1. Introduction
The exactly solvable model has many applications in condensed matter physics, theoretical and mathematical physics [1, 2]. The typical quantization conditions of the energy spectrum is the periodic boundary conditions [3, 4]. In 1988, Sklyanin proposed the reflection matrix and reflection equations to describe the integrable open boundary conditions [5]. From then on, the exactly solvable models with boundary reflection have attracted a lot of interest.
The integrable quantization conditions are very important. Many interesting phenomena such as the boundary bound states and novel elementary excitations are found in the integrable models with open boundary conditions. For this purpose, the first step during the study is to obtain the solution of the reflection equation. From the reflection matrix, one can construct the corresponding exactly solvable models. Meanwhile, if the reflection matrix has the off-diagonal elements, the U(1)-symmetry of the system is broken and the number of particles with an intrinsic degree of freedom are not conserved, which induces many interesting phenomena such as the spin-flipped effect and helical elementary excitations. Motivated by these aspects, many works have been done on the solving the reflection equation [6–14].
The Bn vertex model can be related to the $O(2n+1)$ quantum spin chain and have many applications in the $O(2n+1)$ -sigma models. For an example, the O(3) vertex model is equivalent to the isotropic spin-1 XXX model or the O(3)-sigma model, which has been well studied. For the higher rank case, the model with period and diagonal boundary have been solved by using the analytical or algebraic Bethe ansatz method. However, the integrable Bn open chain associated with off-diagonal reflection matrices has more integrable boundary interactions terms and the corresponding eigen-states involve more structures. Therefore, it is interesting to investigate the most generic (or off-diagonal) reflection matrices of the Bn models.
In this paper, we study the integrable quantization conditions of the B2 model by using the method of fusion. Starting from the fundamental spinorial representation, we obtain the general solutions of reflection equations. We find that the reflection matrix has 7 free boundary parameters, which are used to describe the degree of freedom of boundary couplings, without breaking the integrability of the system. Based on them, we give the model Hamiltonian of integrable B2 model with off-diagonal boundary reflections. The new quantization conditions will induce the novel structure of the energy spectrum.
The advantage of method suggested in this paper is as follows. The symmetry of the system can not be broken by the fusion. Thus, the number of free boundary parameters remain unchanged during the fusion. The number can be calculated easily from the fundamental representation because the dimension of the fundamental one is lower. All the algebras of the fused representations are the same. Second, all the parameterizations are uniform by using the fusion. Third, the fusion can also be used to solve the energy spectrum. For example, the fusion can supply the closed operator production identities of the fused transfer matrices, which are necessary to construct the T − Q relations. Forth, the results can be generalized to the Bn case easily.
This paper is organized as follows. In section 2 , we introduce the spinorial R-matrix of the B2 model. The general solution of the spinorial reflection equation is also obtained. In section 3 , we generate the vectorial R-matrix of the B2 model by using the fusion. In section 4 , we calculate the general solution of the vectorial reflection matrix. In section 5 , we discuss the degenerate cases. In section 6 , we construct the integrable model Hamiltonian based on all the above ingredients. Concluding remarks are given in section 7 .
2. Spinorial representation
The fundamental R-matrix of the B2 model is the spinorial one defined in the space ${V}_{1^{\prime} }\otimes {V}_{2^{\prime} }$ with the form [15] $ \begin{eqnarray}\begin{array}{rcl} & & {R}_{1^{\prime} 2^{\prime} }^{{ss}}{\left(u\right)}_{{ii}}^{{ii}}={a}_{2}(u)=\left(u+\displaystyle \frac{1}{2}\right)\left(u+\displaystyle \frac{3}{2}\right),\\ & & {R}_{1^{\prime} 2^{\prime} }^{{ss}}{\left(u\right)}_{{ij}}^{{ij}}={b}_{2}(u)=u\left(u+\displaystyle \frac{3}{2}\right),\quad i\ne j,\bar{j},\\ & & {R}_{1^{\prime} 2^{\prime} }^{{ss}}{\left(u\right)}_{\bar{i}i}^{i\bar{i}}={c}_{2}(u)=u+\displaystyle \frac{3}{4},\\ & & {\xi }_{i}{\xi }_{\bar{j}}{R}_{1^{\prime} 2^{\prime} }^{{ss}}{\left(u\right)}_{j\bar{j}}^{i\bar{i}}={d}_{2}(u)=-\displaystyle \frac{u}{2},\quad i\ne j,\bar{j},\\ & & {R}_{1^{\prime} 2^{\prime} }^{{ss}}{\left(u\right)}_{i\bar{i}}^{i\bar{i}}={e}_{2}(u)=u(u+1),\\ & & {R}_{1^{\prime} 2^{\prime} }^{{ss}}{\left(u\right)}_{{ji}}^{{ij}}={g}_{2}(u)=\displaystyle \frac{u}{2}+\displaystyle \frac{3}{4},\quad i\ne j,\bar{j},\end{array}\end{eqnarray}$ where $\{i,j\}=\{1,2,3,4\},i+\bar{i}=5,{\xi }_{i}=1$ and ξi = −1 if i ∈ {3, 4}. The ${R}_{1^{\prime} 2^{\prime} }^{{ss}}$ matrix has the following properties
$ \begin{eqnarray*}\begin{array}{rcl}{regularity} & : & {R}_{1^{\prime} 2^{\prime} }^{{ss}}(0)={\rho }_{1}{\left(0\right)}^{\tfrac{1}{2}}{{ \mathcal P }}_{1^{\prime} 2^{\prime} },\\ {unitarity} & : & {R}_{1^{\prime} 2^{\prime} }^{{ss}}(u){R}_{2^{\prime} 1^{\prime} }^{{ss}}(-u)={\rho }_{1}(u),\\ {crossing}\ {unitarity} & : & {R}_{1^{\prime} 2^{\prime} }^{{ss}}{\left(u\right)}^{{t}_{1}}{R}_{2^{\prime} 1^{\prime} }^{{ss}}{\left(-u-3\right)}^{{t}_{1}}={\rho }_{1}\left(u+\displaystyle \frac{3}{2}\right),\end{array}\end{eqnarray*}$
