Integrable anyon chains: from fusion rules to face models to effective field theories

Starting from the fusion rules for the algebra $SO(5)_2$ we construct one-dimensional lattice models of interacting anyons with commuting transfer matrices of `interactions round the face' (IRF) type. The conserved topological charges of the anyon chain are recovered from the transfer matrices in the limit of large spectral parameter. The properties of the models in the thermodynamic limit and the low energy excitations are studied using Bethe ansatz methods. Two of the anyon models are critical at zero temperature. From the analysis of the finite size spectrum we find that they are effectively described by rational conformal field theories invariant under extensions of the Virasoro algebra, namely $\mathcal{W}B_2$ and $\mathcal{W}D_5$, respectively. The latter contains primaries with half and quarter spin. The modular partition function and fusion rules are derived and found to be consistent with the results for the lattice model.


I. INTRODUCTION
Starting with Bethe's study of the spin- 1 2 Heisenberg chain integrable models in low dimensions have provided important insights into the peculiarities of correlated many-body systems subject to strong quantum fluctuations [1][2][3][4], e.g. the appearance of quasi-particles with exotic properties such as fractional quantum numbers and with unusual braiding statistics. In a wider context these anyons appear as excitations in topologically ordered systems without a local order parameter. Realizations for such topological quantum liquids are the fractional Hall states [5] and two-dimensional frustrated quantum magnets [6][7][8]. Non-Abelian anyons can also be realized as local modes with zero energy at junctions of spin-orbit coupled quantum wires through the 'topological' Kondo effect [9][10][11]. Additional interest in these objects arises from the fact that non-Abelian anyons are protected by their topological charge which makes them potentially interesting as resources for quantum computation [12,13].
A class of integrable models describing interacting non-Abelian anyons in one dimension can be obtained from two dimensional classical lattice systems with interactions round the face (IRF) such as the restricted solid on solid (RSOS) models [14] in their Hamiltonian limit. The critical arXiv:1408.1282v2 [cond-mat.str-el] 2 Nov 2014 RSOS models are lattice realizations of the unitary minimal models [15,16] and the family of commuting operators generated by their transfer matrices contains Hamiltonians for Fibonacci or, more generally, SU (2) k anyons with nearest neighbor interactions [17]. Recently, extensive studies of these anyons and their higher spin variants subject to different interactions or in different geometries have led to the identification of a variety of critical phases and the corresponding conformal field theories from finite size spectra obtained numerically [18][19][20][21][22].
Several approaches exist for the construction of integrable IRF models: integrable generalizations of the RSOS models have been obtained by means of the fusion procedure [23]. Alternatively, the correspondence between integrable IRF models and vertex models [24,25] (or, equivalently, anyon chains and quantum spin chains) with an underlying symmetry of a quasi-triangular Hopf algebra can be exploited to construct anyonic quantum chains [26,27]. Finally, new models can be defined by labeling the local height variables on neighboring sites of the lattice by adjacent roots in the A-D-E Dynkin diagrams or, more generally by primary fields of a general rational conformal field theory related through the corresponding fusion algebra [28,29]. For these models to be integrable one has to find a parameterization of the Boltzmann weights for the configuration allowed around a face of the lattice which satisfies the Yang-Baxter equation. This can be achieved by Baxterization of a representation of the braid group associated to the symmetries of the underlying system.
In the following we apply this last approach to construct four integrable quantum chains of a particular type of anyons responsible for the non-Fermi liquid correlations predicted in the topological Kondo effect [10,11]. Specifically, we consider anyons satisfying the fusion rules of SO(5) 2 .
Below we will see that -as might be expected for a model based on SO(5) = B 2 -the anyonic quantum chains are related to IRF models with height variables outside the A-D-E classification.
The integrability of the models is established by construction of representations of the Birman-Murakami-Wenzl (BMW) algebra in terms of anyonic projection operators. The spectra of these models are determined by Bethe ansatz methods and we identify the ground state and low lying excitations. Two of the models can be related to the six-vertex model in the ferromagnetic and anti-ferromagnetic gapped regime, respectively, by means of a Temperley-Lieb equivalence. From the analysis of the finite size spectrum the other two integrable points are found to be critical with the low energy sector described by unitary rational conformal field theories (RCFTs) with extended symmetries associated to the Lie algebras B 2 = SO(5) and D 5 = SO(10), respectively.
To present our results in a self-contained way, some general facts on the classification and spectral data of the relevant minimal models of Casimir-type W-algebras are included in Appendix B. For the anti-ferromagnetic anyon chain we propose a modular invariant partition function in terms of the Virasoro characters of the irreducible highest weight representations of the WD 5 (9, 10) RCFT.
The S-matrix and fusion rules of this CFT are given in Appendix C.

