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Spin-Adapted DMRG Quantum Chemistry Hamiltonian

Partitioning in SU(2)

The partitioning of Hamiltonian in left (\(L\)) and right (\(R\)) blocks is given by

\[\begin{split}(\hat{H})^{[0]} =&\ \big( \hat{H}^{L} \big)^{[0]} \otimes_{[0]} \big( \hat{1}^{R} \big)^{[0]} + \big( \hat{1}^{L} \big)^{[0]} \otimes_{[0]} \big( \hat{H}^{R} \big)^{[0]} \\ &\ + \sqrt{2} \sum_{i\in L} \left[ \big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( \hat{S}_{i}^{R} \big)^{[\frac{1}{2}]} + h.c. \right] \\ &\ + 2 \sum_{i\in L} \left[ \big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( \hat{R}_{i}^{R} \big)^{[\frac{1}{2}]} + h.c. \right] + 2 \sum_{i\in R} \left[ \big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( \hat{R}_{i}^{L} \big)^{[\frac{1}{2}]} + h.c. \right] \\ &\ - \frac{1}{2} \sum_{ik\in L} \left[ \sqrt{3} \big(\hat{A}_{ik} \big)^{[1]} \otimes_{[0]} \big(\hat{P}_{ik}^{R} \big)^{[1]} + \big(\hat{A}_{ik} \big)^{[0]} \otimes_{[0]} \big(\hat{P}_{ik}^{R} \big)^{[0]} + h.c. \right] \\ &\ +\sum_{ij\in L} \left[ \big( \hat{B}_{ij} \big)^{[0]} \otimes_{[0]} \left( 2\big( \hat{Q}_{ij}^{R} \big)^{[0]} - \big( {\hat{Q}}_{ij}^{\prime R} \big)^{[0]} \right) + \sqrt{3} \big( {\hat{B}'}_{ij} \big)^{[1]} \otimes_{[0]} \big( {\hat{Q}}_{ij}^{\prime R} \big)^{[1]} \right]\end{split}\]

where the normal and complementary operators are defined by

\[\begin{split}\big( \hat{S}_{i}^{L/R} \big)^{[\frac{1}{2}]} =&\ \sum_{j\in L/R} t_{ij} \big( a_{j} \big)^{[\frac{1}{2}]} \\ \big( \hat{R}_{i}^{L/R} \big)^{[\frac{1}{2}]} =&\ \sum_{jkl\in L/R} v_{ijkl} \left[ \Big( a_{k}^\dagger \Big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{l} \big)^{[\frac{1}{2}]} \right] \otimes_{[\frac{1}{2}]} \big( a_{j} \big)^{[\frac{1}{2}]} \\ \big( \hat{A}_{ik} \big)^{[0/1]} =&\ \big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0/1]} \big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \\ \big( \hat{P}_{ik}^{R} \big)^{[0/1]} =&\ \sum_{jl\in R} v_{ijkl} \big( a_{j} \big)^{[\frac{1}{2}]} \otimes_{[0/1]} \big( a_{l} \big)^{[\frac{1}{2}]} \\ \big( \hat{B}_{ij} \big)^{[0]} =&\ \big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{j} \big)^{[\frac{1}{2}]} \\ \big( {\hat{B}'}_{ij} \big)^{[1]} =&\ \big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[1]} \big( a_{j} \big)^{[\frac{1}{2}]}\\ \big( \hat{Q}_{ij}^{R} \big)^{[0]} =&\ \sum_{kl\in R} v_{ijkl} \big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{l} \big)^{[\frac{1}{2}]} \\ \big( {\hat{Q}}_{ij}^{\prime R} \big)^{[0/1]} =&\ \sum_{kl\in R} v_{ilkj} \big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0/1]} \big( a_{l} \big)^{[\frac{1}{2}]} \\ \big( {\hat{Q}}_{ij}^{\prime \prime R} \big)^{[0]} :=&\ 2 \big( {\hat{Q}}_{ij}^{R} \big)^{[0]} - \big( {\hat{Q}}_{ij}^{\prime R} \big)^{[0]} = \sum_{kl\in R} (2v_{ijkl} - v_{ilkj}) \big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{l} \big)^{[\frac{1}{2}]}\end{split}\]

Derivation

CG Factors

From \(j_2 = 1/2\) CG factors

\[\begin{split}\bigg\langle j_1\ \left(M - \frac{1}{2} \right)\ \frac{1}{2}\ \frac{1}{2} \bigg| \left( j_1 \pm \frac{1}{2} \right)\ M \bigg\rangle =&\ \pm \sqrt{\frac{1}{2} \left( 1 \pm \frac{M}{j_1 + \frac{1}{2}} \right)} \\ \bigg\langle j_1\ \left(M + \frac{1}{2} \right)\ \frac{1}{2}\ \left( -\frac{1}{2}\right) \bigg| \left( j_1 \pm \frac{1}{2} \right)\ M \bigg\rangle =&\ \sqrt{\frac{1}{2} \left( 1 \mp \frac{M}{j_1 + \frac{1}{2}} \right)}\end{split}\]

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

\[\begin{split}\bigg\langle \frac{1}{2}\ \frac{1}{2}\ j_1\ \left(M - \frac{1}{2} \right) \bigg| \left( j_1 \pm \frac{1}{2} \right)\ M \bigg\rangle =&\ \sqrt{\frac{1}{2} \left( 1 \pm \frac{M}{j_1 + \frac{1}{2}} \right)} \\ \bigg\langle \frac{1}{2}\ \left( -\frac{1}{2}\right)\ j_1\ \left(M + \frac{1}{2} \right) \bigg| \left( j_1 \pm \frac{1}{2} \right)\ M \bigg\rangle =&\ \pm \sqrt{\frac{1}{2} \left( 1 \mp \frac{M}{j_1 + \frac{1}{2}} \right)}\end{split}\]

let \(j_1 = 1\), we have

\[\begin{split}\langle \tfrac{1}{2}\ \tfrac{1}{2}\ 1\ (M - \tfrac{1}{2}) | \tfrac{1}{2}\ M \rangle =&\ \sqrt{\tfrac{1}{2} ( 1-\frac{M}{\tfrac{3}{2}} )} \\ \langle \tfrac{1}{2}\ (-\tfrac{1}{2})\ 1\ (M + \tfrac{1}{2}) | \tfrac{1}{2}\ M \rangle =&\ -\sqrt{\tfrac{1}{2} ( 1+\frac{M}{\tfrac{3}{2}} )}\end{split}\]

So the coefficients for \([\tfrac{1}{2}] \otimes_{[\tfrac{1}{2}]} [1]\) are

\[\begin{split}[\tfrac{1}{2} + 0 = \tfrac{1}{2}] = \sqrt{\tfrac{1}{3}},\quad [-\tfrac{1}{2} + 1 = \tfrac{1}{2}] = -\sqrt{\tfrac{2}{3}} \\ [\tfrac{1}{2} + (-1) = -\tfrac{1}{2}] = \sqrt{\tfrac{2}{3}},\quad [-\tfrac{1}{2} + 0 = -\tfrac{1}{2}] = -\sqrt{\tfrac{1}{3}}\end{split}\]

The coefficients for \([1] \otimes_{[\tfrac{1}{2}]} [\tfrac{1}{2}]\) are

\[\begin{split}[0 + \tfrac{1}{2} = \tfrac{1}{2}] = -\sqrt{\tfrac{1}{3}},\quad [1 -\tfrac{1}{2} = \tfrac{1}{2}] = \sqrt{\tfrac{2}{3}} \\ [(-1) + \tfrac{1}{2} = -\tfrac{1}{2}] = -\sqrt{\tfrac{2}{3}},\quad [0 -\tfrac{1}{2} = -\tfrac{1}{2}] = \sqrt{\tfrac{1}{3}}\end{split}\]

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 \(-1\).

