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The partitioning of Hamiltonian in left (L) and right (R) blocks is given by
(\hat{H})^{[0]} =&\ \big( \hat{H}^{L} \big)^{[0]} \otimes_{[0]} \big( \hat{1}^{R} \big)^{[0]}
+ \big( \hat{1}^{L} \big)^{[0]} \otimes_{[0]} \big( \hat{H}^{R} \big)^{[0]} \\
&\ + \sqrt{2} \sum_{i\in L} \left[ \big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( \hat{S}_{i}^{R} \big)^{[\frac{1}{2}]}
+ h.c. \right] \\
&\ + 2 \sum_{i\in L} \left[ \big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( \hat{R}_{i}^{R} \big)^{[\frac{1}{2}]}
+ h.c. \right]
+ 2 \sum_{i\in R} \left[ \big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( \hat{R}_{i}^{L} \big)^{[\frac{1}{2}]}
+ h.c. \right] \\
&\ - \frac{1}{2} \sum_{ik\in L} \left[
\sqrt{3}
\big(\hat{A}_{ik} \big)^{[1]} \otimes_{[0]}
\big(\hat{P}_{ik}^{R} \big)^{[1]}
+ \big(\hat{A}_{ik} \big)^{[0]} \otimes_{[0]}
\big(\hat{P}_{ik}^{R} \big)^{[0]} + h.c. \right] \\
&\ +\sum_{ij\in L} \left[
\big( \hat{B}_{ij} \big)^{[0]} \otimes_{[0]} \left( 2\big( \hat{Q}_{ij}^{R} \big)^{[0]}
- \big( {\hat{Q}}_{ij}^{\prime R} \big)^{[0]} \right)
+ \sqrt{3} \big( {\hat{B}'}_{ij} \big)^{[1]} \otimes_{[0]} \big( {\hat{Q}}_{ij}^{\prime R} \big)^{[1]}
\right]
where the normal and complementary operators are defined by
\big( \hat{S}_{i}^{L/R} \big)^{[\frac{1}{2}]} =&\ \sum_{j\in L/R} t_{ij} \big( a_{j} \big)^{[\frac{1}{2}]} \\
\big( \hat{R}_{i}^{L/R} \big)^{[\frac{1}{2}]} =&\ \sum_{jkl\in L/R} v_{ijkl}
\left[ \Big( a_{k}^\dagger \Big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{l} \big)^{[\frac{1}{2}]} \right]
\otimes_{[\frac{1}{2}]} \big( a_{j} \big)^{[\frac{1}{2}]} \\
\big( \hat{A}_{ik} \big)^{[0/1]} =&\
\big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0/1]} \big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \\
\big( \hat{P}_{ik}^{R} \big)^{[0/1]} =&\
\sum_{jl\in R} v_{ijkl} \big( a_{j} \big)^{[\frac{1}{2}]} \otimes_{[0/1]} \big( a_{l} \big)^{[\frac{1}{2}]} \\
\big( \hat{B}_{ij} \big)^{[0]} =&\
\big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{j} \big)^{[\frac{1}{2}]} \\
\big( {\hat{B}'}_{ij} \big)^{[1]} =&\
\big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[1]} \big( a_{j} \big)^{[\frac{1}{2}]}\\
\big( \hat{Q}_{ij}^{R} \big)^{[0]} =&\
\sum_{kl\in R} v_{ijkl}
\big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{l} \big)^{[\frac{1}{2}]} \\
\big( {\hat{Q}}_{ij}^{\prime R} \big)^{[0/1]} =&\
\sum_{kl\in R} v_{ilkj}
\big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0/1]} \big( a_{l} \big)^{[\frac{1}{2}]} \\
\big( {\hat{Q}}_{ij}^{\prime \prime R} \big)^{[0]} :=&\
2 \big( {\hat{Q}}_{ij}^{R} \big)^{[0]} - \big( {\hat{Q}}_{ij}^{\prime R} \big)^{[0]}
= \sum_{kl\in R} (2v_{ijkl} - v_{ilkj})
\big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{l} \big)^{[\frac{1}{2}]}
From j_2 = 1/2 CG factors
\bigg\langle j_1\ \left(M - \frac{1}{2} \right)\ \frac{1}{2}\ \frac{1}{2} \bigg| \left( j_1 \pm \frac{1}{2} \right)\ M
\bigg\rangle =&\ \pm \sqrt{\frac{1}{2} \left( 1 \pm \frac{M}{j_1 + \frac{1}{2}} \right)} \\
\bigg\langle j_1\ \left(M + \frac{1}{2} \right)\ \frac{1}{2}\ \left( -\frac{1}{2}\right) \bigg| \left( j_1 \pm \frac{1}{2} \right)\ M
\bigg\rangle =&\ \sqrt{\frac{1}{2} \left( 1 \mp \frac{M}{j_1 + \frac{1}{2}} \right)}
and symmetry relation
\langle j_1\ m_1\ j_2\ m_2 |J\ M\rangle = (-1)^{j_1+j_2-J} \langle j_2\ m_2\ j_1\ m_1 |J\ M\rangle
and
(-1)^{j_1+\frac{1}{2}-j_1\mp\frac{1}{2}} = (-1)^{\frac{1}{2}\mp\frac{1}{2}} = \pm 1
we have
\bigg\langle \frac{1}{2}\ \frac{1}{2}\ j_1\ \left(M - \frac{1}{2} \right) \bigg| \left( j_1 \pm \frac{1}{2} \right)\ M
\bigg\rangle =&\ \sqrt{\frac{1}{2} \left( 1 \pm \frac{M}{j_1 + \frac{1}{2}} \right)} \\
\bigg\langle \frac{1}{2}\ \left( -\frac{1}{2}\right)\ j_1\ \left(M + \frac{1}{2} \right) \bigg| \left( j_1 \pm \frac{1}{2} \right)\ M
\bigg\rangle =&\ \pm \sqrt{\frac{1}{2} \left( 1 \mp \frac{M}{j_1 + \frac{1}{2}} \right)}
let j_1 = 1, we have
\langle \tfrac{1}{2}\ \tfrac{1}{2}\ 1\ (M - \tfrac{1}{2}) | \tfrac{1}{2}\ M \rangle =&\ \sqrt{\tfrac{1}{2} ( 1-\frac{M}{\tfrac{3}{2}} )} \\
\langle \tfrac{1}{2}\ (-\tfrac{1}{2})\ 1\ (M + \tfrac{1}{2}) | \tfrac{1}{2}\ M \rangle =&\ -\sqrt{\tfrac{1}{2} ( 1+\frac{M}{\tfrac{3}{2}} )}
So the coefficients for [\tfrac{1}{2}] \otimes_{[\tfrac{1}{2}]} [1] are
[\tfrac{1}{2} + 0 = \tfrac{1}{2}] = \sqrt{\tfrac{1}{3}},\quad [-\tfrac{1}{2} + 1 = \tfrac{1}{2}] = -\sqrt{\tfrac{2}{3}} \\
[\tfrac{1}{2} + (-1) = -\tfrac{1}{2}] = \sqrt{\tfrac{2}{3}},\quad [-\tfrac{1}{2} + 0 = -\tfrac{1}{2}] = -\sqrt{\tfrac{1}{3}}
The coefficients for [1] \otimes_{[\tfrac{1}{2}]} [\tfrac{1}{2}] are
[0 + \tfrac{1}{2} = \tfrac{1}{2}] = -\sqrt{\tfrac{1}{3}},\quad [1 -\tfrac{1}{2} = \tfrac{1}{2}] = \sqrt{\tfrac{2}{3}} \\
[(-1) + \tfrac{1}{2} = -\tfrac{1}{2}] = -\sqrt{\tfrac{2}{3}},\quad [0 -\tfrac{1}{2} = -\tfrac{1}{2}] = \sqrt{\tfrac{1}{3}}
This means that the SU(2) operator exchange factor for [\tfrac{1}{2}] \otimes_{[\tfrac{1}{2}]} [1] \to [1] \otimes_{[\tfrac{1}{2}]} [\tfrac{1}{2}] is -1. The fermion factor is +1. So the overall exchange factor for this case is -1.