where${\rho }_{1}(u)=\left({u}^{2}-\tfrac{1}{4}\right)\left({u}^{2}-\tfrac{9}{4}\right),{{ \mathcal P }}_{1^{\prime} 2^{\prime} }$ is the exchange operator with the definition $[{{ \mathcal P }}_{1^{\prime} 2^{\prime} }{]}_{{kl}}^{{ij}}={\delta }_{{il}}{\delta }_{{jk}},\{i,j,k,l\}\,=\{1,2,3,4\}$ and ti denotes the transposition in the space Vi. The spinorial R-matrix satisfies the Yang-Baxter equation $ \begin{eqnarray}\begin{array}{l}{R}_{1^{\prime} 2^{\prime} }^{{ss}}(u-v){R}_{1^{\prime} 3^{\prime} }^{{ss}}(u){R}_{2^{\prime} 3^{\prime} }^{{ss}}(v)\\ \quad =\,{R}_{2^{\prime} 3^{\prime} }^{{ss}}(v){R}_{1^{\prime} 3^{\prime} }^{{ss}}(u){R}_{1^{\prime} 2^{\prime} }^{{ss}}(u-v).\end{array}\end{eqnarray}$
where
In order to describe the boundary reflection, we should introduce the reflection matrix ${K}^{s-}(u)$ defined in the auxiliary space. The spinorial R-matrix (1 ) and the reflection equation ${K}^{s-}(u)$ satisfy the reflection equation $ \begin{eqnarray}\begin{array}{l}{R}_{1^{\prime} 2^{\prime} }^{{ss}}(u-v){K}_{1^{\prime} }^{s-}(u){R}_{2^{\prime} 1^{\prime} }^{{ss}}(u+v){K}_{2^{\prime} }^{s-}(v)\\ \quad =\,{K}_{2^{\prime} }^{s-}(v){R}_{1^{\prime} 2^{\prime} }^{{ss}}(u+v){K}_{1^{\prime} }^{s-}(u){R}_{2^{\prime} 1^{\prime} }^{{ss}}(u-v),\end{array}\end{eqnarray}$ where ${K}_{1^{\prime} }^{s-}(u)={K}^{s-}(u)\otimes {id},{K}_{2^{\prime} }^{s-}(u)={id}\otimes {K}^{s-}(u)$ and id means the unitary matrix. we get a Ks-matrix: $ \begin{eqnarray}{K}^{s-}(u)=\left(\begin{array}{cccc}1-\zeta u & {c}_{1}u & {c}_{2}u & {c}_{3}u\\ {c}_{4}u & 1-\eta u & {c}_{7}u & -{c}_{2}u\\ {c}_{5}u & {c}_{8}u & 1+\eta u & {c}_{1}u\\ {c}_{6}u & -{c}_{5}u & {c}_{4}u & 1+\zeta u\end{array}\right),\end{eqnarray}$ where η, ζ and ci, (i = 1, 2,⋯, 8) are the boundary parameters with the constrains $ \begin{eqnarray}\begin{array}{rcl} & & {c}_{2}{c}_{8}={c}_{3}{c}_{5}+{c}_{1}(\zeta +\eta ),\\ & & {c}_{7}{c}_{1}=-{c}_{3}{c}_{4}+{c}_{2}(\zeta -\eta ),\\ & & {c}_{2}{c}_{6}={c}_{5}{c}_{7}-{c}_{4}(\zeta +\eta ).\end{array}\end{eqnarray}$
The dual reflection matrix ${K}^{s+}$ is constructed by the reflection matrix ${K}^{s-}$ as $ \begin{eqnarray}{K}^{s+}(u)={K}^{s-}\left(-u-\displaystyle \frac{3}{2}\right){| }_{\zeta ,\eta ,{c}_{i}\to \tilde{\zeta },\tilde{\eta },{\tilde{c}}_{i}},\end{eqnarray}$ where $\tilde{\eta },\tilde{\zeta }$ and ${\tilde{c}}_{i},(i=1,2,\cdots ,\,8)$ are the boundary parameters characterizing the boundary couplings at the other side. They are not independent and satisfy the constrains $ \begin{eqnarray}\begin{array}{rcl} & & {\tilde{c}}_{2}{\tilde{c}}_{8}={\tilde{c}}_{3}{\tilde{c}}_{5}+{\tilde{c}}_{1}(\tilde{\zeta }+\tilde{\eta }),\\ & & {\tilde{c}}_{7}{\tilde{c}}_{1}=-{\tilde{c}}_{3}{\tilde{c}}_{4}+{\tilde{c}}_{2}(\tilde{\zeta }-\tilde{\eta }),\\ & & {\tilde{c}}_{2}{\tilde{c}}_{6}={\tilde{c}}_{5}{\tilde{c}}_{7}-{\tilde{c}}_{4}(\tilde{\zeta }+\tilde{\eta }).\end{array}\end{eqnarray}$ The dual reflection matrix satisfies the dual reflection equation $ \begin{eqnarray}\begin{array}{rcl} & & {R}_{1^{\prime} 2^{\prime} }^{{ss}}(-u+v){K}_{1^{\prime} }^{s+}(u){R}_{2^{\prime} 1^{\prime} }^{{ss}}(-u-v-3){K}_{2^{\prime} }^{s+}(v)\\ & & ={K}_{2^{\prime} }^{s+}(v){R}_{1^{\prime} 2^{\prime} }^{{ss}}(-u-v-3){K}_{1^{\prime} }^{s+}(u){R}_{2^{\prime} 1^{\prime} }^{{ss}}(-u+v),\end{array}\end{eqnarray}$ where ${K}_{1^{\prime} }^{s+}(u)={K}^{s+}(u)\otimes {id}$ and ${K}_{2^{\prime} }^{s+}(u)={id}\otimes {K}^{s+}(u)$ .
Next, we consider the degenerate cases of the general solutions (4 ) and (6 ). If the boundary parameters $ \begin{eqnarray}\begin{array}{rcl}{c}_{1} & = & {c}_{2}\,=\,\cdots \,=\,{c}_{8}=0,\quad \zeta \,=\,\eta ,\\ {\tilde{c}}_{1} & = & {\tilde{c}}_{2}\,=\,\cdots \,=\,{\tilde{c}}_{8}\,=\,0,\quad \tilde{\zeta }=\tilde{\eta },\end{array}\end{eqnarray}$ the reflection matrices (A.1 ) and (6 ) degenerate into the diagonal ones $ \begin{eqnarray}\begin{array}{rcl}{\bar{K}}^{s-}(u) & = & {diag}\left[\zeta ^{\prime} -u,\zeta ^{\prime} -u,\zeta ^{\prime} +u,\zeta ^{\prime} +u\right],\\ {\bar{K}}^{s+}(u) & = & {diag}\left[\tilde{\zeta }^{\prime} +u+\displaystyle \frac{3}{2},\tilde{\zeta }^{\prime} +u+\displaystyle \frac{3}{2},\tilde{\zeta }^{\prime} -u\right.\\ & & \left.-\displaystyle \frac{3}{2},\tilde{\zeta }^{\prime} -u-\displaystyle \frac{3}{2}\right],\end{array}\end{eqnarray}$ where $\tilde{\zeta }^{\prime} ={\tilde{\zeta }}^{-1}$ and $\tilde{\zeta }^{\prime} ={\tilde{\zeta }}^{-1}$ .
The second degenerate case is that the reflection matrices (4 ) and (6 ) have the off-diagonal constant solution. Rewrite reflection matrices (4 ) as $ \begin{eqnarray}{K}^{s-}(u)=\displaystyle \frac{1}{u}{\tilde{K}}^{s-}(u).\end{eqnarray}$ Taking the limit of $u\to \infty $ and neglecting the common factor of the reflection matrix, we obtain a constant reflection matrix $ \begin{eqnarray}{\tilde{K}}^{s-}=\left(\begin{array}{cccc}-\hat{\zeta } & {\hat{c}}_{1} & {\hat{c}}_{2} & {\hat{c}}_{3}\\ {\hat{c}}_{4} & -\hat{\eta } & {\hat{c}}_{7} & -{\hat{c}}_{2}\\ {\hat{c}}_{5} & {\hat{c}}_{8} & \hat{\eta } & {\hat{c}}_{1}\\ {\hat{c}}_{6} & -{\hat{c}}_{5} & {\hat{c}}_{4} & \hat{\zeta }\end{array}\right),\end{eqnarray}$ where $\hat{\eta },\hat{\zeta }$ and ${\hat{c}}_{i},(i=1,2,\cdots ,\,8)$ are the new parameterization of boundary parameters with constrains $ \begin{eqnarray}\begin{array}{rcl} & & {\hat{c}}_{2}{\hat{c}}_{8}={\hat{c}}_{3}{\hat{c}}_{5}+{\hat{c}}_{1}(\hat{\zeta }+\hat{\eta }),\\ & & {\hat{c}}_{7}{\hat{c}}_{1}=-{\hat{c}}_{3}{\hat{c}}_{4}+{\hat{c}}_{2}(\hat{\zeta }-\hat{\eta }),\\ & & {\hat{c}}_{2}{\hat{c}}_{6}={\hat{c}}_{5}{\hat{c}}_{7}-{\hat{c}}_{4}(\hat{\zeta }+\hat{\eta }).\end{array}\end{eqnarray}$ The dual transformation between ${\tilde{K}}^{s+}$ and ${\tilde{K}}^{s-}$ remains unchanged, $ \begin{eqnarray}{\tilde{K}}^{s+}={\tilde{K}}^{s-}{| }_{\hat{\zeta },\hat{\eta },{\hat{c}}_{i}\to \tilde{\hat{\zeta }},\tilde{\hat{\eta }},{\tilde{\hat{c}}}_{i}}.\end{eqnarray}$ We note that the number of free parameter in (12 ) and (14 ) is 7.