II. SO(5) 2 ANYONS
Algebraically, anyonic theories can be described by braided tensor categories [8,30,31]. A braided tensor category consists of a collection of objects {ψ i } (including an identity) equipped with a tensor product (fusion rules), where N c ab are non-negative integers. In the special case N c ab ∈ {0, 1} it is called multiplicity free. Below we shall use a graphical representation of 'fusion states' of the anyon model where vertices a b c may occur provided that ψ c appears in the fusion of ψ a and ψ b . We require associativity in our fusion i.e.
which is governed by F -moves, also referred to as generalized 6j-symbols, There also must be a mapping that braids two objects, Note that while fusion is commutative, states of the system may pick up a phase R ab c under braiding. Graphically this is represented by For a consistent anyon theory braids have to commute with fusion and satisfy the Yang-Baxter relation. Both properties follow from the Hexagon equation for F -and R-moves.
In this paper we consider a system of particles which satisfy the fusion rules for SO(5) 2 shown in Table I. This fusion algebra is the truncation of the category of irreducible representations of the quantum group U q (so (5)) where q = e 2iπ 5 [30]. For the SO(5) 2 fusion rules there exist four known sets of inequivalent unitary F -moves, which can be found by solving the pentagon equations directly or by utilizing the representation theory of U q (so (5)). Associated with each set of F -moves is a modular S-matrix which diagonalizes the fusion rules (note that the latter may depend on the choice of F -moves [30]). We have selected the F -moves corresponding to the S-matrix With this choice of F -moves we have four possible sets (in two mirror pairs) of R-moves. For the purposes of this article, however, we do not need to explicitly choose one.
Given a consistent set of rules for the fusion and braiding we can construct a one-dimensional chain of interacting 'anyons' with topological charge ψ j . First a suitable basis has to be built using fusion paths [31,32]: starting with an auxiliary anyon ψ a 0 we choose one of the objects appearing in the fusion ψ a 0 ⊗ ψ j , say ψ a 1 . The latter is fused to another ψ j , and so forth. Recording the irreducible subspace of the auxiliary space and the subsequent irreducible subspaces which appear after fusion we construct basis states Note that the construction implies that any two neighboring labels, a i a i+1 must satisfy a local conditions, namely N a i+1 a i ,j = 0. This local condition can be presented in the form of a graph in which labels which are allowed to be neighbors correspond to adjacent vertices.
For a system with periodic boundary conditions we have to identify a 0 and a L (which allows to remove the label a 0 from the basis states (2.4)). In this case the Hilbert space of the model can be further decomposed into sectors labeled by topological charges. To measure these charges one inserts an additional anyon of type ψ into the system which is then moved around the chain using (2.1) and finally removed again [17]. The corresponding topological operator Y has matrix elements The spectrum of these operators is known: their eigenvalues are given in terms of the matrix elements of the modular S-matrix as S k S 1 [8]. Local operators acting on the space of fusion paths of the ψ j anyons can be written in terms of projection operators. In terms of the F -moves (2.1) they can be introduced in the following way [17], In the second line we have used that the F -moves are unitary. For our choice of SO(5) 2 F -moves this can be ensured by choosing a suitable gauge. In terms of the basis states (2.4) the p i can be written alternatively as (2.6) Note that the matrix elements of these operators depend on triples of neighboring labels a i−1 a i a i+1 in the fusion path but only the middle one may change under the action of the p FIG. 1. A graphical representation of allowed neighboring labels in the fusion paths of the ψ 3 anyon chain.
The vertices/nodes of the graphs are the labels of anyons which are connected via an edge if and only if the two anyon labels can appear next to each other.

III. THE ψ 3 -ANYON CHAIN
Based on the SO(5) 2 fusion rules, Table I, non-trivial models can be defined for ψ 2 (or, equivalently, ψ 5 ) and ψ 3 (ψ 4 ) anyons. In this paper we construct integrable chains of ψ 3 anyons. The selection rules for neighboring labels in the fusion path (2.4) displayed in Figure 1. This defines the Hilbert space of the anyon chains.

A. Local Hamiltonians
We shall concentrate on models with local interactions given in terms of the projection operators (2.6). The resulting Hamiltonians are the anyon chain (or face model) analogue of spin chains with nearest neighbor interactions [24,26]. There are only three non-zero linearly independent projection operators which can be defined on the tensor product ψ 3 ⊗ ψ 3 = ψ 1 ⊕ ψ 2 ⊕ ψ 5 , i.e.
Therefore the most general local interaction has a single free coupling parameter h i (θ) = cos π 4 + θ p (2) i + sin and the general global Hamiltonian for the anyon chain can be written as Recently, the phase diagram of this model as a function of the parameter θ has been investigated numerically [33].
From Table I, we see that there exists an automorphism of the fusion rules exchanging ψ 2 and ψ 5 which allows the construction of a non-local unitary transformation A two-dimensional lattice of heights with weights placed on faces.

