1. Introduction
The nineteen-vertex models with open boundaries have been discussed by many authors. Most of them are based on a boson basis. It is still important to study supersymmetric systems even though there have been certain transformations from a bosonic basis to a fermionic one. Such boson-fermion transformation may bring extra boundary effects. Moreover, the fermionic basis can provide a clear picture to describe the meaning of conserved quantities.
The typical supersymmetric nineteen-vertex model is the graded Osp(1∣2) spin chain. The periodic boundary case was first proposed and solved in [1]. A non-graded version of this model was proposed and solved in [2]. The difference between the graded and the non-graded periodic Osp(1∣2) quantum spin chain was discussed in [3]. However, there are only a few works on supersymmetric Osp(1∣2) spin chains with open boundaries. The eigenvalues and the eigenvectors of the Osp(1∣2) model with diagonal reflecting boundary conditions in the Fermi–Bose–Fermi (FBF) background were studied in [4].
In this paper, we apply the Lax pair formulation to Osp(1∣2) spin chain for the open boundary case. The procedure provides an alternative and straightforward proof for the quantum integrability of the model. We calculate the corresponding boundary K-matrices for the non-graded and the graded Osp(1∣2) spin chains using Lax pair operators. We use the diagonal K-matrices solution as an example to illustrate our method. After tedious calculation, the non-diagonal K-matrices are also obtained. We argue that the results also satisfy Sklyanin’s reflection equations (RE) [5]. Finally, we make an underlying comparison of the calculation process between the Lax pair formulation to RE.
The paper is organized as follows. Section 2 shows the Lax pair formulation for the quantum spin chain with open boundary conditions. In sections 3 and 4 , we calculate the Lax pair of Osp(1∣2) spin chain in bulk and boundaries for the non-graded and graded versions, respectively. Then we calculate the corresponding K-matrices with diagonal and non-diagonal boundary conditions. Section 5 is devoted to the conclusion.
2. Lax pair formulation
$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Phi }}}_{j+1} & = & {L}_{j}(\lambda ){{\rm{\Phi }}}_{j},\quad j=1,2,\ldots ,N,\\ \displaystyle \frac{{\rm{d}}}{{\rm{d}}t}{{\rm{\Phi }}}_{j} & = & {M}_{j}(\lambda ){{\rm{\Phi }}}_{j},\quad j=2,3,\ldots ,N,\end{array}\end{eqnarray}$
with equations on the boundaries
$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{\rm{d}}}{{\rm{d}}t}{{\rm{\Phi }}}_{1} & = & {M}_{-}(\lambda ){{\rm{\Phi }}}_{1},\\ \displaystyle \frac{{\rm{d}}}{{\rm{d}}t}{{\rm{\Phi }}}_{N+1} & = & {M}_{+}(\lambda ){{\rm{\Phi }}}_{N+1}.\end{array}\end{eqnarray}$
Here Lj(λ), Mj(λ) and M±(λ) are operators that have matrices form. These operators depend on the spectral parameter λ, while λ does not depend on the time t and dynamical variables. The consistency conditions for equations (2.1 ) and (2.2 ) give rise to the following Lax equations
$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{\rm{d}}}{{\rm{d}}t}{L}_{j}(\lambda ) & = & {M}_{j+1}(\lambda ){L}_{j}(\lambda )-{L}_{j}(\lambda ){M}_{j}(\lambda ),\\ j & = & 2,3,\ldots ,N-1,\end{array}\end{eqnarray}$
with boundary terms
$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{\rm{d}}}{{\rm{d}}t}{L}_{1}(\lambda ) & = & {M}_{2}(\lambda ){L}_{1}(\lambda )-{L}_{1}(\lambda ){M}_{-}(\lambda ),\\ \displaystyle \frac{{\rm{d}}}{{\rm{d}}t}{L}_{N}(\lambda ) & = & {M}_{+}(\lambda ){L}_{N}(\lambda )-{L}_{N}(\lambda ){M}_{N}(\lambda ).\end{array}\end{eqnarray}$
Thus if the equations of motion have the alternative Lax equations (2.3 ) and (2.4 ) form, and the boundary K-matrices are the solutions to constraint equations (2.7 ) and (2.8 ), one can conclude that the open boundary system is completely integrable. Define the one row monodromy matrix and the transfer matrix of the system as the following
$\begin{eqnarray}\begin{array}{rcl}T(\lambda ) & = & {L}_{N}(\lambda )\cdots {L}_{2}(\lambda ){L}_{1}(\lambda ),\\ t(\lambda ) & = & {{tr}}_{0}\{T(\lambda )\}.\end{array}\end{eqnarray}$
Suppose K−(λ), K+(λ) are the matrices of the left and right boundaries, respectively. In the following form, we define the double row transfer matrix τ(λ) for open chain
$\begin{eqnarray}\tau (\lambda )={{tr}}_{0}[{K}_{+}(\lambda )T(\lambda ){K}_{-}(\lambda ){T}^{-1}(-\lambda )].\end{eqnarray}$
The form of Lax equations (2.3 ) and (2.4 ) show that the transfer matrix τ(λ) is independent of time t. The boundary matrices must, nevertheless, satisfy the requirements of the constraints
$\begin{eqnarray}{K}_{-}(\lambda ){M}_{-}(-\lambda )={M}_{-}(\lambda ){K}_{-}(\lambda ),\end{eqnarray}$
$\begin{eqnarray}{{tr}}_{0}[{K}_{+}(\lambda ){M}_{+}(\lambda )\tilde{T}(\lambda )]={{tr}}_{0}[{K}_{+}(\lambda )\tilde{T}(\lambda ){M}_{+}(-\lambda )],\end{eqnarray}$
where $\tilde{T}(\lambda )=T(\lambda ){K}_{-}(\lambda ){T}^{-1}(-\lambda )$. The two equations above can be considered as RE for K-matrices. It suggests that the double row transfer matrices with different spectral parameters commute with each other
$\begin{eqnarray}[\tau (\lambda ),\tau (\mu )]=0.\end{eqnarray}$
As a result, the system with open boundaries has an infinite amount of conserved charges. The model is completely integrable.
3. The non-graded version
Unless otherwise stated, we will use the standard notation for the algebraic Bethe Ansatz approach here and throughout the rest of the article: for every matrix A ∈ End(V), Aj is an embedding operator in the tensor space V ⨂ V ⨂ ⋯, which works as A on the jth space and as an identity on the other factor spaces.
The Osp(1∣2) spin chain has a three-dimensional vector space at each site. For the bosonic (non-graded) formulation case, the R-matrix is given by
$\begin{eqnarray}R(\lambda )=\left(\begin{array}{ccccccccc}a & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & b & 0 & c & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & d & 0 & e & 0 & f & 0 & 0\\ 0 & c & 0 & b & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & e & 0 & g & 0 & -e & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & b & 0 & c & 0\\ 0 & 0 & f & 0 & -e & 0 & d & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & c & 0 & b & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & a\end{array}\right),\end{eqnarray}$
where
$\begin{eqnarray}\begin{array}{rcl}a & = & \eta -\lambda ,\quad b=\lambda ,\quad c=\eta ,\\ d & = & -\displaystyle \frac{\lambda (2\lambda -\eta )}{2\lambda -3\eta },\\ e & = & \displaystyle \frac{2\eta \lambda }{2\lambda -3\eta },\quad f=\displaystyle \frac{4\eta \lambda -3{\eta }^{2}}{2\lambda -3\eta },\\ g & = & \lambda -\displaystyle \frac{3{\eta }^{2}}{2\lambda -3\eta }.\end{array}\end{eqnarray}$
For R ∈ End(V ⨂ V), Rij(λ) is an embedding operator of R-matrix (3.1 ) in the tensor space. With the exception of the ith and jth factor spaces, it serves as an identity on all factor spaces. The R-matrix satisfies the non-graded quantum Yang–Baxter equation
$\begin{eqnarray}\begin{array}{l}{R}_{12}(\lambda -\nu ){R}_{13}(\lambda ){R}_{23}(\nu )\\ \quad =\,{R}_{23}(\nu ){R}_{13}(\lambda ){R}_{12}(\lambda -\nu ),\end{array}\end{eqnarray}$
and possesses the following properties3(The superscript ti denotes the transpose in the ith space.)
$\begin{eqnarray}{\rm{Initial}}\,{\rm{condition:}}\,\,{R}_{12}(0)=\eta {P}_{12},\end{eqnarray}$
$\begin{eqnarray}{\rm{Unitary}}\,{\rm{relation:}}\,\,{R}_{12}(\lambda ){R}_{21}(-\lambda )={\rho }_{1}(\lambda )\times {id},\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{\rm{Crossing}}\,{\rm{Unitary}}\,{\rm{relation:}}\,\,{R}_{12}^{{t}_{1}}(\lambda ){U}_{1}{R}_{21}^{{t}_{1}}(-\lambda +3\eta ){U}_{1}^{-1}\\ \quad ={\rho }_{2}(\lambda )\times {id}.\end{array}\end{eqnarray}$
P12 is the (non-graded) permutation operator defined by the standard basis
$\begin{eqnarray}{P}_{12}=\sum _{a,b=1}^{3}{e}^{{ab}}\bigotimes {e}^{{ba}},\quad {\left({e}^{{ab}}\right)}_{{ij}}={\delta }_{a,i}{\delta }_{b,j}.\end{eqnarray}$
δa,i is the Kronecker delta. Here the subscripts ij of eab indicate the row and column. Ordinarily, we use subscript for space index and superscript for row and column. The functions ρ1(λ) and ρ2(λ) are given by
$\begin{eqnarray}\begin{array}{rcl}{\rho }_{1}(\lambda ) & = & {\eta }^{2}-{\lambda }^{2},\\ {\rho }_{2}(\lambda ) & = & \displaystyle \frac{\lambda (3\eta -\lambda )(5\eta -2\lambda )(\eta -2\lambda )}{{\left(3\eta -2\lambda \right)}^{2}}.\end{array}\end{eqnarray}$
U is a diagonal matrix
$\begin{eqnarray}U=\left(\begin{array}{ccc}-1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & -1\end{array}\right).\end{eqnarray}$
The corresponding Hamiltonian with periodic boundary conditions can be expressed as
$\begin{eqnarray}H=-\eta \displaystyle \frac{{\rm{d}}}{{\rm{d}}\lambda }\mathrm{log}t(\lambda ){| }_{\lambda =0}.\end{eqnarray}$
The transfer matrix t(λ) and the spin operators have SU(2) symmetry [3]
$\begin{eqnarray}\begin{array}{rcl}\left[t(\lambda ),{S}^{z}\right] & = & [t(\lambda ),{S}^{\pm }]=0,\\ \left[{S}^{z},{S}^{\pm }\right] & = & \pm {S}^{\pm },\quad [{S}^{+},{S}^{-}]=2{S}^{z},\end{array}\end{eqnarray}$
in which
$\begin{eqnarray}{S}^{z}=\sum _{n=1}^{N}{s}_{n}^{z},\quad {S}^{\pm }=\sum _{n=1}^{N}{s}_{n}^{\pm },\end{eqnarray}$
and $\begin{eqnarray}{s}^{z}=\displaystyle \frac{1}{2}({e}^{11}-{e}^{33}),\quad {s}^{+}={e}^{13},\quad {s}^{-}={e}^{31}.\end{eqnarray}$
3.1. The Lax pair operators for the bulk and boundaries
It is easy to check that the equations of motion derived from the Hamiltonian (3.10 ) can be recast to the Lax form. Indeed, in the non-graded version, the L operators take the form
$\begin{eqnarray}{L}_{j}(\lambda )={R}_{0j}(\lambda ).\end{eqnarray}$
Following the method in the [12–14], the M operators for the bulk can be derived from the Lax equations (2.1 ), the Yang–Baxter equation (3.3 ), and the periodic Hamiltonian (3.10 ). The M-matrix Mj(λ) has the following form4(For convenience, we normalize the entries of Mj(λ) as polynomial by multiplying a common factor on it.)in auxiliary space 0,
$\begin{eqnarray}{M}_{j}(\lambda )=\left(\begin{array}{ccc}{M}_{j-1,j}^{11} & {M}_{j-1,j}^{12} & {M}_{j-1,j}^{13}\\ {M}_{j-1,j}^{21} & {M}_{j-1,j}^{22} & {M}_{j-1,j}^{23}\\ {M}_{j-1,j}^{31} & {M}_{j-1,j}^{32} & {M}_{j-1,j}^{33}\end{array}\right),\qquad j=2,\cdots ,N.\end{eqnarray}$
The concrete components of the non-graded version M operator are illustrated in section A.1 . We remark that the matrix Mj(λ) acts non-trivially on spaces 0, j − 1, and j. Here we only use j as a subscript of M to follow the conventions in the previous papers mentioned above. Mj(λ) is the operator for the bulk (of the open boundary case) and also for the periodic case.