3. Fusion
In this section, we consider the fusion [16–23], which can be carried out because the spinorial R-matrix (1 ) degenerates into the projector operator at the point of $u=-\tfrac{1}{2}$ , $ \begin{eqnarray}{R}_{1^{\prime} 2^{\prime} }^{{ss}}\left(-\displaystyle \frac{1}{2}\right)={P}_{1^{\prime} 2^{\prime} }^{{ss}(5)}\times S,\end{eqnarray}$ where S is a constant matrix (we omit its expression because we do not need it), and ${P}_{1^{\prime} 2^{\prime} }^{{ss}(5)}$ is the 5-dimensional projector $ \begin{eqnarray}{P}_{1^{\prime} 2^{\prime} }^{{ss}(5)}=\displaystyle \sum _{i=1}^{5}| {\tilde{\psi }}_{i}\rangle \langle {\tilde{\psi }}_{i}| ,\end{eqnarray}$ with
$ \begin{eqnarray*}\begin{array}{rcl} & & | {\tilde{\psi }}_{1}\rangle =\displaystyle \frac{1}{\sqrt{2}}(| 12\rangle -| 21\rangle ),\quad | {\tilde{\psi }}_{2}\rangle =\displaystyle \frac{1}{\sqrt{2}}(| 31\rangle -| 13\rangle ),\\ & & | {\tilde{\psi }}_{3}\rangle =\displaystyle \frac{1}{2}(| 14\rangle -| 41\rangle +| 23\rangle -| 32\rangle ),\\ & & | {\tilde{\psi }}_{4}\rangle =\displaystyle \frac{1}{\sqrt{2}}(| 24\rangle -| 42\rangle ),\quad | {\tilde{\psi }}_{5}\rangle =\displaystyle \frac{1}{\sqrt{2}}(| 34\rangle -| 43\rangle ).\end{array}\end{eqnarray*}$
Taking fusion in the quantum space, we have $ \begin{eqnarray}\begin{array}{l}{P}_{2^{\prime} 3^{\prime} }^{{ss}(5)}{R}_{1^{\prime} 2^{\prime} }^{{ss}}\left(u+\displaystyle \frac{1}{2}\right){R}_{1^{\prime} 3^{\prime} }^{{ss}}(u){P}_{2^{\prime} 3^{\prime} }^{{ss}(5)}\\ \quad =\,u(u+1)(u+2){R}_{1^{\prime} \langle 2^{\prime} 3^{\prime} \rangle }^{{sv}}\left(u+\displaystyle \frac{1}{4}\right).\end{array}\end{eqnarray}$ Denote $\langle 1^{\prime} 2^{\prime} \rangle \equiv 1$ and the dimension of fused space V1 is 5. The fused matrix ${R}_{1^{\prime} 2}^{{sv}}$ has the following properties $ \begin{eqnarray}\begin{array}{rcl} & & {R}_{1^{\prime} 2}^{{sv}}(u){R}_{21^{\prime} }^{{vs}}(-u)={\rho }_{2}(u),\\ & & {R}_{1^{\prime} 2}^{{sv}}{\left(u\right)}^{{t}_{1}}{R}_{21^{\prime} }^{{vs}}{\left(-u-3\right)}^{{t}_{1}}={\rho }_{2}\left(u+\displaystyle \frac{3}{2}\right),\\ & & {R}_{1^{\prime} 2^{\prime} }^{{ss}}(u-v){R}_{1^{\prime} 3}^{{sv}}(u){R}_{2^{\prime} 3}^{{sv}}(v)={R}_{2^{\prime} 3}^{{sv}}(v){R}_{1^{\prime} 3}^{{sv}}(u){R}_{1^{\prime} 2^{\prime} }^{{ss}}(u-v),\end{array}\end{eqnarray}$ where ${\rho }_{2}(u)=\tfrac{25}{16}-{u}^{2}$ .
Next, taking the fusion in the auxiliary space, we obtain the vectorial R-matrix of B2 model $ \begin{eqnarray}{P}_{1^{\prime} 2^{\prime} }^{{ss}(5)}{R}_{2^{\prime} 3}^{{sv}}(u){R}_{1^{\prime} 3}^{{sv}}\left(u-\displaystyle \frac{1}{2}\right){P}_{1^{\prime} 2^{\prime} }^{{ss}(5)}={R}_{\langle 1^{\prime} 2^{\prime} \rangle 3}^{{vv}}\left(u-\displaystyle \frac{1}{4}\right),\end{eqnarray}$ which is defined in the 5-dimensional auxiliary space and 5-dimensional quantum space. Detailed calculation shows that the elements of the vectorial R-matrix read [24] $ \begin{eqnarray}{R}_{12}^{{vv}}{\left(u\right)}_{{kl}}^{{ij}}=u\left(u+\displaystyle \frac{3}{2}\right){\delta }_{{ik}}{\delta }_{{jl}}+\left(u+\displaystyle \frac{3}{2}\right){\delta }_{{il}}{\delta }_{{jk}}-u{\delta }_{j\bar{i}}{\delta }_{k\bar{l}},\end{eqnarray}$
where {i, j, k, l} = {1, 2, 3, 4, 5} and$i+\bar{i}=6$ . The vectorial R-matrix has the following properties
$ \begin{eqnarray*}\begin{array}{rcl}{regularity} & : & {R}_{12}^{{vv}}(0)={\rho }_{3}{\left(0\right)}^{\tfrac{1}{2}}{{ \mathcal P }}_{12},\\ {unitarity} & : & {R}_{12}^{{vv}}(u){R}_{21}^{{vv}}(-u)={\rho }_{3}(u),\\ {crossing}\ {unitarity} & : & {R}_{12}^{{vv}}{\left(u\right)}^{{t}_{1}}{R}_{21}^{{vv}}{\left(-u-3\right)}^{{t}_{1}}={\rho }_{3}\left(u+\displaystyle \frac{3}{2}\right),\end{array}\end{eqnarray*}$
where${\rho }_{3}(u)=({u}^{2}-1)\left({u}^{2}-\tfrac{9}{4}\right)$ . The Rvv12 matrix satisfies the Yang-Baxter equation $ \begin{eqnarray}\begin{array}{rcl} & & {R}_{1^{\prime} 2}^{{sv}}(u-v){R}_{1^{\prime} 3}^{{sv}}(u){R}_{23}^{{vv}}(v)={R}_{23}^{{vv}}(v){R}_{1^{\prime} 3}^{{sv}}(u){R}_{1^{\prime} 2}^{{sv}}(u-v),\\ & & {R}_{12}^{{vv}}(u-v){R}_{13}^{{vv}}(u){R}_{23}^{{vv}}(v)={R}_{23}^{{vv}}(v){R}_{13}^{{vv}}(u){R}_{12}^{{vv}}(u-v).\end{array}\end{eqnarray}$
where {i, j, k, l} = {1, 2, 3, 4, 5} and
where
4. General solution of vectorial reflection matrix
Now, it is clear that the vectorial R-matrix (20 ) of B2 model can be obtained by the spinorial one (1 ) by using the fusion. The integrable model Hamiltonian is constructed by the vectorial representation. For the open boundary case, we should introduce the vectorial reflection matrix ${K}^{v-}$ , which satisfies the reflection equations $ \begin{eqnarray}\begin{array}{l}{R}_{1^{\prime} 2}^{{sv}}(u-v){K}_{1^{\prime} }^{s-}(u){R}_{21^{\prime} }^{{vs}}(u+v){K}_{2}^{v-}(v)\\ \ =\,{K}_{2}^{v-}(v){R}_{1^{\prime} 2}^{{sv}}(u+v){K}_{1^{\prime} }^{s-}(u){R}_{21^{\prime} }^{{vs}}(u-v),\\ {R}_{12}^{{vv}}(u-v){K}_{1}^{v-}(u){R}_{21}^{{vv}}(u+v){K}_{2}^{v-}(v)\\ \ =\,{K}_{2}^{v-}(v){R}_{12}^{{vv}}(u+v){K}_{1}^{v-}(u){R}_{21}^{{vv}}(u-v).\end{array}\end{eqnarray}$
The general solution of the reflection matrix can be obtained by solving the above reflection equations directly. In this paper, we use the fusion technique to obtain the solution. We first note that the dimension of the auxiliary space of the vectorial R-matrix is 5, thus the corresponding reflection matrix defined in the auxiliary space should be 5 × 5. That is to say, we need to fuse the 4 × 4 spinorial reflection matrices ${K}^{s\pm }$ into the 5 × 5 vectorial reflection matrices ${K}^{v\pm }$ .