Tensor Product Formulas

Singlet

\[\begin{split}\big(a_p^\dagger\big)^{[1/2]} \otimes_{[0]} \big(a_q^\dagger\big)^{[1/2]} =&\ \begin{pmatrix} a_{p\alpha}^\dagger \\ a_{p\beta}^\dagger \end{pmatrix}^{[1/2]} \otimes_{[0]} \begin{pmatrix} a_{q\alpha}^\dagger \\ a_{q\beta}^\dagger \end{pmatrix}^{[1/2]} = \frac{1}{\sqrt{2}} \begin{pmatrix} a_{p\alpha}^\dagger a_{q\beta}^\dagger - a_{p\beta}^\dagger a_{q\alpha}^\dagger \end{pmatrix}^{[0]} \\ \big(a_p^\dagger\big)^{[1/2]} \otimes_{[0]} \big(a_q\big)^{[1/2]} =&\ \begin{pmatrix} a_{p\alpha}^\dagger \\ a_{p\beta}^\dagger \end{pmatrix}^{[1/2]} \otimes_{[0]} \begin{pmatrix} -a_{q\beta} \\ a_{q\alpha} \end{pmatrix}^{[1/2]} = \frac{1}{\sqrt{2}} \begin{pmatrix} a_{p\alpha}^\dagger a_{q\alpha}+ a_{p\beta}^\dagger a_{q\beta} \end{pmatrix}^{[0]} \\ \big(a_p\big)^{[1/2]} \otimes_{[0]} \big(a_q\big)^{[1/2]} =&\ \begin{pmatrix} -a_{p\beta} \\ a_{p\alpha} \end{pmatrix}^{[1/2]} \otimes_{[0]} \begin{pmatrix} -a_{q\beta} \\ a_{q\alpha} \end{pmatrix}^{[1/2]} = \frac{1}{\sqrt{2}} \begin{pmatrix} -a_{p\beta} a_{q\alpha} + a_{p\alpha} a_{q\beta} \end{pmatrix}^{[0]}\end{split}\]

Triplet

\[\begin{split}\big(a_p^\dagger\big)^{[1/2]} \otimes_{[1]} \big(a_q^\dagger\big)^{[1/2]} =&\ \begin{pmatrix} a_{p\alpha}^\dagger \\ a_{p\beta}^\dagger \end{pmatrix}^{[1/2]} \otimes_{[1]} \begin{pmatrix} a_{q\alpha}^\dagger \\ a_{q\beta}^\dagger \end{pmatrix}^{[1/2]} = \begin{pmatrix} a_{p\alpha}^\dagger a_{q\alpha}^\dagger \\ \frac{1}{\sqrt{2}} \Big( a_{p\alpha}^\dagger a_{q\beta}^\dagger + a_{p\beta}^\dagger a_{q\alpha}^\dagger \Big) \\ a_{p\beta}^\dagger a_{q\beta}^\dagger \end{pmatrix}^{[1]} \\ \big(a_p^\dagger\big)^{[1/2]} \otimes_{[1]} \big(a_q\big)^{[1/2]} =&\ \begin{pmatrix} a_{p\alpha}^\dagger \\ a_{p\beta}^\dagger \end{pmatrix}^{[1/2]} \otimes_{[1]} \begin{pmatrix} -a_{q\beta} \\ a_{q\alpha} \end{pmatrix}^{[1/2]} = \begin{pmatrix} -a_{p\alpha}^\dagger a_{q\beta} \\ \frac{1}{\sqrt{2}} \Big( a_{p\alpha}^\dagger a_{q\alpha} - a_{p\beta}^\dagger a_{q\beta} \Big) \\ a_{p\beta}^\dagger a_{q\alpha} \end{pmatrix}^{[1]} \\ \big(a_p\big)^{[1/2]} \otimes_{[1]} \big(a_q\big)^{[1/2]} =&\ \begin{pmatrix} -a_{p\beta} \\ a_{p\alpha} \end{pmatrix}^{[1/2]} \otimes_{[1]} \begin{pmatrix} -a_{q\beta} \\ a_{q\alpha} \end{pmatrix}^{[1/2]} = \begin{pmatrix} a_{p\beta} a_{q\beta} \\ -\frac{1}{\sqrt{2}} \Big( a_{p\beta} a_{q\alpha} + a_{p\alpha} a_{q\beta} \Big) \\ a_{p\alpha} a_{q\alpha} \end{pmatrix}^{[1]}\end{split}\]

Doublet times singlet/triplet

\[\begin{split}U^{[1/2]} = &\ \big(a_p^\dagger\big)^{[1/2]} \otimes_{[1/2]} \Big[ \big(a_r\big)^{[1/2]} \otimes_{[1]} \big(a_s\big)^{[1/2]} \Big] = \begin{pmatrix} a_{p\alpha}^\dagger \\ a_{p\beta}^\dagger \end{pmatrix}^{[1/2]} \otimes_{[1/2]} \begin{pmatrix} a_{r\beta} a_{s\beta} \\ -\frac{1}{\sqrt{2}} \Big( a_{r\beta} a_{s\alpha} + a_{r\alpha} a_{s\beta} \Big) \\ a_{r\alpha} a_{s\alpha} \end{pmatrix}^{[1]} \\ =&\ \begin{pmatrix} -\frac{1}{\sqrt{2}}\frac{1}{\sqrt{3}} a_{p\alpha}^\dagger \Big( a_{r\beta} a_{s\alpha} + a_{r\alpha} a_{s\beta} \Big) -\frac{\sqrt{2}}{\sqrt{3}} a_{p\beta}^\dagger a_{r\beta} a_{s\beta} \\ \frac{\sqrt{2}}{\sqrt{3}} a_{p\alpha}^\dagger a_{r\alpha} a_{s\alpha} +\big( -\frac{1}{\sqrt{3}}\big) \big( -\frac{1}{\sqrt{2}} \big) a_{p\beta}^\dagger \Big( a_{r\beta} a_{s\alpha} + a_{r\alpha} a_{s\beta} \Big) \end{pmatrix}^{[1/2]} = \frac{1}{\sqrt{6}} \begin{pmatrix} - a_{p\alpha}^\dagger a_{r\beta} a_{s\alpha} - a_{p\alpha}^\dagger a_{r\alpha} a_{s\beta} -2 a_{p\beta}^\dagger a_{r\beta} a_{s\beta} \\ 2 a_{p\alpha}^\dagger a_{r\alpha} a_{s\alpha} +a_{p\beta}^\dagger a_{r\beta} a_{s\alpha} + a_{p\beta}^\dagger a_{r\alpha} a_{s\beta} \end{pmatrix}^{[1/2]} \\ V^{[1/2]} =&\ \big(a_p^\dagger\big)^{[1/2]} \otimes_{[1/2]} \Big[ \big(a_r\big)^{[1/2]} \otimes_{[0]} \big(a_s\big)^{[1/2]} \Big] = \frac{1}{\sqrt{2}} \begin{pmatrix} a_{p\alpha}^\dagger \\ a_{p\beta}^\dagger \end{pmatrix}^{[1/2]} \otimes_{[1/2]} \begin{pmatrix} -a_{r\beta} a_{s\alpha} + a_{r\alpha} a_{s\beta} \end{pmatrix}^{[0]} \\ =&\ \frac{1}{\sqrt{2}} \begin{pmatrix} -a_{p\alpha}^\dagger a_{r\beta} a_{s\alpha} + a_{p\alpha}^\dagger a_{r\alpha} a_{s\beta}\\ -a_{p\beta}^\dagger a_{r\beta} a_{s\alpha} + a_{p\beta}^\dagger a_{r\alpha} a_{s\beta}\end{pmatrix}^{[1/2]}\end{split}\]