Singlet
\big(a_p^\dagger\big)^{[1/2]} \otimes_{[0]} \big(a_q^\dagger\big)^{[1/2]}
=&\ \begin{pmatrix} a_{p\alpha}^\dagger \\ a_{p\beta}^\dagger \end{pmatrix}^{[1/2]}
\otimes_{[0]}
\begin{pmatrix} a_{q\alpha}^\dagger \\ a_{q\beta}^\dagger \end{pmatrix}^{[1/2]}
= \frac{1}{\sqrt{2}} \begin{pmatrix} a_{p\alpha}^\dagger a_{q\beta}^\dagger - a_{p\beta}^\dagger a_{q\alpha}^\dagger
\end{pmatrix}^{[0]} \\
\big(a_p^\dagger\big)^{[1/2]} \otimes_{[0]} \big(a_q\big)^{[1/2]}
=&\ \begin{pmatrix} a_{p\alpha}^\dagger \\ a_{p\beta}^\dagger \end{pmatrix}^{[1/2]}
\otimes_{[0]}
\begin{pmatrix} -a_{q\beta} \\ a_{q\alpha} \end{pmatrix}^{[1/2]}
= \frac{1}{\sqrt{2}} \begin{pmatrix} a_{p\alpha}^\dagger a_{q\alpha}+ a_{p\beta}^\dagger a_{q\beta}
\end{pmatrix}^{[0]} \\
\big(a_p\big)^{[1/2]} \otimes_{[0]} \big(a_q\big)^{[1/2]}
=&\ \begin{pmatrix} -a_{p\beta} \\ a_{p\alpha} \end{pmatrix}^{[1/2]}
\otimes_{[0]}
\begin{pmatrix} -a_{q\beta} \\ a_{q\alpha} \end{pmatrix}^{[1/2]}
= \frac{1}{\sqrt{2}} \begin{pmatrix} -a_{p\beta} a_{q\alpha} + a_{p\alpha} a_{q\beta}
\end{pmatrix}^{[0]}
Triplet
\big(a_p^\dagger\big)^{[1/2]} \otimes_{[1]} \big(a_q^\dagger\big)^{[1/2]}
=&\ \begin{pmatrix} a_{p\alpha}^\dagger \\ a_{p\beta}^\dagger \end{pmatrix}^{[1/2]}
\otimes_{[1]}
\begin{pmatrix} a_{q\alpha}^\dagger \\ a_{q\beta}^\dagger \end{pmatrix}^{[1/2]}
= \begin{pmatrix}
a_{p\alpha}^\dagger a_{q\alpha}^\dagger \\
\frac{1}{\sqrt{2}} \Big(
a_{p\alpha}^\dagger a_{q\beta}^\dagger + a_{p\beta}^\dagger a_{q\alpha}^\dagger \Big) \\
a_{p\beta}^\dagger a_{q\beta}^\dagger
\end{pmatrix}^{[1]} \\
\big(a_p^\dagger\big)^{[1/2]} \otimes_{[1]} \big(a_q\big)^{[1/2]}
=&\ \begin{pmatrix} a_{p\alpha}^\dagger \\ a_{p\beta}^\dagger \end{pmatrix}^{[1/2]}
\otimes_{[1]}
\begin{pmatrix} -a_{q\beta} \\ a_{q\alpha} \end{pmatrix}^{[1/2]}
= \begin{pmatrix}
-a_{p\alpha}^\dagger a_{q\beta} \\
\frac{1}{\sqrt{2}} \Big(
a_{p\alpha}^\dagger a_{q\alpha} - a_{p\beta}^\dagger a_{q\beta} \Big) \\
a_{p\beta}^\dagger a_{q\alpha}
\end{pmatrix}^{[1]} \\
\big(a_p\big)^{[1/2]} \otimes_{[1]} \big(a_q\big)^{[1/2]}
=&\ \begin{pmatrix} -a_{p\beta} \\ a_{p\alpha} \end{pmatrix}^{[1/2]}
\otimes_{[1]}
\begin{pmatrix} -a_{q\beta} \\ a_{q\alpha} \end{pmatrix}^{[1/2]}
= \begin{pmatrix}
a_{p\beta} a_{q\beta} \\
-\frac{1}{\sqrt{2}} \Big( a_{p\beta} a_{q\alpha} + a_{p\alpha} a_{q\beta} \Big) \\
a_{p\alpha} a_{q\alpha}
\end{pmatrix}^{[1]}
Doublet times singlet/triplet
U^{[1/2]} = &\ \big(a_p^\dagger\big)^{[1/2]} \otimes_{[1/2]} \Big[ \big(a_r\big)^{[1/2]} \otimes_{[1]} \big(a_s\big)^{[1/2]} \Big]
= \begin{pmatrix} a_{p\alpha}^\dagger \\ a_{p\beta}^\dagger \end{pmatrix}^{[1/2]} \otimes_{[1/2]} \begin{pmatrix}
a_{r\beta} a_{s\beta} \\
-\frac{1}{\sqrt{2}} \Big( a_{r\beta} a_{s\alpha} + a_{r\alpha} a_{s\beta} \Big) \\
a_{r\alpha} a_{s\alpha}
\end{pmatrix}^{[1]} \\
=&\ \begin{pmatrix}
-\frac{1}{\sqrt{2}}\frac{1}{\sqrt{3}} a_{p\alpha}^\dagger \Big( a_{r\beta} a_{s\alpha} + a_{r\alpha} a_{s\beta} \Big)
-\frac{\sqrt{2}}{\sqrt{3}} a_{p\beta}^\dagger a_{r\beta} a_{s\beta} \\
\frac{\sqrt{2}}{\sqrt{3}} a_{p\alpha}^\dagger a_{r\alpha} a_{s\alpha}
+\big( -\frac{1}{\sqrt{3}}\big) \big( -\frac{1}{\sqrt{2}} \big) a_{p\beta}^\dagger \Big( a_{r\beta} a_{s\alpha} + a_{r\alpha} a_{s\beta} \Big)
\end{pmatrix}^{[1/2]}
= \frac{1}{\sqrt{6}} \begin{pmatrix}
- a_{p\alpha}^\dagger a_{r\beta} a_{s\alpha} - a_{p\alpha}^\dagger a_{r\alpha} a_{s\beta}
-2 a_{p\beta}^\dagger a_{r\beta} a_{s\beta} \\
2 a_{p\alpha}^\dagger a_{r\alpha} a_{s\alpha}
+a_{p\beta}^\dagger a_{r\beta} a_{s\alpha} + a_{p\beta}^\dagger a_{r\alpha} a_{s\beta} \end{pmatrix}^{[1/2]} \\
V^{[1/2]} =&\ \big(a_p^\dagger\big)^{[1/2]} \otimes_{[1/2]} \Big[ \big(a_r\big)^{[1/2]} \otimes_{[0]} \big(a_s\big)^{[1/2]} \Big]
= \frac{1}{\sqrt{2}} \begin{pmatrix} a_{p\alpha}^\dagger \\ a_{p\beta}^\dagger \end{pmatrix}^{[1/2]} \otimes_{[1/2]}
\begin{pmatrix} -a_{r\beta} a_{s\alpha} + a_{r\alpha} a_{s\beta}
\end{pmatrix}^{[0]} \\
=&\ \frac{1}{\sqrt{2}}
\begin{pmatrix} -a_{p\alpha}^\dagger a_{r\beta} a_{s\alpha} + a_{p\alpha}^\dagger a_{r\alpha} a_{s\beta}\\
-a_{p\beta}^\dagger a_{r\beta} a_{s\alpha} + a_{p\beta}^\dagger a_{r\alpha} a_{s\beta}\end{pmatrix}^{[1/2]}
Therefore,
\sqrt{3} U^{[1/2]} - V^{[1/2]} =&\ \frac{1}{\sqrt{2}} \begin{pmatrix}
- a_{p\alpha}^\dagger a_{r\beta} a_{s\alpha} - a_{p\alpha}^\dagger a_{r\alpha} a_{s\beta}
-2 a_{p\beta}^\dagger a_{r\beta} a_{s\beta} \\
2 a_{p\alpha}^\dagger a_{r\alpha} a_{s\alpha}
+a_{p\beta}^\dagger a_{r\beta} a_{s\alpha} + a_{p\beta}^\dagger a_{r\alpha} a_{s\beta} \end{pmatrix}^{[1/2]}
- \frac{1}{\sqrt{2}}
\begin{pmatrix} -a_{p\alpha}^\dagger a_{r\beta} a_{s\alpha} + a_{p\alpha}^\dagger a_{r\alpha} a_{s\beta}\\
-a_{p\beta}^\dagger a_{r\beta} a_{s\alpha} + a_{p\beta}^\dagger a_{r\alpha} a_{s\beta}\end{pmatrix}^{[1/2]} \\
=&\ \frac{1}{\sqrt{2}}
\begin{pmatrix}
-a_{p\alpha}^\dagger a_{r\beta} a_{s\alpha} - a_{p\alpha}^\dagger a_{r\alpha} a_{s\beta} -2 a_{p\beta}^\dagger a_{r\beta} a_{s\beta}
+a_{p\alpha}^\dagger a_{r\beta} a_{s\alpha} - a_{p\alpha}^\dagger a_{r\alpha} a_{s\beta}\\
2 a_{p\alpha}^\dagger a_{r\alpha} a_{s\alpha} +a_{p\beta}^\dagger a_{r\beta} a_{s\alpha} + a_{p\beta}^\dagger a_{r\alpha} a_{s\beta}
+a_{p\beta}^\dagger a_{r\beta} a_{s\alpha} - a_{p\beta}^\dagger a_{r\alpha} a_{s\beta}\end{pmatrix}^{[1/2]} \\
=&\ \sqrt{2}
\begin{pmatrix}
- a_{p\alpha}^\dagger a_{r\alpha} a_{s\beta} - a_{p\beta}^\dagger a_{r\beta} a_{s\beta} \\
a_{p\alpha}^\dagger a_{r\alpha} a_{s\alpha} + a_{p\beta}^\dagger a_{r\beta} a_{s\alpha}
\end{pmatrix}^{[1/2]}
Another case
S^{[1/2]} = &\ \big(a_r\big)^{[1/2]} \otimes_{[1/2]} \Big[ \big(a_p^\dagger \big)^{[1/2]} \otimes_{[1]} \big(a_q\big)^{[1/2]} \Big]
= \begin{pmatrix} -a_{r\beta} \\ a_{r\alpha} \end{pmatrix}^{[1/2]} \otimes_{[1/2]}
\begin{pmatrix}
-a_{p\alpha}^\dagger a_{q\beta} \\
\frac{1}{\sqrt{2}} \Big( a_{p\alpha}^\dagger a_{q\alpha} - a_{p\beta}^\dagger a_{q\beta} \Big) \\
a_{p\beta}^\dagger a_{q\alpha}
\end{pmatrix}^{[1]} \\
=&\ \begin{pmatrix}
\frac{1}{\sqrt{2}} \frac{1}{\sqrt{3}} (-a_{r\beta}) \Big( a_{p\alpha}^\dagger a_{q\alpha} - a_{p\beta}^\dagger a_{q\beta} \Big)
+\frac{\sqrt{2}}{\sqrt{3}} a_{r\alpha} a_{p\alpha}^\dagger a_{q\beta} \\
-\frac{\sqrt{2}}{\sqrt{3}} a_{r\beta} a_{p\beta}^\dagger a_{q\alpha}
-\frac{1}{\sqrt{2}} \frac{1}{\sqrt{3}} a_{r\alpha} \Big( a_{p\alpha}^\dagger a_{q\alpha} - a_{p\beta}^\dagger a_{q\beta} \Big)
\end{pmatrix}^{[1/2]}
= \frac{1}{\sqrt{6}} \begin{pmatrix}
-a_{r\beta} a_{p\alpha}^\dagger a_{q\alpha} + a_{r\beta} a_{p\beta}^\dagger a_{q\beta} +2 a_{r\alpha} a_{p\alpha}^\dagger a_{q\beta}\\
-2a_{r\beta} a_{p\beta}^\dagger a_{q\alpha} -a_{r\alpha} a_{p\alpha}^\dagger a_{q\alpha} + a_{r\alpha} a_{p\beta}^\dagger a_{q\beta}
\end{pmatrix}^{[1/2]} \\
T^{[1/2]} = &\ \big(a_r\big)^{[1/2]} \otimes_{[1/2]} \Big[ \big(a_p^\dagger \big)^{[1/2]} \otimes_{[0]} \big(a_q\big)^{[1/2]} \Big]
= \frac{1}{\sqrt{2}} \begin{pmatrix} -a_{r\beta} \\ a_{r\alpha} \end{pmatrix}^{[1/2]} \otimes_{[1/2]}
\begin{pmatrix} a_{p\alpha}^\dagger a_{q\alpha}+ a_{p\beta}^\dagger a_{q\beta} \end{pmatrix}^{[0]} \\
=&\ \frac{1}{\sqrt{2}}
\begin{pmatrix} -a_{r\beta} a_{p\alpha}^\dagger a_{q\alpha} - a_{r\beta}a_{p\beta}^\dagger a_{q\beta} \\
a_{r\alpha} a_{p\alpha}^\dagger a_{q\alpha} + a_{r\alpha}a_{p\beta}^\dagger a_{q\beta}\end{pmatrix}^{[1/2]}
Therefore,
\sqrt{3} S^{[1/2]} - T^{[1/2]} =&\
\frac{1}{\sqrt{6}} \begin{pmatrix}
