positive integers. Each representation Λ corresponds to a representation Λ = (Λ 1 , . . . , Λ n ) = i Λ i ω i of su (n + 1) (here ω i are the fundamental weights). Therefore, the possible representations Λ of su (n + 1) k can be represented by the Λ labels alone. We adopt this convention in the following, but it should be kept in mind that the proper labels carry an additional To each representation Λ (strictly speaking Λ ) of su (n + 1) k corresponds a collection of fields G Λ . The 'components' correspond to the weights λ in the representation Λ, and are denoted G Λ λ . Thus, the field G Λ can be thought of as a vector of size dim Λ (note that we only consider one chiral half of the theory). The field G Λ satisfies the OPE with respect to the currents J a , where t a Λ is the generator t a in the representation Λ. As a simple example, the representations of su (2) 2 are (2; 0) , (1; 1) and (0; 2). The associated Λ labels correspond to the trivial representation Λ = 0, the fundamental representation Λ = 1, and the adjoint representation Λ = 2 of su (2). The primary fields corresponding to these representations In the general case, the weights λ = (λ 1 , . . . , λ n ) in the representation Λ are obtained by subtracting simple roots 2, −1, 0 . . . , 0) , . . . , α n = (0, 0, . . . , −1, 2) (A.9) from Λ (see e.g. [48,56]).
The WZW primary fields G Λ λ can also be represented in terms of primary fields Φ Λ λ in the corresponding parafermion CFT: The parafermion CFTs are discussed in appendix B. The quasihole operators may be identified with particular WZW primary fields; the corresponding primary fields are the spin fields σ µ .
where α is an element of the root lattice Q = Zα 1 + · · · + Zα n , as well as The parafermion CFT corresponding to the su (2) 2 model, which is the Ising CFT su (2) 2 /u (1), has the primary fields Φ 0 0 , Φ 1 1 , Φ 1 −1 , Φ 2 2 , Φ 2 0 and Φ 2 −2 prior to field identifications. One then identifies Φ 2 −2 ∼ Φ 2 2 via equation (B.1) (the simple root is α = 2 in this case) and 2). We are left with the three well-known primary fields 1 = Φ 0 0 , σ = Φ 1 1 and ψ = Φ 2 0 of the Ising CFT. The conformal dimensions of the primary fields follow from equation (A.10) and the conformal dimensions of the WZW primary fields [47]; in the following we simply list the results. The braiding calculation further relies on the precise operator product expansions between primary fields, which are used to take limits of the master formulas. In general, the OPE between primary fields φ i with conformal dimension ∆ i reads Here C c ab denotes an OPE coefficient, and it is non-zero only if φ c appears in the fusion between φ a and φ b . For k = 2, each primary field fuses with itself to the identity, so that C 1 aa = 1 for all a. For the remaining OPEs, we use the general expression for a three-point function of conformal fields: with structure constants C abc = C c ab = C b ac = C a bc . These can be determined by performing contractions of the ground state and the quasihole wave functions. 3 parafermion CFT has the following twenty primary fields Here we have used the shorthand 0 = (0, 0, 0). The most important fields are the parafermions ψ 1 , ψ 2 , ψ 3 and the spin fields σ 0 , σ 1 , σ 2 , σ 3 , which are used to define the electron and fundamental quasihole operators in equations (10) and (14) for n = 3. Note that the λ labels of the ψ α are in terms of the simple roots equation (B.6). We denote these vectors by Similarly, denoting the λ labels of the spin fields by q 0 = ω 1 , These vectors v α and q µ obey the correct inner products, where the inner product should be taken with respect to the quadratic form matrix of su (4) 2 .
The conformal dimensions of the primary fields are