Therefore,

\[\begin{split}\sqrt{3} U^{[1/2]} - V^{[1/2]} =&\ \frac{1}{\sqrt{2}} \begin{pmatrix} - a_{p\alpha}^\dagger a_{r\beta} a_{s\alpha} - a_{p\alpha}^\dagger a_{r\alpha} a_{s\beta} -2 a_{p\beta}^\dagger a_{r\beta} a_{s\beta} \\ 2 a_{p\alpha}^\dagger a_{r\alpha} a_{s\alpha} +a_{p\beta}^\dagger a_{r\beta} a_{s\alpha} + a_{p\beta}^\dagger a_{r\alpha} a_{s\beta} \end{pmatrix}^{[1/2]} - \frac{1}{\sqrt{2}} \begin{pmatrix} -a_{p\alpha}^\dagger a_{r\beta} a_{s\alpha} + a_{p\alpha}^\dagger a_{r\alpha} a_{s\beta}\\ -a_{p\beta}^\dagger a_{r\beta} a_{s\alpha} + a_{p\beta}^\dagger a_{r\alpha} a_{s\beta}\end{pmatrix}^{[1/2]} \\ =&\ \frac{1}{\sqrt{2}} \begin{pmatrix} -a_{p\alpha}^\dagger a_{r\beta} a_{s\alpha} - a_{p\alpha}^\dagger a_{r\alpha} a_{s\beta} -2 a_{p\beta}^\dagger a_{r\beta} a_{s\beta} +a_{p\alpha}^\dagger a_{r\beta} a_{s\alpha} - a_{p\alpha}^\dagger a_{r\alpha} a_{s\beta}\\ 2 a_{p\alpha}^\dagger a_{r\alpha} a_{s\alpha} +a_{p\beta}^\dagger a_{r\beta} a_{s\alpha} + a_{p\beta}^\dagger a_{r\alpha} a_{s\beta} +a_{p\beta}^\dagger a_{r\beta} a_{s\alpha} - a_{p\beta}^\dagger a_{r\alpha} a_{s\beta}\end{pmatrix}^{[1/2]} \\ =&\ \sqrt{2} \begin{pmatrix} - a_{p\alpha}^\dagger a_{r\alpha} a_{s\beta} - a_{p\beta}^\dagger a_{r\beta} a_{s\beta} \\ a_{p\alpha}^\dagger a_{r\alpha} a_{s\alpha} + a_{p\beta}^\dagger a_{r\beta} a_{s\alpha} \end{pmatrix}^{[1/2]}\end{split}\]

Another case

\[\begin{split}S^{[1/2]} = &\ \big(a_r\big)^{[1/2]} \otimes_{[1/2]} \Big[ \big(a_p^\dagger \big)^{[1/2]} \otimes_{[1]} \big(a_q\big)^{[1/2]} \Big] = \begin{pmatrix} -a_{r\beta} \\ a_{r\alpha} \end{pmatrix}^{[1/2]} \otimes_{[1/2]} \begin{pmatrix} -a_{p\alpha}^\dagger a_{q\beta} \\ \frac{1}{\sqrt{2}} \Big( a_{p\alpha}^\dagger a_{q\alpha} - a_{p\beta}^\dagger a_{q\beta} \Big) \\ a_{p\beta}^\dagger a_{q\alpha} \end{pmatrix}^{[1]} \\ =&\ \begin{pmatrix} \frac{1}{\sqrt{2}} \frac{1}{\sqrt{3}} (-a_{r\beta}) \Big( a_{p\alpha}^\dagger a_{q\alpha} - a_{p\beta}^\dagger a_{q\beta} \Big) +\frac{\sqrt{2}}{\sqrt{3}} a_{r\alpha} a_{p\alpha}^\dagger a_{q\beta} \\ -\frac{\sqrt{2}}{\sqrt{3}} a_{r\beta} a_{p\beta}^\dagger a_{q\alpha} -\frac{1}{\sqrt{2}} \frac{1}{\sqrt{3}} a_{r\alpha} \Big( a_{p\alpha}^\dagger a_{q\alpha} - a_{p\beta}^\dagger a_{q\beta} \Big) \end{pmatrix}^{[1/2]} = \frac{1}{\sqrt{6}} \begin{pmatrix} -a_{r\beta} a_{p\alpha}^\dagger a_{q\alpha} + a_{r\beta} a_{p\beta}^\dagger a_{q\beta} +2 a_{r\alpha} a_{p\alpha}^\dagger a_{q\beta}\\ -2a_{r\beta} a_{p\beta}^\dagger a_{q\alpha} -a_{r\alpha} a_{p\alpha}^\dagger a_{q\alpha} + a_{r\alpha} a_{p\beta}^\dagger a_{q\beta} \end{pmatrix}^{[1/2]} \\ T^{[1/2]} = &\ \big(a_r\big)^{[1/2]} \otimes_{[1/2]} \Big[ \big(a_p^\dagger \big)^{[1/2]} \otimes_{[0]} \big(a_q\big)^{[1/2]} \Big] = \frac{1}{\sqrt{2}} \begin{pmatrix} -a_{r\beta} \\ a_{r\alpha} \end{pmatrix}^{[1/2]} \otimes_{[1/2]} \begin{pmatrix} a_{p\alpha}^\dagger a_{q\alpha}+ a_{p\beta}^\dagger a_{q\beta} \end{pmatrix}^{[0]} \\ =&\ \frac{1}{\sqrt{2}} \begin{pmatrix} -a_{r\beta} a_{p\alpha}^\dagger a_{q\alpha} - a_{r\beta}a_{p\beta}^\dagger a_{q\beta} \\ a_{r\alpha} a_{p\alpha}^\dagger a_{q\alpha} + a_{r\alpha}a_{p\beta}^\dagger a_{q\beta}\end{pmatrix}^{[1/2]}\end{split}\]