The vectorial reflection matrix ${K}^{v-}$ can be obtained by taking the fusion in the auxiliary space, $ \begin{eqnarray}\begin{array}{l}{P}_{1^{\prime} 2^{\prime} }^{5}{K}_{2^{\prime} }^{s-}\left(u+\displaystyle \frac{1}{4}\right){R}_{1^{\prime} 2^{\prime} }^{{ss}}(2u){K}_{1^{\prime} }^{s-}\left(u-\displaystyle \frac{1}{4}\right){P}_{1^{\prime} 2^{\prime} }^{5}\\ \quad =\,\left(u-\displaystyle \frac{1}{4}\right)\left(u+\displaystyle \frac{3}{4}\right){K}_{\langle 1^{\prime} 2^{\prime} \rangle }^{v-}(u).\end{array}\end{eqnarray}$ Substituting the general solution of spinorial reflection matrices ${K}^{s-}$ with the parameterization (A.20 ) into (23 ) and after some calculations, we obtain the explicit matrix form of ${K}^{v-}$ as $ \begin{eqnarray}{K}^{v-}(u)=\left(\begin{array}{ccccc}{K}_{11}^{v-}(u) & {K}_{12}^{v-}(u) & {K}_{13}^{v-}(u) & {K}_{14}^{v-}(u) & {K}_{15}^{v-}(u)\\ {K}_{21}^{v-}(u) & {K}_{22}^{v-}(u) & {K}_{23}^{v-}(u) & {K}_{24}^{v-}(u) & {K}_{25}^{v-}(u)\\ {K}_{31}^{v-}(u) & {K}_{32}^{v-}(u) & {K}_{33}^{v-}(u) & {K}_{34}^{v-}(u) & {K}_{35}^{v-}(u)\\ {K}_{41}^{v-}(u) & {K}_{42}^{v-}(u) & {K}_{43}^{v-}(u) & {K}_{44}^{v-}(u) & {K}_{45}^{v-}(u)\\ {K}_{51}^{v-}(u) & {K}_{52}^{v-}(u) & {K}_{53}^{v-}(u) & {K}_{54}^{v-}(u) & {K}_{55}^{v-}(u)\end{array}\right),\end{eqnarray}$ where $ \begin{eqnarray}\begin{array}{rcl}{K}_{11}^{v-}(u) & = & -\displaystyle \frac{1}{8}\{32(\eta u+\zeta u-1)+(4u+1)\\ & & \times [{c}_{3}{c}_{6}+{c}_{7}{c}_{8}+{\eta }^{2}+{\zeta }^{2}\\ & & +2{c}_{2}{c}_{5}-8\eta \zeta u+2{c}_{1}{c}_{4}(4u+1)]\},\\ {K}_{12}^{v-}(u) & = & u[({c}_{2}{c}_{4}+{c}_{7}\zeta )(4u+1)-4{c}_{7}],\\ {K}_{13}^{v-}(u) & = & \displaystyle \frac{u}{\sqrt{2}}[({c}_{1}{c}_{7}-{c}_{3}{c}_{4}+{c}_{2}\zeta +{c}_{2}\eta )(4u+1)-8{c}_{2}],\\ {K}_{14}^{v-}(u) & = & u[({c}_{3}\eta -{c}_{1}{c}_{2})(4u+1)-4{c}_{3}],\\ {K}_{15}^{v-}(u) & = & -({c}_{2}^{2}+{c}_{3}{c}_{7})u(4u+1),\\ {K}_{21}^{v-}(u) & = & u[({c}_{1}{c}_{5}+{c}_{8}\zeta )(4u+1)-4{c}_{8}],\\ {K}_{22}^{v-}(u) & = & -\displaystyle \frac{1}{8}\{32(\zeta u-\eta u-1)+(4u+1)\\ & & \times [{c}_{3}{c}_{6}+{c}_{7}{c}_{8}+{\eta }^{2}+{\zeta }^{2}\\ & & +2{c}_{1}{c}_{4}+8\eta \zeta u+2{c}_{2}{c}_{5}(4u+1)]\},\\ {K}_{23}^{v-}(u) & = & \displaystyle \frac{u}{\sqrt{2}}[({c}_{3}{c}_{5}+{c}_{2}{c}_{8}+{c}_{1}\zeta -{c}_{1}\eta )(4u+1)-8{c}_{1}],\\ {K}_{24}^{v-}(u) & = & ({c}_{3}{c}_{8}-{c}_{1}^{2})u(4u+1),\\ {K}_{25}^{v-}(u) & = & u[({c}_{3}\eta -{c}_{1}{c}_{2})(4u+1)+4{c}_{3}],\\ {K}_{31}^{v-}(u) & = & \displaystyle \frac{u}{\sqrt{2}}[({c}_{4}{c}_{8}-{c}_{1}{c}_{6}+{c}_{5}\zeta +{c}_{5}\eta )(4u+1)-8{c}_{5}],\\ {K}_{32}^{v-}(u) & = & \displaystyle \frac{u}{\sqrt{2}}[({c}_{2}{c}_{6}+{c}_{5}{c}_{7}+{c}_{4}\zeta -{c}_{4}\eta )(4u+1)-8{c}_{4}],\\ {K}_{33}^{v-}(u) & = & -\displaystyle \frac{1}{8}[({c}_{7}{c}_{8}+{c}_{3}{c}_{6}+{\zeta }^{2}+{\eta }^{2}){\left(4u+1\right)}^{2}\\ & & -2({c}_{1}{c}_{4}+{c}_{2}{c}_{5})(16{u}^{2}-1)-32],\\ {K}_{34}^{v-}(u) & = & \displaystyle \frac{u}{\sqrt{2}}[({c}_{3}{c}_{5}+{c}_{2}{c}_{8}+{c}_{1}\zeta -{c}_{1}\eta )(4u+1)+8{c}_{1}],\\ {K}_{35}^{v-}(u) & = & \displaystyle \frac{u}{\sqrt{2}}[({c}_{1}{c}_{7}-{c}_{3}{c}_{4}+{c}_{2}\zeta +{c}_{2}\eta )(4u+1)+8{c}_{1}],\\ {K}_{41}^{v-}(u) & = & u[({c}_{6}\eta -{c}_{4}{c}_{5})(4u+1)-4{c}_{6}],\\ {K}_{42}^{v-}(u) & = & ({c}_{6}{c}_{7}-{c}_{4}^{2})u(4u+1),\\ {K}_{43}^{v-}(u) & = & \displaystyle \frac{u}{\sqrt{2}}[({c}_{2}{c}_{6}+{c}_{5}{c}_{7}+{c}_{4}\zeta -{c}_{4}\eta )(4u+1)+8{c}_{4}],\\ {K}_{44}^{v-}(u) & = & -\displaystyle \frac{1}{8}\{32(\eta u-\zeta u-1)+(4u+1)\\ & & \times [{c}_{3}{c}_{6}+{c}_{7}{c}_{8}+{\eta }^{2}+{\zeta }^{2}\\ & & +2{c}_{1}{c}_{4}+8\eta \zeta u+2{c}_{2}{c}_{5}(4u+1)]\},\\ {K}_{45}^{v-}(u) & = & u[({c}_{2}{c}_{4}+{c}_{7}\zeta )(4u+1)+4{c}_{7}],\\ {K}_{51}^{v-}(u) & = & -({c}_{6}{c}_{8}+{c}_{5}^{2})u(4u+1),\\ {K}_{52}^{v-}(u) & = & u[({c}_{6}\eta -{c}_{4}{c}_{5})(4u+1)+4{c}_{6}],\\ {K}_{53}^{v-}(u) & = & \displaystyle \frac{u}{\sqrt{2}}[({c}_{4}{c}_{8}-{c}_{1}{c}_{6}+{c}_{5}\zeta +{c}_{5}\eta )(4u+1)+8{c}_{5}],\\ {K}_{54}^{v-}u) & = & u[({c}_{1}{c}_{5}+{c}_{8}\zeta )(4u+1)+4{c}_{8}],\\ {K}_{55}^{v-}(u) & = & -\displaystyle \frac{1}{8}\{-32(\eta u+\zeta u+1)\\ & & +(4u+1)[{c}_{3}{c}_{6}+{c}_{7}{c}_{8}+{\eta }^{2}+{\zeta }^{2}\\ & & +2{c}_{2}{c}_{5}-8\eta \zeta u+2{c}_{1}{c}_{4}(4u+1)]\}.\end{array}\end{eqnarray}$
The corresponding dual reflection matrix is $ \begin{eqnarray}{K}^{v+}(u)={K}^{v-}\left(-u-\displaystyle \frac{3}{2}\right){| }_{\zeta ,\eta ,{c}_{i}\to \tilde{\zeta },\tilde{\eta },{\tilde{c}}_{i}},\end{eqnarray}$ which satisfies the dual reflection equations $ \begin{eqnarray}\begin{array}{l}{R}_{1^{\prime} 2}^{{sv}}(-u+v){K}_{1^{\prime} }^{s+}(u){R}_{21^{\prime} }^{{vs}}(-u-v-3){K}_{2}^{v+}(v)\\ \quad =\,{K}_{2}^{v+}(v){R}_{1^{\prime} 2}^{{sv}}(-u-v-3){K}_{1^{\prime} }^{s+}(u){R}_{21^{\prime} }^{{vs}}(-u+v),\\ {R}_{12}^{{vv}}(-u+v){K}_{1}^{v+}(u){R}_{21}^{{vv}}(-u-v-3){K}_{2}^{v+}(v)\\ \quad =\,{K}_{2}^{v+}(v){R}_{12}^{{vv}}(-u-v-3){K}_{1}^{v+}(u){R}_{21}^{{vv}}(-u+v).\end{array}\end{eqnarray}$
5. Degenerate cases
Now, we consider some special cases of the general solutions (24 ) and (26 ). If the boundary parameters satisfy $ \begin{eqnarray}\begin{array}{rcl}{c}_{1} & = & {c}_{2}\,=\,\cdots \,=\,{c}_{8}=0,\quad \zeta =\eta ,\\ {\tilde{c}}_{1} & = & {\tilde{c}}_{2}\,=\,\cdots \,=\,{\tilde{c}}_{8}\,=\,0,\quad \tilde{\zeta }=\tilde{\eta },\end{array}\end{eqnarray}$ the reflection matrices (24 ) and (26 ) degenerate into the diagonal ones [25] $ \begin{eqnarray}\begin{array}{rcl}{\bar{K}}^{v-}(u) & = & {diag}\left[(\zeta ^{\prime} +1-4u)(\zeta ^{\prime} -1-4u),\right.\\ & & (\zeta ^{\prime} +1+4u)(\zeta ^{\prime} -1-4u),\\ & & (\zeta ^{\prime} +1+4u)(\zeta ^{\prime} -1-4u),\\ & & (\zeta ^{\prime} +1+4u)(\zeta ^{\prime} -1-4u),\\ & & \left(\zeta ^{\prime} +1+4u)(\zeta ^{\prime} -1+4u)\right],\end{array}\end{eqnarray}$ $ \begin{eqnarray}\begin{array}{rcl}{\bar{K}}^{v+}(u) & = & {diag}\left[(\tilde{\zeta }^{\prime} +7+4u)(\tilde{\zeta }^{\prime} +5+4u),\right.\\ & & (\tilde{\zeta }^{\prime} -5-4u)(\tilde{\zeta }^{\prime} +5+4u),\\ & & (\tilde{\zeta }^{\prime} -5-4u)(\tilde{\zeta }^{\prime} +5+4u),\\ & & (\tilde{\zeta }^{\prime} -5-4u)(\tilde{\zeta }^{\prime} +5+4u),\\ & & \left.(\tilde{\zeta }^{\prime} -7-4u)(\tilde{\zeta }^{\prime} -5-4u)\right],\end{array}\end{eqnarray}$ where $\zeta ^{\prime} ={\zeta }^{-1}$ and $\tilde{\zeta }^{\prime} ={\tilde{\zeta }}^{-1}$ .