B. Points of integrability
In order to establish integrability of the model (3.2) for particular choices of the parameter θ it has to be shown that the Hamiltonian is a member of a complete set of commuting operators.
This can be done starting from R-matrices depending on a parameter acting on the tensor product which satisfy the Yang-Baxter equation (YBE)  [28,29]. Likewise, it is known that models of this type can be mapped to loop models [28,34], see Figure 3.
As a consequence of the Yang-Baxter equation the face model can be solved using the commuting transfer matrices on the Hilbert space of the periodic anyon chain Similar to the topological Y -operators, the transfer matrix can be viewed as describing the process of moving an auxiliary anyon around the chain. In this case, however, the braiding is governed by the parameter-dependent R-matrix (or, equivalently, the Boltzmann weights) rather than by constants of the underlying category.
Just as there is a natural relationship between two-dimensional vertex models and spin chains there exists a correspondence between this two-dimensional interaction round the face model (IRF) and an anyon chain. Provided that the R-matrix degenerates to the identity operator for a particular value u 0 of the spectral parameter it follows that at u = u 0 the transfer matrix becomes the shift operator, This motivates the definition of a momentum operator and Hamiltonian by expanding the transfer matrix around u 0 : For a proper choice of the parameters α and α the local Hamiltonian coincides with (3.1).
To construct points of integrability we construct representations of the Birman-Murakami-Wenzl (BMW) algebra [35] using the local projection operators (2.6) occurring in the ψ 3 -anyon model.
We define the operators where the R ab c belong to one of the sets of R-moves (2.2). These operators satisfy the relations: for |j − j | > 1. As mentioned above there exist three other sets of R-moves consistent with our choice of the modular S-matrix (2.3). Using one of these in (3.9) would be equivalent to replacing The BMW algebra contains a copy of the Temperley-Lieb algebra [36] as a subalgebra, from which we can construct the solution to the YBE Expanding of the resulting transfer matrix around u 0 = 0 one obtains the anyon model (3.2) with (α , α ) = ±( 1 2 √ 2 , 1 2 √ 10 ) for θ = 0 and π. As for other Temperley-Lieb models it can be written as Using the full BMW algebra one can find two additional solutions to the YBE [37]. One of them is [38,39] R j (u) = sinh 7iπ 10 sinh (3.10) This solution leads to integrable points at θ = η, η + π where η = atan 1+ The other R-matrix obtained by Baxterization of the BMW representation is related to (3.10) through the transformation (3.3) and gives rise to integrable points at θ = −η, −η+π. However, these integrable points are equivalent to the ones they are mapped from and are subsequently omitted from further analysis.
As stated earlier, the weights given by (3.4) will lead to a solution of the YBE and a commuting transfer matrix (3.6). It turns out that this transfer matrix belongs to a family of commuting transfer matrices. To define these transfer matrices we present the YBE for face weights The general form we use for these face-weights is (3.12) In the case α = β = γ = 3 and w 3,3 l (u) = w l (u), equations (3.5) and (3.11) are equivalent. We are able to determine the required set of weights (see Appendix A), such that we have a family of commuting transfer matrices for , = 1, . . . , 6 and u, v ∈ C. An additional important feature of these transfer matrices is that the limit of each is also a Y -operator (2.5), i.e.
The appearance of the topological operators as a limit of these transfer matrices is not unexpected as the braiding governed by these weights and the underlying category coincide when u → ±∞.
where the number of roots is related to the total magnetization of the spin chain through S z = L 2 −n. In general the twist ζ can take different values for different states depending on on the symmetry sector of the Temperley-Lieb equivalent model [41]. To determine the allowed values of the twist the Bethe equations (4.1) should be obtained directly from the spectral problem of the model considered, e.g. by deriving functional equations for the eigenvalues of the transfer matrices from the set of fusion relations between the finite set (3.13) [42,43]. In the case of Temperley-Lieb type models the fusion hierarchy does not close in general. Motivated by the functional equations for the XXZ chain we therefore introduce an infinite set of functions Λ d (u) starting from a given eigenvalue Λ(u) ≡ Λ 2 (u) of the transfer matrix t (3) (u) of the anyon chain (we set Λ 1 (u) ≡ 1). 1 Computing this hierarchy for anyon chains of length up to L = 4 we find that the to exist (corresponding to a cycle with p = 1) and is related to the eigenvalues of the so-called Q-operator [44,45]. The periodicity in d allows to rewrite the recursion relation (4.2) in terms As a result the spectral problem can be formulated as a so-called T Q-equation As a consequence of the analyticity of the transfer matrix eigenvalues Λ(u) the zeroes {u } n =1 of the q-functions are solutions to the Bethe equations (4.1). In terms of these Bethe roots energy and momentum of the corresponding state is given by eigenvalues λ ± of the operators for chains with up to L = 10 sites we find that they take values We find that t + and t − are real matrices and Consequently, if λ ± are eigenvalues sharing the same eigenvector then λ + = λ * − . Using this in the T Q-equation (4.4) we can relate these eigenvalues to the twist Note that changing the twist ζ → ζ −1 does not affect the spectrum of the anyon chain: a solution {u k } n k=1 of the Bethe equations (4.1) at twist ζ with energy E and momentum P can be mapped to the solution {−u k } n k=1 for twist ζ −1 having the same energy but momentum (−P ), see (4.5). Additional constraints on the allowed twists follow from the translational invariance of the system: taking the product over all Bethe equations (4.1) we obtain Taking the logarithm of this equation the Bethe roots {u j } can be eliminated from the expression for the momentum eigenvalue (4.5) giving LP = 2πk + i(L − 2n) log ζ for some integer k. As a consequence of periodic boundary conditions the momentum eigenvalues are quantized, LP ∈ 2π Z.