-a_{r\beta} a_{p\alpha}^\dagger a_{q\alpha} + a_{r\beta} a_{p\beta}^\dagger a_{q\beta} +2 a_{r\alpha} a_{p\alpha}^\dagger a_{q\beta}\\
-2a_{r\beta} a_{p\beta}^\dagger a_{q\alpha} -a_{r\alpha} a_{p\alpha}^\dagger a_{q\alpha} + a_{r\alpha} a_{p\beta}^\dagger a_{q\beta}
\end{pmatrix}^{[1/2]}-\frac{1}{\sqrt{2}}
\begin{pmatrix} -a_{r\beta} a_{p\alpha}^\dagger a_{q\alpha} - a_{r\beta}a_{p\beta}^\dagger a_{q\beta} \\
a_{r\alpha} a_{p\alpha}^\dagger a_{q\alpha} + a_{r\alpha}a_{p\beta}^\dagger a_{q\beta}\end{pmatrix}^{[1/2]} \\
=&\ \frac{1}{\sqrt{2}}
\begin{pmatrix}
-a_{r\beta} a_{p\alpha}^\dagger a_{q\alpha} + a_{r\beta} a_{p\beta}^\dagger a_{q\beta} +2 a_{r\alpha} a_{p\alpha}^\dagger a_{q\beta}
+a_{r\beta} a_{p\alpha}^\dagger a_{q\alpha} + a_{r\beta}a_{p\beta}^\dagger a_{q\beta} \\
-2a_{r\beta} a_{p\beta}^\dagger a_{q\alpha} -a_{r\alpha} a_{p\alpha}^\dagger a_{q\alpha} + a_{r\alpha} a_{p\beta}^\dagger a_{q\beta}
-a_{r\alpha} a_{p\alpha}^\dagger a_{q\alpha} - a_{r\alpha}a_{p\beta}^\dagger a_{q\beta}
\end{pmatrix}^{[1/2]} \\
=&\ \sqrt{2}
\begin{pmatrix}
a_{r\beta}a_{p\beta}^\dagger a_{q\beta} +a_{r\alpha} a_{p\alpha}^\dagger a_{q\beta} \\
-a_{r\beta} a_{p\beta}^\dagger a_{q\alpha} -a_{r\alpha} a_{p\alpha}^\dagger a_{q\alpha}
\end{pmatrix}^{[1/2]}
Triplet times triplet
X^{[0]} = &\ \Big[ \big(a_p^\dagger\big)^{[1/2]} \otimes_{[1]} \big(a_q^\dagger\big)^{[1/2]} \Big]
\otimes_{[0]}
\Big[ \big(a_r\big)^{[1/2]} \otimes_{[1]} \big(a_s\big)^{[1/2]} \Big] \\
=&\ \begin{pmatrix}
a_{p\alpha}^\dagger a_{q\alpha}^\dagger \\
\frac{1}{\sqrt{2}} \Big(
a_{p\alpha}^\dagger a_{q\beta}^\dagger + a_{p\beta}^\dagger a_{q\alpha}^\dagger \Big) \\
a_{p\beta}^\dagger a_{q\beta}^\dagger
\end{pmatrix}^{[1]}
\otimes_{[0]}
\begin{pmatrix}
a_{r\beta} a_{s\beta} \\
-\frac{1}{\sqrt{2}} \Big( a_{r\beta} a_{s\alpha} + a_{r\alpha} a_{s\beta} \Big) \\
a_{r\alpha} a_{s\alpha}
\end{pmatrix}^{[1]} \\
=&\ \frac{1}{\sqrt{3}} \begin{pmatrix}
a_{p\alpha}^\dagger a_{q\alpha}^\dagger a_{r\alpha} s_{s\alpha}
+ \frac{1}{2} \Big(
a_{p\alpha}^\dagger a_{q\beta}^\dagger + a_{p\beta}^\dagger a_{q\alpha}^\dagger \Big)
\Big( a_{r\beta} a_{s\alpha} + a_{r\alpha} a_{s\beta} \Big)
+ a_{p\beta}^\dagger a_{q\beta}^\dagger a_{r\beta} a_{s\beta}
\end{pmatrix} \\
Y^{[0]} = &\ \Big[ \big(a_p^\dagger\big)^{[1/2]} \otimes_{[0]} \big(a_q^\dagger\big)^{[1/2]} \Big]
\otimes_{[0]}
\Big[ \big(a_r\big)^{[1/2]} \otimes_{[0]} \big(a_s\big)^{[1/2]} \Big] \\
=&\ \frac{1}{\sqrt{2}} \begin{pmatrix} a_{p\alpha}^\dagger a_{q\beta}^\dagger - a_{p\beta}^\dagger a_{q\alpha}^\dagger
\end{pmatrix}^{[0]} \otimes_{[0]}
\frac{1}{\sqrt{2}} \begin{pmatrix} -a_{r\beta} a_{s\alpha} + a_{r\alpha} a_{s\beta}
\end{pmatrix}^{[0]} \\
=&\ \frac{1}{2} \Big( a_{p\alpha}^\dagger a_{q\beta}^\dagger - a_{p\beta}^\dagger a_{q\alpha}^\dagger \Big)
\Big( -a_{r\beta} a_{s\alpha} + a_{r\alpha} a_{s\beta} \Big)
Using
(a+b)(c+d) + (a-b)(-c+d) = (a+b)(2d) -2b(-c+d) = 2 (ad+bc)
we have
\sqrt{3} X^{[0]} + Y^{[0]} =&\
a_{p\alpha}^\dagger a_{q\alpha}^\dagger a_{r\alpha} s_{s\alpha}
+ a_{p\beta}^\dagger a_{q\beta}^\dagger a_{r\beta} a_{s\beta}
+ a_{p\alpha}^\dagger a_{q\beta}^\dagger a_{r\alpha} a_{s\beta}
+ a_{p\beta}^\dagger a_{q\alpha}^\dagger a_{r\beta} a_{s\alpha} \\
=&\ \sum_{\sigma\sigma'} a_{p\sigma}^\dagger a_{q\sigma'}^\dagger a_{r\sigma} s_{s\sigma'}
Another case
Z^{[0]} = &\ \Big[ \big(a_p^\dagger\big)^{[1/2]} \otimes_{[1]} \big(a_q\big)^{[1/2]} \Big]
\otimes_{[0]}
\Big[ \big(a_r^\dagger \big)^{[1/2]} \otimes_{[1]} \big(a_s\big)^{[1/2]} \Big] \\
=&\ \begin{pmatrix}
-a_{p\alpha}^\dagger a_{q\beta} \\
\frac{1}{\sqrt{2}} \Big(
a_{p\alpha}^\dagger a_{q\alpha} - a_{p\beta}^\dagger a_{q\beta} \Big) \\
a_{p\beta}^\dagger a_{q\alpha}
\end{pmatrix}^{[1]}
\otimes_{[0]}
\begin{pmatrix}
-a_{r\alpha}^\dagger a_{s\beta} \\
\frac{1}{\sqrt{2}} \Big(
a_{r\alpha}^\dagger a_{s\alpha} - a_{r\beta}^\dagger a_{s\beta} \Big) \\
a_{r\beta}^\dagger a_{s\alpha}
\end{pmatrix}^{[1]} \\
=&\ \frac{1}{\sqrt{3}} \begin{pmatrix}
-a_{p\alpha}^\dagger a_{q\beta} a_{r\beta}^\dagger a_{s\alpha}
-\frac{1}{2} \Big(
a_{p\alpha}^\dagger a_{q\alpha} - a_{p\beta}^\dagger a_{q\beta} \Big)
\Big(
a_{r\alpha}^\dagger a_{s\alpha} - a_{r\beta}^\dagger a_{s\beta} \Big)
- a_{p\beta}^\dagger a_{q\alpha} a_{r\alpha}^\dagger a_{s\beta}
\end{pmatrix} \\
W^{[0]} =&\
\Big[ \big(a_p^\dagger\big)^{[1/2]} \otimes_{[0]} \big(a_q\big)^{[1/2]} \Big]
\otimes_{[0]}
\Big[ \big(a_r^\dagger \big)^{[1/2]} \otimes_{[0]} \big(a_s\big)^{[1/2]} \Big] \\
=&\ \frac{1}{\sqrt{2}} \begin{pmatrix} a_{p\alpha}^\dagger a_{q\alpha}+ a_{p\beta}^\dagger a_{q\beta}
\end{pmatrix}^{[0]} \otimes_{[0]}
\frac{1}{\sqrt{2}} \begin{pmatrix} a_{r\alpha}^\dagger a_{s\alpha}+ a_{r\beta}^\dagger a_{s\beta}
\end{pmatrix}^{[0]} \\
=&\ \frac{1}{2} \Big( a_{p\alpha}^\dagger a_{q\alpha}+ a_{p\beta}^\dagger a_{q\beta}\Big)
\Big( a_{r\alpha}^\dagger a_{s\alpha}+ a_{r\beta}^\dagger a_{s\beta} \Big)
Using
(a-b)(c-d) + (a+b)(c+d) = (a+b)(2c) - (2b)(c-d) = 2(ac+bd)
we have
-\sqrt{3} Z^{[0]} + W^{[0]} =&\
a_{p\alpha}^\dagger a_{q\beta} a_{r\beta}^\dagger a_{s\alpha}
+ a_{p\beta}^\dagger a_{q\alpha} a_{r\alpha}^\dagger a_{s\beta}
+ a_{p\alpha}^\dagger a_{q\alpha} a_{r\alpha}^\dagger a_{s\alpha}
+ a_{p\beta}^\dagger a_{q\beta} a_{r\beta}^\dagger a_{s\beta} \\
=&\ \sum_{\sigma\sigma'} a_{p\sigma}^\dagger a_{q\sigma'} a_{r\sigma'}^\dagger a_{s\sigma}
From second singlet formula we have
\sqrt{2} \sum_{i\in L} \big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( \hat{S}_{i}^{R} \big)^{[\frac{1}{2}]}
= \sum_{i\in L} \big( t_{ij} a_{i\alpha}^\dagger a_{j\alpha} + t_{ij} a_{i\beta}^\dagger a_{j\beta} \big)
This is the same as the S term. Note that in the expression for \hat{R}, we have a \otimes_{[0]}, this is because in the original spatial expression there is a summation over \sigma. Then there is a [0] \otimes_{[1/2]} [1/2], which will not produce any extra coefficients.
Using definition
\big( \hat{A}_{ik} \big)^{[0/1]} =&\
\big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0/1]} \big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \\
\big( \hat{P}_{ik}^{R} \big)^{[0/1]} =&\
-\sum_{jl\in R} v_{ijkl} \big( a_{j} \big)^{[\frac{1}{2}]} \otimes_{[0/1]} \big( a_{l} \big)^{[\frac{1}{2}]}
We have
&\ \sum_{ik\in L} \left[ \sqrt{3} \big(\hat{A}_{ik} \big)^{[1]} \otimes_{[0]}
\big(\hat{P}_{ik}^{R} \big)^{[1]} + \big(\hat{A}_{ik} \big)^{[0]} \otimes_{[0]} \big(\hat{P}_{ik}^{R} \big)^{[0]} \right] \\
=&\ \sum_{ik\in L,jl\in R} v_{ijkl} \left[ \sqrt{3}
\left[ \big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[1]} \big( a_{k}^\dagger \big)^{[\frac{1}{2}]}\right]
\otimes_{[0]} \left[ \big( a_{j} \big)^{[\frac{1}{2}]} \otimes_{[1]} \big( a_{l} \big)^{[\frac{1}{2}]} \right]
+ \left[ \big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{k}^\dagger \big)^{[\frac{1}{2}]}\right]
\otimes_{[0]} \left[ \big( a_{j} \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{l} \big)^{[\frac{1}{2}]} \right]
\right] \\
=&\ \sum_{ik\in L,jl\in R} v_{ijkl} \left[ \sum_{\sigma\sigma'} a_{i\sigma}^\dagger a_{k\sigma'}^\dagger
a_{j\sigma} a_{l\sigma'} \right]
= -\sum_{ik\in L,jl\in R,\sigma\sigma'} v_{ijkl} a_{i\sigma}^\dagger a_{k\sigma'}^\dagger a_{l\sigma'} a_{j\sigma}
Note that in last step, we can anticommute a_{l\sigma'}, a_{j\sigma} because it's assumed that in the \sigma summation, when j=l, \sigma \neq \sigma'. Otherwise there will be two a operators acting on the same site and the contribution is zero.
