Exact solution of the quantum spin chains associated with the $sp(4)$ algebra

The off-diagonal Bethe ansatz method is generalized to the integrable model associated with the $sp(4)$ (or $C_2$) Lie algebra. By using the fusion technique, we obtain the complete operator product identities among the fused transfer matrices. These relations, together with some asymptotic behaviors and values of the transfer matrices at certain points, enable us to determine the eigenvalues of the transfer matrices completely. For the model with the periodic boundary condition, the eigenvalues are described by homogeneous $T-Q$ relations, which coincides with those obtained by the conventional Bethe ansatz methods. For the model with the off-diagonal boundary condition, the eigenvalues are given in terms of inhomogeneous $T-Q$ relations, which is due to the fact of the $U(1)$-symmetry-broken and also has failed to be obtained by the conventional Bethe ansatz methods for many years. The method and the results in this paper can be used to study other integrable models associated with the $sp(2n)$ (i.e., $C_n$) algebra with a generic $n$.


Introduction
Quantum integrable models play important roles in a variety of fields such as quantum field theory, condensed matter physics and statistical physics, because they can provide solid benchmarks for understanding the many-body effects and new physical concepts in corresponding universal classes [1,2,3,4,5].
Recently, a generic method (the off-diagonal Bethe ansatz (ODBA)), for solving the integrable models with or without obvious reference states is proposed [6]. With the help of the proposed inhomogeneous T − Q relations, several typical models without U(1) symmetry are solved exactly [7]. Based on the ODBA solution to the eigenvalues, the corresponding Bethe-type states are also retrieved [8,9]. We note that the integrable models without U (1) symmetry is a very hot issue and many interesting efforts have been done [10,11,12,13,14,15,16,17,18,19].
In order to solve the models associated with high rank algebras, the nested ODBA is proposed and the model related with A n algebra was solved exactly [20]. However, for the models related with other Lie algebras such as B n , C n and D n , the corresponding results are still missing. In this paper, we generalize the ODBA method to the integrable models related with the C 2 Lie algebra (or the sp(4) algebra). We study the C 2 vertex model with periodic and open boundary conditions. For the present case, due to eigenvalues of transfer matrix is expressed by a polynomial where the degree is higher than the A n case, thus we need more functional relations to determine it. By using the fusion technique, we obtain the closed operator product identities of the fused transfer matrices. After analyzing the asymptotic behaviors and the values of the fused transfer matrices at certain special points, we obtain the eigenvalues which is described by the inhomogeneous T − Q relations [6]. The method and the results in this paper can be used to study other integrable models associated with the sp(2n) algebra with a generic n.
The paper is organized as follows. In section 2, we study the C 2 model with the periodic boundary condition. The closed functional relations are obtained by the fusion. Based on them, we obtain the exact solution of the system, which coincides with those obtained by the conventional Bethe ansatz methods [21,22]. In section 3, we study the C 2 model with integrable open boundary conditions. We provide the closed operator product identities, the asymptotic behaviors and the values at certain points of the fused transfer matrices, which enable us give the eigenvalue of the transfer matrix in terms of an inhomogeneous T − Q relation. Concluding remarks are given in section 4.
2 Closed chain with the sp(4) invariant

The system
Let V denote a 4-dimensional linear space with an orthonormal basis {|i |i = 1, · · · , 4} which endows the fundamental representation of the C 2 algebra. The sp(4)-invariant R-matrix R(u) ∈ End(V ⊗ V) is given by its matrix elements [22] where the indexī is defined by i +ī = 5, us take the notations for simplicity The R-matrix (2.1) enjoys the properties: where ρ 1 (u) = a(u)a(−u), P is the permutation operator with the elements P ij kl = δ il δ jk , and t i denotes the transposition in i-th space, R 21 (u) = P 12 R 12 (u)P 12 . 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. Moreover, the R-matrix satisfies the quantum Yang-Baxter equation (QYBE) Let us introduce the "row-to-row" (or one-row ) monodromy matrix T (u), which is a 4 × 4 matrix with operator-valued elements acting on V ⊗N , where {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 associated spin chain with the periodic boundary condition is given by [5] t p (u) = tr 0 T 0 (u). (2.5) The QYBE (2.3) of the R-matrix implies that one-row monodromy matrix T (u) satisfies the Yang-Baxter relation From the above relation, one can prove 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 sp(4)-invariant spin chain with the periodic boundary condition, which is described by the Hamiltonian where H kk+1 = P kk+1 R ′ kk+1 (u)| u=0 . The periodic boundary condition implies H N N +1 = H N 1 .

