Factorization identities and algebraic Bethe ansatz for $D^{(2)}_{2}$ models

We express $D^{(2)}_{2}$ transfer matrices as products of $A^{(1)}_{1}$ transfer matrices, for both closed and open spin chains. We use these relations, which we call factorization identities, to solve the models by algebraic Bethe ansatz. We also formulate and solve a new integrable XXZ-like open spin chain that depends on a continuous parameter, which we interpret as the rapidity of the boundary.

1 transfer matrices, for both closed and open spin chains. We use these relations, which we call factorization identities, to solve the models by algebraic Bethe ansatz. We also formulate and solve a new integrable XXZ-like open spin chain that depends on a continuous parameter, which we interpret as the rapidity of the boundary.
Here we express the D (2) 2 transfer matrices for the open spin chains considered in [12] and [16] as products of A (1) 1 transfer matrices. We then use these relations, which we call factorization identities, to solve the models by algebraic Bethe ansatz. In particular, we construct the models' Bethe states, which had not been known, that would be needed to compute scalar products and correlation functions. Moreover, we prove previously-proposed expressions for the models' eigenvalues and Bethe equations [16,[25][26][27][28]. The interesting degeneracies exhibited by these models are also explained.
In the course of this work, we also formulate and solve a new integrable XXZ-like open spin chain, which depends on a continuous parameter. We interpret this parameter as the rapidity of the boundary. We conjecture that this model, like the one in [16], has a noncompact continuum limit. This paper is structured as follows. In Sec. 2, we give an exact formulation (2.11)-(2.12) of the factorization [12] of the D 2 spin chain. We use the factorization of the R-matrix to derive the factorization identity (3.9)-(3.10), which expresses the D (2) 2 transfer matrix as a product of A (1) 1 transfer matrices. We then use this identity to solve the model by means of algebraic Bethe ansatz. Since these computations are straightforward, they may serve as a warm-up exercise for the parallel -but technically more complicated -computations that follow.
The heart of this paper is Sec. 4, where we consider open D (2) 2 chains with two different sets of integrable boundary conditions, corresponding to the two possible values (namely, 0 and 1) of a certain parameter ε. We consider first the case ε = 1, which was studied in [16]. The factorization identity (4.10)-(4.11), whose derivation is presented in Appendix A, involves a novel A (1) 1 transfer matrix (4.12). It is a special case of the more general transfer matrix (4.15), which depends on an arbitrary parameter u 0 that (as remarked above) we interpret as the rapidity of the boundary. We solve the general model by algebraic Bethe ansatz, from which we then extract the solution for the case ε = 1. We treat the case ε = 0, which was studied in [12], in a similar way. Its factorization identity (4.55)-(4.56), whose derivation is also presented in Appendix A, involves a conventional A (1) 1 transfer matrix (4.57), corresponding to u 0 = 0. In Sec. 5, we point out a special case of the model (4.15) with a local Hamiltonian for general values of u 0 . We conclude with a brief discussion of our results in Sec. 6.

Product-form R-matrices
We begin this section by reviewing in Sec. 2.1 a well-known general recipe for constructing an R-matrix by forming suitable tensor products of multiple copies of a more elementary R-matrix. We actually need a (perhaps less familiar) generalization of this construction, namely (2.6). Indeed, in Sec. 2.2, we see that the recent factorization [12] of the D 1 R-matrices is precisely of this type, up to a similarity transformation. The result (2.11)-(2.12) is the basis for all the factorization identities that we will derive in this paper, which express D

Generalities
Consider a solution R(u) of the Yang-Baxter equation (YBE) where here I is the identity matrix on V (below, by abuse of notation, I may denote the identity matrix on more than one copy of V, depending on the context), and P is the permutation matrix on where e ab are the d × d elementary matrices with elements (e ab ) ij = δ a,i δ b,j . As is well known, the R-matrix can be usefully represented graphically by one pair of lines that cross, as shown in Fig. 1; hence the YBE (2.1) is represented using three lines, as shown in Fig. 2. We assume that the R-matrix is regular and unitary where R 21 = P 12 R 12 P 12 . We use the symbol ∝ to denote equality up to a scalar factor. The latter can be represented graphically as in Fig. 3.
is given by the following product of four R-matrices which is a d 4 × d 4 matrix. This R-matrix can be represented graphically by two pairs of lines that cross, as shown in Fig. 4. The corresponding YBE for R, represented in Fig. 5, follows from the YBE for R shown in Fig. 2. A review of models constructed with R-matrices of this type can be found in [29]. We will need a generalization of the construction (2.5), namely, where θ is an arbitrary constant, see Fig. 6. Indeed, using the regularity property (2.3), the construction (2.6) reduces to (2.5) for θ = 0. The proof that (2.6) satisfies the YBE, which requires unitarity (2.4) as well as the YBE (2.1), can also be performed graphically (see Fig.  7), or by a straightforward but long explicit computation.