In spatial expression, this term is BQ - B'Q'. Now -\sqrt{3} Z^{[0]} + W^{[0]} gives B'Q'. And 2 W^{[0]} gives BQ. Therefore,
2 W^{[0]} - \big(-\sqrt{3} Z^{[0]} + W^{[0]}\big) = \sqrt{3} Z^{[0]} + W^{[0]}
This looks like \hat{A}\hat{P} term, but without \frac{1}{2} and h.c.. But this is not correct, because the definition of Q, Q' is not equivalent due to the index order in v_{ijkl}. So they will give different W^{[0]}. Instead we have (note that \big( \hat{B}_{ij} \big)^{[0]} = \big( {\hat{B}'}_{ij} \big)^{[0]})
&\ \sum_{ij\in L} \left[
2\Big( \hat{B}_{ij} \Big)^{[0]} \otimes_{[0]} \Big( \hat{Q}_{ij}^{R} \Big)^{[0]}
- \Big( {\hat{B}'}_{ij} \Big)^{[0]} \otimes_{[0]} \Big( {\hat{Q}'}_{ij}^{R} \Big)^{[0]}
+ \sqrt{3} \Big( {\hat{B}'}_{ij} \Big)^{[1]} \otimes_{[0]} \Big( {\hat{Q}'}_{ij}^{R} \Big)^{[1]}
\right] \\
=&\ \sum_{ij\in L} \left[
\Big( \hat{B}_{ij} \Big)^{[0]} \otimes_{[0]} \left( \Big( 2\hat{Q}_{ij}^{R} \Big)^{[0]}
- \Big( {\hat{Q}'}_{ij}^{R} \Big)^{[0]} \right)
+ \sqrt{3} \Big( {\hat{B}'}_{ij} \Big)^{[1]} \otimes_{[0]} \Big( {\hat{Q}'}_{ij}^{R} \Big)^{[1]}
\right]
Note that B, Q do not have [1] form.
Note that
\sqrt{2} \sum_{i\in L} \left[ \big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( \hat{S}_{i}^{R} \big)^{[\frac{1}{2}]}
+ h.c. \right]
= \sqrt{2} \sum_{i\in R} \left[ \big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( \hat{S}_{i}^{L} \big)^{[\frac{1}{2}]}
+ h.c. \right]
Therefore,
&\ \sqrt{2} \sum_{i\in L} \left[ \big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( \hat{S}_{i}^{R} \big)^{[\frac{1}{2}]}
+ h.c. \right]
+ 2 \sum_{i\in L} \left[ \big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( \hat{R}_{i}^{R} \big)^{[\frac{1}{2}]}
+ h.c. \right]
+ 2 \sum_{i\in R} \left[ \big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( \hat{R}_{i}^{L} \big)^{[\frac{1}{2}]}
+ h.c. \right] \\
=&\ \frac{\sqrt{2}}{2} \sum_{i\in L} \left[ \big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( \hat{S}_{i}^{R} \big)^{[\frac{1}{2}]}
+ h.c. \right]
+ \frac{\sqrt{2}}{2} \sum_{i\in R} \left[ \big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( \hat{S}_{i}^{L} \big)^{[\frac{1}{2}]}
+ h.c. \right] \\
&\ + 2 \sum_{i\in L} \left[ \big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( \hat{R}_{i}^{R} \big)^{[\frac{1}{2}]}
+ h.c. \right]
+ 2 \sum_{i\in R} \left[ \big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( \hat{R}_{i}^{L} \big)^{[\frac{1}{2}]}
+ h.c. \right] \\
=&\ 2 \sum_{i\in L} \left[ \big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]}
\Big[ \big( \hat{R}_{i}^{R} \big)^{[\frac{1}{2}]} + \frac{\sqrt{2}}{4}
\big( \hat{S}_{i}^{R} \big)^{[\frac{1}{2}]} \Big]
+ h.c. \right]
+ 2 \sum_{i\in R} \left[ \big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]}
\Big[ \big( \hat{R}_{i}^{L} \big)^{[\frac{1}{2}]} + \frac{\sqrt{2}}{4}
\big( \hat{S}_{i}^{L} \big)^{[\frac{1}{2}]} \Big]
+ h.c. \right]
So define
\big( \hat{R}_{i}^{\prime L/R} \big)^{[\frac{1}{2}]} :=
\frac{\sqrt{2}}{4} \big( \hat{S}_{i}^{L} \big)^{[\frac{1}{2}]}
+ \big( \hat{R}_{i}^{L} \big)^{[\frac{1}{2}]} =
\frac{\sqrt{2}}{4} \sum_{j\in L/R} t_{ij} \big( a_{j} \big)^{[\frac{1}{2}]} + \sum_{jkl\in L/R} v_{ijkl}
\left[ \Big( a_{k}^\dagger \Big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{l} \big)^{[\frac{1}{2}]} \right]
\otimes_{[\frac{1}{2}]} \big( a_{j} \big)^{[\frac{1}{2}]}
Here \frac{\sqrt{2}}{4} should be understood as \frac{1}{2} \cdot \frac{1}{\sqrt{2}}. The \frac{1}{2} is the same as spatial case, and \frac{1}{\sqrt{2}} is because the expected \sqrt{2} factor is not added for the \hat{R} term.
Here we consider fermion and SU(2) exchange factors together. From j_2 = 1/2 CG factors
\bigg\langle j_1\ \left(M - \frac{1}{2} \right)\ \frac{1}{2}\ \frac{1}{2} \bigg| \left( j_1 \pm \frac{1}{2} \right)\ M
\bigg\rangle =&\ \pm \sqrt{\frac{1}{2} \left( 1 \pm \frac{M}{j_1 + \frac{1}{2}} \right)} \\
\bigg\langle j_1\ \left(M + \frac{1}{2} \right)\ \frac{1}{2}\ \left( -\frac{1}{2}\right) \bigg| \left( j_1 \pm \frac{1}{2} \right)\ M
\bigg\rangle =&\ \sqrt{\frac{1}{2} \left( 1 \mp \frac{M}{j_1 + \frac{1}{2}} \right)}
Let j_1 = \frac{1}{2} we have
\bigg\langle \frac{1}{2}\ \left( - \frac{1}{2} \right)\ \frac{1}{2}\ \frac{1}{2} \bigg| \left( \frac{1}{2} \pm \frac{1}{2} \right)\ 0
\bigg\rangle =&\ \pm \sqrt{\frac{1}{2} } \\
\bigg\langle \frac{1}{2} \ \frac{1}{2} \ \frac{1}{2}\ \left( -\frac{1}{2}\right) \bigg| \left( \frac{1}{2} \pm \frac{1}{2} \right)\ 0
\bigg\rangle =&\ \sqrt{\frac{1}{2} }
The exchange factor formula is
\left( \hat{X}_1^{[S_1]} \otimes_{[S]} \hat{X}_2^{[S_2]} \right)^{[S_z]}
=&\ \sum_{S_{1z},S_{2z}} \hat{X}_1^{[S_1][S_{1z}]} \hat{X}_2^{[S_2][S_{2z}]}
\langle SS_z| S_1S_{1z},\ S_2 S_{2z} \rangle \\
=&\ \mathrm{P}_{\mathrm{fermi}}^{\mathrm{exchange}}(N_1,N_2)
\sum_{S_{1z},S_{2z}} \hat{X}_2^{[S_2][S_{2z}]} \hat{X}_1^{[S_1][S_{1z}]}
\langle SS_z| S_1S_{1z},\ S_2 S_{2z} \rangle \\
=&\ \mathrm{P}_{\mathrm{fermi}}^{\mathrm{exchange}}(N_1,N_2)
\frac{\langle SS_z| S_1S_{1z},\ S_2 S_{2z} \rangle}
{\langle SS_z| S_2S_{2z},\ S_1 S_{1z} \rangle}
\left( \hat{X}_2^{[S_2]} \otimes_{[S]} \hat{X}_1^{[S_1]} \right)^{[S_z]} \\
\hat{X}_1^{[S_1]} \otimes_{[S]} \hat{X}_2^{[S_2]}
=&\ \mathrm{P}_{\mathrm{fermi}}^{\mathrm{exchange}}(N_1,N_2)
\mathrm{P}_{\mathrm{SU(2)}}^{\mathrm{exchange}}(S_1, S_2, S)
\hat{X}_2^{[S_2]} \otimes_{[S]} \hat{X}_1^{[S_1]}
For [1/2] \otimes_{[0]} [1/2], this is
\mathrm{P}^{\mathrm{exchange}}(\tfrac{1}{2}, \tfrac{1}{2}, 0) = (-1) \frac{\big\langle \frac{1}{2} \ \frac{1}{2} \ \frac{1}{2}\ \left( -\frac{1}{2}\right) \big| 0\ 0
\big\rangle}{\big\langle \frac{1}{2} \ \left( -\frac{1}{2}\right) \ \frac{1}{2}\ \frac{1}{2} \big| 0\ 0
\big\rangle} = (-1) \frac{\sqrt{\frac{1}{2}}}{-\sqrt{\frac{1}{2}}} = 1
For [1/2] \otimes_{[1]} [1/2], this is
\mathrm{P}^{\mathrm{exchange}}(\tfrac{1}{2}, \tfrac{1}{2}, 1) = (-1) \frac{\big\langle \frac{1}{2} \ \frac{1}{2} \ \frac{1}{2}\ \left( -\frac{1}{2}\right) \big| 1\ 0
\big\rangle}{\big\langle \frac{1}{2} \ \left( -\frac{1}{2}\right) \ \frac{1}{2}\ \frac{1}{2} \big| 1\ 0
\big\rangle} = (-1) \frac{\sqrt{\frac{1}{2}}}{\sqrt{\frac{1}{2}}} = -1