Therefore,

\[\begin{split}\sqrt{3} S^{[1/2]} - T^{[1/2]} =&\ \frac{1}{\sqrt{6}} \begin{pmatrix} -a_{r\beta} a_{p\alpha}^\dagger a_{q\alpha} + a_{r\beta} a_{p\beta}^\dagger a_{q\beta} +2 a_{r\alpha} a_{p\alpha}^\dagger a_{q\beta}\\ -2a_{r\beta} a_{p\beta}^\dagger a_{q\alpha} -a_{r\alpha} a_{p\alpha}^\dagger a_{q\alpha} + a_{r\alpha} a_{p\beta}^\dagger a_{q\beta} \end{pmatrix}^{[1/2]}-\frac{1}{\sqrt{2}} \begin{pmatrix} -a_{r\beta} a_{p\alpha}^\dagger a_{q\alpha} - a_{r\beta}a_{p\beta}^\dagger a_{q\beta} \\ a_{r\alpha} a_{p\alpha}^\dagger a_{q\alpha} + a_{r\alpha}a_{p\beta}^\dagger a_{q\beta}\end{pmatrix}^{[1/2]} \\ =&\ \frac{1}{\sqrt{2}} \begin{pmatrix} -a_{r\beta} a_{p\alpha}^\dagger a_{q\alpha} + a_{r\beta} a_{p\beta}^\dagger a_{q\beta} +2 a_{r\alpha} a_{p\alpha}^\dagger a_{q\beta} +a_{r\beta} a_{p\alpha}^\dagger a_{q\alpha} + a_{r\beta}a_{p\beta}^\dagger a_{q\beta} \\ -2a_{r\beta} a_{p\beta}^\dagger a_{q\alpha} -a_{r\alpha} a_{p\alpha}^\dagger a_{q\alpha} + a_{r\alpha} a_{p\beta}^\dagger a_{q\beta} -a_{r\alpha} a_{p\alpha}^\dagger a_{q\alpha} - a_{r\alpha}a_{p\beta}^\dagger a_{q\beta} \end{pmatrix}^{[1/2]} \\ =&\ \sqrt{2} \begin{pmatrix} a_{r\beta}a_{p\beta}^\dagger a_{q\beta} +a_{r\alpha} a_{p\alpha}^\dagger a_{q\beta} \\ -a_{r\beta} a_{p\beta}^\dagger a_{q\alpha} -a_{r\alpha} a_{p\alpha}^\dagger a_{q\alpha} \end{pmatrix}^{[1/2]}\end{split}\]

Triplet times triplet

\[\begin{split}X^{[0]} = &\ \Big[ \big(a_p^\dagger\big)^{[1/2]} \otimes_{[1]} \big(a_q^\dagger\big)^{[1/2]} \Big] \otimes_{[0]} \Big[ \big(a_r\big)^{[1/2]} \otimes_{[1]} \big(a_s\big)^{[1/2]} \Big] \\ =&\ \begin{pmatrix} a_{p\alpha}^\dagger a_{q\alpha}^\dagger \\ \frac{1}{\sqrt{2}} \Big( a_{p\alpha}^\dagger a_{q\beta}^\dagger + a_{p\beta}^\dagger a_{q\alpha}^\dagger \Big) \\ a_{p\beta}^\dagger a_{q\beta}^\dagger \end{pmatrix}^{[1]} \otimes_{[0]} \begin{pmatrix} a_{r\beta} a_{s\beta} \\ -\frac{1}{\sqrt{2}} \Big( a_{r\beta} a_{s\alpha} + a_{r\alpha} a_{s\beta} \Big) \\ a_{r\alpha} a_{s\alpha} \end{pmatrix}^{[1]} \\ =&\ \frac{1}{\sqrt{3}} \begin{pmatrix} a_{p\alpha}^\dagger a_{q\alpha}^\dagger a_{r\alpha} s_{s\alpha} + \frac{1}{2} \Big( a_{p\alpha}^\dagger a_{q\beta}^\dagger + a_{p\beta}^\dagger a_{q\alpha}^\dagger \Big) \Big( a_{r\beta} a_{s\alpha} + a_{r\alpha} a_{s\beta} \Big) + a_{p\beta}^\dagger a_{q\beta}^\dagger a_{r\beta} a_{s\beta} \end{pmatrix} \\ Y^{[0]} = &\ \Big[ \big(a_p^\dagger\big)^{[1/2]} \otimes_{[0]} \big(a_q^\dagger\big)^{[1/2]} \Big] \otimes_{[0]} \Big[ \big(a_r\big)^{[1/2]} \otimes_{[0]} \big(a_s\big)^{[1/2]} \Big] \\ =&\ \frac{1}{\sqrt{2}} \begin{pmatrix} a_{p\alpha}^\dagger a_{q\beta}^\dagger - a_{p\beta}^\dagger a_{q\alpha}^\dagger \end{pmatrix}^{[0]} \otimes_{[0]} \frac{1}{\sqrt{2}} \begin{pmatrix} -a_{r\beta} a_{s\alpha} + a_{r\alpha} a_{s\beta} \end{pmatrix}^{[0]} \\ =&\ \frac{1}{2} \Big( a_{p\alpha}^\dagger a_{q\beta}^\dagger - a_{p\beta}^\dagger a_{q\alpha}^\dagger \Big) \Big( -a_{r\beta} a_{s\alpha} + a_{r\alpha} a_{s\beta} \Big)\end{split}\]

Using

\[(a+b)(c+d) + (a-b)(-c+d) = (a+b)(2d) -2b(-c+d) = 2 (ad+bc)\]

we have

\[\begin{split}\sqrt{3} X^{[0]} + Y^{[0]} =&\ a_{p\alpha}^\dagger a_{q\alpha}^\dagger a_{r\alpha} s_{s\alpha} + a_{p\beta}^\dagger a_{q\beta}^\dagger a_{r\beta} a_{s\beta} + a_{p\alpha}^\dagger a_{q\beta}^\dagger a_{r\alpha} a_{s\beta} + a_{p\beta}^\dagger a_{q\alpha}^\dagger a_{r\beta} a_{s\alpha} \\ =&\ \sum_{\sigma\sigma'} a_{p\sigma}^\dagger a_{q\sigma'}^\dagger a_{r\sigma} s_{s\sigma'}\end{split}\]