The D
(2) 2 R-matrix The D (2) 2 R-matrix, following a hint from [30,31], has recently been shown [12] to be of product form, up to a similarity transformation. Indeed, let us write the D (2) 2 R-matrix from [14] as in Appendix A of [27], with spectral parameter u and anisotropy parameter η, and denote it byR(u). Theñ where R(u) is given by (2.6), with R(u) given by the A and θ = iπ. Moreover, the similarity transformation is given by Following [5,7], we define the matrix C by (2.10) Using this notation, the result (2.6)-(2.7) for the D 2 R-matrix takes the final form where R(u) has been redefined (by a simple rescaling) as Note that we use a tilde to denote similarity-transformed quantities. Eqs. (2.11)-(2.12) are an exact formulation, in our notation, of the factorization discovered in [12]. In the isotropic limit η → 0, this result reduces to the fact (see e.g. [32]) that the D 2 (i.e. SO(4)) R-matrix factorizes into a product of two A 1 (i.e. SU (2)) R-matrices, up to a similarity transformation.
For future reference, we note here some useful properties of the R-matrix (2.8) in addition to (2.1)-(2.4): quasi-periodicity PT-symmetry R t 1 t 2 12 (u) = R 21 (u) (2.14) (where t i denotes transposition in the i th vector space), and crossing-unitarity The closed D 2 spin chain We begin with the simplest case, namely, the closed periodic D 2 spin chain. In Sec. 3.1, we use the factorization of the R-matrix (2.11)-(2.12) to derive the factorization identity (3.9)-(3.10) that expresses the D

Factorization identity
The monodromy matrix for a chain of length N is defined bỹ R-matrix. In order to exploit the factorization (2.11)-(2.12), it is convenient to replace each index j in (3.1) (which corresponds to a 4-dimensional vector space) by a pair of indicesj ,j (each of which corresponds to a 2-dimensional vector space). In this way, the monodromy matrix takes the form The relation (2.11) impliesT00 where T00(u) is defined in terms of R's as in (3.2) except without tildes, and B is the quantum-space operator B = B11 . . . BNN . Using (2.12), we obtain where T0(u) is defined by and C is the quantum-space operator Note that T0(u) is a monodromy matrix on 2N sites, with iπ shifts on alternating sites; T0(u) is given by the same expression (3.6), except with0 replaced by0. Note also the periodicity T0(u + 2iπ) = T0(u) as a consequence of (2.13).
The transfer matrix for the closed periodic spin chain is obtained by tracing the monodromy matrix over the auxiliary spacẽ t(u) = tr 0T0 (u) = tr00T00(u) . (3.8) where t(u) is defined in terms of T(u) as in (3.8) except without tildes. Using (3.5), we immediately obtain the result where t(u) is an A 1 closed-chain transfer matrix defined by The result (3.9)-(3.10), which we call a factorization identity, shows that, up to similarity transformations, the D 2 closed-chain transfer matrix is given by a product of A 1 closedchain transfer matrices with twice as many sites.

Algebraic Bethe ansatz
We now proceed to determine the eigenvectors and eigenvalues of the D To this end, we recall (see e.g. [33]) that the A (1) 1 transfer matrix can be diagonalized by algebraic Bethe ansatz. Indeed, consider the general inhomogeneous monodromy matrix with length L where R(u) is given by (2.8), and {θ l } are arbitrary inhomogeneities. (The indices here correspond to 2-dimensional vector spaces, i.e., the same asj andj in (3.6).) We denote the corresponding closed-chain transfer matrix by The operator B(u; {θ l }) in (3.12) serves as a creation operator on the reference state The Bethe states defined by can be shown to obey the following off-shell equation where the variable with a hat is omitted, and χ(u; {θ l }) is given by Moreover, χ j is given by .

(3.19)
Our original monodromy matrix (3.6) corresponds to setting L = 2N in (3.12), and choosing the inhomogeneities as follows It follows that the Bethe states (3.15) with these inhomogeneities are eigenstates of our original transfer matrix (3.11), with corresponding eigenvalues given by provided that {v k } satisfy the Bethe equations .