From CG factors
\langle 1\ m_1 \ 1 \ (-m_1) | 0 \ 0 \rangle = \frac{(-1)^{1-m_1}}{\sqrt{3}}
we have
\mathrm{P}^{\mathrm{exchange}}(1, 1, 0) = (+1) \frac{\big\langle 1 \ 1 \ \ 1\ -1 \big| 0\ 0
\big\rangle}{\big\langle 1 \ -1 \ 1\ 1 \big| 0\ 0
\big\rangle} = (+1) \frac{\frac{(-1)^{0}}{\sqrt{3}}}{\frac{(-1)^{2}}{\sqrt{3}}} = 1
we have
(\hat{H})^{[0], NC} =&\ \big( \hat{H}^{L} \big)^{[0]} \otimes_{[0]} \big( \hat{1}^{R} \big)^{[0]}
+ \big( \hat{1}^{L} \big)^{[0]} \otimes_{[0]} \big( \hat{H}^{R} \big)^{[0]} \\
&\ + 2 \sum_{i\in L} \left[ \big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( \hat{R}_{i}^{\prime R} \big)^{[\frac{1}{2}]}
+ \big( a_{i}\big)^{[\frac{1}{2}]} \otimes_{[0]} \big( \hat{R}_{i}^{\prime R\dagger} \big)^{[\frac{1}{2}]} \right]
+ 2 \sum_{i\in R} \left[ \big( \hat{R}_{i}^{\prime L\dagger} \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{i} \big)^{[\frac{1}{2}]}
+ \big( \hat{R}_{i}^{\prime L} \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{i}^\dagger \big)^{[\frac{1}{2}]}\right] \\
&\ - \frac{1}{2} \sum_{ik\in L} \left[
\big(\hat{A}_{ik} \big)^{[0]} \otimes_{[0]} \big(\hat{P}_{ik}^{R} \big)^{[0]}
+ \sqrt{3} \big(\hat{A}_{ik} \big)^{[1]} \otimes_{[0]} \big(\hat{P}_{ik}^{R} \big)^{[1]}
+ \big(\hat{A}_{ik}^\dagger \big)^{[0]} \otimes_{[0]} \big(\hat{P}_{ik}^{R\dagger} \big)^{[0]}
+ \sqrt{3} \big(\hat{A}_{ik}^\dagger \big)^{[1]} \otimes_{[0]} \big(\hat{P}_{ik}^{R\dagger} \big)^{[1]}
\right] \\
&\ +\sum_{ij\in L} \left[
\big( \hat{B}_{ij} \big)^{[0]} \otimes_{[0]} \big( {\hat{Q}}_{ij}^{\prime\prime R} \big)^{[0]}
+ \sqrt{3} \big( {\hat{B}'}_{ij} \big)^{[1]} \otimes_{[0]} \big( {\hat{Q}}_{ij}^{\prime R} \big)^{[1]}
\right]
With this normal/complementary partitioning, the operators required in left block are
\big\{ \big( \hat{H}^L \big)^{[0]}, \big( \hat{1}^{L} \big)^{[0]}, \big( a_{i}^\dagger \big)^{[\frac{1}{2}]}, \big( a_{i} \big)^{[\frac{1}{2}]},
\big( \hat{R}_{k}^{\prime L\dagger} \big)^{[\frac{1}{2}]}, \big( \hat{R}_{k}^{\prime L} \big)^{[\frac{1}{2}]},
\big(\hat{A}_{ij} \big)^{[0]}, \big(\hat{A}_{ij} \big)^{[1]}, \big(\hat{A}_{ij}^\dagger \big)^{[0]}, \big(\hat{A}_{ij}^\dagger \big)^{[1]},
\big( \hat{B}_{ij} \big)^{[0]}, \big( {\hat{B}'}_{ij} \big)^{[1]}
\big\}\quad (i,j\in L, k\in R)
The operators required in right block are
\big\{ \big( \hat{1}^{R} \big)^{[0]}, \big( \hat{H}^{R} \big)^{[0]}, \big( \hat{R}_{i}^{\prime R} \big)^{[\frac{1}{2}]},
\big( \hat{R}_{i}^{\prime R\dagger} \big)^{[\frac{1}{2}]}, \big( a_{k} \big)^{[\frac{1}{2}]}, \big( a_{k}^\dagger \big)^{[\frac{1}{2}]},
\big(\hat{P}_{ij}^{R} \big)^{[0]}, \big(\hat{P}_{ij}^{R} \big)^{[1]}, \big(\hat{P}_{ij}^{R\dagger} \big)^{[0]},
\big(\hat{P}_{ij}^{R\dagger} \big)^{[1]}, \big( {\hat{Q}}_{ij}^{\prime\prime R} \big)^{[0]}, \big( {\hat{Q}}_{ij}^{\prime R} \big)^{[1]}
\big\}\quad (i,j\in L, k\in R)
Assuming that there are K sites in total, and K_L/K_R sites in left/right block (optimally, K_L \le K_R), the total number of operators (and also the number of terms in Hamiltonian with partition) in left or right block is
N_{NC} = 1 + 1 + 2K_L + 2K_R + 4K_L^2 + 2K_L^2 = 6K_L^2 + 2K + 2
Note that due the CG factors, exchange any \otimes_{[0]} product will not produce extra sign.
(\hat{H})^{[0], CN} =&\ \big( \hat{H}^{L} \big)^{[0]} \otimes_{[0]} \big( \hat{1}^{R} \big)^{[0]}
+ \big( \hat{1}^{L} \big)^{[0]} \otimes_{[0]} \big( \hat{H}^{R} \big)^{[0]} \\
&\ + 2 \sum_{i\in L} \left[ \big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( \hat{R}_{i}^{\prime R} \big)^{[\frac{1}{2}]}
+ \big( a_{i}\big)^{[\frac{1}{2}]} \otimes_{[0]} \big( \hat{R}_{i}^{\prime R\dagger} \big)^{[\frac{1}{2}]} \right]
+ 2 \sum_{i\in R} \left[ \big( \hat{R}_{i}^{\prime L\dagger} \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{i} \big)^{[\frac{1}{2}]}
+ \big( \hat{R}_{i}^{\prime L} \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{i}^\dagger \big)^{[\frac{1}{2}]}\right] \\
&\ - \frac{1}{2} \sum_{jl\in R} \left[
\big(\hat{P}_{jl}^{L} \big)^{[0]} \otimes_{[0]} \big(\hat{A}_{jl} \big)^{[0]}
+ \sqrt{3} \big(\hat{P}_{jl}^{L} \big)^{[1]} \otimes_{[0]} \big(\hat{A}_{jl} \big)^{[1]}
+ \big(\hat{P}_{jl}^{L\dagger} \big)^{[0]} \otimes_{[0]} \big(\hat{A}_{jl}^\dagger \big)^{[0]}
+ \sqrt{3} \big(\hat{P}_{jl}^{L\dagger} \big)^{[1]} \otimes_{[0]} \big(\hat{A}_{jl}^\dagger \big)^{[1]}
\right] \\
&\ +\sum_{kl\in R} \left[
\big( {\hat{Q}}_{kl}^{\prime\prime L} \big)^{[0]} \otimes_{[0]} \big( \hat{B}_{kl} \big)^{[0]}
+ \sqrt{3} \big( {\hat{Q}}_{kl}^{\prime L} \big)^{[1]} \otimes_{[0]} \big( {\hat{B}'}_{kl} \big)^{[1]}
\right]
Now the operators required in left block are
\big\{ \big( \hat{H}^L \big)^{[0]}, \big( \hat{1}^{L} \big)^{[0]}, \big( a_{i}^\dagger \big)^{[\frac{1}{2}]}, \big( a_{i} \big)^{[\frac{1}{2}]},
\big( \hat{R}_{k}^{\prime L\dagger} \big)^{[\frac{1}{2}]}, \big( \hat{R}_{k}^{\prime L} \big)^{[\frac{1}{2}]},
\big(\hat{P}_{kl}^{L} \big)^{[0]}, \big(\hat{P}_{kl}^{L} \big)^{[1]}, \big(\hat{P}_{kl}^{L\dagger} \big)^{[0]},
\big(\hat{P}_{kl}^{L\dagger} \big)^{[1]}, \big( {\hat{Q}}_{kl}^{\prime\prime L} \big)^{[0]}, \big( {\hat{Q}}_{kl}^{\prime L} \big)^{[1]}
\big\}\quad (k,l\in R, i\in L)
The operators required in right block are
\big\{ \big( \hat{1}^{R} \big)^{[0]}, \big( \hat{H}^{R} \big)^{[0]}, \big( \hat{R}_{i}^{\prime R} \big)^{[\frac{1}{2}]},
\big( \hat{R}_{i}^{\prime R\dagger} \big)^{[\frac{1}{2}]}, \big( a_{k} \big)^{[\frac{1}{2}]}, \big( a_{k}^\dagger \big)^{[\frac{1}{2}]},
\big(\hat{A}_{kl} \big)^{[0]}, \big(\hat{A}_{kl} \big)^{[1]}, \big(\hat{A}_{kl}^\dagger \big)^{[0]}, \big(\hat{A}_{kl}^\dagger \big)^{[1]},
\big( \hat{B}_{kl} \big)^{[0]}, \big( {\hat{B}'}_{kl} \big)^{[1]}
\big\}\quad (k,l\in R, i\in L)
The total number of operators (and also the number of terms in Hamiltonian with partition) in left or right block is
N_{CN} = 1 + 1 + 2K_L + 2K_R + 4K_R^2 + 2K_R^2 = 6K_R^2 + 2K + 2
The enlarged left/right block is denoted as L*/R*. Make sure that all L operators are to the left of * operators. (The exchange factor for this is -1 for doublet \otimes triplet and +1 doublet \otimes singlet.)
