The $D(D_{3})$-anyon chain: integrable boundary conditions and excitation spectra

Chains of interacting non-Abelian anyons with local interactions invariant under the action of the Drinfeld double of the dihedral group $D_3$ are constructed. Formulated as a spin chain the Hamiltonians are generated from commuting transfer matrices of an integrable vertex model for periodic and braided as well as open boundaries. A different anyonic model with the same local Hamiltonian is obtained within the fusion path formulation. This model is shown to be related to an integrable fusion interaction round the face model. Bulk and surface properties of the anyon chain are computed from the Bethe equations for the spin chain. The low energy effective theories and operator content of the models (in both the spin chain and fusion path formulation) are identified from analytical and numerical studies of the finite size spectra. For all boundary conditions considered the continuum theory is found to be a product of two conformal field theories. Depending on the coupling constants the factors can be a $Z_4$ parafermion or a $\mathcal{M}_{(5,6)}$ minimal model.


Introduction
In recent years there has been a surge of attention directed towards the understanding of many-particle systems exhibiting topological order, i.e. phases which cannot be characterised by a local order parameter. Possible realisations of such topological quantum liquids in condensed matter physics are the fractional quantum Hall (FQH) states [44,50] and certain twodimensional frustrated quantum magnets [6,39,49]. The excitations in these systems display anyonic statistics and an understanding of their collective behaviour is essential for the classification of topological phase transitions. Particularly interesting are non-Abelian anyons where the interchange of two particles is described by non-trivial representations of the braid group complemented by fusion rules for the decomposition of product states. The fact that these non-Abelian anyons are protected by their topological charge has led to proposals for the use of such systems in universal quantum computation [40,51].
Some insight into the peculiar properties of many interacting anyons can be obtained in the context of simple model systems: such models can be obtained by associating anyonic degrees of freedom with each site of a lattice and defining interactions compatible with their braiding and fusion rules [20]. The phase diagram of the resulting lattice models can be studied based on the numerical computation of finite size spectra. This approach is particularly powerful for anyonic chains, i.e. one-dimensional lattices, where the numerical data can be compared against predictions from conformal field theory (CFT). Another approach, also in one dimension, makes use of the fact that the lattice model may become integrable for particular choices of coupling constants. For the solution of such models various analytical methods, e.g. in the framework of the Quantum Inverse Scattering Method (QISM), have been established which allow the study of the spectrum of their low-energy excitations, their thermodynamical properties including the long-distance asymptotics of correlation functions and even form factors [41,42].
So far much of the work on such lattice models has been focussed on systems of the non-Abelian Ising or Fibonacci anyons related to the quasiparticles in certain FQH states and their generalisations appearing in su(2) k Chern-Simons theories [5,20,31,45,59,60]. These anyons have relatively simple fusion rules which allows for tractable computation of systems with nearest and next-nearest neighbour interactions. Integrable points within the one-dimensional versions of these models have been identified as restricted solid-on-solid (RSOS) or interaction round the face (IRF) models constructed from representations of Temperley-Lieb algebras [20,37,36]. An alternative method to define an anyonic theory is via the Drinfeld doubles of a finite group algebra, D(G), and its representations [15]. The quasi-particles in these systems are irreducible representations (irreps) of D(G) labelled by their flux, i.e. an element of h ∈ G, and their topological charge determined by the transformation properties under the residual global symmetry commuting with the flux h.
Being a quasi-triangular Hopf algebra the quantum double allows for a direct construction of integrable quantum chains with nearest neighbour interactions described by a local Hamiltonian which is invariant under the corresponding symmetry [14,21]: within the QISM one obtains quantum spin chains on a Hilbert space being a tensor product of the finite-dimensional local spaces corresponding to a spin S, a qudit or a more general n-state quantum system. On the other hand, it is already known that for any given model whose local Hamiltonian has the symmetry of a quasi-triangular Hopf algebra associated with an anyonic theory, it is possible to construct quantum chains using the fusion path formalism [22]. Here the basis vectors are composed of sequences of anyons and we shall refer to this as a fusion path chain. The local Hamiltonian is formally identical in the spin and the fusion path formalism. Therefore, one should expect the bulk properties of the spin and the fusion path model to be the same. The finite size spectrum of low energy excitations, however, is known to depend on boundary conditions [4,11] and therefore should differ between the two realisations.
In this paper we study this problem for a specific one-dimensional anyon chain with nearestneighbour interactions. The underlying symmetry of the Hamiltonian is that of the Drinfeld double of a dihedral group, specifically D(D 3 ). In the following section we define this algebra and recall its irreps and the corresponding fusion rules. Then, using the spin basis, integrable models are constructed subject to periodic, braided and open boundary conditions, all of which being based on the usual QISM transfer matrix [24]. While for the fusion path basis we construct a fusion IRF transfer matrix whose series expansion contains the global one-dimensional Hamiltonian. In Section 3 we compute the bulk and surface properties of the model from the Bethe ansatz formulation of the spectral properties for the spin chain. The conformal field theory and operator content for the periodic spin chain version of the D(D 3 ) model has been identified previously [23]. In Section 4 we expand this work providing more details on the analysis as well as extending the study of the finite-size spectrum to the spin chain with braided and open boundary conditions. In addition we present results for the fusion path chain in support of the expectation that the low energy excitations of the D(D 3 )-anyon chain are described by the same CFT for all types of boundary conditions studied here, namely products of Z 4 parermion and M (5,6) minimal models.

The D(D 3 ) algebra
The model we consider in this article has the underlying symmetry of the Drinfeld (or quantum) double of a finite group algebra. The finite group we utilise is the dihedral group of order six, D 3 , and is isomorphic to the group of permutations on three elements, S 3 . This group is based upon the symmetries of an equilateral triangle and has the presentation, where e is the identity element of the group, σ is a rotation and τ is a flip. The Drinfeld double of this group is defined as the vector space, where * denotes an element from the dual space of CD 3 . This space forms a quasi-triangular Hopf algebra when equipped with the multiplication and coproduct, The remaining structure is uniquely determined by these relations [12,46]. This algebra has an associated universal R-matrix i.e. an algebraic solution to the Yang-Baxter equation.