Now we can calculate the M−(λ) and M+(λ) operators with the help of the following boundary terms in the open chain Hamiltonian
$\begin{eqnarray}{H}_{1}=\left(\begin{array}{ccc}{l}_{11} & & \\ & {l}_{22} & \\ & & {l}_{33}\end{array}\right),\quad {H}_{N}=\left(\begin{array}{ccc}{r}_{11} & & \\ & {r}_{22} & \\ & & {r}_{33}\end{array}\right).\end{eqnarray}$
Here we choose H1 and HN as diagonal matrices. They correspond to the diagonal boundary K-matrices. From the equations (2.3 ) and (2.4 ), it follows that the M−(λ) has the form
$\begin{eqnarray}{M}_{-}(\lambda )=\left(\begin{array}{ccccccccc}{D}_{1}^{(1)} & & & & & & & & \\ & {D}_{1}^{(2)} & & {A}_{+} & & & & & \\ & & {D}_{1}^{(3)} & & {B}_{+} & & {C}_{+} & & \\ & {A}_{-} & & {D}_{1}^{(4)} & & & & & \\ & & {B}_{-} & & {D}_{1}^{(5)} & & {F}_{+} & & \\ & & & & & {D}_{1}^{(6)} & & {G}_{+} & \\ & & {C}_{-} & & {F}_{-} & & {D}_{1}^{(7)} & & \\ & & & & & {G}_{-} & & {D}_{1}^{(8)} & \\ & & & & & & & & {D}_{1}^{(9)}\end{array}\right),\end{eqnarray}$
where
$\begin{eqnarray}\begin{array}{rcl}{D}_{1}^{(1)} & = & {D}_{1}^{(9)}=-3{\rm{i}}{\eta }^{2}(4{\lambda }^{2}-9{\eta }^{2}),\\ {D}_{1}^{(2)} & = & -{D}_{1}^{(4)}=3{\rm{i}}{\eta }^{2}({l}_{11}-{l}_{22})(4{\lambda }^{2}-9{\eta }^{2}),\\ {D}_{1}^{(6)} & = & -{D}_{1}^{(8)}=3{\rm{i}}{\eta }^{2}({l}_{22}-{l}_{33})(4{\lambda }^{2}-9{\eta }^{2}),\\ {D}_{1}^{(3)} & = & 3{\rm{i}}{\eta }^{2}(4{\lambda }^{2}+16{l}_{11}{\lambda }^{2}-4{l}_{22}{\lambda }^{2}-12{l}_{33}{\lambda }^{2}\\ & & -9{l}_{11}{\eta }^{2}+9{l}_{33}{\eta }^{2}+6{\eta }^{2}),\\ {D}_{1}^{(7)} & = & 3{\rm{i}}{\eta }^{2}(4{\lambda }^{2}-12{l}_{11}{\lambda }^{2}-4{l}_{22}{\lambda }^{2}+16{l}_{33}{\lambda }^{2}\\ & & +9{l}_{11}{\eta }^{2}-9{l}_{33}{\eta }^{2}+6{\eta }^{2}),\\ {D}_{1}^{(5)} & = & -3{\rm{i}}{\eta }^{2}(4{l}_{11}{\lambda }^{2}-8{l}_{22}{\lambda }^{2}+4{l}_{33}{\lambda }^{2}+15{\eta }^{2}),\\ {A}_{+} & = & 3{\rm{i}}\eta ({l}_{11}\lambda -{l}_{22}\lambda +\eta )(4{\lambda }^{2}-9{\eta }^{2}),\\ {A}_{-} & = & 3{\rm{i}}\eta ({l}_{22}\lambda -{l}_{11}\lambda +\eta )(4{\lambda }^{2}-9{\eta }^{2}),\\ {B}_{+} & = & 6{\rm{i}}\eta (2{l}_{33}{\lambda }^{3}-2{l}_{22}{\lambda }^{3}-2\eta {\lambda }^{2}-4{l}_{11}\eta {\lambda }^{2}\\ & & +3{l}_{22}\eta {\lambda }^{2}+{l}_{33}\eta {\lambda }^{2}-3{l}_{11}{\eta }^{2}\lambda +3{l}_{22}{\eta }^{2}\lambda -3{\eta }^{3}),\\ {B}_{-} & = & 6{\rm{i}}\eta (2{l}_{33}{\lambda }^{3}-2{l}_{22}{\lambda }^{3}+2\eta {\lambda }^{2}+4{l}_{11}\eta {\lambda }^{2}\\ & & -3{l}_{22}\eta {\lambda }^{2}-{l}_{33}\eta {\lambda }^{2}-3{l}_{11}{\eta }^{2}\lambda +3{l}_{22}{\eta }^{2}\lambda +3{\eta }^{3}),\\ {C}_{+} & = & 3{\rm{i}}\eta (8{l}_{33}{\lambda }^{3}-8{l}_{11}{\lambda }^{3}-8\eta {\lambda }^{2}-2{l}_{11}\eta {\lambda }^{2}+4{l}_{22}\eta {\lambda }^{2}\\ & & -2{l}_{33}\eta {\lambda }^{2}+3{l}_{11}{\eta }^{2}\lambda -3{l}_{33}{\eta }^{2}\lambda +3{\eta }^{3}),\\ {C}_{-} & = & 3{\rm{i}}\eta (8{l}_{11}{\lambda }^{3}-8{l}_{33}{\lambda }^{3}-8\eta {\lambda }^{2}-2{l}_{11}\eta {\lambda }^{2}\\ & & +4{l}_{22}\eta {\lambda }^{2}-2{l}_{33}\eta {\lambda }^{2}-3{l}_{11}{\eta }^{2}\lambda +3{l}_{33}{\eta }^{2}\lambda +3{\eta }^{3}),\\ {F}_{+} & = & 6{\rm{i}}\eta (2{l}_{22}{\lambda }^{3}-2{l}_{11}{\lambda }^{3}-2\eta {\lambda }^{2}+{l}_{11}\eta {\lambda }^{2}\\ & & +3{l}_{22}\eta {\lambda }^{2}-4{l}_{33}\eta {\lambda }^{2}-3{l}_{22}{\eta }^{2}\lambda +3{l}_{33}{\eta }^{2}\lambda -3{\eta }^{3}),\\ {F}_{-} & = & 6{\rm{i}}\eta (2{l}_{22}{\lambda }^{3}-2{l}_{11}{\lambda }^{3}+2\eta {\lambda }^{2}-{l}_{11}\eta {\lambda }^{2}\\ & & -3{l}_{22}\eta {\lambda }^{2}+4{l}_{33}\eta {\lambda }^{2}-3{l}_{22}{\eta }^{2}\lambda +3{l}_{33}{\eta }^{2}\lambda +3{\eta }^{3}),\\ {G}_{+} & = & 3{\rm{i}}\eta ({l}_{22}\lambda -{l}_{33}\lambda +\eta )(4{\lambda }^{2}-9{\eta }^{2}),\\ {G}_{-} & = & 3{\rm{i}}\eta ({l}_{33}\lambda -{l}_{22}\lambda +\eta )(4{\lambda }^{2}-9{\eta }^{2}).\end{array}\end{eqnarray}$
The M+ matrix has the form
$\begin{eqnarray}{M}_{+}(\lambda )=\left(\begin{array}{ccccccccc}{D}_{N}^{(1)} & & & & & & & & \\ & {D}_{N}^{(2)} & & {A}_{+}^{{\prime} } & & & & & \\ & & {D}_{N}^{(3)} & & {B}_{+}^{{\prime} } & & {C}_{+}^{{\prime} } & & \\ & {A}_{-}^{{\prime} } & & {D}_{N}^{(4)} & & & & & \\ & & {B}_{-}^{{\prime} } & & {D}_{N}^{(5)} & & {F}_{+}^{{\prime} } & & \\ & & & & & {D}_{N}^{(6)} & & {G}_{+}^{{\prime} } & \\ & & {C}_{-}^{{\prime} } & & {F}_{-}^{{\prime} } & & {D}_{N}^{(7)} & & \\ & & & & & {G}_{-}^{{\prime} } & & {D}_{N}^{(8)} & \\ & & & & & & & & {D}_{N}^{(9)}\end{array}\right),\end{eqnarray}$
where
$\begin{eqnarray}\begin{array}{rcl}{D}_{N}^{(1)} & = & {D}_{N}^{(9)}=-3{\rm{i}}{\eta }^{2}(4{\lambda }^{2}-9{\eta }^{2}),\\ {D}_{N}^{(2)} & = & -{D}_{N}^{(4)}=3{\rm{i}}{\eta }^{2}({r}_{11}-{r}_{22})(4{\lambda }^{2}-9{\eta }^{2}),\\ {D}_{N}^{(6)} & = & -{D}_{N}^{(8)}=3{\rm{i}}{\eta }^{2}({r}_{22}-{r}_{33})(4{\lambda }^{2}-9{\eta }^{2}),\\ {D}_{N}^{(3)} & = & 3{\rm{i}}{\eta }^{2}(4{\lambda }^{2}+16{r}_{11}{\lambda }^{2}-4{r}_{22}{\lambda }^{2}\\ & & -12{r}_{33}{\lambda }^{2}-9{r}_{11}{\eta }^{2}+9{r}_{33}{\eta }^{2}+6{\eta }^{2}),\\ {D}_{N}^{(7)} & = & 3{\rm{i}}{\eta }^{2}(4{\lambda }^{2}-12{r}_{11}{\lambda }^{2}-4{r}_{22}{\lambda }^{2}\\ & & +16{r}_{33}{\lambda }^{2}+9{r}_{11}{\eta }^{2}-9{r}_{33}{\eta }^{2}+6{\eta }^{2}),\\ {D}_{N}^{(5)} & = & -3{\rm{i}}{\eta }^{2}(4{r}_{11}{\lambda }^{2}-8{r}_{22}{\lambda }^{2}+4{r}_{33}{\lambda }^{2}+15{\eta }^{2}),\\ {A}_{+}^{{\prime} } & = & 3{\rm{i}}\eta ({r}_{22}\lambda -{r}_{11}\lambda +\eta )(4{\lambda }^{2}-9{\eta }^{2}),\\ {A}_{-}^{{\prime} } & = & 3{\rm{i}}\eta ({r}_{11}\lambda -{r}_{22}\lambda +\eta )(4{\lambda }^{2}-9{\eta }^{2}),\\ {B}_{+}^{{\prime} } & = & 6{\rm{i}}\eta (2{r}_{33}{\lambda }^{3}-2{r}_{22}{\lambda }^{3}+2\eta {\lambda }^{2}+4{r}_{11}\eta {\lambda }^{2}\\ & & -3{r}_{22}\eta {\lambda }^{2}-{r}_{33}\eta {\lambda }^{2}-3{r}_{11}{\eta }^{2}\lambda +3{r}_{22}{\eta }^{2}\lambda +3{\eta }^{3}),\\ {B}_{-}^{{\prime} } & = & 6{\rm{i}}\eta (2{r}_{33}{\lambda }^{3}-2{r}_{22}{\lambda }^{3}-2\eta {\lambda }^{2}-4{r}_{11}\eta {\lambda }^{2}\\ & & +3{r}_{22}\eta {\lambda }^{2}+{r}_{33}\eta {\lambda }^{2}-3{r}_{11}{\eta }^{2}\lambda +3{r}_{22}{\eta }^{2}\lambda -3{\eta }^{3}),\\ {C}_{+}^{{\prime} } & = & 3{\rm{i}}\eta (8{r}_{11}{\lambda }^{3}-8{r}_{33}{\lambda }^{3}-8\eta {\lambda }^{2}-2{r}_{11}\eta {\lambda }^{2}\\ & & +4{r}_{22}\eta {\lambda }^{2}-2{r}_{33}\eta {\lambda }^{2}-3{r}_{11}{\eta }^{2}\lambda +3{r}_{33}{\eta }^{2}\lambda +3{\eta }^{3}),\\ {C}_{-}^{{\prime} } & = & 3{\rm{i}}\eta (8{r}_{33}{\lambda }^{3}-8{r}_{11}{\lambda }^{3}-8\eta {\lambda }^{2}-2{r}_{11}\eta {\lambda }^{2}+4{r}_{22}\eta {\lambda }^{2}\\ & & -2{r}_{33}\eta {\lambda }^{2}+3{r}_{11}{\eta }^{2}\lambda -3{r}_{33}{\eta }^{2}\lambda +3{\eta }^{3}),\\ {F}_{+}^{{\prime} } & = & 6{\rm{i}}\eta (2{r}_{22}{\lambda }^{3}-2{r}_{11}{\lambda }^{3}+2\eta {\lambda }^{2}-{r}_{11}\eta {\lambda }^{2}-3{r}_{22}\eta {\lambda }^{2}\\ & & +4{r}_{33}\eta {\lambda }^{2}-3{r}_{22}{\eta }^{2}\lambda +3{r}_{33}{\eta }^{2}\lambda +3{\eta }^{3}),\\ {F}_{-}^{{\prime} } & = & 6{\rm{i}}\eta (2{r}_{22}{\lambda }^{3}-2{r}_{11}{\lambda }^{3}-2\eta {\lambda }^{2}+{r}_{11}\eta {\lambda }^{2}+3{r}_{22}\eta {\lambda }^{2}\\ & & -4{r}_{33}\eta {\lambda }^{2}-3{r}_{22}{\eta }^{2}\lambda +3{r}_{33}{\eta }^{2}\lambda -3{\eta }^{3}),\\ {G}_{+}^{{\prime} } & = & 3{\rm{i}}\eta ({r}_{33}\lambda -{r}_{22}\lambda +\eta )(4{\lambda }^{2}-9{\eta }^{2}),\\ {G}_{-}^{{\prime} } & = & 3{\rm{i}}\eta ({r}_{22}\lambda -{r}_{33}\lambda +\eta )(4{\lambda }^{2}-9{\eta }^{2}).\end{array}\end{eqnarray}$
We remark that the matrix M−(λ) acts non-trivially on spaces 0, and 1, and the matrix M+(λ) acts non-trivially on spaces 0, and N. So far, we have obtained all the Lax pair operators for the non-graded Osp(1∣2) spin chain with open boundaries.