For states corresponding to root configurations {u j } L/2 j=1 this is always true. From our numerical data we find, however, that the corresponding eigenvalues of the transfer matrix take asymptotic values λ ± ∈ {0, ±1, ± √ 5} only. Configurations with n = L 2 roots, on the other hand, are only possible for twists ζ L−2n = 1 all of which are all allowed by (4.7) with (4.6).
In summary we have used the Temperley-Lieb equivalence to relate the complete spectrum of the anyon chain to those found in the S z = L 2 − n eigenspaces of XXZ spin-1/2 chains with anisotropy |∆| = √ 5/2 and periodic boundaries twisted by (4.8) Since the spectral properties of the model should not depend on the boundary conditions in the thermodynamic limit this relation to the XXZ spin chain implies that the anyon chain has gapped excitations over a possibly degenerate ground state for θ = 0 and π.
1. Ground state and low lying excitations for θ = 0 For θ = 0 the ground state of the related anti-ferromagnetic XXZ chain is a completely filled band of n = L/2 imaginary Bethe roots −π/2 < iu j ≤ π/2. We have solved the Bethe equations The other excitations are separated from the ground states by a gap. From the solution of the XXZ model this gap is [47] ∆E = 2 π √ 10 K(k) 1 − k 2 2.9 10 −4 . (4.10) Here K(k) is the complete elliptic integral with modulus defined by

Ground state and low lying excitations for θ = π
For coupling θ = π the Temperley-Lieb equivalent spin chain is the ferromagnetic XXZ model.
The ground states appear in the sector with n = 0 Bethe roots. They have energy E , 0 ≤ k ≤ L − 1 (4.11) We have omitted data for the 5th lowest excitation due to insufficient solutions of the Bethe equations. The ground state has multiplicity two with states in symmetry sectors ψ 1 and ψ 6 . The next three energies have multiplicities two, four and two, respectively, and are in sectors ψ 3 ⊕ ψ 4 , ψ 2 ⊕ ψ 5 and ψ 3 ⊕ ψ 4 , respectively.
where, as a consequence of (4.8), ζ can take values ζ = exp( 2πi L−2 ) for integer . From (4.5) energy and momentum of these solutions are obtained to be (4.12) The lowest of these excitations is the one with quantum numbers k = 0 = , implying a gap In our numerical analysis of the spectrum we find for θ = π that the ground state and excitations with finite energy show degeneracies growing with the system size L.

B. The BMW points
The existence of a family of commuting transfer matrices for the integrable points θ = η, η + π, allows for a straightforward construction of Bethe equations which are necessarily complete.
Specifically, we can find functional relations (A1) for the transfer matrices resembling the SO(5) 2 fusion rules ) .
where y 6 = ±1 is the eigenvalue of t (6) (u) = Y 6 . By inspection we see that the Bethe equations are equivalent to the ones appearing in the Z 5 Fateev-Zamolodchikov (FZ) model, sometimes referred to as the self-dual chiral Potts model, up to a twist [48,49]. This connection to the Z 5 FZ model is not unexpected as the family of commuting transfer matrices implies that this SO(5) 2 model is also a descendant of the zero-field six-vertex model, like the FZ model [50].
Given a set of solutions to the Bethe equations, {u j }, we are able to determine energy and momentum of the corresponding state from (3.7) .
we find that states in the sectors ψ 1 or ψ 6 are parametrized by n = 2L Bethe roots and have a constant c 2 = 5 appearing in the momentum. Similarly, for states in the ψ 3 or ψ 4 sector we find n = 2L and c 2 = 1 while for the states in the ψ 2 or ψ 5 sector we find n < 2L and c ∈ C (in our numerical studies we find n = 2L − 4 or n = 2L − 2 for the latter). We can gain additional information from the asymptotics of the transfer matrices (which become the Yoperators) appearing in the fusion relations (4.13), determining the relation Thus simply knowing y 6 and how many finite Bethe roots there are determines the eigenvalue of Y 2 . This information is summarized in table II.