(3.22)
These equations take a symmetric form in terms of u j ≡ v j − η, namely, .
the expression for the eigenvalues (3.21) of the A 1 closed-chain transfer matrix t(u) (3.11) take the final form Coming back to the D 2 closed-chain transfer matrixt(u) (3.8), we conclude from the factorization identity (3.9)-(3.10) that its Bethe states are given by where the vectors |v 1 · · · v m are given by (3.15), and B and C are given respectively by (3.4) and (3.7), see [30] for an alternative approach. Moreover, the corresponding eigenvalues Λ(u) are given by where χ(u) is given by (3.25), and the associated Bethe equations are given by (3.23). The latter results agree with expressions obtained by Reshetikhin using analytical Bethe ansatz [25].

Z 2 symmetry
The transfer matrix t(u) (3.11) has the property where C (3.7) is defined in terms of C (2.10). The proof is short: the fact that the R-matrix satisfies the identity implies that the monodromy matrix (3.6) satisfies the corresponding identity By tracing over the auxiliary space0, we obtain (3.28).
The property (3.28) implies that the D 2 transfer matrix t(u) (3.10) can also be written in the form and therefore it has the Z 2 symmetry The Z 2 symmetry of the staggered six-vertex model was noted already in [7].

Degeneracies
For real values of η, each of the eigenvalues of t(u) (3.11) is either a singlet or a doublet (2-fold degenerate). However, as the result of the Z 2 symmetry, some of the degeneracies of t(u) (3.10) become doubled, leading to doublets or quartets.
The key point is that the Z 2 symmetry shifts the argument of the B-operator by iπ as follows from (3.12) and (3.30). The Bethe states (3.15) therefore transform as follows since the reference state remains invariant C |0 = |0 . In other words, under the Z 2 symmetry, each of the Bethe roots v k (or, equivalently, u k ) is shifted by iπ. If Q(u + iπ) = ±Q(u), then the Bethe states corresponding to Q(u) and Q(u + iπ) are mapped into each other by the Z 2 symmetry C. (The argument is the same as for the open chain, which is presented in Sec. 4.2.4.) It follows from (3.32) that the two Bethe states have the same eigenvalue of t(u), which means that they are degenerate.
Our goal in the remainder of this paper is to obtain factorization identities analogous to (3.9)-(3.10) for D 2 spin chain. We will consider two different sets of integrable boundary conditions, corresponding to the two possible values (namely, 0 and 1) of a certain parameter ε. As before, our strategy will be to use factorization identities to solve the models. After introducing the transfer matrix in Sec. 4.1, we consider the case ε = 1 in Sec. 4.2, followed by case ε = 0 in Sec. 4.3.

Transfer matrix
In order to construct an integrable open-chain transfer matrix [34], we need not only an R-matrix, but also a K-matrix, i.e., a solution of the corresponding boundary Yang-Baxter equation [34][35][36]. For D (2) n+1 , such K-matrices have been found in [26,37]. The K-matrices in [37] depend on two discrete parameters: p (which can take n + 1 different values, namely, p = 0, 1, . . . , n) and ε (which can take two different values, namely, ε = 0, 1). We consider here n = 1 (corresponding to D 2 ); and, for concreteness, we set p = 0. (The case p = 1 is simply related to the case p = 0 by a p ↔ n − p duality symmetry [37,38].) The right K-matrix, which we denote here byK R (u), is then given bỹ with ε = 0, 1. For the left K-matrix, we take [37] where M is defined in (2.15), so that the transfer matrix has quantum-group symmetry, see Sec. 4.2.3.
The D 2 open-chain transfer matrix for a chain with N sites is given by [34] t whereT 0 (u) is given by (3.1) and (3.2). Similarly, T 0 (u) is given by or equivalently T00 (u) =R11 ,00 (u) . . .RNN ,00 (u) , (4.6) where we have replaced (as we did forT 0 (u) in Sec. 3.1) each index j in (4.5) by a pair of indicesj ,j. Eq. (2.11) then implies where T00(u) is defined in terms of R's as in (4.6) except without tildes. Using (2.12), we obtain where T0(u) is defined by and T0(u) is given by the same expression (4.9), except with0 replaced by0.