3.2. Boundary reflection matrices
We can now calculate the boundary K-matrices from the constraint equations (2.7 ) and (2.8 ). Let us set
$\begin{eqnarray}\begin{array}{rcl}{K}_{-}(\lambda ) & = & \left(\begin{array}{ccc}{K}_{-}^{11} & & \\ & {K}_{-}^{22} & \\ & & {K}_{-}^{33}\end{array}\right),\\ {K}_{+}(\lambda ) & = & \left(\begin{array}{ccc}{K}_{+}^{11} & & \\ & {K}_{+}^{22} & \\ & & {K}_{+}^{33}\end{array}\right).\end{array}\end{eqnarray}$
Substituting (3.17 ) into (2.7 ), we have the following constraints for K−
$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{K}_{-}^{11}}{{K}_{-}^{22}} & = & \displaystyle \frac{(\eta +{l}_{11}\lambda -{l}_{22}\lambda )}{(\eta -{l}_{11}\lambda +{l}_{22}\lambda )},\\ \displaystyle \frac{{K}_{-}^{33}}{{K}_{-}^{22}} & = & \displaystyle \frac{(\eta -{l}_{22}\lambda +{l}_{33}\lambda )}{(\eta +{l}_{22}\lambda -{l}_{33}\lambda )},\end{array}\end{eqnarray}$
and
$\begin{eqnarray}{l}_{11}={l}_{33},\quad {l}_{22}={l}_{33}+\displaystyle \frac{4}{3}.\end{eqnarray}$
Then we have
$\begin{eqnarray}{K}_{-}(\lambda )=\left(\begin{array}{ccc}-4\lambda +3\eta & & \\ & 4\lambda +3\eta & \\ & & -4\lambda +3\eta \end{array}\right).\end{eqnarray}$
Substituting (3.19 ) into (2.8 ), we obtain
$\begin{eqnarray}{r}_{11}={r}_{33},\quad {r}_{22}={r}_{33}+\displaystyle \frac{4}{3},\end{eqnarray}$
and
$\begin{eqnarray}{K}_{+}(\lambda )=\left(\begin{array}{ccc}-4\lambda +3\eta & & \\ & -4\lambda +9\eta & \\ & & -4\lambda +3\eta \end{array}\right).\end{eqnarray}$
As for non-diagonal boundary conditions, we list a set of non-trivial solutions below without showing the intermediate process and M± matrices. Let us consider the following boundary terms in Hamiltonian
$\begin{eqnarray}{H}_{1}=\left(\begin{array}{ccc}{l}_{11} & & {l}_{13}\\ & {l}_{22} & \\ {l}_{31} & & {l}_{33}\end{array}\right),\,{H}_{N}=\left(\begin{array}{ccc}{r}_{11} & & {r}_{13}\\ & {r}_{22} & \\ {r}_{31} & & {r}_{33}\end{array}\right).\end{eqnarray}$
After tedious calculation, we obtain the K− and K+ matrices in the same way
$\begin{eqnarray}{K}_{-}(\lambda )=\left(\begin{array}{ccc}({l}_{11}-{l}_{33})\lambda +\eta & & -\displaystyle \frac{{\left({l}_{11}-{l}_{33}\right)}^{2}}{2{l}_{31}}\lambda \\ & \eta & \\ 2{l}_{31}\lambda & & ({l}_{33}-{l}_{11})\lambda +\eta \end{array}\right),\end{eqnarray}$
$\begin{eqnarray}{K}_{+}(\lambda )=\left(\begin{array}{ccc}({r}_{11}-{r}_{33})(-\lambda +\displaystyle \frac{3}{2}\eta )-\eta & & \displaystyle \frac{{\left({r}_{11}-{r}_{33}\right)}^{2}}{2{r}_{31}}(\lambda -\frac{3}{2}\eta )\\ & \eta & \\ {r}_{31}(-2\lambda +3\eta ) & & ({r}_{33}-{r}_{11})(-\lambda +\displaystyle \frac{3}{2}\eta )-\eta \end{array}\right),\end{eqnarray}$
where
$\begin{eqnarray}{l}_{22}=\displaystyle \frac{{l}_{11}+{l}_{33}}{2},\quad {l}_{13}=-\displaystyle \frac{{\left({l}_{11}-{l}_{33}\right)}^{2}}{4{l}_{31}},\end{eqnarray}$
$\begin{eqnarray}{r}_{22}=\displaystyle \frac{{r}_{11}+{r}_{33}}{2},\,{r}_{13}=-\displaystyle \frac{{\left({r}_{11}-{r}_{33}\right)}^{2}}{4{r}_{31}}.\end{eqnarray}$
Each of the K− ( $\begin{eqnarray}\begin{array}{l}{R}_{12}(\lambda -v){\mathop{K}\limits^{1}}_{-}(\lambda ){R}_{21}(\lambda +v){\mathop{K}\limits^{2}}_{-}(v)\\ \quad =\,{\mathop{K}\limits^{2}}_{-}(v){R}_{12}(\lambda +v){\mathop{K}\limits^{1}}_{-}(\lambda ){R}_{21}(\lambda -v),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{R}_{21}^{-1}(\lambda -v){\mathop{K}\limits^{1}}_{+}(\lambda ){R}_{12}(\lambda +v){\mathop{K}\limits^{2}}_{+}(v)\\ \quad =\,{\mathop{K}\limits^{2}}_{+}(v){R}_{21}(\lambda +v){\mathop{K}\limits^{1}}_{+}(\lambda ){R}_{12}^{-1}(\lambda -v).\end{array}\end{eqnarray}$
The results ensure the integrability of the model.
4. The graded version
Now let us discuss the graded version of the open Osp(1∣2) spin chain. The graded R-matrix is given by
$\begin{eqnarray}R(\lambda )=\left(\begin{array}{ccccccccc}a & 0 & 0 & 0 & 0 & e & 0 & -e & 0\\ 0 & b & 0 & c & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & b & 0 & 0 & 0 & c & 0 & 0\\ 0 & c & 0 & b & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & d & 0 & 0 & 0 & 0\\ -e & 0 & 0 & 0 & 0 & g & 0 & f & 0\\ 0 & 0 & c & 0 & 0 & 0 & b & 0 & 0\\ e & 0 & 0 & 0 & 0 & f & 0 & g & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & d\end{array}\right),\end{eqnarray}$
where
$\begin{eqnarray}\begin{array}{rcl}a & = & \lambda +\eta -\displaystyle \frac{2\eta \lambda }{2\lambda -3\eta },\quad b=\lambda ,\quad c=\eta ,\quad d=\lambda -\eta ,\\ e & = & \displaystyle \frac{2\eta \lambda }{2\lambda -3\eta },\quad f=-\displaystyle \frac{4\eta \lambda -3{\eta }^{2}}{2\lambda -3\eta },\quad g=\displaystyle \frac{\lambda (2\lambda -\eta )}{2\lambda -3\eta }.\end{array}\end{eqnarray}$
For R ∈ End(V ⨂ V), Rij(λ) is a super embedding operator of R-matrix (4.1 ) in the Z2 graded tensor space. With the exception of the ith and jth factor spaces, the embedding R-matrix (4.1 ) in the tensor space functions as an identity. The super tensor product of two operators is defined as ${\left(A\bigotimes B\right)}_{\beta \delta }^{\alpha \gamma }={\left(-1\right)}^{[p(\alpha )+p(\beta )]p(\gamma )}{A}_{\beta }^{\alpha }{B}_{\delta }^{\gamma }$.
The R-matrix satisfies the graded quantum Yang–Baxter equation
$\begin{eqnarray}{R}_{12}(\lambda -\nu ){R}_{13}(\lambda ){R}_{23}(\nu )={R}_{23}(\nu ){R}_{13}(\lambda ){R}_{12}(\lambda -\nu ).\end{eqnarray}$
In terms of matrix entries, it can be written explicitly as
$\begin{eqnarray}\begin{array}{l}R{\left(v-\lambda \right)}_{{\beta }_{1}{\beta }_{2}}^{{\alpha }_{1}{\alpha }_{2}}R{\left(v\right)}_{{\gamma }_{1}{\beta }_{3}}^{{\beta }_{1}{\alpha }_{3}}R{\left(\lambda \right)}_{{\gamma }_{2}{\gamma }_{3}}^{{\beta }_{2}{\beta }_{3}}{\left(-1\right)}^{(p({\beta }_{1})+p({\gamma }_{1}))p({\beta }_{2})}\\ \quad =R{\left(\lambda \right)}_{{\beta }_{2}{\beta }_{3}}^{{\alpha }_{2}{\alpha }_{3}}R{\left(v\right)}_{{\beta }_{1}{\gamma }_{3}}^{{\alpha }_{1}{\beta }_{3}}R{\left(v-\lambda \right)}_{{\gamma }_{1}{\gamma }_{2}}^{{\beta }_{1}{\beta }_{2}}{\left(-1\right)}^{(p({\alpha }_{1})+p({\beta }_{1}))p({\beta }_{2})}.\end{array}\end{eqnarray}$
The R-matrix possesses the following properties
$\begin{eqnarray}{\rm{Initial}}\,{\rm{condition:}}\,\,{R}_{12}(0)=\eta {P}_{12},\end{eqnarray}$
$\begin{eqnarray}{\rm{Unitary}}\,{\rm{relation:}}\,\,{R}_{12}(\lambda ){R}_{21}(-\lambda )={\rho }_{1}(\lambda )\times {id},\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{\rm{Crossing}}\,{\rm{Unitary}}\,{\rm{relation:}}\,\,{R}_{12}^{{{st}}_{1}}(\lambda ){R}_{21}^{{{st}}_{1}}(-\lambda +3\eta )\\ \quad =\,{\rho }_{2}(\lambda )\times {id}.\end{array}\end{eqnarray}$
The superscript st1 denotes the super transpose in the 1st quantum space. P12 is the graded permutation operator defined as
$\begin{eqnarray}{P}_{{\beta }_{1}{\beta }_{2}}^{{\alpha }_{1}{\alpha }_{2}}={\left(-1\right)}^{p({\alpha }_{1})p({\alpha }_{2})}{\delta }_{{\alpha }_{1}{\beta }_{2}}{\delta }_{{\beta }_{1}{\alpha }_{2}},\end{eqnarray}$
in which p(αi) is the Grassmann parity. It takes 1 for fermions and 0 for bosons. In this paper, we choose Bose–Fermi–Fermi (BFF) grading, which means p(1) = 0 (boson) and p(2) =p(3) = 1 (fermions). The above ρ1(λ) and ρ2(λ) are scalar functions
$\begin{eqnarray}\begin{array}{rcl}{\rho }_{1}(\lambda ) & = & {\eta }^{2}-{\lambda }^{2},\\ {\rho }_{2}(\lambda ) & = & \displaystyle \frac{\lambda (3\eta -\lambda )(5\eta -2\lambda )(\eta -2\lambda )}{{\left(3\eta -2\lambda \right)}^{2}}.\end{array}\end{eqnarray}$
The associated Hamiltonian is given by
$\begin{eqnarray}H={H}_{1}+\sum _{i=1}^{N}[{P}_{i,i+1}+\displaystyle \frac{2}{3}{E}_{i,i+1}]+{H}_{N},\end{eqnarray}$
where ${\left({E}_{i,i+1}\right)}_{{ab}}^{{cd}}={\alpha }_{{ab}}{\alpha }_{{cd}}^{{st}}$ is the Osp(1∣2) Temperley–Lieb invariant operator. Here the superscript st denotes the super transpose. α is given by
$\begin{eqnarray}\alpha =\left(\begin{array}{ccc}1 & 0 & 0\\ 0 & 0 & 1\\ 0 & -1 & 0\end{array}\right).\end{eqnarray}$
Compared with SU(2) symmetry in the non-graded version, the graded version transfer matrix defined by (2.6 ) with supertrace ${\rm{str}}{X}={\sum }_{a=1}^{3}{\left(-1\right)}^{p(a)}{X}_{{aa}}$ has the Osp(1∣2) symmetry
$\begin{eqnarray}\begin{array}{rcl}[t(\lambda ),{S}^{z}] & = & [t(\lambda ),{S}^{\pm }]=[t(\lambda ),{J}^{\pm }]=0,\\ \left[{S}^{z},{S}^{\pm }\right] & = & \pm {S}^{\pm },\quad [{S}^{z},{J}^{\pm }]=\pm \displaystyle \frac{1}{2}{J}^{\pm },\quad [{S}^{+},{S}^{-}]=2{S}^{z},\\ \{{J}^{+},{J}^{-}\} & = & 2{S}^{z},\quad \{{J}^{\pm },{J}^{\pm }\}=\pm 2{S}^{\pm },\quad \{{S}^{\pm },{J}^{\mp }\}=-{J}^{\pm },\end{array}\end{eqnarray}$
in which
$\begin{eqnarray}\begin{array}{rcl}{S}^{z} & = & \displaystyle \sum _{n=1}^{N}{s}_{n}^{z},\quad {S}^{\pm }=\displaystyle \sum _{n=1}^{N}{s}_{n}^{\pm },\\ {J}^{\pm } & = & \displaystyle \sum _{n=1}^{N}{j}_{n}^{\pm }{Q}_{n+1}\cdots {Q}_{N},\end{array}\end{eqnarray}$
where
$\begin{eqnarray}\begin{array}{rcl}{s}^{z} & = & \displaystyle \frac{1}{2}({e}^{33}-{e}^{22}),\quad {s}^{+}={e}^{32}\ ,\quad {s}^{-}={e}^{23},\\ {j}^{+} & = & -{e}^{12}-{e}^{31},\quad {j}^{-}=-{e}^{13}+{e}^{21}\ ,\\ Q & = & {e}^{11}-{e}^{22}-{e}^{33}.\end{array}\end{eqnarray}$
4.1. The Lax pair for bulk and boundaries
Similarly, the equations of motion derived from the Hamiltonian (4.10 ) can be recast to the Lax form. In the graded case, we also choose the R matrix as the L operator, Lj(λ) = R0j(λ). Then the M operators for the bulk can be derived from the Lax equations (2.1 ) and the graded quantum Yang–Baxter equation (4.3 ). The M-matrix Mj(λ) also has the following 3 × 3 matrix form in auxiliary space 0,
$\begin{eqnarray}{M}_{j}(\lambda )=\left(\begin{array}{ccc}{M}_{j-1,j}^{11} & {M}_{j-1,j}^{12} & {M}_{j-1,j}^{13}\\ {M}_{j-1,j}^{21} & {M}_{j-1,j}^{22} & {M}_{j-1,j}^{23}\\ {M}_{j-1,j}^{31} & {M}_{j-1,j}^{32} & {M}_{j-1,j}^{33}\end{array}\right),\qquad j=2,\cdots ,N.\end{eqnarray}$
We show the concrete components of the graded M operator in section A.2 .
