Nested off-diagonal Bethe ansatz and exact solutions of the su(n) spin chain with generic integrable boundaries

The nested off-diagonal Bethe ansatz method is proposed to diagonalize multi-component integrable models with generic integrable boundaries. As an example, the exact solutions of the su(n)-invariant spin chain model with both periodic and non-diagonal boundaries are derived by constructing the nested T-Q relations based on the operator product identities among the fused transfer matrices and the asymptotic behavior of the transfer matrices.

fields, the XXZ spin torus, the closed XYZ chain with odd site number and other models with general boundary terms [40,41]. With the help of the Hirota equation, Nepomechie [42] generalized the results of [10] to the arbitrary spin XXX open chain with general boundary terms. An expression for the corresponding eigenvectors was also proposed recently in [43].
The central idea of the method in [10] is to construct a proper T − Q ansatz with an extra off-diagonal term (comparing with the ordinary ones [15]) based on the functional relations between the transfer matrix (the trace of the monodromy matrix) and the quantum determinant ∆ q (u), at some special points of the spectral parameter u = θ j , i.e., t(θ j )t(θ j − η) ∼ ∆ q (θ j ).
( one are given in Section 5 and Section 6, respectively. We summarize our results and give some discussions in Section 7. Some detailed technical proof is given in Appendix A. 2 su(n)-invariant spin chain with periodic boundary conditions

Transfer matrix
Let V denote an n-dimensional linear space. The Hamiltonian of su(n)-invariant quantum spin system with periodic boundary condition is given by [44,45] H = N j=1 P j,j+1 , (2.1) where N is the number of sites, P j,j+1 is permutation operator, P bd ac = δ ad δ bc with a, b, c, d = 1, · · · , n. The integrability of the system (2.1) is guaranteed by the su(n)-invariant R-matrix R(u) ∈ End(V ⊗ V) [46,47] R ij (u) = n α,β=1 ue αα i ⊗ e ββ j + n α,β=1 ηe αβ i ⊗ e βα j , (2.2) where e αβ is the n×n Weyl matrix with the definition (e αβ ) µν = δ αµ δ βν , α, β, µ, ν = 1, · · · , n, u is the spectral parameter and η is the crossing parameter, respectively. The R-matrix can be expressed in terms of the permutation operator P as R 12 (u) = u + ηP 1,2 . (2. 3) The R-matrix satisfies the quantum Yang-Baxter equation (QYBE) (2.4) and possesses the following properties: Initial condition : R 12 (0) = ηP 1,2 , Here R 21 (u) = P 1,2 R 12 (u)P 1,2 , P (∓) 1,2 = 1 2 {1 ∓ P 1,2 } is anti-symmetric (symmetric) project operator in the tensor product space V ⊗ V, and t i denotes the transposition in the i-th space. Here and below we adopt the standard notation: 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 an identity on the other factor spaces; R ij (u) is an embedding operator of R-matrix in the tensor space, which acts as an identity on the factor spaces except for the i-th and j-th ones.
Let us introduce the "row-to-row" (or one-row ) monodromy matrix T (u), which is an n × n matrix with operator-valued elements acting on V ⊗N , (2.9) Here {θ j |j = 1, · · · , N} are arbitrary free complex parameters which are usually called as inhomogeneous parameters.
The transfer matrix t (p) (u) of the spin chain with periodic boundary condition (or closed chain) is given by [19] t (p) (u) = tr 0 T 0 (u). (2.10) The QYBE implies that one-row monodromy matrix T (u) satisfies the following relation
Finally, we take the homogeneous limit θ j → 0. In this case, the eigenvalue of the Hamiltonian (2.1) can be expressed in terms of the Bethe roots 3 su(n)-invariant spin chain with general open boundary conditions

