Exact solutions of the $C_n$ quantum spin chain

We study the exact solutions of quantum integrable model associated with the $C_n$ Lie algebra, with either a periodic or an open one with off-diagonal boundary reflections, by generalizing the nested off-diagonal Bethe ansatz method. Taking the $C_3$ as an example we demonstrate how the generalized method works. We give the fusion structures of the model and provide a way to close fusion processes. Based on the resulted operator product identities among fused transfer matrices and some necessary additional constraints such as asymptotic behaviors and relations at some special points, we obtain the eigenvalues of transfer matrices and parameterize them as homogeneous $T-Q$ relations in the periodic case or inhomogeneous ones in the open case. We also give the exact solutions of the $C_n$ model with an off-diagonal open boundary condition. The method and results in this paper can be generalized to other high rank integrable models associated with other Lie algebras.


Introduction
Quantum integrable models have many applications in the fields of quantum field theory, condensed matter physics, string theory and mathematical physics. The algebraic/coordinate Bethe ansatz and T −Q relations are the very powerful methods to obtain the exact solutions of integrable models with periodic or diagonal open boundary conditions [1][2][3][4][5]. Focusing on the boundary integrable models, it is well-known that some reflection matrices including the off-diagonal elements also satisfy the reflection equations, which implies that the systems are still integrable even with off-diagonal boundary reflections. However, due to the existence of off-diagonal elements, it is quite difficult to calculated the exact solutions of this kind of systems because that the reflection matrices at two boundaries cannot be diagonalized simultaneously. We also note that the models with off-diagonal boundary reflections are very important and have many applications in many issues such as the open AdS/CFT theory, edge states and topological physics. Therefore, many interesting methods such as q-Qnsager alegebra [6][7][8][9], separation of variables [10][11][12], modified algebraic Bethe ansatz [13][14][15][16] and off-diagonal Bethe ansatz (ODBA) [17,18] are proposed to study this kind of systems.
The ODBA is an universal method to solve the models with generic integrable boundary conditions. With the help of proposed inhomogeneous T − Q relations, the exact solutions of some typical models with off-diagonal boundary reflections are obtained [18]. Furthermore, in order to solve the models with high ranks [19][20][21][22][23][24][25], the nested ODBA is proposed and the exact solutions of models associated with A n [26,27], A (2) 2 [28], B 2 [29], C 2 [30] and D (1) 3 [29] Lie algebras were obtained. One important property of high rank integrable models is that the eigenvalue of transfer matrix is a polynomial where the degree is higher, thus we need more functional relations to determine it completely. Meanwhile, due to the different algebraic structures, the closing conditions of these functional relations are different.
In this paper, we study the functional relations of the integrable C n vertex model by using the fusion technique [31][32][33][34][35][36][37][38] and the nested ODBA [18]. We systemically analyze the fusion behaviors and obtain recursive fusion relations among the fused transfer matrices.
The fusion relations with periodic boundary conditions are different from those with open boundaries. We provide a way to close these recursive fusion relations. Based on them and asymptotic behaviors as well as values at certain points, we obtain the eigenvalues of transfer matrices and parameterize them as the homogeneous or inhomogeneous T −Q relations. The associated Bethe ansatz equations are also given. Then we generalize these results to the C n model with off-diagonal open boundary condition. We expect that the method and results provided in this work can be applied to other high rank integrable models associated with other Lie algebras.
The plan of the paper is as follows. In section 2, we study the model with periodic boundary condition. The fusion structures of integrable C 3 vertex model is shown in detailed. The closed recursive fusion relations among fused transfer matrices are given. By constructing the T − Q relations, we obtain the eigenvalues and associated Bethe ansatz equations of the system. In section 3, we diagonalize the model with off-diagonal boundary reflections.
The reflection matrices with off-diagonal elements and corresponding fusion behavior are introduced. Based on the closed operators product identities, we obtain the eigenvalues of transfer matrices and expressed them as the inhomogeneous T − Q relations. These results are also generalized to the C n model, which are listed in section 4. The summary of main results and some concluding remarks are presented in section 5.