Representations
The representation theory of the Drinfeld doubles of finite group algebras are well known [17,33]. The irreducible representations of D(D 3 ) are classified by the conjugacy classes of D 3 . For a given conjugacy class a representative element is chosen and the representations of the centraliser subgroup of this element are determined. An action on conjugacy class is defined and then combined in prescribed manner with an irrep of the centraliser. This yields an irrep of D(D 3 ) labelled by both the representative element and the irrep of the centraliser. The irreps associated with the conjugacy class {e} are: where ω = e 2iπ 3 . The irreps associated with the conjugacy class {σ, σ 2 } are: where k ∈ {0, 1, 2}. The irreps associated with the conjugacy class {τ, στ, σ 2 τ } are: The anyonic theory corresponding with D(D 3 ) associates an irrep with an anyon [15]. For convenience it is simpler to denote each irreps by a single letter, a, ..., h. We equate a = (e, +), b = (e, −), c = (e, 1), d = (σ, 0), e = (σ, 1), f = (σ, 2), g = (τ, +), h = (τ, −) Properties of the anyons are inherited from their associated irreps, e.g. the dimension of an anyon equals the dimension of its corresponding irrep.

Fusion Rules
Required for an anyonic theory are the fusion rules of particles. These rules are defined by the tensor product decompositions of the associated irreps [33]: It follows that the local Hamiltonians commute with each other and with the action of the  algebra: h (1) , h (2) = 0 and (π g ⊗ π g )∆(a), h (k) = 0, for all a ∈ D(D 3 ). Therefore they have the underlying symmetry of D(D 3 ) as the R-matrix did. From explicit calculation of h (1) and h (2) we find, where Π is usual two-site permutation operator and * is, and herein reserved for, complex conjugation. Both local Hamiltonians (5) are self-adjoint.

Global Hamiltonian
In the following we shall consider a variety of models with interactions described by the local operators (5) but subject to different boundary conditions. As a consequence of the existence of two distinct local Hamiltonians the global Hamiltonian is comprised of two terms weighted by a free coupling parameter θ ∈ [0, 2π] as For all boundary conditions considered below these models are integrable thanks to the existence of a commuting transfer matrix. Furthermore the two components of the global Hamiltonian will commute, This commutativity will be particularly useful in the investigation of the models as it allows us to sutdy the spectra of H (1) and H (2) separately. Typically the spectra of H (1) and H (2) will be identical or of a related form.

Periodic Spin Chain
We begin by considering the D(D 3 ) model as a spin chain with periodic boundary conditions: its global Hamiltonians are defined by for k ∈ {1, 2}. Note that the periodic closure by the term h By construction this transfer matrix is a polynomial of degree L in the variables z 1 and z * 2 . It has been observed that this transfer matrix factorises and that its eigenvalues are always of the form [24] Λ(z 1 , Therefore the eigenvalues can be conveniently described in terms of their zeroes z k, ≡ iωe x k, , for k = 1, 2 and = 1, . . . , L. Furthermore, starting from the D(D 3 ) fusion rules functional relations satisfied by the transfer matrices (or equivalently their eigenvalues) can be derived [21]: where λ k (z) are analytic functions. This implies that the two sets of parameters {x k, } L =1 must independently satisfy the Bethe equations [24] (−1) It is important to note that while there are exactly L Bethe roots in each set {x k, } L =1 , they are allowed to be at ±∞, but at most one at each. The energy eigenvalue of H (k) corresponding to the set of Bethe roots {x k, } is given by Here we have used the property that sets of Bethe roots are invariant under complex con- Since the local Hamiltonians are Hermitian by construction, the energies (11) must be real. This reality of the energy imposes an additional physicality constraint on solutions to the Bethe equation (care must be taken to deal with roots at ±∞ appropriately). We note that all the root configurations considered below in the discussion of the spectrum of the system do satisfy this condition.
Let us remark that the Bethe equations (10) and the corresponding energies (11) of the D(D 3 ) spin chain of even length L coincide with those of the three-state Potts spin chain with L/2 sites [3]. We shall use this equivalence below to identify some of the thermodynamical properties of the D(D 3 ) chain.
The energy eigenvalues of the complete Hamiltonian are characterised by two solutions to the Bethe equations (10). As a consequence of Equations (6), (7) and along with (11) they are given by provided these energies (or the corresponding root configurations) pair. Specifically, levels are said to pair if the two corresponding sets of Bethe roots form an eigenvalue of the transfer matrix, see Equation (8). As the two sets of Bethe roots need not correspond to a unique eigenvalue of the transfer matrix, e.g. there may be two eigenvalues that differ by a constant factor or an eigenvalue might be degenerate, we refer to the total number of eigenvalues, including degeneracies, as the pairing multiplicity.
The total momentum of the corresponding state can also be given in terms of the two sets of Bethe roots: at z 1 = 1 = z 2 the transfer matrix becomes a shift operator by one site. Therefore the momentum operator is P = −i ln t(1, 1). By construction the eigenvalues of this operator are real (2π/L times an integer for periodic boundary conditions considered here). Unlike for the energy (11) it is not possible to identify partial momentum contribution from one of the participating Bethe configurations uniquely [24]. Using the invariance of the sets of Bethe roots under complex conjugation we use as definition of the partial momenta P (k) . For later use we note that the second expression is half of the momentum of the three-state Potts spin chain [3]. Consistency with Eq. (8) implies that the complete momentum is related to the partial ones as Again, roots x k, = ±∞ have to be taken into account to ensure finite (partial) momentum. The total momentum is given by the difference of partial momenta reflecting the fact H (2) is the spatial inversion of H (1) . The remaining constant represents a macroscopic effect, details of which have been discussed in earlier works [24].

