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.


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 [2,3]. Then it was shown [4,5] that the computing of the anomalous dimensions of determinant-like operators of N = 4 SYM can be mapped to the eigenvalue problem of certain integrable open spin chain ( spin chain with boundary condition specified by reflection K-matrices or boundary scattering matrices) Hamiltonians [2,6,7], while by AdS/CFT the K-matrices of the open chain correspond to open strings attached to maximal giant gravitons [5,8]. 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,

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we develop the nested off-diagonal Bethe ansatz method, a generalization of the method proposed in [10][11][12][13], 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 [14][15][16], the T-Q approach [17][18][19][20][21], the algebraic BA [22][23][24][25][26][27], the analytic BA [28], the functional BA [29,30] or the separation of variables method [31][32][33][34] and many others . 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 [25].
Recently, a systematic method [10][11][12][13] 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 [56,57]. With the help of the Hirota equation, Nepomechie [58] generalized the results of [10][11][12][13] to the arbitrary spin XXX open chain with general boundary terms. An expression for the corresponding eigenvectors was also proposed recently in [59].
The central idea of the method in [10][11][12][13] is to construct a proper T −Q ansatz with an extra off-diagonal term (comparing with the ordinary ones [20]) 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., 1 t(θ j )t(θ j − η) ∼ ∆ q (θ j ). (1.1) In this paper, we generalize the off-diagonal Bethe ansatz method to the multi-component integrable models (integrable spin chains associated with higher rank algebras). This generalization allows us to construct the nested T − Q relations based on the recursive operator product identities and the asymptotic behavior of the transfer matrices for the systems with both periodic and arbitrary integrable open boundary conditions. We elucidate our method with the su(n) spin chain (both periodic and open) model as an example. Our method might be used to the integrable systems associated with B n , C n and D n algebras. The paper is organized as follows. Section 2 serves as an introduction of our notations and some basic ingredients. We briefly describe the inhomogeneous su(n)-invariant spin chain with periodic boundary condition. Based on some operator product relations for the antisymmetric fused transfer matrices and their asymptotic behaviors, the nested T − Q ansatz of their eigenvalues and the corresponding Bethe ansatz equations (BAEs) are constructed. In section 3, we study the su(n)-invariant open spin chains with general

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open boundary integrable conditions. Based on some properties of the R-matrix and Kmatrices, we obtain the important operator product identities among the fused transfer matrices of the open chains and their asymptotic behaviors when u −→ ∞. In section 4, we focus on the su(3)-invariant open spin chain with the most general non-diagonal boundary terms. The nested Bethe ansatz solution for the eigenvalues of the transfer matrix and the corresponding Bethe ansatz equations (BAEs) are given in detail based on the operator product identities of the transfer matrix and their asymptotic behaviors and values of the transfer matrices 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 [60,61] 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- 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: 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.

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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 , 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 [24] t (p) (u) = tr 0 T 0 (u). (2.9) The QYBE implies that one-row monodromy matrix T (u) satisfies the following relation (2.10) The above equation leads to the fact that the transfer matrices with different spectral parameters commute with each other: [t (p) (u), t (p) (v)] = 0. Then t (p) (u) serves as the generating functional of the conserved quantities, which ensures the integrability of the closed spin chain. The Hamiltonian (2.1) can be obtained from the transfer matrix as following

Operator product identities
Our main tool is the so-called fusion technique [65][66][67][68][69]. 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 We introduce further the fused one-row monodromy matrices T 1,...,m (u) (cf. (2.8)) 12) and the associated fused transfer matrices t which includes the fundamental transfer matrix t (p) (u) given by (2.9) as the first one, i.e., t (p) (u) = t   (2.14)

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We note that t (p) n (u) is the quantum determinant (proportional to the identity operator for generic u and {θ j }), 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 initial condition (2.4), the properties (2.7) of the R-matrix (e.g, R ij (η)R ij (−η) = 0) and the properties (A.7) (see below) imply that the fused monodromy matrices T 1,...,m (u) given by (2.12) vanishes at some special points, We use the convention: τ (p) (u) = τ  The operator identities (2.16) implies that these operators satisfy the following functional relations

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The analyticity of the R-matrix implies that the eigenvalues Λ , l = 1, . . . n, (2.21) where the functions Q (l) p (u) are given by where {L r |r = 1, . . . n − 1} are some non-negative integers and the parameters {λ 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 [62][63][64].