C 3 model with periodic boundary condition 2.1 Integrability
Through this paper, we adopt following standard notations. Let V denote a 6-dimensional linear space with orthogonal bases {|i |i = 1, · · · , 6}. 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. For a matrix R ∈ End(V ⊗V ), R ij is an embedding operator defined in the same tensor space, which acts as an identity on the factor spaces except for the i-th and j-th ones. The quantum integrable system associated with C 3 Lie algebra is described by a 36 × 36 R-matrix R 12 (u) with the elements [22] R 12 (u) ij kl = u(u + 4)δ ik δ jl + (u + 4)δ il δ jk − uξ i ξ k δ jī δ kl , (2.1) where u is the spectral parameter, i +ī = 7, 4,6]. For the simplicity, we introduce following notations The R-matrix (2.1) has following properties regularity : where ρ v (u) = a(u)a(−u), P 12 is the permutation operator with the matrix elements [P 12 ] ij kl = δ il δ jk , t i denotes the transposition in the i-th space, and R 21 = P 12 R 12 P 12 . The R-matrix (2.1) satisfies the Yang-Baxter equation The monodromy matrix of the system is constructed by the R-matrix (2.1) as where the subscript 0 means the auxiliary space, the other tensor space V ⊗N is the physical or quantum space, N is the number of sites and {θ j |j = 1, · · · , N} are the inhomogeneous parameters. The monodromy matrix satisfies the Yang-Baxter relation Taking the partial trace of monodromy matrix in the auxiliary space, we arrive at the transfer matrix of the system with periodic boundary condition From the Yang-Baxter relation (2.6), one can prove that the transfer matrices with different spectral parameters commutate with each other, i.e., [t (p) (u), t (p) (v)] = 0. Therefore, t (p) (u) serves as the generating function of all the conserved quantities of the system. The model Hamiltonian with C 3 -invariant is given by

Fusion
One wonderful property of R-matrix is that the R-matrix may degenerate into the projection operators at some special points, which makes it possible for us to do the fusion. Focus on the C 3 model, the elements of R-matrix (2.1) are the polynomials of u with degree two. Thus there are two degenerate points. One is u = −4. At which we have Here P 12 is a one-dimensional projection operator with the form is a one-dimensional vector in the product space V 1 ⊗V 2 and S ′ 12 is a constant matrix (we omit its expression because we do not need it). Obviously, P 12 . From the Yang-Baxter equation (2.4), the one-dimensional fusion associated with projector (2.10) leads to We see that the result is also a one-dimensional vector.
The other degenerate point of R-matrix (2.1) is u = −1. At which we have 12 × S 12 . (2.12) Here S 12 is a constant matrix and P The projector P 21 can be obtained from P 12 by exchanging the bases of V1 and V 2 . The projector (2.17) shows that we can fuse the spaces V1 and V 2 , and the result is that we obtain a new fused R-matrix, we find that the fused R-matrix (2.18) is the same as the original one (2.1), i.e., We remark that from the way of above fusion, the auxiliary space cannot be enlarged anymore. However, both the orders of elements of R-matrix (2.1) and that of the fused one (2.14) are two. Therefore, the above fusion processes indeed are not closed and we should go further.
In order to obtain the closed fusion relations among fused R-matrices, we have to consider the degenerations of fused R-matrix (2.14) at the other degenerate point, u = −3/2. At which, the fused R-matrix (2.14) has a 14-dimensional projected subspace, which can be seen from the identity where S ′ 12 is an irrelevant constant matrix, P  21) and the corresponding bases are It is obvious that the projector P can be obtained from P By carefully analyzing the fusion structure, We find that the 14-dimensional projected space defined by (2.21) can also be obtained from the product of three R-matrices (2.1) at certain points with the following way 123 × S 123 , (2.22) where S 123 is constant matrix, P 123 is a 14-dimensional projector defined in the spaces     Taking the fusion with projector (2.23), we construct another fused R-matrix We note that the dimension of the fused space V 123 = V1 is 14. In the above construction, we have used the relation (2.22). The fused R-matrix (2.24) has following properties where ρṽ(u) = −(u + 3)(u − 3) andρṽ(u) = −(u + 1)(u + 7).
The elements of fused R-matrix (2.24) are the polynomials of u with degree one. Thus there is only one degenerate point u = −3. At which, we have 12 × S1 2 , where S1 2 is an irrelevant constant matrix omitted here, P  27) and the corresponding bases are Again, the projector P can be obtained from P Taking the fusion of R-matrix (2.24) in the auxiliary space by using the 14-dimensional projector P 12 , we obtain a fused R-matrix 12 12 . (2.28) The dimension of fused space V 1 2 is 14, which equals to the dimension of fused space V1.
After taking the correspondence we find the fused R-matrix (2.28) is the same as the fused one (2.14), i.e., Eq.(2.30) gives another intrinsic relation to close the fusion processes.
Taking the fusion of R-matrix (2.24) in the quantum space by using the 14-dimensional projector P (14) 23 given by (2.13), we obtain a fused R-matrix The fused R-matrix (2.31) is defined in the tensor space V1 ⊗ V2 and has following properties . Last, we remark that the following identity holds which can be checked by direct calculation. Eq.(2.33) implies that we can not obtain more nontrivial fused R-matrix if we take fusion only in the auxiliary spaces.