Braided Chain
One closed chain proposed as an alternative to the periodic chain is the braided chain [25,34,38]. In this model, translational invariance is replaced by invariance under a global braiding operator. As a consequence the underlying symmetry of the model will not be broken, i.e. it has the full global D(D 3 ) symmetry. The global Hamiltonians for these boundary conditions are defined by, There also exist different possible definitions for the braiding operator b i relating to other limits of R(z 1 , z 2 ) [21]. The integrability of the braided model is ensured by the existence of a transfer matrix t(z 1 , z 2 ), which can be found in [21]. Eigenstates of this model are again characterised by the Bethe Equations (10). As in the periodic case there must be L Bethe roots, this time with one Bethe root allowed at +∞ but none allowed at −∞.
As mentioned above the Hamiltonian of this model is invariant under the action of the global braiding operator. Specifically, we find that B can be realized by the transfer matrix of the braided model as B = t(1, 1). Furthermore, the braiding operator is idempotent: from the analysis of small systems we find that B n = I, when n = 3L, L even, n = 2L, L odd.
This allows us to define an analog of the momentum operator as the generator of braiding operations by P b = −i ln B. The eigenvalues of P b are restricted to integer multiples of either π L or 2π 3L depending on the parity of L.

Open Boundary Conditions
We also consider spin chains with open boundary conditions. In this case integrable models derive from representations of Sklyanin's reflection algebra [57]. c-number representations of this algebra define possible boundary terms. For the present model these K-matrices are found to be the same as have been determined for the D(D 3 ) one-parameter R-matrix [13]. The global Hamiltonians are, Here, the boundary operators B (k)− (B (k)+ ) act on the first (last) site of the chain, respectively. There exist three possible (independent) options for each of these operators, namely The real boundary amplitudes where χ ± k , k ∈ {1, 2}, have to satisfy χ + 1 χ + 2 = χ − 1 χ − 2 = 0. Like the periodic and braided models integrability is derived from the existence of a transfer matrix (see [21,24] for the open D(D 3 ) transfer matrix) and the eigenstates of the Hamiltonian are classified by sets of Bethe roots. The Bethe equations for the Hamiltonian H (k) are independent of the choice of the boundary operators, with Bethe roots always appearing in pairs of ±x and where In addition to the explicit dependence of the Bethe equations on the boundary amplitudes χ ± k the latter determine the number d k of Bethe roots x k,j : for the open chain with free ends, i.e. χ ± k = 0, there are d k = 2L Bethe roots with at most one pair of roots at ±∞ [24]. A non-zero boundary term at one end of the chain, i.e. choosing one of either χ + k or χ − k non-zero, changes the number of roots to d k = 2L+2, while for boundary terms at both ends there are d k = 2L+4 Bethe roots. We refer to the extra pairs of roots appearing as compared to the free-ends case as boundary roots. 1 These boundary roots are finite for non-zero boundary amplitudes χ ± k but approach ±∞ in the limit of free-ends.
The energy of the open boundary Hamiltonian H (k) corresponding to a solution of Eqs. (17) is given by Again we note that the energy eigenvalues are real as the global Hamiltonian is Hermitian. Additionally, it is important to note that the presence of boundary interactions breaks the D(D 3 ) invariance of the model. Only for free ends the open model has this invariance.

Fusion Path Analogues
As the local Hamiltonians have the symmetry of D(D 3 ) it is possible to create fusion path analogues [22]. Depending on the boundary conditions imposed the global Hamiltonians may or may not be equivalent to their spin formalism counterparts discussed above. The construction of the analogous fusion path chains uses the Pasquier's method of representation theory reliant face-vertex correspondence [54]. This allows the fusion path analogues to be considered as the Hamiltonian limits of RSOS models and proves their integrability. The connection between fusion IRF models and many other physical systems has already been established [28].
We first define the fusion path basis. Basis vectors of the fusion path space are of the form, where a i ∈ {a, b, c, d, e, f, g, h} and neighbouring labels satisfy the condition, = {ag, bh, cg, ch, dg, dh, eg, eh, fg, fh, ga, gc, gd, ge, gf, hb, hc, hd, he, hf}.
Thus a i+1 must appear in the fusion of a i and g. Diagrammatically a basis vector corresponds to the figure below, where the joining of two lines indicates fusion which occurs from left to right and top to bottom, To construct local operators on this space we utilise F -moves (generalised 6-j symbols), which allow the temporary re-ording of fusion, b c In terms of these F -moves we can define two-site projection operators which act non-trivially on a single link i of the fusion path lattice 2 , The unitary F -moves associated with D(D 3 ) can be calculated explicitly from the representation theory of D(D 3 ) [56]. Below we shall consider local interactions defined analogously to the nearest neighbour spin chain ones, Eq. (5): Note that these Hamiltonians act on three consecutive labels of the fusion path basis but only can change the middle label. As a consequence of the equivalence of the local interactions between the spin and fusion path formalisms the global models in the two formalisms may differ only by boundary conditions. The open model with free ends and braided model both have D(D 3 ) invariance which means that the fusion path and spin versions of these chains are equivalent. This implies that the energy spectra are identical and the denegeracies appearing in each formalism are related via a mapping. For convenience we use the degeneracy of the spin chain formalism for these choices of boundary conditions. For periodic boundary conditions, however, neither the spin chain nor fusion path model has the complete D(D 3 ) invariance and thus the two models, while sharing bulk properties, are distinct [22].
To construct a periodic model in the fusion path basis we need to consider the space spanned by the basis vectors satisfying a 0 = a L . As a consequence of the fusion rules this periodic closure is possible only for lattices of even length L. Furthermore, they lead to the decomposition of the Hilbert space where each of these subspaces has dimension 3 L + 1. The global Hamiltonians are, The integrability of this model can be established based on the existence of an R-matrix connected to an RSOS model whose heights correspond to the labels of the irreps of D(D 3 ). As the R-matrix of Eq. (4) is expressible in terms of the D(D 3 ) projection operators it follows that there exists an equivalent operator in the fusion path basis [54], satisfying a face Yang-Baxter equatioñ The weights appearing inR(z 1 , z 2 ) are used to construct the commuting transfer matrix [32], Once again this transfer matrix generates a family of commuting operators, including the global HamiltoniansH (1) andH (2) , implying integrability. Again, the dependence of the transfer matrix on two spectral parameters guarantees commutativity of the two components Through the analysis of small finite size systems we are able to make two important observations. Firstly, we find that the eigenvaluesΛ(z 1 , z 2 ) of the fusion path transfer matrix factorize into two polynomials of degree L in the same manner as those in the spin chain case (8). Secondly, we find that the eigenvalues satisfy functional relations similar to the Eqs. (9), i.e.
Again, λ k (z) are analytic functions. The ± sign depends upon the eigenvalue in question.
Preliminary calculations indicate that this relation can be obtained using the fusion procedure for RSOS models [7,43]. Like the periodic spin chain case these functional relations lead to Bethe Equations which have to be satisfied by the zeroes of the transfer matrix eigenvalues, where η = ±1 corresponds to the allowed signs in the functional relations (21). This sign was not present in the Bethe equations for the spin chain (10) indicating the likely presence of different excitations. As before, every set of roots {x k, } solving the Bethe equations (22) parametrizes an eigenvalue of the fusion path model. The energy eigenvalue ofH (k) is again given by Equation (11).
In previous studies of anyonic fusion path models [20,37] integrability was also observed by relating them to transfer matrices associated with RSOS models. However, in these instances the fusion path R-matrices corresponded to representations of the Temperley-Lieb algebra. Every RSOS model can be naturally associated with a graph where nodes represent the labels of anyons in the theory which are connected if they can appear next to each other in the fusion path basis as given by Eq. (18), see [53]. For the D(D 3 ) model considered here we obtain the graph given in Figure 1. This graph is equivalent to McKay's representation graph for the representation π g of D(D 3 ) [48]. We note that this graph shows that the D(D 3 ) fusion path model does not correspond to any of the known RSOS models associated with Dynkin diagrams [53,52,55,62]. It also does not appear amongst the more general graphs associated with other RSOS models [16].