Another case

\[\begin{split}Z^{[0]} = &\ \Big[ \big(a_p^\dagger\big)^{[1/2]} \otimes_{[1]} \big(a_q\big)^{[1/2]} \Big] \otimes_{[0]} \Big[ \big(a_r^\dagger \big)^{[1/2]} \otimes_{[1]} \big(a_s\big)^{[1/2]} \Big] \\ =&\ \begin{pmatrix} -a_{p\alpha}^\dagger a_{q\beta} \\ \frac{1}{\sqrt{2}} \Big( a_{p\alpha}^\dagger a_{q\alpha} - a_{p\beta}^\dagger a_{q\beta} \Big) \\ a_{p\beta}^\dagger a_{q\alpha} \end{pmatrix}^{[1]} \otimes_{[0]} \begin{pmatrix} -a_{r\alpha}^\dagger a_{s\beta} \\ \frac{1}{\sqrt{2}} \Big( a_{r\alpha}^\dagger a_{s\alpha} - a_{r\beta}^\dagger a_{s\beta} \Big) \\ a_{r\beta}^\dagger a_{s\alpha} \end{pmatrix}^{[1]} \\ =&\ \frac{1}{\sqrt{3}} \begin{pmatrix} -a_{p\alpha}^\dagger a_{q\beta} a_{r\beta}^\dagger a_{s\alpha} -\frac{1}{2} \Big( a_{p\alpha}^\dagger a_{q\alpha} - a_{p\beta}^\dagger a_{q\beta} \Big) \Big( a_{r\alpha}^\dagger a_{s\alpha} - a_{r\beta}^\dagger a_{s\beta} \Big) - a_{p\beta}^\dagger a_{q\alpha} a_{r\alpha}^\dagger a_{s\beta} \end{pmatrix} \\ W^{[0]} =&\ \Big[ \big(a_p^\dagger\big)^{[1/2]} \otimes_{[0]} \big(a_q\big)^{[1/2]} \Big] \otimes_{[0]} \Big[ \big(a_r^\dagger \big)^{[1/2]} \otimes_{[0]} \big(a_s\big)^{[1/2]} \Big] \\ =&\ \frac{1}{\sqrt{2}} \begin{pmatrix} a_{p\alpha}^\dagger a_{q\alpha}+ a_{p\beta}^\dagger a_{q\beta} \end{pmatrix}^{[0]} \otimes_{[0]} \frac{1}{\sqrt{2}} \begin{pmatrix} a_{r\alpha}^\dagger a_{s\alpha}+ a_{r\beta}^\dagger a_{s\beta} \end{pmatrix}^{[0]} \\ =&\ \frac{1}{2} \Big( a_{p\alpha}^\dagger a_{q\alpha}+ a_{p\beta}^\dagger a_{q\beta}\Big) \Big( a_{r\alpha}^\dagger a_{s\alpha}+ a_{r\beta}^\dagger a_{s\beta} \Big)\end{split}\]

Using

\[(a-b)(c-d) + (a+b)(c+d) = (a+b)(2c) - (2b)(c-d) = 2(ac+bd)\]

we have

\[\begin{split}-\sqrt{3} Z^{[0]} + W^{[0]} =&\ a_{p\alpha}^\dagger a_{q\beta} a_{r\beta}^\dagger a_{s\alpha} + a_{p\beta}^\dagger a_{q\alpha} a_{r\alpha}^\dagger a_{s\beta} + a_{p\alpha}^\dagger a_{q\alpha} a_{r\alpha}^\dagger a_{s\alpha} + a_{p\beta}^\dagger a_{q\beta} a_{r\beta}^\dagger a_{s\beta} \\ =&\ \sum_{\sigma\sigma'} a_{p\sigma}^\dagger a_{q\sigma'} a_{r\sigma'}^\dagger a_{s\sigma}\end{split}\]

S Term

From second singlet formula we have

\[\sqrt{2} \sum_{i\in L} \big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( \hat{S}_{i}^{R} \big)^{[\frac{1}{2}]} = \sum_{i\in L} \big( t_{ij} a_{i\alpha}^\dagger a_{j\alpha} + t_{ij} a_{i\beta}^\dagger a_{j\beta} \big)\]

R Term

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.

AP Term

Using definition

\[\begin{split}\big( \hat{A}_{ik} \big)^{[0/1]} =&\ \big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0/1]} \big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \\ \big( \hat{P}_{ik}^{R} \big)^{[0/1]} =&\ -\sum_{jl\in R} v_{ijkl} \big( a_{j} \big)^{[\frac{1}{2}]} \otimes_{[0/1]} \big( a_{l} \big)^{[\frac{1}{2}]}\end{split}\]

We have

\[\begin{split}&\ \sum_{ik\in L} \left[ \sqrt{3} \big(\hat{A}_{ik} \big)^{[1]} \otimes_{[0]} \big(\hat{P}_{ik}^{R} \big)^{[1]} + \big(\hat{A}_{ik} \big)^{[0]} \otimes_{[0]} \big(\hat{P}_{ik}^{R} \big)^{[0]} \right] \\ =&\ \sum_{ik\in L,jl\in R} v_{ijkl} \left[ \sqrt{3} \left[ \big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[1]} \big( a_{k}^\dagger \big)^{[\frac{1}{2}]}\right] \otimes_{[0]} \left[ \big( a_{j} \big)^{[\frac{1}{2}]} \otimes_{[1]} \big( a_{l} \big)^{[\frac{1}{2}]} \right] + \left[ \big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{k}^\dagger \big)^{[\frac{1}{2}]}\right] \otimes_{[0]} \left[ \big( a_{j} \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{l} \big)^{[\frac{1}{2}]} \right] \right] \\ =&\ \sum_{ik\in L,jl\in R} v_{ijkl} \left[ \sum_{\sigma\sigma'} a_{i\sigma}^\dagger a_{k\sigma'}^\dagger a_{j\sigma} a_{l\sigma'} \right] = -\sum_{ik\in L,jl\in R,\sigma\sigma'} v_{ijkl} a_{i\sigma}^\dagger a_{k\sigma'}^\dagger a_{l\sigma'} a_{j\sigma}\end{split}\]

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.

BQ Term

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 \(\big( \hat{B}_{ij} \big)^{[0]} = \big( {\hat{B}'}_{ij} \big)^{[0]}\))

\[\begin{split}&\ \sum_{ij\in L} \left[ 2\Big( \hat{B}_{ij} \Big)^{[0]} \otimes_{[0]} \Big( \hat{Q}_{ij}^{R} \Big)^{[0]} - \Big( {\hat{B}'}_{ij} \Big)^{[0]} \otimes_{[0]} \Big( {\hat{Q}'}_{ij}^{R} \Big)^{[0]} + \sqrt{3} \Big( {\hat{B}'}_{ij} \Big)^{[1]} \otimes_{[0]} \Big( {\hat{Q}'}_{ij}^{R} \Big)^{[1]} \right] \\ =&\ \sum_{ij\in L} \left[ \Big( \hat{B}_{ij} \Big)^{[0]} \otimes_{[0]} \left( \Big( 2\hat{Q}_{ij}^{R} \Big)^{[0]} - \Big( {\hat{Q}'}_{ij}^{R} \Big)^{[0]} \right) + \sqrt{3} \Big( {\hat{B}'}_{ij} \Big)^{[1]} \otimes_{[0]} \Big( {\hat{Q}'}_{ij}^{R} \Big)^{[1]} \right]\end{split}\]

Note that \(B, Q\) do not have \([1]\) form.