As this integrable point is related to the Z 5 FZ model and the Bethe equations are equivalent up to an allowed phase factor, we can use the Bethe root classification of Albertini [49]. Thus one expects that the Bethe roots come in the following varieties We have confirmed this by exact diagonalization of the transfer matrix for systems with small chain lengths, L ≤ 12.
1. Thermodynamic limit of the BMW model at θ = η In the thermodynamic limit the ground state for the θ = η point consists of L 2 4-strings with only even chain lengths admissible. The corresponding energy density can be computed using the root density formalism [51] where the configuration of Bethe roots is described by the density of 4-strings. The latter is given as the solution of the linear integral equation .  Eliminating the rapidity u we obtain the dispersion relation (p) of the elementary excitations in the system and the Fermi velocity is obtained to be v F = 5 cos( π 4 + η) 4 cos( π 10 ) ≈ 1.021807736 .

Thermodynamic limit of the BMW model at θ = η + π
For θ = η + π the ground state in the thermodynamic limit consists of only +-and −-strings appearing in a ratio of five to three. For finite size systems we find that the ground state is only realized when L is a multiple of 8. As we have two different types of string configurations appearing we have two root densities defined by coupled integral equations ) .
Again, these integral equations are straightforward to solve by Fourier transform giving .
As was the case with the previous critical integrable point, the dressed energies of ±-strings are scalar multiples of the corresponding root densities, ρ ± (u) .
Again the system has massless excitations with linear dispersion near its Fermi points, which in this case are at multiples of π 4 , see Fig. 6. The ground state energy density and Fermi velocity are ≈ 0.2554519349 .

V. FINITE SIZE SPECTRA AND CONFORMAL FIELD THEORY
The models at θ = η, π + η derived from the BMW transfer matrices constructed above are critical and therefore expected to be described by conformal field theories (CFTs). As a consequence of conformal invariance in the continuum limit the scaling behavior of the ground state energy is predicted to be [52,53] This sequence is then extrapolated to obtain a numerical approximation X ext A. The BMW model at θ = η Solving the Bethe equations for the ground state configuration of 4-strings in systems with even L we find the scaling of the ground state energy This implies that the central charge is c = 1.14285(1), predicted to be actually 8 7 in agreement with the ferromagnetic Z 5 Fateev-Zamolodchikov model. This central charge appears in several rational CFTs from the minimal series of Casimir-type W-algebras, see Appendix B. Restricting ourselves to unitary ones these are Note that A 4 = SL(5) with real form SU (5) contains B 2 = SO(5) and C 2 = Sp(4) as a subalgebra.
Since B 2 ∼ = C 2 we only consider the former. 3 To identify the low energy effective theory of this anyon model we extend our analysis to the scaling dimensions obtained from the finite size spectrum of excited states, see Table III. Clearly our data rule out the Z 5 parafermions since the conformal weights observed in the sector ψ 3 ⊕ ψ 4 , e.g. 1 28 , 3 28 and 15 28 , are not present in the spectrum (B5) for k = 5. On the other hand the finite size data are compatible with the WB 2 (5, 7) minimal model (B6): the conformal weights up to h = 2/7 are complete, the first one missing is h = 17/35. This is not surprising as the numerical analysis gets more demanding for higher values of the conformal weight. Therefore, we conclude that if the continuum limit of this BMW model is described by a rational conformal field theory, it presumably is the (5, 7) minimal model of the WB 2 series.
B. The BMW model at θ = π + η Using the finite size data for the ground state we find the scaling behavior,  To identify which c = 1 CFT is realized we can look at the finite size properties of low-lying excitations presented in Tables IV-VII. Clearly, the observed spectrum contains conformal Sector(s) Note that among the states with scaling dimension X = 1 1 8 we have only identified the (1, 0) and (0, 1) descendents of the primaries (h,h) = ( 1 16 , 1 16 ) at the Fermi points π 4 and 3π 4 , respectively. For the other two descendents as well as the primary fields with ( 9 16 , 9 16 ) (with momenta P (0) = π 4 , 3π 4 ) we have not identified the corresponding root configuration.
weights not present in the Z 4 parafermion theory, thereby excluding the A 3 -model. Furthermore, we can rule out the WC 3 (4, 6) model: even without explicit computation of the spectrum we just know from the general formula (B3) for the conformal weights, that their denominators are of the form 2pq. Therefore, to yield the denominators 5 and 20 observed numerically in the anyon model 2pq has to be divisible by 5.
