Panpan Xue¹, Guang-Liang Li¹, Junpeng Cao^,2,3,4,8, Kun Hao^5,6, Tao Yang^5,6,7, Wen-Li Yang^,5,6,7,8, Kangjie Shi⁵¹Shaanxi Province Key Laboratory of Quantum Information and Quantum Optoelectronic Devices, Xian Jiaotong University, Xian 710049, China
²Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
³School of Physical Sciences, University of Chinese Academy of Sciences, Beijing, China
⁴Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China
⁵Institute of Modern Physics, Northwest University, Xian 710069, China
⁶Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xian 710069, China
⁷School of Physics, Northwest University, Xian 710069, China

First author contact: ⁸Authors to whom any correspondence should be addressed.
Received:2019-09-16Revised:2019-10-19Accepted:2019-10-31Online:2019-12-31

Fund supported:

*National Program for Basic Research of MOST.2016YFA0300600 National Program for Basic Research of MOST.2016YFA0302104 National Natural Science Foundation of China.11934015 National Natural Science Foundation of China.11975183 National Natural Science Foundation of China.11547045 National Natural Science Foundation of China.11774397 National Natural Science Foundation of China.11775178 National Natural Science Foundation of China.11775177 National Natural Science Foundation of China.91536115 Australian Research Council.190101529 Major Basic Research Program of Natural Science of Shaanxi Province.2017KCT-12 Major Basic Research Program of Natural Science of Shaanxi Province.2017ZDJC-32 Double First-Class University Construction Project of Northwest University

Abstract
New integrable B₂ model with off-diagonal boundary reflections is proposed. The general solutions of the reflection matrix for the B₂ model are obtained by using the fusion technique. 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. The new quantization conditions will induce the novel structure of the energy spectrum and the boundary states. The corresponding boundary effects can be studied based on the results in this paper. Meanwhile, the reflection matrix of high rank models associated with B_n algebra can also be obtained by using the method suggested in this paper.
Keywords： Bethe Ansatz;lattice integrable models;Yang-Baxter equation;reflection equation


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Cite this article
Panpan Xue, Guang-Liang Li, Junpeng Cao, Kun Hao, Tao Yang, Wen-Li Yang, Kangjie Shi. Integrable boundary conditions for the B₂ model^*. Communications in Theoretical Physics, 2020, 72(1): 015001- doi:10.1088/1572-9494/ab5448

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 B_n 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 B_n 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 B_n 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 B_n 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](1)$ \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 ξ_i=−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 t_i denotes the transposition in the space V_i. The spinorial R-matrix satisfies the Yang-Baxter equation(2)$ \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}$

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(3)$ \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 K^s-matrix:(4)$ \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}$where η, ζ and c_i, (i=1, 2,⋯, 8) are the boundary parameters with the constrains(5)$ \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(6)$ \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(7)$ \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}$The dual reflection matrix satisfies the dual reflection equation(8)$ \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(9)$ \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}$the reflection matrices (A.1) and (6) degenerate into the diagonal ones(10)$ \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(11)$ \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(12)$ \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(13)$ \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,(14)$ \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 $u=-\tfrac{1}{2}$,(15)$ \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(16)$ \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(17)$ \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(18)$ \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(19)$ \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}$which is defined in the 5-dimensional auxiliary space and 5-dimensional quantum space. Detailed calculation shows that the elements of the vectorial R-matrix read [24](20)$ \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 R^vv₁₂ matrix satisfies the Yang-Baxter equation(21)$ \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}$

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(22)$ \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,(23)$ \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(24)$ \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}$where(25)$ \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(26)$ \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}$which satisfies the dual reflection equations(27)$ \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(28)$ \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}$the reflection matrices (24) and (26) degenerate into the diagonal ones [25](29)$ \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}$(30)$ \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(31)$ \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(32)$ \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(33)$ \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(34)$ \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 5^N-dimensional physical space. Considering the boundary reflection, the transfer matrix of the system is(35)$ \begin{eqnarray}t(u)={{tr}}_{0}\{{K}_{0}^{v+}(u){T}_{0}(u){K}_{0}^{v-}(u){T}_{0}^{-1}(-u)\}.\end{eqnarray}$From the Yang-Baxter equation (21), reflection equation (22) and dual reflection equation (27), one can prove that the transfer matrices with different spectral parameters commute with each other, [t(u), t(v)]=0. Therefore, t(u) serves as the generating function of all the conserved quantities of the system. The model Hamiltonian can be obtained by taking the derivative of the logarithm of the transfer matrix(36)$ \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(A.1)$ \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}$where c_ij are the non-zero boundary parameters. Substituting equation (4) into the reflection equation (3), the left hand side of equation (3) should be equal to the right hand side of equation (3), which give the constraints among the boundary parameters. For simplicity, we use the index method(A.2)$ \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(A.3)$ \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(A.4)$ \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(A.5)$ \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(A.6)$ \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(A.7)$ \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}$Substituting relations (A.3) and (A.4) into (A.7), we obtain(A.8)$ \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(A.9)$ \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}$Substituting relations (A.3) and (A.6) into (A.9), we have(A.10)$ \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(A.11)$ \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}$Substituting relations (A.6), (A.8) and (A.10) into (A.11), we have(A.12)$ \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}$which means(A.13)$ \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(A.14)$ \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}$With the help of relations (A.4), (A.8), (A.10) and (A.12), we obtain(A.15)$ \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}$Then we have(A.16)$ \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(A.17)$ \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}$With the help of relations (A.5), (A.8), (A.12) and (A.15), we get(A.18)$ \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}$which gives(A.19)$ \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(A.20)$ \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).