For the constant solution (12 ) of spinorial reflection matrix, the fusion relation (23 ) degenerates $ \begin{eqnarray}\begin{array}{l}{P}_{1^{\prime} 2^{\prime} }^{5}{\tilde{K}}_{2^{\prime} }^{s-}\left(u+\displaystyle \frac{1}{4}\right){R}_{1^{\prime} 2^{\prime} }^{{ss}}(2u){\tilde{K}}_{1^{\prime} }^{s-}\left(u-\displaystyle \frac{1}{4}\right){P}_{1^{\prime} 2^{\prime} }^{5}\\ \quad =\,\left(u+\displaystyle \frac{3}{4}\right){\tilde{K}}_{\langle 1^{\prime} 2^{\prime} \rangle }^{v-}(u),\end{array}\end{eqnarray}$ which gives another degenerated vectorial reflection matrix ${\tilde{K}}^{v-}$ with the elements $ \begin{eqnarray}\begin{array}{rcl}{\tilde{K}}_{11}^{v-}(u) & = & \displaystyle \frac{1}{2}[-2{\hat{c}}_{2}{\hat{c}}_{5}-{\hat{c}}_{7}{\hat{c}}_{8}-{\hat{c}}_{3}{\hat{c}}_{6}\\ & & -2{\hat{c}}_{1}{\hat{c}}_{4}(4u+1)-{\hat{\zeta }}^{2}+8u\hat{\zeta }\hat{\eta }-{\hat{\eta }}^{2}],\\ {\tilde{K}}_{12}^{v-}(u) & = & 4u({\hat{c}}_{2}{\hat{c}}_{4}+{\hat{c}}_{7}\hat{\zeta }),\\ {\tilde{K}}_{13}^{v-}(u) & = & -2\sqrt{2}u[{\hat{c}}_{3}{\hat{c}}_{4}-{\hat{c}}_{1}{\hat{c}}_{7}-{\hat{c}}_{2}(\hat{\zeta }+\hat{\eta })],\\ {\tilde{K}}_{14}^{v-}(u) & = & -4u(-{\hat{c}}_{3}\hat{\eta }+{\hat{c}}_{1}{\hat{c}}_{2}),\\ {\tilde{K}}_{15}^{v-}(u) & = & -4u({\hat{c}}_{2}^{2}+{\hat{c}}_{3}{\hat{c}}_{7}),\\ {\tilde{K}}_{21}^{v-}(u) & = & 4u({\hat{c}}_{1}{\hat{c}}_{5}+{\hat{c}}_{8}\hat{\zeta }),\\ {\tilde{K}}_{22}^{v-}(u) & = & \displaystyle \frac{1}{2}[-2{\hat{c}}_{1}{\hat{c}}_{4}-{\hat{c}}_{7}{\hat{c}}_{8}-{\hat{c}}_{3}{\hat{c}}_{6}\\ & & -2{\hat{c}}_{2}{\hat{c}}_{5}(4u+1)-{\hat{\zeta }}^{2}-8u\hat{\zeta }\hat{\eta }-{\hat{\eta }}^{2}],\\ {\tilde{K}}_{23}^{v-}(u) & = & 2\sqrt{2}u[{\hat{c}}_{3}{\hat{c}}_{5}+{\hat{c}}_{2}{\hat{c}}_{8}+{\hat{c}}_{1}(\hat{\zeta }-\hat{\eta })],\\ {\tilde{K}}_{24}^{v-}(u) & = & ({\hat{c}}_{3}{\hat{c}}_{8}-{\hat{c}}_{1}^{2})4u,\\ {\tilde{K}}_{25}^{v-}(u) & = & -4u(-{\hat{c}}_{3}\hat{\eta }+{\hat{c}}_{1}{\hat{c}}_{2}),\\ {\tilde{K}}_{31}^{v-}(u) & = & 2\sqrt{2}u[{\hat{c}}_{4}{\hat{c}}_{8}-{\hat{c}}_{1}{\hat{c}}_{6}+{\hat{c}}_{5}(\hat{\zeta }+\hat{\eta })],\\ {\tilde{K}}_{32}^{v-}(u) & = & 2\sqrt{2}u[{\hat{c}}_{5}{\hat{c}}_{7}+{\hat{c}}_{2}{\hat{c}}_{6}-{\hat{c}}_{4}(-\hat{\zeta }+\hat{\eta })],\\ {\tilde{K}}_{33}^{v-}(u) & = & \displaystyle \frac{1}{2}[2({\hat{c}}_{1}{\hat{c}}_{4}+{\hat{c}}_{2}{\hat{c}}_{5})(4u-1)\\ & & -(4u+1)({\hat{\zeta }}^{2}+{\hat{\eta }}^{2}+{\hat{c}}_{7}{\hat{c}}_{8}+{\hat{c}}_{3}{\hat{c}}_{6})],\\ {\tilde{K}}_{34}^{v-}(u) & = & 2\sqrt{2}u[{\hat{c}}_{3}{\hat{c}}_{5}+{\hat{c}}_{2}{\hat{c}}_{8}-{\hat{c}}_{1}(-\hat{\zeta }+\hat{\eta })],\\ {\tilde{K}}_{35}^{v-}(u) & = & -2\sqrt{2}u[{\hat{c}}_{3}{\hat{c}}_{4}-{\hat{c}}_{1}{\hat{c}}_{7}-{\hat{c}}_{2}(\hat{\zeta }+\hat{\eta })],\\ {\tilde{K}}_{41}^{v-}(u) & = & -4u({\hat{c}}_{4}{\hat{c}}_{5}-{\hat{c}}_{6}\hat{\eta }),\\ {\tilde{K}}_{42}^{v-}(u) & = & ({\hat{c}}_{6}{\hat{c}}_{7}-{\hat{c}}_{4}^{2})4u,\\ {\tilde{K}}_{43}^{v-}(u) & = & 2\sqrt{2}u[{\hat{c}}_{5}{\hat{c}}_{7}+{\hat{c}}_{2}{\hat{c}}_{6}-{\hat{c}}_{4}(-\hat{\zeta }+\hat{\eta })],\\ {\tilde{K}}_{44}^{v-}(u) & = & \displaystyle \frac{1}{2}[-2{\hat{c}}_{1}{\hat{c}}_{4}-{\hat{c}}_{7}{\hat{c}}_{8}-{\hat{c}}_{3}{\hat{c}}_{6}\\ & & -2{\hat{c}}_{2}{\hat{c}}_{5}(4u+1)-{\hat{\zeta }}^{2}-8u\hat{\zeta }\hat{\eta }-{\hat{\eta }}^{2}],\\ {\tilde{K}}_{45}^{v-}(u) & = & 4u({\hat{c}}_{2}{\hat{c}}_{4}+{\hat{c}}_{7}\hat{\zeta }),\\ {\tilde{K}}_{51}^{v-}(u) & = & -({\hat{c}}_{6}{\hat{c}}_{8}+{\hat{c}}_{5}^{2})4u,\\ {\tilde{K}}_{52}^{v-}(u) & = & -4u({\hat{c}}_{4}{\hat{c}}_{5}-{\hat{c}}_{6}\hat{\eta }),\\ {\tilde{K}}_{53}^{v-}(u) & = & 2\sqrt{2}u[{\hat{c}}_{4}{\hat{c}}_{8}-{\hat{c}}_{1}{\hat{c}}_{6}+{\hat{c}}_{5}(\hat{\zeta }+\hat{\eta })],\\ {\tilde{K}}_{54}^{v-}(u) & = & 4u({\hat{c}}_{1}{\hat{c}}_{5}+{\hat{c}}_{8}\hat{\zeta }),\\ {\tilde{K}}_{55}^{v-}(u) & = & \displaystyle \frac{1}{2}[-2{\hat{c}}_{2}{\hat{c}}_{5}-{\hat{c}}_{7}{\hat{c}}_{8}-{\hat{c}}_{3}{\hat{c}}_{6}\\ & & -2{\hat{c}}_{1}{\hat{c}}_{4}(4u+1)-{\hat{\zeta }}^{2}+8u\hat{\zeta }\hat{\eta }-{\hat{\eta }}^{2}].\end{array}\end{eqnarray}$ All the matrix elements are the polynomials of u. We note that the degree of polynomials in equation (32 ) is lower than that in equation (25 ). Again, the dual transformation (26 ) between ${\tilde{K}}^{v+}$ and ${\tilde{K}}^{v-}$ remains unchanged $ \begin{eqnarray}{\tilde{K}}^{v+}(u)={\tilde{K}}^{v-}\left(-u-\displaystyle \frac{3}{2}\right){| }_{\hat{\zeta },\hat{\eta },{\hat{c}}_{i}\to \tilde{\hat{\zeta }},\tilde{\hat{\eta }},{\tilde{\hat{c}}}_{i}}.\end{eqnarray}$
6. The B2 model with integrable open boundaries