In the same way, let us set
$\begin{eqnarray}\begin{array}{rcl}{H}_{1} & = & \left(\begin{array}{ccc}{l}_{11} & & \\ & {l}_{22} & \\ & & {l}_{33}\end{array}\right),\\ {H}_{N} & = & \left(\begin{array}{ccc}{r}_{11} & & \\ & {r}_{22} & \\ & & {r}_{33}\end{array}\right).\end{array}\end{eqnarray}$
From the equations (2.3 ) and (2.4 ) , we obtain the associated M− matrix
$\begin{eqnarray}{M}_{-}(\lambda )=\left(\begin{array}{ccccccccc}{D}_{1}^{(1)} & & & & & {B}_{+} & & {F}_{+} & \\ & {D}_{1}^{(2)} & & {A}_{+} & & & & & \\ & & {D}_{1}^{(3)} & & & & {C}_{+} & & \\ & {A}_{-} & & {D}_{1}^{(4)} & & & & & \\ & & & & {D}_{1}^{(5)} & & & & \\ {B}_{-} & & & & & {D}_{1}^{(6)} & & {G}_{+} & \\ & & {C}_{-} & & & & {D}_{1}^{(7)} & & \\ {F}_{-} & & & & & {G}_{-} & & {D}_{1}^{(8)} & \\ & & & & & & & & {D}_{1}^{(9)}\end{array}\right),\end{eqnarray}$
where
$\begin{eqnarray}\begin{array}{rcl}{D}_{1}^{(1)} & = & 3{\rm{i}}{\eta }^{2}(8{l}_{11}{\lambda }^{2}-4{l}_{22}{\lambda }^{2}-4{l}_{33}{\lambda }^{2}-15{\eta }^{2}),\\ {D}_{1}^{(2)} & = & -{D}_{1}^{(4)}=3{\rm{i}}{\eta }^{2}({l}_{11}-{l}_{22})(4{\lambda }^{2}-9{\eta }^{2}),\\ {D}_{1}^{(3)} & = & -{D}_{1}^{(7)}=3{\rm{i}}{\eta }^{2}({l}_{11}-{l}_{33})(4{\lambda }^{2}-9{\eta }^{2}),\\ {D}_{1}^{(5)} & = & {D}_{1}^{(9)}=-3{\rm{i}}{\eta }^{2}(4{\lambda }^{2}-9{\eta }^{2}),\\ {D}_{1}^{(6)} & = & 3{\rm{i}}{\eta }^{2}(4{\lambda }^{2}-4{l}_{11}{\lambda }^{2}+16{l}_{22}{\lambda }^{2}-12{l}_{33}{\lambda }^{2}\\ & & -9{l}_{22}{\eta }^{2}+9{l}_{33}{\eta }^{2}+6{\eta }^{2}),\\ {D}_{1}^{(8)} & = & 3{\rm{i}}{\eta }^{2}(4{\lambda }^{2}-4{l}_{11}{\lambda }^{2}-12{l}_{22}{\lambda }^{2}+16{l}_{33}{\lambda }^{2}\\ & & +9{l}_{22}{\eta }^{2}-9{l}_{33}{\eta }^{2}+6{\eta }^{2}),\\ {A}_{+} & = & 3{\rm{i}}\eta (4{\lambda }^{2}-9{\eta }^{2})({l}_{11}\lambda -{l}_{22}\lambda +\eta ),\\ {A}_{-} & = & 3{\rm{i}}\eta (4{\lambda }^{2}-9{\eta }^{2})({l}_{22}\lambda -{l}_{11}\lambda +\eta ),\\ {B}_{+} & = & 6{\rm{i}}\eta (2{l}_{33}{\lambda }^{3}-2{l}_{11}{\lambda }^{3}+2\eta {\lambda }^{2}-3{l}_{11}\eta {\lambda }^{2}\\ & & +4{l}_{22}\eta {\lambda }^{2}-{l}_{33}\eta {\lambda }^{2}+3{l}_{11}{\eta }^{2}\lambda -3{l}_{22}{\eta }^{2}\lambda +3{\eta }^{3}),\\ {B}_{-} & = & 6{\rm{i}}\eta (2{l}_{33}{\lambda }^{3}-2{l}_{11}{\lambda }^{3}-2\eta {\lambda }^{2}+3{l}_{11}\eta {\lambda }^{2}\\ & & -4{l}_{22}\eta {\lambda }^{2}+{l}_{33}\eta {\lambda }^{2}+3{l}_{11}{\eta }^{2}\lambda -3{l}_{22}{\eta }^{2}\lambda -3{\eta }^{3}),\\ {C}_{+} & = & 3{\rm{i}}\eta (4{\lambda }^{2}-9{\eta }^{2})({l}_{11}\lambda -{l}_{33}\lambda +\eta ),\\ {C}_{-} & = & 3{\rm{i}}\eta (4{\lambda }^{2}-9{\eta }^{2})({l}_{33}\lambda -{l}_{11}\lambda +\eta ),\\ {F}_{+} & = & 6{\rm{i}}\eta (2{l}_{11}{\lambda }^{3}-2{l}_{22}{\lambda }^{3}-2\eta {\lambda }^{2}+3{l}_{11}\eta {\lambda }^{2}\\ & & +{l}_{22}\eta {\lambda }^{2}-4{l}_{33}\eta {\lambda }^{2}-3{l}_{11}{\eta }^{2}\lambda +3{l}_{33}{\eta }^{2}\lambda -3{\eta }^{3}),\\ {F}_{-} & = & 6{\rm{i}}\eta (2{l}_{11}{\lambda }^{3}-2{l}_{22}{\lambda }^{3}+2\eta {\lambda }^{2}-3{l}_{11}\eta {\lambda }^{2}\\ & & -{l}_{22}\eta {\lambda }^{2}+4{l}_{33}\eta {\lambda }^{2}-3{l}_{11}{\eta }^{2}\lambda +3{l}_{33}{\eta }^{2}\lambda +3{\eta }^{3}),\\ {G}_{+} & = & 3{\rm{i}}\eta (8{l}_{33}{\lambda }^{3}-8{l}_{22}{\lambda }^{3}-8\eta {\lambda }^{2}+4{l}_{11}\eta {\lambda }^{2}\\ & & -2{l}_{22}\eta {\lambda }^{2}-2{l}_{33}\eta {\lambda }^{2}+3{l}_{22}{\eta }^{2}\lambda -3{l}_{33}{\eta }^{2}\lambda +3{\eta }^{3}),\\ {G}_{-} & = & 3{\rm{i}}\eta (8{l}_{22}{\lambda }^{3}-8{l}_{33}{\lambda }^{3}-8\eta {\lambda }^{2}+4{l}_{11}\eta {\lambda }^{2}\\ & & -2{l}_{22}\eta {\lambda }^{2}-2{l}_{33}\eta {\lambda }^{2}-3{l}_{22}{\eta }^{2}\lambda +3{l}_{33}{\eta }^{2}\lambda +3{\eta }^{3}),\end{array}\end{eqnarray}$
and the M+(λ) matrix
$\begin{eqnarray}{M}_{+}(\lambda )=\left(\begin{array}{ccccccccc}{D}_{N}^{(1)} & & & & & {B}_{+}^{{\prime} } & & {F}_{+}^{{\prime} } & \\ & {D}_{N}^{(2)} & & {A}_{+}^{{\prime} } & & & & & \\ & & {D}_{N}^{(3)} & & & & {C}_{+}^{{\prime} } & & \\ & {A}_{-}^{{\prime} } & & {D}_{N}^{(4)} & & & & & \\ & & & & {D}_{N}^{(5)} & & & & \\ {B}_{-}^{{\prime} } & & & & & {D}_{N}^{(6)} & & {G}_{+}^{{\prime} } & \\ & & {C}_{-}^{{\prime} } & & & & {D}_{N}^{(7)} & & \\ {F}_{-}^{{\prime} } & & & & & {G}_{-}^{{\prime} } & & {D}_{N}^{(8)} & \\ & & & & & & & & {D}_{N}^{(9)}\end{array}\right),\end{eqnarray}$
where
$\begin{eqnarray}\begin{array}{rcl}{D}_{N}^{(1)} & = & 3{\rm{i}}{\eta }^{2}(8{r}_{11}{\lambda }^{2}-4{r}_{22}{\lambda }^{2}-4{r}_{33}{\lambda }^{2}-15{\eta }^{2}),\\ {D}_{N}^{(2)} & = & -{D}_{N}^{(4)}=3{\rm{i}}{\eta }^{2}({r}_{11}-{r}_{22})(4{\lambda }^{2}-9{\eta }^{2}),\\ {D}_{N}^{(3)} & = & -{D}_{N}^{(7)}=3{\rm{i}}{\eta }^{2}({r}_{11}-{r}_{33})(4{\lambda }^{2}-9{\eta }^{2}),\\ {D}_{N}^{(5)} & = & {D}_{N}^{(9)}=-3{\rm{i}}{\eta }^{2}(4{\lambda }^{2}-9{\eta }^{2}),\\ {D}_{N}^{(6)} & = & 3{\rm{i}}{\eta }^{2}(4{\lambda }^{2}-4{r}_{11}{\lambda }^{2}+16{r}_{22}{\lambda }^{2}\\ & & -12{r}_{33}{\lambda }^{2}-9{r}_{22}{\eta }^{2}+9{r}_{33}{\eta }^{2}+6{\eta }^{2}),\\ {D}_{N}^{(8)} & = & 3{\rm{i}}{\eta }^{2}(4{\lambda }^{2}-4{r}_{11}{\lambda }^{2}-12{r}_{22}{\lambda }^{2}\\ & & +16{r}_{33}{\lambda }^{2}+9{r}_{22}{\eta }^{2}-9{r}_{33}{\eta }^{2}+6{\eta }^{2}),\\ {A}_{+}^{{\prime} } & = & 3{\rm{i}}\eta (4{\lambda }^{2}-9{\eta }^{2})({r}_{22}\lambda -{r}_{11}\lambda +\eta ),\\ {A}_{-}^{{\prime} } & = & 3{\rm{i}}\eta (4{\lambda }^{2}-9{\eta }^{2})({r}_{11}\lambda -{r}_{22}\lambda +\eta ),\\ {B}_{+}^{{\prime} } & = & 6{\rm{i}}\eta (2{r}_{11}{\lambda }^{3}-2{r}_{33}{\lambda }^{3}+2\eta {\lambda }^{2}-3{r}_{11}\eta {\lambda }^{2}\\ & & +4{r}_{22}\eta {\lambda }^{2}-{r}_{33}\eta {\lambda }^{2}-3{r}_{11}{\eta }^{2}\lambda +3{r}_{22}{\eta }^{2}\lambda +3{\eta }^{3}),\\ {B}_{-}^{{\prime} } & = & 6{\rm{i}}\eta (2{r}_{11}{\lambda }^{3}-2{r}_{33}{\lambda }^{3}-2\eta {\lambda }^{2}+3{r}_{11}\eta {\lambda }^{2}\\ & & -4{r}_{22}\eta {\lambda }^{2}+{r}_{33}\eta {\lambda }^{2}-3{r}_{11}{\eta }^{2}\lambda +3{r}_{22}{\eta }^{2}\lambda -3{\eta }^{3}),\\ {C}_{+}^{{\prime} } & = & 3{\rm{i}}\eta ({r}_{33}\lambda -{r}_{11}\lambda +\eta )(4{\lambda }^{2}-9{\eta }^{2}),\\ {C}_{-}^{{\prime} } & = & 3{\rm{i}}\eta ({r}_{11}\lambda -{r}_{33}\lambda +\eta )(4{\lambda }^{2}-9{\eta }^{2}),\\ {F}_{+}^{{\prime} } & = & 6{\rm{i}}\eta (2{r}_{22}{\lambda }^{3}-2{r}_{11}{\lambda }^{3}-2\eta {\lambda }^{2}+3{r}_{11}\eta {\lambda }^{2}\\ & & +{r}_{22}\eta {\lambda }^{2}-4{r}_{33}\eta {\lambda }^{2}+3{r}_{11}{\eta }^{2}\lambda -3{r}_{33}{\eta }^{2}\lambda -3{\eta }^{3}),\\ {F}_{-}^{{\prime} } & = & 6{\rm{i}}\eta (2{r}_{22}{\lambda }^{3}-2{r}_{11}{\lambda }^{3}+2\eta {\lambda }^{2}-3{r}_{11}\eta {\lambda }^{2}\\ & & -{r}_{22}\eta {\lambda }^{2}+4{r}_{33}\eta {\lambda }^{2}+3{r}_{11}{\eta }^{2}\lambda -3{r}_{33}{\eta }^{2}\lambda +3{\eta }^{3}),\\ {G}_{+}^{{\prime} } & = & 3{\rm{i}}\eta (8{r}_{22}{\lambda }^{3}-8{r}_{33}{\lambda }^{3}-8\eta {\lambda }^{2}+4{r}_{11}\eta {\lambda }^{2}\\ & & -2{r}_{22}\eta {\lambda }^{2}-2{r}_{33}\eta {\lambda }^{2}-3{r}_{22}{\eta }^{2}\lambda +3{r}_{33}{\eta }^{2}\lambda +3{\eta }^{3}),\\ {G}_{-}^{{\prime} } & = & 3{\rm{i}}\eta (8{r}_{33}{\lambda }^{3}-8{r}_{22}{\lambda }^{3}-8\eta {\lambda }^{2}+4{r}_{11}\eta {\lambda }^{2}\\ & & -2{r}_{22}\eta {\lambda }^{2}-2{r}_{33}\eta {\lambda }^{2}+3{r}_{22}{\eta }^{2}\lambda -3{r}_{33}{\eta }^{2}\lambda +3{\eta }^{3}).\end{array}\end{eqnarray}$