The Bethe Equations and Exact Results for Spin Chains
As a consequence of Eq. (12) the ground state energy of the model is always obtained by the following combinations (NB: provided that these states are allowed to pair) [24] Figure 1: A graphical representation of allowed neighbouring labels in the fusion path 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 as given by Equation (18).
h is the highest. An immediate implication of this form of the ground state energy is that there are level-crossings for θ being integer multiples of π 2 leading to first order quantum phase transitions. Here we will use a different consequence of (23): the complete spectrum of the model can be obtained from an analysis at these particular points in combination with the implementation of the pairing rules [24]. An additional simplification arises from the fact that the spectra of H (1) and H (2) are identical for most boundary conditions considered in this paper: this allows us to restrict the analysis of the low-energy spectrum to those of the Hamiltonians H θ=0 = H (1) and H θ=π = −H (1) whose ground state energies are E h , respectively. Only in the case of open boundary conditions with D(D 3 )-symmetry breaking boundary fields the spectra of ±H (1) and ±H (2) are independent.

Energy Density in the Thermodynamical Limit
The study of the excitation spectrum of a model requires knowledge of the bulk properties. As such we recall previously obtained results and present them here for completeness [24]. In the thermodynamic limit L → ∞ bulk properties of the system are independent of the boundary conditions imposed. Therefore we can compute the energy density from the Bethe equations (10) for the periodic spin chain. To this end the solutions to the Bethe equations need to be classified and the root configurations corresponding to the ground state and low energy excitations have to the identified. As mentioned above, the Bethe equations for the periodic D(D 3 ) spin chain arise also in the context of the 3-state Potts model. For the latter the classification of solutions has been obtained by Albertini et al. [3,2], see also Ref. [24]. In particular, numerical diagonalisation of the transfer matrix shows that the lowest energy states of H θ=π consist of three different Bethe root (z l ≡ z 1,l ) types: 3. 2-strings, where the Bethe roots come in pairs, z l = iωe x s l + 2iπ as well as a limited number of Bethe roots at ±∞. Letting N + , N − and N 2 be the number of +-strings, −-strings and 2-strings respectively and setting n ±∞ ∈ {0, 1} to be the number of Bethe roots at ±∞ then we have the constraint In the 3-states Potts model these root types were also identified, along with a few other which we don't consider, with the additional constraint n +∞ = n −∞ [3].
Allowing combinations of these roots we then find that the for the periodic Hamiltonian we can take the logarithm of the Bethe equations and define the following set of counting functions: In the thermodynamical limit we find that the ground state for H θ=π consists entirely of 2strings [3,24]. For finite size systems this configuration is only realised when L is even. The lowest energy Bethe root configuration for odd L is given by L−1 2 2-strings and one ±-string. Similarly we find that the lowest energy states of H θ=0 consist of the same three Bethe root types and hence we have the same counting functions. In the thermodynamical limit the Bethe root configuration of the ground state consists of only negative and positive Bethe roots appearing in the ratio of three −-strings to one +-string. For finite size systems this configuration is only realised when L is a multiple of four. The lowest energy Bethe root configuration for the other chain lengths also consists of only negative and positive Bethe roots appearing approximately in the ratio 3:1.
Based on these observations the root density formalism [63] can be applied to compute the corresponding energy densities: the density of 2-strings in the thermodynamic ground state of H θ=π and their dressed energies H θ=π are determined by linear integral equations These equations (and the corresponding ones for H θ=0 ) can be solved by Fourier transformation giving the ground state energy densities [35,3,24] 1

Fermi-velocity
The low-energy excitations over these ground states have a linear dispersion and their Fermi velocities have been computed within the root density formalism in the context of the threestate Potts model [3]. As discussed above, it is possible to identify energy and momentum eigenvalues of this model with those of the partial Hamiltonians H (k) (11) and corresponding momenta (13) using the equivalence of the corresponding Bethe equations.
Noting that the contribution of a single 2-string to the partial momentum can be expressed in terms of their density in the thermodynamic limit we can elimininate the rapidity x from Eqs. (24) and (26) to obtain the dispersion relation (p) of 2-strings. Therefore the Fermi velocity of low lying excitations of H θ=π is found to be in agreement with the finite size analysis of the spectrum performed in [23].
Similarly we can compute the Fermi velocity of gapless excitations for the Hamiltonian H θ=0 . Again the result is twice than what has been found for the three-state Potts chain [3], i.e. v F = 3 2 .