Now, we have all the ingredients to construct the integrable B2 model with open boundary conditions. The interaction in the bulk is described by the monodromy matrix constructed by the vectorial R-matrices as $ \begin{eqnarray}{T}_{0}(u)={R}_{01}^{{vv}}(u){R}_{02}^{{vv}}\cdots {R}_{0N}^{{vv}}(u),\end{eqnarray}$ where V0 is the 5-dimensional auxiliary space and ${V}_{1}\otimes {V}_{2}\cdots \otimes {V}_{N}$ is the 5N-dimensional physical space. Considering the boundary reflection, the transfer matrix of the system is $ \begin{eqnarray}t(u)={{tr}}_{0}\{{K}_{0}^{v+}(u){T}_{0}(u){K}_{0}^{v-}(u){T}_{0}^{-1}(-u)\}.\end{eqnarray}$ From the Yang-Baxter equation (21 ), reflection equation (22 ) and dual reflection equation (27 ), one can prove that the transfer matrices with different spectral parameters commute with each other, [t(u), t(v)] = 0. Therefore, t(u) serves as the generating function of all the conserved quantities of the system. The model Hamiltonian can be obtained by taking the derivative of the logarithm of the transfer matrix $ \begin{eqnarray}\begin{array}{rcl}H & = & \displaystyle \frac{\partial \mathrm{ln}t(u)}{\partial u}{| }_{u=0}\\ & = & \displaystyle \sum _{k=1}^{N-1}{H}_{{kk}+1}+\displaystyle \frac{1}{2\xi }{K}_{1}^{v-{\prime}} (0)\\ & & +\displaystyle \frac{{{tr}}_{0}\{{K}_{0}^{v+}(0){H}_{N0}\}}{{{tr}}_{0}{K}_{0}^{v+}(0)}+{constant}.\end{array}\end{eqnarray}$ Here the first term ${H}_{{kk}+1}={{ \mathcal P }}_{{kk}+1}[{R}_{{kk}+1}^{{vv}}(u)]^{\prime} {| }_{u=0}$ describes the coupling in the bulk. The second and third terms describe the external magnetic fields at two boundaries with ${K}_{1}^{v-}(0)=\xi \times {id}$ . Substituting the values of vectorial R-matrix (20 ), the reflection matrix (24 ) and the dual one (26 ) into the Hamiltonian (36 ) and choosing the suitable parameterization, the explicit form of the Hamiltonian can be obtained. The corresponding eigen-energy could be calculated by the off-diagonal Bethe ansatz method [26–28].
7. Discussion
In this paper, we study the integrable quantization conditions of the B2 model. Starting from the fundamental spinorial R-matrix, we obtain the vectorial R-matrix by using the fusion technique. The general solutions of the vectorial reflection equations are obtained and the corresponding structures are analyzed. We find that the reflection matrix has 7 free boundary parameters, which are used to describe the degree of freedom of boundary couplings, without breaking the integrability of the system. Based on them, we give the model Hamiltonian of integrable B2 model with off-diagonal boundary reflections.
Appendix
The general solution of the matrix ${K}^{s-}(u)$ may have 16 non-zero elements $ \begin{eqnarray}{K}^{s-}(u)=\left(\begin{array}{cccc}1+{c}_{11}u & {c}_{12}u & {c}_{13}u &a {c}_{14}u\\ {c}_{21}u & 1+{c}_{22}u & {c}_{23}u & {c}_{24}u\\ {c}_{31}u & {c}_{32}u & 1+{c}_{33}u & {c}_{34}u\\ {c}_{41}u & {c}_{42}u & {c}_{43}u & 1+{c}_{44}u\end{array}\right),\end{eqnarray}$ where cij are the non-zero boundary parameters. Substituting equation (4 ) into the reflection equation (3 ), the left hand side of equation (3 ) should be equal to the right hand side of equation (3 ), which give the constraints among the boundary parameters. For simplicity, we use the index method $ \begin{eqnarray}\begin{array}{l}[{\rm{left}}\,{\rm{hand}}\,{\rm{side}}\,{\rm{of}}\,{\rm{equation(3)}}\,{]}_{{jl}}^{{ik}}\\ =[{\rm{right}}\,{\rm{hand}}\,{\rm{side}}\,{\rm{of}}\,{\rm{equation(3)}}{]}_{{jl}}^{{ik}},\end{array}\end{eqnarray}$ where i, k are the row indices and j, l are column indices. Considering the matrix element $\left({}_{24}^{11}\right)$ , we have $ \begin{eqnarray}\displaystyle \frac{1}{4}{c}_{14}({c}_{12}-{c}_{34}){uv}(u-v)(u+v)(3+2u-2v)=0.\end{eqnarray}$ Because equation (A.3 ) is valid for all the values of u and v, thus the coefficient must be zero, which gives c34=c12. We note that ${c}_{14}\ne 0$ . From the condition that the matrix element $\left({}_{34}^{11}\right)$ in both sides of equation (3 ) should be equal, we have $ \begin{eqnarray}\displaystyle \frac{1}{4}{c}_{14}({c}_{13}+{c}_{24}){uv}(u-v)(u+v)(3+2u-2v)=0,\end{eqnarray}$ which gives c24=-c13. From the matrix element $\left({}_{13}^{22}\right)$ , we have $ \begin{eqnarray}\displaystyle \frac{1}{4}{c}_{23}({c}_{21}-{c}_{43}){uv}(u-v)(u+v)(3+2u-2v)=0,\end{eqnarray}$ which gives c43=c21. The matrix element $\left({}_{33}^{12}\right)$ gives $ \begin{eqnarray}\displaystyle \frac{1}{4}{c}_{32}({c}_{31}+{c}_{42}){uv}(u-v)(u+v)(3+2u-2v)=0,\end{eqnarray}$ which reads ${c}_{42}=-{c}_{31}$ . From the element $\left({}_{12}^{24}\right)$ , we have $ \begin{eqnarray}\begin{array}{l}\displaystyle \frac{1}{8}{uv}(u-v)(3+2u-2v)(3{c}_{11}{c}_{14}+3{c}_{12}{c}_{24}+3{c}_{13}{c}_{34}\\ \quad +\,3{c}_{14}{c}_{44}+2{c}_{11}{c}_{14}u+4{c}_{12}{c}_{24}u\\ \quad +\,4{c}_{13}{c}_{34}u+2{c}_{14}{c}_{44}u+2{c}_{11}{c}_{14}v\\ \quad +\,2{c}_{44}{c}_{14}v+4{c}_{12}{c}_{24}v+4{c}_{13}{c}_{34}v)=0.