Transfer matrix
Integrable open chain can be constructed as follows [15,25]. Let us introduce a pair of K-matrices K − (u) and K + (u). The former satisfies the reflection equation (RE)

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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) Then the double-row transfer matrix t(u) of the open spin chain is given by From the QYBE and the (dual) RE, one may check that the transfer matrices with different spectral parameters commute with each other: [t(u), t(v)] = 0. Thus t(u) serves as the generating functional of the conserved quantities, which ensures the integrability of the system. In this paper, we consider a generic solution K − (u) to the RE associated with the 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 JHEP04(2014)143

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 [65][66][67][68][69] and for K-matrices [75,76]. 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 [75,76], let us introduce the fused K-matrices and double-row monodromy matrices by the following recursive relations where the fused one-row monodromy matrix T 1,...,m (u) is given by (2.12) and For the open spin chain, the m-th fused transfer matrix t m (u) constructed by the antisymmetric fusion procedure is given by It follows from the fusion of the R-matrix [65][66][67][68][69] and that of the K-matrices [75,76] 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, 2) allow us to rewrite the fused K-matrix in another form The above equation, (3.13)-(3.14), (2.5)-(2.7) and the degenerate properties of the Rmatrix and the K-matrices: (3.24) implies that at the following 2m special points 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 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 matrixT 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 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)

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It follows from the definitions of the fused transfer matrices (3.18) that the eigenvaluē Λ m (u) of the resulting commutative operator τ m (u), as a function of u, is a polynomial of degree 2N + 2m. The functional relations (3.27) give rise to that the eigenvalueΛ m (u) of τ m (u) satisfies the following relations where the function ρ 0 (u) is given by Then τ n (u) is proportional to identity operator with a known coefficientΛ n (u) where ϑ j are some continuous free parameters which are related to boundary interaction terms (e.g., the boundary magnetic fields). The maximum number of the continuous parameters is n/2 if n is even and is (n − 1)/2 if n is odd. Some remarks are in order. When M andM commute with each other and thus can be diagonalized simultaneously by some gauge transformation, the corresponding open spin JHEP04(2014)143 chain can be diagonalized by the algebraic Bethe ansatz method after a global gauge transformation [71]. 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 [41,77,78] can be used to obtain the Bethe ansatz solutions of the associated open spin chains [79][80][81]. However, the results in [10][11][12][13] 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

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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. 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 solution of the su (3) as an example to demonstrate our method in detail. Let us introduce 3 functions {K (l) |l = 1, 2, 3} as follows

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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 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 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. with the help of (3.22), by putting u = ±θ j , ±θ j − η in (4.21) and (4.22) we can easily show that the ansatz (4.21)-(4.22) make the very functional relations (3.28) with n = 3 fulfilled. The regular property of Λ(u) and Λ 2 (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 (4.29)-(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) and (4.7). To show the procedure clearly, we construct 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, we may derive the solutions of the su(n)-invariant quantum spin chain with general open boundary conditions. Here we present the final result. The functions z m (u) now read where {K (l) (u)|l = 1, . . . , n} satisfy (3.37)-(3.38) and {Q (m) (u)|m = 0, 1, . . . , n} are given by (4.14)-(4.16). In principle, K (l) (u) could be any decomposition of (3.37). For simplicity,

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we parameterize them satisfying the following relations Similarly as (4.18), let us introduce the functions {z i (u)|i = 1, . . . , n} bỹ where the functions x i (u) are and l = 1, 2, . . . , n 2 if n is even, l = 1, 2, . . . , n−1 2 . 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 (u) 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 if m is odd and n is even, if m is even and n is odd, if m is odd and n is odd.  l } satisfy the associated Bethe ansatz equations j ) = 0, j = 1, . . . , L 1 .
, k = 1, . . . , L 2l , (5.17) Some remarks about the existence of solutions of (5.9) are in order. Due to the construction rule (5.15) for Λ m (u), the asymptotic behaviors of Λ m (u) and Λ n−m (u) give the same condition in (5.9), namely, the total number of the independent equations in (5.9) is n/2 if n is even and is (n − 1)/2 if n is odd. Thus, the equations (5.9)-(5.13) will fix the constants {c i } in (5.7). We have checked that for n = 3 there exists an unique solution to (5.9)-(5.13) (e.g. (4.26)) and that for n > 4 there are more solutions (actually there are two solutions for n = 4) to (5.9)-(5.13). For an example, the independent equations for n = 4 and p =p = 2 are However, the different solutions only give different parameterizations of Λ i (u) in the T-Q-type. It has been shown in [10][11][12][13] that even for the open chains related to the su(2) case there indeed exist different T-Q ansatz for the eigenvalue of the transfer matrix. The numerical check [10][11][12][13]58] for the small sites of lattice shows that any of them gives the complete set of eigenvalues. The eigenvalue of the Hamiltonian (3.10) is where the parameters {λ (1) l } are the roots of the BAEs (5.16)-(5.18) 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 [77,78,[82][83][84][85][86][87][88].

Conclusions
In this paper, we propose the nested off-diagonal Bethe ansatz method for solving the multi-component integrable models with generic integrable boundaries, a generalization of the method proposed in [10][11][12][13] (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 (2.16) for the closed chain or  (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 other algebras.

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China (Grant No. 20116101110017) and BCMIIS are gratefully acknowledged. We would like to thanks for the anonymous referees for their valuable suggestions for revising the manuscript. Two of the authors (W. -L. Yang and K. Shi) would like to thank IoP, CAS for the hospitality and they enjoyed during their visit there. W.-L. Yang also would like to thank KITPC and MCM for their hospitalities where some parts of work have been done during his visiting.