Boundary Fields
We can also determine the exact expressions for the surface energy, i.e. the L 0 contributions to the energy, for the open model with interacting boundary fields. Firstly we find the ground energy for the Hamiltonians H θ=π and H θ=0 , and then extend this result to generic θ using Eq. (23). Starting with the Bethe equations (17) we can apply the same method that was used to calculate bulk energy density in Section 3.1. For the open spin chain with free ends the surface energy has been computed previously [24]. Due to the symmetry of the Bethe equations the general case can be studied in the context of the open chain with a single boundary field present, e.g. χ + 1 = 0 and all other boundary amplitudes vanishing, and compute the correction to case of the free ends.
For even L and χ + 1 not too large the Bethe root configuration corresponding to the ground state of H θ=π consists of L 2-strings, distributed symmetrically around the imaginary axis (just as in the free-ends case [24]) and, in addition two boundary Bethe roots. The latter are found to be either ±-strings depending on the sign of χ + 1 . As the magnitude of χ + 1 is decreased these boundary Bethe roots tend towards ±∞ to recover the Bethe configurations of the free-ends model. 3 We find that the correction to surface energy for the Hamiltonian H θ=π with one interacting boundary field, compared to the free ends case, is Similarly, we find the surface energy correction for H θ=0 , Formally, the calculation of these corrections requires − 1 . The restriction to this interval is due to the change in the analytical behaviour of the Bethe equations which is reflected by the presence of poles in the above expressions.
Putting these results to together we are able to determine the ground state energies of H θ=π and H θ=0 up to order L 0 In Figure 2 we plot the predicted functions g θ=π (χ) and g θ=0 (χ) compared to the equivalent numerical values obtained by solving the the Bethe equations (with χ − 1 = χ ± 2 = 0).  The numerical solution of the Bethe equations for larger boundary amplitudes is limited by numerical instabilities resulting from, e.g., boundary Bethe roots passing the other Bethe roots. The asymptotic behaviour of the surface energy corrections for large χ does, however, coincide with what is expected from the corresponding eigenvalues of the boundary operators B (k)± (16), i.e. −1 and 2. This leads us to conjecture that the energies are analytically correct for all values of χ ± k ∈ R. Using Equation (23) along with the relations, we are able to determine the ground state energy for of H θ for generic θ. We should again recall that we have the constraint χ + 1 χ + 2 = χ − 1 χ − 2 = 0 and that the interacting boundary terms break the D(D 3 ) invariance of the model.

Excitations and Conformal Field Theories
The presence of a coupling parameter makes the identification of a conformal field theory a difficult task. Every energy level of the Hamiltonian depends on the coupling parameter θ, moreover which of the energies is the lowest will change with θ. This implies that the model can not be described by a single conformal field theory but rather but multiple ones. To simplify the issue we first study the model at the level crossings, i.e. for coupling parameters being integer multiples of π 2 . As discussed above the energy spectrum at these points is that of the partial Hamiltonians H θ=0 = H (1) and H θ=π = −H (1) (up to degeneracies). For these we can use powerful machinery to accurately describe the model at these special points. It turns out that the low energy effective theories of the partial models are given by minimal models.
Once the critical theories at these special couplings have been identified it is a relatively straight-forward, albeit non-trivial, task to obtain the conformal operator content of the complete model for general coupling θ: the main difficulties come with identifying the previously mentioned pairing rules and connecting them with some conserved quantity of the model.
To make the presentation self-contained we begin by presenting our results on the low energy spectrum of the periodic spin chain reported previously [24,23] withan extended discussion of the residual D(D 3 ) symmetry under these boundary conditions and the pairing rules. This section is followed by new results of our studies of the critical properties of the periodic fusion path chain and the models with braided and open boundary conditions.

Periodic Spin Chain
As discussed above the spectrum at the the level crossings is expected to be described by a single conformal field theory. As a consequence of conformal invariance the scaling behaviour of the ground state energy is predicted to be [9,1] where c is the central charge of the underlying Virasoro algebra. For a given realization of the CFT its operator content is constrained by modular invariance of the partition function and the particular choice of boundary conditions [11,10]. Further constraints are imposed by locality of the physical fields. The primary fields present in the critical model determine the finite size energies and partial momenta of the excitated states: This allows us to determine the scaling dimensions X = h +h and conformal spins s = h −h of the primary fields (n,n are non-negative integers) from numerical finite size data obtained by solution to the Bethe equations along with Equations (11) and (13). Note that due to the massive degeneracies appearing in the spectrum for couplings θ being integer multiples of π/2 the complete momenta are not unique. The partial ones entering (29), however, are. This allows the use of finite size data at the level crossings for the identification of the critical theory.