4.2. Boundary reflection matrices
We can now calculate the boundary K-matrices from the constraint equations (2.7 ) and (2.8 ). Let us assume
$\begin{eqnarray}\begin{array}{rcl}{K}_{-}(\lambda ) & = & \left(\begin{array}{ccc}{K}_{-}^{11} & & \\ & {K}_{-}^{22} & \\ & & {K}_{-}^{33}\end{array}\right),\\ {K}_{+}(\lambda ) & = & \left(\begin{array}{ccc}{K}_{+}^{11} & & \\ & {K}_{+}^{22} & \\ & & {K}_{+}^{33}\end{array}\right).\end{array}\end{eqnarray}$
Substituting (4.17 ) into (2.7 ), we have the following constraints for K−
$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{K}_{-}^{22}}{{K}_{-}^{11}} & = & \displaystyle \frac{(\eta -{l}_{11}\lambda +{l}_{22}\lambda )}{(\eta +{l}_{11}\lambda -{l}_{22}\lambda )},\\ \displaystyle \frac{{K}_{-}^{33}}{{K}_{-}^{11}} & = & \displaystyle \frac{(\eta -{l}_{11}\lambda +{l}_{33}\lambda )}{(\eta +{l}_{11}\lambda -{l}_{33}\lambda )},\end{array}\end{eqnarray}$
and
$\begin{eqnarray}{l}_{22}={l}_{33},\quad {l}_{11}={l}_{33}+\displaystyle \frac{4}{3}.\end{eqnarray}$
Then we get
$\begin{eqnarray}{K}_{-}(\lambda )=\left(\begin{array}{ccc}4\lambda +3\eta & & \\ & -4\lambda +3\eta & \\ & & -4\lambda +3\eta \end{array}\right),\end{eqnarray}$
Substituting ( $\begin{eqnarray}{r}_{22}={r}_{33},\quad {r}_{11}={r}_{33}+\displaystyle \frac{4}{3},\end{eqnarray}$
and
$\begin{eqnarray}{K}_{+}(\lambda )=\left(\begin{array}{ccc}-4\lambda +9\eta & & \\ & 4\lambda -3\eta & \\ & & 4\lambda -3\eta \end{array}\right).\end{eqnarray}$
The above K-matrices are equivalent to those obtained by Yue and Xiong [4] with FBF grading via solving the RE.As for non-diagonal boundary conditions, the non-diagonal elements should have the same parity. So we set
$\begin{eqnarray}\begin{array}{rcl}{H}_{1} & = & \left(\begin{array}{ccc}{l}_{11} & & \\ & {l}_{22} & {l}_{23}\\ & {l}_{32} & {l}_{33}\end{array}\right),\\ {H}_{N} & = & \left(\begin{array}{ccc}{r}_{11} & & \\ & {r}_{22} & {r}_{23}\\ & {r}_{32} & {r}_{33}\end{array}\right).\end{array}\end{eqnarray}$
After tedious calculation, we obtain the following K− and K+ matrices
$\begin{eqnarray}{K}_{-}(\lambda )=\left(\begin{array}{ccc}\eta & & \\ & ({l}_{22}-{l}_{33})\lambda +\eta & -\displaystyle \frac{{\left({l}_{22}-{l}_{33}\right)}^{2}}{2{l}_{32}}\lambda \\ & 2{l}_{32}\lambda & ({l}_{33}-{l}_{22})\lambda +\eta \end{array}\right),\end{eqnarray}$
$\begin{eqnarray}{K}_{+}(\lambda )=\left(\begin{array}{ccc}\eta & & \\ & ({r}_{22}-{r}_{33})(\lambda -\displaystyle \frac{3}{2}\eta )+\eta & \displaystyle \frac{{\left({r}_{22}-{r}_{33}\right)}^{2}}{4{r}_{32}}(-2\lambda +3\eta )\\ & {r}_{32}(2\lambda -3\eta ) & ({r}_{33}-{r}_{22})(\lambda -\displaystyle \frac{3}{2}\eta )+\eta \end{array}\right),\end{eqnarray}$
where
$\begin{eqnarray}{l}_{11}=\displaystyle \frac{{l}_{22}+{l}_{33}}{2},\quad {l}_{23}=-\displaystyle \frac{{\left({l}_{22}-{l}_{33}\right)}^{2}}{4{l}_{32}},\end{eqnarray}$
$\begin{eqnarray}{r}_{11}=\displaystyle \frac{{r}_{22}+{r}_{33}}{2},\quad {r}_{23}=-\displaystyle \frac{{\left({r}_{22}-{r}_{33}\right)}^{2}}{4{r}_{32}}.\end{eqnarray}$
Again, we omit the form of M± matrices in this case.The K− and K+ matrices above also satisfy the graded version RE5(The symbol sti denotes the super transpose in the ith space ${\left({A}^{{st}}\right)}_{{ij}}={A}_{{ji}}{\left(-1\right)}^{p(i)[p(i)+p(j)]}$, and isti denotes the inverse operation to sti.)[4]
$\begin{eqnarray}\begin{array}{l}{R}_{12}(\lambda -v){\mathop{K}\limits^{1}}_{-}(\lambda ){R}_{21}(\lambda +v){\mathop{K}\limits^{2}}_{-}(v)\\ \quad ={\mathop{K}\limits^{2}}_{-}(v){R}_{12}(\lambda +v){\mathop{K}\limits^{1}}_{-}(\lambda ){R}_{21}(\lambda -v),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{R}_{12}(v-\lambda ){\mathop{K}\limits^{1}}_{+}(\lambda ){\tilde{\tilde{R}}}_{21}^{{{ist}}_{1}\ {{st}}_{2}}(-\lambda -v){\mathop{K}\limits^{2}}_{+}(v)\\ \quad =\mathop{{K}_{+}}\limits^{2}(v){\tilde{R}}_{12}^{{{ist}}_{1}\ {{st}}_{2}}(-\lambda -v){\mathop{K}\limits^{1}}_{+}(\lambda ){R}_{21}(v-\lambda ),\end{array}\end{eqnarray}$
where
$\begin{eqnarray}\begin{array}{rcl}{\tilde{\tilde{R}}}_{21}^{{{ist}}_{1}\ {{st}}_{2}}(-\lambda -v) & = & {\left({\left({\left({R}_{21}{\left(-\lambda -v\right)}^{-1}\right)}^{{{ist}}_{2}}\right)}^{-1}\right)}^{{{st}}_{2}},\\ {\tilde{R}}_{12}^{{{ist}}_{1}\ {{st}}_{2}}(-\lambda -v) & = & {\left({\left({\left({R}_{12}{\left(-\lambda -v\right)}^{-1}\right)}^{{{st}}_{1}}\right)}^{-1}\right)}^{{{ist}}_{1}}.\end{array}\end{eqnarray}$
5. Conclusion
For the open boundary Osp(1∣2) spin chain, we proposed the Lax pair formulation. The associated Lax pair operators were calculated. Based on that, we obtained the boundary K-matrices for both the non-graded and graded versions. In addition, the double row monodromy matrix and the transfer matrix of the spin chain have been built. The results obtained from the Lax pair formulation coincide with those from RE in finding the boundary K-matrices.
Both the RE scheme and the Lax pair formulation can demonstrate the integrability of the model. There is one special case where only the Lax pair formulation works for the model without crossing unitarity [15], while the RE scheme does not. When solving the RE, one has to take into account the spectral parameter and the crossing parameter. In the last step, a thorough classified discussion of the spectral parameter is needed to determine the concrete entries of the K-matrices. The Lax pair formulation starts with the designed boundaries Hamiltonians H1 and HN. If the assumed boundary Hamiltonians happen to guarantee the integrability of the system, then one can definitely get the correct M operators and K-matrices by finishing certain procedures. Although the whole calculation is lengthy, we do not have to consider the spectral parameters in the procedures, and the boundary Hamiltonians can be seen as constants in the derivations. This means that the Lax pair formulation has advantages in dealing with supersymmetric models, especially for non-diagonal boundary cases.
Acknowledgments
The work of KH was supported by the National Natural Science Foundation of China (Grant Nos. 12275214, 11805152, 12047502 and 11947301), the Natural Science Basic Research Program of Shaanxi Province Grant Nos. 2021JCW-19 and 2019JQ-107, and Shaanxi Key Laboratory for Theoretical Physics Frontiers in China. XZ would like to thank Prof Xiaotian Xu for the helpful discussions.