Normal/Complementary Partitioning

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,

\[\begin{split}&\ \sqrt{2} \sum_{i\in L} \left[ \big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( \hat{S}_{i}^{R} \big)^{[\frac{1}{2}]} + h.c. \right] + 2 \sum_{i\in L} \left[ \big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( \hat{R}_{i}^{R} \big)^{[\frac{1}{2}]} + h.c. \right] + 2 \sum_{i\in R} \left[ \big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( \hat{R}_{i}^{L} \big)^{[\frac{1}{2}]} + h.c. \right] \\ =&\ \frac{\sqrt{2}}{2} \sum_{i\in L} \left[ \big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( \hat{S}_{i}^{R} \big)^{[\frac{1}{2}]} + h.c. \right] + \frac{\sqrt{2}}{2} \sum_{i\in R} \left[ \big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( \hat{S}_{i}^{L} \big)^{[\frac{1}{2}]} + h.c. \right] \\ &\ + 2 \sum_{i\in L} \left[ \big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( \hat{R}_{i}^{R} \big)^{[\frac{1}{2}]} + h.c. \right] + 2 \sum_{i\in R} \left[ \big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( \hat{R}_{i}^{L} \big)^{[\frac{1}{2}]} + h.c. \right] \\ =&\ 2 \sum_{i\in L} \left[ \big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \Big[ \big( \hat{R}_{i}^{R} \big)^{[\frac{1}{2}]} + \frac{\sqrt{2}}{4} \big( \hat{S}_{i}^{R} \big)^{[\frac{1}{2}]} \Big] + h.c. \right] + 2 \sum_{i\in R} \left[ \big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \Big[ \big( \hat{R}_{i}^{L} \big)^{[\frac{1}{2}]} + \frac{\sqrt{2}}{4} \big( \hat{S}_{i}^{L} \big)^{[\frac{1}{2}]} \Big] + h.c. \right]\end{split}\]

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.

Operator Exchange factors

Here we consider fermion and SU(2) exchange factors together. From \(j_2 = 1/2\) CG factors

\[\begin{split}\bigg\langle j_1\ \left(M - \frac{1}{2} \right)\ \frac{1}{2}\ \frac{1}{2} \bigg| \left( j_1 \pm \frac{1}{2} \right)\ M \bigg\rangle =&\ \pm \sqrt{\frac{1}{2} \left( 1 \pm \frac{M}{j_1 + \frac{1}{2}} \right)} \\ \bigg\langle j_1\ \left(M + \frac{1}{2} \right)\ \frac{1}{2}\ \left( -\frac{1}{2}\right) \bigg| \left( j_1 \pm \frac{1}{2} \right)\ M \bigg\rangle =&\ \sqrt{\frac{1}{2} \left( 1 \mp \frac{M}{j_1 + \frac{1}{2}} \right)}\end{split}\]

Let \(j_1 = \frac{1}{2}\) we have

\[\begin{split}\bigg\langle \frac{1}{2}\ \left( - \frac{1}{2} \right)\ \frac{1}{2}\ \frac{1}{2} \bigg| \left( \frac{1}{2} \pm \frac{1}{2} \right)\ 0 \bigg\rangle =&\ \pm \sqrt{\frac{1}{2} } \\ \bigg\langle \frac{1}{2} \ \frac{1}{2} \ \frac{1}{2}\ \left( -\frac{1}{2}\right) \bigg| \left( \frac{1}{2} \pm \frac{1}{2} \right)\ 0 \bigg\rangle =&\ \sqrt{\frac{1}{2} }\end{split}\]

The exchange factor formula is

\[\begin{split}\left( \hat{X}_1^{[S_1]} \otimes_{[S]} \hat{X}_2^{[S_2]} \right)^{[S_z]} =&\ \sum_{S_{1z},S_{2z}} \hat{X}_1^{[S_1][S_{1z}]} \hat{X}_2^{[S_2][S_{2z}]} \langle SS_z| S_1S_{1z},\ S_2 S_{2z} \rangle \\ =&\ \mathrm{P}_{\mathrm{fermi}}^{\mathrm{exchange}}(N_1,N_2) \sum_{S_{1z},S_{2z}} \hat{X}_2^{[S_2][S_{2z}]} \hat{X}_1^{[S_1][S_{1z}]} \langle SS_z| S_1S_{1z},\ S_2 S_{2z} \rangle \\ =&\ \mathrm{P}_{\mathrm{fermi}}^{\mathrm{exchange}}(N_1,N_2) \frac{\langle SS_z| S_1S_{1z},\ S_2 S_{2z} \rangle} {\langle SS_z| S_2S_{2z},\ S_1 S_{1z} \rangle} \left( \hat{X}_2^{[S_2]} \otimes_{[S]} \hat{X}_1^{[S_1]} \right)^{[S_z]} \\ \hat{X}_1^{[S_1]} \otimes_{[S]} \hat{X}_2^{[S_2]} =&\ \mathrm{P}_{\mathrm{fermi}}^{\mathrm{exchange}}(N_1,N_2) \mathrm{P}_{\mathrm{SU(2)}}^{\mathrm{exchange}}(S_1, S_2, S) \hat{X}_2^{[S_2]} \otimes_{[S]} \hat{X}_1^{[S_1]}\end{split}\]

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

\[\begin{split}(\hat{H})^{[0], NC} =&\ \big( \hat{H}^{L} \big)^{[0]} \otimes_{[0]} \big( \hat{1}^{R} \big)^{[0]} + \big( \hat{1}^{L} \big)^{[0]} \otimes_{[0]} \big( \hat{H}^{R} \big)^{[0]} \\ &\ + 2 \sum_{i\in L} \left[ \big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( \hat{R}_{i}^{\prime R} \big)^{[\frac{1}{2}]} + \big( a_{i}\big)^{[\frac{1}{2}]} \otimes_{[0]} \big( \hat{R}_{i}^{\prime R\dagger} \big)^{[\frac{1}{2}]} \right] + 2 \sum_{i\in R} \left[ \big( \hat{R}_{i}^{\prime L\dagger} \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{i} \big)^{[\frac{1}{2}]} + \big( \hat{R}_{i}^{\prime L} \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{i}^\dagger \big)^{[\frac{1}{2}]}\right] \\ &\ - \frac{1}{2} \sum_{ik\in L} \left[ \big(\hat{A}_{ik} \big)^{[0]} \otimes_{[0]} \big(\hat{P}_{ik}^{R} \big)^{[0]} + \sqrt{3} \big(\hat{A}_{ik} \big)^{[1]} \otimes_{[0]} \big(\hat{P}_{ik}^{R} \big)^{[1]} + \big(\hat{A}_{ik}^\dagger \big)^{[0]} \otimes_{[0]} \big(\hat{P}_{ik}^{R\dagger} \big)^{[0]} + \sqrt{3} \big(\hat{A}_{ik}^\dagger \big)^{[1]} \otimes_{[0]} \big(\hat{P}_{ik}^{R\dagger} \big)^{[1]} \right] \\ &\ +\sum_{ij\in L} \left[ \big( \hat{B}_{ij} \big)^{[0]} \otimes_{[0]} \big( {\hat{Q}}_{ij}^{\prime\prime R} \big)^{[0]} + \sqrt{3} \big( {\hat{B}'}_{ij} \big)^{[1]} \otimes_{[0]} \big( {\hat{Q}}_{ij}^{\prime R} \big)^{[1]} \right]\end{split}\]

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

\[N_{NC} = 1 + 1 + 2K_L + 2K_R + 4K_L^2 + 2K_L^2 = 6K_L^2 + 2K + 2\]

Complementary/Normal Partitioning

Note that due the CG factors, exchange any \(\otimes_{[0]}\) product will not produce extra sign.