\end{array}\end{eqnarray}$ Substituting relations (A.3 ) and (A.4 ) into (A.7 ), we obtain $ \begin{eqnarray}\begin{array}{l}\displaystyle \frac{1}{8}{c}_{14}({c}_{11}+{c}_{44}){uv}(u-v)\\ \quad \times \,(3+2u+2v)(3+2u-2v)=0,\end{array}\end{eqnarray}$ which gives ${c}_{11}=-{c}_{44}$ . Considering the element $\left({}_{13}^{21}\right)$ , we have $ \begin{eqnarray}\begin{array}{l}-\displaystyle \frac{1}{8}{uv}(u-v)(3+2u-2v)(3{c}_{12}{c}_{31}+3{c}_{22}{c}_{32}\\ \quad +\,3{c}_{32}{c}_{33}+3{c}_{34}{c}_{42}+2{c}_{22}{c}_{32}u+4{c}_{12}{c}_{31}u\\ \quad +\,4{c}_{42}{c}_{34}u+2{c}_{32}{c}_{33}u+2{c}_{22}{c}_{32}v\\ \quad +\,2{c}_{33}{c}_{32}v+4{c}_{12}{c}_{31}v+4{c}_{34}{c}_{42}v)=0.\end{array}\end{eqnarray}$ Substituting relations (A.3 ) and (A.6 ) into (A.9 ), we have $ \begin{eqnarray}\begin{array}{l}-\displaystyle \frac{1}{8}{c}_{32}({c}_{22}+{c}_{33}){uv}(u-v)\\ \quad \times \,(3+2u+2v)(3+2u-2v)=0,\end{array}\end{eqnarray}$ which gives ${c}_{22}=-{c}_{33}$ . From the element $\left({}_{12}^{12}\right)$ , we obtain $ \begin{eqnarray}\begin{array}{l}\displaystyle \frac{1}{8}{uv}(u-v)(3+2u-2v)(3{c}_{11}{c}_{12}+3{c}_{12}{c}_{22}+3{c}_{13}{c}_{32}\\ \quad +\,3{c}_{14}{c}_{42}+2{c}_{11}{c}_{12}u+2{c}_{12}{c}_{22}u\\ \quad +\,2{c}_{14}{c}_{31}u+2{c}_{13}{c}_{32}u+4{c}_{14}{c}_{42}u\\ \quad +2{c}_{11}{c}_{12}v+2{c}_{12}{c}_{22}v+2{c}_{14}{c}_{31}v\\ \quad +\,2{c}_{13}{c}_{32}v+4{c}_{14}{c}_{42}v)=0.\end{array}\end{eqnarray}$ Substituting relations (A.6 ), (A.8 ) and (A.10 ) into (A.11 ), we have $ \begin{eqnarray}\begin{array}{l}-\displaystyle \frac{1}{8}({c}_{14}{c}_{31}-{c}_{13}{c}_{32}+{c}_{12}{c}_{33}+{c}_{12}{c}_{44}){uv}\\ \quad \times \,(u-v)(3+2u+2v)(3+2u-2v)=0,\end{array}\end{eqnarray}$ which means $ \begin{eqnarray}{c}_{44}=\displaystyle \frac{1}{{c}_{12}}({c}_{13}{c}_{32}-{c}_{12}{c}_{33}-{c}_{14}{c}_{31}).\end{eqnarray}$ The matrix element $\left({}_{12}^{11}\right)$ gives the constraint $ \begin{eqnarray}\begin{array}{l}-\displaystyle \frac{1}{8}{uv}(u-v)(3+2u-2v)(3{c}_{11}{c}_{21}+3{c}_{21}{c}_{22}+3{c}_{23}{c}_{31}\\ \quad +\,3{c}_{24}{c}_{41}+2{c}_{11}{c}_{21}u+2{c}_{21}{c}_{22}u\\ \quad +\,2{c}_{23}{c}_{31}u+2{c}_{13}{c}_{41}u+4{c}_{24}{c}_{41}u\\ \quad +\,2{c}_{11}{c}_{21}v+2{c}_{21}{c}_{22}v+2{c}_{23}{c}_{31}v\\ \quad +\,2{c}_{13}{c}_{41}v+4{c}_{24}{c}_{41}v)=0.\end{array}\end{eqnarray}$ With the help of relations (A.4 ), (A.8 ), (A.10 ) and (A.12 ), we obtain $ \begin{eqnarray}\begin{array}{l}\displaystyle \frac{1}{8{c}_{12}}(-{c}_{14}{c}_{21}{c}_{31}-{c}_{12}{c}_{23}{c}_{31}+{c}_{13}{c}_{21}{c}_{32}+{c}_{12}{c}_{13}{c}_{41})\\ \quad \times \,{uv}(u-v)(3+2u+2v)(3+2u-2v)=0.\end{array}\end{eqnarray}$ Then we have $ \begin{eqnarray}{c}_{32}=\displaystyle \frac{1}{{c}_{13}{c}_{21}}({c}_{14}{c}_{21}{c}_{31}+{c}_{12}{c}_{23}{c}_{31}-{c}_{12}{c}_{13}{c}_{41}).\end{eqnarray}$ Considering the element $\left({}_{11}^{13}\right)$ , we have $ \begin{eqnarray}\begin{array}{l}\displaystyle \frac{1}{8}{uv}(u-v)(3+2u-2v)(3{c}_{11}{c}_{13}+3{c}_{12}{c}_{23}+3{c}_{13}{c}_{33}\\ \quad +\,3{c}_{14}{c}_{43}+2{c}_{11}{c}_{13}u-2{c}_{14}{c}_{21}u\\ \quad +\,2{c}_{12}{c}_{23}u+2{c}_{13}{c}_{33}u+4{c}_{14}{c}_{43}u\\ \quad +\,2{c}_{11}{c}_{13}v-2{c}_{14}{c}_{21}v+2{c}_{12}{c}_{23}v\\ \quad +\,2{c}_{13}{c}_{33}v+4{c}_{14}{c}_{43}v)=0.\end{array}\end{eqnarray}$ With the help of relations (A.5 ), (A.8 ), (A.12 ) and (A.15 ), we get $ \begin{eqnarray}\begin{array}{l}\displaystyle \frac{1}{8{c}_{21}}({c}_{14}{c}_{21}^{2}+{c}_{12}{c}_{21}{c}_{23}-{c}_{13}{c}_{23}{c}_{31}+2{c}_{13}{c}_{21}{c}_{33}+{c}_{13}^{2}{c}_{41})\\ \quad \times \,{uv}(u-v)(3+2u+2v)(3+2u-2v)=0,\end{array}\end{eqnarray}$ which gives $ \begin{eqnarray}{c}_{33}=\displaystyle \frac{1}{2{c}_{13}{c}_{21}}({c}_{13}{c}_{23}{c}_{31}-{c}_{13}^{2}{c}_{41}-{c}_{14}{c}_{21}^{2}-{c}_{12}{c}_{21}{c}_{23}).\end{eqnarray}$
After long calculations of all the other matrix elements, we can not obtain new constraint any more. Therefore, the 16 non-zero elements in the reflection matrix ${K}^{s-}(u)$ must satisfy above 9 constraints and only 7 parameters of them are free. For simplicity and comparing with previous results, we denote $ \begin{eqnarray}\begin{array}{rcl} & & {c}_{11}=-{c}_{44}=-\zeta ,\quad {c}_{22}=-{c}_{33}=-\eta ,\quad {c}_{12}={c}_{34}={c}_{1},\\ & & {c}_{13}=-{c}_{24}={c}_{2},\quad {c}_{21}={c}_{43}={c}_{4},\quad {c}_{31}=-{c}_{42}={c}_{5},\\ & & {c}_{14}={c}_{3},\quad {c}_{41}={c}_{6},\quad {c}_{23}={c}_{7},\quad {c}_{32}={c}_{8}.\end{array}\end{eqnarray}$
Then we arrive at equation (4 ).
Then we arrive at equation (