JHEP03 ( 2022 )

: We generalize the nested oﬀ-diagonal Bethe ansatz method to study the quantum chain associated with the twisted D (2)3 algebra (or the D (2)3 model) with either periodic or integrable open boundary conditions. We obtain the intrinsic operator product identities among the fused transfer matrices and ﬁnd a way to close the recursive fusion relations, which makes it possible to determinate eigenvalues of transfer matrices with an arbitrary anisotropic parameter η . Based on them, and the asymptotic behaviors and values at certain points, we construct eigenvalues of transfer matrices in terms of homogeneous T − Q relations for the periodic case and inhomogeneous ones for the open case with some oﬀ-diagonal boundary reﬂections. The associated Bethe ansatz equations are also given. The method and results in this paper can be generalized to the D (2) n +1 model and other high rank integrable models.


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Abstract: We generalize the nested off-diagonal Bethe ansatz method to study the quantum chain associated with the twisted D (2) 3 algebra (or the D (2) 3 model) with either periodic or integrable open boundary conditions. We obtain the intrinsic operator product identities among the fused transfer matrices and find a way to close the recursive fusion relations, which makes it possible to determinate eigenvalues of transfer matrices with an arbitrary anisotropic parameter η. Based on them, and the asymptotic behaviors and values at certain points, we construct eigenvalues of transfer matrices in terms of homogeneous T − Q relations for the periodic case and inhomogeneous ones for the open case with some offdiagonal boundary reflections. The associated Bethe ansatz equations are also given. The method and results in this paper can be generalized to the D (2) n+1 model and other high rank integrable models.

Introduction
One of the major achievements of one-dimensional quantum integrable systems is that it can supply us some believable results of strong interacting quantum many-body systems. The coordinate [1,2] and algebraic Bethe ansatz [3][4][5][6][7] as well as the T − Q relations [8][9][10] are the powerful methods to obtain the exact solutions of the systems and have made great achievements in the past several decades. Based on the exact solution, many interesting physical concepts and mathematical structures in the quantum field theory, quantum group and quantum algebra are obtained [11].
Motivated by the applications in AdS/CFT, string theory and conformal field theory, the study of quantum integrable models with high rank Lie algebras becomes a very important issue [29][30][31]. Many efforts have been made to investigate this kind of systems.
The D (2) n+1 is a typical twisted Lie affine algebra and the related quantum integrable models have been attracted many attentions [38-42, 44, 46]. Reshetikhin obtained the Bethe ansatz solutions of the D (2) n+1 model with periodic boundary condition [38]. Martins and Guan studied the integrability of the D (2) n+1 model with open boundary condition [39]. Further, Nepomechie, Pimenta and Retore studied the integrable quantum group invariant and D (2) n+1 open spin chains, where an interesting result is that the R-matrix can be constructed by two six-vertex R-matrices [40]. The integrable D (2) n+1 reflection matrices with quantum group symmetry and with other integrable boundary conditions were given in [41][42][43][44][45]. Another important progress is the Bethe ansatz solution of the D (2) 2 model [43-46], which has application in the black hole theory.
In this paper, we study the quantum integrable spin chain associated with the twisted D (2) 3 . We generalize the nested ODBA method to the chain with either the periodic or the open boundary condition. By using the fusion technique [47][48][49][50][51][52], we systematically analyze the fusion structure of the system. We provide a way to close the recursive fusion relations, which make the fusion relations can be used to construct the energy spectrum without any additional constraints. We obtain the closed intrinsic operator product identities among the fused transfer matrices. Based on them, and the asymptotic behaviors and values at certain points, we obtain the eigenvalues of fused transfer matrices, which are expressed as the homogeneous T − Q relations for the periodic case and inhomogeneous ones for the open case with off-diagonal reflection matrices. The associated Bethe ansatz equations are also given.
The plan of the paper is as follows. In section 2, we study the D (2) 3 spin chain with the periodic boundary condition. The closed operator product identities among the fused transfer matrices are given. By constructing the homogeneous T − Q relations, we obtain the eigenvalues and Bethe ansatz equations of the system. Section 3 is devoted to diagonalize the model with some non-diagonal boundary reflections. We obtain the recursive fusion relations, the eigenvalues of transfer matrices in terms of inhomogeneous T − Q relations, and the Bethe ansatz equations. The summary of main results and some concluding remarks are presented in section 4. Some details deriving the fusions of the R-matrices and related K-matrices are given in appendices A and B.

model with the periodic boundary condition
Throughout this paper, we adopt the standard notations. V denotes a n-dimensional linear space with the orthogonal basis {|i , i = 1, 2, · · · , n}. For any matrix A ∈ End(V), A j is an embedding operator in the tensor space V ⊗ V ⊗ · · · , which acts as A on the j-th space and as identity on the other factor spaces. For any matrix B ∈ End(V ⊗ V), B ij is an embedding operator of B in the tensor space, which acts as identity on the factor spaces except for the i-th and j-th ones.