Transfer matrix
Integrable open chain can be constructed as follows [12,20]. Let us introduce a pair of K-matrices K − (u) and K + (u). The former satisfies the reflection equation (RE) 1) and the latter satisfies the dual RE For open spin-chains, instead of the standard "row-to-row" monodromy matrix T (u) (2.9), one needs to consider the "double-row" monodromy matrix J (u) Then the double-row transfer matrix t(u) of the open spin chain is given by where ξ is a boundary parameter and M is an n × n constant matrix (only depends on boundary parameters). Besides the RE, the K-matrix satisfies the following properties Since the second power of M becomes the n × n identity matrix, the eigenvalues of M must be ±1. Suppose that there are p positive eigenvalues and q negative eigenvalues, then we have p + q = n and trM = p − q. At the same time, we introduce the corresponding dual K-matrix K + (u) which is a generic solution of the dual RE (3.2) whereξ is a boundary parameter andM is an n × n boundary parameter dependent matrix, whose eigenvalues are ±1. Again, we suppose that there arep positive eigenvalues andq negative eigenvalues, then we havep +q = n and trM =p −q. Besides the dual RE, the K-matrix also satisfies the following properties The Hamiltonian of the open spin chain specified by the K-matrices K ± (u) (3.6) and (3.8) can be expressed in terms of the transfer matrix (3.5) as

Operator product identities
Similar to the closed spin chain case in the previous section, we apply the fusion technique to study the open spin chain. In this case, we need to use the fusion techniques both for R-matrices [48] and for K-matrices [54,55]. We only consider the antisymmetric fusion procedure which leads to the desired operator identities to determine the spectrum of the transfer matrix t(u) given by (3.5).
Following [54,55], let us introduce the fused K-matrices and double-row monodromy matrices by the following recursive relations 14) where the fused one-row monodromy matrix T 1,...,m (u) is given by (2.13) and which includes the fundamental transfer matrix t(u) given by (3.5) as the first one, i.e., t(u) = t 1 (u). It follows from the fusion of the R-matrix [48] and that of the K-matrices [54,55] that the fused transfer matrices constitute commutative families, namely, Moreover, we remark that t n (u) is the so-called quantum determinant and that for generic u and {θ j } it is proportional to the identity operator, namely, The commutativity of the transfer matrices with different spectral parameters implies that Now let us evaluate the product of the fundamental transfer matrix and the fused ones at some special points According to the definition (3.18), we thus have the following functional relations among the transfer matrices In terms of the corresponding eigenvalues, the above relations become One may check that the fused transfer matrices t m (u) have some zero points, which allows us to rewrite the transfer matrices as Since the operator τ m (u) is proportional to the transfer matrix t m (u) by c-number coefficient, the corresponding eigenvalueΛ m (u) has the following relation with Λ m (u) where the function ρ 0 (u) is given by Then τ n (u) is proportional to identity operator with a known coefficientΛ n (u) formation [50]. In case of the boundary parameters (which are related to the matrices M andM ) have some constraints so that a proper "local vacuum state" exists, the generalized algebraic Bethe ansatz method [29,56] can be used to obtain the Bethe ansatz solutions of the associated open spin chains [57,58]. However, the results in [10] strongly suggest that for generic M andM such a simple "local vacuum state" do not exist even for the su(2)