\[\begin{split}(\hat{H})^{[0], CN} =&\ \big( \hat{H}^{L} \big)^{[0]} \otimes_{[0]} \big( \hat{1}^{R} \big)^{[0]} + \big( \hat{1}^{L} \big)^{[0]} \otimes_{[0]} \big( \hat{H}^{R} \big)^{[0]} \\ &\ + 2 \sum_{i\in L} \left[ \big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( \hat{R}_{i}^{\prime R} \big)^{[\frac{1}{2}]} + \big( a_{i}\big)^{[\frac{1}{2}]} \otimes_{[0]} \big( \hat{R}_{i}^{\prime R\dagger} \big)^{[\frac{1}{2}]} \right] + 2 \sum_{i\in R} \left[ \big( \hat{R}_{i}^{\prime L\dagger} \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{i} \big)^{[\frac{1}{2}]} + \big( \hat{R}_{i}^{\prime L} \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{i}^\dagger \big)^{[\frac{1}{2}]}\right] \\ &\ - \frac{1}{2} \sum_{jl\in R} \left[ \big(\hat{P}_{jl}^{L} \big)^{[0]} \otimes_{[0]} \big(\hat{A}_{jl} \big)^{[0]} + \sqrt{3} \big(\hat{P}_{jl}^{L} \big)^{[1]} \otimes_{[0]} \big(\hat{A}_{jl} \big)^{[1]} + \big(\hat{P}_{jl}^{L\dagger} \big)^{[0]} \otimes_{[0]} \big(\hat{A}_{jl}^\dagger \big)^{[0]} + \sqrt{3} \big(\hat{P}_{jl}^{L\dagger} \big)^{[1]} \otimes_{[0]} \big(\hat{A}_{jl}^\dagger \big)^{[1]} \right] \\ &\ +\sum_{kl\in R} \left[ \big( {\hat{Q}}_{kl}^{\prime\prime L} \big)^{[0]} \otimes_{[0]} \big( \hat{B}_{kl} \big)^{[0]} + \sqrt{3} \big( {\hat{Q}}_{kl}^{\prime L} \big)^{[1]} \otimes_{[0]} \big( {\hat{B}'}_{kl} \big)^{[1]} \right]\end{split}\]

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

\[N_{CN} = 1 + 1 + 2K_L + 2K_R + 4K_R^2 + 2K_R^2 = 6K_R^2 + 2K + 2\]

Blocking

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

\[\begin{split}\big( \hat{R}_{i}^{L/R} \big)^{[1/2]} =&\ \sum_{jkl\in L/R} v_{ijkl} \left[ \big( a_{k}^\dagger \big)^{[1/2]} \otimes_{[0]} \big( a_{l} \big)^{[1/2]} \right] \otimes_{[1/2]} \big( a_{j} \big)^{[1/2]} \\ =&\ \frac{1}{\sqrt{2}} \sum_{jkl\in L/R} v_{ijkl} \begin{pmatrix} a_{k\alpha}^\dagger a_{l\alpha}+ a_{k\beta}^\dagger a_{l\beta} \end{pmatrix}^{[0]} \otimes_{[1/2]} \big( a_{j} \big)^{[1/2]} \\ =&\ \frac{1}{\sqrt{2}} \sum_{jkl\in L/R} v_{ijkl} \begin{pmatrix} -a_{k\alpha}^\dagger a_{l\alpha}a_{j\beta} - a_{k\beta}^\dagger a_{l\beta}a_{j\beta} \\ a_{k\alpha}^\dagger a_{l\alpha}a_{j\alpha}+ a_{k\beta}^\dagger a_{l\beta}a_{j\alpha} \end{pmatrix}^{[1/2]}\end{split}\]

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

\[\begin{split}\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]}\end{split}\]

After reordering of terms

\[\begin{split}\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]}\end{split}\]

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)

\[\begin{split}\big( \hat{A}_{ik} \big)^{[0/1]} =&\ \big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0/1]} \big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \\ \big( \hat{A}_{ik}^\dagger \big)^{[0]} =&\ \big( a_{i} \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{k} \big)^{[\frac{1}{2}]} = \big( a_{k} \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{i} \big)^{[\frac{1}{2}]} \\ \big( \hat{A}_{ik}^\dagger \big)^{[1]} =&\ -\big( a_{i} \big)^{[\frac{1}{2}]} \otimes_{[1]} \big( a_{k} \big)^{[\frac{1}{2}]} = \big( a_{k} \big)^{[\frac{1}{2}]} \otimes_{[1]} \big( a_{i} \big)^{[\frac{1}{2}]} \\ \big( \hat{P}_{ik}^{R} \big)^{[0/1]} =&\ \sum_{jl\in R} v_{ijkl} \big( a_{l} \big)^{[\frac{1}{2}]} \otimes_{[0/1]} \big( a_{j} \big)^{[\frac{1}{2}]} \\ \big( \hat{B}_{ij} \big)^{[0]} =&\ \big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{j} \big)^{[\frac{1}{2}]} \\ \big( {\hat{B}'}_{ij} \big)^{[1]} =&\ \big( a_{i}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[1]} \big( a_{j} \big)^{[\frac{1}{2}]}\\ \big( {\hat{Q}}_{ij}^{\prime R} \big)^{[1]} =&\ \sum_{kl\in R} v_{ilkj} \big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[1]} \big( a_{l} \big)^{[\frac{1}{2}]} \\ \big( {\hat{Q}}_{ij}^{\prime \prime R} \big)^{[0]} =&\ \sum_{kl\in R} (2v_{ijkl} - v_{ilkj}) \big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{l} \big)^{[\frac{1}{2}]}\end{split}\]

we have

\[\begin{split}\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]}\end{split}\]

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,

\[\begin{split}\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]}\end{split}\]

Number of terms

\[\begin{split}N_{R',NC} =&\ (2 + 4K_L + 4K_L^2) K_R + (2 + 4 + 4K_R) K_L = 4K_L^2 K_R + 8K_L K_R + 2K + 4 K_L \\ N_{R',CN} =&\ (2 + 4K_L + 4) K_R + (2 + 4K_R^2 + 4K_R) K_L = 4K_R^2 K_L + 8K_R K_L + 2K + 4 K_R\end{split}\]