The case ε = 1
For the case ε = 1, the transfer matrixt(u) (4.4) satisfies where t(u) satisfies the remarkable factorization identity where t(u) is an A 1 open-chain transfer matrix defined by and T0(u) and T0(u) are defined in (3.6) and (4.9), respectively. The proof of this factorization identity is presented in Appendix A. Note the periodicity t(u + 2iπ) = t(u) as a consequence of (2.13).
Notice the shift by iπ in the argument of T (compared with T ) in the transfer matrix (4.12). While this shift may appear innocuous, its effects are profound. To our knowledge, open-chain transfer matrices with such shifts have not been considered before; a priori, it is not even clear whether such transfer matrices commute for different values of the spectral parameter.
We will interpret such a shift as the rapidity of the boundary; or equivalently, as a boundary inhomogeneity. We will then proceed to diagonalize the transfer matrix.

Transfer matrix with a moving boundary
As in the closed-chain case (see (3.12)), it is convenient to consider a slightly more general problem, namely, a chain of length L with arbitrary inhomogeneities at each site. The monodromy matrices are therefore given by where R(u) is given by (2.8), and {θ l } are arbitrary inhomogeneities, cf. (3.6) and (4.9). These monodromy matrices satisfy the familiar fundamental relations where the shift u 0 in the argument of T is arbitrary. The transfer matrix for our problem (4.12) is clearly a special case of (4.15).
It is straightforward to show using (4.14) and Ř (u) ,Ř(v) = 0 (whereŘ(u) ≡ PR(u)), that the double-row monodromy matrix U(u; {θ l }) (4.15) obeys the following boundary Yang-Baxter equation (BYBE) (4.16) Note the shift by u 0 in the R-matrix whose argument has the sum of rapidities. It implies that if a "particle" approaches the boundary with rapidity u, then after reflection the particle has rapidity −u − u 0 . We can attribute this shift to a moving boundary, with rapidity u 0 . Equivalently, this shift can be regarded as a boundary inhomogeneity, as opposed to the bulk inhomogeneities {θ l }.
Despite the presence of a shift in the BYBE, the transfer matrix nevertheless has the crucial commutativity property Indeed, the commutativity proof in [34] can be readily generalized to accommodate this shift, for arbitrary values of u 0 .

Algebraic Bethe ansatz
We now proceed to diagonalize the transfer matrix (4.15) by algebraic Bethe ansatz. Following [34], we set and act with B(u; {θ l }) on the reference state (3.14) to create the Bethe states which obey the following off-shell equation Here, χ(u; {θ l }) is given by sinh( 1 2 (u − θ l )) sinh( 1 2 (u + u 0 + θ l )) , and χ j is given by .

(4.23)
Note that a nonzero value of u 0 indeed profoundly affects the solution.
Our original monodromy matrices (3.6) and (4.9) correspond to setting L = 2N in (4.13), and choosing the inhomogeneities {θ l } as in (3.20). Moreover, our original transfer matrix (4.12) corresponds to setting the shift u 0 = iπ in (4.15). It follows that the Bethe states (4.19) with these parameter values are eigenstates of our original transfer matrix (4.12), with corresponding eigenvalues given by provided that {v k } satisfy the Bethe equations .

(4.26)
These equations take a symmetric form in terms of u j ≡ v j − η, namely, . sinh( 1 2 (u − u k )) cosh( 1 2 (u + u k )) = q(u + η) , (4.28) the expression for the eigenvalues (4.24) of the A 1 open-chain transfer matrix t(u) (4.12) take the final form (4.29) Returning to the D 2 open-chain transfer matrixt(u) (4.4) with ε = 1, we conclude from the factorization identity (4.10)-(4.11) that its Bethe states are given by where the vectors |v 1 · · · v m are given by (4.19), and B is given by (3.4), which is a new result. Moreover, the corresponding eigenvalues Λ(u) are given by where χ(u) is given by (4.29), and the associated Bethe equations are given by (4.27). The latter results agree with the recent proposal in [16], which improved on an earlier proposal [38].

Symmetries
We briefly discuss here the quantum group (QG) and Z 2 symmetries of the transfer matrix, which we will then use to understand the degeneracies of the spectrum.