First we have
\big( \hat{R}_{i}^{L/R} \big)^{[1/2]} =&\ \sum_{jkl\in L/R} v_{ijkl}
\left[ \big( a_{k}^\dagger \big)^{[1/2]} \otimes_{[0]} \big( a_{l} \big)^{[1/2]} \right]
\otimes_{[1/2]} \big( a_{j} \big)^{[1/2]} \\
=&\ \frac{1}{\sqrt{2}} \sum_{jkl\in L/R} v_{ijkl} \begin{pmatrix} a_{k\alpha}^\dagger a_{l\alpha}+ a_{k\beta}^\dagger a_{l\beta}
\end{pmatrix}^{[0]} \otimes_{[1/2]} \big( a_{j} \big)^{[1/2]} \\
=&\ \frac{1}{\sqrt{2}} \sum_{jkl\in L/R} v_{ijkl} \begin{pmatrix}
-a_{k\alpha}^\dagger a_{l\alpha}a_{j\beta} - a_{k\beta}^\dagger a_{l\beta}a_{j\beta} \\
a_{k\alpha}^\dagger a_{l\alpha}a_{j\alpha}+ a_{k\beta}^\dagger a_{l\beta}a_{j\alpha}
\end{pmatrix}^{[1/2]}
From the formula \sqrt{3} U^{[1/2]} - V^{[1/2]} we have
\big( \hat{R}_{i}^{L/R} \big)^{[1/2]} = \frac{\sqrt{3}}{2} \sum_{jkl\in L/R} v_{ijkl}
\big(a_k^\dagger\big)^{[1/2]} \otimes_{[1/2]} \Big[ \big(a_l\big)^{[1/2]} \otimes_{[1]} \big(a_j\big)^{[1/2]} \Big]
- \frac{1}{2} \sum_{jkl\in L/R} v_{ijkl}
\big(a_k^\dagger\big)^{[1/2]} \otimes_{[1/2]} \Big[ \big(a_l\big)^{[1/2]} \otimes_{[0]} \big(a_j\big)^{[1/2]} \Big]
From the formula \sqrt{3} S^{[1/2]} - T^{[1/2]} we have (for k\neq l)
\big( \hat{R}_{i}^{L/R} \big)^{[1/2]} = \frac{\sqrt{3}}{2} \sum_{jkl\in L/R} v_{ijkl}
\big(a_l\big)^{[1/2]} \otimes_{[1/2]} \Big[ \big(a_k^\dagger\big)^{[1/2]} \otimes_{[1]} \big(a_j\big)^{[1/2]} \Big]
- \frac{1}{2} \sum_{jkl\in L/R} v_{ijkl}
\big(a_l\big)^{[1/2]} \otimes_{[1/2]} \Big[ \big(a_k^\dagger\big)^{[1/2]} \otimes_{[0]} \big(a_j\big)^{[1/2]} \Big]
We have
\big( \hat{R}_{i}^{\prime L*} \big)^{[1/2]} =&\
\big( \hat{R}_{i}^{\prime L} \big)^{[1/2]} \otimes_{[1/2]} \big( \hat{1}^* \big)^{[0]}
+ \big( \hat{1}^L \big)^{[0]} \otimes_{[1/2]} \big( \hat{R}_{i}^{\prime *} \big)^{[1/2]} \\
&\ + \sum_{j \in L} \left[ \sum_{kl\in *} v_{ijkl} \big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{l} \big)^{[\frac{1}{2}]} \right]
\otimes_{[\frac{1}{2}]} \big( a_{j} \big)^{[\frac{1}{2}]}
+ \sum_{j \in *} \left[ \sum_{kl\in L} v_{ijkl} \big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{l} \big)^{[\frac{1}{2}]} \right]
\otimes_{[\frac{1}{2}]} \big( a_{j} \big)^{[\frac{1}{2}]} \\
&\ - \frac{1}{2} \sum_{k \in L}
\big(a_k^\dagger\big)^{[1/2]} \otimes_{[1/2]} \left[ \sum_{jl\in *} v_{ijkl} \big(a_l\big)^{[1/2]} \otimes_{[0]} \big(a_j\big)^{[1/2]} \right]
+\frac{\sqrt{3}}{2} \sum_{k \in L}
\big(a_k^\dagger\big)^{[1/2]} \otimes_{[1/2]} \left[ \sum_{jl\in *} v_{ijkl} \big(a_l\big)^{[1/2]} \otimes_{[1]} \big(a_j\big)^{[1/2]} \right] \\
&\ - \frac{1}{2} \sum_{k \in *}
\big(a_k^\dagger\big)^{[1/2]} \otimes_{[1/2]} \left[ \sum_{jl\in L} v_{ijkl} \big(a_l\big)^{[1/2]} \otimes_{[0]} \big(a_j\big)^{[1/2]} \right]
+\frac{\sqrt{3}}{2} \sum_{k \in *}
\big(a_k^\dagger\big)^{[1/2]} \otimes_{[1/2]} \left[ \sum_{jl\in L} v_{ijkl} \big(a_l\big)^{[1/2]} \otimes_{[1]} \big(a_j\big)^{[1/2]} \right]\\
&\ - \frac{1}{2} \sum_{l\in L}
\big(a_l\big)^{[1/2]} \otimes_{[1/2]} \left[ \sum_{jk\in *} v_{ijkl} \big(a_k^\dagger\big)^{[1/2]} \otimes_{[0]} \big(a_j\big)^{[1/2]} \right]
+\frac{\sqrt{3}}{2} \sum_{l\in L}
\big(a_l\big)^{[1/2]} \otimes_{[1/2]} \left[ \sum_{jk\in *} v_{ijkl} \big(a_k^\dagger\big)^{[1/2]} \otimes_{[1]} \big(a_j\big)^{[1/2]} \right]\\
&\ - \frac{1}{2} \sum_{l\in *}
\big(a_l\big)^{[1/2]} \otimes_{[1/2]} \left[ \sum_{jk\in L} v_{ijkl} \big(a_k^\dagger\big)^{[1/2]} \otimes_{[0]} \big(a_j\big)^{[1/2]} \right]
+\frac{\sqrt{3}}{2} \sum_{l\in *}
\big(a_l\big)^{[1/2]} \otimes_{[1/2]} \left[ \sum_{jk\in L} v_{ijkl} \big(a_k^\dagger\big)^{[1/2]} \otimes_{[1]} \big(a_j\big)^{[1/2]} \right] \\
=&\ \big( \hat{R}_{i}^{\prime L} \big)^{[1/2]} \otimes_{[1/2]} \big( \hat{1}^* \big)^{[0]}
+ \big( \hat{1}^L \big)^{[0]} \otimes_{[1/2]} \big( \hat{R}_{i}^{\prime *} \big)^{[1/2]} \\
&\ + \sum_{j \in L} \big( a_{j} \big)^{[\frac{1}{2}]} \otimes_{[\frac{1}{2}]}
\left[ \sum_{kl\in *} v_{ijkl} \big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{l} \big)^{[\frac{1}{2}]} \right]
+ \sum_{j \in *} \left[ \sum_{kl\in L} v_{ijkl} \big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{l} \big)^{[\frac{1}{2}]} \right]
\otimes_{[\frac{1}{2}]} \big( a_{j} \big)^{[\frac{1}{2}]} \\
&\ - \frac{1}{2} \sum_{k \in L}
\big(a_k^\dagger\big)^{[1/2]} \otimes_{[1/2]} \left[ \sum_{jl\in *} v_{ijkl} \big(a_l\big)^{[1/2]} \otimes_{[0]} \big(a_j\big)^{[1/2]} \right]
+\frac{\sqrt{3}}{2} \sum_{k \in L}
\big(a_k^\dagger\big)^{[1/2]} \otimes_{[1/2]} \left[ \sum_{jl\in *} v_{ijkl} \big(a_l\big)^{[1/2]} \otimes_{[1]} \big(a_j\big)^{[1/2]} \right] \\
&\ - \frac{1}{2} \sum_{k \in *} \left[ \sum_{jl\in L} v_{ijkl} \big(a_l\big)^{[1/2]} \otimes_{[0]} \big(a_j\big)^{[1/2]} \right]
\otimes_{[1/2]} \big(a_k^\dagger\big)^{[1/2]}
-\frac{\sqrt{3}}{2} \sum_{k \in *} \left[ \sum_{jl\in L} v_{ijkl} \big(a_l\big)^{[1/2]} \otimes_{[1]} \big(a_j\big)^{[1/2]} \right]
\otimes_{[1/2]} \big(a_k^\dagger\big)^{[1/2]} \\
&\ - \frac{1}{2} \sum_{l\in L}
\big(a_l\big)^{[1/2]} \otimes_{[1/2]} \left[ \sum_{jk\in *} v_{ijkl} \big(a_k^\dagger\big)^{[1/2]} \otimes_{[0]} \big(a_j\big)^{[1/2]} \right]
+\frac{\sqrt{3}}{2} \sum_{l\in L}
\big(a_l\big)^{[1/2]} \otimes_{[1/2]} \left[ \sum_{jk\in *} v_{ijkl} \big(a_k^\dagger\big)^{[1/2]} \otimes_{[1]} \big(a_j\big)^{[1/2]} \right]\\
&\ - \frac{1}{2} \sum_{l\in *} \left[ \sum_{jk\in L} v_{ijkl} \big(a_k^\dagger\big)^{[1/2]} \otimes_{[0]} \big(a_j\big)^{[1/2]} \right]
\otimes_{[1/2]} \big(a_l\big)^{[1/2]}
-\frac{\sqrt{3}}{2} \sum_{l\in *} \left[ \sum_{jk\in L} v_{ijkl} \big(a_k^\dagger\big)^{[1/2]} \otimes_{[1]} \big(a_j\big)^{[1/2]} \right]
\otimes_{[1/2]} \big(a_l\big)^{[1/2]}
After reordering of terms
\big( \hat{R}_{i}^{\prime L*} \big)^{[1/2]} =&\
\big( \hat{R}_{i}^{\prime L} \big)^{[1/2]} \otimes_{[1/2]} \big( \hat{1}^* \big)^{[0]}
+ \big( \hat{1}^L \big)^{[0]} \otimes_{[1/2]} \big( \hat{R}_{i}^{\prime *} \big)^{[1/2]} \\
&\ - \frac{1}{2} \sum_{k \in L}
\big(a_k^\dagger\big)^{[1/2]} \otimes_{[1/2]} \left[ \sum_{jl\in *} v_{ijkl} \big(a_l\big)^{[1/2]} \otimes_{[0]} \big(a_j\big)^{[1/2]} \right]
+\frac{\sqrt{3}}{2} \sum_{k \in L}
\big(a_k^\dagger\big)^{[1/2]} \otimes_{[1/2]} \left[ \sum_{jl\in *} v_{ijkl} \big(a_l\big)^{[1/2]} \otimes_{[1]} \big(a_j\big)^{[1/2]} \right] \\
&\ + \sum_{j \in L} \big( a_{j} \big)^{[\frac{1}{2}]} \otimes_{[\frac{1}{2}]}
\left[ \sum_{kl\in *} v_{ijkl} \big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{l} \big)^{[\frac{1}{2}]} \right] \\
&\ - \frac{1}{2} \sum_{l\in L}
\big(a_l\big)^{[1/2]} \otimes_{[1/2]} \left[ \sum_{jk\in *} v_{ijkl} \big(a_k^\dagger\big)^{[1/2]} \otimes_{[0]} \big(a_j\big)^{[1/2]} \right]
+\frac{\sqrt{3}}{2} \sum_{l\in L}
\big(a_l\big)^{[1/2]} \otimes_{[1/2]} \left[ \sum_{jk\in *} v_{ijkl} \big(a_k^\dagger\big)^{[1/2]} \otimes_{[1]} \big(a_j\big)^{[1/2]} \right]\\
&\ - \frac{1}{2} \sum_{k \in *} \left[ \sum_{jl\in L} v_{ijkl} \big(a_l\big)^{[1/2]} \otimes_{[0]} \big(a_j\big)^{[1/2]} \right]
\otimes_{[1/2]} \big(a_k^\dagger\big)^{[1/2]}
-\frac{\sqrt{3}}{2} \sum_{k \in *} \left[ \sum_{jl\in L} v_{ijkl} \big(a_l\big)^{[1/2]} \otimes_{[1]} \big(a_j\big)^{[1/2]} \right]
\otimes_{[1/2]} \big(a_k^\dagger\big)^{[1/2]} \\