Asymptotic behaviors of the transfer matrices
case.
The asymptotic behaviors (3.7) and (3.9) enable us to derive that the eigenvalueΛ m From the solution to the above equations, one can construct a nested T-Q ansatz for the eigenvalues Λ m (u). It is remarked that there are some different solutions to the above equations. However, it was shown in [29,42]   as an example to demonstrate our method in detail.
The functional relations (3.28) of the eigenvaluesΛ m (u) now read where the function ρ 0 (u) is given by Let us introduce 3 functions {K (l) |l = 1, 2, 3} as follows which satisfy the following relations From the definitions (3.18) of the fused transfer matrices t m (u) and the asymptotic behaviors of the K-matrices K ± (u), we have that the eigenvalues of the transfer matrices have the following asymptotic behaviors Moreover, the properties of R-matrix (2.5)-(2.8) and K-matrices (3.7) and (3.9) allow us to derive that the fused transfer matrices satisfy the following properties at some special points: The above relations allow us to derive similar relations of the eigenvalues {Λ m (u)}. Then the resulting relations (total number of the conditions is equal to 2 + 4 = 6), the very relations (4.2) and the asymptotic behaviors (4.9)-(4.10) allow us to determine the eigenvaluesΛ m (u) (also Λ m (u) via the relations (3.27)).
Let us define the corresponding Q (r) (u) for the open spin chains where {L r |r = 1, . . . n − 1} are some non-negative integers. In the following part of the paper, we adopt the convention In order to construct the solution of open su(3) spin chain, we introduce threez(u) functions Here z m (u) is defined as and x 1 (u) is defined as The nested functional T-Q ansatz is expressed as Here F 1 (u) is a polynomial of degree 2L 1 − 2N. The consistency of zero residues of Λ(u) at Let all terms with f 1 (u) in Λ m (u) be zero at all the degenerate points considered in (4.11-4.16). f 1 (u) can be given by where c is a constant. This allows us to write down the explicit nested T-Q ansatz (4.24)-(4.25) as follows where the non-negative integers L 1 and L 2 satisfy the relation: the functions {K (l) (u)|l = 1, 2, 3} are given by (4.4)-(4.6) and the parameter c is given by The above relation and the relations (3.27) betweenΛ m (u) and Λ m (u) lead to that the asymptotic behaviors (4.9)-(4.10) of the eigenvaluesΛ m (u) are automatically satisfied. Noticing one can easily show that the ansatz (4.29)-(4.30) also make the very functional relations (4.2) fulfilled. The regular property of Λ(u) leads to the associated Bethe ansatz equations, The eigenvalue of the Hamiltonian (3.10) in the case of n = 3 is given by where the parameters {λ (1) l } are the roots of the BAEs (4.34)-(4.35) in the homogeneous limit θ j = 0.

su(4)-invariant spin chain with non-diagonal boundary magnetic fields
In this section, we use the method outlined in Section 3 to give the Bethe ansatz solution of the su (4)-invariant open spin chain with generic boundary terms. The model may include two free continuous parameters ϑ 1 and ϑ 2 defined in (3.32), which is the first non-trivial case to study the multi-components models beyond su(2) case. Without loss of generality, we consider the matrices M andM with p = 2 andp = 2.
Let us introduce 4 functions {K (l) (u)|l = 1, . . . 4} Then the asymptotic behaviors of the eigenvalues of the transfer matrices read Moreover, we can derive the following relations among the fused transfer matrices at some special points: and The above relations allow us to derive similar relations of the eigenvalues {Λ m (u)|m = 1, 2, 3}.
Then the resulting relations (total number of the conditions is equal to 2+4+6 = 12), the very relations (3.28) with n = 4 and the asymptotic behaviors (5.6)-(5.8) allow us to determine the eigenvaluesΛ m (u) (also Λ m (u) via the relations (3.27)).
For the su (4) open spin chain, thez(u) functions arẽ Here the function z m (u) is defined in (4.22) with m = 1, · · · , 4 and Q (4) (u) ≡ 1. The function The nested T-Q ansatz can be constructed as With the similar analysis used for the su(3) case, we have The eigenvalues of the transfer matrices of the su (4)-invariant open chain with the most general non-diagonal boundary terms are thus given by The above relation and the relations (3.27) betweenΛ m (u) and Λ m (u) lead to that the asymptotic behaviors (5.6)-(5.8) of the eigenvaluesΛ m (u) are automatically satisfied. One can easily show that the ansatz (5.31)-(5.33) also make the very functional relations (3.28) fulfilled. The regular property of Λ(u) leads to the associated BAEs It can be shown that the BAEs (5.38)-(5.40) also guarantee the regularity of the ansatz Λ 2 (u) and Λ 3 (u) given by (5.32) and (5.33), respectively. Moreover, the ansatz (5.31)-(5.33) indeed satisfy the relations (5.9)-(5.20). Finally, we conclude that the ansatz Λ m (u) given by where the parameters {λ where {K (l) (u)|l = 1, . . . , n} satisfy (3.35)-(3.36). In principle, K (l) (u) could be any decomposition of (3.35). For simplicity, we parameterize them satisfying the property The function x m (u) is and l = 1, 2, . . . , n 2 if n is even, l = 1, 2, . . . , n−1 2 andz n (u) = z n (u) if n is odd. The functions {F 2l−1 (u)} are given by where l = 2, . . . , n 2 if n is even and l = 2, . . . , n−1 2 if n is odd, and The functions f 2l−1 has the crossing symmetry Here the parameters {c 2l−1 } are determined, with helps of the asymptotic behaviors of the eigenvalues of the transfer matrices, by the following relations , if m is even and n is even; (6.7) if m is odd and n is even, (6.8) if m is even and n is odd, (6.9) if m is odd and n is odd. (6.10) Then the nested T-Q ansatz of the eigenvalues Λ(u) of the transfer matrix t(u) is All the eigenvalues Λ m (u) of the fused transfer matrix t m (u) are given by where the m 1,··· ,4 is the same as that in Eqs. (6.7)-(6.10). The parameters {λ (r) l } satisfy the associated Bethe ansatz equations 14) where the parameters {λ (1) l } are the roots of the BAEs (6.13)-(6.15) with θ j = 0.