Spectrum of H θ=π
The ground state energy of H θ=π is known to be [3,24] Using the Fermi-velocity (27) computed before the central charge of the effective field theory for the low energy degrees of freedom in H θ=π is identified to be c = 4/5. Hence, this sector of the model is in the universality class of the minimal model M (5,6)  To identify the operator content of the periodic spin chain we have solved the Bethe equations (10) for lattice sizes up to a minimum of L = 40, although over 100 sites were considered whenever possible. The sequence of finite size estimations for the scaling dimensions has then be extrapolated to get a numerical approximation X ext θ to the scaling dimension which can then be identified with a pair (h,h) of conformal weights from the Kac table. In Table 1 we present our data for the low lying excitations appearing in the θ = π sector of the periodic spin chain for even chain lengths. For the analysis of the finite size spectrum it is convenient to classify excitations of the model in terms of symmetry sectors. For even L the Hilbert space of the spin chain can be decomposed as However, as periodic closure of the system breaks the D(D 3 ) invariance of the model we can no longer use this decomposition directly. Instead we find it useful define four residual symmetry sectors, i.e. π a ⊕ π b , π c , π d and π e ⊕ π f . Here, the symmetry sector π a ⊕ π b is defined to Table 1: Scaling dimensions X π extrapolated from the finite size behaviour of the ground state and low energy excitations of H θ=π (periodic) for even L. (h,h) are the predictions from the M (5,6) minimal model. We have also indicated the D(D 3 ) sector in which the state appears and its conjectured degeneracy. The operator content of the sector π d is obtained from that of π c by interchanging h andh.
be the subspace of π ⊗L g composed of all one-dimensional irreps (π a and π b ) appearing in its decomposition, likewise for the other sectors. Under this definition we find that all of these (non-intersecting) symmetry sectors are invariant under the action of global Hamiltonian of the periodic spin chain with generic θ.
Apart from fixing transformation properties of the excited states their classification according to symmetry counting arguments can be used to conjecture the degeneracy of each excitation for large finite systems based on the finite-size spectra, also shown in Table 1. Furthermore, it has been noticed [23] that the symmetry of the eigenstates is connected to the number of Bethe roots at ±∞, see Table 2. Finally, we observe that every excitation (h,h) appearing in the sector π c can be related to an excitation (h, h) appearing in sector π d via a mapping of Bethe roots, {x j } → {−x j }.
The excitations appearing here for even L coincide with those from the self-dual ferromagnetic 3-state Potts quantum chain subjected to either periodic of twisted boundary conditions [61]. This stems from both models being constructed from the same set of solutions to the star-triangle equation [19], albeit with different limits applied. The excitations in the π a ⊕ π b and π e ⊕ π f sectors have been observed in the charge Q = 0 and Q = 1 sector of the periodic 3-state Potts chain, respectively. The excitations in the π c and π d sectors correspond to the 3-state Potts chain with twisted boundary conditions allowing for the non-half-integer spins observed. The excitations appearing in the periodic Potts chain have also been determined using Bethe ansatz methods by Albertini et al. [3].
We have also determined the excitations for chains of odd length. In this case the states can be classified by the eigenvalue of the state under the D 3 rotation operator, σ. The excitations are given Table 3. Table 3: As Table 1 for H θ=π (periodic) when is L odd. Symmetry is classified by the action of the D 3 rotation σ. σ X ext.
These excitations are not present in the 3-state Potts model since the equivalence to this model is restricted to L even. We were able to again classify the symmetry sectors in terms of Bethe roots as presented in Table 4. We found that n −∞ = 0 in the case of odd length chains.
The ground state energy is known to be [3,24], Using the Fermi-velocity we find that the central charge is 1, which does not uniquely define a conformal field theory. The field content of the theory is obtained from the finite size spectrum. One method of determining the the finite size spectrum is using the dressed charge formalism (see appendix) leading to the identification of the Z 4 parafermion theory [64,29] coinciding with the anti-ferromagnetic 3-state Potts model. The allowed conformal weights for this theory are [29,47] h,h ∈ l(l + 2) Alternately we can solve the Bethe equations directly and determine the scaling behaviour of the low-lying excitations. Here we must consider L = 0, 1, 2, 3 (mod 4) separately, see Tables 5-7 below. In particular, we find that the finite size gap of the lowest states for = L (mod 4) = 0 is determined by an (anti-)chiral Z k=4 spin field with conformal weight h = (k − )/(2k(k + 2)). Table 5: Scaling dimensions X 0 extrapolated from the finite size behaviour of the ground state and low energy excitations of H θ=0 (periodic) for L = 0 (mod 4) (the error of the extrapolation is smaller than the last displayed digit). (h,h) are the predictions from the Z 4 parafermionic CFT. For the other columns, see Table 1.  Table 5 for H θ=0 (periodic) when L = 1 (mod 4). The excitations for chains with length L = 3 (mod 4) have the same exponents but the opposite spin. Symmetry is classified by the action of the D 3 rotation σ. σ X ext.  Table 7: As Table 5 but for H θ=0 (periodic) when L = 2 (mod 4).
As with the previous case we again can partition the excitations according to the residual symmetry sectors. These sectors are still characterised by the number of Bethe roots at ±∞ as described in Table 2.
Comparing these excitations to the anti-ferromagnetic 3-state Potts chain [3,47] (for L even), we find that the excitations in the π a ⊕ π b and π e ⊕ π f were previously identified and restricted to the n +∞ = n −∞ case. We were unable to find any literature dealing with the anti-ferromagnetic end of the twisted 3-Potts model. We expect, however, that the excitations in that case to match those appearing in the π c and π d sectors. Similarly the excitations for odd L have not previously been studied.

Pairing rules and discussion
The results on the low energy spectra H θ for θ = 0, π completely determine the critical behaviour (up to degeneracies) of the periodic D(D 3 ) spin chain at all the level crossings, i.e. when θ is a multiple of π 2 : this is a consequence of the fact that the partial energies, i.e. eigenvalues (11) of H (1) and H (2) , and momenta (13) corresponding to a given Bethe root configuration are identical (although the partial momenta enter in the definition of the total momentum (14) with opposite signs).
For generic values of θ the energy eigenvalues of the periodic spin chain are given by (12) in terms of two root configurations of the Bethe equations (10) provided that these configurations pair. Based on studies of small system sizes it has been observed that two sets of Bethe roots pair to form an eigenvalue of the transfer matrix, Equation (8), if and only if they have matching number of roots at ±∞ [24]. As a consequence of the classification of excitations according to their symmetry above, see Tables 2 and 4, this is equivalent to saying Bethe root configurations (and their corresponding energies) pair if and only if they belong to the same symmetry sector. Furthermore, we have observed that within a symmetry sector pairing is uniform in the sense that every two sets of Bethe root configurations within a symmetry sector pair the same number of times, a quantity referred to as the pairing multiplicity. The relationship between pairing multiplicity and symmetry sectors is documented in Table 8. Table 8: The pairing multiplicities, m p , of the periodic spin chain for any two energies of H (1) and H (2) that belong to the same symmetry sector. If the energies do not belong to the same symmetry sector then they do not pair, i.e. the pairing multiplicity is zero.
This information, along with Equations (12) and (29) and the relevant tables, is sufficient to determine the energies and degeneracies of the ground state and low lying excitations of the model for generic θ. The resulting spectrum is that of a direct product of two conformal field theories. The physical fields appearing in the combined theory are composite operators with scaling dimension X tot = X (1) + X (2) = 2 k=1 (h (k) +h (k) ) with (h (k) ,h (k) ) being the conformal weights from the two components. Similarly, the total spin of an excitation can be calculated from Equations (14) and (29) (2) . Note that application of the pairing rules to the conformal dimensions identified for the periodic spin chain above ensure that this total spin is always either an integer or half-integer which guarantees locality of the physical fields.
The phase diagram of the complete model can be summarised by Figure 3.