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Besides the above properties, the R-matrix (2.1) satisfies the Yang-Baxter equation The monodromy matrix of the system is constructed by the vectorial R-matrix (2.1) as where the index 0 indicates the auxiliary space and the indices {1, · · · , N } denote the physical or quantum spaces, N is the number of sites and {θ j |j = 1, · · · , N } are the inhomogeneous parameters. The monodromy matrix satisfies the Yang-Baxter relation The transfer matrix is defined as the partial trace of monodromy matrix in the auxiliary space where the up-index (p) of the transfer matrix t (p) (u) just stands for the periodic boundary condition (or the closed chain). From the Yang-Baxter relation (2.15), one can prove that the transfer matrices with different spectral parameters commute with each other, i.e., [t (p) (u), t (p) (v)] = 0. Therefore, t (p) (u) serves as the generating functional of the conserved quantities of the system. The Hamiltonian of the D spin chain with the periodic boundary condition can be given in terms of the transfer matrix (2.16) as with The periodic boundary condition reads

Operator product identities among the fused transfer matrices
In order to obtain eigenvalues of the fundamental transfer matrix t (p) (u), we need further to introduce some fused transfer matrices [24]. For the D 3 spin chain with the periodic boundary condition, we introduce the fused monodromy matrices via the fused R-matrices R s ± ṽ 0 j (u) given by (A.1) and (A.2) in the appendix A, (2.20) Here and after,0 = 0 denotes the auxiliary space for the spinorial representation s + and 0 =0 denotes the auxiliary space for the spinorial representation s − . We note that the JHEP03(2022)175 quantum spaces of T + 0 (u) and T − 0 (u) are the same, which are also the quantum spaces of T 0 (u). The fused R-matrices R where R s + s − 0 0 (u) is defined by eq. (B.5). Taking the partial trace in the auxiliary spaces, we obtain the fused transfer matrices t (p) From the Yang-Baxter relation (2.22), we can prove that the transfer matrices t (p) (u) and t (p) ± (u) commute with each other, i.e., Thus they have the common eigenstates. From the Yang-Baxter relations (2.15), (2.21) and (2.22) at certain points and using the properties of projectors, we obtain Taking the partial trace of eq. (2.26) in the auxiliary spaces and using the relation (2.25), we obtain the operator product identities

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Next, we consider the asymptotic behaviors of fused transfer matrices. According to the definitions, the direct calculation gives α α are the conserved quantities acting on the quantum space V ⊗ V ⊗ · · · ⊗ V. The related operators are defined as Here the repeated indicators should be summarized. R

vv(±) 0j
and R are the leading terms of e ∓2u R vv 0j (u) and e ∓u R s ± ṽ 0 j (u) with u → ±∞, respectively. From the direct calculation, we find that the eigenvalues of conserved quantities 6 Then the asymptotic behaviors of fused transfer matrices read (2.30) Acting the transfer matrices on the common eigenstate, we obtain the corresponding eigenvalues. Denote the eigenvalues of t (p) (u) and t (p) ± (u) as Λ (p) (u) and Λ (p) ± (u), respectively. As mentioned previously, the eigenvalues Λ (p) (u) and Λ (p) ± (u) are the trigonometric polynomials of u with degrees 2N and N , respectively. Therefore, we need 4N +3 conditions to determine the values of Λ (p) (u) and Λ (2.31)

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The corresponding asymptotic behaviors are Then we arrive at that 4N functional relations (2.31) together with 6 asymptotic behaviors (2.32) give us sufficient conditions to determine the eigenvalues of transfer matrices.

T − Q relations for eigenvalues
The function relations (2.31) and asymptotic behaviors (2.32) allow us to parameterize the eigenvalues Λ (p) (u) and Λ (p) k } and L 2 is the number of Bethe roots {µ (2) l }. Because the eigenvalues Λ (p) (u) and Λ Some remarks are in order. We note that the BAEs (2.37) and (2.38) are homogeneous. This is because the periodic boundary condition does not break the U(1) symmetry of the system. The BAEs obtained from the regularity of Λ (p) (u) are the same as those obtained from the regularities of Λ The existence of two good quantum numbers consists with the fact that there are two sets of homogeneous BAEs. It is easy to check that Λ (p) (u) and Λ (p) ± (u) satisfy the functional relations (2.31) and the asymptotic behaviors (2.32). Therefore, we conclude that Λ (p) (u) and Λ (p) ± (u) are the eigenvalues of the transfer matrices t (p) (u) and t (p) ± (u), respectively. We should note that the T − Q relations (2.33)-(2.35) and associated BAEs (2.37)-(2.38) have the well-defined homogeneous limit. These results with the constraint {θ j } = 0 coincide with the previous results [29,38].
The eigenvalues of the Hamiltonian (2.17) can be obtained by the Λ (p) (u) as

model with non-diagonal boundary condition
In this section, we study the system with general integrable open boundary condition. The boundary reflection at one side is quantified by the reflection matrix K v (u) which satisfies the reflection equation The boundary reflection at the other side is described by the dual reflection matrixK v (u), which satisfies the dual reflection equation