Reduction to the diagonal boundary terms
When two K-matrices K + (u) and K − (u) are both diagonal matrices, or they can be diagonalized simultaneously by some gauge transformation, all the parameters c 2l−1 vanish, leading to F 2l−1 (u) = 0. The nested T-Q ansatz of the Λ(u) in this case becomes For example, K (l) (u) can be parameterized as where ε l = ±1 andε l = ±1. The regularity of Λ(u) leads to the associated BAEs , j = 1, . . . , L 1 , (6.21) The BAEs (6.21)-(6.22) guarantee the regularities of the expressions (6.24) of the eigenvalues Λ m (u) for the higher fused transfer matrices.

Conclusions
In this paper, we propose the nested off-diagonal Bethe ansatz method for solving the multicomponent integrable models with generic integrable boundaries, a generalization of the method proposed in [10] (related to su(2) algebra) for integrable models associated with higher rank algebras. In the method some functional relations (for the su(n) case such as The above relation and the fusion condition (2.8) allow one to derive the following identity Let us evaluate the product of the operators T1(θ j ) and T2(θ j − η) namely, we have Similarly, we havê Due to the fact that R 12 (−η) is proportional to the antisymmetric projector (2.8), the relation (A.3) also implies Using similar method to derive the above relation and following the procedure [48], we can derive the following relations Combining the above relation with (A.4), we can show that Then we can conclude that T 1 (θ j )T 2,3,...,m (θ j − η) satisfy the relation (A.1).
With the similar method used to prove (A.4) and the reflection equation (3.1), we can obtain the following relations:

Introduction
The appearance of integrability in planar AdS/CFT [1] is a rather unexpected occurrence and has led to many remarkable results [2] (see also references therein) and even ultimately to the exact solution of planar N = 4 supersymmetric Yang-Mills (SYM) theory. The anomalous dimensions of single-trace operators of N = 4 SYM are given by the eigenvalues of certain integrable closed spin chain Hamiltonians [3,2]. Then it was shown [4,5] [8,5]. Therefore spin chain model has played an important role in understanding the physical contents of planar N = 4 SYM theory and planar AdS/CFT. Moreover, it has already provided valuable insight into the important universality class of boundary quantum physical systems in condensed matter physics [9]. Motivated by the above great applications, in this paper, we develop the nested off-diagonal Bethe ansatz method, a generalization of the method proposed in [10], to solve the eigenvalue problem of multi-component spin chains with the most general integrable boundary terms.
So far, there have been several well-known methods for deriving the Bethe ansatz (BA) solutions of quantum integrable models: the coordinate BA [11,12,13], the T-Q approach [14,15,16], the algebraic BA [17,18,19,20,21,22], the analytic BA [23], the functional BA [24] or the separation of variables method [25,27,26] and many others [28,29,30,31,32,33,34,35,36,37,38,39,40,41]. However, there exists a quite unusual class of integrable models which do not possess the U(1) symmetry (whose transfer matrices contain not only the diagonal elements but also some off-diagonal elements of the monodromy matrix and the usual U(1) symmetry is broken, i.e., the total spin is no longer conserved). Normally, most of the conventional methods do not work for these models even though their integrability has been proven for many years [20].
Recently, a systematic method [10] for dealing with such kind of models associated with su(2) algebra was proposed by the present authors, which had been shown successfully to construct the exact solutions of the open Heisenberg spin chain with unparallel boundary fields, the XXZ spin torus, the closed XYZ chain with odd site number and other models with general boundary terms [42,43]. With the help of the Hirota equation, Nepomechie [44] generalized the results of [10] to the arbitrary spin XXX open chain with general boundary terms. An expression for the corresponding eigenvectors was also proposed recently in [45].
The central idea of the method in [10] is to construct a proper T − Q ansatz with an extra off-diagonal term (comparing with the ordinary ones [15]) based on the functional relations between the transfer matrix (the trace of the monodromy matrix) and the quantum determinant ∆ q (u), at some special points of the spectral parameter u = θ j , i.e., 3 (1.1) In this paper, we generalize the off-diagonal Bethe ansatz method to the multi-component at some special points. The results for the su(n)-invariant case is given in Section 5. We summarize our results and give some discussions in Section 6. Some detailed technical proof is given in Appendices A and B.
2 su(n)-invariant spin chain with periodic boundary conditions