&\ + \sum_{j \in *} \left[ \sum_{kl\in L} v_{ijkl} \big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{l} \big)^{[\frac{1}{2}]} \right]
\otimes_{[\frac{1}{2}]} \big( a_{j} \big)^{[\frac{1}{2}]} \\
&\ - \frac{1}{2} \sum_{l\in *} \left[ \sum_{jk\in L} v_{ijkl} \big(a_k^\dagger\big)^{[1/2]} \otimes_{[0]} \big(a_j\big)^{[1/2]} \right]
\otimes_{[1/2]} \big(a_l\big)^{[1/2]}
-\frac{\sqrt{3}}{2} \sum_{l\in *} \left[ \sum_{jk\in L} v_{ijkl} \big(a_k^\dagger\big)^{[1/2]} \otimes_{[1]} \big(a_j\big)^{[1/2]} \right]
\otimes_{[1/2]} \big(a_l\big)^{[1/2]} \\
=&\ \big( \hat{R}_{i}^{\prime L} \big)^{[1/2]} \otimes_{[1/2]} \big( \hat{1}^* \big)^{[0]}
+ \big( \hat{1}^L \big)^{[0]} \otimes_{[1/2]} \big( \hat{R}_{i}^{\prime *} \big)^{[1/2]} \\
&\ - \frac{1}{2} \sum_{k \in L}
\big(a_k^\dagger\big)^{[1/2]} \otimes_{[1/2]} \left[ \sum_{jl\in *} v_{ijkl} \big(a_l\big)^{[1/2]} \otimes_{[0]} \big(a_j\big)^{[1/2]} \right]
+\frac{\sqrt{3}}{2} \sum_{k \in L}
\big(a_k^\dagger\big)^{[1/2]} \otimes_{[1/2]} \left[ \sum_{jl\in *} v_{ijkl} \big(a_l\big)^{[1/2]} \otimes_{[1]} \big(a_j\big)^{[1/2]} \right] \\
&\ + \frac{1}{2} \sum_{j\in L} \big(a_j\big)^{[1/2]} \otimes_{[1/2]} \left[ \sum_{kl\in *} (2 v_{ijkl} - v_{ilkj}) \big(a_k^\dagger\big)^{[1/2]} \otimes_{[0]} \big(a_l\big)^{[1/2]} \right]
+\frac{\sqrt{3}}{2} \sum_{l\in L}
\big(a_l\big)^{[1/2]} \otimes_{[1/2]} \left[ \sum_{jk\in *} v_{ijkl} \big(a_k^\dagger\big)^{[1/2]} \otimes_{[1]} \big(a_j\big)^{[1/2]} \right]\\
&\ - \frac{1}{2} \sum_{k \in *} \left[ \sum_{jl\in L} v_{ijkl} \big(a_l\big)^{[1/2]} \otimes_{[0]} \big(a_j\big)^{[1/2]} \right]
\otimes_{[1/2]} \big(a_k^\dagger\big)^{[1/2]}
-\frac{\sqrt{3}}{2} \sum_{k \in *} \left[ \sum_{jl\in L} v_{ijkl} \big(a_l\big)^{[1/2]} \otimes_{[1]} \big(a_j\big)^{[1/2]} \right]
\otimes_{[1/2]} \big(a_k^\dagger\big)^{[1/2]} \\
&\ + \frac{1}{2} \sum_{j\in *} \left[ \sum_{kl\in L} (2v_{ijkl} - v_{ilkj}) \big(a_k^\dagger\big)^{[1/2]} \otimes_{[0]} \big(a_l\big)^{[1/2]} \right]
\otimes_{[1/2]} \big(a_j\big)^{[1/2]}
-\frac{\sqrt{3}}{2} \sum_{l\in *} \left[ \sum_{jk\in L} v_{ijkl} \big(a_k^\dagger\big)^{[1/2]} \otimes_{[1]} \big(a_j\big)^{[1/2]} \right]
\otimes_{[1/2]} \big(a_l\big)^{[1/2]}
By definition (The overall exchange factor for [1/2] \otimes_{[0]} [1/2] is 1, and for [1/2] \otimes_{[1]} [1/2] is -1)
\big( \hat{A}_{ik} \big)^{[0/1]} =&\ \big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0/1]} \big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \\
\big( \hat{A}_{ik}^\dagger \big)^{[0]} =&\ \big( a_{i} \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{k} \big)^{[\frac{1}{2}]}
= \big( a_{k} \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{i} \big)^{[\frac{1}{2}]} \\
\big( \hat{A}_{ik}^\dagger \big)^{[1]} =&\ -\big( a_{i} \big)^{[\frac{1}{2}]} \otimes_{[1]} \big( a_{k} \big)^{[\frac{1}{2}]}
= \big( a_{k} \big)^{[\frac{1}{2}]} \otimes_{[1]} \big( a_{i} \big)^{[\frac{1}{2}]} \\
\big( \hat{P}_{ik}^{R} \big)^{[0/1]} =&\
\sum_{jl\in R} v_{ijkl} \big( a_{l} \big)^{[\frac{1}{2}]} \otimes_{[0/1]} \big( a_{j} \big)^{[\frac{1}{2}]} \\
\big( \hat{B}_{ij} \big)^{[0]} =&\
\big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{j} \big)^{[\frac{1}{2}]} \\
\big( {\hat{B}'}_{ij} \big)^{[1]} =&\
\big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[1]} \big( a_{j} \big)^{[\frac{1}{2}]}\\
\big( {\hat{Q}}_{ij}^{\prime R} \big)^{[1]} =&\
\sum_{kl\in R} v_{ilkj}
\big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[1]} \big( a_{l} \big)^{[\frac{1}{2}]} \\
\big( {\hat{Q}}_{ij}^{\prime \prime R} \big)^{[0]} =&\ \sum_{kl\in R} (2v_{ijkl} - v_{ilkj})
\big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{l} \big)^{[\frac{1}{2}]}
we have
\big( \hat{R}_{i}^{\prime L*,NC} \big)^{[1/2]} =&\
\big( \hat{R}_{i}^{\prime L} \big)^{[1/2]} \otimes_{[1/2]} \big( \hat{1}^* \big)^{[0]}
+ \big( \hat{1}^L \big)^{[0]} \otimes_{[1/2]} \big( \hat{R}_{i}^{\prime *} \big)^{[1/2]} \\
&\ - \frac{1}{2} \sum_{k \in L} \big(a_k^\dagger\big)^{[1/2]} \otimes_{[1/2]} \big( \hat{P}_{ik}^{*} \big)^{[0]}
+\frac{\sqrt{3}}{2} \sum_{k \in L} \big(a_k^\dagger\big)^{[1/2]} \otimes_{[1/2]} \big( \hat{P}_{ik}^{*} \big)^{[1]} \\
&\ + \frac{1}{2} \sum_{j\in L} \big(a_j\big)^{[1/2]} \otimes_{[1/2]} \big( {\hat{Q}}_{ij}^{\prime \prime *} \big)^{[0]}
+\frac{\sqrt{3}}{2} \sum_{l\in L} \big(a_l\big)^{[1/2]} \otimes_{[1/2]} \big( {\hat{Q}}_{il}^{\prime *} \big)^{[1]}\\
&\ - \frac{1}{2} \sum_{k \in *,jl\in L} v_{ijkl} \big( \hat{A}_{jl}^\dagger \big)^{[0]} \otimes_{[1/2]} \big(a_k^\dagger\big)^{[1/2]}
-\frac{\sqrt{3}}{2} \sum_{k \in *,jl\in L} v_{ijkl} \big( \hat{A}_{jl}^\dagger \big)^{[1]} \otimes_{[1/2]} \big(a_k^\dagger\big)^{[1/2]} \\
&\ + \frac{1}{2} \sum_{j\in *,kl\in L} (2v_{ijkl} - v_{ilkj}) \big( \hat{B}_{kl} \big)^{[0]} \otimes_{[1/2]} \big(a_j\big)^{[1/2]}
-\frac{\sqrt{3}}{2} \sum_{l\in *,jk\in L} v_{ijkl} \big( {\hat{B}'}_{kj} \big)^{[1]} \otimes_{[1/2]} \big(a_l\big)^{[1/2]} \\
\big( \hat{R}_{i}^{\prime L*,CN} \big)^{[1/2]} =&\
\big( \hat{R}_{i}^{\prime L} \big)^{[1/2]} \otimes_{[1/2]} \big( \hat{1}^* \big)^{[0]}
+ \big( \hat{1}^L \big)^{[0]} \otimes_{[1/2]} \big( \hat{R}_{i}^{\prime *} \big)^{[1/2]} \\
&\ - \frac{1}{2} \sum_{k \in L,jl\in *} v_{ijkl} \big(a_k^\dagger\big)^{[1/2]} \otimes_{[1/2]} \big( \hat{A}_{jl}^\dagger \big)^{[0]}
+\frac{\sqrt{3}}{2} \sum_{k \in L,jl\in *} v_{ijkl} \big(a_k^\dagger\big)^{[1/2]} \otimes_{[1/2]} \big( \hat{A}_{jl}^\dagger \big)^{[1]} \\
&\ + \frac{1}{2} \sum_{j\in L,kl\in *} (2 v_{ijkl} - v_{ilkj}) \big(a_j\big)^{[1/2]} \otimes_{[1/2]} \big( \hat{B}_{kl} \big)^{[0]}
+\frac{\sqrt{3}}{2} \sum_{l\in L,jk\in *} v_{ijkl} \big(a_l\big)^{[1/2]} \otimes_{[1/2]} \big( {\hat{B}'}_{kj} \big)^{[1]} \\
&\ - \frac{1}{2} \sum_{k \in *} \big( \hat{P}_{ik}^{L} \big)^{[0]} \otimes_{[1/2]} \big(a_k^\dagger\big)^{[1/2]}
-\frac{\sqrt{3}}{2} \sum_{k \in *} \big( \hat{P}_{ik}^{L} \big)^{[1]} \otimes_{[1/2]} \big(a_k^\dagger\big)^{[1/2]} \\
&\ + \frac{1}{2} \sum_{j\in *} \big( {\hat{Q}}_{ij}^{\prime \prime L} \big)^{[0]} \otimes_{[1/2]} \big(a_j\big)^{[1/2]}
-\frac{\sqrt{3}}{2} \sum_{l\in *} \big( {\hat{Q}}_{il}^{ \prime L} \big)^{[1]} \otimes_{[1/2]} \big(a_l\big)^{[1/2]}
To generate symmetrized P, we need to change the A line to the following
- \frac{1}{4} \sum_{k \in *,jl\in L} (v_{ijkl} + v_{ilkj}) \big( \hat{A}_{jl}^\dagger \big)^{[0]} \otimes_{[1/2]} \big(a_k^\dagger\big)^{[1/2]}
-\frac{\sqrt{3}}{4} \sum_{k \in *,jl\in L} (v_{ijkl} - v_{ilkj}) \big( \hat{A}_{jl}^\dagger \big)^{[1]} \otimes_{[1/2]} \big(a_k^\dagger\big)^{[1/2]}
Similarly,
\big( \hat{R}_{i}^{\prime R*,NC} \big)^{[1/2]} =&\
\big( \hat{R}_{i}^{\prime *} \big)^{[1/2]} \otimes_{[1/2]} \big( \hat{1}^R \big)^{[0]}
+ \big( \hat{1}^* \big)^{[0]} \otimes_{[1/2]} \big( \hat{R}_{i}^{\prime R} \big)^{[1/2]} \\
&\ - \frac{1}{2} \sum_{k \in *} \big(a_k^\dagger\big)^{[1/2]} \otimes_{[1/2]} \big( \hat{P}_{ik}^{R} \big)^{[0]}
+\frac{\sqrt{3}}{2} \sum_{k \in *} \big(a_k^\dagger\big)^{[1/2]} \otimes_{[1/2]} \big( \hat{P}_{ik}^{R} \big)^{[1]} \\
&\ + \frac{1}{2} \sum_{j\in *} \big(a_j\big)^{[1/2]} \otimes_{[1/2]} \big( {\hat{Q}}_{ij}^{\prime \prime R} \big)^{[0]}