Fusion
The R-matrix (2.1) can degenerate into the projector operators at some special points, which make it possible for us to do the fusion procedure [23,24,25,26,27,28]. For example, if u = −3, we have 12 × S 1 . (2.8) Here P 12 is an one-dimensional projector operator with the form where |ψ 0 = 1 2 (|14 + |23 − |32 − |41 ) is a vector in the space V ⊗ V and S 1 is a constant matrix which we do not present here for simplicity. If u = −1, then (2.10) Here P 12 is a five-dimensional projector operator with the form where the corresponding vectors are and S 2 is a constant matrix.
From the QYBE (2.3), the one-dimensional fusion associated with the projector (2.9) leads to 21 . (2.12) From the five-dimensional fusion associated with the projector (2.11), we obtain a new fused 12 , whereρ 0 (u) = (u − 1)(u + 3). For simplicity, let us denote the resulting five-dimensional fusion space byV1 which is spanned by {|ψ (5) i |i = 1, · · · , 5}. It is easy to check that the matrix elements of the fused R-matrixR1 3 (u) ≡R 12 3 (u) [orR 31 (u) ≡R 3 12 (u)], as function of u, are polynomials of u with degree one. Moreover, we havē At the point of u = − 5 2 , the fusedR-matrix degenerates into the four-dimensional projector whereS is a constant matrix and the four-dimensional projector P 12 takes the form of 16) and the corresponding vectors are According the property (2.15), we can do the fusion by the four-dimensional projector P 12 again, which gives After taking the correspondence we have the very identifications where the R-matrices R 13 (u) and R 31 (u) are given by (2.1).

T − Q relations
From the fusedR-matrix, we can define the fused monodromy matrix which is a 5 × 5 matrix with operator-valued elements acting on V ⊗N . The fusedR-matrix and the fused monodromy matrixT (u) satisfy the Yang-Baxter relation The fused transfer matrix is given bȳ Using fusion relations (2.12), (2.13) and (2.17), we have Following the method developed in [20] and using the identification (2.19), we can show that the identities hold Considering the relations (2.19) and (2.23)-(2.28), we obtain the operator production identities among the fused transfer matrices as The commutativity of the transfer matrices t p (u) andt p (u) with different spectral parameters implies that they have common eigenstates (namely, the common eigenstates do not depend on the spetrum parameter u). Let us denote the eigenvalues of the transfer matrices t p (u) andt p (u) as Λ p (u) andΛ p (u), respectively. From the operator production identities The eigenvalue Λ p (u) of the transfer matrix t p (u) is a polynomial of u with the degree 2N, which can be completely determined by 2N + 1 conditions. Besides the functional relations (2.32)-(2.34), we still need one condition which can be obtained by analyzing the asymptotic behavior of t p (u). Form the definition, the asymptotic behavior of t p (u) can be calculated as which leads to that The eigenvalueΛ p (u) of the fused transfer matrixt p (u) is a polynomial of u with the degree N, which can be completely determined by the functional relations (2.32)-(2.34) and the asymptotic behavior oft p (u) given bȳ giving rise to thatΛ (2.36) Then the 3N + 2 relations (2.32)-(2.36) completely determine the eigenvalues Λ p (u) and Λ p (u), which are given in terms of the homogeneous T − Q relations: We note that the Bethe ansatz equations obtained from the regularity of Λ p (u) are the same as those obtained from the regularity ofΛ p (u). It is easy to check that Λ p (u) andΛ p (u) satisfy the functional relations (2.32)-(2.34) and the asymptotic behaviors (2.35) and (2.36).
Therefore, we conclude that Λ p (u) andΛ p (u) are the eigenvalues of the transfer matrices t p (u) andt p (u), respectively. It is remarked that the T −Q relation (2.37) and the associated Bethe ansatz equations (2.40)-(2.41) coincide with those [21,22] obtained previously by the the conventional Bethe ansatz methods.
The eigenvalues of the Hamiltonian (2.7) then can be expressed in terms of the Bethe roots as 3 Open chain with the integrable boundary terms
In this paper, we consider the open chain with the off-diagonal K-matrix K − (u) [29,30,31] while the dual reflection matrix K + (u) is Here ζ, c 1 , c 2 andζ,c 1 ,c 2 are the boundary parameters which describe the boundary interactions. For the generic values of these parameters, one is easy to check that [K − (u), K + (v)] = 0, which implies that the K ± (u) matrices cannot be diagonalized simultaneously. This gives rise to that the conventional Bethe ansatz methods [5,7] would fail to get the spectrum of the transfer matrix t(u) specified by the K-matrices given by (3.7) and (3.8), because of lacking the reference state. We will generalize the method developed in section 2 to get eigenvalues of the transfer matrix t(u) (3.5) specified by the K-matrices (3.7) and (3.8) in the following subsections.