Blocking of other complementary operators is straightforward

\[\begin{split}\big( \hat{P}_{ik}^{L*,CN} \big)^{[0/1]} =&\ \big( \hat{P}_{ik}^{L} \big)^{[0/1]} \otimes_{[0/1]} \big( \hat{1}^* \big)^{[0]} + \big( \hat{1}^L \big)^{[0]} \otimes_{[0/1]} \big( \hat{P}_{ik}^{*} \big)^{[0/1]} + \sum_{j \in L, l \in *} v_{ijkl} \big( a_{l} \big)^{[\frac{1}{2}]} \otimes_{[0/1]} \big( a_{j} \big)^{[\frac{1}{2}]} + \sum_{j \in *, l \in L} v_{ijkl} \big( a_{l} \big)^{[\frac{1}{2}]} \otimes_{[0/1]} \big( a_{j} \big)^{[\frac{1}{2}]} \\ =&\ \big( \hat{P}_{ik}^{L} \big)^{[0/1]} \otimes_{[0/1]} \big( \hat{1}^* \big)^{[0]} + \big( \hat{1}^L \big)^{[0]} \otimes_{[0/1]} \big( \hat{P}_{ik}^{*} \big)^{[0/1]} \pm \sum_{j \in L, l \in *} v_{ijkl} \big( a_{j} \big)^{[\frac{1}{2}]} \otimes_{[0/1]} \big( a_{l} \big)^{[\frac{1}{2}]} + \sum_{j \in *, l \in L} v_{ijkl} \big( a_{l} \big)^{[\frac{1}{2}]} \otimes_{[0/1]} \big( a_{j} \big)^{[\frac{1}{2}]} \\ \big( \hat{P}_{ik}^{R*,NC} \big)^{[0/1]} =&\ \big( \hat{P}_{ik}^{*} \big)^{[0/1]} \otimes_{[0/1]} \big( \hat{1}^R \big)^{[0]} + \big( \hat{1}^* \big)^{[0]} \otimes_{[0/1]} \big( \hat{P}_{ik}^{R} \big)^{[0/1]} \pm \sum_{j \in *, l \in R} v_{ijkl} \big( a_{j} \big)^{[\frac{1}{2}]} \otimes_{[0/1]} \big( a_{l} \big)^{[\frac{1}{2}]} + \sum_{j \in R, l \in *} v_{ijkl} \big( a_{l} \big)^{[\frac{1}{2}]} \otimes_{[0/1]} \big( a_{j} \big)^{[\frac{1}{2}]}\end{split}\]

and

\[\begin{split}\big( {\hat{Q}}_{ij}^{\prime \prime L*,CN} \big)^{[0]} =&\ \big( {\hat{Q}}_{ij}^{\prime \prime L} \big)^{[0]} \otimes_{[0]} \big( \hat{1}^* \big)^{[0]} + \big( \hat{1}^L \big)^{[0]} \otimes_{[0]} \big( {\hat{Q}}_{ij}^{\prime \prime *} \big)^{[0]} + \sum_{k\in L, l \in *} (2v_{ijkl} - v_{ilkj}) \big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{l} \big)^{[\frac{1}{2}]} + \sum_{k\in *, l \in L} (2v_{ijkl} - v_{ilkj}) \big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{l} \big)^{[\frac{1}{2}]} \\ =&\ \big( {\hat{Q}}_{ij}^{\prime \prime L} \big)^{[0]} \otimes_{[0]} \big( \hat{1}^* \big)^{[0]} + \big( \hat{1}^L \big)^{[0]} \otimes_{[0]} \big( {\hat{Q}}_{ij}^{\prime \prime *} \big)^{[0]} + \sum_{k\in L, l \in *} (2v_{ijkl} - v_{ilkj}) \big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{l} \big)^{[\frac{1}{2}]} + \sum_{k\in *, l \in L} (2v_{ijkl} - v_{ilkj}) \big( a_{l} \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \\ \big( {\hat{Q}}_{ij}^{\prime \prime R*,NC} \big)^{[0]} =&\ \big( {\hat{Q}}_{ij}^{\prime \prime *} \big)^{[0]} \otimes_{[0]} \big( \hat{1}^R \big)^{[0]} + \big( \hat{1}^* \big)^{[0]} \otimes_{[0]} \big( {\hat{Q}}_{ij}^{\prime \prime R} \big)^{[0]} + \sum_{k\in *, l \in R} (2v_{ijkl} - v_{ilkj}) \big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{l} \big)^{[\frac{1}{2}]} + \sum_{k\in R, l \in *} (2v_{ijkl} - v_{ilkj}) \big( a_{l} \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{k}^\dagger \big)^{[\frac{1}{2}]}\end{split}\]

and

\[\begin{split}\big( {\hat{Q}}_{ij}^{\prime L*,CN} \big)^{[1]} =&\ \big( {\hat{Q}}_{ij}^{\prime L} \big)^{[1]} \otimes_{[1]} \big( \hat{1}^* \big)^{[0]} + \big( \hat{1}^L \big)^{[0]} \otimes_{[1]} \big( {\hat{Q}}_{ij}^{\prime *} \big)^{[1]} + \sum_{k\in L, l \in *} v_{ilkj} \big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[1]} \big( a_{l} \big)^{[\frac{1}{2}]} + \sum_{k\in *, l \in L} v_{ilkj} \big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[1]} \big( a_{l} \big)^{[\frac{1}{2}]} \\ =&\ \big( {\hat{Q}}_{ij}^{\prime L} \big)^{[1]} \otimes_{[1]} \big( \hat{1}^* \big)^{[0]} + \big( \hat{1}^L \big)^{[0]} \otimes_{[1]} \big( {\hat{Q}}_{ij}^{\prime *} \big)^{[1]} + \sum_{k\in L, l \in *} v_{ilkj} \big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[1]} \big( a_{l} \big)^{[\frac{1}{2}]} - \sum_{k\in *, l \in L} v_{ilkj} \big( a_{l} \big)^{[\frac{1}{2}]} \otimes_{[1]} \big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \\ \big( {\hat{Q}}_{ij}^{\prime R*,CN} \big)^{[1]} =&\ \big( {\hat{Q}}_{ij}^{\prime *} \big)^{[1]} \otimes_{[1]} \big( \hat{1}^R \big)^{[0]} + \big( \hat{1}^* \big)^{[0]} \otimes_{[1]} \big( {\hat{Q}}_{ij}^{\prime R} \big)^{[1]} + \sum_{k\in *, l \in R} v_{ilkj} \big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[1]} \big( a_{l} \big)^{[\frac{1}{2}]} - \sum_{k\in R, l \in *} v_{ilkj} \big( a_{l} \big)^{[\frac{1}{2}]} \otimes_{[1]} \big( a_{k}^\dagger \big)^{[\frac{1}{2}]}\end{split}\]

Middle-Site Transformation

\[\begin{split}\big( \hat{P}_{ik}^{L,NC\to CN} \big)^{[0/1]} =&\ \sum_{jl\in L} v_{ijkl} \big( a_{l} \big)^{[\frac{1}{2}]} \otimes_{[0/1]} \big( a_{j} \big)^{[\frac{1}{2}]} = \sum_{jl\in L} v_{ijkl} \big( \hat{A}_{jl}^\dagger \big)^{[0/1]} \\ \big( {\hat{Q}}_{ij}^{\prime \prime L,NC\to CN} \big)^{[0]} =&\ \sum_{kl\in R} (2v_{ijkl} - v_{ilkj}) \big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[0]} \big( a_{l} \big)^{[\frac{1}{2}]} = \sum_{kl\in R} (2v_{ijkl} - v_{ilkj}) \big( \hat{B}_{kl} \big)^{[0]} \\ \big( {\hat{Q}}_{ij}^{\prime L,NC\to CN} \big)^{[1]} =&\ \sum_{kl\in R} v_{ilkj} \big( a_{k}^\dagger \big)^{[\frac{1}{2}]} \otimes_{[1]} \big( a_{l} \big)^{[\frac{1}{2}]} = \sum_{kl\in R} v_{ilkj} \big( {\hat{B}'}_{kl} \big)^{[1]}\end{split}\]