Off-diagonal Bethe Ansatz on the $so(5)$ spin chain

The $so(5)$ (i.e., $B_2$) quantum integrable spin chains with both periodic and non-diagonal boundaries are studied via the off-diagonal Bethe Ansatz method. By using the fusion technique, sufficient operator product identities (comparing to those in [1]) to determine the spectrum of the transfer matrices are derived. For the periodic case, we recover the results obtained in [1], while for the non-diagonal boundary case, a new inhomogeneous $T-Q$ relation is constructed. The present method can be directly generalized to deal with the $so(2n+1)$ (i.e., $B_n$) quantum integrable spin chains with general boundaries.

The nested ODBA was initially proposed in studying the su(n) (i.e., A n ) spin chain with generic boundaries [30,31]. However, ODBA to approach high-rank quantum integrable models associated with B n , C n and D n Lie algebras is still missing. We note that such kind of models with obvious U(1)-symmetry has been studied extensively. For example, with some functional relations and algebraic Bethe Ansatz analysis (the analytic Bethe Ansatz method), Reshetikhin derived the energy spectrum of the periodic quantum spin chains associated with B n , C n , D n and other Lie algebras [1,32]. The algebraic Bethe Ansatz for those models with periodic boundary condition was constructed by Martins and Ramos [33], while the method for approaching such kind of models with diagonal open boundaries was developed by Li, Shi and Yue [34,35].
In this paper, we develop a nested ODBA method to approach the quantum integrable so(5) (i.e., B 2 ) spin chain with either periodic or non-diagonal open boundary condition.
This method can be generalized to so(2n + 1) (i.e., B n ) case directly. The paper is organized as follows. In section 2, we study the so(5) model with periodic boundary condition. Closed functional relations among the transfer matrices to determine the eigenvalues are constructed with fusion techniques. In section 3, we study the so(5) model with an off-diagonal open boundary condition. By constructing some operator product identities, we derive the exact eigenvalues of the transfer matrix in terms of an inhomogeneous T − Q relation. Section 4 is attributed to concluding remarks. Some detailed calculations are listed in Appendices A-C.

.1 The model
Let V denote a 5-dimensional linear space with an orthonormal basis {|i |i = 1, · · · , 5} which endows the fundamental representation of the so(5) (or B 2 ) algebra. The quantum spin chain associated with the B 2 algebra is described by a 25 × 25 R-matrix R vv 12 (u) defined in the V ⊗ V space with the matrix elements [33] R vv 12 (u) ij kl = u(u + 3 2 )δ ik δ jl + (u + 3 2 )δ il δ jk − uδ jī δ kl , (2.1) where {i, j, k, l} = {1, 2, 3, 4, 5}, i +ī = 6. We introduce the notation for simplicity where 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 . 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. The R-matrix satisfies the Yang-Baxter equation For the periodic boundary condition, we introduce the monodromy matrix where the index 0 indicates the auxiliary space and the other tensor space V ⊗N is the physical or quantum space, N is the number of sites and {θ j } are the inhomogeneous parameters.
The monodromy matrix satisfies the Yang-Baxter relation The transfer matrix is the trace of monodromy matrix in the auxiliary space From the Yang-Baxter relation, one can prove that the transfer matrices with different spectral parameters commute with each other, [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 Hamiltonian is given by

Spinorial R-matrix and the fused ones
In order to obtain closed operator product identities (see (2.43)-(2.49) below) which allow one to completely determine the eigenvalues of the transfer matrix t (p) (u), we need further an R-matrix associated with the spinorial representation of the so(5) algebra. Let us denote the spinorial representation by V (s) with an orthonormal basis {|i (s) |i = 1, · · · , 4}. The spinorial 16 × 16 R-matrix has the following non-zero matrix elements [36] R ss 12 (u) ii ii = a 2 (u) = (u + The spinorial R-matrix satisfies the properties regularity : R ss where P (s) 12 is the permutation operator among the spinorial representation space (c.f., P 12 in (2.3) ).
Following the fusion procedure [37][38][39][40][41][42][43], we can construct another R-matrix R sv where S is some non-degenerate constant matrix, and P ss(5) 12 is a 5-dimensional projector operator with the form P ss(5) 12 where the corresponding vectors are 3 Let V ( ss ) denote the projected subspace of V (s) ⊗V (s) by the projector P one has the identification: V ≡ V ( ss ) , which leads to an R-matrix R sv 12 (u) defined in V (s) ⊗V . The non-vanishing matrix elements of the resulting R-matrix are 4 R sv (u) 11 11 = R sv (u) 12 12 (2.14) It is easily to check that the fused R sv 12 matrix has the properties unitarity : R sv Moreover, the fused R sv 1 2 also satisfy the Yang-Baxter equations We have checked that the R-matrices R sv 12 (u) and R vv 12 (u) enjoy the properties: Here we have usedV (resp.Ṽ ) to denote the projected subspace in V ⊗ V by P 21 (resp. the projected subspace in V ⊗ V ⊗ V by P 321 ) and adopted the convention:1 ≡ 12 and 1 ≡ 123 .
Some remarks are in order. It is shown that each matrix elements of the above fused R-matrices, as a function of u, is a polynomial with degree up to two. Due to the fact that