Among the remaining rational CFTs the smallest ones respecting the five-fold discrete symmetry of the BMW model are the WB 0,2 (4, 5) model and the WD 5 (9, 10) model. Comparing their spectra (B7) and (B8) with the numerical data we find that the former does not accommodate all observed weights while all weights of the (9, 10) model of the A peculiar feature of the low energy spectrum observed numerically is the identification of levels corresponding to operators with non-integer spin s, see Tables V-VII. For the sectors ψ 2 and ψ 5 ,  Sector(s) 0.62500(1)  where the total momentum is not determined uniquely in terms of the Bethe roots, see Eq. (4.15) one can use this fact to determine the Fermi point P (0) at which these levels occur: for example, the levels (h,h) ∈ {( 1 5 , 9 20 ), ( 4 5 , 1 20 )} observed for L mod 4 = 2 can only appear at P (0) = π 4 . Changing the length of the system to L mod 4 = 6 they are observed at P (0) = 3π 4 , indicating that they have to appear with a multiplicity of two in the full partition function.
To provide additional evidence for our proposal that the anyon model is a lattice regularization of a WD 5 (9, 10) rational CFT the modular invariant partition function of the model has to be expressed in terms of the Virasoro characters of the relevant representations of the W-algebra. All Sector(s) This insight, together with similar considerations for the other representations, is sufficient to set up a differential equation from which the characters of the rational CFT can be computed to arbitrary powers without additional prior knowledge of the detailed structure of the representations: the modular differential equation encodes the fact that the characters of the nine inequivalent irreps of the CFT form a nine-dimensional representation of the modular group in terms of modular functions with asymptotic behavior fixed by the numbers h i − c 24 [54]. Assuming that all characters can be written as modular forms of weight 1/2 divided by the Dedekind eta-function we find χ 0 (q) = 1 2η(q) Θ 0,5 (q) + Θ 0,4 (q) − Θ 4,4 (q) , Here we have used the Jacobi-Riemann Theta-functions . Note that we have introduced an explicit factor of two in the definition of χ 1/16 (q) and χ 9/16 (q).
In fact, these characters as well as χ 5/4 (q) = 2 q −1/24+5/4 1 + q + 2q 2 + 3q 3 + 5q 4 + . . . should be read as χ h (q) = χ + h (q) + χ − h (q) being the sum of two identical characters for the two equivalent representations appearing in the Kac-table for h ∈ { 5 4 , 1 16 , 9 16 }, see (B8). For the present discussion of the conformal spectrum this is sufficient. In Appendix C, where we compute the fusion rules for the WD 5 ((9, 10) rational CFT, these representations have to be disentangled in a consistent way. With the S-matrix (C1) at hand this condition can be solved and we find From the finite size spectrum of the anyon model we know that M hh = 1 2 for h,h ∈ { 1 16 , 9 16 } which implies m = 1, q = 2. Similarly, as discussed above, the quarter spin fields in the symmetry sectors ψ 2 and ψ 5 appear twice (with different momentum) in the low energy spectrum, hence p = 2. As a result the smallest off-diagonal partition function which incorporates all non-diagonal combinations found in our model is M ij (4, 1, 2, 2). This leads to a fourfold degenerate ground state in the model while it is unique in the lattice model with periodic boundary conditions studied here.
We expect that the other ground states (corresponding to the degenerate minima of the dispersion at momenta P (0) = k π 4 , k = 1, 2, 3) are realized when considering more general (twisted) boundary conditions for the anyon chain.
Finally, we note that in the CFT describing the collective behavior of the BMW model at θ = π + η the symmetry of the underlying anyon model appears to be modified. If we understand the SO(5) symmetry of the anyons as a discrete one (similar as the cyclic subgroup Z 5 of SU (2)), however, SO(10) could correspond to the dihedral subgroup D 5 of SU (2). This is appealing for two reasons: firstly, it would fit with the well known A-D-E classification of rational conformal field theories with c = 1 as conformal field theories with extended symmetries given by modding out discrete symmetries of SU (2) [55]. Secondly, a similar construction with D(D 3 ) anyons also has a c = 1 point, where it coincides with the Z 4 parafermions [27,56]. These, however, are SU (4) ∼ = SO(6), i.e.Â 3 ∼ =D 3 .

VI. DISCUSSION
In this paper we have constructed several integrable one-dimensional models of interacting anyons satisfying the fusion rules for SO (5)  By solving the Bethe equations we have identified the ground states and low energy excitations of these models. The TL-models are equivalent to the six-vertex model in its (anti-)ferromagnetic massive phases. The other two integrable models are critical at zero temperature. From the finite size spectrum we have been able to identify the conformal field theories describing their scaling limit. Based on the Bethe ansatz solution we find that these systems are closely related to the parafermionic theory. For the WD 5 (9, 10) rational CFT we have computed the characters of the Virasoro representations, S-matrix and fusion rules. Based on the data obtained from the finite size spectrum, an off-diagonal modular invariant partition function has been proposed. This is a first step towards the construction of lattice operators corresponding to the fields appearing in the continuum theory (see e.g. [57]).