Transfer matrix
Let V denote an n-dimensional linear space. The Hamiltonian of su(n)-invariant quantum spin system with periodic boundary condition is given by [46,47] where N is the number of sites, P j,j+1 is permutation operator, P bd ac = δ ad δ bc with a, b, c, d = 1, · · · , n. The integrability of the system (2.1) is guaranteed by the su(n)-invariant R-matrix R(u) ∈ End(V ⊗ V) [48,49] R 12 (u) = u + ηP 1,2 , where u is the spectral parameter and η is the crossing parameter. The R-matrix satisfies the quantum Yang-Baxter equation (QYBE) and possesses the following properties: Initial condition :

4)
Unitarity : Fusion conditions : Here R 21 (u) = P 1,2 R 12 (u)P 1,2 , P 1,2 = 1 2 {1 ∓ P 1,2 } is anti-symmetric (symmetric) project operator in the tensor product space V ⊗ V, and t i denotes the transposition in the i-th space. Here and below we adopt the standard notation: 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 an identity on the other factor spaces; R ij (u) is an embedding operator of R-matrix in the tensor space, which acts as an identity on the factor spaces except for the i-th and j-th ones.
Let us introduce the "row-to-row" (or one-row ) monodromy matrix T (u), which is an n × n matrix with operator-valued elements acting on V ⊗N , (2.8) Here {θ j |j = 1, · · · , N} are arbitrary free complex parameters which are usually called as inhomogeneous parameters.
The transfer matrix t (p) (u) of the spin chain with periodic boundary condition (or closed chain) is given by [19] t (p) (u) = tr 0 T 0 (u). (2.9) The QYBE implies that one-row monodromy matrix T (u) satisfies the following relation

Operator product identities
Our main tool is the so-called fusion technique [51]. We shall only consider the antisymmetric fusion procedure which leads to the desired operator identities to determine the spectrum of the transfer matrix t (p) (u) given by (2.9).
For this purpose, let us introduce the anti-symmetric projectors which are determined by the following induction relations (2.14) We note that t Let us evaluate the product of the fundamental transfer matrix and the fused ones at some special points θ j and θ j − η. According to the definition (2.13), we thus have the following functional relations among the transfer matrices The operator identities (2.16) implies that these operators satisfy the following functional relations

Nested T-Q relation
The explicit expression (2.15) of the quantum determinant, the asymptotic behaviors (2.19) and where the functions Q The regular property of Λ (p) (u) implies that the residues of Λ (p) (u) at each apparent simple pole λ (r) l have to vanish. This leads to the associated BAEs, By taking the limit θ j = 0, the above BAEs are readily reduced to those previously obtained by other Bethe ansatz methods [48,49,50].