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In the open boundary condition, besides the monodromy matrix T v 0 (u) given by (2.14), we should also consider the reflecting monodromy matrix which satisfies the Yang-Baxter relation The transfer matrix t(u) of the model with boundary reflections is defined as [14] t where H kk+1 is given by (2.18).

Reflection matrix
In this paper, we consider the integrable open boundary condition where the reflection matrices have the non-diagonal elements, which break the U(1)-symmetry of the system. The non-diagonal reflection matrix for D (2) n+1 vertex model has been constructed by Malara et al.

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where the non-zero matrix elements are , (3.8) and c is a free boundary parameter. The dual reflection matrixK v (u) is obtained by the where c is the boundary parameters at the other side. For a generic choice of the boundary Thus the U(1) symmetry of the system is broken while the integrability is still held. Substituting the expressions of R-matrix (2.1) and reflection matrices (3.7) and (3.9) into (3.6), we obtain the integrable Hamiltonian of D (2) 3 model with the non-diagonal boundary reflections given by (3.7) and (3.9).

Operators product relations
Similarly as the periodic case in the previous section, we need further to introduce some fused transfer matrices (see below (3.13)) besides the fundamental one t(u). Due to the boundary reflection, besides the fused monodromy matrices T ± 0 (u) given by (2.20), we should define the reflecting fused monodromy which satisfy the Yang-Baxter relations The fused transfer matrices are constructed as (3.14) Thus t(u) and t ± (u) have the common eigenstates.
where j = 1, · · · , N . The values of transfer matrices t(u) and t ± (u) at the point of u = 0 can be calculated directly In the derivation, we have used the relations The asymptotic behaviors of t(u) and t ± (u) read where Q v ± and Q s ± ± are the conserved quantities with the definitions and R vs ± (±) j0 are the leading terms of e ∓2u R vv j0 (u) and e ∓u R vs ± j0 (u) with u → ±∞, respectively, and the repeated indicators should be summarized. The detailed calculation shows that the eigenvalues of conserved quantities Q v ± and Q s ± ± can be characterized by the quantum number m as where m ∈ [1, N + 1]. Then we obtain the asymptotic behaviors of t(u) and t ± (u) as Acting the fused transfer matrices t(u) and t ± (u) on a common eigenstate, we obtain the eigenvalues. Denote the eigenvalues of t(u) and t ± (u) as Λ(u) and Λ ± (u), respectively. From the operators product identities (3.20), we obtain the functional relations among the eigenvalues Λ(u) and Λ ± (u) as a1(±θj − θ l )e1(±θj − θ l + 4η)a1(±θj + θ l )e1(±θj + θ l + 4η),

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where the numbers of Bethe roots should satisfy the constraint L 1 = L 2 + N + 1, (3.36) and the parameter h is Some remarks are in order. The BAEs (3.34) are inhomogeneous while the BAEs (3.35) are homogeneous. This is because the reflection matrices (3.7) and (3.9) can be divided into the direct summation of a 4 × 4 non-diagonal and a 2 × 2 diagonal submatrices. The boundary reflection in the non-diagonal subspace breaks the U(1) symmetry of the system. While in the diagonal subspace, there exists a conserved charge, which leads to the homogeneous BAEs (3.35). From the asymptotic behavior of the eigenvalues, we obtain that the quantum number m of conserved quantities Q v ± and Q s ± ± is related with the number of Bethe roots {µ (2) l } as m = L 2 − N , which is consistent with the conclusion that BAEs (3.35) are homogeneous. We shall also note that the BAEs obtained from the regularities of Λ(u) are the same as those obtained from the regularities of Λ ± (u). The function Q (l) (u) has two sets of zero roots, i.e., {µ (l) k +lη|k = 1, · · · , L l } and {−µ (l) k +lη | k = 1, · · · , L l }, where l = 1, 2. The BAEs obtained from these two sets of zero roots are also the same. It is easy to check that Λ(u) (3.30) and Λ ± (u) (3.31)-(3.32) satisfy the functional relations (3.27), the values at the special points (3.28) and the asymptotic behaviors (3.29). Therefore, we conclude that the inhomogeneous T − Q relations (3.30)- (3.32) give the eigenvalues of the transfer matrices t(u) and t ± (u). All the eigenvalues (3.30)-(3.32) and BAEs (3.34)-(3.35) have the well-defined homogeneous limit.
The energy spectrum of the Hamiltonian (3.6) can be obtained by Λ(u) as (3.38)