.1. Fusion rules and OPEs.
We list the full set of fusion rules between the primary fields in equation (B.7). In general, the fusion rules read where Λ × Λ denotes the fusion of the representations Λ × Λ , which may be obtained by the Littlewood-Richardson rule. Note that the field identifications equations (B.1) and (B.2) may need to be used on the fusion outcomes. The parafermions have simple, Abelian fusion rules: they have Λ = 0, so their λ labels add modulo 2Q by virtue of equation (B.12). We reminder the reader that Q is the root lattice, and k = 2 in this case . The fusion table is For the remaining fusion rules, we first note the following: (B.13) By associativity of the fusion rules, the σ fusion rules encode all τ fusion rules as well. Then, the following fusion tables encode all fusion rules: In this particular CFT, the fusion rule equation (B.13) implies that the braiding properties of the fields τ are closely related to those of the fields σ. In particular the difference is a sign: braiding two τ fields is equivalent to braiding a pair of σ and ψ 123 around another pair, which is seen to give a relative minus sign compared to the braiding of the σ fields alone. The corre sponding WZW primary fields T µ = τ µ e iqµφ/ √ 2 yield the same braid matrices as the H µ , again up to a sign. We have verified this by explicitly calculating the F and R symbols for the representations Λ = (0, 1, 1) and Λ = (1, 1, 0) to which the τ fields correspond, using the quantum group approach [49].
We turn to the coefficients appearing in the operator product expansions of the field. By performing contractions of the ground state and quasihole wave functions, we reduce the correlators to three point functions, which determines several OPE coefficients. The coefficients for parafermions read: ψ 1 + ψ 23 + γ 1 1 + ψ 123 + ρ γ 2 ψ 2 + ψ 13 + γ 2 ψ 3 + ψ 12 + γ 3 1 + ψ 123 + ρ γ 3 ψ 3 + ψ 12 + γ 3 ψ 2 + ψ 13 + γ 2 ψ 1 + ψ 23 + γ 1 (B.14) The structure of the remaining relevant OPE coefficients is The sector ρ. The weight (0, 0, 0) in the adjoint representation Λ = (1, 0, 1) has multiplicity three-this means that the field ρ = Φ (1,0,1) (0,0,0) actually consists of three independent Virasoro primary fields. A similar feature was noted in in the NASS case [33], where the equivalent sector splits up into two independent Virasoro primary fields. We proceed in a similar way as in that paper, defining fields ρ µ by This distinction between the sector ρ and the fields ρ µ is necessary to ensure consistency of the four-point functions of spin fields: studying their behavior also leads to the choice of OPE coefficient 3 √ h above-see appendix C. Additionally one finds the OPEs i.e. the fields ρ µ are not independent. They may be written in terms of the three independent fields ρ c , ρ s , ρ t as ρ 0 = −ρ c We provide the details on the CFT su (n + 1) 2 /u (1) n needed to perform the braiding calcul ation. The primary fields are labeled by the representations Λ = (Λ 0 ; Λ 1 , ..., Λ n ) with µ Λ µ = 2, and weights λ obtained by subtracting the simple roots equation (A.9). The important fields after the field identifications are where p = 0, 1 denotes the fusion channel and ∆ = n(n+2) 2(n+1)(n+3) is the conformal dimension of g. We remind the reader of the notation w ij = w i − w j and x = w12w34 w13w24 . Additionally Γ( 2 n+3 ) Γ( n+1 n+3 ) and the F p i are the following functions in terms of the hypergeometric functions 2 F 1 (a, b; c; x): , n + 2 n + 3 ; 1 + n + 1 n + 3 ; x , n + 2 n + 3 ; n + 1 n + 3 ; x . (C.5) For the transformation w 1 w 2 , corresponding to x → −x 1−x , we have For the transformation w 2 w 3 , corresponding to x → 1 x : .

(D.3)
Finally, for the transformation w 1 w 3 , corresponding to

(D.4)
To obtain the braid behavior of the fundamental quasiholes, the following identities are also useful: (D.5)

Appendix E. Rank level duality
We comment on the consequences of rank-level duality, which relates the su (n + 1) k and su (k) n+1 WZW theories. In particular, we consider the consequences for the correlators, and thereby the braiding behavior of the quasiholes. In [50], the relation between the correlators of WZW primary fields in the dual WZW theories was derived. For the present purposes, we only consider the su (n + 1) 2 and su (2) n+1 cases. The correlators of four primary fields of the former theory are given in equations (C.2)-(C.4). The equivalent correlators for the later theory are stated here, using the convention x = w12w34 w13w24 , which differs from the one used in [34], where these correlators were derived. The correlators C ( p) a of the fields g, corresponding to the fundamental representation of the su (2) n+1 WZW theory read where µ, µ label the weights of the fundamental (i.e. two-dimensional) representation of su (2). The p = 0 channel corresponds to the trivial intermediate channel, (0), while the p = 1 channel corresponds to (2), the adjoint (i.e. three dimensional) representation. The tilde indicates that we deal with the su (2) n+1 quantities instead of the su (n + 1) 2 version (for the general su (n + 1) k results, see [34]), that is ∆ = 3 2(n+3) , h = For the correlators of the su (n + 1) 2 and su (2) n+1 WZW theories, rank level duality takes the following form [50]
(E.8) Before we comment on the consequences for the braid matrices, we note that we obtained the results for the correlators C ( p) a by taking the result from [34], and transforming x → − x 1−x , to take the different choices for the anharmonic ratios into account. This leads to the fact that for the su (2) 2 correlators, i.e. either C The duality relation between the correlators equation (E.8), implies that the braid matrices are also related. To avoid clutter in the notation, we denote braid matrices derived from the WZW correlators by W (n+1,k) ij