Reference View Option By original order
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[1]KorepinV EBogoliubovN MIzerginA G1993 Quantum Inverse Scattering Method and Correlation FunctionCambridgeCambridge University Press
[Cited within: 1]

[2]AlcarazF CBarberM NBatchelorM TBaxterR JQuispelG R W1987 J. Phys. A 20 6397
DOI:10.1088/0305-4470/20/18/038 [Cited within: 1]

[3]SklyaninE KFaddeevL D1978 Sov. Phys. Dokl 23 902
DOI:10.1037/016700 [Cited within: 1]

[4]TakhtadzhanL AFaddeevL D1979 Rush. Math. Surveys 34 11
DOI:10.1070/RM1979v034n05ABEH003909 [Cited within: 1]

[5]SklyaninE K1988 J. Phys. A 21 2375
DOI:10.1088/0305-4470/21/10/015 [Cited within: 1]

[6]MalaraRLima-SantosA2006 J. Stat. Mech. 2006 P09013
DOI:10.1088/1742-5468/2006/09/P09013 [Cited within: 1]

[7]de VegaH JGonzález-RuizA1993 J. Phys. A 26 L519
DOI:10.1088/0305-4470/26/12/007

[8]de VegaH JGonzález-RuizA1994 J. Phys. A 27 6129
DOI:10.1088/0305-4470/27/18/021

[9]Lima-SantosA1999 Nucl. Phys. B 558 637
DOI:10.1016/S0550-3213(99)00456-3

[10]ShiKLiG-LFanHHouB Y2000 High Energy Phys. and Nucl. Phys. 24 11(in Chinese)


[11]de VegaH JGonzález-RuizA1994 Nucl. Phys. B 417 553
DOI:10.1016/0550-3213(94)90484-7

[12]de VegaH JGonzález-RuizA1994 Mod. Phys. Lett. A 09 2207
DOI:10.1142/S0217732394002069

[13]LiG-LYueR HHouB Y2000 Nucl. Phys. B 586 711
DOI:10.1016/S0550-3213(00)00416-8

[14]Lima-santosAMalaraR2003 Nucl. Phys. B 675 661
DOI:10.1016/j.nuclphysb.2003.09.037 [Cited within: 1]

[15]ChicherinDDerkachovSIsaevA P2013 J. Phys. A 46 485201
DOI:10.1088/1751-8113/46/48/485201 [Cited within: 1]

[16]FrahmHSeelAWirthT2008 Nucl. Phys. B 802 351
DOI:10.1016/j.nuclphysb.2008.04.008 [Cited within: 1]

[17]KarowskiM1979 Nucl. Phys. B 153 244
DOI:10.1016/0550-3213(79)90600-X

[18]KulishP PReshetikhinN YSklyaninE K1981 Lett. Math. Phys. 5 393
DOI:10.1007/BF02285311

[19]KulishP PSklyaninE K1982 Lect. Notes Phys. 151 61


[20]KirillovA NReshetikhinN Y1986 J. Sov. Math. 35 2627
DOI:10.1007/BF01083768

[21]KirillovA NReshetikhinN Y1987 J. Phys. A 20 1565
DOI:10.1088/0305-4470/20/6/038

[22]MezincescuLNepomechieR I1992 Nucl. Phys. B 372 597
DOI:10.1016/0550-3213(92)90367-K

[23]ZhouY-K1996 Nucl. Phys. B 458 504
DOI:10.1016/0550-3213(95)00553-6 [Cited within: 1]

[24]MartinsM JRamosP B1997 Nucl. Phys. B 500 579
DOI:10.1016/S0550-3213(97)00342-8 [Cited within: 1]

[25]Lima-santosA2003 Nucl. Phys. B 654 466
DOI:10.1016/S0550-3213(03)00042-7 [Cited within: 1]

[26]WangYYangW-LCaoJShiK2015 Off-Diagonal Bethe Ansatz for Exactly Solvable ModelsBerlinSpringer
[Cited within: 1]

[27]CaoJYangW-LShiKWangY2014 JHEP 04 143
DOI:10.1007/JHEP04(2014)143

[28]CaoJCuiSYangW-LShiKWangY2015 JHEP 04 036
DOI:10.1007/JHEP02(2015)036 [Cited within: 1]