Fusion of the reflection matrices
In order to obtain the closed operator production identities, we should do the fusion for the reflection matrices. The one-dimensional fusion for the reflection matrices gives where From the five-dimensional fusion, we obtain a new fused reflection matricesK as where the projector P 12 is given by (2.11). Due to the dimension of the fused spaceV is 5, the correspondingK ± 1 (u) both are the 5 × 5 matrices. Moreover, we have checked that the matrix elements ofK ± 1 (u), as function of u, are all polynomials of u with degree two. The fusedR-matrix and the fused reflection matrixK ± (u) satisfy the reflection equations (3.14) Next, we do the fusion between the reflection matrices K ± (u) andK ± (u) by the fourdimensional projector P 12 given by (2.16), which gives 12 .
It is easy to check that the matrix elements of K ± 1 2 (u) are 4 × 4 matrices, whose matrix elements, as function of u, are all polynomials of u with degree one. Moreover, keeping the correspondence (2.18) in mind, we have the identifications where the K-matrices K ± (u) are given by (3.7) and (3.8).

Operator production identities
Again, besides the fused monodromy matrixT0(u) given by (2.20), we also need the reflecting fused monodromy matrixT (u) given bŷ T0(u) =R N0 (u + θ N ) · · ·R 20 (u + θ 2 )R 10 (u + θ 1 ), (3.16) where the dimension of auxiliary spaceV is 5 and the quantum space keeps unchanged. The matrixT0 satisfies the Yang-Baxter relation The fused transfer matrixt(u) is Some remarks are in order. From the definitions (3.5) and (3.18), the transfer matrix t(u) (resp.t(u)), as a function of u, is a polynomial with degree of 4N + 2 (resp. a polynomial with degree of 2N + 4). Hence in order to determine the eigenvalues of the transfer matrices t(u) andt(u), one needs at least 6N + 8 conditions. Using the similar method developed in previous section, we will look for the 6N + 8 conditions.
Using fusion relations (2.12), (2.13) and (2.17), we have 12T 1 (u)T 2 (u − 1)P We can show that Yang-Baxter relations (3.4) and (3.17) at certain points also givê Form the definition, the asymptotic behavior of t(u) can be calculated as Besides, we also know the values of t(u) at the points of 0 and −3, The asymptotic behavior oft(u) reads t(u)| u→∞ = tr 12 P Using the method developed in [20], we can evaluate the values oft(u) at some special points as follows:t (3.40) The 6N +8 relations (3.37)-(3.47) enable us completely to determine the eigenvalues Λ(u) andΛ(u) which are given in terms of some inhomogeneous T − Q relations. For simplicity, we first define some functions: where x is a constant which is related with the boundary parameters (see below (3.54)) and {h i (u),h(u)|i = 1, 2} are some functions given by Then the eigenvalues Λ(u) andΛ(u) can be written as the form of inhomogeneous T − Q relation Λ(u) = Z 1 (u) + Z 2 (u) + Z 3 (u) + Z 4 (u) + f 1 (u) + f 2 (u), (3.50) where the non-negative integers L 1 and L 2 satisfy L 1 = 2L 2 + N + 1.

Discussion
In this paper, we generalize the ODBA method to the integrable models related with the sp(4) Lie algebra. By using the fusion technique, we obtain the closed operator product identities of the fused transfer matrices. Based on them and the asymptotic behaviors as well as the special values, we obtain the exact solution of the system with the periodic and off-diagonal open boundary conditions. The method and the results in this paper can be generalized to the high rank C n (i.e., the sp(2n)) case directly.