Transfer matrix
Integrable open chain can be constructed as follows [12,20]. Let us introduce a pair of K-matrices K − (u) and K + (u). The former satisfies the reflection equation (RE) and the latter satisfies the dual RE For open spin-chains, instead of the standard "row-to-row" monodromy matrix T (u) (2.8), one needs to consider the "double-row" monodromy matrix J (u) 3) where ξ is a boundary parameter and M is an n × n constant matrix (only depends on boundary parameters). Besides the RE, the K-matrix satisfies the following properties Since the second power of M becomes the n × n identity matrix, the eigenvalues of M must be ±1. Suppose that there are p positive eigenvalues and q negative eigenvalues, then we have p + q = n and trM = p − q. At the same time, we introduce the corresponding dual K-matrix K + (u) which is a generic solution of the dual RE (3.2) whereξ is a boundary parameter andM is an n × n boundary parameter dependent matrix, whose eigenvalues are ±1. Again, we suppose that there arep positive eigenvalues andq negative eigenvalues, then we havep +q = n and trM =p −q. Besides the dual RE, the K-matrix also satisfies the following properties The Hamiltonian of the open spin chain specified by the K-matrices K ± (u) (3.6) and (3.8) can be expressed in terms of the transfer matrix (3.5) as

Operator product identities
Similar to the closed spin chain case in the previous section, we apply the fusion technique to study the open spin chain. In this case, we need to use the fusion techniques both for R-matrices [51] and for K-matrices [57,58]. We only consider the antisymmetric fusion procedure which leads to the desired operator identities to determine the spectrum of the transfer matrix t(u) given by (3.5).
Following [57,58], let us introduce the fused K-matrices and double-row monodromy matrices by the following recursive relations 14)  It follows from the fusion of the R-matrix [51] and that of the K-matrices [57,58] that the fused transfer matrices constitute commutative families, namely, Moreover, we remark that t n (u) is the so-called quantum determinant and that for generic u and {θ j } it is proportional to the identity operator, namely, The above equation, (3.13)-(3.14), (2.5)-(2.7) and the degenerate properties of the R-matrix and the K-matrices: (3.24) implies that at the following 2m special points   In terms of the corresponding eigenvalues, the above relations become Using the similar method that we have derived the zero points (2.17) of the fused monodromy matrix T 1,...,m (u), we can figure out the zero points of the fused monodromy matrix T 1,...,m (u) and those of the fused K-matrices K ± 1,...,m (u) respectively. Thanks to the alternative expression (3.19) of the fused transfer matrix t m (u), we know that these zero points all together constitute the zero points of the transfer matrix, which allows us to rewrite the transfer matrix as (3.29) Since the operator τ m (u) is proportional to the transfer matrix t m (u) by c-number coefficient, the corresponding eigenvalueΛ m (u) has the following relation with Λ m (u)  where the function ρ 0 (u) is given by Then τ n (u) is proportional to identity operator with a known coefficientΛ n (u) formation [53]. In case of the boundary parameters (which are related to the matrices M andM ) have some constraints so that a proper "local vacuum state" exists, the generalized algebraic Bethe ansatz method [31,59] can be used to obtain the Bethe ansatz solutions of the associated open spin chains [60,61]. However, the results in [10] strongly suggest that for generic M andM such a simple "local vacuum state" do not exist even for the su (2) case.

Asymptotic behaviors of the transfer matrices
The asymptotic behaviors (3.7) and (3.9) enable us to derive that the eigenvalueΛ m (u) of the operators {τ m (u)} given by (3.29) have the following asymptotic behaviors  Before closing this section, let us give a summary of the set of properties which characterize the eigenvalue of the transfer matrix t m (u): • Explicit expression of t n (u) or the quantum determinant (3.22).
• Analytical property and asymptotical behaviors (3.36) of the transfer matrices.
• Functional relations (3.27) for the fused transfer matrices.
• The above condition are believed to determine the eigenvalues of the transfer matrices t m (u).