Periodic Fusion Path Chain
By construction this chain differs from the periodic spin chain discussed in the previous section only through boundary conditions. Therefore, the two models share their bulk properties, including energy per unit lattice site, Fermi velocity and central charge. To identify the operator content of the low energy effective theory for the fusion path model we have computed the complete spectrum of the Hamiltonian numerically for up to L = 10 sites. We find that part of the spectrum coincides (numerically exact) with eigenvalues of the periodic spin chain (although the corresponding degeneracies in the different formalisms do not match). Specifically, this applies to the energies which have been associated to the π a ⊕ π b and π e ⊕ π f symmetry sectors above (we emphasize that the periodic fusion path model can only be constructed for L even). In addition we have diagonalized the transfer matrix for up to L = 8 sites. From these results we find that the transfer matrix eigenvalues factorize into two polynomials as in (8). The corresponding roots of these polynomials can be used to parametrize the eigenvalues and are conjectured to be given by the Bethe equations (22).
We have verified this conjecture by comparing the energies obtained from the Bethe equations with those obtained by numerical diagonalization. The eigenvalues appearing in both the spin chain and the fusion path formalism are described by root configurations solving (22) with η = +1. This includes the ground states of H θ=π and H θ=0 with energies given by Equations (30) and (31), respectively. Generally, we find that each set of Bethe roots corresponding to an eigenvalue of the periodic fusion path chain contains either only finite roots or, for η = +1, exactly two roots at +∞ and −∞.

Spectrum ofH θ=π
As this model shares bulk properties with the periodic spin chain this model lies in the M (5,6) universality class with central charge c = 4 5 . From the diagonalization of the transfer matrix we can associate root configurations to the ground state and low lying excitations ofH θ=π which consist of ≈ 1 2 L 2-strings and a few ±-strings. Solving the conjectured Bethe equations for systems of up to 100 sites we have found the excitations given in Table 9. Table 9: Scaling dimensions X π extrapolated from the finite size behaviour of the ground state and low energy excitations ofH θ=π (fusion path) for L = 0 (mod 2). From the numerical diagonalisation of the Hamiltonian for finite size systems we conjecture the degeneracy of each pair of conformal weights, (h,h). The value η is from the Bethe Equations (22).
Note that the new excitations corresponding to roots of the Bethe equations (22) with η = −1 do not correspond to excitations of the 3-state Potts chains subject to the boundary conditions studied previously [61]. Moreover, because the formulation of this chain is reliant on D(D 3 ) symmetry, which is not present in the usual 3-state Potts local Hamiltonian, it is reasonable to expect that these excitations won't appear for any other formulation of the 3-state Potts model.

Spectrum ofH θ=0
Using similar methods we can determine the low energy excitations of the fusion path model for θ = 0. This is done for even L where, as in the spin chain case, we have to discuss the cases of L 2 even or odd separately. The conformal dimensions identified from the low-lying excitations are shown in Table 10.
As was the case with theH θ=π model we find new excitations, again characterised by η = −1, which do not appear in 3-state Potts chains subject to the boundary conditions studied previously. Due to commensurability conditions the spectra can only be compared for lattices with lengths differing by multiples of 4. Therefore we do not have sufficient numerical data from exact diagonalization of the Hamiltonian to present any conjectures concerning the degeneracy of the excitations.

Pairing rules and discussion
In contrast to the periodic spin formulation the pairing rules for the periodic fusion path chain could not be easily determined. The residual symmetry sectors used in the previous section are no longer present. It is possible, however, to define a conserved topological charge for the fusion path model based of the D(D 3 ) F-moves aforementioned [20,30]. This charge allows to differentiate between different topological sectors which are labelled by the irreps of D(D 3 ). A Table 10: Scaling dimensions X 0 extrapolated from the finite size behaviour of the ground state and low energy excitations ofH θ=0 (fusion path) for L = 0, 2 (mod 4). We have omitted the column regarding the degeneracy of the excitation due to a lack of data. The error of the extrapolation is smaller than the last displayed digit. The association of a ×2 with a pair of conformal weights implies that there are two excitations with those conformal weights that have distinct energies for finite size systems.
better understanding of these topological symmetry sectors will be necessary to gain insight into the pairing rules for this model. Finally we want to stress that while the computation of scaling dimensions is based on the solution of the Bethe equations (22), which can be achieved for relatively large L, the identification of allowed Bethe root configurations still relied upon the explicit diagonalisation of the transfer matrices of small systems. Thus while we have a high level of confidence in the accuracy of the scaling dimension it is not clear whether all primary excitations have been identified. In particular, the existence of additional primary operators with larger scaling dimensions cannot be ruled out.

Braided Chain
As discussed above, the D(D 3 ) quantum chains in the spin chain and the fusion path formalism are equivalent for braided and open boundary conditions. Therefore we discuss of their critical properties in the former.
Since the braided chain has the full D(D 3 ) symmetry it would be possible to classify all eigenstates of this model in terms of the irreps. We find, however, that the enlarged symmetry gives rise to additional degeneracies in the spectrum which extend across multiple symmetry sectors of the braided spin chain. This implies that for these boundary conditions a classification of symmetry sectors based on the presence of infinite Bethe roots is no longer possible. In particular, this leads to higher degeneracies in the spectrum of the braided spin chain model as compared to the periodic one.