Operator product identities
Based on the obtained fused R-matrices, we define the fused monodromy matrices We note that the quantum spaces of the above monodromy matrices are the same, which is V ⊗N , and the corresponding auxiliary spaces are V0 and V0 with dimension 14. The fused monodromy matrices (2.34) satisfy the Yang-Baxter relations Therefore, they have common eigenstates and can be diagonalized simultaneously.
By using the above fusion relations of R-matrices and the definitions (2.5) and (2.34), we obtain the fusion behavior of monodromy matrices 12 Here the subscripts 1 and 2 mean the original 6-dimensional auxiliary spaces V 1 and V 2 , thē 1 means the 14-dimensional fused auxiliary space V1 by the operators P (14) 21 or P 12 , and1 means the 14-dimensional fused auxiliary space V1 by the operator P (14) 321 . Next, we calculate the products of two monodromy matrices with special spectral parameters. By using the property of permutation operator, we obtain where δ is the degenerate point of R ab (u) and P (d) ba is the corresponding d-dimensional project operator. The product of three monodromy matrices at fixed points is 32 Taking the partial traces of Eq.(2.41) in the auxiliary spaces and using the correspondences (2.20) and (2.30), we obtain the closed operator product identities among transfer matrices In the derivation, we have used the property of projector P 32 P 32 P 32 P 321 . (2.43) The asymptotic behaviors of the fused transfer matrices can be calculated directly Denote the eigenvalues of the transfer matrices t (p) (u), t The asymptotic behaviors (2.44) of the fused transfer matrices lead to From the definitions (2.7) and (2.36), we know that the eigenvalues Λ (p) (u) and Λ
Then the eigenvalues of the Hamiltonian (2.8) reads integrable requires that K − satisfies the reflection equation while K + satisfies the dual reflection equation In this paper, we consider the case that the reflection matrices have off-diagonal elements, thus the numbers of quasi-particles with different intrinsic degrees of freedom are not conserved during the reflection processes. The reflection matrix K − 0 (u) defined in the space V 0 takes the form of [39][40][41] where ζ, c 1 and c 2 are the arbitrary boundary parameters. The dual reflection matrix K + 0 (u) is defined as whereζ andc i (i = 1, 2) are the boundary parameters.
Due to the boundary reflection, besides the monodromy matrix T 0 (u) given by (2.5), we also need the reflecting monodromy matrix T 0 (u) = R N 0 (u + θ N ) · · · R 20 (u + θ 2 )R 10 (u + θ 1 ), (3.5) which satisfies the Yang-Baxter relation The transfer matrix t(u) of the system with open boundary condition is In the Hamiltonian (3.8), because two boundary reflection matrices K − 0 (u) (3.3) and K + 0 (u) (3.4) do not commutate with each other, i.e., [K − 0 (u), K + 0 (v)] = 0, the K ± 0 (u) cannot be diagonalized simultaneously. Then it is quite hard to derive the exact solutions of the system via the conventional Bethe Ansatz due to the absence of a proper reference state. We will generalize the method developed in section 2 to calculate the eigenvalues of transfer matrix (3.7) and that of Hamiltonian (3.8) in the following subsections.

Fusion
12 , P 123 , P 12 , P where Det q (K ± (u)) are the quantum determinants of reflection matrices K ± (u), We note that the reflection equation and dual one require that the inserted R-matrices in (3.9) with determined spectral parameters are necessary.
Using the 14-dimensional projector P 12 , we construct the 14 × 14 fused K-matrices The fused reflection matrices (3.11) satisfy the reflection equations which means that the fusion does not break the integrability.
The 14-dimensional projector P (14) 123 allows us to construct the 14 × 14 fused K-matrices 123 ≡ K − 1 (u), The fused reflection matrix (3.13) satisfy the reflection equations (3.14) Using the 6-dimensional projectors P (6) 12 and the correspondence (2.19), we have We note the fused reflection matrices (3.15) are the same as the original ones given by (3.3) and (3.4). Similarly, with the help of the 14-dimensional projectors P (14) 12 and the correspondence (2.29), we have We note that the fused reflection matrices (3.16) are the same as the fused ones (3.11). Now we have obtain all the necessary fused reflection matrices, which are used to construct the conserved quantities and fusion relations of the system with open boundary conditions.