+\frac{\sqrt{3}}{2} \sum_{l\in *} \big(a_l\big)^{[1/2]} \otimes_{[1/2]} \big( {\hat{Q}}_{il}^{\prime R} \big)^{[1]}\\
&\ - \frac{1}{2} \sum_{k \in R,jl\in *} v_{ijkl} \big( \hat{A}_{jl}^\dagger \big)^{[0]} \otimes_{[1/2]} \big(a_k^\dagger\big)^{[1/2]}
-\frac{\sqrt{3}}{2} \sum_{k \in R,jl\in *} v_{ijkl} \big( \hat{A}_{jl}^\dagger \big)^{[1]} \otimes_{[1/2]} \big(a_k^\dagger\big)^{[1/2]} \\
&\ + \frac{1}{2} \sum_{j\in R,kl\in *} (2v_{ijkl} - v_{ilkj}) \big( \hat{B}_{kl} \big)^{[0]} \otimes_{[1/2]} \big(a_j\big)^{[1/2]}
-\frac{\sqrt{3}}{2} \sum_{l\in R,jk\in *} v_{ijkl} \big( {\hat{B}'}_{kj} \big)^{[1]} \otimes_{[1/2]} \big(a_l\big)^{[1/2]} \\
\big( \hat{R}_{i}^{\prime R*,CN} \big)^{[1/2]} =&\
\big( \hat{R}_{i}^{\prime *} \big)^{[1/2]} \otimes_{[1/2]} \big( \hat{1}^R \big)^{[0]}
+ \big( \hat{1}^* \big)^{[0]} \otimes_{[1/2]} \big( \hat{R}_{i}^{\prime R} \big)^{[1/2]} \\
&\ - \frac{1}{2} \sum_{k \in *,jl\in R} v_{ijkl} \big(a_k^\dagger\big)^{[1/2]} \otimes_{[1/2]} \big( \hat{A}_{jl}^\dagger \big)^{[0]}
+\frac{\sqrt{3}}{2} \sum_{k \in *,jl\in R} v_{ijkl} \big(a_k^\dagger\big)^{[1/2]} \otimes_{[1/2]} \big( \hat{A}_{jl}^\dagger \big)^{[1]} \\
&\ + \frac{1}{2} \sum_{j\in *,kl\in R} (2 v_{ijkl} - v_{ilkj}) \big(a_j\big)^{[1/2]} \otimes_{[1/2]} \big( \hat{B}_{kl} \big)^{[0]}
+\frac{\sqrt{3}}{2} \sum_{l\in *,jk\in R} v_{ijkl} \big(a_l\big)^{[1/2]} \otimes_{[1/2]} \big( {\hat{B}'}_{kj} \big)^{[1]} \\
&\ - \frac{1}{2} \sum_{k \in R} \big( \hat{P}_{ik}^{*} \big)^{[0]} \otimes_{[1/2]} \big(a_k^\dagger\big)^{[1/2]}
-\frac{\sqrt{3}}{2} \sum_{k \in R} \big( \hat{P}_{ik}^{*} \big)^{[1]} \otimes_{[1/2]} \big(a_k^\dagger\big)^{[1/2]} \\
&\ + \frac{1}{2} \sum_{j\in R} \big( {\hat{Q}}_{ij}^{\prime \prime *} \big)^{[0]} \otimes_{[1/2]} \big(a_j\big)^{[1/2]}
-\frac{\sqrt{3}}{2} \sum_{l\in R} \big( {\hat{Q}}_{il}^{ \prime *} \big)^{[1]} \otimes_{[1/2]} \big(a_l\big)^{[1/2]}
Number of terms
N_{R',NC} =&\ (2 + 4K_L + 4K_L^2) K_R + (2 + 4 + 4K_R) K_L = 4K_L^2 K_R + 8K_L K_R + 2K + 4 K_L \\
N_{R',CN} =&\ (2 + 4K_L + 4) K_R + (2 + 4K_R^2 + 4K_R) K_L = 4K_R^2 K_L + 8K_R K_L + 2K + 4 K_R
Blocking of other complementary operators is straightforward
\big( \hat{P}_{ik}^{L*,CN} \big)^{[0/1]} =&\ \big( \hat{P}_{ik}^{L} \big)^{[0/1]} \otimes_{[0/1]} \big( \hat{1}^* \big)^{[0]}
+ \big( \hat{1}^L \big)^{[0]} \otimes_{[0/1]} \big( \hat{P}_{ik}^{*} \big)^{[0/1]}
+ \sum_{j \in L, l \in *} v_{ijkl} \big( a_{l} \big)^{[\frac{1}{2}]} \otimes_{[0/1]} \big( a_{j} \big)^{[\frac{1}{2}]}
+ \sum_{j \in *, l \in L} v_{ijkl} \big( a_{l} \big)^{[\frac{1}{2}]} \otimes_{[0/1]} \big( a_{j} \big)^{[\frac{1}{2}]} \\
=&\ \big( \hat{P}_{ik}^{L} \big)^{[0/1]} \otimes_{[0/1]} \big( \hat{1}^* \big)^{[0]}
+ \big( \hat{1}^L \big)^{[0]} \otimes_{[0/1]} \big( \hat{P}_{ik}^{*} \big)^{[0/1]}
\pm \sum_{j \in L, l \in *} v_{ijkl} \big( a_{j} \big)^{[\frac{1}{2}]} \otimes_{[0/1]} \big( a_{l} \big)^{[\frac{1}{2}]}
+ \sum_{j \in *, l \in L} v_{ijkl} \big( a_{l} \big)^{[\frac{1}{2}]} \otimes_{[0/1]} \big( a_{j} \big)^{[\frac{1}{2}]} \\
\big( \hat{P}_{ik}^{R*,NC} \big)^{[0/1]} =&\ \big( \hat{P}_{ik}^{*} \big)^{[0/1]} \otimes_{[0/1]} \big( \hat{1}^R \big)^{[0]}
+ \big( \hat{1}^* \big)^{[0]} \otimes_{[0/1]} \big( \hat{P}_{ik}^{R} \big)^{[0/1]}
\pm \sum_{j \in *, l \in R} v_{ijkl} \big( a_{j} \big)^{[\frac{1}{2}]} \otimes_{[0/1]} \big( a_{l} \big)^{[\frac{1}{2}]}
+ \sum_{j \in R, l \in *} v_{ijkl} \big( a_{l} \big)^{[\frac{1}{2}]} \otimes_{[0/1]} \big( a_{j} \big)^{[\frac{1}{2}]}
and
\big( {\hat{Q}}_{ij}^{\prime \prime L*,CN} \big)^{[0]} =&\ \big( {\hat{Q}}_{ij}^{\prime \prime L} \big)^{[0]} \otimes_{[0]} \big( \hat{1}^* \big)^{[0]}
+ \big( \hat{1}^L \big)^{[0]} \otimes_{[0]} \big( {\hat{Q}}_{ij}^{\prime \prime *} \big)^{[0]}
+ \sum_{k\in L, l \in *} (2v_{ijkl} - v_{ilkj}) \big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{l} \big)^{[\frac{1}{2}]}
+ \sum_{k\in *, l \in L} (2v_{ijkl} - v_{ilkj}) \big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{l} \big)^{[\frac{1}{2}]} \\
=&\ \big( {\hat{Q}}_{ij}^{\prime \prime L} \big)^{[0]} \otimes_{[0]} \big( \hat{1}^* \big)^{[0]}
+ \big( \hat{1}^L \big)^{[0]} \otimes_{[0]} \big( {\hat{Q}}_{ij}^{\prime \prime *} \big)^{[0]}
+ \sum_{k\in L, l \in *} (2v_{ijkl} - v_{ilkj}) \big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{l} \big)^{[\frac{1}{2}]}
+ \sum_{k\in *, l \in L} (2v_{ijkl} - v_{ilkj}) \big( a_{l} \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \\
\big( {\hat{Q}}_{ij}^{\prime \prime R*,NC} \big)^{[0]} =&\ \big( {\hat{Q}}_{ij}^{\prime \prime *} \big)^{[0]} \otimes_{[0]} \big( \hat{1}^R \big)^{[0]}
+ \big( \hat{1}^* \big)^{[0]} \otimes_{[0]} \big( {\hat{Q}}_{ij}^{\prime \prime R} \big)^{[0]}
+ \sum_{k\in *, l \in R} (2v_{ijkl} - v_{ilkj}) \big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{l} \big)^{[\frac{1}{2}]}
+ \sum_{k\in R, l \in *} (2v_{ijkl} - v_{ilkj}) \big( a_{l} \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{k}^\dagger \big)^{[\frac{1}{2}]}
and
\big( {\hat{Q}}_{ij}^{\prime L*,CN} \big)^{[1]} =&\ \big( {\hat{Q}}_{ij}^{\prime L} \big)^{[1]} \otimes_{[1]} \big( \hat{1}^* \big)^{[0]}
+ \big( \hat{1}^L \big)^{[0]} \otimes_{[1]} \big( {\hat{Q}}_{ij}^{\prime *} \big)^{[1]}
+ \sum_{k\in L, l \in *} v_{ilkj} \big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[1]} \big( a_{l} \big)^{[\frac{1}{2}]}
+ \sum_{k\in *, l \in L} v_{ilkj} \big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[1]} \big( a_{l} \big)^{[\frac{1}{2}]} \\
=&\ \big( {\hat{Q}}_{ij}^{\prime L} \big)^{[1]} \otimes_{[1]} \big( \hat{1}^* \big)^{[0]}
+ \big( \hat{1}^L \big)^{[0]} \otimes_{[1]} \big( {\hat{Q}}_{ij}^{\prime *} \big)^{[1]}
+ \sum_{k\in L, l \in *} v_{ilkj} \big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[1]} \big( a_{l} \big)^{[\frac{1}{2}]}
- \sum_{k\in *, l \in L} v_{ilkj} \big( a_{l} \big)^{[\frac{1}{2}]} \otimes_{[1]} \big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \\
\big( {\hat{Q}}_{ij}^{\prime R*,CN} \big)^{[1]} =&\ \big( {\hat{Q}}_{ij}^{\prime *} \big)^{[1]} \otimes_{[1]} \big( \hat{1}^R \big)^{[0]}
+ \big( \hat{1}^* \big)^{[0]} \otimes_{[1]} \big( {\hat{Q}}_{ij}^{\prime R} \big)^{[1]}
+ \sum_{k\in *, l \in R} v_{ilkj} \big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[1]} \big( a_{l} \big)^{[\frac{1}{2}]}
- \sum_{k\in R, l \in *} v_{ilkj} \big( a_{l} \big)^{[\frac{1}{2}]} \otimes_{[1]} \big( a_{k}^\dagger \big)^{[\frac{1}{2}]}
\big( \hat{P}_{ik}^{L,NC\to CN} \big)^{[0/1]} =&\
\sum_{jl\in L} v_{ijkl} \big( a_{l} \big)^{[\frac{1}{2}]} \otimes_{[0/1]} \big( a_{j} \big)^{[\frac{1}{2}]}
= \sum_{jl\in L} v_{ijkl} \big( \hat{A}_{jl}^\dagger \big)^{[0/1]} \\
\big( {\hat{Q}}_{ij}^{\prime \prime L,NC\to CN} \big)^{[0]} =&\
\sum_{kl\in R} (2v_{ijkl} - v_{ilkj}) \big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{l} \big)^{[\frac{1}{2}]}
= \sum_{kl\in R} (2v_{ijkl} - v_{ilkj}) \big( \hat{B}_{kl} \big)^{[0]} \\
\big( {\hat{Q}}_{ij}^{\prime L,NC\to CN} \big)^{[1]} =&\
\sum_{kl\in R} v_{ilkj} \big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[1]} \big( a_{l} \big)^{[\frac{1}{2}]}
= \sum_{kl\in R} v_{ilkj} \big( {\hat{B}'}_{kl} \big)^{[1]}