Operator product identities
Besides the transfer matrix t (p) 1 (u) given by (2.6), let us introduce 3 fused transfer matrices: whereT v 12···m (u) are the fused monodromy matrices where the spinoral R-matrix R sv 12 (u) is given by (2.14). It is easily to show that all the transfer matrices constitute a commutative family, namely, We have used the convention:t where we have introduced some normalized monodromy matrices: It is remarked that the quantum spaces of the above monodromy matrices are the same (i.e., V ⊗N ) and that the corresponding auxiliary spaces areV andṼ with dimensions 11 and 15.
Then the associated transfer matrices are given by The equivalence (2.29) and the relations (2.35)-(2.37) imply that Fowllowing the method developed in [30], we obtain the identities and (A.13), we have that the transfer matrices satisfy the operator product identities: Now, we consider the asymptotic behaviors of the fused transfer matrices. Direct calculation shows Let us denote the eigenvalues of the transfer matrices t (p) (u), t s (u), respectively. From the operator product identities (2.43)-(2.49), we have the functional relations among the eigenvalues 5 : 5 It is remarked that only (2.51) and (2.57) were used to obtain Λ (p) (u) and Λ (p) s (u) for the closed B n chain [1,32]. Since that Λ (p) (u) (resp. Λ (p) s (u)) is a polynomial of u with degree 2N (resp. degree N ), one needs 3N + 2 conditions to determine them completely while (2.51) and (2.57), together with the asymptotic behaviors, only give 2N + 2 conditions. Therefore, in order to close the functional relations, (2.51)-(2.57) are necessary.
The asymptotic behaviors (2.50) of the fused transfer matrices lead to the corresponding asymptotic behaviors of their eigenvalues:

.1 Open chain
Integrable open chain can be constructed as follows [6,7]. Let us introduce a pair of Kmatrices K v− (u) and K v+ (u). The former satisfies the reflection equation (RE) and the latter satisfies the dual RE For open spin chains, instead of the "row-to-row" monodromy matrix T v 0 (u) (2.4), one needs to consider the "double-row" monodromy matrix as follows. Let us introduce another "rowto-row" monodromy matrix which satisfies the Yang-Baxter relation The transfer matrix t(u) is defined as In this paper, we consider an open chain associated with the off-diagonal K-matrix where the non-vanishing matrix elements are Here c 1 and c 2 are arbitrary boundary parameters. The dual reflection matrix K v+ (u) is also an off-diagonal one and given by

Operator product relations
We define the dual fused monodromy matrices aŝ where the R-matrices R vs 21 (u) is defined by (2.15) and the others are defined by the relations We have checked that the R-matrices also enjoy the properties (3.14) which are used to derive the functional relations (3.17) below.

Inhomogeneous T − Q relations
Define and where x is a parameter which will be determined later (see (3.46) All the eigenvalues are polynomials of u, the residues of right hand sides of Eq.(3.42) should be zero, which gives rise to the BAEs , k = 1, 2, · · · , L 1 , (λ (2) l − θ j )(λ (2) l + θ j ), l = 1, 2, · · · , L 2 . (3.45) Considering the asymptotic behaviors of Λ(u), Λ 2 (u), Λ 3 (u) and Λ s (u), we obtain a constraint L 2 = 2L 1 + N + 1 and the value of parameter x in the functions {f m (u)} given in (3.41) as Due to the choice of reflection matrices (3.7) and (3.9), one conserved charge is survived, which means that the number of Bethe roots L 1 or L 2 could be any non-negative integer but the constraint L 2 = 2L 1 + N + 1 must be hold. We have checked that the BAEs (3.44)-

Discussion
In this paper, we study the so(5) quantum spin chains with integrable boundary conditions. By using the fusion technique, we obtain the closed operator product identities of the The method and the results in this paper can be generalized to the so(2n + 1) (i.e., B n ) case directly.

Acknowledgments
The financial supports from the National Program for Basic Research of MOST (Grant Nos. 2016YFA0300600 and 2016YFA0302104), the National Natural Science Foundation of China (Grant Nos. 11434013, 11425522, 11547045, 11774397, 11775178, 11775177 and 91536115 and after some calculations, we arrive at (2.29), namely, . (A.13)

A.2: Projectors
Here we list the projectors used in this paper. P