Operator product identities
The fused reflecting monodromy matrices are defined aŝ where R 21 (u) and R 21 (u) can be obtained from the first relations in (2.15) and (2.25), respectively. The fused reflecting monodromy matrices satisfy the Yang-Baxter relations The fused transfer matrices are the partial traces of fused monodromy matrices (3.19) where the fused reflection matrices K ± 0 (u) and K ± 0 (u) are given by (3.11) and (3.13), respectively. From the Yang-Baxter relations (2.35), (3.18) and reflection equations (3.12), (3.14), one can prove that the transfer matrices t(u), t 2 (u) and t 3 (u) commutate with each other (3.20) Thus these transfer matrices have common eigenstates and can be diagonalized simultaneously.
In order to solve these transfer matrices, we should seek the constraints they satisfied.
The method is fusion. The fusions of reflecting monodromy matrices read 12T 1 (u)T 2 (u − 1)P (14) 12 123T 1 (u)T 2 (u − 1)T 3 (u − 2)P (14) 123 21T 2 (u)T1(u − 3)P Meanwhile, the products of reflecting monodromy matrices at two special points satisfŷ 12T 1 (−θ j )T 2 (−θ j − 1), Then, we are read to consider the constraints of transfer matrices. Direct calculation In the derivation, we have used the relations The following relation also holds where The asymptotic of transfer matrices can be derived directly According to the definitions, we also know In the derivation, we have used the relations From the construction of transfer matrices, we know that t(u), t 2 (u) and t 3 (u) are the operator polynomials of u with degrees 4N + 2, 4N + 4 and 2N + 6, respectively. Thus we need 10N + 15 independent conditions to determine their eigenvalues.

Functional relations
We have proved that the transfer matrices t(u), t 2 (u) and t 3 (u) have common eigenstates.
Acting the transfer matrices on the common eigenstates, we obtain the corresponding eigenvalues. Denote the eigenvalues of t(u), t 2 (u) and t 3 (u) as Λ(u), Λ 2 (u) and Λ 3 (u), respectively.

Inhomogeneous T − Q relations
For the simplicity, we define some functions 32) and the numbers of Bethe roots satisfy the constraints L 1 = L 2 + N and L 3 = L 2 . By using these functions, we construct the eigenvalues of transfer matrices as Z 6 (u) = Z 6 (u) + f 3 (u),Z 5 (u) = Z 5 (u). (3.34) All the eigenvalues are the polynomials, thus the residues of right hand sides of Eq. (3.33) should be zero, which gives the Bethe ansatz equations l + 1 2 , l = 1, 2, · · · , L 2 , = −x, m = 1, 2, · · · , L 3 . (3. 35) We note that from the regularity analysis of any Λ(u), Λ 2 (u) or Λ 3 (u), one can obtain the complete set of Bethe ansatz equations. The Bethe ansatz equations obtained from Λ(u) are the same as those obtained from Λ 2 (u) and Λ 3 (u). Meanwhile, the function Q (m) (u) has two zero points, namely, λ If c 1 = c 2 =c 1 =c 2 = 0, the boundary reflection matrices degenerate into the diagonal ones and our results cover that obtained by the algebraic Bethe ansatz [42].

C n model
In this section, we generalize above results to the C n model. The C n model with periodic boundary condition has been studied in reference [22]. Thus we focus on the open boundary conditions. The main idea is the same as before. Here we only list the results. The R-matrix of the C n model is a (2n) 2 × (2n) 2 one with the elements R(u) ij kl = u(u + n + 1)δ ik δ jl + (u + n + 1)δ il δ jk − uξ i ξ k δ jī δ kl , (4.1) where i, j, k, l = 1, · · · , 2n, i +ī = 2n + 1, ξ i = 1 if i ∈ [1, n] and ξ i = −1 if i ∈ [n + 1, 2n].
The off-diagonal boundary reflection matrices is chosen as whereζ,c 1 andc 2 are the free boundary parameters and I is a n × n unitary matrix. The dual reflection matrixK + 0 (u) is determined by the mappinḡ whereζ,c 1 andc 2 are the boundary parameters.
From the singularities analysis of inhomogeneous T − Q relations (4.6), we obtain the Bethe ansatz equations, which also depend on the parity of n. If n is odd, the Bethe ansatz equations are 1 λ (1) = −x, k = 1, 2, · · · , L l , l ∈ odd in [2, n − 2], Q (l−1) (λ the asymptotic behaviors and the relations at some special points, we obtain the eigenvalues (3.33) of the system and give the associated Bethe ansatz equations (3.35). Moreover, we also generalize these results (4.6)- (4.18) to the C n model with off-diagonal boundary reflections (4.2)-(4.3). The method and results given in this paper can be generalized to other high rank quantum integrable systems.