Discussion
In this paper, we have studied the quantum integrable model associated with the twisted D 3 Lie algebra by generalizing the nested off-diagonal Bethe ansatz. We obtain the exact solutions of the system with either periodic or non-diagonal open boundary conditions. With the help of fusion, we obtain the closed recursive operator product identities among the fused transfer matrices. Based on them and the asymptotic behaviors as well as the special values at certain points, we obtain the eigen-spectrum and Bethe ansatz equations. For the periodic case, the eigenvalues of the transfer matrices are described by the homogeneous T − Q relations. While for the open boundary case, the eigenvalues are characterized by the inhomogeneous T − Q relations due to the off-diagonal K-matrices (3.7)-(3.9). The method can be generalized to the models with other high rank twisted algebras.

A.1 Spinorial R-matrix
In this appendix, we shall give the R-matrices R s ± v (u) which has the same quantum space as that of (2.1) but the spinorial representations of D (2) 3 as their auxiliary spaces. Namely, We remark that the fused R-matrices R s ± ṽ 1 2 (u) are necessary to derive the exact solution of the D (2) 3 model. For simplicity, throughout this paper, we denote1 = 1 for the representation s + and1 =1 for the representation s − . Here we list some useful relations among the R-matrices: whereM is the diagonal matrix given bȳ and the Yang-Baxter relations

A.2 Fusion and the fused R-matrices
By using the fusion technique [47][48][49][50][51][52], we systematically analyze the fusion structure of the R-matrices. For this purpose, we consider the fusion of R with the basis vectors x 1 = e 3η (2 cosh 2η + cosh 4η) and S (+) 1 2 is a constant matrix omitted here. Exchanging the two spaces V 1 and V 2 , we obtain the fused R-matrix R vs + 21 (u). From it, we deduce another 4-dimensional projector Taking the fusion of R whereρ 0 (u) = 4 sinh(u+2η) sinh(u−4η), 1 2 denotes the fused space V 1 2 , and R s − v 1 2 3 (u) and R vs − 3 1 2 (u) are the new fused R-matrices. For simplicity, we define1 ≡ 1 2 . We shall note that although the dimension of fused space V1 is 4, the V1 is not the original spinorial representation space V 1 , i.e., V 1 = V 1 2 . In fact, V1 is the space of another spinorial representation s − of D with the basis vectors and S (−) 1 2 is a constant matrix omitted here. From eq. (A.11), we know that the fused R-matrix R vs − 21 (3η + iπ) degenerates into the 4-dimensional projector (A.14) Taking the fusion of R  with the basis vector , (A.20) and S 12 is a constant omitted here. Exchanging two spaces V 1 and V 2 , we obtain The fusion of two vectorial R-matrices by using the projectors P with the bases |52 + e η (2e −η + cosh 5η + sinh 3η) √ cosh η cosh 3η Here, two 16-dimensional spaces V 1 and V 2 are fused into a 16-dimensional fused space V 12 . We find that the fused space V 12 can be divided into two 4-dimensional spaces V 1 and V2 , where V 1 is the space of spinorial representation s + and V2 is the space of spinorial representation s − . The S 1 2 is a 16 × 16 matrix defined in the tensor space where the nonzero matrix elements are The matrixS 1 2 can be obtained from the S 1 2 by using the mappinḡ

B Fusion of the reflection matrices
In this appendix, we shall give the related fusion of the reflection matrices [53,54] associated with those of the R-matrices in appendix A. By using the one-dimensional projector P vv (1)  12 , we obtain the fusion relation of reflection matrix K v (u) and that ofK v (u) as P vv (1)   and satisfies the Yang-Baxter equation The matrices K s ± (u) satisfy the reflection equation The matricesK s ± (u) satisfy the dual reflection equation Taking the fusion of reflection matrices K v (u) and K s ± (u) by using the 4-dimensional projector P From eqs. (B.13) and (B.14), we see that the reflection matrix K s ∓ (u) can be obtained from K v (u) and K s ± (u). Similarly, the fusion of dual reflection matricesK v (u) andK s ± (u) by using the 4-dimensional projector P , we see that the dual reflection matrixK s ∓ (u) can be obtained fromK v (u) andK s ± (u), which means that the fusion process now is closed.

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