Appendix. The M-matrix for bulk and periodic case
A.1. The non-graded version M operator
For the non-graded version, the elements of M in (3.15 ) can be expressed as
$\begin{eqnarray*}\begin{array}{rcl}{M}_{j-1,j}^{11} & = & -3{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda +\eta )\\ & & \times \,({e}_{j-1}^{11}{e}_{j}^{11}+{e}_{j-1}^{11}{e}_{j}^{22}\\ & & +{e}_{j-1}^{11}{e}_{j}^{33}+{e}_{j-1}^{22}{e}_{j}^{11}+{e}_{j-1}^{33}{e}_{j}^{11})\\ & & +{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(3\lambda +5\eta ){e}_{j-1}^{22}{e}_{j}^{22}\\ & & +{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(3\lambda +2\eta )\\ & & \times ({e}_{j-1}^{22}{e}_{j}^{33}+{e}_{j-1}^{33}{e}_{j}^{22})\\ & & +{\rm{i}}\eta (12{\lambda }^{3}-16\eta {\lambda }^{2}-27{\eta }^{2}\lambda -9{\eta }^{3}){e}_{j-1}^{33}{e}_{j}^{33}\\ & & +3{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(2\lambda -\eta )(\lambda +\eta ){e}_{j-1}^{21}{e}_{j}^{12}\\ & & +3{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(2\lambda +\eta )(\lambda -\eta ){e}_{j-1}^{12}{e}_{j}^{21}\\ & & -2{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(2\lambda -\eta )(\lambda +\eta )\\ & & \times ({e}_{j-1}^{21}{e}_{j}^{23}+{e}_{j-1}^{32}{e}_{j}^{12})\\ & & +2{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(2\lambda +\eta )(\lambda -\eta )\\ & & \times ({e}_{j-1}^{12}{e}_{j}^{32}+{e}_{j-1}^{23}{e}_{j}^{21})\\ & & +{\rm{i}}\eta (2\lambda +3\eta )(4\lambda -3\eta )(\lambda +\eta ){e}_{j-1}^{31}{e}_{j}^{13}\\ & & -{\rm{i}}\eta (2\lambda +3\eta )(4\lambda +3\eta )(\lambda -\eta ){e}_{j-1}^{13}{e}_{j}^{31}\\ & & +{\rm{i}}\eta (2\lambda +3\eta )(12{\lambda }^{3}-12\eta {\lambda }^{2}\\ & & -10{\eta }^{2}\lambda +9{\eta }^{3}){e}_{j-1}^{32}{e}_{j}^{23}\\ & & +{\rm{i}}\eta (2\lambda -3\eta )(12{\lambda }^{3}+12\eta {\lambda }^{2}\\ & & -10{\eta }^{2}\lambda -9{\eta }^{3}){e}_{j-1}^{23}{e}_{j}^{32},\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{rcl}{M}_{j-1,j}^{22} & = & -3{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda -\eta )\\ & & \times \,({e}_{j-1}^{11}{e}_{j}^{11}+{e}_{j-1}^{33}{e}_{j}^{33})\\ & & -{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(3\lambda -5\eta )\\ & & \times ({e}_{j-1}^{11}{e}_{j}^{22}+{e}_{j-1}^{22}{e}_{j}^{11}+{e}_{j-1}^{22}{e}_{j}^{33}+{e}_{j-1}^{33}{e}_{j}^{22})\\ & & -{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(3\lambda -2\eta )\\ & & \times ({e}_{j-1}^{11}{e}_{j}^{33}+{e}_{j-1}^{33}{e}_{j}^{11})\\ & & -{\rm{i}}\eta (12{\lambda }^{3}-40\eta {\lambda }^{2}-27{\eta }^{2}\lambda +45{\eta }^{3}){e}_{j-1}^{22}{e}_{j}^{22}\\ & & -{\rm{i}}{\eta }^{2}(2\lambda +3\eta )(\lambda -9\eta ){e}_{j-1}^{21}{e}_{j}^{12}\\ & & -{\rm{i}}{\eta }^{2}(2\lambda -3\eta )(\lambda +9\eta ){e}_{j-1}^{12}{e}_{j}^{21}\\ & & +2{\rm{i}}{\eta }^{2}(4{\lambda }^{2}-9{\eta }^{2})\\ & & \times ({e}_{j-1}^{21}{e}_{j}^{23}-{e}_{j-1}^{12}{e}_{j}^{32}+{e}_{j-1}^{32}{e}_{j}^{12}-{e}_{j-1}^{23}{e}_{j}^{21})\\ & & +{\rm{i}}{\eta }^{2}(4{\lambda }^{2}-9{\eta }^{2})({e}_{j-1}^{31}{e}_{j}^{23}-{e}_{j-1}^{13}{e}_{j}^{32})\\ & & -{\rm{i}}{\eta }^{2}(2\lambda -3\eta )(\lambda +9\eta ){e}_{j-1}^{32}{e}_{j}^{23}\\ & & -{\rm{i}}{\eta }^{2}(2\lambda +3\eta )(\lambda -9\eta ){e}_{j-1}^{23}{e}_{j}^{32},\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{rcl}{M}_{j-1,j}^{33} & = & {\rm{i}}\eta (12{\lambda }^{3}-16\eta {\lambda }^{2}-27{\eta }^{2}\lambda -9{\eta }^{3}){e}_{j-1}^{11}{e}_{j}^{11}\\ & & +{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(3\lambda +2\eta )\\ & & \times ({e}_{j-1}^{11}{e}_{j}^{22}+{e}_{j-1}^{22}{e}_{j}^{11})\\ & & -3{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda +\eta )\\ & & \times ({e}_{j-1}^{11}{e}_{j}^{33}+{e}_{j-1}^{22}{e}_{j}^{33}+{e}_{j-1}^{33}{e}_{j}^{11}+{e}_{j-1}^{33}{e}_{j}^{22}+{e}_{j-1}^{33}{e}_{j}^{33})\\ & & +{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(3\lambda +5\eta ){e}_{j-1}^{22}{e}_{j}^{22}\\ & & +{\rm{i}}\eta (2\lambda -3\eta )(12{\lambda }^{3}+12\eta {\lambda }^{2}-10{\eta }^{2}\lambda \\ & & -9{\eta }^{3}){e}_{j-1}^{21}{e}_{j}^{12}\\ & & +{\rm{i}}\eta (2\lambda +3\eta )(12{\lambda }^{3}-12\eta {\lambda }^{2}\\ & & -10{\eta }^{2}\lambda +9{\eta }^{3}){e}_{j-1}^{12}{e}_{j}^{21}\\ & & -2{\rm{i}}(2\lambda +3\eta )(2\lambda -3\eta )(2\lambda +\eta )(\lambda -\eta )\\ & & \times ({e}_{j-1}^{21}{e}_{j}^{23}+{e}_{j-1}^{32}{e}_{j}^{12})\\ & & +2{\rm{i}}(2\lambda +3\eta )(2\lambda -3\eta )(2\lambda -\eta )(\lambda +\eta )\\ & & \times ({e}_{j-1}^{12}{e}_{j}^{32}+{e}_{j-1}^{23}{e}_{j}^{21})\\ & & -{\rm{i}}\eta (2\lambda -3\eta )(4\lambda +3\eta )(\lambda -\eta ){e}_{j-1}^{31}{e}_{j}^{13}\\ & & +{\rm{i}}\eta (2\lambda +3\eta )(4\lambda -3\eta )(\lambda +\eta ){e}_{j-1}^{13}{e}_{j}^{31}\\ & & +3{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(2\lambda +\eta )\\ & & \times \,(\lambda -\eta ){e}_{j-1}^{32}{e}_{j}^{23}\\ & & +3{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(2\lambda -\eta )\\ & & \times \,(\lambda +\eta ){e}_{j-1}^{23}{e}_{j}^{32},\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{rcl}{M}_{j-1,j}^{21} & = & 3{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda +\eta ){e}_{j-1}^{11}{e}_{j}^{12}\\ & & -2{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda +\eta ){e}_{j-1}^{11}{e}_{j}^{23}\\ & & -2{\rm{i}}\eta \lambda (2\lambda +3\eta )(\lambda +\eta ){e}_{j-1}^{21}{e}_{j}^{13}\\ & & -3{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda -\eta ){e}_{j-1}^{12}{e}_{j}^{11}\\ & & -{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )\\ & & \times \,(5\lambda -3\eta ){e}_{j-1}^{12}{e}_{j}^{22}\\ & & +{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(5\lambda +3\eta ){e}_{j-1}^{22}{e}_{j}^{12}\\ & & -{\rm{i}}\eta (2\lambda +3\eta ){\left(2\lambda -3\eta \right)}^{2}{e}_{j-1}^{12}{e}_{j}^{33}\\ & & -2{\rm{i}}\eta (2\lambda +3\eta )(5\lambda -3\eta \lambda -3{\eta }^{2}){e}_{j-1}^{22}{e}_{j}^{23}\\ & & -{\rm{i}}\eta \lambda (4{\lambda }^{2}-9{\eta }^{2})({e}_{j-1}^{32}{e}_{j}^{13}-{e}_{j-1}^{13}{e}_{j}^{32})\\ & & -2{\rm{i}}\eta (2\lambda +3\eta )(\lambda -\eta ){e}_{j-1}^{13}{e}_{j}^{21}\\ & & -2{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda -\eta ){e}_{j-1}^{23}{e}_{j}^{11}\\ & & -2{\rm{i}}\eta (2\lambda -3\eta )(5\lambda +3\eta \lambda -3{\eta }^{2}){e}_{j-1}^{23}{e}_{j}^{22}\\ & & -{\rm{i}}\eta (2\lambda -3\eta ){\left(2\lambda +3\eta \right)}^{2}{e}_{j-1}^{33}{e}_{j}^{12}\\ & & -2{\rm{i}}\eta (2\lambda +3\eta )(3{\lambda }^{2}-4\eta \lambda +3{\eta }^{2}){e}_{j-1}^{23}{e}_{j}^{33}\\ & & -2{\rm{i}}\eta (2\lambda -3\eta )(3{\lambda }^{2}+4\eta \lambda +3{\eta }^{2}){e}_{j-1}^{33}{e}_{j}^{23},\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{rcl}{M}_{j-1,j}^{12} & = & -3{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda -\eta ){e}_{j-1}^{11}{e}_{j}^{21}\\ & & +3{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda +\eta ){e}_{j-1}^{21}{e}_{j}^{11}\\ & & -2{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda +\eta ){e}_{j-1}^{11}{e}_{j}^{32}\\ & & +{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(5\lambda +3\eta ){e}_{j-1}^{21}{e}_{j}^{22}\\ & & -2{\rm{i}}\eta \lambda (2\lambda +3\eta )(\lambda +\eta ){e}_{j-1}^{31}{e}_{j}^{12}\\ & & +{\rm{i}}\eta (2\lambda -3\eta ){\left(2\lambda +3\eta \right)}^{2}{e}_{j-1}^{21}{e}_{j}^{33}\\ & & -{\rm{i}}\eta \lambda (4{\lambda }^{2}-9{\eta }^{2})({e}_{j-1}^{31}{e}_{j}^{23}-{e}_{j-1}^{23}{e}_{j}^{31})\\ & & -2{\rm{i}}\eta \lambda (2\lambda -3\eta )(\lambda -\eta ){e}_{j-1}^{12}{e}_{j}^{31}\\ & & -{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(5\lambda -3\eta ){e}_{j-1}^{22}{e}_{j}^{21}\\ & & -2{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda +\eta ){e}_{j-1}^{32}{e}_{j}^{11}\\ & & -2{\rm{i}}\eta (2\lambda -3\eta )(5{\lambda }^{2}+3\eta \lambda -3{\eta }^{2}){e}_{j-1}^{22}{e}_{j}^{32}\\ & & -2{\rm{i}}\eta (2\lambda +3\eta )(5{\lambda }^{2}-3\eta \lambda -3{\eta }^{2}){e}_{j-1}^{32}{e}_{j}^{22}\\ & & -2{\rm{i}}\eta (2\lambda -3\eta )(3{\lambda }^{2}+4\eta \lambda -3{\eta }^{2}){e}_{j-1}^{32}{e}_{j}^{33}\\ & & -{\rm{i}}\eta (2\lambda +3\eta ){\left(2\lambda -3\eta \right)}^{2}{e}_{j-1}^{33}{e}_{j}^{21}\\ & & -2{\rm{i}}\eta (2\lambda +3\eta )(3{\lambda }^{2}-4\eta \lambda -3{\eta }^{2}){e}_{j-1}^{33}{e}_{j}^{32},\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{rcl}{M}_{j-1,j}^{31} & = & -{\rm{i}}\eta (2\lambda +3\eta )(4\lambda -3\eta )(\lambda +\eta )\\ & & \times ({e}_{j-1}^{11}{e}_{j}^{13}+{e}_{j-1}^{13}{e}_{j}^{33})\\ & & +4{\rm{i}}\eta \lambda (4{\lambda }^{2}-9{\eta }^{2})({e}_{j-1}^{12}{e}_{j}^{12}-{e}_{j-1}^{23}{e}_{j}^{23})\\ & & -{\rm{i}}\eta \lambda (2\lambda +3\eta )(12\lambda -13\eta ){e}_{j-1}^{12}{e}_{j}^{23}\\ & & -{\rm{i}}{\eta }^{2}(4{\lambda }^{2}-9{\eta }^{2})({e}_{j-1}^{22}{e}_{j}^{13}+{e}_{j-1}^{13}{e}_{j}^{22})\\ & & +{\rm{i}}\eta (2\lambda -3\eta )(4\lambda +3\eta )\\ & & \times \,(\lambda -\eta )({e}_{j-1}^{13}{e}_{j}^{11}+{e}_{j-1}^{33}{e}_{j}^{13})\\ & & +{\rm{i}}\eta \lambda (2\lambda -3\eta )(12\lambda +13\eta ){e}_{j-1}^{23}{e}_{j}^{12},\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{rcl}{M}_{j-1,j}^{13} & = & {\rm{i}}\eta (2\lambda -3\eta )(4\lambda +3\eta )(\lambda -\eta )\\ & & \times ({e}_{j-1}^{11}{e}_{j}^{31}+{e}_{j-1}^{31}{e}_{j}^{33})\\ & & +4{\rm{i}}\eta \lambda (4{\lambda }^{2}-9{\eta }^{2})({e}_{j-1}^{21}{e}_{j}^{21}-{e}_{j-1}^{32}{e}_{j}^{32})\\ & & -{\rm{i}}\eta (2\lambda +3\eta )(4\lambda -3\eta )(\lambda +\eta )\\ & & \times \,({e}_{j-1}^{31}{e}_{j}^{11}+{e}_{j-1}^{33}{e}_{j}^{31})\\ & & +{\rm{i}}\eta \lambda (2\lambda -3\eta )(12\lambda +13\eta ){e}_{j-1}^{21}{e}_{j}^{32}\\ & & -{\rm{i}}{\eta }^{2}(4{\lambda }^{2}-9{\eta }^{2})({e}_{j-1}^{31}{e}_{j}^{22}-{e}_{j-1}^{22}{e}_{j}^{31})\\ & & -{\rm{i}}\eta \lambda (2\lambda +3\eta )(12\lambda -13\eta ){e}_{j-1}^{32}{e}_{j}^{21},\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{rcl}{M}_{j-1,j}^{32} & = & 2{\rm{i}}\eta (2\lambda +3\eta )(3{\lambda }^{2}-4\eta \lambda +3{\eta }^{2}){e}_{j-1}^{11}{e}_{j}^{12}\\ & & -{\rm{i}}\eta (2\lambda +3\eta ){\left(2\lambda -3\eta \right)}^{2}{e}_{j-1}^{11}{e}_{j}^{23}\\ & & +{\rm{i}}\eta \lambda (4{\lambda }^{2}-9{\eta }^{2})({e}_{j-1}^{21}{e}_{j}^{13}-{e}_{j-1}^{13}{e}_{j}^{21})\\ & & +2{\rm{i}}\eta (2\lambda -3\eta )(3{\lambda }^{2}+4\eta \lambda +3{\eta }^{2}){e}_{j-1}^{12}{e}_{j}^{11}\\ & & +2{\rm{i}}\eta (2\lambda +3\eta )(5{\lambda }^{2}-3\eta \lambda -3{\eta }^{2}){e}_{j-1}^{12}{e}_{j}^{22}\\ & & +2{\rm{i}}\eta (2\lambda -3\eta )(5{\lambda }^{2}+3\eta \lambda -3{\eta }^{2}){e}_{j-1}^{22}{e}_{j}^{12}\\ & & +2{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda +\eta ){e}_{j-1}^{12}{e}_{j}^{33}\\ & & -{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(5\lambda -3\eta ){e}_{j-1}^{22}{e}_{j}^{23}\\ & & +2{\rm{i}}\eta \lambda (2\lambda -3\eta )(\lambda -\eta ){e}_{j-1}^{32}{e}_{j}^{13}\\ & & +{\rm{i}}\eta (2\lambda -3\eta ){\left(2\lambda +3\eta \right)}^{2}{e}_{j-1}^{23}{e}_{j}^{11}\\ & & +2{\rm{i}}\eta \lambda (2\lambda +3\eta )(\lambda +\eta ){e}_{j-1}^{13}{e}_{j}^{32}\\ & & +{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(5\lambda +3\eta ){e}_{j-1}^{23}{e}_{j}^{22}\\ & & +2{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda -\eta ){e}_{j-1}^{33}{e}_{j}^{12}\\ & & +3{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda +\eta ){e}_{j-1}^{23}{e}_{j}^{33}\\ & & -3{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda -\eta ){e}_{j-1}^{33}{e}_{j}^{23},\end{array}\end{eqnarray*}$
$\begin{eqnarray}\begin{array}{rcl}{M}_{j-1,j}^{23} & = & 2{\rm{i}}\eta (2\lambda -3\eta )(3{\lambda }^{2}+4\eta \lambda +3{\eta }^{2}){e}_{j-1}^{11}{e}_{j}^{21}\\ & & +2{\rm{i}}\eta (2\lambda +3\eta )(3{\lambda }^{2}-4\eta \lambda +3{\eta }^{2}){e}_{j-1}^{21}{e}_{j}^{11}\\ & & +{\rm{i}}\eta (2\lambda -3\eta ){\left(2\lambda +3\eta \right)}^{2}{e}_{j-1}^{11}{e}_{j}^{32}\\ & & +2{\rm{i}}\eta (2\lambda -3\eta )(5{\lambda }^{2}+3\eta \lambda -3{\eta }^{2}){e}_{j-1}^{21}{e}_{j}^{22}\\ & & +{\rm{i}}\eta \lambda (4{\lambda }^{2}-9{\eta }^{2})({e}_{j-1}^{31}{e}_{j}^{12}-{e}_{j-1}^{12}{e}_{j}^{31})\\ & & +2{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda -\eta ){e}_{j-1}^{21}{e}_{j}^{33}\\ & & +2{\rm{i}}\eta \lambda (2\lambda -3\eta )(\lambda -\eta ){e}_{j-1}^{31}{e}_{j}^{23}\\ & & +2{\rm{i}}\eta (2\lambda +3\eta )(5{\lambda }^{2}-3\eta \lambda -3{\eta }^{2}){e}_{j-1}^{22}{e}_{j}^{21}\\ & & -{\rm{i}}\eta (2\lambda +3\eta ){\left(2\lambda -3\eta \right)}^{2}{e}_{j-1}^{32}{e}_{j}^{11}\\ & & +{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(5\lambda +3\eta ){e}_{j-1}^{22}{e}_{j}^{32}\\ & & -{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(5\lambda -3\eta ){e}_{j-1}^{32}{e}_{j}^{22}\\ & & -3{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda -\eta ){e}_{j-1}^{32}{e}_{j}^{33}\\ & & +2{\rm{i}}\eta \lambda (2\lambda +3\eta )(\lambda +\eta ){e}_{j-1}^{23}{e}_{j}^{31}\\ & & +2{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda +\eta ){e}_{j-1}^{33}{e}_{j}^{21}\\ & & +3{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda +\eta ){e}_{j-1}^{33}{e}_{j}^{32}.\end{array}\end{eqnarray}$
A.2. The graded M operator
For the graded version, the elements in the M operator (4.15 ) can be expressed as
$\begin{eqnarray*}\begin{array}{rcl}{M}_{j-1,j}^{11} & = & -{\rm{i}}\eta (12{\lambda }^{3}-40\eta {\lambda }^{2}\\ & & -27{\eta }^{2}+45{\eta }^{3}){e}_{j-1}^{11}{e}_{j}^{11}\\ & & -{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(3\lambda -5\eta )\\ & & \times ({e}_{j-1}^{11}{e}_{j}^{22}+{e}_{j-1}^{11}{e}_{j}^{33}+{e}_{j-1}^{22}{e}_{j}^{11}+{e}_{j-1}^{33}{e}_{j}^{11})\\ & & -3{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda -\eta )\\ & & \times ({e}_{j-1}^{22}{e}_{j}^{22}+{e}_{j-1}^{33}{e}_{j}^{33})\\ & & -{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(3\lambda -2\eta )\\ & & \times ({e}_{j-1}^{22}{e}_{j}^{33}+{e}_{j-1}^{33}{e}_{j}^{22})\\ & & -2{\rm{i}}{\eta }^{2}(4{\lambda }^{2}-9{\eta }^{2})\\ & & \times ({e}_{j-1}^{21}{e}_{j}^{31}-{e}_{j-1}^{12}{e}_{j}^{13}-{e}_{j-1}^{31}{e}_{j}^{21}+{e}_{j-1}^{13}{e}_{j}^{12})\\ & & -{\rm{i}}{\eta }^{2}(2\lambda +3\eta )(\lambda -9\eta )\\ & & \times ({e}_{j-1}^{21}{e}_{j}^{12}+{e}_{j-1}^{12}{e}_{j}^{21}+{e}_{j-1}^{31}{e}_{j}^{13}+{e}_{j-1}^{13}{e}_{j}^{31})\\ & & +{\rm{i}}{\eta }^{2}(4{\lambda }^{2}-9{\eta }^{2})({e}_{j-1}^{32}{e}_{j}^{23}+{e}_{j-1}^{23}{e}_{j}^{32}),\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{rcl}{M}_{j-1,j}^{22} & = & -{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(3\lambda +5\eta ){e}_{j-1}^{22}{e}_{j}^{11}\\ & & -3{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda +\eta )\\ & & \times \,({e}_{j-1}^{11}{e}_{j}^{22}+{e}_{j-1}^{22}{e}_{j}^{11}\\ & & +{e}_{j-1}^{22}{e}_{j}^{22}+{e}_{j-1}^{22}{e}_{j}^{33}+{e}_{j-1}^{33}{e}_{j}^{22})\\ & & -{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(3\lambda +2\eta )\\ & & \times \,({e}_{j-1}^{11}{e}_{j}^{33}+{e}_{j-1}^{33}{e}_{j}^{11})\\ & & -{\rm{i}}\eta (12{\lambda }^{3}-16\eta {\lambda }^{2}\lambda -9{\eta }^{3}){e}_{j-1}^{33}{e}_{j}^{33}\\ & & +2{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda -\eta )\\ & & \times ({e}_{j-1}^{21}{e}_{j}^{31}+{e}_{j-1}^{13}{e}_{j}^{12})\\ & & +2{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda +\eta )\\ & & \times ({e}_{j-1}^{12}{e}_{j}^{13}+{e}_{j-1}^{31}{e}_{j}^{21})\\ & & +3{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda -\eta ){e}_{j-1}^{21}{e}_{j}^{12}\\ & & -3{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda +\eta ){e}_{j-1}^{12}{e}_{j}^{21}\\ & & -{\rm{i}}\eta (2\lambda +3\eta )(6{\lambda }^{2}+2\eta \lambda -9{\eta }^{2}){e}_{j-1}^{31}{e}_{j}^{13}\\ & & +{\rm{i}}\eta (2\lambda -3\eta )(6{\lambda }^{2}-2\eta \lambda -9{\eta }^{2}){e}_{j-1}^{13}{e}_{j}^{31}\\ & & +{\rm{i}}\eta (2\lambda +3\eta )(4\lambda -3\eta )(\lambda +\eta ){e}_{j-1}^{32}{e}_{j}^{23}\\ & & -{\rm{i}}\eta (2\lambda -3\eta )(4\lambda +3\eta )(\lambda -\eta ){e}_{j-1}^{23}{e}_{j}^{32},\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{rcl}{M}_{j-1,j}^{33} & = & -{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(3\lambda +5\eta ){e}_{j-1}^{11}{e}_{j}^{11}\\ & & -{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(3\lambda +2\eta )\\ & & \times ({e}_{j-1}^{11}{e}_{j}^{22}+{e}_{j-1}^{22}{e}_{j}^{11})\\ & & -3{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda +\eta )\\ & & \times ({e}_{j-1}^{11}{e}_{j}^{33}+{e}_{j-1}^{22}{e}_{j}^{33}+{e}_{j-1}^{33}{e}_{j}^{11}\\ & & +{e}_{j-1}^{33}{e}_{j}^{22}+{e}_{j-1}^{33}{e}_{j}^{33})\\ & & -{\rm{i}}\eta (12{\lambda }^{3}-16\eta {\lambda }^{2}-27{\eta }^{2}\lambda -9{\eta }^{3}){e}_{j-1}^{22}{e}_{j}^{22}\\ & & -2{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda +\eta )\\ & & \times \,({e}_{j-1}^{21}{e}_{j}^{31}+{e}_{j-1}^{13}{e}_{j}^{12})\\ & & -2{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda -\eta )\\ & & \times \,({e}_{j-1}^{12}{e}_{j}^{13}+{e}_{j-1}^{31}{e}_{j}^{21})\\ & & -{\rm{i}}\eta (2\lambda +3\eta )(6{\lambda }^{2}+2\eta \lambda -9{\eta }^{2}){e}_{j-1}^{21}{e}_{j}^{12}\\ & & +{\rm{i}}\eta (2\lambda -3\eta )(6{\lambda }^{2}-2\eta \lambda -9{\eta }^{2}){e}_{j-1}^{12}{e}_{j}^{21}\\ & & +3{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda -\eta ){e}_{j-1}^{31}{e}_{j}^{13}\\ & & -3{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda +\eta ){e}_{j-1}^{13}{e}_{j}^{31}\\ & & -{\rm{i}}\eta (2\lambda -3\eta )(4\lambda +3\eta )(\lambda -\eta ){e}_{j-1}^{32}{e}_{j}^{23}\\ & & +{\rm{i}}\eta (2\lambda +3\eta )(4\lambda -3\eta )(\lambda +\eta ){e}_{j-1}^{23}{e}_{j}^{32},\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{rcl}{M}_{j-1,j}^{21} & = & -2{\rm{i}}\eta (2\lambda -3\eta )(5{\lambda }^{2}+3\eta \lambda -3{\eta }^{2}){e}_{j-1}^{11}{e}_{j}^{31}\\ & & +2{\rm{i}}\eta (2\lambda +3\eta )(5{\lambda }^{2}-3\eta \lambda -3{\eta }^{2}){e}_{j-1}^{31}{e}_{j}^{11}\\ & & -{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(5\lambda -3\eta ){e}_{j-1}^{11}{e}_{j}^{12}\\ & & +2{\rm{i}}\eta \lambda (2\lambda -3\eta )(\lambda -\eta ){e}_{j-1}^{21}{e}_{j}^{32}\\ & & +2{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda +\eta ){e}_{j-1}^{31}{e}_{j}^{22}\\ & & +2{\rm{i}}\eta (2\lambda -3\eta )(3{\lambda }^{2}+4\eta \lambda +3{\eta }^{2}){e}_{j-1}^{31}{e}_{j}^{33}\\ & & +{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(5\lambda +3\eta ){e}_{j-1}^{12}{e}_{j}^{11}\\ & & +2{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda -\eta ){e}_{j-1}^{22}{e}_{j}^{31}\\ & & +2{\rm{i}}\eta \lambda (2\lambda +3\eta )(\lambda +\eta ){e}_{j-1}^{32}{e}_{j}^{21}\\ & & +3{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda +\eta ){e}_{j-1}^{12}{e}_{j}^{22}\\ & & +3{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda -\eta ){e}_{j-1}^{22}{e}_{j}^{12}\\ & & +{\rm{i}}\eta (2\lambda -3\eta ){\left(2\lambda +3\eta \right)}^{2}{e}_{j-1}^{12}{e}_{j}^{33}\\ & & +{\rm{i}}\eta \lambda (4{\lambda }^{2}-9{\eta }^{2})({e}_{j-1}^{32}{e}_{j}^{13}+{e}_{j-1}^{13}{e}_{j}^{32})\\ & & +2{\rm{i}}\eta (2\lambda +3\eta )(3{\lambda }^{2}-4\eta \lambda +3{\eta }^{2}){e}_{j-1}^{33}{e}_{j}^{31}\\ & & +{\rm{i}}\eta (2\lambda +3\eta ){\left(2\lambda -3\eta \right)}^{2}{e}_{j-1}^{33}{e}_{j}^{12},\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{rcl}{M}_{j-1,j}^{12} & = & {\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(5\lambda +3\eta ){e}_{j-1}^{11}{e}_{j}^{21}\\ & & -{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(5\lambda -3\eta ){e}_{j-1}^{21}{e}_{j}^{11}\\ & & -3{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda -\eta ){e}_{j-1}^{21}{e}_{j}^{22}\\ & & -2{\rm{i}}\eta (2\lambda +3\eta )(5{\lambda }^{2}-3\eta \lambda -3{\eta }^{2}){e}_{j-1}^{11}{e}_{j}^{13}\\ & & -{\rm{i}}\eta (2\lambda +3\eta ){\left(2\lambda -3\eta \right)}^{2}{e}_{j-1}^{21}{e}_{j}^{33}\\ & & -{\rm{i}}\eta \lambda (4{\lambda }^{2}-9{\eta }^{2})({e}_{j-1}^{31}{e}_{j}^{23}+{e}_{j-1}^{23}{e}_{j}^{31})\\ & & -3{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda +\eta ){e}_{j-1}^{22}{e}_{j}^{21}\\ & & +2{\rm{i}}\eta \lambda (2\lambda +3\eta )(\lambda +\eta ){e}_{j-1}^{12}{e}_{j}^{23}\\ & & +2{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda +\eta ){e}_{j-1}^{22}{e}_{j}^{13}\\ & & +2{\rm{i}}\eta (2\lambda -3\eta )(5{\lambda }^{2}+3\eta \lambda -3{\eta }^{2}){e}_{j-1}^{13}{e}_{j}^{11}\\ & & -{\rm{i}}\eta (2\lambda -3\eta ){\left(2\lambda +3\eta \right)}^{2}{e}_{j-1}^{33}{e}_{j}^{21}\\ & & +2{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda -\eta ){e}_{j-1}^{13}{e}_{j}^{22}\\ & & +2{\rm{i}}\eta \lambda (2\lambda -3\eta )(\lambda -\eta ){e}_{j-1}^{23}{e}_{j}^{12}\\ & & +2{\rm{i}}\eta (2\lambda +3\eta )(3{\lambda }^{2}-4\eta \lambda +3{\eta }^{2}){e}_{j-1}^{13}{e}_{j}^{33}\\ & & +2{\rm{i}}\eta (2\lambda -3\eta )(3{\lambda }^{2}+4\eta \lambda +3{\eta }^{2}){e}_{j-1}^{33}{e}_{j}^{13},\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{rcl}{M}_{j-1,j}^{31} & = & +2{\rm{i}}\eta (2\lambda -3\eta )(5{\lambda }^{2}+3\eta \lambda -3{\eta }^{2}){e}_{j-1}^{11}{e}_{j}^{21}\\ & & -2{\rm{i}}\eta (2\lambda +3\eta )(5{\lambda }^{2}-3\eta \lambda -3{\eta }^{2}){e}_{j-1}^{21}{e}_{j}^{11}\\ & & -2{\rm{i}}\eta (2\lambda -3\eta )(3{\lambda }^{2}+4\eta \lambda +3{\eta }^{2}){e}_{j-1}^{21}{e}_{j}^{22}\\ & & -{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(5\lambda -3\eta ){e}_{j-1}^{11}{e}_{j}^{13}\\ & & -2{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda +\eta ){e}_{j-1}^{21}{e}_{j}^{33}\\ & & -2{\rm{i}}\eta \lambda (2\lambda -3\eta )(\lambda -\eta ){e}_{j-1}^{31}{e}_{j}^{23}\\ & & -2{\rm{i}}\eta (2\lambda +3\eta )(3{\lambda }^{2}-4\eta \lambda +3{\eta }^{2}){e}_{j-1}^{22}{e}_{j}^{21}\\ & & +{\rm{i}}\eta \lambda (4{\lambda }^{2}-9{\eta }^{2})({e}_{j-1}^{12}{e}_{j}^{23}+{e}_{j-1}^{23}{e}_{j}^{12})\\ & & +{\rm{i}}\eta (2\lambda +3\eta ){\left(2\lambda -3\eta \right)}^{2}{e}_{j-1}^{22}{e}_{j}^{13}\\ & & +{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(5\lambda +3\eta ){e}_{j-1}^{13}{e}_{j}^{11}\\ & & -2{\rm{i}}\eta \lambda (2\lambda +3\eta )(\lambda +\eta ){e}_{j-1}^{23}{e}_{j}^{31}\\ & & -2{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda -\eta ){e}_{j-1}^{33}{e}_{j}^{21}\\ & & +{\rm{i}}\eta (2\lambda -3\eta ){\left(2\lambda +3\eta \right)}^{2}{e}_{j-1}^{13}{e}_{j}^{22}\\ & & +3{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda +\eta ){e}_{j-1}^{13}{e}_{j}^{33}\\ & & +3{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda -\eta ){e}_{j-1}^{33}{e}_{j}^{13},\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{rcl}{M}_{j-1,j}^{13} & = & {\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(5\lambda +3\eta ){e}_{j-1}^{11}{e}_{j}^{31}\\ & & -{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(5\lambda -3\eta ){e}_{j-1}^{31}{e}_{j}^{11}\\ & & +2{\rm{i}}\eta (2\lambda +3\eta )(5{\lambda }^{2}-3\eta \lambda -3{\eta }^{2}){e}_{j-1}^{11}{e}_{j}^{12}\\ & & -{\rm{i}}\eta \lambda (4{\lambda }^{2}-9{\eta }^{2})\\ & & \times ({e}_{j-1}^{21}{e}_{j}^{32}+{e}_{j-1}^{32}{e}_{j}^{21})\\ & & -{\rm{i}}\eta (2\lambda +3\eta ){\left(2\lambda -3\eta \right)}^{2}{e}_{j-1}^{31}{e}_{j}^{22}\\ & & -3{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda -\eta ){e}_{j-1}^{31}{e}_{j}^{33}\\ & & -2{\rm{i}}\eta (2\lambda -3\eta )(5{\lambda }^{2}+3\eta \lambda -3{\eta }^{2}){e}_{j-1}^{12}{e}_{j}^{11}\\ & & -{\rm{i}}\eta (2\lambda -3\eta ){\left(2\lambda +3\eta \right)}^{2}{e}_{j-1}^{22}{e}_{j}^{31}\\ & & -2{\rm{i}}\eta (2\lambda +3\eta )(3{\lambda }^{2}-4\eta \lambda +3{\eta }^{2}){e}_{j-1}^{12}{e}_{j}^{22}\\ & & -2{\rm{i}}\eta (2\lambda -3\eta )(3{\lambda }^{2}+4\eta \lambda +3{\eta }^{2}){e}_{j-1}^{22}{e}_{j}^{12}\\ & & -2{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda -\eta ){e}_{j-1}^{12}{e}_{j}^{33}\\ & & -2{\rm{i}}\eta \lambda (2\lambda -3\eta )(\lambda -\eta ){e}_{j-1}^{32}{e}_{j}^{13}\\ & & -3{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda +\eta ){e}_{j-1}^{33}{e}_{j}^{31}\\ & & -2{\rm{i}}\eta \lambda (2\lambda +3\eta )(\lambda +\eta ){e}_{j-1}^{13}{e}_{j}^{32}\\ & & -2{\rm{i}}\eta (2\lambda +3\eta )(2\lambda -3\eta )(\lambda +\eta ){e}_{j-1}^{33}{e}_{j}^{12},\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{rcl}{M}_{j-1,j}^{32} & = & -4{\rm{i}}\eta \lambda (4{\lambda }^{2}-9{\eta }^{2})({e}_{j-1}^{21}{e}_{j}^{21}-{e}_{j-1}^{13}{e}_{j}^{13})\\ & & -{\rm{i}}{\eta }^{2}(4{\lambda }^{2}-9{\eta }^{2})({e}_{j-1}^{11}{e}_{j}^{23}+{e}_{j-1}^{23}{e}_{j}^{11})\\ & & +{\rm{i}}\eta \lambda (2\lambda +3\eta )(12\lambda -13\eta ){e}_{j-1}^{21}{e}_{j}^{13}\\ & & -{\rm{i}}\eta \lambda (2\lambda +3\eta )(4\lambda -3\eta )(\lambda +\eta )\\ & & \times ({e}_{j-1}^{22}{e}_{j}^{32}+{e}_{j-1}^{23}{e}_{j}^{33})\\ & & -{\rm{i}}\eta \lambda (2\lambda -3\eta )(12\lambda +13\eta ){e}_{j-1}^{13}{e}_{j}^{21}\\ & & +{\rm{i}}\eta \lambda (2\lambda -3\eta )(4\lambda +3\eta )(\lambda -\eta )\\ & & \times \,({e}_{j-1}^{23}{e}_{j}^{22}+{e}_{j-1}^{33}{e}_{j}^{23}),\end{array}\end{eqnarray*}$
$\begin{eqnarray}\begin{array}{rcl}{M}_{j-1,j}^{23} & = & 4{\rm{i}}\eta \lambda (4{\lambda }^{2}-9{\eta }^{2})({e}_{j-1}^{31}{e}_{j}^{31}-{e}_{j-1}^{12}{e}_{j}^{12})\\ & & -{\rm{i}}{\eta }^{2}(4{\lambda }^{2}-9{\eta }^{2})({e}_{j-1}^{11}{e}_{j}^{32}+{e}_{j-1}^{32}{e}_{j}^{11})\\ & & +{\rm{i}}\eta \lambda (2\lambda +3\eta )(12\lambda -13\eta ){e}_{j-1}^{31}{e}_{j}^{12}\\ & & -{\rm{i}}\eta \lambda (2\lambda -3\eta )(12\lambda +13\eta ){e}_{j-1}^{12}{e}_{j}^{31}\\ & & +{\rm{i}}\eta \lambda (2\lambda -3\eta )(4\lambda +3\eta )(\lambda -\eta )\\ & & \times ({e}_{j-1}^{22}{e}_{j}^{32}+{e}_{j-1}^{32}{e}_{j}^{33})\\ & & -{\rm{i}}\eta \lambda (2\lambda +3\eta )(4\lambda -3\eta )(\lambda +\eta )\\ & & \times ({e}_{j-1}^{32}{e}_{j}^{22}+{e}_{j-1}^{33}{e}_{j}^{32}).\end{array}\end{eqnarray}$