Spectrum of H θ=0
As H θ=0 = −H θ=π we can again use our results for the periodic spin chain to discuss the low energy spectrum of the braided one. Just as for θ = π the excitations appearing in the braided chain are those present in the π a ⊕ π b and π c sectors of the periodic chain for even L and in the σ = 1 sector for odd L. Thus we can deduce the excitations appearing in the braided model from Tables 5-7:

Pairing rules and discussion
Like the periodic case the results for the braided chain at θ = 0, π determine the low energy spectrum at all of the level crossings. Again, the full spectrum for generic θ is given by (12) in terms of two Bethe root configurations provided that these configurations pair. For the braided model it has been observed previously, that any two solutions to the Bethe equations can be paired to form an eigenvalue of the transfer matrix given by Equation (8) [24]. This is consistent with statement above, that a given root configuration corresponds to eigenstates in multiple symmetry sectors. Another difference to the periodic spin chain is that the assignment of a root configuration to a particular symmetry sector of the braided chain is different for H (1) or H (2) . Despite these technical differences we find that the pairing multiplicities, i.e. the number of times two solutions to the Bethe equations pair, depends solely on the symmetry sector the eigenvalue of the transfer matrix lies in and is defined by Table 8. As before, pairing determines the physical fields appearing in the low energy effective theory of the braided spin chain. Their scaling dimensions and total spin of these composite operators are related to the ones of their components as before. We note, however, that the total momentum in the braided model is not a multiple of 2π/L but instead constrained by Eq. (15).

Open Chain
As in the case of braided boundary conditions it is sufficient to discuss the low energy behaviour of the open chain in the spin chain formulation. At level crossings conformal invariance predicts that the lowest energies of the open chain with free ends, i.e. χ ± 1 = χ ± 2 = 0, have the scaling behaviour given by [9] where c is the central charge, v F is the Fermi-velocity, h is a conformal weight and n is a non-negative integer.

Spectrum of H θ=π
The ground state energy for θ = π [24] is, As is the case with the periodic chain we have that v F = 3 and c = 4 5 , yielding the same CFT as expected. Using an analogous method to that outlined in Section 4.1.1 to calculate X ext. we can extrapolate values for the conformal weights, h ext. . The values are presented in Table 11.

Spectrum of H θ=0
The ground state of the open chain for θ = 0 [24] is, which gives v F = 3 2 and c = 1, in agreement with the periodic chain. As with the previous section we numerical approximate the conformal weights by solving the Bethe equations. The results are summarised in Table 12. We should remark that we have listed the conformal weight h = 3 4 twice in Table 12 to emphasize that there are two different Bethe root configurations with different finite size energies extrapolating to this conformal dimension for L mod 4 = 2.

Pairing rules and discussion
This model is similar to the braided version and has the full global symmetry of the algebra D(D 3 ). Every pair of solutions to the Bethe equations pair which again implies that conformal weights will appear in multiple symmetry sectors and the number of times they pair depends solely the symmetry sector the eigenvalue of the transfer matrix lies in, Table 8.

Discussion
In this paper we have analysed the low energy spectrum of the integrable D(D 3 ) symmetric chain subject to various boundary conditions. In the spin chain formulation the Hamiltonian derives from a commuting two-parameter transfer matrix of a vertex model and the eigenvalues can be obtained by Bethe ansatz methods. We have constructed a related class of models with local D(D 3 ) symmetry using the fusion path formulation: these models, too, are integrable as they can be obtained from a solution to the Yang-Baxter equation for a face (or RSOS) model. For open and braided boundary conditions the two formulations of the model are equivalent. For periodic closure, however, the fusion path chain differs from the spin chain by boundary terms. Based on studies of small systems we have proposed a set of Bethe equations whose solutions determine the eigenvalues of the fusion path chain. From a finite size analysis of the spectrum of these models we have identified the conformal field theories providing an effective description of the low energy modes to contain two sectors from -depending on the parameter θ -the minimal model M (5,6) (the three-state Potts model) and the Z 4 parafermion. The physical fields are products of operators from these sectors. The individual factors can carry fractional (non-integer or non-(para)fermionic) spins implying the appearance of Virasoro characters in the partition function of the model which have not been discussed in the context of M (5,6) or Z 4 alone.
The locality of physical fields in the model is guaranteed by pairing rules. This situation is similar to other models with several gapless modes propagating with different Fermi velocities [8,26,27]. We have to emphasise, however, that in the present models this factorisation of different modes is exact already for finite chains and on all energy scales, unlike in say the separation of spin and charge degrees of freedom observed within the low energy spectrum of the one-dimensional Hubbard model where the coupling between the sectors becomes manifest in subleading corrections to scaling and at higher energies. Another difference for the model studied here is that the two sectors of the effective theory are not related to subalgebras of the global symmetry of the model. Therefore, to establish the pairing rules and corresponding multiplicities we have resorted to numerical studies of small systems together with counting arguments for the total number of states of the system. For the spin chain formulation we found that the pairing is determined by the boundary conditions and can be related to the residual symmetry of a given eigenstate. It is also reflected by the appearance of infinite roots appearing in the configurations solving the Bethe equations of the model. The spectrum of the periodic D(D 3 ) model in the fusion path formulation also shows pairing on all energy scales. Unlike in the spin chain formulation, however, the pairing is not transitive and cannot be described based on residual symmetries as in the spin chain. The identification of the pairing rules and as to whether these rules are connected to topological invariants involving the D(D 3 ) F -moves remains an interesting open problem for this model. As a first step to address this problem, numerical methods could be used to form conjectures. Ultimately, however, it would be desirable to obtain the complete picture starting from the integrable structures underlying this model. For the topological invariants this requires to relate them to elements of the RSOS Yang Baxter algebra. For the solution of the spectral problem of the anyon chain the functional relations (21) have to be established using the fusion procedure for RSOS transfer matrices. We shall address these questions in future work.
Hence the scaling dimensions and conformal spins of primary operators in the effective field theory for H θ=0 in terms of the quantum numbers ∆N ± and Q ± characterising the corresponding excitation (29) are The ∆N ± ≡ N ± − L 2 ± L 4 correspond to the change in number of ±-strings (subject to the condition that the total number of roots is L) as compared to the thermodynamic ground state while the Q ± determine the momentum of the excitation. For a configuration with n ± Bethe roots at ±∞ they can take discrete values For a given solution {x ± k } of the Bethe equations the Q ± can also be determined numerically using the counting functions defined above: Due to the discrete set of possible values for Q ± the data for systems with L ≤ 10 are sufficient to identify the quantum numbers for the lowest finite size gaps of H θ=0 , see in Table 13. The observed dimensions support our identification of the critical theory with a Z 4 parafermion.