These findings lead us to conjecture that the sequence of c = 1 theories describing the critical properties of the anti-ferromagnetic Z n FZ models for n odd [49] are in fact rational CFTs with WD n symmetry. For the case n = 3, corresponding to the 3-state Potts chain, the modular invariant partition function has been computed [58] leading to the identification of a low energy effective theory with Z 4 parafermions (see also Ref. [27] for another related anyon chain). This CFT happens to have the same spectrum as the WD 3 model. Further support for this conjecture is obtained from the numerical solution of the Bethe equations of the FZ models for n = 7, 9 indicating the presence of WD n primary fields with conformal weight h ∈ { k 2 4n | k = 0 . . . n} ∪ { 1 16 , 9 16 , 1} (note that the exponents 1 16 , 9 16 appear in the spectrum of the FZ model with π-twisted boundary conditions).
Finally, let us remark that different anyon chains can be constructed from the SO(5) 2 fusion rules, Table I: considering fusion paths (2.4) for ψ 2 -anyons the neighboring labels are restricted to be adjacent nodes on the graph shown in Figure 7. In this case the fusion path basis can be decomposed into two decoupled subspaces: the Hilbert space spanned by states with particles {ψ 3 , ψ 4 } is isomorphic to that of a spin-1 2 chain. Hence it can be written as a tensor product of local spaces C 2 . The complementary set of particles {ψ 1 , ψ 2 , ψ 5 , ψ 6 } forms a fusion category equivalent to categories of irreducible representation of the dihedral group of order 10. Using this insight we can expect that the model is connected to the one-dimensional spin-1 2 XXZ model [26]. Indeed we find that the R-matrix of the six-vertex model with anisotropy parameter γ can be expressed in terms of the local projection operators appearing in this ψ 2 -anyon chain: R(u) = (sinh(γ) + sinh(2u)) p (1) + (sinh(γ + 2u)) p (5) + (sinh(γ) − sinh(2u)) p (6) .
The resulting model shares bulk properties with the spin-1 2 XXZ chain. As in the case of the ψ 3model this does not, however, mean that the excitation spectrum or the operator content will be the same. This is again a consequence of the presence of commuting topological charges modifying the boundary conditions in the anyon model. Even more models for anyons satisfying a given set of fusion rules can be obtained by using other sets of F -moves consistent with the pentagon equation. This may lead to different models similar as in the case of SU (2) 3 fusion rules leading to both the Fibonacci and the (non-unitary) Yang-Lee anyon chains [17,59]. This question and the possibility of additional integrable models for interacting SO(5) 2 anyons will be studied in future work.  The additional weights are constructed from (3.10) as descendents [50].
Among the resulting transfer matrices t ( ) (u), the ones for = 1, 5, 6 are found to be independent of the spectral parameter u. For t (1) (u) and t (6) (u) this follows from ψ 1 ⊗ ψ 3 and ψ 6 ⊗ ψ 3 both being isomorphic to simple objects. The absence of a parameter in t (5) (u) is connected to our choice of representation of the BMW algebra. The different representations and their corresponding Rmatrices lead to different sets of transfer matrices and we find that it is always the case that either t (2) (u) or t (5) (u) is parameter independent.
From the analysis of small systems, L ≤ 10, we find that the topological Y -operators (2.5) are obtained as limits of the transfer matrices, i.e.
Given the interpretation of both the t ( ) (u) and the Y as descriptions of certain braiding processes we expect this relation to hold for arbitrary system sizes. Using the fusion procedure [43] for face models we find closed relations for transfer matrices ) . (A1) Examining Table I  give rise to a finite-dimensional representation of the modular group PSL(2, Z). As all fields in the symmetry algebra must be mutually local, the scaling dimensions are restricted to be integers or half-integers, 2 d k ∈ N. Therefore, chiral symmetry algebras constitute meromorphic conformal field theories.
A class of chiral symmetry algebras whose representation theory is particularly well understood are W-algebras associated with Lie-algebras g, where the scaling dimensions d k are related to the exponents of the Lie-algebra g, i.e. where one has one W (d k ) field associated to each independent Casimir operator. For a Lie-algebra of rank , the W-algebra is generated by such fields, one of those always being the Virasoro field associated to the quadratic Casimir operator. For this class of W-algebras, free field realizations are known and hence the algebras can, in principle, be explicitly constructed [63]. For all semi-simple Lie-algebras the corresponding W-algebras are known together with their so-called minimal series of rational theories they admit.
In the following let g denote a simple Lie-algebra of rank , and h a Cartan subalgebra of it.