Z 2 symmetry
The open-chain transfer matrix t(u) (4.12) has the property C t(u) C = t(u + iπ) , (4.40) where C is given by (3.7), similarly to the closed-chain transfer matrix (3.28). Indeed, the monodromy matrix identities (3.30) and Multiplying both sides of (4.42) by M0 and tracing over the auxiliary space0, we obtain the desired result (4.40).
One consequence of the property (4.40) is that the D 2 open-chain transfer matrix t(u) has the Z 2 symmetry C t(u) C = t(u) . Indeed, we see from the factorization identity (4.11) that where we have passed to the second line using (4.40) and the 2iπ-periodicity of t(u); the final equality follows from the commutativity property (4.17). The Z 2 symmetry of the open-chain transfer matrix (4.43) was first noted in [16].
The QG and Z 2 generators commute

Degeneracies
For real values of η, the degeneracies of the D 2 open-chain transfer matrix t(u) are higher than expected from QG symmetry alone, as discussed in [27,37,38]. These higher degeneracies can now be fully explained using the above Z 2 symmetry.
Realizing from (4.15) and (4.18) that the double-row monodromy matrix is given here by we see from (4.42) that the Z 2 symmetry shifts the argument of the B-operator by iπ C B(u) C = B(u + iπ) , (4.47) similarly to the closed-chain case (3.33). The Bethe states (4.19) therefore transform as follows C |v 1 · · · v m = |v 1 + iπ · · · v m + iπ . In other words, under the Z 2 symmetry, each of the Bethe roots v k (or, equivalently, u k ) is shifted by iπ.
The property (4.40) implies that t(u) and t(u+iπ) are related by a unitary transformation (at least for real values of η, since C is involutory and symmetric), and therefore have the same spectrum. Hence, if χ(u) is an eigenvalue of t(u), then χ(u + iπ) is also an eigenvalue of t(u). Thus, if Q(u) satisfies the TQ-equation, then Q(u + iπ) also satisfies the TQ-equation, as follows simply from performing the shift u → u + iπ in (4.29). Hence, given a set of Bethe roots {u k }, there are only two possibilities for the corresponding Q-function (4.28): • Q(u + iπ) = Q(u), in which case the corresponding Bethe state is an eigenstate of the Z 2 symmetry C. The Bethe state is a highest-weight state of a representation of the QG with odd dimension [27,37,38]; hence, the corresponding eigenvalue has odd degeneracy.
• Q(u + iπ) = Q(u), in which case the Bethe states corresponding to Q(u) and Q(u + iπ) are mapped into each other by the Z 2 symmetry C. It follows from (4.43) that the two Bethe states have the same eigenvalue of t(u), which means that they are degenerate. The degeneracy of the corresponding eigenvalue is doubled, and is therefore even.

Hamiltonian
For an open-chain transfer matrix t(u) constructed with a regular R-matrix (2.3) and with all inhomogeneity parameters {θ l } set to zero (i.e., a homogeneous spin chain), a local Hamiltonian can be obtained simply from t (0) [34]. However, since the transfer matrix (4.12) corresponds to a spin chain with inhomogeneities at alternate sites, t (0) is not local. Nevertheless, a local Hamiltonian can be obtained from d du log t(u) u=0 = t −1 (0) t (0), which is the familiar prescription for periodic homogeneous chains.

The case ε = 0
We now consider the case ε = 0, which is similar to the previous case, except for one key difference. The D 2 transfer matrixt(u) (4.4) again satisfies but t(u) now satisfies the factorization identity where t(u) is an A  As before, φ(u) is defined in (4.11), and T0(u) and T0(u) are defined in (3.6) and (4.9), respectively. The proof of this factorization identity is also presented in Appendix A.
Note that the transfer matrix (4.57), in contrast with the previous case (4.12), does not have any shift in the argument of T (compared with T ). Indeed, the transfer matrix (4.57) is of the standard form [34]. This is the key difference, alluded to above, between the ε = 1 and ε = 0 cases.

Algebraic Bethe ansatz
We can immediately diagonalize the transfer matrix (4.57) using our previous results (4.18)-(4.23): simply set (as before) L = 2N and choose the inhomogeneities {θ l } as in (3.20), but now set the shift u 0 = 0. Hence, the Bethe states (4.19) with these parameter values are eigenstates of the transfer matrix (4.57), with corresponding eigenvalues given by provided that u j ≡ v j − η satisfy the Bethe equations . (4.60) Returning to the D 2 open-chain transfer matrixt(u) (4.4) with ε = 0, we conclude from the factorization identity (4.55)-(4.56) that its Bethe states are given by where the vectors |v 1 · · · v m are given by (4.19), and B is given by (3.4), which is a new result. Moreover, the corresponding eigenvalues Λ(u) are given by where χ(u) is given by (4.58), and the associated Bethe equations are given by (4.60). The Bethe equations agree with those obtained by coordinate Bethe ansatz in [26]; the transfermatrix eigenvalues and Bethe equations agree with those obtained by analytical Bethe ansatz in [27,28].
The symmetries and degeneracies for the ε = 0 case are the same as for ε = 1.