su(3)-invariant spin chain with non-diagonal boundary term
In this section, we use the method outlined in the previous section to give the Bethe ansatz as an example to demonstrate our method in detail.
Let us define the corresponding Q (r) (u) for the open spin chains 14) l + rη), r = 1, . . . , n − 1, (4.15) where {L r |r = 1, . . . , n − 1} are some non-negative integers. In the following part of the paper, we adopt the convention In order to construct the solution of the open su(3) spin chain, we introduce three functions Here z m (u) is given by the following relations with {K (m) (u)|m = 1, 2, 3} are given by (4.2)-(4.4) (here we have assumed Q (3) (u) = 1 since that the su(3)-case is considered) and x 1 (u) is defined as The nested functional T-Q ansatz is expressed as Λ 2 (u) = ρ 2 (2u − η) We remark that the extra term x 1 (u) in (4.18) given by (4.20) does not violate the very functional relation (3.28) with n = 3 due to the fact a(±θ j − η) = d(±θ j ) = 0, but it does change the form of the resulting BAEs (see (4.29) below). The function x 1 (u) can be determined by regularity of Λ(u) and Λ 2 (u) given by (4.21) and (4.22) and their asymptotic behaviors as follows. The vanishing of the residues of Λ(u) at λ (1) j and −λ (1) j − η requires with In order not to violate the relations (4.8)-(4.13), let all terms with x 1 (u) in Λ m (u) be zero at all the degenerate points considered in (4.8)-(4.13), then the function f 1 (u) is given by The asymptotic behaviors of Λ(u) and Λ 2 (u) then fix the constant c, c = 2(cos ϑ − 1), (4.26) where ϑ is specified by the eigenvalues of the matrixM M (4.1). It is remarked that F 1 (u) is a polynomial of degree 2L 1 − 2N. Then the above relations lead to the constraint among the non-negative integers L 1 and L 2 L 1 = N + L 2 + 2. The regular property of Λ(u) and Λ 2 (u) leads to the associated Bethe ansatz equations, , l = 1, . . . , L 1 , (4.29) (4.30) The eigenvalue of the Hamiltonian (3.10) in the case of n = 3 is given by where the parameters {λ (1) l } are the roots of the BAEs (4.29)-(4.30) in the homogeneous limit θ j = 0.

Exact solution of su(n)-invariant spin chain with general open boundaries
The analogs of (4.8)-(4.13) for arbitrary n at the special points listed in (3.25) can also be constructed with the properties of (3.24). To show the procedure clearly, we constructed those relations for n = 4 in Appendix B. In fact, those relations are ensured for the diagonal case as already demonstrated by the algebraic Bethe ansatz. For the non-diagonal case, since we put x i (u) to be zero for u at those degenerate points, the relations must also hold no matter how their exact forms are. By following the same procedure as the previous section, (4.14)-(4.16). In principle, K (l) (u) could be any decomposition of (3.37). For simplicity, we parameterize them satisfying the following relations The function x m (u) is and l = 1, 2, . . . , n 2 if n is even, l = 1, 2, . . . , n−1 2 andz n (u) = z n (u) if n is odd. The functions {F 2l−1 (u)} are given by where l = 2, . . . , n 2 if n is even and l = 2, . . . , n−1 2 if n is odd, and The functions f 2l−1 has the following crossing symmetry relation Here the parameters {c 2l−1 } are determined, with helps of the asymptotic behaviors of the eigenvalues of the transfer matrices, by the following relations ansatz of the eigenvalues Λ(u) of the transfer matrix t(u) is All the eigenvalues Λ m (u) of the fused transfer matrix t m (u) are given by where the m 1 , m 2 , m 3 and m 4 are the same as those in the equations (5.9)-(5.12). The parameters {λ (r) l } satisfy the associated Bethe ansatz equations  l + η) where the parameters {λ (1) l } are the roots of the BAEs (5.15)-(5.17) with θ j = 0. When two K-matrices K + (u) and K − (u) are both diagonal matrices, or they can be diagonalized simultaneously by some gauge transformation, all the parameters c 2l−1 vanish, leading to F 2l−1 (u) = 0. The eigenvalue Λ(u) of the transfer matrix t(u) and the BAEs recover those obtained by the other Bethe ansatz methods [59,62,63,64,65,66,67,68].

Conclusions
In this paper, we propose the nested off-diagonal Bethe ansatz method for solving the multicomponent integrable models with generic integrable boundaries, a generalization of the method proposed in [10] (related to su(2) algebra) for integrable models associated with higher rank algebras. In the method some functional relations (for the su(n) case such as  (4)-case) at n−1 m=1 2m special points (3.25), we obtain the eigenvalues of the transfer matrix. When the K-matrices are both diagonal ones, our results can be reduced to those obtained by the conventional Bethe ansatz methods. Therefore, our method provides an unified procedure for approaching the integrable models both with and without U(1) symmetry. We remark that this method might also be applied to other quantum integrable models defined in different algebras.