We denote by d g the dimension of the Lie-algebra and by g * its dual Coxeter number. Let Φ + denote the set of positive roots α, and ht α the height of the root α. The height of a positive root is determined from its unique decomposition α = r i=1 n i α i into a linear combination of the simple roots α i with non-negative integers n i : ht α = r i=1 n i . Finally, the maximal root may be denoted by ρ. Under affinization of the Lie-algebra toĝ, we introduce the level x = 2k/|ρ| 2 , where k is the central extension ofĝ. Choosing the Killing form as our invariant symmetric bilinear form on g, one finds that |ρ| 2 = 1/g * and the central charge for Wĝ at level x is given by [60] We see that the central charge is entirely expressed in terms of basic properties of the Lie-algebra and the sums of the (squares of the) heights of the positive roots. The former are standard and can be found in every text book on Lie-algebras, the latter can easily be computed from the explicit choice of set of simple roots ∆. In fact, the values of these sums do not depend on the choice of ∆ after all. The data needed to explicitely compute the central charge for a simple Lie algebra are given in Table VIII. (2 + 1) 2 − 1 1 6 ( + 1)(4 − 1) 1 6 ( + 1)(2 − 1)(2 + 1) C (2 + 1) + 1 1 6 ( + 1)(4 − 1)  are needed to build a complete, rational conformal field theory. The corresponding L 0 eigenvalues of the highest weight states are all known.
In of the algebras, we also list the central charges of their corresponding series of minimal models for level x + g * = p/q ∈ Q.
For non-simply-laced Lie-algebras alternative constructions of extended chiral symmetries are possible if one allows for generators with half-integer spin. In particular [61], one can construct an alternative W-algebra for the B series, i.e. for SO(2 + 1), which contains precisely one fermionic generator. In the literature, these algebras are often denoted WB f -algebras, and are given as W(2, 4, . . . , 2d, . . . , 2 , 1 2 (2 + 1)). We will denote them as WB 0 -algebras in the following. These algebras also admit a minimal series whose members have the central charges We note that the first member, WB 0,1 = W(2, 3 2 ), yields the minimal series of the N = 1 supersymmetric extension of the Virasoro algebra, when p and q are replaced by half-integers p/2 and q/2.

Spectra of Casimir-type W-algebras
In a similar way as the central charges (B1) the conformal weights appearing in minimal models of Casimir-type W-algebras are essentially determined by data from the underlying Lie-algebra.
Let again denote the rank of a simple Lie-algebra g.
Highest-weight representations of g to weights λ = i=1 r i λ i with λ i denoting the fundamental weights, and r i ∈ N can be labeled by the positive integers r = (r 1 , r 2 , . . . , r ). The weight lattice has an associated dual lattice, spanned by the fundamental co-roots λ * i . A co-weight is then given by λ * = i=1 s i λ * i with s i ∈ N and labeled by the positive integers s = (s 1 , s 2 , . . . , s ). The conformal weights of a minimal model of a Casimir-type W-algebra Wĝ with central charge c g (p, q) are given by [61,64] h r,s (p, q) = where C denotes the Cartan matrix of the simple Lie-algebra g, and where the weights and coweights must fulfill the conditions i=1 r i m i ≤ p − 1 and i=1 s i m * i ≤ q − 1. Here, the m i are the normalized components of the highest root ψ in the directions of the simple roots α i , i.e.
Thus, all we need to know are the integers m i and m * i and the matrices C and D, which give the scalar products To denote them in the following table as concise as possible, we denote by E ij the × matrix with entries (E ij ) kl = δ ik δ jl . Next, we denote by 1 the × identity matrix.
We denote the Cartan-matrix of A as As all Cartan-matrices of simple Lie-algebras are deviations from the A -case, we give all other Cartan-matrices in terms of A up to corrections in terms of some E ij . The matrix D is always diagonal and just the identity in the simply-laced case. As a consequence, Eq. (B3) can be factorized in the simply laced case to A similar formula yields the conformal weights for the series of WB 0 -algebra minimal models, Here, r,s = (r − s mod 2) distinguishes between the Neveu-Schwarz sector (r − s ≡ 0 mod 2) and the Ramond sector (r − s ≡ 1 mod 2). These sectors correspond to periodic or anti-periodic boundary conditions, respectively.

Some examples
In the following we present the spectra of some rational CFTs with central charge c = 1 and c = 8 7 , respectively.
Therefore the true multiplicities of the conformal weights are one except for h ∈ { 1 16 , 9 16 , 5 4 }, which appear with multiplicity two.
Appendix C: S-matrix and fusion rules for WD 5 (9,10) With the characters (5.5) the S-matrix for the modular transformation χ i (e −2πi/τ ) = ij S ij χ j (e 2πiτ ) is easily found to be (φ = 1+ S WD 5 (9,10) = 1 2 √ 10 where rows and columns refer to the representations in the order listed in (5.4). Note that while this S-matrix fulfills S 2 = (ST ) 3 = 1 it is not symmetric. This is to be expected, as we have characters with non-trivial multiplicities.
As a final test for a theory to be a bona fide rational CFT the fusion rules, as computed from the characters and their S-matrix via the Verlinde formula 5 N k ij = r S i,r S j,r S † k,r S vac,r , have to be admissible, i.e. all fusion coefficients must be non-negative integers.