Hamiltonian
From the A 1 transfer matrix (4.57), we can generate two distinct local Hamiltonians, by evaluating its logarithmic derivative at 0 and at iπ where, in terms of the Temperley-Lieb operators (4.50), the Hamiltonians H (1) and H (2) are given by ; j=even (e j e j+1 + e j+1 e j ) , and c = 1 sinh(2η) (1 − 2N cosh(2η)). We can use the factorization identity (4.56) to relate these Hamiltonians to the Hamiltonian H coming from the D 2 transfer matrix t(u). We obtain, up to an additive constant, where u 0 is arbitrary. We then obtain from (4.15) where the Hamiltonian H is given in terms of Temperley-Lieb operators (4.50) by and the constant c(u 0 ) is given by . To obtain the above results, it is helpful to introduce a generalization of the matrix C (2.10), namely, which reduces to C (2.10) for u 0 = ±iπ. Then, similarly to (4.52), we find where For the choice (5.1) of inhomogeneities, the Bethe states (4.19) are eigenstates of the transfer matrix (4.15), with corresponding eigenvalues given by with q(u) given by (4.22), provided that {v j } satisfy the Bethe equations .
In terms of u j ≡ v j − η, these Bethe equations take a more symmetric form .

(5.11)
We note that these Bethe equations are an "open-chain version" of the closed-chain Bethe equations (3.4) in [10]. We also note that the transfer matrix has the QG symmetry (4.36)-(4.39) for any value of u 0 .

Discussion
We have exploited the factorization of the D 2 R-matrix into a product of A 1 R-matrices (2.11)-(2.12) to derive corresponding factorization identities for the transfer matrices of both closed and open spin chains, see (3.9)-(3.10), (4.10)-(4.11) and (4.55)-(4.56). We have used these factorization identities to solve the models by algebraic Bethe ansatz. In particular, we have constructed the Bethe states of these models, which heretofore had not been known. These constructions should be useful for computing scalar products and correlation functions. Moreover, we have proved previously-proposed expressions for the models' eigenvalues and Bethe equations. The interesting degeneracies exhibited by the QG-invariant open chains for real values of η have now also been explained.
In the course of this work, we have uncovered a new integrable XXZ-like open spin chain, with transfer matrix (4.15), which depends on a continuous parameter u 0 . We have interpreted this parameter as the rapidity of the boundary. For inhomogeneities −iπ at alternate sites (3.20), this model continuously interpolates between the cases ε = 0 (u 0 = 0) and ε = 1 (u 0 = iπ). For inhomogeneities −u 0 at alternate sites (5.1), this model has a local Hamiltonian (5.3) for general values of u 0 . We conjecture that, for the parameters η and u 0 in suitable domains, the continuum limit of the latter model is a non-compact boundary conformal field theory, as is the case for u 0 = iπ [16], see also [6][7][8][9][10][11].
A.1 The case ε = 1 We now focus on the case ε = 1. The key step, having already expressed theR's in terms of R's, is to also express theK's in terms of R's. Remarkably, the right K-matrix (4.1) with ε = 1 satisfies the identity Eq. (A.3) therefore further simplifies to The product of terms on the second line of (A.5) can be simplified as follows: where square brackets are used to indicate the terms to be transformed in the subsequent step. In passing to the third line of (A.6), we have used the identity P00 C00 T0(u + iπ) T0(u) = T0(u) T0(u + iπ) P00 C00 , (A.7) which follows from the fact P C ∝ R(iπ) (see (2.10)) and the second relation in (4.14). Eq. (A.5) therefore becomes t(u) = 2 8N sinh(u + η) tr00 B00K L 00 (u) B00 × T0(u + iπ) T0(u) R00(2u) T0(u) T0(u + iπ) P00 . (A.8) Using the third relation in (4.14), we arrive at t(u) = 2 8N sinh(u + η) tr00 P00 B00K L 00 (u) B00 × T0(u + iπ) T0(u) R00(2u) T0(u) T0(u + iπ) . (A.9) The left K-matrix (4.3) satisfies, as a consequence of the identity for the right K-matrix (A.4), the following corresponding identity where F and G are arbitrary, whose proof is as follows: In passing to the third line, we have used the crossing-unitarity (2.15) and PT-symmetry (2.14) of R(u). In the subsequent step, we have repeatedly used the cyclic property of the trace.